diff --git "a/data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Module.LinearMap.Basic.sym.json" "b/data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Module.LinearMap.Basic.sym.json"
new file mode 100644--- /dev/null
+++ "b/data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Module.LinearMap.Basic.sym.json"
@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["LinearMap","instModuleDomMulActOfSMulCommClass","_proof_3"],"typeFallback":"forall {R : Type.{u_3}} {R' : Type.{u_4}} {S : Type.{u_5}} {M : Type.{u_2}} {M' : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 : Semiring.{u_3} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 : Semiring.{u_4} R'] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 : AddCommMonoid.{u_1} M'] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 : Module.{u_3, u_2} R M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23 : Module.{u_4, u_1} R' M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16] {σ₁₂ : RingHom.{u_3, u_4} R R' (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7) (Semiring.toNonAssocSemiring.{u_4} R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10)} [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.32 : Semiring.{u_5} S] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.35 : Module.{u_5, u_2} S M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.32 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.48 : SMulCommClass.{u_3, u_5, u_2} R S M (SMulZeroClass.toSMul.{u_3, 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.Basic.3265207960._hygCtx._hyg.13)))) (DistribSMul.toSMulZeroClass.{u_3, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13)) (DistribMulAction.toDistribSMul.{u_3, u_2} R M (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13) (Module.toDistribMulAction.{u_3, u_2} R M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19)))) (SMulZeroClass.toSMul.{u_5, u_2} S 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.Basic.3265207960._hygCtx._hyg.13)))) (DistribSMul.toSMulZeroClass.{u_5, u_2} S M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13)) (DistribMulAction.toDistribSMul.{u_5, u_2} S M (MonoidWithZero.toMonoid.{u_5} S (Semiring.toMonoidWithZero.{u_5} S inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.32)) (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13) (Module.toDistribMulAction.{u_5, u_2} S M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.32 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.35))))] (x._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.75 : DomMulAct.{u_5} S) (x._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.77 : DomMulAct.{u_5} S) (x._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.79 : LinearMap.{u_3, u_4, u_2, u_1} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23) (x._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.84 : M), Eq.{succ u_1} M' (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 R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23) M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => M') (LinearMap.instFunLike.{u_3, u_4, u_2, u_1} R R' M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23 σ₁₂) (HSMul.hSMul.{u_5, max u_2 u_1, max u_2 u_1} (DomMulAct.{u_5} S) (LinearMap.{u_3, u_4, u_2, u_1} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23) (LinearMap.{u_3, u_4, u_2, u_1} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23) (instHSMul.{u_5, max u_2 u_1} (DomMulAct.{u_5} S) (LinearMap.{u_3, u_4, u_2, u_1} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23) (SemigroupAction.toSMul.{u_5, max u_1 u_2} (DomMulAct.{u_5} S) (LinearMap.{u_3, u_4, u_2, u_1} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23) (Monoid.toSemigroup.{u_5} (DomMulAct.{u_5} S) (DomMulAct.instMonoidOfMulOpposite.{u_5} S (MulOpposite.instMonoid.{u_5} S (MonoidWithZero.toMonoid.{u_5} S (Semiring.toMonoidWithZero.{u_5} S inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.32))))) (MulAction.toSemigroupAction.{u_5, max u_1 u_2} (DomMulAct.{u_5} S) (LinearMap.{u_3, u_4, u_2, u_1} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23) (DomMulAct.instMonoidOfMulOpposite.{u_5} S (MulOpposite.instMonoid.{u_5} S (MonoidWithZero.toMonoid.{u_5} S (Semiring.toMonoidWithZero.{u_5} S inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.32)))) (DistribMulAction.toMulAction.{u_5, max u_1 u_2} (DomMulAct.{u_5} S) (LinearMap.{u_3, u_4, u_2, u_1} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23) (DomMulAct.instMonoidOfMulOpposite.{u_5} S (MulOpposite.instMonoid.{u_5} S (MonoidWithZero.toMonoid.{u_5} S (Semiring.toMonoidWithZero.{u_5} S inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.32)))) (AddCommMonoid.toAddMonoid.{max u_2 u_1} (LinearMap.{u_3, u_4, u_2, u_1} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23) (LinearMap.addCommMonoid.{u_3, u_4, u_2, u_1} R R' M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23 σ₁₂)) (LinearMap.instDistribMulActionDomMulActOfSMulCommClass.{u_3, u_4, u_2, u_1, u_5} R R' M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23 σ₁₂ S (MonoidWithZero.toMonoid.{u_5} S (Semiring.toMonoidWithZero.{u_5} S inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.32)) (Module.toDistribMulAction.{u_5, u_2} S M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.32 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.35) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.48))))) (HAdd.hAdd.{u_5, u_5, u_5} (DomMulAct.{u_5} S) (DomMulAct.{u_5} S) (DomMulAct.{u_5} S) (instHAdd.{u_5} (DomMulAct.{u_5} S) (Distrib.toAdd.{u_5} (DomMulAct.{u_5} S) (NonUnitalNonAssocSemiring.toDistrib.{u_5} (DomMulAct.{u_5} S) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_5} (DomMulAct.{u_5} S) (Semiring.toNonAssocSemiring.{u_5} (DomMulAct.{u_5} S) (DomMulAct.instSemiringOfMulOpposite.{u_5} S (MulOpposite.instSemiring.{u_5} S inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.32))))))) x._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.75 x._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.77) x._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.79) x._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.84) (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 R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23) M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => M') (LinearMap.instFunLike.{u_3, u_4, u_2, u_1} R R' M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23 σ₁₂) (HAdd.hAdd.{max u_2 u_1, max u_2 u_1, max u_2 u_1} (LinearMap.{u_3, u_4, u_2, u_1} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23) (LinearMap.{u_3, u_4, u_2, u_1} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23) (LinearMap.{u_3, u_4, u_2, u_1} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23) (instHAdd.{max u_2 u_1} (LinearMap.{u_3, u_4, u_2, u_1} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23) (AddCommMagma.toAdd.{max u_2 u_1} (LinearMap.{u_3, u_4, u_2, u_1} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23) (AddCommSemigroup.toAddCommMagma.{max u_2 u_1} (LinearMap.{u_3, u_4, u_2, u_1} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23) (AddCommMonoid.toAddCommSemigroup.{max u_2 u_1} (LinearMap.{u_3, u_4, u_2, u_1} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23) (LinearMap.addCommMonoid.{u_3, u_4, u_2, u_1} R R' M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23 σ₁₂))))) (HSMul.hSMul.{u_5, max u_2 u_1, max u_2 u_1} (DomMulAct.{u_5} S) (LinearMap.{u_3, u_4, u_2, u_1} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23) (LinearMap.{u_3, u_4, u_2, u_1} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23) (instHSMul.{u_5, max u_2 u_1} (DomMulAct.{u_5} S) (LinearMap.{u_3, u_4, u_2, u_1} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23) (SemigroupAction.toSMul.{u_5, max u_1 u_2} (DomMulAct.{u_5} S) (LinearMap.{u_3, u_4, u_2, u_1} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23) (Monoid.toSemigroup.{u_5} (DomMulAct.{u_5} S) (DomMulAct.instMonoidOfMulOpposite.{u_5} S (MulOpposite.instMonoid.{u_5} S (MonoidWithZero.toMonoid.{u_5} S (Semiring.toMonoidWithZero.{u_5} S inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.32))))) (MulAction.toSemigroupAction.{u_5, max u_1 u_2} (DomMulAct.{u_5} S) (LinearMap.{u_3, u_4, u_2, u_1} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23) (DomMulAct.instMonoidOfMulOpposite.{u_5} S (MulOpposite.instMonoid.{u_5} S (MonoidWithZero.toMonoid.{u_5} S (Semiring.toMonoidWithZero.{u_5} S inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.32)))) (DistribMulAction.toMulAction.{u_5, max u_1 u_2} (DomMulAct.{u_5} S) (LinearMap.{u_3, u_4, u_2, u_1} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23) (DomMulAct.instMonoidOfMulOpposite.{u_5} S (MulOpposite.instMonoid.{u_5} S (MonoidWithZero.toMonoid.{u_5} S (Semiring.toMonoidWithZero.{u_5} S inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.32)))) (AddCommMonoid.toAddMonoid.{max u_2 u_1} (LinearMap.{u_3, u_4, u_2, u_1} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23) (LinearMap.addCommMonoid.{u_3, u_4, u_2, u_1} R R' M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23 σ₁₂)) (LinearMap.instDistribMulActionDomMulActOfSMulCommClass.{u_3, u_4, u_2, u_1, u_5} R R' M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23 σ₁₂ S (MonoidWithZero.toMonoid.{u_5} S (Semiring.toMonoidWithZero.{u_5} S inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.32)) (Module.toDistribMulAction.{u_5, u_2} S M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.32 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.35) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.48))))) x._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.75 x._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.79) (HSMul.hSMul.{u_5, max u_2 u_1, max u_2 u_1} (DomMulAct.{u_5} S) (LinearMap.{u_3, u_4, u_2, u_1} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23) (LinearMap.{u_3, u_4, u_2, u_1} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23) (instHSMul.{u_5, max u_2 u_1} (DomMulAct.{u_5} S) (LinearMap.{u_3, u_4, u_2, u_1} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23) (SemigroupAction.toSMul.{u_5, max u_1 u_2} (DomMulAct.{u_5} S) (LinearMap.{u_3, u_4, u_2, u_1} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23) (Monoid.toSemigroup.{u_5} (DomMulAct.{u_5} S) (DomMulAct.instMonoidOfMulOpposite.{u_5} S (MulOpposite.instMonoid.{u_5} S (MonoidWithZero.toMonoid.{u_5} S (Semiring.toMonoidWithZero.{u_5} S inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.32))))) (MulAction.toSemigroupAction.{u_5, max u_1 u_2} (DomMulAct.{u_5} S) (LinearMap.{u_3, u_4, u_2, u_1} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23) (DomMulAct.instMonoidOfMulOpposite.{u_5} S (MulOpposite.instMonoid.{u_5} S (MonoidWithZero.toMonoid.{u_5} S (Semiring.toMonoidWithZero.{u_5} S inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.32)))) (DistribMulAction.toMulAction.{u_5, max u_1 u_2} (DomMulAct.{u_5} S) (LinearMap.{u_3, u_4, u_2, u_1} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23) (DomMulAct.instMonoidOfMulOpposite.{u_5} S (MulOpposite.instMonoid.{u_5} S (MonoidWithZero.toMonoid.{u_5} S (Semiring.toMonoidWithZero.{u_5} S inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.32)))) (AddCommMonoid.toAddMonoid.{max u_2 u_1} (LinearMap.{u_3, u_4, u_2, u_1} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23) (LinearMap.addCommMonoid.{u_3, u_4, u_2, u_1} R R' M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23 σ₁₂)) (LinearMap.instDistribMulActionDomMulActOfSMulCommClass.{u_3, u_4, u_2, u_1, u_5} R R' M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23 σ₁₂ S (MonoidWithZero.toMonoid.{u_5} S (Semiring.toMonoidWithZero.{u_5} S inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.32)) (Module.toDistribMulAction.{u_5, u_2} S M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.32 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.35) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.48))))) 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Sᵈᵐᵃ) (x_2 : M →ₛₗ[σ₁₂] M') (x_3 : M), ((x + x_1) • x_2) x_3 = (x • x_2 + x_1 • x_2) 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(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))))] (b : A) (b_1 : A), (Eq.{succ u_15} A b b_1) -> (Eq.{succ u_15} (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) (LinearMap.mulRight.{u_14, u_15} R A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.17 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.27 b) (LinearMap.mulRight.{u_14, u_15} R A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.1264609845._hygCtx._hyg.17 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.27 b_1))","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] (b b_1 : A), b = b_1 → LinearMap.mulRight R b = LinearMap.mulRight R b_1","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] (b b_1 : A), b = b_1 → LinearMap.mulRight R b = LinearMap.mulRight R b_1","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"],["LinearMap","mulRight"],["Eq"],["AddZero","toZero"],["NonUnitalNonAssocSemiring"],["Semiring"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]],"valueReferences":[["Semiring","toNonAssocSemiring"],["RingHom","id"],["Eq","refl"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["LinearMap","mulRight"],["LinearMap"],["Eq"],["Eq","rec"]]},{"isProp":true,"kind":"theorem","name":["Sum","elimZeroLeft","_proof_2"],"typeFallback":"forall {ι : Type.{u_1}} {κ : Type.{u_2}} {R : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3410682868._hygCtx._hyg.10 : Semiring.{u_3} R] 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inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3410682868._hygCtx._hyg.10) (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3410682868._hygCtx._hyg.10)) (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3410682868._hygCtx._hyg.10)) t) (Sum.elim.{u_2, u_1, succ u_3} κ ι R (OfNat.ofNat.{max u_2 u_3} (κ -> R) 0 (Zero.toOfNat0.{max u_2 u_3} (κ -> R) (Pi.instZero.{u_2, u_3} κ (fun (a._@._internal._hyg.0 : κ) => R) (fun (i : κ) => MulZeroClass.toZero.{u_3} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u_3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3410682868._hygCtx._hyg.10))))))) f))","typeFull":"∀ {ι : Type u_1} {κ : Type u_2} {R : Type u_3} [inst : Semiring R] (t : R) (f : ι → R),\n  Sum.elim 0 (t • f) = (RingHom.id R) t • Sum.elim 0 f","typeReadable":"∀ {ι : Type u_1} {κ : Type u_2} {R : Type u_3} [inst : Semiring R] (t : R) (f : ι → R),\n  Sum.elim 0 (t • f) = (RingHom.id R) t • Sum.elim 0 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(R : Type.{u_6}) (A : Type.{u_7}) [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2408345197._hygCtx._hyg.9 : Semiring.{u_6} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2408345197._hygCtx._hyg.12 : NonAssocSemiring.{u_7} A] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2408345197._hygCtx._hyg.15 : Module.{u_6, u_7} R A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2408345197._hygCtx._hyg.9 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2408345197._hygCtx._hyg.12))] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2408345197._hygCtx._hyg.19 : IsScalarTower.{u_6, u_7, u_7} R A A (SMulZeroClass.toSMul.{u_6, u_7} R A (AddZero.toZero.{u_7} A (AddZeroClass.toAddZero.{u_7} A (AddMonoid.toAddZeroClass.{u_7} A (AddMonoidWithOne.toAddMonoid.{u_7} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u_7} A (NonAssocSemiring.toAddCommMonoidWithOne.{u_7} A 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inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2408345197._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2408345197._hygCtx._hyg.15)))) (SMulZeroClass.toSMul.{u_7, u_7} A A (AddZero.toZero.{u_7} A (AddZeroClass.toAddZero.{u_7} A (AddMonoid.toAddZeroClass.{u_7} A (AddMonoidWithOne.toAddMonoid.{u_7} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u_7} A (NonAssocSemiring.toAddCommMonoidWithOne.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2408345197._hygCtx._hyg.12)))))) (DistribSMul.toSMulZeroClass.{u_7, u_7} A A (AddMonoid.toAddZeroClass.{u_7} A (AddMonoidWithOne.toAddMonoid.{u_7} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u_7} A (NonAssocSemiring.toAddCommMonoidWithOne.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2408345197._hygCtx._hyg.12)))) (instDistribSMul.{u_7} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2408345197._hygCtx._hyg.12)))) (SMulZeroClass.toSMul.{u_6, u_7} R A (AddZero.toZero.{u_7} A (AddZeroClass.toAddZero.{u_7} A (AddMonoid.toAddZeroClass.{u_7} A (AddMonoidWithOne.toAddMonoid.{u_7} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u_7} A (NonAssocSemiring.toAddCommMonoidWithOne.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2408345197._hygCtx._hyg.12)))))) (DistribSMul.toSMulZeroClass.{u_6, u_7} R A (AddMonoid.toAddZeroClass.{u_7} A (AddMonoidWithOne.toAddMonoid.{u_7} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u_7} A (NonAssocSemiring.toAddCommMonoidWithOne.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2408345197._hygCtx._hyg.12)))) (DistribMulAction.toDistribSMul.{u_6, u_7} R A (MonoidWithZero.toMonoid.{u_6} R (Semiring.toMonoidWithZero.{u_6} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2408345197._hygCtx._hyg.9)) (AddMonoidWithOne.toAddMonoid.{u_7} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u_7} A (NonAssocSemiring.toAddCommMonoidWithOne.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2408345197._hygCtx._hyg.12))) (Module.toDistribMulAction.{u_6, u_7} R A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2408345197._hygCtx._hyg.9 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2408345197._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2408345197._hygCtx._hyg.15))))], Eq.{succ u_7} (LinearMap.{u_6, u_6, u_7, u_7} R R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2408345197._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2408345197._hygCtx._hyg.9 (RingHom.id.{u_6} R (Semiring.toNonAssocSemiring.{u_6} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2408345197._hygCtx._hyg.9)) A A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2408345197._hygCtx._hyg.12)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2408345197._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2408345197._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2408345197._hygCtx._hyg.15) (LinearMap.mulRight.{u_6, u_7} R A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2408345197._hygCtx._hyg.9 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2408345197._hygCtx._hyg.12) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2408345197._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2408345197._hygCtx._hyg.19 (OfNat.ofNat.{u_7} A 1 (One.toOfNat1.{u_7} A (AddMonoidWithOne.toOne.{u_7} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u_7} A (NonAssocSemiring.toAddCommMonoidWithOne.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2408345197._hygCtx._hyg.12)))))) (LinearMap.id.{u_6, u_7} R A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2408345197._hygCtx._hyg.9 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2408345197._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2408345197._hygCtx._hyg.15)","typeFull":"∀ (R : Type u_6) (A : Type u_7) [inst : Semiring R] [inst_1 : NonAssocSemiring A] [inst_2 : Module R A]\n  [inst_3 : IsScalarTower R A A], LinearMap.mulRight R 1 = LinearMap.id","typeReadable":"∀ (R : Type u_6) (A : Type u_7) [inst : Semiring R] [inst_1 : NonAssocSemiring A] [inst_2 : Module R A]\n  [inst_3 : IsScalarTower R A A], LinearMap.mulRight R 1 = LinearMap.id","typeReferences":[["IsScalarTower"],["Module"],["SMulZeroClass","toSMul"],["AddMonoidWithOne","toAddMonoid"],["instDistribSMul"],["LinearMap","id"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["NonAssocSemiring"],["MonoidWithZero","toMonoid"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Eq"],["NonAssocSemiring","toAddCommMonoidWithOne"],["DistribSMul","toSMulZeroClass"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["LinearMap"],["OfNat","ofNat"],["Module","toDistribMulAction"],["One","toOfNat1"],["AddMonoidWithOne","toOne"],["LinearMap","mulRight"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["NonUnitalNonAssocSemiring","toAddCommMonoid"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["MulZeroOneClass","toMulOneClass"],["mul_one"],["LinearMap","id"],["OfNat","ofNat"],["Semiring","toNonAssocSemiring"],["One","toOfNat1"],["RingHom","id"],["LinearMap","ext"],["AddMonoidWithOne","toOne"],["LinearMap","mulRight"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["NonAssocSemiring","toAddCommMonoidWithOne"],["NonAssocSemiring","toMulZeroOneClass"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Algebra","Module","LinearMap","Basic",0,"LinearMap","instSMulDomMulAct","_simp_2"],"typeFallback":"forall {M : Type.{u_9}} {N : Type.{u_10}} {α : Type.{u_11}} {inst._@.Mathlib.Algebra.Group.Action.Defs.533111421._hygCtx._hyg.21 : SMul.{u_9, u_11} M α} {inst._@.Mathlib.Algebra.Group.Action.Defs.533111421._hygCtx._hyg.25 : SMul.{u_10, u_11} N α} [self : SMulCommClass.{u_9, u_10, u_11} M N α inst._@.Mathlib.Algebra.Group.Action.Defs.533111421._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Group.Action.Defs.533111421._hygCtx._hyg.25] (m : M) (n : N) (a : α), Eq.{succ u_11} α (HSMul.hSMul.{u_10, u_11, u_11} N α α (instHSMul.{u_10, u_11} N α inst._@.Mathlib.Algebra.Group.Action.Defs.533111421._hygCtx._hyg.25) n (HSMul.hSMul.{u_9, u_11, u_11} M α α (instHSMul.{u_9, u_11} M α inst._@.Mathlib.Algebra.Group.Action.Defs.533111421._hygCtx._hyg.21) m a)) (HSMul.hSMul.{u_9, u_11, u_11} M α α (instHSMul.{u_9, u_11} M α inst._@.Mathlib.Algebra.Group.Action.Defs.533111421._hygCtx._hyg.21) m (HSMul.hSMul.{u_10, u_11, u_11} N α α (instHSMul.{u_10, u_11} N α inst._@.Mathlib.Algebra.Group.Action.Defs.533111421._hygCtx._hyg.25) n a))","typeFull":"∀ {M : Type u_9} {N : Type u_10} {α : Type u_11} {inst : SMul M α} {inst_1 : SMul N α} [self : SMulCommClass M N α]\n  (m : M) (n : N) (a : α), n • m • a = m • n • a","typeReadable":"∀ {M : Type u_9} {N : Type u_10} {α : Type u_11} {inst : SMul M α} {inst_1 : SMul N α} [self : SMulCommClass M N α]\n  (m : M) (n : N) (a : α), n • m • a = m • n • a","typeReferences":[["SMulCommClass"],["SMul"],["HSMul","hSMul"],["instHSMul"],["Eq"]],"valueReferences":[["HSMul","hSMul"],["Eq","symm"],["instHSMul"],["SMulCommClass","smul_comm"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","instSMulDomMulAct","_proof_3"],"typeFallback":"forall {R : Type.{u_4}} {R' : Type.{u_5}} {M : Type.{u_3}} {M' : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.7 : Semiring.{u_4} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.10 : Semiring.{u_5} R'] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.13 : AddCommMonoid.{u_3} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.16 : AddCommMonoid.{u_1} M'] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.19 : Module.{u_4, u_3} R M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.13] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.23 : Module.{u_5, u_1} R' M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.16] {σ₁₂ : RingHom.{u_4, u_5} R R' (Semiring.toNonAssocSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.7) (Semiring.toNonAssocSemiring.{u_5} R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.10)} {S' : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.34 : Monoid.{u_2} S'] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.37 : DistribMulAction.{u_2, u_3} S' M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.34 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.13)] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.41 : SMulCommClass.{u_4, u_2, u_3} R S' M (SMulZeroClass.toSMul.{u_4, u_3} R M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.13)))) (DistribSMul.toSMulZeroClass.{u_4, u_3} R M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.13)) (DistribMulAction.toDistribSMul.{u_4, u_3} R M (MonoidWithZero.toMonoid.{u_4} R (Semiring.toMonoidWithZero.{u_4} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.13) (Module.toDistribMulAction.{u_4, u_3} R M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.19)))) (SMulZeroClass.toSMul.{u_2, u_3} S' M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.13)))) (DistribSMul.toSMulZeroClass.{u_2, u_3} S' M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.13)) (DistribMulAction.toDistribSMul.{u_2, u_3} S' M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.34 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.13) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.37)))] (a : DomMulAct.{u_2} S') (f : LinearMap.{u_4, u_5, u_3, u_1} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.23) (c : R) (x : M), Eq.{succ u_1} M' (HSMul.hSMul.{u_2, max u_3 u_1, max u_3 u_1} (DomMulAct.{u_2} S') (M -> M') (M -> M') (instHSMul.{u_2, max u_3 u_1} (DomMulAct.{u_2} S') (M -> M') (DomMulAct.instSMulForall.{u_2, u_1, 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{R : Type.{u_1}} {R' : Type.{u_2}} {S : Type.{u_5}} {M : Type.{u_3}} {M' : Type.{u_4}} [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 : Semiring.{u_2} R'] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 : AddCommMonoid.{u_3} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 : AddCommMonoid.{u_4} M'] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 : Module.{u_1, u_3} R M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23 : Module.{u_2, u_4} R' M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16] {σ₁₂ : RingHom.{u_1, u_2} R R' (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7) (Semiring.toNonAssocSemiring.{u_2} R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10)} [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.32 : Semiring.{u_5} S] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.35 : Module.{u_5, u_3} S M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.32 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.48 : SMulCommClass.{u_1, u_5, u_3} R S M (SMulZeroClass.toSMul.{u_1, u_3} R M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13)))) (DistribSMul.toSMulZeroClass.{u_1, u_3} R M 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(AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13)) (DistribMulAction.toDistribSMul.{u_5, u_3} S M (MonoidWithZero.toMonoid.{u_5} S (Semiring.toMonoidWithZero.{u_5} S inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.32)) (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13) (Module.toDistribMulAction.{u_5, u_3} S M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.32 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.35))))] (x._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.89 : LinearMap.{u_1, u_2, u_3, u_4} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23), Eq.{max (succ u_4) (succ u_3)} (LinearMap.{u_1, u_2, u_3, u_4} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23) (HSMul.hSMul.{u_5, max u_3 u_4, max u_3 u_4} (DomMulAct.{u_5} S) (LinearMap.{u_1, u_2, u_3, u_4} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 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inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23 σ₁₂)))))))","typeFull":"∀ {R : Type u_1} {R' : Type u_2} {S : Type u_5} {M : Type u_3} {M' : Type u_4} [inst : Semiring R]\n  [inst_1 : Semiring R'] [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M'] [inst_4 : Module R M]\n  [inst_5 : Module R' M'] {σ₁₂ : R →+* R'} [inst_6 : Semiring S] [inst_7 : Module S M] [inst_8 : SMulCommClass R S M]\n  (x : M →ₛₗ[σ₁₂] M'), 0 • x = 0","typeReadable":"∀ {R : Type u_1} {R' : Type u_2} {S : Type u_5} {M : Type u_3} {M' : Type u_4} [inst : Semiring R]\n  [inst_1 : Semiring R'] [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M'] [inst_4 : Module R M]\n  [inst_5 : Module R' M'] {σ₁₂ : R →+* R'} [inst_6 : Semiring S] [inst_7 : Module S M] [inst_8 : SMulCommClass R S M]\n  (x : M →ₛₗ[σ₁₂] M'), 0 • x = 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Type.{u_7}} [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2843790991._hygCtx._hyg.9 : Semiring.{u_6} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2843790991._hygCtx._hyg.12 : NonAssocSemiring.{u_7} A] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2843790991._hygCtx._hyg.15 : Module.{u_6, u_7} R A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2843790991._hygCtx._hyg.9 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2843790991._hygCtx._hyg.12))] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2843790991._hygCtx._hyg.19 : IsScalarTower.{u_6, u_7, u_7} R A A (SMulZeroClass.toSMul.{u_6, u_7} R A (AddZero.toZero.{u_7} A (AddZeroClass.toAddZero.{u_7} A (AddMonoid.toAddZeroClass.{u_7} A (AddMonoidWithOne.toAddMonoid.{u_7} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u_7} A (NonAssocSemiring.toAddCommMonoidWithOne.{u_7} A 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inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2843790991._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2843790991._hygCtx._hyg.15)))) (SMulZeroClass.toSMul.{u_7, u_7} A A (AddZero.toZero.{u_7} A (AddZeroClass.toAddZero.{u_7} A (AddMonoid.toAddZeroClass.{u_7} A (AddMonoidWithOne.toAddMonoid.{u_7} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u_7} A (NonAssocSemiring.toAddCommMonoidWithOne.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2843790991._hygCtx._hyg.12)))))) (DistribSMul.toSMulZeroClass.{u_7, u_7} A A (AddMonoid.toAddZeroClass.{u_7} A (AddMonoidWithOne.toAddMonoid.{u_7} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u_7} A (NonAssocSemiring.toAddCommMonoidWithOne.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2843790991._hygCtx._hyg.12)))) (instDistribSMul.{u_7} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2843790991._hygCtx._hyg.12)))) (SMulZeroClass.toSMul.{u_6, u_7} R A 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(Module.toDistribMulAction.{u_6, u_7} R A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2843790991._hygCtx._hyg.9 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2843790991._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2843790991._hygCtx._hyg.15))))] {a : A} {b : A}, Iff (Eq.{succ u_7} (LinearMap.{u_6, u_6, u_7, u_7} R R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2843790991._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2843790991._hygCtx._hyg.9 (RingHom.id.{u_6} R (Semiring.toNonAssocSemiring.{u_6} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2843790991._hygCtx._hyg.9)) A A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2843790991._hygCtx._hyg.12)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A 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inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2843790991._hygCtx._hyg.19 b)) (Eq.{succ u_7} A a b)","typeFull":"∀ {R : Type u_6} {A : Type u_7} [inst : Semiring R] [inst_1 : NonAssocSemiring A] [inst_2 : Module R A]\n  [inst_3 : IsScalarTower R A A] {a b : A}, LinearMap.mulRight R a = LinearMap.mulRight R b ↔ a = b","typeReadable":"∀ {R : Type u_6} {A : Type u_7} [inst : Semiring R] [inst_1 : NonAssocSemiring A] [inst_2 : Module R A]\n  [inst_3 : IsScalarTower R A A] {a b : A}, LinearMap.mulRight R a = LinearMap.mulRight R b ↔ a = 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{R : Type.{u_1}} {M : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Defs.1263408075._hygCtx._hyg.6 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.Defs.1263408075._hygCtx._hyg.9 : AddCommMonoid.{u_3} M] [inst._@.Mathlib.Algebra.Module.Defs.1263408075._hygCtx._hyg.12 : Module.{u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Defs.1263408075._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Module.Defs.1263408075._hygCtx._hyg.9] (r : R) (s : R) (x : M), Eq.{succ u_3} M (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 (AddCommMonoid.toAddCommSemigroup.{u_3} M inst._@.Mathlib.Algebra.Module.Defs.1263408075._hygCtx._hyg.9)))) (HSMul.hSMul.{u_1, u_3, u_3} R M M (instHSMul.{u_1, u_3} R M (SMulZeroClass.toSMul.{u_1, u_3} R M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Defs.1263408075._hygCtx._hyg.9)))) 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(DistribSMul.toSMulZeroClass.{u_1, u_3} R M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Defs.1263408075._hygCtx._hyg.9)) (DistribMulAction.toDistribSMul.{u_1, u_3} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.Defs.1263408075._hygCtx._hyg.6)) (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Defs.1263408075._hygCtx._hyg.9) (Module.toDistribMulAction.{u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Defs.1263408075._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Module.Defs.1263408075._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.Defs.1263408075._hygCtx._hyg.12))))) s x)) (HSMul.hSMul.{u_1, u_3, u_3} R M M (instHSMul.{u_1, u_3} R M (SMulZeroClass.toSMul.{u_1, u_3} R M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Defs.1263408075._hygCtx._hyg.9)))) (DistribSMul.toSMulZeroClass.{u_1, u_3} R M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Defs.1263408075._hygCtx._hyg.9)) (DistribMulAction.toDistribSMul.{u_1, u_3} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.Defs.1263408075._hygCtx._hyg.6)) (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.Defs.1263408075._hygCtx._hyg.9) (Module.toDistribMulAction.{u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Defs.1263408075._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Module.Defs.1263408075._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.Defs.1263408075._hygCtx._hyg.12))))) (HAdd.hAdd.{u_1, u_1, u_1} R R R (instHAdd.{u_1} R (Distrib.toAdd.{u_1} R (NonUnitalNonAssocSemiring.toDistrib.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Defs.1263408075._hygCtx._hyg.6))))) r s) x)","typeFull":"∀ {R : Type u_1} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] (r s : R) (x : M),\n  r • x + s • x = (r + s) • x","typeReadable":"∀ {R : Type u_1} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] (r s : R) (x : M),\n  r • x + s • x = (r + s) • x","typeReferences":[["Module"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["instHSMul"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["Eq"],["DistribSMul","toSMulZeroClass"],["Distrib","toAdd"],["NonUnitalNonAssocSemiring","toDistrib"],["instHAdd"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["HAdd","hAdd"],["Module","toDistribMulAction"],["AddCommMonoid","toAddCommSemigroup"],["HSMul","hSMul"],["AddZero","toZero"],["Semiring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["Distrib","toAdd"],["NonUnitalNonAssocSemiring","toDistrib"],["instHAdd"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["HAdd","hAdd"],["Module","toDistribMulAction"],["Semiring","toNonAssocSemiring"],["AddCommMonoid","toAddCommSemigroup"],["HSMul","hSMul"],["MonoidWithZero","toMonoid"],["Eq","symm"],["instHSMul"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["AddZero","toZero"],["add_smul"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","mulLeft","congr_simp"],"typeFallback":"forall (R : Type.{u_14}) {A : Type.{u_15}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3455109452._hygCtx._hyg.17 : Semiring.{u_14} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3455109452._hygCtx._hyg.20 : NonUnitalNonAssocSemiring.{u_15} A] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3455109452._hygCtx._hyg.23 : Module.{u_14, u_15} R A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3455109452._hygCtx._hyg.17 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3455109452._hygCtx._hyg.20)] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3455109452._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.3455109452._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.3455109452._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.3455109452._hygCtx._hyg.17)) (AddCommMonoid.toAddMonoid.{u_15} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3455109452._hygCtx._hyg.20)) (Module.toDistribMulAction.{u_14, u_15} R A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3455109452._hygCtx._hyg.17 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3455109452._hygCtx._hyg.20) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3455109452._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.3455109452._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.3455109452._hygCtx._hyg.20))) (instDistribSMul.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3455109452._hygCtx._hyg.20)))] (a : A) (a_1 : A), (Eq.{succ u_15} A a a_1) -> (Eq.{succ u_15} (LinearMap.{u_14, u_14, u_15, u_15} R R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3455109452._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3455109452._hygCtx._hyg.17 (RingHom.id.{u_14} R (Semiring.toNonAssocSemiring.{u_14} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3455109452._hygCtx._hyg.17)) A A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3455109452._hygCtx._hyg.20) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_15} A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3455109452._hygCtx._hyg.20) inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3455109452._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3455109452._hygCtx._hyg.23) (LinearMap.mulLeft.{u_14, u_15} R A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3455109452._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3455109452._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3455109452._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3455109452._hygCtx._hyg.27 a) (LinearMap.mulLeft.{u_14, u_15} R A inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3455109452._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3455109452._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3455109452._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.3455109452._hygCtx._hyg.27 a_1))","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_1 : A), a = a_1 → LinearMap.mulLeft R a = LinearMap.mulLeft R a_1","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_1 : A), a = a_1 → LinearMap.mulLeft R a = LinearMap.mulLeft R a_1","typeReferences":[["Module"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["LinearMap"],["instDistribSMul"],["AddZeroClass","toAddZero"],["SMulCommClass"],["Module","toDistribMulAction"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["MonoidWithZero","toMonoid"],["LinearMap","mulLeft"],["Eq"],["AddZero","toZero"],["NonUnitalNonAssocSemiring"],["Semiring"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]],"valueReferences":[["Semiring","toNonAssocSemiring"],["RingHom","id"],["Eq","refl"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["LinearMap","mulLeft"],["LinearMap"],["Eq"],["Eq","rec"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","mulLeft_mul"],"typeFallback":"forall (R : Type.{u_6}) (A : Type.{u_7}) [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.182380114._hygCtx._hyg.9 : Semiring.{u_6} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.182380114._hygCtx._hyg.12 : NonUnitalSemiring.{u_7} A] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.182380114._hygCtx._hyg.15 : Module.{u_6, u_7} R A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.182380114._hygCtx._hyg.9 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A (NonUnitalSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.182380114._hygCtx._hyg.12))] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.182380114._hygCtx._hyg.19 : SMulCommClass.{u_6, u_7, u_7} R A A (SMulZeroClass.toSMul.{u_6, u_7} R A (AddZero.toZero.{u_7} A (AddZeroClass.toAddZero.{u_7} A (AddMonoid.toAddZeroClass.{u_7} A (AddCommMonoid.toAddMonoid.{u_7} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A (NonUnitalSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.182380114._hygCtx._hyg.12)))))) (DistribSMul.toSMulZeroClass.{u_6, u_7} R A (AddMonoid.toAddZeroClass.{u_7} A (AddCommMonoid.toAddMonoid.{u_7} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A (NonUnitalSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.182380114._hygCtx._hyg.12)))) (DistribMulAction.toDistribSMul.{u_6, u_7} R A (MonoidWithZero.toMonoid.{u_6} R (Semiring.toMonoidWithZero.{u_6} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.182380114._hygCtx._hyg.9)) (AddCommMonoid.toAddMonoid.{u_7} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A (NonUnitalSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.182380114._hygCtx._hyg.12))) (Module.toDistribMulAction.{u_6, u_7} R A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.182380114._hygCtx._hyg.9 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A (NonUnitalSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.182380114._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.182380114._hygCtx._hyg.15)))) (SMulZeroClass.toSMul.{u_7, u_7} A A (AddZero.toZero.{u_7} A (AddZeroClass.toAddZero.{u_7} A (AddMonoid.toAddZeroClass.{u_7} A (AddCommMonoid.toAddMonoid.{u_7} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A (NonUnitalSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.182380114._hygCtx._hyg.12)))))) (DistribSMul.toSMulZeroClass.{u_7, u_7} A A (AddMonoid.toAddZeroClass.{u_7} A (AddCommMonoid.toAddMonoid.{u_7} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A (NonUnitalSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.182380114._hygCtx._hyg.12)))) (instDistribSMul.{u_7} A (NonUnitalSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.182380114._hygCtx._hyg.12))))] (a : A) (b : A), Eq.{succ u_7} (LinearMap.{u_6, u_6, u_7, u_7} R R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.182380114._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.182380114._hygCtx._hyg.9 (RingHom.id.{u_6} R (Semiring.toNonAssocSemiring.{u_6} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.182380114._hygCtx._hyg.9)) A A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A (NonUnitalSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.182380114._hygCtx._hyg.12)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A (NonUnitalSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.182380114._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.182380114._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.182380114._hygCtx._hyg.15) (LinearMap.mulLeft.{u_6, u_7} R A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.182380114._hygCtx._hyg.9 (NonUnitalSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.182380114._hygCtx._hyg.12) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.182380114._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.182380114._hygCtx._hyg.19 (HMul.hMul.{u_7, u_7, u_7} A A A (instHMul.{u_7} A (Distrib.toMul.{u_7} A (NonUnitalNonAssocSemiring.toDistrib.{u_7} A (NonUnitalSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.182380114._hygCtx._hyg.12)))) a b)) (LinearMap.comp.{u_6, u_6, u_6, u_7, u_7, u_7} R R R A A A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.182380114._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.182380114._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.182380114._hygCtx._hyg.9 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A (NonUnitalSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.182380114._hygCtx._hyg.12)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A (NonUnitalSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.182380114._hygCtx._hyg.12)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A (NonUnitalSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.182380114._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.182380114._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.182380114._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.182380114._hygCtx._hyg.15 (RingHom.id.{u_6} R (Semiring.toNonAssocSemiring.{u_6} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.182380114._hygCtx._hyg.9)) (RingHom.id.{u_6} R (Semiring.toNonAssocSemiring.{u_6} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.182380114._hygCtx._hyg.9)) (RingHom.id.{u_6} R (Semiring.toNonAssocSemiring.{u_6} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.182380114._hygCtx._hyg.9)) (RingHomCompTriple.ids.{u_6, u_6} R R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.182380114._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.182380114._hygCtx._hyg.9 (RingHom.id.{u_6} R (Semiring.toNonAssocSemiring.{u_6} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.182380114._hygCtx._hyg.9))) (LinearMap.mulLeft.{u_6, u_7} R A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.182380114._hygCtx._hyg.9 (NonUnitalSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.182380114._hygCtx._hyg.12) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.182380114._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.182380114._hygCtx._hyg.19 a) (LinearMap.mulLeft.{u_6, u_7} R A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.182380114._hygCtx._hyg.9 (NonUnitalSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.182380114._hygCtx._hyg.12) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.182380114._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.182380114._hygCtx._hyg.19 b))","typeFull":"∀ (R : Type u_6) (A : Type u_7) [inst : Semiring R] [inst_1 : NonUnitalSemiring A] [inst_2 : Module R A]\n  [inst_3 : SMulCommClass R A A] (a b : A), LinearMap.mulLeft R (a * b) = LinearMap.mulLeft R a ∘ₗ LinearMap.mulLeft R b","typeReadable":"∀ (R : Type u_6) (A : Type u_7) [inst : Semiring R] [inst_1 : NonUnitalSemiring A] [inst_2 : Module R A]\n  [inst_3 : SMulCommClass R A A] (a b : A), LinearMap.mulLeft R (a * b) = LinearMap.mulLeft R a ∘ₗ LinearMap.mulLeft R b","typeReferences":[["RingHomCompTriple","ids"],["Module"],["LinearMap","comp"],["HMul","hMul"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["instDistribSMul"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["MonoidWithZero","toMonoid"],["Eq"],["LinearMap","mulLeft"],["DistribSMul","toSMulZeroClass"],["NonUnitalNonAssocSemiring","toDistrib"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["Distrib","toMul"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["LinearMap"],["SMulCommClass"],["Module","toDistribMulAction"],["NonUnitalSemiring","toNonUnitalNonAssocSemiring"],["NonUnitalSemiring"],["instHMul"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["LinearMap","instFunLike"],["Eq","trans"],["RingHomCompTriple","ids"],["LinearMap","comp"],["HMul","hMul"],["NonUnitalSemiring","toSemigroupWithZero"],["mul_assoc"],["DFunLike","coe"],["Semigroup","toMul"],["congrArg"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["LinearMap","ext"],["congrFun'"],["LinearMap","mulLeft"],["Eq"],["NonUnitalNonAssocSemiring","toDistrib"],["True"],["Distrib","toMul"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["LinearMap"],["eq_self"],["of_eq_true"],["NonUnitalSemiring","toNonUnitalNonAssocSemiring"],["instHMul"],["SemigroupWithZero","toSemigroup"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","instModuleDomMulActOfSMulCommClass","_proof_4"],"typeFallback":"forall {R : Type.{u_3}} {R' : Type.{u_4}} {S : Type.{u_5}} {M : Type.{u_2}} {M' : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 : Semiring.{u_3} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 : Semiring.{u_4} R'] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 : AddCommMonoid.{u_1} M'] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 : Module.{u_3, u_2} R M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23 : Module.{u_4, u_1} R' M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16] {σ₁₂ : RingHom.{u_3, u_4} R R' (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7) (Semiring.toNonAssocSemiring.{u_4} R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10)} [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.32 : Semiring.{u_5} S] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.35 : Module.{u_5, u_2} S M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.32 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.48 : SMulCommClass.{u_3, u_5, u_2} R S M (SMulZeroClass.toSMul.{u_3, 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.Basic.3265207960._hygCtx._hyg.13)))) (DistribSMul.toSMulZeroClass.{u_3, u_2} R M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13)) (DistribMulAction.toDistribSMul.{u_3, u_2} R M (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13) (Module.toDistribMulAction.{u_3, u_2} R M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19)))) (SMulZeroClass.toSMul.{u_5, u_2} S 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.Basic.3265207960._hygCtx._hyg.13)))) (DistribSMul.toSMulZeroClass.{u_5, u_2} S M (AddMonoid.toAddZeroClass.{u_2} M (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13)) (DistribMulAction.toDistribSMul.{u_5, u_2} S M (MonoidWithZero.toMonoid.{u_5} S (Semiring.toMonoidWithZero.{u_5} S inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.32)) (AddCommMonoid.toAddMonoid.{u_2} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13) (Module.toDistribMulAction.{u_5, u_2} S M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.32 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.35))))] (x._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.89 : LinearMap.{u_3, u_4, u_2, u_1} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23) (x._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.94 : M), Eq.{succ u_1} M' (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 R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23) M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => M') (LinearMap.instFunLike.{u_3, u_4, u_2, u_1} R R' M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23 σ₁₂) (HSMul.hSMul.{u_5, max u_2 u_1, max u_2 u_1} (DomMulAct.{u_5} S) (LinearMap.{u_3, u_4, u_2, u_1} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23) (LinearMap.{u_3, u_4, u_2, u_1} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23) (instHSMul.{u_5, max u_2 u_1} (DomMulAct.{u_5} S) (LinearMap.{u_3, u_4, u_2, u_1} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 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y)","typeReferences":[["HAdd","hAdd"],["FunLike"],["instHAdd"],["Add"],["Eq"],["DFunLike","coe"],["AddHomClass"]],"valueReferences":[["map_add"],["HAdd","hAdd"],["instHAdd"],["Eq","symm"],["DFunLike","coe"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","mulLeft_inj"],"typeFallback":"forall {R : Type.{u_6}} {A : Type.{u_7}} [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.374716132._hygCtx._hyg.9 : Semiring.{u_6} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.374716132._hygCtx._hyg.12 : NonAssocSemiring.{u_7} A] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.374716132._hygCtx._hyg.15 : Module.{u_6, u_7} R A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.374716132._hygCtx._hyg.9 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.374716132._hygCtx._hyg.12))] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.374716132._hygCtx._hyg.19 : SMulCommClass.{u_6, u_7, u_7} R A A 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inst._@.Mathlib.Algebra.Module.LinearMap.Basic.374716132._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.374716132._hygCtx._hyg.15) (LinearMap.mulLeft.{u_6, u_7} R A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.374716132._hygCtx._hyg.9 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.374716132._hygCtx._hyg.12) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.374716132._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.374716132._hygCtx._hyg.19 a) (LinearMap.mulLeft.{u_6, u_7} R A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.374716132._hygCtx._hyg.9 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.374716132._hygCtx._hyg.12) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.374716132._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.374716132._hygCtx._hyg.19 b)) (Eq.{succ u_7} A a b)","typeFull":"∀ {R : Type u_6} {A : Type u_7} [inst : Semiring R] [inst_1 : NonAssocSemiring A] [inst_2 : Module R A]\n  [inst_3 : SMulCommClass R A A] {a b : A}, LinearMap.mulLeft R a = LinearMap.mulLeft R b ↔ a = b","typeReadable":"∀ {R : Type u_6} {A : Type u_7} [inst : Semiring R] [inst_1 : NonAssocSemiring A] [inst_2 : Module R A]\n  [inst_3 : SMulCommClass R A A] {a b : A}, LinearMap.mulLeft R a = LinearMap.mulLeft R b ↔ a = b","typeReferences":[["Module"],["SMulZeroClass","toSMul"],["AddMonoidWithOne","toAddMonoid"],["instDistribSMul"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["NonAssocSemiring"],["MonoidWithZero","toMonoid"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Eq"],["LinearMap","mulLeft"],["NonAssocSemiring","toAddCommMonoidWithOne"],["DistribSMul","toSMulZeroClass"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["LinearMap"],["SMulCommClass"],["Module","toDistribMulAction"],["Iff"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["LinearMap","instFunLike"],["Eq","mp"],["Iff","mp"],["MulZeroOneClass","toMulOneClass"],["DFunLike","coe"],["Iff","intro"],["congrArg"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["congr"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["LinearMap","mulLeft"],["Eq"],["Eq","rec"],["NonAssocSemiring","toAddCommMonoidWithOne"],["NonAssocSemiring","toMulZeroOneClass"],["rfl"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["mul_one"],["LinearMap"],["OfNat","ofNat"],["One","toOfNat1"],["AddMonoidWithOne","toOne"],["LinearMap","ext_iff"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","mulLeft_inj","_simp_1"],"typeFallback":"forall {R : Type.{u_6}} {A : Type.{u_7}} [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.374716132._hygCtx._hyg.9 : Semiring.{u_6} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.374716132._hygCtx._hyg.12 : NonAssocSemiring.{u_7} A] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.374716132._hygCtx._hyg.15 : Module.{u_6, u_7} R A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.374716132._hygCtx._hyg.9 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.374716132._hygCtx._hyg.12))] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.374716132._hygCtx._hyg.19 : SMulCommClass.{u_6, u_7, u_7} R A A (SMulZeroClass.toSMul.{u_6, u_7} R A (AddZero.toZero.{u_7} A (AddZeroClass.toAddZero.{u_7} A (AddMonoid.toAddZeroClass.{u_7} A (AddMonoidWithOne.toAddMonoid.{u_7} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u_7} A (NonAssocSemiring.toAddCommMonoidWithOne.{u_7} A 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inst._@.Mathlib.Algebra.Module.LinearMap.Basic.374716132._hygCtx._hyg.12) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.374716132._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.374716132._hygCtx._hyg.19 a) (LinearMap.mulLeft.{u_6, u_7} R A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.374716132._hygCtx._hyg.9 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.374716132._hygCtx._hyg.12) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.374716132._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.374716132._hygCtx._hyg.19 b)) (Eq.{succ u_7} A a b)","typeFull":"∀ {R : Type u_6} {A : Type u_7} [inst : Semiring R] [inst_1 : NonAssocSemiring A] [inst_2 : Module R A]\n  [inst_3 : SMulCommClass R A A] {a b : A}, (LinearMap.mulLeft R a = LinearMap.mulLeft R b) = (a = b)","typeReadable":"∀ {R : Type u_6} {A : Type u_7} [inst : Semiring R] [inst_1 : NonAssocSemiring A] [inst_2 : Module R A]\n  [inst_3 : SMulCommClass R A A] {a b : A}, (LinearMap.mulLeft R a = LinearMap.mulLeft R b) = (a = b)","typeReferences":[["Module"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["LinearMap"],["instDistribSMul"],["AddZeroClass","toAddZero"],["AddMonoidWithOne","toAddMonoid"],["SMulCommClass"],["Module","toDistribMulAction"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["MonoidWithZero","toMonoid"],["NonAssocSemiring"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["LinearMap","mulLeft"],["Eq"],["AddZero","toZero"],["NonAssocSemiring","toAddCommMonoidWithOne"],["DistribSMul","toSMulZeroClass"],["Semiring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["Semiring","toNonAssocSemiring"],["RingHom","id"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["LinearMap","mulLeft"],["LinearMap"],["Eq"],["LinearMap","mulLeft_inj"],["propext"]]},{"isProp":true,"kind":"theorem","name":["Sum","elimZeroRight","_proof_2"],"typeFallback":"forall {ι : Type.{u_1}} {κ : Type.{u_2}} {R : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2630281405._hygCtx._hyg.10 : Semiring.{u_3} R] (t : R) (f : ι -> R), Eq.{succ (max (max u_1 u_2) u_3)} ((Sum.{u_1, u_2} ι κ) -> R) (Sum.elim.{u_1, u_2, succ u_3} ι κ R (HSMul.hSMul.{u_3, max u_1 u_3, max u_1 u_3} R (ι -> R) (ι -> R) (instHSMul.{u_3, max u_1 u_3} R (ι -> R) (SMulZeroClass.toSMul.{u_3, max u_1 u_3} R (ι -> R) (AddZero.toZero.{max u_1 u_3} (ι -> R) (AddZeroClass.toAddZero.{max u_1 u_3} (ι -> R) (AddMonoid.toAddZeroClass.{max u_1 u_3} (ι -> R) (AddCommMonoid.toAddMonoid.{max u_1 u_3} (ι -> R) (Pi.addCommMonoid.{u_1, u_3} ι (fun (a._@._internal._hyg.0 : ι) => R) (fun (i : ι) => NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2630281405._hygCtx._hyg.10)))))))) (DistribSMul.toSMulZeroClass.{u_3, max u_1 u_3} R (ι -> R) (AddMonoid.toAddZeroClass.{max u_1 u_3} (ι -> R) (AddCommMonoid.toAddMonoid.{max u_1 u_3} (ι -> R) (Pi.addCommMonoid.{u_1, u_3} ι (fun (a._@._internal._hyg.0 : ι) => R) (fun (i : ι) => NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2630281405._hygCtx._hyg.10)))))) (DistribMulAction.toDistribSMul.{u_3, max u_1 u_3} R (ι -> R) (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2630281405._hygCtx._hyg.10)) (AddCommMonoid.toAddMonoid.{max u_1 u_3} (ι -> R) (Pi.addCommMonoid.{u_1, u_3} ι (fun (a._@._internal._hyg.0 : ι) => R) (fun (i : ι) => NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2630281405._hygCtx._hyg.10))))) (Module.toDistribMulAction.{u_3, max u_1 u_3} R (ι -> R) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2630281405._hygCtx._hyg.10 (Pi.addCommMonoid.{u_1, u_3} ι (fun (a._@._internal._hyg.0 : ι) => R) (fun (i : ι) => NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2630281405._hygCtx._hyg.10)))) (Pi.Function.module.{u_1, u_3, u_3} ι R R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2630281405._hygCtx._hyg.10 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2630281405._hygCtx._hyg.10))) (Semiring.toModule.{u_3} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2630281405._hygCtx._hyg.10))))))) t f) (OfNat.ofNat.{max u_2 u_3} (κ -> R) 0 (Zero.toOfNat0.{max u_2 u_3} (κ -> R) (Pi.instZero.{u_2, u_3} κ (fun (a._@._internal._hyg.0 : κ) => R) (fun (i : κ) => MulZeroClass.toZero.{u_3} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u_3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2630281405._hygCtx._hyg.10)))))))) (HSMul.hSMul.{u_3, max (max u_1 u_2) u_3, max (max u_1 u_2) u_3} R ((Sum.{u_1, u_2} ι κ) -> R) ((Sum.{u_1, u_2} ι κ) -> R) (instHSMul.{u_3, max (max u_1 u_2) u_3} R ((Sum.{u_1, u_2} ι κ) -> R) (SMulZeroClass.toSMul.{u_3, max (max u_1 u_2) u_3} R ((Sum.{u_1, u_2} ι κ) -> R) (AddZero.toZero.{max (max u_1 u_2) u_3} ((Sum.{u_1, u_2} ι κ) -> R) (AddZeroClass.toAddZero.{max (max u_1 u_2) u_3} ((Sum.{u_1, u_2} ι κ) -> R) (AddMonoid.toAddZeroClass.{max (max u_1 u_2) u_3} ((Sum.{u_1, u_2} ι κ) -> R) (AddCommMonoid.toAddMonoid.{max (max u_1 u_2) u_3} ((Sum.{u_1, u_2} ι κ) -> R) (Pi.addCommMonoid.{max u_1 u_2, u_3} (Sum.{u_1, u_2} ι κ) (fun (a._@._internal._hyg.0 : Sum.{u_1, u_2} ι κ) => R) (fun (i : Sum.{u_1, u_2} ι κ) => NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2630281405._hygCtx._hyg.10)))))))) (DistribSMul.toSMulZeroClass.{u_3, max (max u_1 u_2) u_3} R ((Sum.{u_1, u_2} ι κ) -> R) (AddMonoid.toAddZeroClass.{max (max u_1 u_2) u_3} ((Sum.{u_1, u_2} ι κ) -> R) (AddCommMonoid.toAddMonoid.{max (max u_1 u_2) u_3} ((Sum.{u_1, u_2} ι κ) -> R) (Pi.addCommMonoid.{max u_1 u_2, u_3} (Sum.{u_1, u_2} ι κ) (fun (a._@._internal._hyg.0 : Sum.{u_1, u_2} ι κ) => R) (fun (i : Sum.{u_1, u_2} ι κ) => NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2630281405._hygCtx._hyg.10)))))) (DistribMulAction.toDistribSMul.{u_3, max (max u_1 u_2) u_3} R ((Sum.{u_1, u_2} ι κ) -> R) (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2630281405._hygCtx._hyg.10)) (AddCommMonoid.toAddMonoid.{max (max u_1 u_2) u_3} ((Sum.{u_1, u_2} ι κ) -> R) (Pi.addCommMonoid.{max u_1 u_2, u_3} (Sum.{u_1, u_2} ι κ) (fun (a._@._internal._hyg.0 : Sum.{u_1, u_2} ι κ) => R) (fun (i : Sum.{u_1, u_2} ι κ) => NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2630281405._hygCtx._hyg.10))))) (Module.toDistribMulAction.{u_3, max (max u_1 u_2) u_3} R ((Sum.{u_1, u_2} ι κ) -> R) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2630281405._hygCtx._hyg.10 (Pi.addCommMonoid.{max u_1 u_2, u_3} (Sum.{u_1, u_2} ι κ) (fun (a._@._internal._hyg.0 : Sum.{u_1, u_2} ι κ) => R) (fun (i : Sum.{u_1, u_2} ι κ) => NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2630281405._hygCtx._hyg.10)))) (Pi.Function.module.{max u_1 u_2, u_3, u_3} (Sum.{u_1, u_2} ι κ) R R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2630281405._hygCtx._hyg.10 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2630281405._hygCtx._hyg.10))) (Semiring.toModule.{u_3} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2630281405._hygCtx._hyg.10))))))) (DFunLike.coe.{succ u_3, succ u_3, succ u_3} (RingHom.{u_3, u_3} R R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2630281405._hygCtx._hyg.10) (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2630281405._hygCtx._hyg.10)) R (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : R) => R) (RingHom.instFunLike.{u_3, u_3} R R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2630281405._hygCtx._hyg.10) (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2630281405._hygCtx._hyg.10)) (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2630281405._hygCtx._hyg.10)) t) (Sum.elim.{u_1, u_2, succ u_3} ι κ R f (OfNat.ofNat.{max u_2 u_3} (κ -> R) 0 (Zero.toOfNat0.{max u_2 u_3} (κ -> R) (Pi.instZero.{u_2, u_3} κ (fun (a._@._internal._hyg.0 : κ) => R) (fun (i : κ) => MulZeroClass.toZero.{u_3} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u_3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2630281405._hygCtx._hyg.10)))))))))","typeFull":"∀ {ι : Type u_1} {κ : Type u_2} {R : Type u_3} [inst : Semiring R] (t : R) (f : ι → R),\n  Sum.elim (t • f) 0 = (RingHom.id R) t • Sum.elim f 0","typeReadable":"∀ {ι : Type u_1} {κ : Type u_2} {R : Type u_3} [inst : Semiring R] (t : R) (f : ι → R),\n  Sum.elim (t • f) 0 = (RingHom.id R) t • Sum.elim f 0","typeReferences":[["RingHom"],["Sum","elim"],["Pi","addCommMonoid"],["RingHom","instFunLike"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["DFunLike","coe"],["Pi","Function","module"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["MonoidWithZero","toMonoid"],["instHSMul"],["Zero","toOfNat0"],["Eq"],["Semiring","toModule"],["DistribSMul","toSMulZeroClass"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["Sum"],["Module","toDistribMulAction"],["MulZeroClass","toZero"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["HSMul","hSMul"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Semiring"],["Pi","instZero"]],"valueReferences":[["RingHom"],["Eq","trans"],["Sum","elim"],["Pi","addCommMonoid"],["HMul","hMul"],["RingHom","instFunLike"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["DFunLike","coe"],["congrArg"],["Pi","Function","module"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["funext"],["MonoidWithZero","toMonoid"],["instHSMul"],["Zero","toOfNat0"],["Eq"],["Semiring","toModule"],["DistribSMul","toSMulZeroClass"],["True"],["NonUnitalNonAssocSemiring","toDistrib"],["Distrib","toMul"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["MulZeroClass","mul_zero"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["Sum"],["eq_self"],["Module","toDistribMulAction"],["of_eq_true"],["MulZeroClass","toZero"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["HSMul","hSMul"],["instHMul"],["Sum","casesOn"],["AddZero","toZero"],["Sum","inr"],["AddMonoid","toAddZeroClass"],["Pi","instZero"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","instSMulCommClassDomMulAct"],"typeFallback":"forall {R : Type.{u_1}} {R' : Type.{u_2}} {M : Type.{u_4}} {M' : Type.{u_5}} [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.7 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.10 : Semiring.{u_2} R'] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.13 : AddCommMonoid.{u_4} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.16 : AddCommMonoid.{u_5} M'] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.19 : Module.{u_1, u_4} R M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.13] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.23 : Module.{u_2, u_5} R' M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.16] {σ₁₂ : RingHom.{u_1, u_2} R R' (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.7) (Semiring.toNonAssocSemiring.{u_2} R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.10)} {S' : Type.{u_6}} {T' : Type.{u_7}} [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.34 : Monoid.{u_6} S'] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.37 : DistribMulAction.{u_6, u_4} S' M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.34 (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.13)] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.41 : SMulCommClass.{u_1, u_6, u_4} R S' M (SMulZeroClass.toSMul.{u_1, u_4} R M (AddZero.toZero.{u_4} M (AddZeroClass.toAddZero.{u_4} M (AddMonoid.toAddZeroClass.{u_4} M (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.13)))) (DistribSMul.toSMulZeroClass.{u_1, u_4} R M (AddMonoid.toAddZeroClass.{u_4} M (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.13)) (DistribMulAction.toDistribSMul.{u_1, u_4} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.13) (Module.toDistribMulAction.{u_1, u_4} R M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.19)))) (SMulZeroClass.toSMul.{u_6, u_4} S' M (AddZero.toZero.{u_4} M (AddZeroClass.toAddZero.{u_4} M (AddMonoid.toAddZeroClass.{u_4} M (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.13)))) (DistribSMul.toSMulZeroClass.{u_6, u_4} S' M (AddMonoid.toAddZeroClass.{u_4} M (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.13)) (DistribMulAction.toDistribSMul.{u_6, u_4} S' M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.34 (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.13) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.37)))] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.46 : Monoid.{u_7} T'] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.49 : DistribMulAction.{u_7, u_4} T' M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.46 (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.13)] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.53 : SMulCommClass.{u_1, u_7, u_4} R T' M (SMulZeroClass.toSMul.{u_1, u_4} R M (AddZero.toZero.{u_4} M (AddZeroClass.toAddZero.{u_4} M (AddMonoid.toAddZeroClass.{u_4} M (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.13)))) (DistribSMul.toSMulZeroClass.{u_1, u_4} R M (AddMonoid.toAddZeroClass.{u_4} M (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.13)) (DistribMulAction.toDistribSMul.{u_1, u_4} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.13) (Module.toDistribMulAction.{u_1, u_4} R M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.19)))) (SMulZeroClass.toSMul.{u_7, u_4} T' M (AddZero.toZero.{u_4} M (AddZeroClass.toAddZero.{u_4} M (AddMonoid.toAddZeroClass.{u_4} M (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.13)))) (DistribSMul.toSMulZeroClass.{u_7, u_4} T' M (AddMonoid.toAddZeroClass.{u_4} M (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.13)) (DistribMulAction.toDistribSMul.{u_7, u_4} T' M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.46 (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.13) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.49)))] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.58 : SMulCommClass.{u_6, u_7, u_4} S' T' M (SMulZeroClass.toSMul.{u_6, u_4} S' M (AddZero.toZero.{u_4} M (AddZeroClass.toAddZero.{u_4} M (AddMonoid.toAddZeroClass.{u_4} M (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.13)))) (DistribSMul.toSMulZeroClass.{u_6, u_4} S' M (AddMonoid.toAddZeroClass.{u_4} M (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.13)) (DistribMulAction.toDistribSMul.{u_6, u_4} S' M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.34 (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.13) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.37))) (SMulZeroClass.toSMul.{u_7, u_4} T' M (AddZero.toZero.{u_4} M (AddZeroClass.toAddZero.{u_4} M (AddMonoid.toAddZeroClass.{u_4} M (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.13)))) (DistribSMul.toSMulZeroClass.{u_7, u_4} T' M (AddMonoid.toAddZeroClass.{u_4} M (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.13)) (DistribMulAction.toDistribSMul.{u_7, u_4} T' M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.46 (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.13) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.49)))], SMulCommClass.{u_6, u_7, max u_5 u_4} (DomMulAct.{u_6} S') (DomMulAct.{u_7} T') (LinearMap.{u_1, u_2, u_4, u_5} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.23) (LinearMap.instSMulDomMulAct.{u_1, u_2, u_4, u_5, u_6} R R' M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.23 σ₁₂ S' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.34 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.37 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.41) (LinearMap.instSMulDomMulAct.{u_1, u_2, u_4, u_5, u_7} R R' M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.23 σ₁₂ T' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.46 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.49 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.354753555._hygCtx._hyg.53)","typeFull":"∀ {R : Type u_1} {R' : Type u_2} {M : Type u_4} {M' : Type u_5} [inst : Semiring R] [inst_1 : Semiring R']\n  [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M'] [inst_4 : Module R M] [inst_5 : Module R' M'] {σ₁₂ : R →+* R'}\n  {S' : Type u_6} {T' : Type u_7} [inst_6 : Monoid S'] [inst_7 : DistribMulAction S' M] [inst_8 : SMulCommClass R S' M]\n  [inst_9 : Monoid T'] [inst_10 : DistribMulAction T' M] [inst_11 : SMulCommClass R T' M] [SMulCommClass S' T' M],\n  SMulCommClass S'ᵈᵐᵃ T'ᵈᵐᵃ (M →ₛₗ[σ₁₂] M')","typeReadable":"∀ {R : Type u_1} {R' : Type u_2} {M : Type u_4} {M' : Type u_5} [inst : Semiring R] [inst_1 : Semiring R']\n  [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M'] [inst_4 : Module R M] [inst_5 : Module R' M'] {σ₁₂ : R →+* R'}\n  {S' : Type u_6} {T' : Type u_7} [inst_6 : Monoid S'] [inst_7 : DistribMulAction S' M] [inst_8 : SMulCommClass R S' M]\n  [inst_9 : Monoid T'] [inst_10 : DistribMulAction T' M] [inst_11 : SMulCommClass R T' M] [SMulCommClass S' T' M],\n  SMulCommClass S'ᵈᵐᵃ T'ᵈᵐᵃ (M →ₛₗ[σ₁₂] M')","typeReferences":[["RingHom"],["Module"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["DomMulAct"],["AddCommMonoid","toAddMonoid"],["LinearMap"],["AddZeroClass","toAddZero"],["LinearMap","instSMulDomMulAct"],["SMulCommClass"],["Module","toDistribMulAction"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["Monoid"],["AddZero","toZero"],["DistribMulAction"],["DistribSMul","toSMulZeroClass"],["Semiring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["Equiv","instEquivLike"],["LinearMap","instFunLike"],["Eq","trans"],["SMulCommClass","mk"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["SMulCommClass","smul_comm"],["DFunLike","coe"],["LinearMap","instSMulDomMulAct"],["Equiv"],["congrArg"],["LinearMap","ext"],["EquivLike","toFunLike"],["instHSMul"],["Equiv","symm"],["Eq"],["DistribSMul","toSMulZeroClass"],["True"],["DistribMulAction","toDistribSMul"],["DomMulAct"],["LinearMap"],["AddZeroClass","toAddZero"],["DomMulAct","mk"],["eq_self"],["of_eq_true"],["HSMul","hSMul"],["id"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","ltoFun","_proof_1"],"typeFallback":"forall 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inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.27) (AddCommSemigroup.toAddCommMagma.{max u_2 u_3} (LinearMap.{u_1, u_1, u_2, u_3} R R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11)) M N inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.27) (AddCommMonoid.toAddCommSemigroup.{max u_2 u_3} (LinearMap.{u_1, u_1, u_2, u_3} R R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11 (RingHom.id.{u_1} R 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x_1).toFun","typeReferences":[["instHAdd"],["Module"],["LinearMap","toAddHom"],["LinearMap"],["LinearMap","addCommMonoid"],["HAdd","hAdd"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["AddCommMonoid","toAddCommSemigroup"],["RingHom","id"],["AddHom","toFun"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"],["Eq"],["Semiring"]],"valueReferences":[["LinearMap","addCommMonoid"],["HAdd","hAdd"],["rfl"],["Semiring","toNonAssocSemiring"],["AddCommMonoid","toAddCommSemigroup"],["instHAdd"],["RingHom","id"],["LinearMap","toAddHom"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["AddHom","toFun"],["LinearMap"]]},{"isProp":true,"kind":"theorem","name":["DomMulAct","smul_linearMap_apply"],"typeFallback":"forall {R : Type.{u_1}} {R' : Type.{u_2}} {M : Type.{u_4}} {M' : Type.{u_5}} [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.391921328._hygCtx._hyg.7 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.391921328._hygCtx._hyg.10 : Semiring.{u_2} R'] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.391921328._hygCtx._hyg.13 : AddCommMonoid.{u_4} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.391921328._hygCtx._hyg.16 : AddCommMonoid.{u_5} M'] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.391921328._hygCtx._hyg.19 : Module.{u_1, u_4} R M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.391921328._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.391921328._hygCtx._hyg.13] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.391921328._hygCtx._hyg.23 : Module.{u_2, u_5} R' M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.391921328._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.391921328._hygCtx._hyg.16] {σ₁₂ : RingHom.{u_1, u_2} R R' (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.391921328._hygCtx._hyg.7) (Semiring.toNonAssocSemiring.{u_2} R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.391921328._hygCtx._hyg.10)} {S' : Type.{u_6}} 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(MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.391921328._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.391921328._hygCtx._hyg.13) (Module.toDistribMulAction.{u_1, u_4} R M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.391921328._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.391921328._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.391921328._hygCtx._hyg.19)))) (SMulZeroClass.toSMul.{u_6, u_4} S' M (AddZero.toZero.{u_4} M (AddZeroClass.toAddZero.{u_4} M (AddMonoid.toAddZeroClass.{u_4} M (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.391921328._hygCtx._hyg.13)))) (DistribSMul.toSMulZeroClass.{u_6, u_4} S' M (AddMonoid.toAddZeroClass.{u_4} M (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.391921328._hygCtx._hyg.13)) (DistribMulAction.toDistribSMul.{u_6, u_4} S' M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.391921328._hygCtx._hyg.34 (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.391921328._hygCtx._hyg.13) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.391921328._hygCtx._hyg.37)))] (a : DomMulAct.{u_6} S') (f : LinearMap.{u_1, u_2, u_4, u_5} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.391921328._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.391921328._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.391921328._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.391921328._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.391921328._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.391921328._hygCtx._hyg.23) (x : M), Eq.{succ u_5} M' (DFunLike.coe.{max (succ u_4) (succ u_5), succ u_4, succ u_5} (LinearMap.{u_1, u_2, u_4, u_5} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.391921328._hygCtx._hyg.7 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(instHSMul.{u_6, max u_4 u_5} (DomMulAct.{u_6} S') (LinearMap.{u_1, u_2, u_4, u_5} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.391921328._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.391921328._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.391921328._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.391921328._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.391921328._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.391921328._hygCtx._hyg.23) (LinearMap.instSMulDomMulAct.{u_1, u_2, u_4, u_5, u_6} R R' M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.391921328._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.391921328._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.391921328._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.391921328._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.391921328._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.391921328._hygCtx._hyg.23 σ₁₂ S' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.391921328._hygCtx._hyg.34 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.391921328._hygCtx._hyg.37 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.391921328._hygCtx._hyg.41)) a f) x) (DFunLike.coe.{max (succ u_4) (succ u_5), succ u_4, succ u_5} (LinearMap.{u_1, u_2, u_4, u_5} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.391921328._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.391921328._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.391921328._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.391921328._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.391921328._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.391921328._hygCtx._hyg.23) M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => M') (LinearMap.instFunLike.{u_1, u_2, u_4, u_5} R R' M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.391921328._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.391921328._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.391921328._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.391921328._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.391921328._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.391921328._hygCtx._hyg.23 σ₁₂) f (HSMul.hSMul.{u_6, u_4, u_4} S' M M (instHSMul.{u_6, u_4} S' M (SMulZeroClass.toSMul.{u_6, u_4} S' M (AddZero.toZero.{u_4} M (AddZeroClass.toAddZero.{u_4} M (AddMonoid.toAddZeroClass.{u_4} M (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.391921328._hygCtx._hyg.13)))) (DistribSMul.toSMulZeroClass.{u_6, u_4} S' M (AddMonoid.toAddZeroClass.{u_4} M (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.391921328._hygCtx._hyg.13)) (DistribMulAction.toDistribSMul.{u_6, u_4} S' M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.391921328._hygCtx._hyg.34 (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.391921328._hygCtx._hyg.13) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.391921328._hygCtx._hyg.37)))) (DFunLike.coe.{succ u_6, succ u_6, succ u_6} (Equiv.{succ u_6, succ u_6} (DomMulAct.{u_6} S') S') (DomMulAct.{u_6} S') (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : DomMulAct.{u_6} S') => S') (EquivLike.toFunLike.{succ u_6, succ u_6, succ u_6} (Equiv.{succ u_6, succ u_6} (DomMulAct.{u_6} S') S') (DomMulAct.{u_6} S') S' (Equiv.instEquivLike.{succ u_6, succ u_6} (DomMulAct.{u_6} S') S')) (Equiv.symm.{succ u_6, succ u_6} S' (DomMulAct.{u_6} S') (DomMulAct.mk.{u_6} S')) a) x))","typeFull":"∀ {R : Type u_1} {R' : Type u_2} {M : Type u_4} {M' : Type u_5} [inst : Semiring R] [inst_1 : Semiring R']\n  [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M'] [inst_4 : Module R M] [inst_5 : Module R' M'] {σ₁₂ : R →+* R'}\n  {S' : Type u_6} [inst_6 : Monoid S'] [inst_7 : DistribMulAction S' M] [inst_8 : SMulCommClass R S' M] (a : S'ᵈᵐᵃ)\n  (f : M →ₛₗ[σ₁₂] M') (x : M), (a • f) x = f (DomMulAct.mk.symm a • x)","typeReadable":"∀ {R : Type u_1} {R' : Type u_2} {M : Type u_4} {M' : Type u_5} [inst : Semiring R] [inst_1 : Semiring R']\n  [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M'] [inst_4 : Module R M] [inst_5 : Module R' M'] {σ₁₂ : R →+* R'}\n  {S' : Type u_6} [inst_6 : Monoid S'] [inst_7 : DistribMulAction S' M] [inst_8 : SMulCommClass R S' M] (a : S'ᵈᵐᵃ)\n  (f : M →ₛₗ[σ₁₂] M') (x : M), (a • f) x = f (DomMulAct.mk.symm a • x)","typeReferences":[["RingHom"],["Equiv","instEquivLike"],["LinearMap","instFunLike"],["Module"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["DFunLike","coe"],["Equiv"],["LinearMap","instSMulDomMulAct"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["EquivLike","toFunLike"],["MonoidWithZero","toMonoid"],["Equiv","symm"],["instHSMul"],["Eq"],["DistribMulAction"],["DistribSMul","toSMulZeroClass"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["DomMulAct"],["AddZeroClass","toAddZero"],["LinearMap"],["DomMulAct","mk"],["SMulCommClass"],["Module","toDistribMulAction"],["HSMul","hSMul"],["Monoid"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["rfl"],["LinearMap","instFunLike"],["HSMul","hSMul"],["instHSMul"],["DomMulAct"],["LinearMap"],["DFunLike","coe"],["LinearMap","instSMulDomMulAct"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","mulRight_inj","_simp_1"],"typeFallback":"forall {R : Type.{u_6}} {A : Type.{u_7}} [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2843790991._hygCtx._hyg.9 : Semiring.{u_6} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2843790991._hygCtx._hyg.12 : NonAssocSemiring.{u_7} A] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2843790991._hygCtx._hyg.15 : Module.{u_6, u_7} R A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2843790991._hygCtx._hyg.9 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2843790991._hygCtx._hyg.12))] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2843790991._hygCtx._hyg.19 : IsScalarTower.{u_6, u_7, u_7} R A A (SMulZeroClass.toSMul.{u_6, u_7} R A (AddZero.toZero.{u_7} A (AddZeroClass.toAddZero.{u_7} A (AddMonoid.toAddZeroClass.{u_7} A (AddMonoidWithOne.toAddMonoid.{u_7} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u_7} A (NonAssocSemiring.toAddCommMonoidWithOne.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2843790991._hygCtx._hyg.12)))))) (DistribSMul.toSMulZeroClass.{u_6, u_7} R A (AddMonoid.toAddZeroClass.{u_7} A (AddMonoidWithOne.toAddMonoid.{u_7} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u_7} A (NonAssocSemiring.toAddCommMonoidWithOne.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2843790991._hygCtx._hyg.12)))) (DistribMulAction.toDistribSMul.{u_6, u_7} R A (MonoidWithZero.toMonoid.{u_6} R (Semiring.toMonoidWithZero.{u_6} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2843790991._hygCtx._hyg.9)) (AddMonoidWithOne.toAddMonoid.{u_7} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u_7} A (NonAssocSemiring.toAddCommMonoidWithOne.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2843790991._hygCtx._hyg.12))) (Module.toDistribMulAction.{u_6, u_7} R A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2843790991._hygCtx._hyg.9 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2843790991._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2843790991._hygCtx._hyg.15)))) (SMulZeroClass.toSMul.{u_7, u_7} A A (AddZero.toZero.{u_7} A (AddZeroClass.toAddZero.{u_7} A (AddMonoid.toAddZeroClass.{u_7} A (AddMonoidWithOne.toAddMonoid.{u_7} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u_7} A (NonAssocSemiring.toAddCommMonoidWithOne.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2843790991._hygCtx._hyg.12)))))) (DistribSMul.toSMulZeroClass.{u_7, u_7} A A (AddMonoid.toAddZeroClass.{u_7} A (AddMonoidWithOne.toAddMonoid.{u_7} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u_7} A (NonAssocSemiring.toAddCommMonoidWithOne.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2843790991._hygCtx._hyg.12)))) (instDistribSMul.{u_7} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2843790991._hygCtx._hyg.12)))) (SMulZeroClass.toSMul.{u_6, u_7} R A (AddZero.toZero.{u_7} A (AddZeroClass.toAddZero.{u_7} A (AddMonoid.toAddZeroClass.{u_7} A (AddMonoidWithOne.toAddMonoid.{u_7} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u_7} A (NonAssocSemiring.toAddCommMonoidWithOne.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2843790991._hygCtx._hyg.12)))))) (DistribSMul.toSMulZeroClass.{u_6, u_7} R A (AddMonoid.toAddZeroClass.{u_7} A (AddMonoidWithOne.toAddMonoid.{u_7} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u_7} A (NonAssocSemiring.toAddCommMonoidWithOne.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2843790991._hygCtx._hyg.12)))) (DistribMulAction.toDistribSMul.{u_6, u_7} R A (MonoidWithZero.toMonoid.{u_6} R (Semiring.toMonoidWithZero.{u_6} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2843790991._hygCtx._hyg.9)) (AddMonoidWithOne.toAddMonoid.{u_7} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u_7} A (NonAssocSemiring.toAddCommMonoidWithOne.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2843790991._hygCtx._hyg.12))) (Module.toDistribMulAction.{u_6, u_7} R A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2843790991._hygCtx._hyg.9 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2843790991._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2843790991._hygCtx._hyg.15))))] {a : A} {b : A}, Eq.{1} Prop (Eq.{succ u_7} (LinearMap.{u_6, u_6, u_7, u_7} R R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2843790991._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2843790991._hygCtx._hyg.9 (RingHom.id.{u_6} R (Semiring.toNonAssocSemiring.{u_6} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2843790991._hygCtx._hyg.9)) A A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2843790991._hygCtx._hyg.12)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2843790991._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2843790991._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2843790991._hygCtx._hyg.15) (LinearMap.mulRight.{u_6, u_7} R A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2843790991._hygCtx._hyg.9 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2843790991._hygCtx._hyg.12) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2843790991._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2843790991._hygCtx._hyg.19 a) (LinearMap.mulRight.{u_6, u_7} R A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2843790991._hygCtx._hyg.9 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2843790991._hygCtx._hyg.12) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2843790991._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2843790991._hygCtx._hyg.19 b)) (Eq.{succ u_7} A a b)","typeFull":"∀ {R : Type u_6} {A : Type u_7} [inst : Semiring R] [inst_1 : NonAssocSemiring A] [inst_2 : Module R A]\n  [inst_3 : IsScalarTower R A A] {a b : A}, (LinearMap.mulRight R a = LinearMap.mulRight R b) = (a = b)","typeReadable":"∀ {R : Type u_6} {A : Type u_7} [inst : Semiring R] [inst_1 : NonAssocSemiring A] [inst_2 : Module R A]\n  [inst_3 : IsScalarTower R A A] {a b : A}, (LinearMap.mulRight R a = LinearMap.mulRight R b) = (a = b)","typeReferences":[["Module"],["IsScalarTower"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["LinearMap"],["instDistribSMul"],["AddZeroClass","toAddZero"],["AddMonoidWithOne","toAddMonoid"],["Module","toDistribMulAction"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["MonoidWithZero","toMonoid"],["NonAssocSemiring"],["LinearMap","mulRight"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Eq"],["AddZero","toZero"],["NonAssocSemiring","toAddCommMonoidWithOne"],["DistribSMul","toSMulZeroClass"],["Semiring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["Semiring","toNonAssocSemiring"],["LinearMap","mulRight_inj"],["RingHom","id"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["LinearMap","mulRight"],["LinearMap"],["Eq"],["propext"]]},{"isProp":true,"kind":"theorem","name":["Sum","elimZeroRight","_proof_1"],"typeFallback":"forall {ι : Type.{u_1}} {κ : Type.{u_2}} {R : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2630281405._hygCtx._hyg.10 : Semiring.{u_3} R] (f : ι -> R) (g : ι -> R), Eq.{succ (max (max u_1 u_2) u_3)} ((Sum.{u_1, u_2} ι κ) -> R) (Sum.elim.{u_1, u_2, succ u_3} ι κ R (HAdd.hAdd.{max u_1 u_3, max u_1 u_3, max u_1 u_3} (ι -> R) (ι -> R) (ι -> R) (instHAdd.{max u_1 u_3} (ι -> R) (AddCommMagma.toAdd.{max u_1 u_3} (ι -> R) (AddCommSemigroup.toAddCommMagma.{max u_1 u_3} (ι -> R) (AddCommMonoid.toAddCommSemigroup.{max u_1 u_3} (ι -> R) (Pi.addCommMonoid.{u_1, u_3} ι (fun (a._@._internal._hyg.0 : ι) => R) (fun (i : ι) => NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2630281405._hygCtx._hyg.10)))))))) f g) (OfNat.ofNat.{max u_2 u_3} (κ -> R) 0 (Zero.toOfNat0.{max u_2 u_3} (κ -> R) (Pi.instZero.{u_2, u_3} κ (fun (a._@._internal._hyg.0 : κ) => R) (fun (i : κ) => MulZeroClass.toZero.{u_3} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u_3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2630281405._hygCtx._hyg.10)))))))) (HAdd.hAdd.{max (max u_1 u_2) u_3, max (max u_1 u_2) u_3, max (max u_1 u_2) u_3} ((Sum.{u_1, u_2} ι κ) -> R) ((Sum.{u_1, u_2} ι κ) -> R) ((Sum.{u_1, u_2} ι κ) -> R) (instHAdd.{max (max u_1 u_2) u_3} ((Sum.{u_1, u_2} ι κ) -> R) (AddCommMagma.toAdd.{max (max u_1 u_2) u_3} ((Sum.{u_1, u_2} ι κ) -> R) (AddCommSemigroup.toAddCommMagma.{max (max u_1 u_2) u_3} ((Sum.{u_1, u_2} ι κ) -> R) (AddCommMonoid.toAddCommSemigroup.{max (max u_1 u_2) u_3} ((Sum.{u_1, u_2} ι κ) -> R) (Pi.addCommMonoid.{max u_1 u_2, u_3} (Sum.{u_1, u_2} ι κ) (fun (a._@._internal._hyg.0 : Sum.{u_1, u_2} ι κ) => R) (fun (i : Sum.{u_1, u_2} ι κ) => NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} R 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inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2630281405._hygCtx._hyg.10)))))))))","typeFull":"∀ {ι : Type u_1} {κ : Type u_2} {R : Type u_3} [inst : Semiring R] (f g : ι → R),\n  Sum.elim (f + g) 0 = Sum.elim f 0 + Sum.elim g 0","typeReadable":"∀ {ι : Type u_1} {κ : Type u_2} {R : Type u_3} [inst : Semiring R] (f g : ι → R),\n  Sum.elim (f + g) 0 = Sum.elim f 0 + Sum.elim g 0","typeReferences":[["Sum","elim"],["instHAdd"],["Pi","addCommMonoid"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["OfNat","ofNat"],["Sum"],["HAdd","hAdd"],["Semiring","toNonAssocSemiring"],["AddCommMonoid","toAddCommSemigroup"],["MulZeroClass","toZero"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["Zero","toOfNat0"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"],["Eq"],["Pi","instZero"],["Semiring"]],"valueReferences":[["Eq","trans"],["Sum","elim"],["Pi","addCommMonoid"],["AddCommMonoid","toAddMonoid"],["AddMonoidWithOne","toAddMonoid"],["congrArg"],["Semiring","toNonAssocSemiring"],["funext"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["Zero","toOfNat0"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Eq"],["NonAssocSemiring","toAddCommMonoidWithOne"],["AddSemigroup","toAdd"],["True"],["instHAdd"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["add_zero"],["OfNat","ofNat"],["Sum"],["HAdd","hAdd"],["eq_self"],["AddCommMonoid","toAddCommSemigroup"],["of_eq_true"],["MulZeroClass","toZero"],["AddMonoid","toAddSemigroup"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["Sum","casesOn"],["Sum","inr"],["AddMonoid","toAddZeroClass"],["Pi","instZero"]]},{"isProp":true,"kind":"theorem","name":["DomMulAct","coe_smul_linearMap"],"typeFallback":"forall {R : Type.{u_1}} {R' : Type.{u_2}} {M : Type.{u_4}} {M' : Type.{u_5}} [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2760390637._hygCtx._hyg.7 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2760390637._hygCtx._hyg.10 : Semiring.{u_2} R'] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2760390637._hygCtx._hyg.13 : AddCommMonoid.{u_4} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2760390637._hygCtx._hyg.16 : AddCommMonoid.{u_5} M'] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2760390637._hygCtx._hyg.19 : Module.{u_1, u_4} R M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2760390637._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2760390637._hygCtx._hyg.13] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2760390637._hygCtx._hyg.23 : Module.{u_2, u_5} R' M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2760390637._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2760390637._hygCtx._hyg.16] {σ₁₂ : RingHom.{u_1, u_2} R R' (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2760390637._hygCtx._hyg.7) (Semiring.toNonAssocSemiring.{u_2} R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2760390637._hygCtx._hyg.10)} {S' : Type.{u_6}} [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2760390637._hygCtx._hyg.34 : Monoid.{u_6} S'] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2760390637._hygCtx._hyg.37 : DistribMulAction.{u_6, u_4} S' M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2760390637._hygCtx._hyg.34 (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2760390637._hygCtx._hyg.13)] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2760390637._hygCtx._hyg.41 : SMulCommClass.{u_1, u_6, u_4} R S' M (SMulZeroClass.toSMul.{u_1, u_4} R M (AddZero.toZero.{u_4} M (AddZeroClass.toAddZero.{u_4} M (AddMonoid.toAddZeroClass.{u_4} M (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2760390637._hygCtx._hyg.13)))) (DistribSMul.toSMulZeroClass.{u_1, u_4} R M (AddMonoid.toAddZeroClass.{u_4} M (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2760390637._hygCtx._hyg.13)) (DistribMulAction.toDistribSMul.{u_1, u_4} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2760390637._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2760390637._hygCtx._hyg.13) (Module.toDistribMulAction.{u_1, u_4} R M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2760390637._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2760390637._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2760390637._hygCtx._hyg.19)))) (SMulZeroClass.toSMul.{u_6, u_4} S' M (AddZero.toZero.{u_4} M (AddZeroClass.toAddZero.{u_4} M (AddMonoid.toAddZeroClass.{u_4} M (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2760390637._hygCtx._hyg.13)))) (DistribSMul.toSMulZeroClass.{u_6, u_4} S' M (AddMonoid.toAddZeroClass.{u_4} M (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2760390637._hygCtx._hyg.13)) (DistribMulAction.toDistribSMul.{u_6, u_4} S' M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2760390637._hygCtx._hyg.34 (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2760390637._hygCtx._hyg.13) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2760390637._hygCtx._hyg.37)))] (a : DomMulAct.{u_6} S') (f : LinearMap.{u_1, u_2, u_4, u_5} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2760390637._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2760390637._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2760390637._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2760390637._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2760390637._hygCtx._hyg.19 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Module R M] [inst_5 : Module R' M'] {σ₁₂ : R →+* R'}\n  {S' : Type u_6} [inst_6 : Monoid S'] [inst_7 : DistribMulAction S' M] [inst_8 : SMulCommClass R S' M] (a : S'ᵈᵐᵃ)\n  (f : M →ₛₗ[σ₁₂] M'), ⇑(a • f) = a • ⇑f","typeReadable":"∀ {R : Type u_1} {R' : Type u_2} {M : Type u_4} {M' : Type u_5} [inst : Semiring R] [inst_1 : Semiring R']\n  [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M'] [inst_4 : Module R M] [inst_5 : Module R' M'] {σ₁₂ : R →+* R'}\n  {S' : Type u_6} [inst_6 : Monoid S'] [inst_7 : DistribMulAction S' M] [inst_8 : SMulCommClass R S' M] (a : S'ᵈᵐᵃ)\n  (f : M →ₛₗ[σ₁₂] M'), ⇑(a • f) = a • ⇑f","typeReferences":[["RingHom"],["LinearMap","instFunLike"],["Module"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["DFunLike","coe"],["LinearMap","instSMulDomMulAct"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["instHSMul"],["Eq"],["DistribMulAction"],["DistribSMul","toSMulZeroClass"],["DomMulAct","instSMulForall"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["DomMulAct"],["AddZeroClass","toAddZero"],["LinearMap"],["SMulCommClass"],["Module","toDistribMulAction"],["HSMul","hSMul"],["Monoid"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["rfl"],["LinearMap","instFunLike"],["HSMul","hSMul"],["instHSMul"],["DomMulAct"],["LinearMap"],["DFunLike","coe"],["LinearMap","instSMulDomMulAct"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","instSMulDomMulAct","_proof_1"],"typeFallback":"forall {R : Type.{u_4}} {R' : Type.{u_5}} {M : Type.{u_3}} {M' : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.7 : Semiring.{u_4} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.10 : Semiring.{u_5} R'] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.13 : AddCommMonoid.{u_3} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.16 : AddCommMonoid.{u_1} M'] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.19 : Module.{u_4, u_3} R M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.13] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.23 : Module.{u_5, u_1} R' M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.16] {σ₁₂ : RingHom.{u_4, u_5} R R' (Semiring.toNonAssocSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.7) (Semiring.toNonAssocSemiring.{u_5} R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.10)} {S' : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.34 : Monoid.{u_2} S'] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.37 : DistribMulAction.{u_2, u_3} S' M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.34 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.13)] (a : DomMulAct.{u_2} S') (f : LinearMap.{u_4, u_5, u_3, u_1} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.23) (x : M) (y : M), Eq.{succ u_1} M' (HSMul.hSMul.{u_2, max u_3 u_1, max u_3 u_1} (DomMulAct.{u_2} S') (M -> M') (M -> M') (instHSMul.{u_2, max u_3 u_1} (DomMulAct.{u_2} S') (M -> M') (DomMulAct.instSMulForall.{u_2, u_1, u_3} S' M' M (SMulZeroClass.toSMul.{u_2, u_3} S' M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.13)))) (DistribSMul.toSMulZeroClass.{u_2, u_3} S' M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.13)) (DistribMulAction.toDistribSMul.{u_2, u_3} S' M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.34 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.13) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.37))))) a (DFunLike.coe.{max (succ u_3) (succ u_1), succ u_3, succ u_1} (LinearMap.{u_4, u_5, u_3, u_1} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.23) M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => M') (LinearMap.instFunLike.{u_4, u_5, u_3, u_1} R R' M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.23 σ₁₂) 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 (AddCommMonoid.toAddCommSemigroup.{u_3} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.13)))) x y)) (HAdd.hAdd.{u_1, u_1, u_1} M' M' M' (instHAdd.{u_1} M' (AddCommMagma.toAdd.{u_1} M' (AddCommSemigroup.toAddCommMagma.{u_1} M' (AddCommMonoid.toAddCommSemigroup.{u_1} M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.16)))) (HSMul.hSMul.{u_2, max u_3 u_1, max u_3 u_1} (DomMulAct.{u_2} S') (M -> M') (M -> M') (instHSMul.{u_2, max u_3 u_1} (DomMulAct.{u_2} S') (M -> M') (DomMulAct.instSMulForall.{u_2, u_1, u_3} S' M' M 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inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.23) M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => M') (LinearMap.instFunLike.{u_4, u_5, u_3, u_1} R R' M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.23 σ₁₂) f) x) (HSMul.hSMul.{u_2, max u_3 u_1, max u_3 u_1} (DomMulAct.{u_2} S') (M -> M') (M -> M') (instHSMul.{u_2, max u_3 u_1} (DomMulAct.{u_2} S') (M -> M') (DomMulAct.instSMulForall.{u_2, u_1, u_3} S' M' M (SMulZeroClass.toSMul.{u_2, u_3} S' M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.13)))) (DistribSMul.toSMulZeroClass.{u_2, u_3} S' M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.13)) (DistribMulAction.toDistribSMul.{u_2, u_3} S' M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.34 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.13) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.37))))) a (DFunLike.coe.{max (succ u_3) (succ u_1), succ u_3, succ u_1} (LinearMap.{u_4, u_5, u_3, u_1} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.23) M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => M') (LinearMap.instFunLike.{u_4, u_5, u_3, u_1} R R' M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.23 σ₁₂) f) y))","typeFull":"∀ {R : Type u_4} {R' : Type u_5} {M : Type u_3} {M' : Type u_1} [inst : Semiring R] [inst_1 : Semiring R']\n  [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M'] [inst_4 : Module R M] [inst_5 : Module R' M'] {σ₁₂ : R →+* R'}\n  {S' : Type u_2} [inst_6 : Monoid S'] [inst_7 : DistribMulAction S' M] (a : S'ᵈᵐᵃ) (f : M →ₛₗ[σ₁₂] M') (x y : M),\n  (a • ⇑f) (x + y) = (a • ⇑f) x + (a • ⇑f) y","typeReadable":"∀ {R : Type u_4} {R' : Type u_5} {M : Type u_3} {M' : Type u_1} [inst : Semiring R] [inst_1 : Semiring R']\n  [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M'] [inst_4 : Module R M] [inst_5 : Module R' M'] {σ₁₂ : R →+* R'}\n  {S' : Type u_2} [inst_6 : Monoid S'] [inst_7 : DistribMulAction S' M] (a : S'ᵈᵐᵃ) (f : M →ₛₗ[σ₁₂] M') (x y : M),\n  (a • ⇑f) (x + y) = (a • ⇑f) x + (a • ⇑f) y","typeReferences":[["RingHom"],["LinearMap","instFunLike"],["Module"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["DFunLike","coe"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["instHSMul"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["Eq"],["DistribMulAction"],["DistribSMul","toSMulZeroClass"],["DomMulAct","instSMulForall"],["instHAdd"],["DistribMulAction","toDistribSMul"],["DomMulAct"],["LinearMap"],["AddZeroClass","toAddZero"],["HAdd","hAdd"],["AddCommMonoid","toAddCommSemigroup"],["HSMul","hSMul"],["Monoid"],["AddZero","toZero"],["Semiring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["LinearMap","map_add"],["Equiv","instEquivLike"],["Eq","trans"],["LinearMap","instFunLike"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["DFunLike","coe"],["Equiv"],["congrArg"],["EquivLike","toFunLike"],["instHSMul"],["Equiv","symm"],["congrFun'"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"],["Eq"],["DistribSMul","toSMulZeroClass"],["DomMulAct","instSMulForall"],["True"],["instHAdd"],["DistribMulAction","toDistribSMul"],["DomMulAct"],["AddZero","toAdd"],["AddZeroClass","toAddZero"],["LinearMap"],["DomMulAct","mk"],["HAdd","hAdd"],["eq_self"],["of_eq_true"],["AddCommMonoid","toAddCommSemigroup"],["HSMul","hSMul"],["smul_add"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]]},{"isProp":false,"kind":"definition","name":["LinearMap","instSMulDomMulAct"],"typeFallback":"forall {R : Type.{u_1}} {R' : Type.{u_2}} {M : Type.{u_4}} {M' : Type.{u_5}} [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.7 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.10 : Semiring.{u_2} R'] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.13 : AddCommMonoid.{u_4} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.16 : AddCommMonoid.{u_5} M'] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.19 : Module.{u_1, u_4} R M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.13] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.23 : Module.{u_2, u_5} R' M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.16] {σ₁₂ : RingHom.{u_1, u_2} R R' (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.7) (Semiring.toNonAssocSemiring.{u_2} R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.10)} {S' : Type.{u_6}} [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.34 : Monoid.{u_6} S'] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.37 : DistribMulAction.{u_6, u_4} S' M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.34 (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.13)] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.41 : SMulCommClass.{u_1, u_6, u_4} R S' M (SMulZeroClass.toSMul.{u_1, u_4} R M (AddZero.toZero.{u_4} M (AddZeroClass.toAddZero.{u_4} M (AddMonoid.toAddZeroClass.{u_4} M (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.13)))) (DistribSMul.toSMulZeroClass.{u_1, u_4} R M (AddMonoid.toAddZeroClass.{u_4} M (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.13)) (DistribMulAction.toDistribSMul.{u_1, u_4} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.13) (Module.toDistribMulAction.{u_1, u_4} R M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.19)))) (SMulZeroClass.toSMul.{u_6, u_4} S' M (AddZero.toZero.{u_4} M (AddZeroClass.toAddZero.{u_4} M (AddMonoid.toAddZeroClass.{u_4} M (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.13)))) (DistribSMul.toSMulZeroClass.{u_6, u_4} S' M (AddMonoid.toAddZeroClass.{u_4} M (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.13)) (DistribMulAction.toDistribSMul.{u_6, u_4} S' M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.34 (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.13) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.37)))], SMul.{u_6, max u_5 u_4} (DomMulAct.{u_6} S') (LinearMap.{u_1, u_2, u_4, u_5} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3852489289._hygCtx._hyg.23)","typeFull":"{R : Type u_1} →\n  {R' : Type u_2} →\n    {M : Type u_4} →\n      {M' : Type u_5} →\n        [inst : Semiring R] →\n          [inst_1 : Semiring R'] →\n            [inst_2 : AddCommMonoid M] →\n              [inst_3 : AddCommMonoid M'] →\n                [inst_4 : Module R M] →\n                  [inst_5 : Module R' M'] →\n                    {σ₁₂ : R →+* R'} →\n                      {S' : Type u_6} →\n                        [inst_6 : Monoid S'] →\n                          [inst_7 : DistribMulAction S' M] → [SMulCommClass R S' M] → SMul S'ᵈᵐᵃ (M →ₛₗ[σ₁₂] M')","typeReadable":"{R : Type u_1} →\n  {R' : Type u_2} →\n    {M : Type u_4} →\n      {M' : Type u_5} →\n        [inst : Semiring R] →\n          [inst_1 : Semiring R'] →\n            [inst_2 : AddCommMonoid M] →\n              [inst_3 : AddCommMonoid M'] →\n                [inst_4 : Module R M] →\n                  [inst_5 : Module R' M'] →\n                    {σ₁₂ : R →+* R'} →\n                      {S' : Type u_6} →\n                        [inst_6 : Monoid S'] →\n                          [inst_7 : DistribMulAction S' M] → [SMulCommClass R S' M] → SMul S'ᵈᵐᵃ (M →ₛₗ[σ₁₂] M')","typeReferences":[["RingHom"],["Module"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["DomMulAct"],["AddCommMonoid","toAddMonoid"],["LinearMap"],["AddZeroClass","toAddZero"],["SMulCommClass"],["SMul"],["Module","toDistribMulAction"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["Monoid"],["AddZero","toZero"],["DistribMulAction"],["DistribSMul","toSMulZeroClass"],["Semiring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["SMul","mk"],["DomMulAct","instSMulForall"],["LinearMap","instFunLike"],["LinearMap","mk"],["AddHom","mk"],["DistribMulAction","toDistribSMul"],["LinearMap","instSMulDomMulAct","_proof_3"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["DomMulAct"],["AddZeroClass","toAddZero"],["LinearMap"],["LinearMap","instSMulDomMulAct","_proof_1"],["DFunLike","coe"],["AddCommMonoid","toAddCommSemigroup"],["HSMul","hSMul"],["instHSMul"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"],["AddZero","toZero"],["DistribSMul","toSMulZeroClass"],["AddMonoid","toAddZeroClass"]]},{"isProp":false,"kind":"definition","name":["LinearMap","instDistribMulActionDomMulActOfSMulCommClass"],"typeFallback":"forall {R : Type.{u_1}} {R' : Type.{u_2}} {M : Type.{u_4}} {M' : Type.{u_5}} [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.7 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.10 : Semiring.{u_2} R'] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13 : AddCommMonoid.{u_4} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.16 : AddCommMonoid.{u_5} M'] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.19 : Module.{u_1, u_4} R M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.23 : Module.{u_2, u_5} R' M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.16] {σ₁₂ : RingHom.{u_1, u_2} R R' (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.7) (Semiring.toNonAssocSemiring.{u_2} R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.10)} {S' : Type.{u_6}} [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.33 : Monoid.{u_6} S'] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.36 : DistribMulAction.{u_6, u_4} S' M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.33 (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13)] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.40 : SMulCommClass.{u_1, u_6, u_4} R S' M (SMulZeroClass.toSMul.{u_1, u_4} R M (AddZero.toZero.{u_4} M (AddZeroClass.toAddZero.{u_4} M (AddMonoid.toAddZeroClass.{u_4} M (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13)))) (DistribSMul.toSMulZeroClass.{u_1, u_4} R M (AddMonoid.toAddZeroClass.{u_4} M (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13)) (DistribMulAction.toDistribSMul.{u_1, u_4} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13) (Module.toDistribMulAction.{u_1, u_4} R M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.19)))) (SMulZeroClass.toSMul.{u_6, u_4} S' M (AddZero.toZero.{u_4} M (AddZeroClass.toAddZero.{u_4} M (AddMonoid.toAddZeroClass.{u_4} M (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13)))) (DistribSMul.toSMulZeroClass.{u_6, u_4} S' M (AddMonoid.toAddZeroClass.{u_4} M (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13)) (DistribMulAction.toDistribSMul.{u_6, u_4} S' M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.33 (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.36)))], DistribMulAction.{u_6, max u_5 u_4} (DomMulAct.{u_6} S') (LinearMap.{u_1, u_2, u_4, u_5} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.23) (DomMulAct.instMonoidOfMulOpposite.{u_6} S' (MulOpposite.instMonoid.{u_6} S' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.33)) (AddCommMonoid.toAddMonoid.{max u_4 u_5} (LinearMap.{u_1, u_2, u_4, u_5} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.23) (LinearMap.addCommMonoid.{u_1, u_2, u_4, u_5} R R' M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.23 σ₁₂))","typeFull":"{R : Type u_1} →\n  {R' : Type u_2} →\n    {M : Type u_4} →\n      {M' : Type u_5} →\n        [inst : Semiring R] →\n          [inst_1 : Semiring R'] →\n            [inst_2 : AddCommMonoid M] →\n              [inst_3 : AddCommMonoid M'] →\n                [inst_4 : Module R M] →\n                  [inst_5 : Module R' M'] →\n                    {σ₁₂ : R →+* R'} →\n                      {S' : Type u_6} →\n                        [inst_6 : Monoid S'] →\n                          [inst_7 : DistribMulAction S' M] →\n                            [SMulCommClass R S' M] → DistribMulAction S'ᵈᵐᵃ (M →ₛₗ[σ₁₂] M')","typeReadable":"{R : Type u_1} →\n  {R' : Type u_2} →\n    {M : Type u_4} →\n      {M' : Type u_5} →\n        [inst : Semiring R] →\n          [inst_1 : Semiring R'] →\n            [inst_2 : AddCommMonoid M] →\n              [inst_3 : AddCommMonoid M'] →\n                [inst_4 : Module R M] →\n                  [inst_5 : Module R' M'] →\n                    {σ₁₂ : R →+* R'} →\n                      {S' : Type u_6} →\n                        [inst_6 : Monoid S'] →\n                          [inst_7 : DistribMulAction S' M] →\n                            [SMulCommClass R S' M] → DistribMulAction S'ᵈᵐᵃ (M →ₛₗ[σ₁₂] M')","typeReferences":[["RingHom"],["Module"],["DistribMulAction","toDistribSMul"],["MulOpposite","instMonoid"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["DomMulAct"],["AddCommMonoid","toAddMonoid"],["LinearMap"],["AddZeroClass","toAddZero"],["DomMulAct","instMonoidOfMulOpposite"],["LinearMap","addCommMonoid"],["SMulCommClass"],["Module","toDistribMulAction"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["Monoid"],["AddZero","toZero"],["DistribMulAction"],["DistribSMul","toSMulZeroClass"],["Semiring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["LinearMap","instDistribMulActionDomMulActOfSMulCommClass","_proof_3"],["LinearMap","instDistribMulActionDomMulActOfSMulCommClass","_proof_4"],["SemigroupAction","mk"],["DistribMulAction","mk"],["MulOpposite","instMonoid"],["DomMulAct"],["AddCommMonoid","toAddMonoid"],["LinearMap"],["DomMulAct","instMonoidOfMulOpposite"],["LinearMap","instSMulDomMulAct"],["LinearMap","addCommMonoid"],["LinearMap","instDistribMulActionDomMulActOfSMulCommClass","_proof_1"],["LinearMap","instDistribMulActionDomMulActOfSMulCommClass","_proof_2"],["Monoid","toSemigroup"],["MulAction","mk"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","instModuleDomMulActOfSMulCommClass","_proof_1"],"typeFallback":"forall {R : Type.{u_2}} {R' : Type.{u_3}} {S : Type.{u_1}} {M : Type.{u_4}} {M' : Type.{u_5}} [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 : Semiring.{u_3} R'] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 : AddCommMonoid.{u_4} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 : AddCommMonoid.{u_5} M'] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 : Module.{u_2, u_4} R M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23 : Module.{u_3, u_5} R' M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16] {σ₁₂ : RingHom.{u_2, u_3} R R' (Semiring.toNonAssocSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7) (Semiring.toNonAssocSemiring.{u_3} R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10)} [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.32 : Semiring.{u_1} S] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.35 : Module.{u_1, u_4} S M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.32 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.48 : SMulCommClass.{u_2, u_1, u_4} R S M (SMulZeroClass.toSMul.{u_2, u_4} R M (AddZero.toZero.{u_4} M (AddZeroClass.toAddZero.{u_4} M (AddMonoid.toAddZeroClass.{u_4} M (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13)))) (DistribSMul.toSMulZeroClass.{u_2, u_4} R M (AddMonoid.toAddZeroClass.{u_4} M (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13)) (DistribMulAction.toDistribSMul.{u_2, u_4} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13) (Module.toDistribMulAction.{u_2, u_4} R M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19)))) (SMulZeroClass.toSMul.{u_1, u_4} S M (AddZero.toZero.{u_4} M (AddZeroClass.toAddZero.{u_4} M (AddMonoid.toAddZeroClass.{u_4} M (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13)))) (DistribSMul.toSMulZeroClass.{u_1, u_4} S M (AddMonoid.toAddZeroClass.{u_4} M (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13)) (DistribMulAction.toDistribSMul.{u_1, u_4} S M (MonoidWithZero.toMonoid.{u_1} S (Semiring.toMonoidWithZero.{u_1} S inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.32)) (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13) (Module.toDistribMulAction.{u_1, u_4} S M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.32 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.35))))] (x._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.75 : DomMulAct.{u_1} S) (x._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.77 : DomMulAct.{u_1} S) (x._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.79 : LinearMap.{u_2, u_3, u_4, u_5} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23), Eq.{max (succ u_5) (succ u_4)} (LinearMap.{u_2, u_3, u_4, u_5} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23) (HSMul.hSMul.{u_1, max u_4 u_5, max u_4 u_5} (DomMulAct.{u_1} S) (LinearMap.{u_2, u_3, u_4, u_5} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23) (LinearMap.{u_2, u_3, u_4, u_5} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 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inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23) (Monoid.toSemigroup.{u_1} (DomMulAct.{u_1} S) (DomMulAct.instMonoidOfMulOpposite.{u_1} S (MulOpposite.instMonoid.{u_1} S (MonoidWithZero.toMonoid.{u_1} S (Semiring.toMonoidWithZero.{u_1} S inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.32))))) (MulAction.toSemigroupAction.{u_1, max u_5 u_4} (DomMulAct.{u_1} S) (LinearMap.{u_2, u_3, u_4, u_5} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23) (DomMulAct.instMonoidOfMulOpposite.{u_1} S (MulOpposite.instMonoid.{u_1} S (MonoidWithZero.toMonoid.{u_1} S (Semiring.toMonoidWithZero.{u_1} S inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.32)))) (DistribMulAction.toMulAction.{u_1, max u_5 u_4} (DomMulAct.{u_1} S) (LinearMap.{u_2, u_3, u_4, u_5} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23) (DomMulAct.instMonoidOfMulOpposite.{u_1} S (MulOpposite.instMonoid.{u_1} S (MonoidWithZero.toMonoid.{u_1} S (Semiring.toMonoidWithZero.{u_1} S inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.32)))) (AddCommMonoid.toAddMonoid.{max u_4 u_5} (LinearMap.{u_2, u_3, u_4, u_5} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23) (LinearMap.addCommMonoid.{u_2, u_3, u_4, u_5} R R' M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23 σ₁₂)) (LinearMap.instDistribMulActionDomMulActOfSMulCommClass.{u_2, u_3, u_4, u_5, u_1} R R' M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23 σ₁₂ S (MonoidWithZero.toMonoid.{u_1} S (Semiring.toMonoidWithZero.{u_1} S inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.32)) (Module.toDistribMulAction.{u_1, u_4} S M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.32 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.35) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.48))))) (HAdd.hAdd.{u_1, u_1, u_1} (DomMulAct.{u_1} S) (DomMulAct.{u_1} S) (DomMulAct.{u_1} S) (instHAdd.{u_1} (DomMulAct.{u_1} S) (Distrib.toAdd.{u_1} (DomMulAct.{u_1} S) (NonUnitalNonAssocSemiring.toDistrib.{u_1} (DomMulAct.{u_1} S) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} (DomMulAct.{u_1} S) (Semiring.toNonAssocSemiring.{u_1} (DomMulAct.{u_1} S) (DomMulAct.instSemiringOfMulOpposite.{u_1} S (MulOpposite.instSemiring.{u_1} S inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.32))))))) x._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.75 x._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.77) x._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.79) (HAdd.hAdd.{max u_4 u_5, max u_4 u_5, max 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(LinearMap.{u_2, u_3, u_4, u_5} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23) (instHAdd.{max u_4 u_5} (LinearMap.{u_2, u_3, u_4, u_5} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 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inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23 σ₁₂))))) (HSMul.hSMul.{u_1, max u_4 u_5, max u_4 u_5} (DomMulAct.{u_1} S) (LinearMap.{u_2, u_3, u_4, u_5} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23) (LinearMap.{u_2, u_3, u_4, u_5} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23) (instHSMul.{u_1, max u_4 u_5} (DomMulAct.{u_1} S) (LinearMap.{u_2, u_3, u_4, u_5} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23) (SemigroupAction.toSMul.{u_1, max u_5 u_4} (DomMulAct.{u_1} S) (LinearMap.{u_2, u_3, u_4, u_5} R R' 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inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23) (DomMulAct.instMonoidOfMulOpposite.{u_1} S (MulOpposite.instMonoid.{u_1} S (MonoidWithZero.toMonoid.{u_1} S (Semiring.toMonoidWithZero.{u_1} S inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.32)))) (DistribMulAction.toMulAction.{u_1, max u_5 u_4} (DomMulAct.{u_1} S) (LinearMap.{u_2, u_3, u_4, u_5} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23) (DomMulAct.instMonoidOfMulOpposite.{u_1} S (MulOpposite.instMonoid.{u_1} S (MonoidWithZero.toMonoid.{u_1} S (Semiring.toMonoidWithZero.{u_1} S inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.32)))) (AddCommMonoid.toAddMonoid.{max u_4 u_5} (LinearMap.{u_2, u_3, u_4, u_5} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23) (LinearMap.addCommMonoid.{u_2, u_3, u_4, u_5} R R' M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23 σ₁₂)) (LinearMap.instDistribMulActionDomMulActOfSMulCommClass.{u_2, u_3, u_4, u_5, u_1} R R' M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23 σ₁₂ S (MonoidWithZero.toMonoid.{u_1} S (Semiring.toMonoidWithZero.{u_1} S inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.32)) (Module.toDistribMulAction.{u_1, u_4} S M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.32 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.35) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.48))))) x._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.75 x._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.79) (HSMul.hSMul.{u_1, max u_4 u_5, max u_4 u_5} (DomMulAct.{u_1} S) (LinearMap.{u_2, u_3, u_4, u_5} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23) (LinearMap.{u_2, u_3, u_4, u_5} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23) (instHSMul.{u_1, max u_4 u_5} (DomMulAct.{u_1} S) (LinearMap.{u_2, u_3, u_4, u_5} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23) (SemigroupAction.toSMul.{u_1, max u_5 u_4} (DomMulAct.{u_1} S) (LinearMap.{u_2, u_3, u_4, u_5} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23) (Monoid.toSemigroup.{u_1} (DomMulAct.{u_1} S) (DomMulAct.instMonoidOfMulOpposite.{u_1} S (MulOpposite.instMonoid.{u_1} S (MonoidWithZero.toMonoid.{u_1} S (Semiring.toMonoidWithZero.{u_1} S inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.32))))) (MulAction.toSemigroupAction.{u_1, max u_5 u_4} (DomMulAct.{u_1} S) (LinearMap.{u_2, u_3, u_4, u_5} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23) (DomMulAct.instMonoidOfMulOpposite.{u_1} S (MulOpposite.instMonoid.{u_1} S (MonoidWithZero.toMonoid.{u_1} S (Semiring.toMonoidWithZero.{u_1} S inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.32)))) (DistribMulAction.toMulAction.{u_1, max u_5 u_4} (DomMulAct.{u_1} S) (LinearMap.{u_2, u_3, u_4, u_5} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23) (DomMulAct.instMonoidOfMulOpposite.{u_1} S (MulOpposite.instMonoid.{u_1} S (MonoidWithZero.toMonoid.{u_1} S (Semiring.toMonoidWithZero.{u_1} S inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.32)))) (AddCommMonoid.toAddMonoid.{max u_4 u_5} (LinearMap.{u_2, u_3, u_4, u_5} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23) (LinearMap.addCommMonoid.{u_2, u_3, u_4, u_5} R R' M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23 σ₁₂)) (LinearMap.instDistribMulActionDomMulActOfSMulCommClass.{u_2, u_3, u_4, u_5, u_1} R R' M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23 σ₁₂ S (MonoidWithZero.toMonoid.{u_1} S (Semiring.toMonoidWithZero.{u_1} S inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.32)) (Module.toDistribMulAction.{u_1, u_4} S M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.32 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.35) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.48))))) x._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.77 x._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.79))","typeFull":"∀ {R : Type u_2} {R' : Type u_3} {S : Type u_1} {M : Type u_4} {M' : Type u_5} [inst : Semiring R]\n  [inst_1 : Semiring R'] [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M'] [inst_4 : Module R M]\n  [inst_5 : Module R' M'] {σ₁₂ : R →+* R'} [inst_6 : Semiring S] [inst_7 : Module S M] [inst_8 : SMulCommClass R S M]\n  (x x_1 : Sᵈᵐᵃ) (x_2 : M →ₛₗ[σ₁₂] M'), (x + x_1) • x_2 = x • x_2 + x_1 • x_2","typeReadable":"∀ {R : Type u_2} {R' : Type u_3} {S : Type u_1} {M : Type u_4} {M' : Type u_5} [inst : Semiring R]\n  [inst_1 : Semiring R'] [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M'] [inst_4 : Module R M]\n  [inst_5 : Module R' M'] {σ₁₂ : R →+* R'} [inst_6 : Semiring S] [inst_7 : Module S M] [inst_8 : SMulCommClass R S M]\n  (x x_1 : Sᵈᵐᵃ) (x_2 : M →ₛₗ[σ₁₂] M'), (x + x_1) • x_2 = x • x_2 + x_1 • 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inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2254470004._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2254470004._hygCtx._hyg.15)))) (SMulZeroClass.toSMul.{u_7, u_7} A A (AddZero.toZero.{u_7} A (AddZeroClass.toAddZero.{u_7} A (AddMonoid.toAddZeroClass.{u_7} A (AddMonoidWithOne.toAddMonoid.{u_7} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u_7} A (NonAssocSemiring.toAddCommMonoidWithOne.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2254470004._hygCtx._hyg.12)))))) (DistribSMul.toSMulZeroClass.{u_7, u_7} A A (AddMonoid.toAddZeroClass.{u_7} A (AddMonoidWithOne.toAddMonoid.{u_7} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u_7} A (NonAssocSemiring.toAddCommMonoidWithOne.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2254470004._hygCtx._hyg.12)))) (instDistribSMul.{u_7} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2254470004._hygCtx._hyg.12))))], Eq.{succ u_7} (LinearMap.{u_6, u_6, u_7, u_7} R R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2254470004._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2254470004._hygCtx._hyg.9 (RingHom.id.{u_6} R (Semiring.toNonAssocSemiring.{u_6} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2254470004._hygCtx._hyg.9)) A A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2254470004._hygCtx._hyg.12)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2254470004._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2254470004._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2254470004._hygCtx._hyg.15) (LinearMap.mulLeft.{u_6, u_7} R A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2254470004._hygCtx._hyg.9 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2254470004._hygCtx._hyg.12) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2254470004._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2254470004._hygCtx._hyg.19 (OfNat.ofNat.{u_7} A 1 (One.toOfNat1.{u_7} A (AddMonoidWithOne.toOne.{u_7} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u_7} A (NonAssocSemiring.toAddCommMonoidWithOne.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2254470004._hygCtx._hyg.12)))))) (LinearMap.id.{u_6, u_7} R A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2254470004._hygCtx._hyg.9 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2254470004._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2254470004._hygCtx._hyg.15)","typeFull":"∀ (R : Type u_6) (A : Type u_7) [inst : Semiring R] [inst_1 : NonAssocSemiring A] [inst_2 : Module R A]\n  [inst_3 : SMulCommClass R A A], LinearMap.mulLeft R 1 = LinearMap.id","typeReadable":"∀ (R : Type u_6) (A : Type u_7) [inst : Semiring R] [inst_1 : NonAssocSemiring A] [inst_2 : Module R A]\n  [inst_3 : SMulCommClass R A A], LinearMap.mulLeft R 1 = LinearMap.id","typeReferences":[["Module"],["SMulZeroClass","toSMul"],["AddMonoidWithOne","toAddMonoid"],["instDistribSMul"],["LinearMap","id"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["NonAssocSemiring"],["MonoidWithZero","toMonoid"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Eq"],["LinearMap","mulLeft"],["NonAssocSemiring","toAddCommMonoidWithOne"],["DistribSMul","toSMulZeroClass"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["LinearMap"],["OfNat","ofNat"],["SMulCommClass"],["Module","toDistribMulAction"],["One","toOfNat1"],["AddMonoidWithOne","toOne"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["NonUnitalNonAssocSemiring","toAddCommMonoid"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["MulZeroOneClass","toMulOneClass"],["LinearMap","id"],["OfNat","ofNat"],["Semiring","toNonAssocSemiring"],["One","toOfNat1"],["RingHom","id"],["LinearMap","ext"],["AddMonoidWithOne","toOne"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["LinearMap","mulLeft"],["NonAssocSemiring","toAddCommMonoidWithOne"],["one_mul"],["NonAssocSemiring","toMulZeroOneClass"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","instDistribMulActionDomMulActOfSMulCommClass","_proof_4"],"typeFallback":"forall {R : Type.{u_2}} {R' : Type.{u_3}} {M : Type.{u_4}} {M' : Type.{u_5}} [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.7 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.10 : Semiring.{u_3} R'] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13 : AddCommMonoid.{u_4} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.16 : AddCommMonoid.{u_5} M'] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.19 : Module.{u_2, u_4} R M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.23 : Module.{u_3, u_5} R' M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.16] {σ₁₂ : RingHom.{u_2, u_3} R R' (Semiring.toNonAssocSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.7) (Semiring.toNonAssocSemiring.{u_3} R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.10)} {S' : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.33 : Monoid.{u_1} S'] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.36 : DistribMulAction.{u_1, u_4} S' M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.33 (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13)] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.40 : SMulCommClass.{u_2, u_1, u_4} R S' M (SMulZeroClass.toSMul.{u_2, u_4} R M (AddZero.toZero.{u_4} M (AddZeroClass.toAddZero.{u_4} M (AddMonoid.toAddZeroClass.{u_4} M (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13)))) (DistribSMul.toSMulZeroClass.{u_2, u_4} R M (AddMonoid.toAddZeroClass.{u_4} M (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13)) (DistribMulAction.toDistribSMul.{u_2, u_4} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13) (Module.toDistribMulAction.{u_2, u_4} R M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.19)))) (SMulZeroClass.toSMul.{u_1, u_4} S' M (AddZero.toZero.{u_4} M (AddZeroClass.toAddZero.{u_4} M (AddMonoid.toAddZeroClass.{u_4} M (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13)))) (DistribSMul.toSMulZeroClass.{u_1, u_4} S' M (AddMonoid.toAddZeroClass.{u_4} M (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13)) (DistribMulAction.toDistribSMul.{u_1, u_4} S' M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.33 (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.36)))] (x._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.114 : DomMulAct.{u_1} S') (x._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.116 : LinearMap.{u_2, u_3, u_4, u_5} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.23) (x._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.118 : LinearMap.{u_2, u_3, u_4, u_5} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.23), Eq.{max (succ u_5) (succ u_4)} (LinearMap.{u_2, u_3, u_4, u_5} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.7 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inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.23 σ₁₂))))) x._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.116 x._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.118)) (HAdd.hAdd.{max u_4 u_5, max u_4 u_5, max u_4 u_5} (LinearMap.{u_2, u_3, u_4, u_5} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.23) (LinearMap.{u_2, u_3, u_4, u_5} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.23) (LinearMap.{u_2, u_3, u_4, u_5} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.23) (instHAdd.{max u_4 u_5} (LinearMap.{u_2, u_3, u_4, u_5} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.10 σ₁₂ M 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x._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.116) (HSMul.hSMul.{u_1, max u_4 u_5, max u_4 u_5} (DomMulAct.{u_1} S') (LinearMap.{u_2, u_3, u_4, u_5} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.23) (LinearMap.{u_2, u_3, u_4, u_5} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.16 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x_2) = x • x_1 + x • x_2","typeReadable":"∀ {R : Type u_2} {R' : Type u_3} {M : Type u_4} {M' : Type u_5} [inst : Semiring R] [inst_1 : Semiring R']\n  [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M'] [inst_4 : Module R M] [inst_5 : Module R' M'] {σ₁₂ : R →+* R'}\n  {S' : Type u_1} [inst_6 : Monoid S'] [inst_7 : DistribMulAction S' M] [inst_8 : SMulCommClass R S' M] (x : S'ᵈᵐᵃ)\n  (x_1 x_2 : M →ₛₗ[σ₁₂] M'), x • (x_1 + x_2) = x • x_1 + x • x_2","typeReferences":[["RingHom"],["Module"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["LinearMap","instSMulDomMulAct"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["instHSMul"],["Eq"],["DistribMulAction"],["AddSemigroup","toAdd"],["DistribSMul","toSMulZeroClass"],["instHAdd"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["DomMulAct"],["AddZeroClass","toAddZero"],["LinearMap"],["LinearMap","addCommMonoid"],["HAdd","hAdd"],["SMulCommClass"],["Module","toDistribMulAction"],["AddMonoid","toAddSemigroup"],["HSMul","hSMul"],["Monoid"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["rfl"],["LinearMap","instFunLike"],["instHAdd"],["DomMulAct"],["AddCommMonoid","toAddMonoid"],["LinearMap"],["DFunLike","coe"],["LinearMap","instSMulDomMulAct"],["HAdd","hAdd"],["LinearMap","addCommMonoid"],["LinearMap","ext"],["AddMonoid","toAddSemigroup"],["HSMul","hSMul"],["instHSMul"],["AddSemigroup","toAdd"]]},{"isProp":true,"kind":"theorem","name":["DomMulAct","mk_smul_linearMap_apply"],"typeFallback":"forall {R : Type.{u_1}} {R' : Type.{u_2}} {M : Type.{u_4}} {M' : Type.{u_5}} [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.7 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.10 : Semiring.{u_2} R'] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.13 : AddCommMonoid.{u_4} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.16 : AddCommMonoid.{u_5} M'] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.19 : Module.{u_1, u_4} R M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.13] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.23 : Module.{u_2, u_5} R' M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.16] {σ₁₂ : RingHom.{u_1, u_2} R R' (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.7) (Semiring.toNonAssocSemiring.{u_2} R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.10)} {S' : Type.{u_6}} [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.34 : Monoid.{u_6} S'] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.37 : DistribMulAction.{u_6, u_4} S' M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.34 (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.13)] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.41 : SMulCommClass.{u_1, u_6, u_4} R S' M (SMulZeroClass.toSMul.{u_1, u_4} R M (AddZero.toZero.{u_4} M (AddZeroClass.toAddZero.{u_4} M (AddMonoid.toAddZeroClass.{u_4} M (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.13)))) (DistribSMul.toSMulZeroClass.{u_1, u_4} R M (AddMonoid.toAddZeroClass.{u_4} M (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.13)) (DistribMulAction.toDistribSMul.{u_1, u_4} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.13) (Module.toDistribMulAction.{u_1, u_4} R M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.19)))) (SMulZeroClass.toSMul.{u_6, u_4} S' M (AddZero.toZero.{u_4} M (AddZeroClass.toAddZero.{u_4} M (AddMonoid.toAddZeroClass.{u_4} M (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.13)))) (DistribSMul.toSMulZeroClass.{u_6, u_4} S' M (AddMonoid.toAddZeroClass.{u_4} M (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.13)) (DistribMulAction.toDistribSMul.{u_6, u_4} S' M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.34 (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.13) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.37)))] (a : S') (f : LinearMap.{u_1, u_2, u_4, u_5} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.23) (x : M), Eq.{succ u_5} M' (DFunLike.coe.{max (succ u_4) (succ u_5), succ u_4, succ u_5} (LinearMap.{u_1, u_2, u_4, u_5} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.23) M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => M') (LinearMap.instFunLike.{u_1, u_2, u_4, u_5} R R' M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.23 σ₁₂) (HSMul.hSMul.{u_6, max u_4 u_5, max u_4 u_5} (DomMulAct.{u_6} S') (LinearMap.{u_1, u_2, u_4, u_5} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.23) (LinearMap.{u_1, u_2, u_4, u_5} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.23) (instHSMul.{u_6, max u_4 u_5} (DomMulAct.{u_6} S') (LinearMap.{u_1, u_2, u_4, u_5} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.23) (LinearMap.instSMulDomMulAct.{u_1, u_2, u_4, u_5, u_6} R R' M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.23 σ₁₂ S' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.34 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.37 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.41)) (DFunLike.coe.{succ u_6, succ u_6, succ u_6} (Equiv.{succ u_6, succ u_6} S' (DomMulAct.{u_6} S')) S' (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : S') => DomMulAct.{u_6} S') (EquivLike.toFunLike.{succ u_6, succ u_6, succ u_6} (Equiv.{succ u_6, succ u_6} S' (DomMulAct.{u_6} S')) S' (DomMulAct.{u_6} S') (Equiv.instEquivLike.{succ u_6, succ u_6} S' (DomMulAct.{u_6} S'))) (DomMulAct.mk.{u_6} S') a) f) x) (DFunLike.coe.{max (succ u_4) (succ u_5), succ u_4, succ u_5} (LinearMap.{u_1, u_2, u_4, u_5} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.23) M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => M') (LinearMap.instFunLike.{u_1, u_2, u_4, u_5} R R' M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.23 σ₁₂) f (HSMul.hSMul.{u_6, u_4, u_4} S' M M (instHSMul.{u_6, u_4} S' M (SMulZeroClass.toSMul.{u_6, u_4} S' M (AddZero.toZero.{u_4} M (AddZeroClass.toAddZero.{u_4} M (AddMonoid.toAddZeroClass.{u_4} M (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.13)))) (DistribSMul.toSMulZeroClass.{u_6, u_4} S' M (AddMonoid.toAddZeroClass.{u_4} M (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.13)) (DistribMulAction.toDistribSMul.{u_6, u_4} S' M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.34 (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.13) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.2543075141._hygCtx._hyg.37)))) a x))","typeFull":"∀ {R : Type u_1} {R' : Type u_2} {M : Type u_4} {M' : Type u_5} [inst : Semiring R] [inst_1 : Semiring R']\n  [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M'] [inst_4 : Module R M] [inst_5 : Module R' M'] {σ₁₂ : R →+* R'}\n  {S' : Type u_6} [inst_6 : Monoid S'] [inst_7 : DistribMulAction S' M] [inst_8 : SMulCommClass R S' M] (a : S')\n  (f : M →ₛₗ[σ₁₂] M') (x : M), (DomMulAct.mk a • f) x = f (a • x)","typeReadable":"∀ {R : Type u_1} {R' : Type u_2} {M : Type u_4} {M' : Type u_5} [inst : Semiring R] [inst_1 : Semiring R']\n  [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M'] [inst_4 : Module R M] [inst_5 : Module R' M'] {σ₁₂ : R →+* R'}\n  {S' : Type u_6} [inst_6 : Monoid S'] [inst_7 : DistribMulAction S' M] [inst_8 : SMulCommClass R S' M] (a : S')\n  (f : M →ₛₗ[σ₁₂] M') (x : M), (DomMulAct.mk a • f) x = f (a • x)","typeReferences":[["RingHom"],["Equiv","instEquivLike"],["LinearMap","instFunLike"],["Module"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["DFunLike","coe"],["Equiv"],["LinearMap","instSMulDomMulAct"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["EquivLike","toFunLike"],["MonoidWithZero","toMonoid"],["instHSMul"],["Eq"],["DistribMulAction"],["DistribSMul","toSMulZeroClass"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["DomMulAct"],["AddZeroClass","toAddZero"],["LinearMap"],["DomMulAct","mk"],["SMulCommClass"],["Module","toDistribMulAction"],["HSMul","hSMul"],["Monoid"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["DomMulAct","mk"],["rfl"],["Equiv","instEquivLike"],["LinearMap","instFunLike"],["EquivLike","toFunLike"],["HSMul","hSMul"],["instHSMul"],["DomMulAct"],["LinearMap"],["DFunLike","coe"],["Equiv"],["LinearMap","instSMulDomMulAct"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","instDistribMulActionDomMulActOfSMulCommClass","_proof_3"],"typeFallback":"forall {R : Type.{u_4}} {R' : Type.{u_5}} {M : Type.{u_3}} {M' : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.7 : Semiring.{u_4} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.10 : Semiring.{u_5} R'] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13 : AddCommMonoid.{u_3} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.16 : AddCommMonoid.{u_2} M'] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.19 : Module.{u_4, u_3} R M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.23 : Module.{u_5, u_2} R' M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.16] {σ₁₂ : RingHom.{u_4, u_5} R R' (Semiring.toNonAssocSemiring.{u_4} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.7) (Semiring.toNonAssocSemiring.{u_5} R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.10)} {S' : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.33 : Monoid.{u_1} S'] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.36 : DistribMulAction.{u_1, u_3} S' M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.33 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13)] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.40 : SMulCommClass.{u_4, u_1, u_3} R S' M (SMulZeroClass.toSMul.{u_4, u_3} R M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13)))) (DistribSMul.toSMulZeroClass.{u_4, u_3} R M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13)) (DistribMulAction.toDistribSMul.{u_4, u_3} R M (MonoidWithZero.toMonoid.{u_4} R (Semiring.toMonoidWithZero.{u_4} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13) (Module.toDistribMulAction.{u_4, u_3} R M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.19)))) (SMulZeroClass.toSMul.{u_1, u_3} S' M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13)))) (DistribSMul.toSMulZeroClass.{u_1, u_3} S' M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13)) (DistribMulAction.toDistribSMul.{u_1, u_3} S' M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.33 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.36)))] (x._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.104 : DomMulAct.{u_1} S'), Eq.{max (succ u_2) (succ u_3)} (LinearMap.{u_4, u_5, u_3, u_2} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.16 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inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.23) (instHSMul.{u_1, max u_3 u_2} (DomMulAct.{u_1} S') (LinearMap.{u_4, u_5, u_3, u_2} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.23) (LinearMap.instSMulDomMulAct.{u_4, u_5, u_3, u_2, u_1} R R' M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.23 σ₁₂ S' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.40)) x._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.104 (OfNat.ofNat.{max u_3 u_2} (LinearMap.{u_4, u_5, u_3, u_2} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.16 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inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.23) (AddZero.toZero.{max u_3 u_2} (LinearMap.{u_4, u_5, u_3, u_2} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.23) (AddZeroClass.toAddZero.{max u_3 u_2} (LinearMap.{u_4, u_5, u_3, u_2} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.23) (AddMonoid.toAddZeroClass.{max u_3 u_2} (LinearMap.{u_4, u_5, u_3, u_2} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.23) (AddCommMonoid.toAddMonoid.{max u_3 u_2} (LinearMap.{u_4, u_5, u_3, u_2} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.23) (LinearMap.addCommMonoid.{u_4, u_5, u_3, u_2} R R' M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.23 σ₁₂)))))))","typeFull":"∀ {R : Type u_4} {R' : Type u_5} {M : Type u_3} {M' : Type u_2} [inst : Semiring R] [inst_1 : Semiring R']\n  [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M'] [inst_4 : Module R M] [inst_5 : Module R' M'] {σ₁₂ : R →+* R'}\n  {S' : Type u_1} [inst_6 : Monoid S'] [inst_7 : DistribMulAction S' M] [inst_8 : SMulCommClass R S' M] (x : S'ᵈᵐᵃ),\n  x • 0 = 0","typeReadable":"∀ {R : Type u_4} {R' : Type u_5} {M : Type u_3} {M' : Type u_2} [inst : Semiring R] [inst_1 : Semiring R']\n  [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M'] [inst_4 : Module R M] [inst_5 : Module R' M'] {σ₁₂ : R →+* R'}\n  {S' : Type u_1} [inst_6 : Monoid S'] [inst_7 : DistribMulAction S' M] [inst_8 : SMulCommClass R S' M] (x : S'ᵈᵐᵃ),\n  x • 0 = 0","typeReferences":[["RingHom"],["Module"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["LinearMap","instSMulDomMulAct"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["instHSMul"],["Zero","toOfNat0"],["Eq"],["DistribMulAction"],["DistribSMul","toSMulZeroClass"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["DomMulAct"],["AddZeroClass","toAddZero"],["LinearMap"],["OfNat","ofNat"],["LinearMap","addCommMonoid"],["SMulCommClass"],["Module","toDistribMulAction"],["HSMul","hSMul"],["Monoid"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["rfl"],["LinearMap","instFunLike"],["DomMulAct"],["AddCommMonoid","toAddMonoid"],["LinearMap"],["AddZeroClass","toAddZero"],["DFunLike","coe"],["LinearMap","instSMulDomMulAct"],["OfNat","ofNat"],["LinearMap","addCommMonoid"],["LinearMap","ext"],["HSMul","hSMul"],["instHSMul"],["Zero","toOfNat0"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","mulRight_eq_zero_iff"],"typeFallback":"forall (R : Type.{u_6}) (A : Type.{u_7}) [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.9 : Semiring.{u_6} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.12 : NonAssocSemiring.{u_7} A] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.15 : Module.{u_6, u_7} R A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.9 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.12))] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.19 : IsScalarTower.{u_6, u_7, u_7} R A A (SMulZeroClass.toSMul.{u_6, u_7} R A (AddZero.toZero.{u_7} A (AddZeroClass.toAddZero.{u_7} A (AddMonoid.toAddZeroClass.{u_7} A (AddMonoidWithOne.toAddMonoid.{u_7} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u_7} A (NonAssocSemiring.toAddCommMonoidWithOne.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.12)))))) (DistribSMul.toSMulZeroClass.{u_6, u_7} R A (AddMonoid.toAddZeroClass.{u_7} A (AddMonoidWithOne.toAddMonoid.{u_7} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u_7} A (NonAssocSemiring.toAddCommMonoidWithOne.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.12)))) (DistribMulAction.toDistribSMul.{u_6, u_7} R A (MonoidWithZero.toMonoid.{u_6} R (Semiring.toMonoidWithZero.{u_6} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.9)) (AddMonoidWithOne.toAddMonoid.{u_7} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u_7} A (NonAssocSemiring.toAddCommMonoidWithOne.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.12))) (Module.toDistribMulAction.{u_6, u_7} R A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.9 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.15)))) (SMulZeroClass.toSMul.{u_7, u_7} A A (AddZero.toZero.{u_7} A (AddZeroClass.toAddZero.{u_7} A (AddMonoid.toAddZeroClass.{u_7} A (AddMonoidWithOne.toAddMonoid.{u_7} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u_7} A (NonAssocSemiring.toAddCommMonoidWithOne.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.12)))))) (DistribSMul.toSMulZeroClass.{u_7, u_7} A A (AddMonoid.toAddZeroClass.{u_7} A (AddMonoidWithOne.toAddMonoid.{u_7} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u_7} A (NonAssocSemiring.toAddCommMonoidWithOne.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.12)))) (instDistribSMul.{u_7} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.12)))) (SMulZeroClass.toSMul.{u_6, u_7} R A (AddZero.toZero.{u_7} A (AddZeroClass.toAddZero.{u_7} A (AddMonoid.toAddZeroClass.{u_7} A (AddMonoidWithOne.toAddMonoid.{u_7} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u_7} A (NonAssocSemiring.toAddCommMonoidWithOne.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.12)))))) (DistribSMul.toSMulZeroClass.{u_6, u_7} R A (AddMonoid.toAddZeroClass.{u_7} A (AddMonoidWithOne.toAddMonoid.{u_7} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u_7} A (NonAssocSemiring.toAddCommMonoidWithOne.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.12)))) (DistribMulAction.toDistribSMul.{u_6, u_7} R A (MonoidWithZero.toMonoid.{u_6} R (Semiring.toMonoidWithZero.{u_6} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.9)) (AddMonoidWithOne.toAddMonoid.{u_7} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u_7} A (NonAssocSemiring.toAddCommMonoidWithOne.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.12))) (Module.toDistribMulAction.{u_6, u_7} R A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.9 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.15))))] (a : A), Iff (Eq.{succ u_7} (LinearMap.{u_6, u_6, u_7, u_7} R R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.9 (RingHom.id.{u_6} R (Semiring.toNonAssocSemiring.{u_6} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.9)) A A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.12)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.15) (LinearMap.mulRight.{u_6, u_7} R A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.9 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.12) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.19 a) (OfNat.ofNat.{u_7} (LinearMap.{u_6, u_6, u_7, u_7} R R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.9 (RingHom.id.{u_6} R (Semiring.toNonAssocSemiring.{u_6} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.9)) A A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.12)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.15) 0 (Zero.toOfNat0.{u_7} (LinearMap.{u_6, u_6, u_7, u_7} R R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.9 (RingHom.id.{u_6} R (Semiring.toNonAssocSemiring.{u_6} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.9)) A A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.12)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.15) (LinearMap.instZero.{u_6, u_6, u_7, u_7} R R A A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.9 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.12)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.15 (RingHom.id.{u_6} R (Semiring.toNonAssocSemiring.{u_6} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.9)))))) (Eq.{succ u_7} A a (OfNat.ofNat.{u_7} A 0 (Zero.toOfNat0.{u_7} A (MulZeroClass.toZero.{u_7} A (NonUnitalNonAssocSemiring.toMulZeroClass.{u_7} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.12))))))","typeFull":"∀ (R : Type u_6) (A : Type u_7) [inst : Semiring R] [inst_1 : NonAssocSemiring A] [inst_2 : Module R A]\n  [inst_3 : IsScalarTower R A A] (a : A), LinearMap.mulRight R a = 0 ↔ a = 0","typeReadable":"∀ (R : Type u_6) (A : Type u_7) [inst : Semiring R] [inst_1 : NonAssocSemiring A] [inst_2 : Module R A]\n  [inst_3 : IsScalarTower R A A] (a : A), LinearMap.mulRight R a = 0 ↔ a = 0","typeReferences":[["IsScalarTower"],["Module"],["SMulZeroClass","toSMul"],["AddMonoidWithOne","toAddMonoid"],["instDistribSMul"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["NonAssocSemiring"],["MonoidWithZero","toMonoid"],["Zero","toOfNat0"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Eq"],["NonAssocSemiring","toAddCommMonoidWithOne"],["DistribSMul","toSMulZeroClass"],["LinearMap","instZero"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["LinearMap"],["OfNat","ofNat"],["Module","toDistribMulAction"],["MulZeroClass","toZero"],["Iff"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["LinearMap","mulRight"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["LinearMap","instZero"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["LinearMap","mulRight_zero_eq_zero"],["LinearMap"],["OfNat","ofNat"],["Semiring","toNonAssocSemiring"],["LinearMap","mulRight_inj"],["RingHom","id"],["MulZeroClass","toZero"],["Iff"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["LinearMap","mulRight"],["Zero","toOfNat0"],["Eq","rec"],["Eq"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","instDistribMulActionDomMulActOfSMulCommClass","_proof_2"],"typeFallback":"forall {R : Type.{u_1}} {R' : Type.{u_2}} {M : Type.{u_3}} {M' : Type.{u_4}} [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.7 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.10 : Semiring.{u_2} R'] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13 : AddCommMonoid.{u_3} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.16 : AddCommMonoid.{u_4} M'] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.19 : Module.{u_1, u_3} R M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.23 : Module.{u_2, u_4} R' M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.16] {σ₁₂ : RingHom.{u_1, u_2} R R' (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.7) (Semiring.toNonAssocSemiring.{u_2} R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.10)} {S' : Type.{u_5}} [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.33 : Monoid.{u_5} S'] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.36 : DistribMulAction.{u_5, u_3} S' M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.33 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13)] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.40 : SMulCommClass.{u_1, u_5, u_3} R S' M (SMulZeroClass.toSMul.{u_1, u_3} R M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13)))) (DistribSMul.toSMulZeroClass.{u_1, u_3} R M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13)) (DistribMulAction.toDistribSMul.{u_1, u_3} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13) (Module.toDistribMulAction.{u_1, u_3} R M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.19)))) (SMulZeroClass.toSMul.{u_5, u_3} S' M (AddZero.toZero.{u_3} M (AddZeroClass.toAddZero.{u_3} M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13)))) (DistribSMul.toSMulZeroClass.{u_5, u_3} S' M (AddMonoid.toAddZeroClass.{u_3} M (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13)) (DistribMulAction.toDistribSMul.{u_5, u_3} S' M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.33 (AddCommMonoid.toAddMonoid.{u_3} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.36)))] (x._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.88 : LinearMap.{u_1, u_2, u_3, u_4} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.19 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inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.23) (LinearMap.{u_1, u_2, u_3, u_4} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.23) (instHSMul.{u_5, max u_3 u_4} (DomMulAct.{u_5} S') (LinearMap.{u_1, u_2, u_3, u_4} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.23) (LinearMap.instSMulDomMulAct.{u_1, u_2, u_3, u_4, u_5} R R' M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.23 σ₁₂ S' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.40)) (OfNat.ofNat.{u_5} (DomMulAct.{u_5} S') 1 (One.toOfNat1.{u_5} (DomMulAct.{u_5} S') (MulOne.toOne.{u_5} (DomMulAct.{u_5} S') (MulOneClass.toMulOne.{u_5} (DomMulAct.{u_5} S') (Monoid.toMulOneClass.{u_5} (DomMulAct.{u_5} S') (DomMulAct.instMonoidOfMulOpposite.{u_5} S' (MulOpposite.instMonoid.{u_5} S' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.33))))))) x._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.88) x._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.88","typeFull":"∀ {R : Type u_1} {R' : Type u_2} {M : Type u_3} {M' : Type u_4} [inst : Semiring R] [inst_1 : Semiring R']\n  [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M'] [inst_4 : Module R M] [inst_5 : Module R' M'] {σ₁₂ : R →+* R'}\n  {S' : Type u_5} [inst_6 : Monoid S'] [inst_7 : DistribMulAction S' M] [inst_8 : SMulCommClass R S' M]\n  (x : M →ₛₗ[σ₁₂] M'), 1 • x = x","typeReadable":"∀ {R : Type u_1} {R' : Type u_2} {M : Type u_3} {M' : Type u_4} [inst : Semiring R] [inst_1 : Semiring R']\n  [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M'] [inst_4 : Module R M] [inst_5 : Module R' M'] {σ₁₂ : R →+* R'}\n  {S' : Type u_5} [inst_6 : Monoid S'] [inst_7 : DistribMulAction S' M] [inst_8 : SMulCommClass R S' M]\n  (x : M →ₛₗ[σ₁₂] M'), 1 • x = x","typeReferences":[["MulOneClass","toMulOne"],["RingHom"],["Module"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["LinearMap","instSMulDomMulAct"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["Monoid","toMulOneClass"],["MonoidWithZero","toMonoid"],["instHSMul"],["Eq"],["DistribMulAction"],["DistribSMul","toSMulZeroClass"],["MulOne","toOne"],["DistribMulAction","toDistribSMul"],["MulOpposite","instMonoid"],["Semiring","toMonoidWithZero"],["DomMulAct"],["AddZeroClass","toAddZero"],["LinearMap"],["DomMulAct","instMonoidOfMulOpposite"],["OfNat","ofNat"],["SMulCommClass"],["Module","toDistribMulAction"],["One","toOfNat1"],["HSMul","hSMul"],["Monoid"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["MulOneClass","toMulOne"],["MulOne","toOne"],["congr_arg"],["LinearMap","toAddHom"],["SemigroupAction","toSMul"],["MulOpposite","instMonoid"],["AddCommMonoid","toAddMonoid"],["DomMulAct"],["LinearMap"],["DomMulAct","instMonoidOfMulOpposite"],["LinearMap","instSMulDomMulAct"],["OfNat","ofNat"],["One","toOfNat1"],["LinearMap","ext"],["one_smul"],["HSMul","hSMul"],["Monoid","toMulOneClass"],["DistribMulAction","toMulAction"],["instHSMul"],["Monoid","toSemigroup"],["MulAction","toSemigroupAction"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","ltoFun","_proof_2"],"typeFallback":"forall (R : Type.{u_1}) (M : Type.{u_2}) (N : Type.{u_3}) (A : Type.{u_4}) [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.14 : Semiring.{u_4} A] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.17 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.20 : Module.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.17] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.24 : AddCommMonoid.{u_3} N] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.27 : Module.{u_1, u_3} R N inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.24] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.31 : Module.{u_4, u_3} A N inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.24] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.35 : SMulCommClass.{u_1, u_4, u_3} R A N (SMulZeroClass.toSMul.{u_1, u_3} R N (AddZero.toZero.{u_3} N (AddZeroClass.toAddZero.{u_3} N (AddMonoid.toAddZeroClass.{u_3} N (AddCommMonoid.toAddMonoid.{u_3} N inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.24)))) (DistribSMul.toSMulZeroClass.{u_1, u_3} R N (AddMonoid.toAddZeroClass.{u_3} N (AddCommMonoid.toAddMonoid.{u_3} N inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.24)) (DistribMulAction.toDistribSMul.{u_1, u_3} R N (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11)) (AddCommMonoid.toAddMonoid.{u_3} N inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.24) (Module.toDistribMulAction.{u_1, u_3} R N inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.27)))) (SMulZeroClass.toSMul.{u_4, u_3} A N (AddZero.toZero.{u_3} N (AddZeroClass.toAddZero.{u_3} N (AddMonoid.toAddZeroClass.{u_3} N (AddCommMonoid.toAddMonoid.{u_3} N inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.24)))) (DistribSMul.toSMulZeroClass.{u_4, u_3} A N (AddMonoid.toAddZeroClass.{u_3} N (AddCommMonoid.toAddMonoid.{u_3} N inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.24)) (DistribMulAction.toDistribSMul.{u_4, u_3} A N (MonoidWithZero.toMonoid.{u_4} A (Semiring.toMonoidWithZero.{u_4} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.14)) 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inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.27 (HSMul.hSMul.{u_4, max u_2 u_3, max u_2 u_3} A (LinearMap.{u_1, u_1, u_2, u_3} R R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11)) M N inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.27) (LinearMap.{u_1, u_1, u_2, u_3} R R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11 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inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.27) (AddMonoid.toAddZeroClass.{max u_2 u_3} (LinearMap.{u_1, u_1, u_2, u_3} R R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11)) M N inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.27) (AddCommMonoid.toAddMonoid.{max u_2 u_3} (LinearMap.{u_1, u_1, u_2, u_3} R R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11)) M N inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.27) (LinearMap.addCommMonoid.{u_1, u_1, u_2, u_3} R R M N inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.27 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11))))))) (DistribSMul.toSMulZeroClass.{u_4, max u_2 u_3} A (LinearMap.{u_1, u_1, u_2, u_3} R R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11)) M N inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.27) (AddMonoid.toAddZeroClass.{max u_2 u_3} (LinearMap.{u_1, u_1, u_2, u_3} R R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11)) M N inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.27) (AddCommMonoid.toAddMonoid.{max u_2 u_3} (LinearMap.{u_1, u_1, u_2, u_3} R R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11)) M N inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.27) (LinearMap.addCommMonoid.{u_1, u_1, u_2, u_3} R R M N inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.27 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11))))) (DistribMulAction.toDistribSMul.{u_4, max u_2 u_3} A (LinearMap.{u_1, u_1, u_2, 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(Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11)) M N inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.27) (LinearMap.addCommMonoid.{u_1, u_1, u_2, u_3} R R M N inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.27 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R 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inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.27 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11))) (LinearMap.module.{u_1, u_1, u_4, u_2, u_3} R R A M N inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.27 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inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11)) M N inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.27 (HSMul.hSMul.{u_4, max u_2 u_3, max u_2 u_3} A (LinearMap.{u_1, u_1, u_2, u_3} R R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11)) M N inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.27) (LinearMap.{u_1, u_1, u_2, u_3} R R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11)) M N inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.27) (instHSMul.{u_4, max u_2 u_3} A (LinearMap.{u_1, u_1, u_2, u_3} R R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11)) M N inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.27) (SMulZeroClass.toSMul.{u_4, max u_2 u_3} A (LinearMap.{u_1, u_1, u_2, u_3} R R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11)) M N inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.27) (AddZero.toZero.{max u_2 u_3} (LinearMap.{u_1, u_1, u_2, u_3} R R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11)) M N inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.27) (AddZeroClass.toAddZero.{max u_2 u_3} (LinearMap.{u_1, u_1, u_2, u_3} R R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11)) M N inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.27) (AddMonoid.toAddZeroClass.{max u_2 u_3} (LinearMap.{u_1, u_1, u_2, u_3} R R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11)) M N inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.27) (AddCommMonoid.toAddMonoid.{max u_2 u_3} (LinearMap.{u_1, u_1, u_2, u_3} R R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11)) M N inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.27) (LinearMap.addCommMonoid.{u_1, u_1, u_2, u_3} R R M N inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.27 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11))))))) (DistribSMul.toSMulZeroClass.{u_4, max u_2 u_3} A (LinearMap.{u_1, u_1, u_2, u_3} R R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11)) M N inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.27) (AddMonoid.toAddZeroClass.{max u_2 u_3} (LinearMap.{u_1, u_1, u_2, u_3} R R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11)) M N inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.27) (AddCommMonoid.toAddMonoid.{max u_2 u_3} (LinearMap.{u_1, u_1, u_2, u_3} R R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11)) M N inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.27) (LinearMap.addCommMonoid.{u_1, u_1, u_2, u_3} R R M N inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.24 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inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.27 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11)))) (Module.toDistribMulAction.{u_4, max u_2 u_3} A (LinearMap.{u_1, u_1, u_2, u_3} R R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11)) M N inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.27) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.14 (LinearMap.addCommMonoid.{u_1, u_1, u_2, u_3} R R M N inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.27 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11))) (LinearMap.module.{u_1, u_1, u_4, u_2, u_3} R R A M N inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.27 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11)) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.31 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.35)))))) x._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.83 x._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.85)))","typeFull":"∀ (R : Type u_1) (M : Type u_2) (N : Type u_3) (A : Type u_4) [inst : Semiring R] [inst_1 : Semiring A]\n  [inst_2 : AddCommMonoid M] [inst_3 : Module R M] [inst_4 : AddCommMonoid N] [inst_5 : Module R N]\n  [inst_6 : Module A N] [inst_7 : SMulCommClass R A N] (x : A) (x_1 : M →ₗ[R] N), (x • x_1).toFun = (x • x_1).toFun","typeReadable":"∀ (R : Type u_1) (M : Type u_2) (N : Type u_3) (A : Type u_4) [inst : Semiring R] [inst_1 : Semiring A]\n  [inst_2 : AddCommMonoid M] [inst_3 : Module R M] [inst_4 : AddCommMonoid N] [inst_5 : Module R N]\n  [inst_6 : Module A N] [inst_7 : SMulCommClass R A N] (x : A) (x_1 : M →ₗ[R] N), (x • x_1).toFun = (x • x_1).toFun","typeReferences":[["Module"],["LinearMap","toAddHom"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["MonoidWithZero","toMonoid"],["instHSMul"],["AddHom","toFun"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"],["Eq"],["DistribSMul","toSMulZeroClass"],["LinearMap","module"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["LinearMap"],["LinearMap","addCommMonoid"],["SMulCommClass"],["Module","toDistribMulAction"],["AddCommMonoid","toAddCommSemigroup"],["HSMul","hSMul"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["rfl"],["LinearMap","module"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["LinearMap","toAddHom"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["LinearMap"],["LinearMap","addCommMonoid"],["Module","toDistribMulAction"],["Semiring","toNonAssocSemiring"],["AddCommMonoid","toAddCommSemigroup"],["RingHom","id"],["MonoidWithZero","toMonoid"],["HSMul","hSMul"],["instHSMul"],["AddHom","toFun"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"],["AddZero","toZero"],["DistribSMul","toSMulZeroClass"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","mulLeft_eq_zero_iff"],"typeFallback":"forall (R : Type.{u_6}) (A : Type.{u_7}) [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1652332528._hygCtx._hyg.9 : Semiring.{u_6} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1652332528._hygCtx._hyg.12 : NonAssocSemiring.{u_7} A] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1652332528._hygCtx._hyg.15 : Module.{u_6, u_7} R A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1652332528._hygCtx._hyg.9 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1652332528._hygCtx._hyg.12))] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1652332528._hygCtx._hyg.19 : SMulCommClass.{u_6, u_7, u_7} R A A (SMulZeroClass.toSMul.{u_6, u_7} R A (AddZero.toZero.{u_7} A (AddZeroClass.toAddZero.{u_7} A (AddMonoid.toAddZeroClass.{u_7} A (AddMonoidWithOne.toAddMonoid.{u_7} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u_7} A (NonAssocSemiring.toAddCommMonoidWithOne.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1652332528._hygCtx._hyg.12)))))) (DistribSMul.toSMulZeroClass.{u_6, u_7} R A (AddMonoid.toAddZeroClass.{u_7} A (AddMonoidWithOne.toAddMonoid.{u_7} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u_7} A (NonAssocSemiring.toAddCommMonoidWithOne.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1652332528._hygCtx._hyg.12)))) (DistribMulAction.toDistribSMul.{u_6, u_7} R A (MonoidWithZero.toMonoid.{u_6} R (Semiring.toMonoidWithZero.{u_6} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1652332528._hygCtx._hyg.9)) (AddMonoidWithOne.toAddMonoid.{u_7} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u_7} A (NonAssocSemiring.toAddCommMonoidWithOne.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1652332528._hygCtx._hyg.12))) (Module.toDistribMulAction.{u_6, u_7} R A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1652332528._hygCtx._hyg.9 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1652332528._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1652332528._hygCtx._hyg.15)))) (SMulZeroClass.toSMul.{u_7, u_7} A A (AddZero.toZero.{u_7} A (AddZeroClass.toAddZero.{u_7} A (AddMonoid.toAddZeroClass.{u_7} A (AddMonoidWithOne.toAddMonoid.{u_7} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u_7} A (NonAssocSemiring.toAddCommMonoidWithOne.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1652332528._hygCtx._hyg.12)))))) (DistribSMul.toSMulZeroClass.{u_7, u_7} A A (AddMonoid.toAddZeroClass.{u_7} A (AddMonoidWithOne.toAddMonoid.{u_7} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u_7} A (NonAssocSemiring.toAddCommMonoidWithOne.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1652332528._hygCtx._hyg.12)))) (instDistribSMul.{u_7} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1652332528._hygCtx._hyg.12))))] (a : A), Iff (Eq.{succ u_7} (LinearMap.{u_6, u_6, u_7, u_7} R R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1652332528._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1652332528._hygCtx._hyg.9 (RingHom.id.{u_6} R (Semiring.toNonAssocSemiring.{u_6} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1652332528._hygCtx._hyg.9)) A A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1652332528._hygCtx._hyg.12)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1652332528._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1652332528._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1652332528._hygCtx._hyg.15) (LinearMap.mulLeft.{u_6, u_7} R A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1652332528._hygCtx._hyg.9 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1652332528._hygCtx._hyg.12) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1652332528._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1652332528._hygCtx._hyg.19 a) (OfNat.ofNat.{u_7} (LinearMap.{u_6, u_6, u_7, u_7} R R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1652332528._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1652332528._hygCtx._hyg.9 (RingHom.id.{u_6} R (Semiring.toNonAssocSemiring.{u_6} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1652332528._hygCtx._hyg.9)) A A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1652332528._hygCtx._hyg.12)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1652332528._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1652332528._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1652332528._hygCtx._hyg.15) 0 (Zero.toOfNat0.{u_7} (LinearMap.{u_6, u_6, u_7, u_7} R R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1652332528._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1652332528._hygCtx._hyg.9 (RingHom.id.{u_6} R (Semiring.toNonAssocSemiring.{u_6} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1652332528._hygCtx._hyg.9)) A A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1652332528._hygCtx._hyg.12)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1652332528._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1652332528._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1652332528._hygCtx._hyg.15) (LinearMap.instZero.{u_6, u_6, u_7, u_7} R R A A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1652332528._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1652332528._hygCtx._hyg.9 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1652332528._hygCtx._hyg.12)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1652332528._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1652332528._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1652332528._hygCtx._hyg.15 (RingHom.id.{u_6} R (Semiring.toNonAssocSemiring.{u_6} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1652332528._hygCtx._hyg.9)))))) (Eq.{succ u_7} A a (OfNat.ofNat.{u_7} A 0 (Zero.toOfNat0.{u_7} A (MulZeroClass.toZero.{u_7} A (NonUnitalNonAssocSemiring.toMulZeroClass.{u_7} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1652332528._hygCtx._hyg.12))))))","typeFull":"∀ (R : Type u_6) (A : Type u_7) [inst : Semiring R] [inst_1 : NonAssocSemiring A] [inst_2 : Module R A]\n  [inst_3 : SMulCommClass R A A] (a : A), LinearMap.mulLeft R a = 0 ↔ a = 0","typeReadable":"∀ (R : Type u_6) (A : Type u_7) [inst : Semiring R] [inst_1 : NonAssocSemiring A] [inst_2 : Module R A]\n  [inst_3 : SMulCommClass R A A] (a : A), LinearMap.mulLeft R a = 0 ↔ a = 0","typeReferences":[["Module"],["SMulZeroClass","toSMul"],["AddMonoidWithOne","toAddMonoid"],["instDistribSMul"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["NonAssocSemiring"],["MonoidWithZero","toMonoid"],["Zero","toOfNat0"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Eq"],["LinearMap","mulLeft"],["NonAssocSemiring","toAddCommMonoidWithOne"],["DistribSMul","toSMulZeroClass"],["LinearMap","instZero"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["LinearMap"],["OfNat","ofNat"],["SMulCommClass"],["Module","toDistribMulAction"],["MulZeroClass","toZero"],["Iff"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["LinearMap","instZero"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["LinearMap","mulLeft_zero_eq_zero"],["LinearMap"],["LinearMap","mulLeft_inj"],["OfNat","ofNat"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["MulZeroClass","toZero"],["Iff"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["Zero","toOfNat0"],["Eq","rec"],["LinearMap","mulLeft"],["Eq"]]},{"isProp":false,"kind":"definition","name":["LinearMap","instModuleDomMulActOfSMulCommClass"],"typeFallback":"forall {R : Type.{u_1}} {R' : Type.{u_2}} {S : Type.{u_3}} {M : Type.{u_4}} {M' : Type.{u_5}} [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 : Semiring.{u_2} R'] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 : AddCommMonoid.{u_4} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 : AddCommMonoid.{u_5} M'] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 : Module.{u_1, u_4} R M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23 : Module.{u_2, u_5} R' M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16] {σ₁₂ : RingHom.{u_1, u_2} R R' (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7) (Semiring.toNonAssocSemiring.{u_2} R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10)} [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.32 : Semiring.{u_3} S] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.35 : Module.{u_3, u_4} S M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.32 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.48 : SMulCommClass.{u_1, u_3, u_4} R S M (SMulZeroClass.toSMul.{u_1, u_4} R M (AddZero.toZero.{u_4} M (AddZeroClass.toAddZero.{u_4} M (AddMonoid.toAddZeroClass.{u_4} M (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13)))) (DistribSMul.toSMulZeroClass.{u_1, u_4} R M (AddMonoid.toAddZeroClass.{u_4} M (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13)) (DistribMulAction.toDistribSMul.{u_1, u_4} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13) (Module.toDistribMulAction.{u_1, u_4} R M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19)))) (SMulZeroClass.toSMul.{u_3, u_4} S M (AddZero.toZero.{u_4} M (AddZeroClass.toAddZero.{u_4} M (AddMonoid.toAddZeroClass.{u_4} M (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13)))) (DistribSMul.toSMulZeroClass.{u_3, u_4} S M (AddMonoid.toAddZeroClass.{u_4} M (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13)) (DistribMulAction.toDistribSMul.{u_3, u_4} S M (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.32)) (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13) (Module.toDistribMulAction.{u_3, u_4} S M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.32 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.35))))], Module.{u_3, max u_5 u_4} (DomMulAct.{u_3} S) (LinearMap.{u_1, u_2, u_4, u_5} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23) (DomMulAct.instSemiringOfMulOpposite.{u_3} S (MulOpposite.instSemiring.{u_3} S inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.32)) (LinearMap.addCommMonoid.{u_1, u_2, u_4, u_5} R R' M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.3265207960._hygCtx._hyg.23 σ₁₂)","typeFull":"{R : Type u_1} →\n  {R' : Type u_2} →\n    {S : Type u_3} →\n      {M : Type u_4} →\n        {M' : Type u_5} →\n          [inst : Semiring R] →\n            [inst_1 : Semiring R'] →\n              [inst_2 : AddCommMonoid M] →\n                [inst_3 : AddCommMonoid M'] →\n                  [inst_4 : Module R M] →\n                    [inst_5 : Module R' M'] →\n                      {σ₁₂ : R →+* R'} →\n                        [inst_6 : Semiring S] →\n                          [inst_7 : Module S M] → [SMulCommClass R S M] → Module Sᵈᵐᵃ (M →ₛₗ[σ₁₂] M')","typeReadable":"{R : Type u_1} →\n  {R' : Type u_2} →\n    {S : Type u_3} →\n      {M : Type u_4} →\n        {M' : Type u_5} →\n          [inst : Semiring R] →\n            [inst_1 : Semiring R'] →\n              [inst_2 : AddCommMonoid M] →\n                [inst_3 : AddCommMonoid M'] →\n                  [inst_4 : Module R M] →\n                    [inst_5 : Module R' M'] →\n                      {σ₁₂ : R →+* R'} →\n                        [inst_6 : Semiring S] →\n                          [inst_7 : Module S M] → [SMulCommClass R S M] → Module Sᵈᵐᵃ (M →ₛₗ[σ₁₂] M')","typeReferences":[["RingHom"],["Module"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["DomMulAct"],["AddCommMonoid","toAddMonoid"],["LinearMap"],["AddZeroClass","toAddZero"],["LinearMap","addCommMonoid"],["SMulCommClass"],["Module","toDistribMulAction"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["AddZero","toZero"],["MulOpposite","instSemiring"],["DomMulAct","instSemiringOfMulOpposite"],["Semiring"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]],"valueReferences":[["LinearMap","addCommMonoid"],["Module","toDistribMulAction"],["Module","mk"],["MonoidWithZero","toMonoid"],["Semiring","toMonoidWithZero"],["DomMulAct"],["LinearMap","instModuleDomMulActOfSMulCommClass","_proof_1"],["LinearMap"],["LinearMap","instModuleDomMulActOfSMulCommClass","_proof_2"],["LinearMap","instDistribMulActionDomMulActOfSMulCommClass"],["MulOpposite","instSemiring"],["DomMulAct","instSemiringOfMulOpposite"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","instDistribMulActionDomMulActOfSMulCommClass","_proof_1"],"typeFallback":"forall {R : Type.{u_2}} {R' : Type.{u_3}} {M : Type.{u_4}} {M' : Type.{u_5}} [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.7 : Semiring.{u_2} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.10 : Semiring.{u_3} R'] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13 : AddCommMonoid.{u_4} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.16 : AddCommMonoid.{u_5} M'] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.19 : Module.{u_2, u_4} R M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.23 : Module.{u_3, u_5} R' M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.16] {σ₁₂ : RingHom.{u_2, u_3} R R' (Semiring.toNonAssocSemiring.{u_2} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.7) (Semiring.toNonAssocSemiring.{u_3} R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.10)} {S' : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.33 : Monoid.{u_1} S'] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.36 : DistribMulAction.{u_1, u_4} S' M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.33 (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13)] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.40 : SMulCommClass.{u_2, u_1, u_4} R S' M (SMulZeroClass.toSMul.{u_2, u_4} R M (AddZero.toZero.{u_4} M (AddZeroClass.toAddZero.{u_4} M (AddMonoid.toAddZeroClass.{u_4} M (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13)))) (DistribSMul.toSMulZeroClass.{u_2, u_4} R M (AddMonoid.toAddZeroClass.{u_4} M (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13)) (DistribMulAction.toDistribSMul.{u_2, u_4} R M (MonoidWithZero.toMonoid.{u_2} R (Semiring.toMonoidWithZero.{u_2} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.7)) (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13) (Module.toDistribMulAction.{u_2, u_4} R M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.19)))) (SMulZeroClass.toSMul.{u_1, u_4} S' M (AddZero.toZero.{u_4} M (AddZeroClass.toAddZero.{u_4} M (AddMonoid.toAddZeroClass.{u_4} M (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13)))) (DistribSMul.toSMulZeroClass.{u_1, u_4} S' M (AddMonoid.toAddZeroClass.{u_4} M (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13)) (DistribMulAction.toDistribSMul.{u_1, u_4} S' M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.33 (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.36)))] (x._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.67 : DomMulAct.{u_1} S') (x._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.69 : DomMulAct.{u_1} S') (x._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.71 : LinearMap.{u_2, u_3, u_4, u_5} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.23), Eq.{max (succ u_5) (succ u_4)} (LinearMap.{u_2, u_3, u_4, u_5} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.23) (HSMul.hSMul.{u_1, max u_4 u_5, max u_4 u_5} (DomMulAct.{u_1} S') (LinearMap.{u_2, u_3, u_4, u_5} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.23) (LinearMap.{u_2, u_3, u_4, u_5} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1041117916._hygCtx._hyg.23) (instHSMul.{u_1, max u_4 u_5} (DomMulAct.{u_1} S') (LinearMap.{u_2, u_3, u_4, u_5} R R' 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[inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.9 : Semiring.{u_6} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.12 : NonAssocSemiring.{u_7} A] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.15 : Module.{u_6, u_7} R A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.9 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.12))] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.19 : IsScalarTower.{u_6, u_7, u_7} R A A (SMulZeroClass.toSMul.{u_6, u_7} R A (AddZero.toZero.{u_7} A (AddZeroClass.toAddZero.{u_7} A (AddMonoid.toAddZeroClass.{u_7} A (AddMonoidWithOne.toAddMonoid.{u_7} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u_7} A (NonAssocSemiring.toAddCommMonoidWithOne.{u_7} A 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inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.15)))) (SMulZeroClass.toSMul.{u_7, u_7} A A (AddZero.toZero.{u_7} A (AddZeroClass.toAddZero.{u_7} A (AddMonoid.toAddZeroClass.{u_7} A (AddMonoidWithOne.toAddMonoid.{u_7} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u_7} A (NonAssocSemiring.toAddCommMonoidWithOne.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.12)))))) (DistribSMul.toSMulZeroClass.{u_7, u_7} A A (AddMonoid.toAddZeroClass.{u_7} A (AddMonoidWithOne.toAddMonoid.{u_7} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u_7} A (NonAssocSemiring.toAddCommMonoidWithOne.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.12)))) (instDistribSMul.{u_7} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.12)))) (SMulZeroClass.toSMul.{u_6, u_7} R A (AddZero.toZero.{u_7} A (AddZeroClass.toAddZero.{u_7} A (AddMonoid.toAddZeroClass.{u_7} A (AddMonoidWithOne.toAddMonoid.{u_7} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u_7} A (NonAssocSemiring.toAddCommMonoidWithOne.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.12)))))) (DistribSMul.toSMulZeroClass.{u_6, u_7} R A (AddMonoid.toAddZeroClass.{u_7} A (AddMonoidWithOne.toAddMonoid.{u_7} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u_7} A (NonAssocSemiring.toAddCommMonoidWithOne.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.12)))) (DistribMulAction.toDistribSMul.{u_6, u_7} R A (MonoidWithZero.toMonoid.{u_6} R (Semiring.toMonoidWithZero.{u_6} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.9)) (AddMonoidWithOne.toAddMonoid.{u_7} A (AddCommMonoidWithOne.toAddMonoidWithOne.{u_7} A (NonAssocSemiring.toAddCommMonoidWithOne.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.12))) 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(NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.15) (LinearMap.mulRight.{u_6, u_7} R A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.9 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.12) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.19 a) (OfNat.ofNat.{u_7} (LinearMap.{u_6, u_6, u_7, u_7} R R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.9 (RingHom.id.{u_6} R (Semiring.toNonAssocSemiring.{u_6} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.9)) A A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.12)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.15) 0 (Zero.toOfNat0.{u_7} (LinearMap.{u_6, u_6, u_7, u_7} R R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.9 (RingHom.id.{u_6} R (Semiring.toNonAssocSemiring.{u_6} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.9)) A A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.12)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.15) (LinearMap.instZero.{u_6, u_6, u_7, u_7} R R A A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.9 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.12)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.15 (RingHom.id.{u_6} R (Semiring.toNonAssocSemiring.{u_6} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.9)))))) (Eq.{succ u_7} A a (OfNat.ofNat.{u_7} A 0 (Zero.toOfNat0.{u_7} A (MulZeroClass.toZero.{u_7} A (NonUnitalNonAssocSemiring.toMulZeroClass.{u_7} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1565215323._hygCtx._hyg.12))))))","typeFull":"∀ (R : Type u_6) (A : Type u_7) [inst : Semiring R] [inst_1 : NonAssocSemiring A] [inst_2 : Module R A]\n  [inst_3 : IsScalarTower R A A] (a : A), (LinearMap.mulRight R a = 0) = (a = 0)","typeReadable":"∀ (R : Type u_6) (A : Type u_7) [inst : Semiring R] [inst_1 : NonAssocSemiring A] [inst_2 : Module R A]\n  [inst_3 : IsScalarTower R A A] (a : A), (LinearMap.mulRight R a = 0) = (a = 0)","typeReferences":[["IsScalarTower"],["Module"],["SMulZeroClass","toSMul"],["AddMonoidWithOne","toAddMonoid"],["instDistribSMul"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["NonAssocSemiring"],["MonoidWithZero","toMonoid"],["Zero","toOfNat0"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Eq"],["NonAssocSemiring","toAddCommMonoidWithOne"],["DistribSMul","toSMulZeroClass"],["LinearMap","instZero"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["LinearMap"],["OfNat","ofNat"],["Module","toDistribMulAction"],["MulZeroClass","toZero"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["LinearMap","mulRight"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["LinearMap","instZero"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["LinearMap"],["OfNat","ofNat"],["LinearMap","mulRight_eq_zero_iff"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["MulZeroClass","toZero"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["LinearMap","mulRight"],["Zero","toOfNat0"],["Eq"],["propext"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","instIsTorsionFree"],"typeFallback":"forall {R : Type.{u_1}} {R' : Type.{u_2}} {S : Type.{u_3}} {M : Type.{u_4}} {M' : Type.{u_5}} [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1562582104._hygCtx._hyg.7 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1562582104._hygCtx._hyg.10 : Semiring.{u_2} R'] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1562582104._hygCtx._hyg.13 : AddCommMonoid.{u_4} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1562582104._hygCtx._hyg.16 : AddCommMonoid.{u_5} M'] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1562582104._hygCtx._hyg.19 : Module.{u_1, u_4} R M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1562582104._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1562582104._hygCtx._hyg.13] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1562582104._hygCtx._hyg.23 : Module.{u_2, u_5} R' M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1562582104._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1562582104._hygCtx._hyg.16] {σ₁₂ : RingHom.{u_1, u_2} R R' (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1562582104._hygCtx._hyg.7) (Semiring.toNonAssocSemiring.{u_2} R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1562582104._hygCtx._hyg.10)} [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1562582104._hygCtx._hyg.32 : Semiring.{u_3} S] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1562582104._hygCtx._hyg.39 : Module.{u_3, u_5} S M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1562582104._hygCtx._hyg.32 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1562582104._hygCtx._hyg.16] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1562582104._hygCtx._hyg.43 : SMulCommClass.{u_2, u_3, u_5} R' S M' (SMulZeroClass.toSMul.{u_2, u_5} R' M' (AddZero.toZero.{u_5} M' (AddZeroClass.toAddZero.{u_5} M' (AddMonoid.toAddZeroClass.{u_5} M' (AddCommMonoid.toAddMonoid.{u_5} M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1562582104._hygCtx._hyg.16)))) (DistribSMul.toSMulZeroClass.{u_2, u_5} R' M' (AddMonoid.toAddZeroClass.{u_5} M' (AddCommMonoid.toAddMonoid.{u_5} M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1562582104._hygCtx._hyg.16)) (DistribMulAction.toDistribSMul.{u_2, u_5} R' M' (MonoidWithZero.toMonoid.{u_2} R' (Semiring.toMonoidWithZero.{u_2} R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1562582104._hygCtx._hyg.10)) (AddCommMonoid.toAddMonoid.{u_5} M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1562582104._hygCtx._hyg.16) (Module.toDistribMulAction.{u_2, u_5} R' M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1562582104._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1562582104._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1562582104._hygCtx._hyg.23)))) (SMulZeroClass.toSMul.{u_3, u_5} S M' (AddZero.toZero.{u_5} M' (AddZeroClass.toAddZero.{u_5} M' (AddMonoid.toAddZeroClass.{u_5} M' (AddCommMonoid.toAddMonoid.{u_5} M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1562582104._hygCtx._hyg.16)))) (DistribSMul.toSMulZeroClass.{u_3, u_5} S M' (AddMonoid.toAddZeroClass.{u_5} M' (AddCommMonoid.toAddMonoid.{u_5} M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1562582104._hygCtx._hyg.16)) (DistribMulAction.toDistribSMul.{u_3, u_5} S M' (MonoidWithZero.toMonoid.{u_3} S (Semiring.toMonoidWithZero.{u_3} S inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1562582104._hygCtx._hyg.32)) (AddCommMonoid.toAddMonoid.{u_5} M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1562582104._hygCtx._hyg.16) (Module.toDistribMulAction.{u_3, u_5} S M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1562582104._hygCtx._hyg.32 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1562582104._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1562582104._hygCtx._hyg.39))))] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1562582104._hygCtx._hyg.48 : Module.IsTorsionFree.{u_3, u_5} S M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1562582104._hygCtx._hyg.32 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1562582104._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1562582104._hygCtx._hyg.39], Module.IsTorsionFree.{u_3, max u_5 u_4} S (LinearMap.{u_1, u_2, u_4, u_5} R R' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1562582104._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1562582104._hygCtx._hyg.10 σ₁₂ M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1562582104._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1562582104._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1562582104._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1562582104._hygCtx._hyg.23) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1562582104._hygCtx._hyg.32 (LinearMap.addCommMonoid.{u_1, u_2, u_4, u_5} R R' M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1562582104._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1562582104._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1562582104._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1562582104._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1562582104._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1562582104._hygCtx._hyg.23 σ₁₂) (LinearMap.module.{u_1, u_2, u_3, u_4, u_5} R R' S M M' inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1562582104._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1562582104._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1562582104._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1562582104._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1562582104._hygCtx._hyg.19 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1562582104._hygCtx._hyg.23 σ₁₂ inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1562582104._hygCtx._hyg.32 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1562582104._hygCtx._hyg.39 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.1562582104._hygCtx._hyg.43)","typeFull":"∀ {R : Type u_1} {R' : Type u_2} {S : Type u_3} {M : Type u_4} {M' : Type u_5} [inst : Semiring R]\n  [inst_1 : Semiring R'] [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M'] [inst_4 : Module R M]\n  [inst_5 : Module R' M'] {σ₁₂ : R →+* R'} [inst_6 : Semiring S] [inst_7 : Module S M'] [inst_8 : SMulCommClass R' S M']\n  [Module.IsTorsionFree S M'], Module.IsTorsionFree S (M →ₛₗ[σ₁₂] M')","typeReadable":"∀ {R : Type u_1} {R' : Type u_2} {S : Type u_3} {M : Type u_4} {M' : Type u_5} [inst : Semiring R]\n  [inst_1 : Semiring R'] [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M'] [inst_4 : Module R M]\n  [inst_5 : Module R' M'] {σ₁₂ : R →+* R'} [inst_6 : Semiring S] [inst_7 : Module S M'] [inst_8 : SMulCommClass R' S M']\n  [Module.IsTorsionFree S M'], Module.IsTorsionFree S (M →ₛₗ[σ₁₂] M')","typeReferences":[["Module","IsTorsionFree"],["RingHom"],["Module"],["LinearMap","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","coe_injective"],["LinearMap","instFunLike"],["LinearMap","module"],["Pi","addCommMonoid"],["Function","Injective","moduleIsTorsionFree"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddCommMonoid","toAddMonoid"],["LinearMap"],["DFunLike","coe"],["Pi","Function","module"],["LinearMap","addCommMonoid"],["Module","toDistribMulAction"],["LinearMap","coe_smul"],["MonoidWithZero","toMonoid"],["Pi","instModuleIsTorsionFree"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","mulRight_mul"],"typeFallback":"forall (R : Type.{u_6}) (A : Type.{u_7}) [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.26182231._hygCtx._hyg.9 : Semiring.{u_6} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.26182231._hygCtx._hyg.12 : NonUnitalSemiring.{u_7} A] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.26182231._hygCtx._hyg.15 : Module.{u_6, u_7} R A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.26182231._hygCtx._hyg.9 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A (NonUnitalSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.26182231._hygCtx._hyg.12))] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.26182231._hygCtx._hyg.19 : IsScalarTower.{u_6, u_7, u_7} R A A (SMulZeroClass.toSMul.{u_6, u_7} R A (AddZero.toZero.{u_7} A (AddZeroClass.toAddZero.{u_7} A (AddMonoid.toAddZeroClass.{u_7} A (AddCommMonoid.toAddMonoid.{u_7} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A (NonUnitalSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.26182231._hygCtx._hyg.12)))))) (DistribSMul.toSMulZeroClass.{u_6, u_7} R A (AddMonoid.toAddZeroClass.{u_7} A (AddCommMonoid.toAddMonoid.{u_7} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A (NonUnitalSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.26182231._hygCtx._hyg.12)))) (DistribMulAction.toDistribSMul.{u_6, u_7} R A (MonoidWithZero.toMonoid.{u_6} R (Semiring.toMonoidWithZero.{u_6} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.26182231._hygCtx._hyg.9)) (AddCommMonoid.toAddMonoid.{u_7} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A (NonUnitalSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.26182231._hygCtx._hyg.12))) (Module.toDistribMulAction.{u_6, u_7} R A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.26182231._hygCtx._hyg.9 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A (NonUnitalSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.26182231._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.26182231._hygCtx._hyg.15)))) (SMulZeroClass.toSMul.{u_7, u_7} A A (AddZero.toZero.{u_7} A (AddZeroClass.toAddZero.{u_7} A (AddMonoid.toAddZeroClass.{u_7} A (AddCommMonoid.toAddMonoid.{u_7} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A (NonUnitalSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.26182231._hygCtx._hyg.12)))))) (DistribSMul.toSMulZeroClass.{u_7, u_7} A A (AddMonoid.toAddZeroClass.{u_7} A (AddCommMonoid.toAddMonoid.{u_7} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A (NonUnitalSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.26182231._hygCtx._hyg.12)))) (instDistribSMul.{u_7} A (NonUnitalSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.26182231._hygCtx._hyg.12)))) (SMulZeroClass.toSMul.{u_6, u_7} R A (AddZero.toZero.{u_7} A (AddZeroClass.toAddZero.{u_7} A (AddMonoid.toAddZeroClass.{u_7} A (AddCommMonoid.toAddMonoid.{u_7} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A (NonUnitalSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.26182231._hygCtx._hyg.12)))))) (DistribSMul.toSMulZeroClass.{u_6, u_7} R A (AddMonoid.toAddZeroClass.{u_7} A (AddCommMonoid.toAddMonoid.{u_7} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A (NonUnitalSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.26182231._hygCtx._hyg.12)))) (DistribMulAction.toDistribSMul.{u_6, u_7} R A (MonoidWithZero.toMonoid.{u_6} R (Semiring.toMonoidWithZero.{u_6} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.26182231._hygCtx._hyg.9)) (AddCommMonoid.toAddMonoid.{u_7} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A (NonUnitalSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.26182231._hygCtx._hyg.12))) (Module.toDistribMulAction.{u_6, u_7} R A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.26182231._hygCtx._hyg.9 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A (NonUnitalSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.26182231._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.26182231._hygCtx._hyg.15))))] (a : A) (b : A), Eq.{succ u_7} (LinearMap.{u_6, u_6, u_7, u_7} R R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.26182231._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.26182231._hygCtx._hyg.9 (RingHom.id.{u_6} R (Semiring.toNonAssocSemiring.{u_6} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.26182231._hygCtx._hyg.9)) A A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A (NonUnitalSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.26182231._hygCtx._hyg.12)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A (NonUnitalSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.26182231._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.26182231._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.26182231._hygCtx._hyg.15) (LinearMap.mulRight.{u_6, u_7} R A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.26182231._hygCtx._hyg.9 (NonUnitalSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.26182231._hygCtx._hyg.12) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.26182231._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.26182231._hygCtx._hyg.19 (HMul.hMul.{u_7, u_7, u_7} A A A (instHMul.{u_7} A (Distrib.toMul.{u_7} A (NonUnitalNonAssocSemiring.toDistrib.{u_7} A (NonUnitalSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.26182231._hygCtx._hyg.12)))) a b)) (LinearMap.comp.{u_6, u_6, u_6, u_7, u_7, u_7} R R R A A A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.26182231._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.26182231._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.26182231._hygCtx._hyg.9 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A (NonUnitalSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.26182231._hygCtx._hyg.12)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A (NonUnitalSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.26182231._hygCtx._hyg.12)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_7} A (NonUnitalSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.26182231._hygCtx._hyg.12)) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.26182231._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.26182231._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.26182231._hygCtx._hyg.15 (RingHom.id.{u_6} R (Semiring.toNonAssocSemiring.{u_6} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.26182231._hygCtx._hyg.9)) (RingHom.id.{u_6} R (Semiring.toNonAssocSemiring.{u_6} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.26182231._hygCtx._hyg.9)) (RingHom.id.{u_6} R (Semiring.toNonAssocSemiring.{u_6} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.26182231._hygCtx._hyg.9)) (RingHomCompTriple.ids.{u_6, u_6} R R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.26182231._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.26182231._hygCtx._hyg.9 (RingHom.id.{u_6} R (Semiring.toNonAssocSemiring.{u_6} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.26182231._hygCtx._hyg.9))) (LinearMap.mulRight.{u_6, u_7} R A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.26182231._hygCtx._hyg.9 (NonUnitalSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.26182231._hygCtx._hyg.12) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.26182231._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.26182231._hygCtx._hyg.19 b) (LinearMap.mulRight.{u_6, u_7} R A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.26182231._hygCtx._hyg.9 (NonUnitalSemiring.toNonUnitalNonAssocSemiring.{u_7} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.26182231._hygCtx._hyg.12) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.26182231._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.26182231._hygCtx._hyg.19 a))","typeFull":"∀ (R : Type u_6) (A : Type u_7) [inst : Semiring R] [inst_1 : NonUnitalSemiring A] [inst_2 : Module R A]\n  [inst_3 : IsScalarTower R A A] (a b : A),\n  LinearMap.mulRight R (a * b) = LinearMap.mulRight R b ∘ₗ LinearMap.mulRight R a","typeReadable":"∀ (R : Type u_6) (A : Type u_7) [inst : Semiring R] [inst_1 : NonUnitalSemiring A] [inst_2 : Module R A]\n  [inst_3 : IsScalarTower R A A] (a b : A),\n  LinearMap.mulRight R (a * b) = LinearMap.mulRight R b ∘ₗ LinearMap.mulRight R a","typeReferences":[["RingHomCompTriple","ids"],["IsScalarTower"],["Module"],["LinearMap","comp"],["HMul","hMul"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["instDistribSMul"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["MonoidWithZero","toMonoid"],["Eq"],["DistribSMul","toSMulZeroClass"],["NonUnitalNonAssocSemiring","toDistrib"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["Distrib","toMul"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["LinearMap"],["Module","toDistribMulAction"],["NonUnitalSemiring","toNonUnitalNonAssocSemiring"],["NonUnitalSemiring"],["instHMul"],["LinearMap","mulRight"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["LinearMap","instFunLike"],["Eq","trans"],["RingHomCompTriple","ids"],["LinearMap","comp"],["HMul","hMul"],["NonUnitalSemiring","toSemigroupWithZero"],["mul_assoc"],["DFunLike","coe"],["Semigroup","toMul"],["congrArg"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["LinearMap","ext"],["Eq"],["NonUnitalNonAssocSemiring","toDistrib"],["True"],["Distrib","toMul"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["LinearMap"],["eq_self"],["of_eq_true"],["NonUnitalSemiring","toNonUnitalNonAssocSemiring"],["instHMul"],["LinearMap","mulRight"],["SemigroupWithZero","toSemigroup"]]},{"isProp":false,"kind":"definition","name":["LinearMap","ltoFun"],"typeFallback":"forall (R : Type.{u_6}) (M : Type.{u_7}) (N : Type.{u_8}) (A : Type.{u_9}) [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11 : Semiring.{u_6} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.14 : Semiring.{u_9} A] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.17 : AddCommMonoid.{u_7} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.20 : Module.{u_6, u_7} R M inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.17] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.24 : AddCommMonoid.{u_8} N] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.27 : Module.{u_6, u_8} R N inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.24] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.31 : Module.{u_9, u_8} A N inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.24] [inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.35 : SMulCommClass.{u_6, u_9, u_8} R A N (SMulZeroClass.toSMul.{u_6, u_8} R N (AddZero.toZero.{u_8} N (AddZeroClass.toAddZero.{u_8} N (AddMonoid.toAddZeroClass.{u_8} N (AddCommMonoid.toAddMonoid.{u_8} N inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.24)))) (DistribSMul.toSMulZeroClass.{u_6, u_8} R N (AddMonoid.toAddZeroClass.{u_8} N (AddCommMonoid.toAddMonoid.{u_8} N inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.24)) (DistribMulAction.toDistribSMul.{u_6, u_8} R N (MonoidWithZero.toMonoid.{u_6} R (Semiring.toMonoidWithZero.{u_6} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11)) (AddCommMonoid.toAddMonoid.{u_8} N inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.24) (Module.toDistribMulAction.{u_6, u_8} R N inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.27)))) (SMulZeroClass.toSMul.{u_9, u_8} A N (AddZero.toZero.{u_8} N (AddZeroClass.toAddZero.{u_8} N (AddMonoid.toAddZeroClass.{u_8} N (AddCommMonoid.toAddMonoid.{u_8} N inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.24)))) (DistribSMul.toSMulZeroClass.{u_9, u_8} A N (AddMonoid.toAddZeroClass.{u_8} N (AddCommMonoid.toAddMonoid.{u_8} N inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.24)) (DistribMulAction.toDistribSMul.{u_9, u_8} A N (MonoidWithZero.toMonoid.{u_9} A (Semiring.toMonoidWithZero.{u_9} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.14)) (AddCommMonoid.toAddMonoid.{u_8} N inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.24) (Module.toDistribMulAction.{u_9, u_8} A N inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.31))))], LinearMap.{u_9, u_9, max u_8 u_7, max u_7 u_8} A A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.14 (RingHom.id.{u_9} A (Semiring.toNonAssocSemiring.{u_9} A inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.14)) (LinearMap.{u_6, u_6, u_7, u_8} R R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11 (RingHom.id.{u_6} R (Semiring.toNonAssocSemiring.{u_6} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11)) M N inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.27) (M -> N) (LinearMap.addCommMonoid.{u_6, u_6, u_7, u_8} R R M N inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.27 (RingHom.id.{u_6} R (Semiring.toNonAssocSemiring.{u_6} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11))) (Pi.addCommMonoid.{u_7, u_8} M (fun (a._@._internal._hyg.0 : M) => N) (fun (i : M) => inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.24)) (LinearMap.module.{u_6, u_6, u_9, u_7, u_8} R R A M N inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.27 (RingHom.id.{u_6} R (Semiring.toNonAssocSemiring.{u_6} R inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.11)) inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.31 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.35) (Pi.Function.module.{u_7, u_9, u_8} M A N inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.24 inst._@.Mathlib.Algebra.Module.LinearMap.Basic.850839177._hygCtx._hyg.31)","typeFull":"(R : Type u_6) →\n  (M : Type u_7) →\n    (N : Type u_8) →\n      (A : Type u_9) →\n        [inst : Semiring R] →\n          [inst_1 : Semiring A] →\n            [inst_2 : AddCommMonoid M] →\n              [inst_3 : Module R M] →\n                [inst_4 : AddCommMonoid N] →\n                  [inst_5 : Module R N] →\n                    [inst_6 : Module A N] → [inst_7 : SMulCommClass R A N] → (M →ₗ[R] N) →ₗ[A] M → N","typeReadable":"(R : Type u_6) →\n  (M : Type u_7) →\n    (N : Type u_8) →\n      (A : Type u_9) →\n        [inst : Semiring R] →\n          [inst_1 : Semiring A] →\n            [inst_2 : AddCommMonoid M] →\n              [inst_3 : Module R M] →\n                [inst_4 : AddCommMonoid N] →\n                  [inst_5 : Module R N] →\n                    [inst_6 : Module A N] → [inst_7 : SMulCommClass R A N] → (M →ₗ[R] N) →ₗ[A] M → 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