File: Mathlib/MeasureTheory/Function/AEEqFun/DomAct.lean

full_name: DomMulAct.instSMulAEEqFun
code: @[to_additive]
instance : SMul Mᵈᵐᵃ (α →ₘ[μ] β) 
goal: SMul Mᵈᵐᵃ (α →ₘ[μ] β)

full_name: DomAddAct.instVAddAEEqFun
code: @[to_additive]
instance : SMul Mᵈᵐᵃ (α →ₘ[μ] β) 
goal: VAdd Mᵈᵃᵃ (α →ₘ[μ] β)

full_name: DomMulAct.smul_aeeqFun_aeeq
code: @[to_additive]
theorem smul_aeeqFun_aeeq (c : Mᵈᵐᵃ) (f : α →ₘ[μ] β) :
    c • f =ᵐ[μ] (f <| mk.symm c • ·) 
goal: ↑(c • f) =ᵐ[μ] fun x => ↑f (DomMulAct.mk.symm c • x)

full_name: DomAddAct.vadd_aeeqFun_aeeq
code: @[to_additive]
theorem smul_aeeqFun_aeeq (c : Mᵈᵐᵃ) (f : α →ₘ[μ] β) :
    c • f =ᵐ[μ] (f <| mk.symm c • ·) 
goal: ↑(c +ᵥ f) =ᵐ[μ] fun x => ↑f (DomAddAct.mk.symm c +ᵥ x)

full_name: DomMulAct.mk_smul_mk_aeeqFun
code: @[to_additive (attr := simp)]
theorem mk_smul_mk_aeeqFun (c : M) (f : α → β) (hf : AEStronglyMeasurable f μ) :
    mk c • AEEqFun.mk f hf = AEEqFun.mk (f <| c • ·)
      (hf.comp_measurePreserving (measurePreserving_smul _ _)) 
goal: DomMulAct.mk c • MeasureTheory.AEEqFun.mk f hf = MeasureTheory.AEEqFun.mk (fun x => f (c • x)) ⋯

full_name: DomAddAct.mk_vadd_mk_aeeqFun
code: @[to_additive (attr := simp)]
theorem mk_smul_mk_aeeqFun (c : M) (f : α → β) (hf : AEStronglyMeasurable f μ) :
    mk c • AEEqFun.mk f hf = AEEqFun.mk (f <| c • ·)
      (hf.comp_measurePreserving (measurePreserving_smul _ _)) 
goal: DomAddAct.mk c +ᵥ MeasureTheory.AEEqFun.mk f hf = MeasureTheory.AEEqFun.mk (fun x => f (c +ᵥ x)) ⋯

full_name: DomAddAct.vadd_aeeqFun_const
code: @[to_additive (attr := simp)]
theorem smul_aeeqFun_const (c : Mᵈᵐᵃ) (b : β) :
    c • (AEEqFun.const α b : α →ₘ[μ] β) = AEEqFun.const α b 
goal: c +ᵥ MeasureTheory.AEEqFun.const α b = MeasureTheory.AEEqFun.const α b

full_name: DomMulAct.smul_aeeqFun_const
code: @[to_additive (attr := simp)]
theorem smul_aeeqFun_const (c : Mᵈᵐᵃ) (b : β) :
    c • (AEEqFun.const α b : α →ₘ[μ] β) = AEEqFun.const α b 
goal: c • MeasureTheory.AEEqFun.const α b = MeasureTheory.AEEqFun.const α b

full_name: DomMulAct.instSMulCommClassAEEqFun
code: instance [SMul N β] [ContinuousConstSMul N β] : SMulCommClass Mᵈᵐᵃ N (α →ₘ[μ] β) 
goal: SMulCommClass Mᵈᵐᵃ N (α →ₘ[μ] β)

full_name: DomMulAct.instSMulCommClassAEEqFun_1
code: instance [SMul N β] [ContinuousConstSMul N β] : SMulCommClass N Mᵈᵐᵃ (α →ₘ[μ] β) 
goal: SMulCommClass N Mᵈᵐᵃ (α →ₘ[μ] β)

full_name: DomAddAct.instVAddCommClassAEEqFun_2
code: @[to_additive]
instance [SMul N α] [MeasurableSMul N α] [SMulInvariantMeasure N α μ] [SMulCommClass M N α] :
    SMulCommClass Mᵈᵐᵃ Nᵈᵐᵃ (α →ₘ[μ] β) 
goal: VAddCommClass Mᵈᵃᵃ Nᵈᵃᵃ (α →ₘ[μ] β)

full_name: DomMulAct.instSMulCommClassAEEqFun_2
code: @[to_additive]
instance [SMul N α] [MeasurableSMul N α] [SMulInvariantMeasure N α μ] [SMulCommClass M N α] :
    SMulCommClass Mᵈᵐᵃ Nᵈᵐᵃ (α →ₘ[μ] β) 
goal: SMulCommClass Mᵈᵐᵃ Nᵈᵐᵃ (α →ₘ[μ] β)

full_name: DomMulAct.instSMulZeroClassAEEqFun
code: instance [Zero β] : SMulZeroClass Mᵈᵐᵃ (α →ₘ[μ] β) 
goal: SMulZeroClass Mᵈᵐᵃ (α →ₘ[μ] β)

full_name: DomMulAct.instDistribSMulAEEqFun
code: instance [AddMonoid β] [ContinuousAdd β] : DistribSMul Mᵈᵐᵃ (α →ₘ[μ] β) 
goal: DistribSMul Mᵈᵐᵃ (α →ₘ[μ] β)

full_name: DomAddAct.instAddActionAEEqFun
code: @[to_additive]
instance : MulAction Mᵈᵐᵃ (α →ₘ[μ] β) 
goal: AddAction Mᵈᵃᵃ (α →ₘ[μ] β)

full_name: DomMulAct.instMulActionAEEqFun
code: @[to_additive]
instance : MulAction Mᵈᵐᵃ (α →ₘ[μ] β) 
goal: MulAction Mᵈᵐᵃ (α →ₘ[μ] β)

full_name: DomMulAct.instMulDistribMulActionAEEqFun
code: instance [Monoid β] [ContinuousMul β] : MulDistribMulAction Mᵈᵐᵃ (α →ₘ[μ] β) 
goal: MulDistribMulAction Mᵈᵐᵃ (α →ₘ[μ] β)

full_name: DomMulAct.instDistribMulActionAEEqFun
code: instance [AddMonoid β] [ContinuousAdd β] : DistribMulAction Mᵈᵐᵃ (α →ₘ[μ] β) 
goal: DistribMulAction Mᵈᵐᵃ (α →ₘ[μ] β)