File: Mathlib/Algebra/Module/Hom.lean

full_name: AddMonoidHom.instDistribSMul
code: instance instDistribSMul [AddZeroClass A] [AddCommMonoid B] [DistribSMul M B] :
    DistribSMul M (A →+ B) 
goal: DistribSMul M (A →+ B)

full_name: AddMonoidHom.instDistribMulAction
code: instance instDistribMulAction : DistribMulAction R (A →+ B) 
goal: DistribMulAction R (A →+ B)

full_name: AddMonoidHom.coe_smul
code: @[simp] theorem coe_smul (r : R) (f : A →+ B) : ⇑(r • f) = r • ⇑f 
goal: ⇑(r • f) = r • ⇑f

full_name: AddMonoidHom.smul_apply
code: theorem smul_apply (r : R) (f : A →+ B) (x : A) : (r • f) x = r • f x 
goal: (r • f) x = r • f x

full_name: AddMonoidHom.smulCommClass
code: instance smulCommClass [SMulCommClass R S B] : SMulCommClass R S (A →+ B) 
goal: SMulCommClass R S (A →+ B)

full_name: AddMonoidHom.isScalarTower
code: instance isScalarTower [SMul R S] [IsScalarTower R S B] : IsScalarTower R S (A →+ B) 
goal: IsScalarTower R S (A →+ B)

full_name: AddMonoidHom.isCentralScalar
code: instance isCentralScalar [DistribMulAction Rᵐᵒᵖ B] [IsCentralScalar R B] :
    IsCentralScalar R (A →+ B) 
goal: IsCentralScalar R (A →+ B)

full_name: AddMonoidHom.instModule
code: instance instModule [Semiring R] [AddMonoid A] [AddCommMonoid B] [Module R B] : Module R (A →+ B) 
goal: Module R (A →+ B)

full_name: AddMonoidHom.instDomMulActModule
code: instance instDomMulActModule
    {S M M₂ : Type*} [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] [Module S M] :
    Module Sᵈᵐᵃ (M →+ M₂) 
goal: Module Sᵈᵐᵃ (M →+ M₂)

full_name: AddMonoid.End.instDistribSMul
code: instance instDistribSMul [DistribSMul M A] : DistribSMul M (AddMonoid.End A) 
goal: DistribSMul M (AddMonoid.End A)

full_name: AddMonoid.End.instDistribMulAction
code: instance instDistribMulAction : DistribMulAction R (AddMonoid.End A) 
goal: DistribMulAction R (AddMonoid.End A)

full_name: AddMonoid.End.coe_smul
code: @[simp] theorem coe_smul (r : R) (f : AddMonoid.End A) : ⇑(r • f) = r • ⇑f 
goal: ⇑(r • f) = r • ⇑f

full_name: AddMonoid.End.smul_apply
code: theorem smul_apply (r : R) (f : AddMonoid.End A) (x : A) : (r • f) x = r • f x 
goal: (r • f) x = r • f x

full_name: AddMonoid.End.smulCommClass
code: instance smulCommClass [SMulCommClass R S A] : SMulCommClass R S (AddMonoid.End A) 
goal: SMulCommClass R S (AddMonoid.End A)

full_name: AddMonoid.End.isScalarTower
code: instance isScalarTower [SMul R S] [IsScalarTower R S A] : IsScalarTower R S (AddMonoid.End A) 
goal: IsScalarTower R S (AddMonoid.End A)

full_name: AddMonoid.End.isCentralScalar
code: instance isCentralScalar [DistribMulAction Rᵐᵒᵖ A] [IsCentralScalar R A] :
    IsCentralScalar R (AddMonoid.End A) 
goal: IsCentralScalar R (AddMonoid.End A)

full_name: AddMonoid.End.instModule
code: instance instModule [Semiring R] [AddCommMonoid A] [Module R A] : Module R (AddMonoid.End A) 
goal: Module R (AddMonoid.End A)

full_name: AddMonoid.End.applyModule
code: /-- The tautological action by `AddMonoid.End α` on `α`.

This generalizes `AddMonoid.End.applyDistribMulAction`. -/
instance applyModule [AddCommMonoid A] : Module (AddMonoid.End A) A 
goal: Module (AddMonoid.End A) A

full_name: AddMonoidHom.smulLeft_apply
code: /-- Scalar multiplication on the left as an additive monoid homomorphism. -/
@[simps! (config := .asFn)]
protected def smulLeft [Monoid M] [AddMonoid A] [DistribMulAction M A] (c : M) : A →+ A 
goal: ⇑(AddMonoidHom.smulLeft c) = fun x => c • x

full_name: AddMonoidHom.smulLeft
code: /-- Scalar multiplication on the left as an additive monoid homomorphism. -/
@[simps! (config := .asFn)]
protected def smulLeft [Monoid M] [AddMonoid A] [DistribMulAction M A] (c : M) : A →+ A 
goal: A →+ A

full_name: AddMonoidHom.smul
code: /-- Scalar multiplication as a biadditive monoid homomorphism. We need `M` to be commutative
to have addition on `M →+ M`. -/
protected def smul [Semiring R] [AddCommMonoid M] [Module R M] : R →+ M →+ M 
goal: R →+ M →+ M

full_name: AddMonoidHom.coe_smul'
code: @[simp] theorem coe_smul' [Semiring R] [AddCommMonoid M] [Module R M] :
    ⇑(.smul : R →+ M →+ M) = AddMonoidHom.smulLeft 
goal: ⇑AddMonoidHom.smul = AddMonoidHom.smulLeft