File: Mathlib/MeasureTheory/Function/LpSpace/DomAct/Basic.lean

full_name: DomMulAct.instSMulSubtypeAEEqFunMemAddSubgroupLp
code: @[to_additive]
instance : SMul Mᵈᵐᵃ (Lp E p μ) 
goal: SMul Mᵈᵐᵃ ↥(MeasureTheory.Lp E p μ)

full_name: DomAddAct.instVAddSubtypeAEEqFunMemAddAddSubgroupLp
code: @[to_additive]
instance : SMul Mᵈᵐᵃ (Lp E p μ) 
goal: VAdd Mᵈᵃᵃ ↥(MeasureTheory.Lp E p μ)

full_name: DomMulAct.smul_Lp_val
code: @[to_additive (attr := simp)]
theorem smul_Lp_val (c : Mᵈᵐᵃ) (f : Lp E p μ) : (c • f).1 = c • f.1 
goal: ↑(c • f) = c • ↑f

full_name: DomAddAct.vadd_Lp_val
code: @[to_additive (attr := simp)]
theorem smul_Lp_val (c : Mᵈᵐᵃ) (f : Lp E p μ) : (c • f).1 = c • f.1 
goal: ↑(c +ᵥ f) = c +ᵥ ↑f

full_name: DomMulAct.smul_Lp_ae_eq
code: @[to_additive]
theorem smul_Lp_ae_eq (c : Mᵈᵐᵃ) (f : Lp E p μ) : 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_Lp_ae_eq
code: @[to_additive]
theorem smul_Lp_ae_eq (c : Mᵈᵐᵃ) (f : Lp E p μ) : 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: DomAddAct.mk_vadd_toLp
code: @[to_additive]
theorem mk_smul_toLp (c : M) {f : α → E} (hf : Memℒp f p μ) :
    mk c • hf.toLp f =
      (hf.comp_measurePreserving <| measurePreserving_smul c μ).toLp (f <| c • ·) 
goal: DomAddAct.mk c +ᵥ MeasureTheory.Memℒp.toLp f hf = MeasureTheory.Memℒp.toLp (fun x => f (c +ᵥ x)) ⋯

full_name: DomMulAct.mk_smul_toLp
code: @[to_additive]
theorem mk_smul_toLp (c : M) {f : α → E} (hf : Memℒp f p μ) :
    mk c • hf.toLp f =
      (hf.comp_measurePreserving <| measurePreserving_smul c μ).toLp (f <| c • ·) 
goal: DomMulAct.mk c • MeasureTheory.Memℒp.toLp f hf = MeasureTheory.Memℒp.toLp (fun x => f (c • x)) ⋯

full_name: DomMulAct.smul_Lp_const
code: @[to_additive (attr := simp)]
theorem smul_Lp_const [IsFiniteMeasure μ] (c : Mᵈᵐᵃ) (a : E) :
    c • Lp.const p μ a = Lp.const p μ a 
goal: c • (MeasureTheory.Lp.const p μ) a = (MeasureTheory.Lp.const p μ) a

full_name: DomAddAct.vadd_Lp_const
code: @[to_additive (attr := simp)]
theorem smul_Lp_const [IsFiniteMeasure μ] (c : Mᵈᵐᵃ) (a : E) :
    c • Lp.const p μ a = Lp.const p μ a 
goal: c +ᵥ (MeasureTheory.Lp.const p μ) a = (MeasureTheory.Lp.const p μ) a

full_name: DomMulAct.mk_smul_indicatorConstLp
code: @[to_additive]
theorem mk_smul_indicatorConstLp (c : M)
    {s : Set α} (hs : MeasurableSet s) (hμs : μ s ≠ ∞) (b : E) :
    mk c • indicatorConstLp p hs hμs b =
      indicatorConstLp p (hs.preimage <| measurable_const_smul c)
        (by rwa [SMulInvariantMeasure.measure_preimage_smul c hs]) b 
goal: DomMulAct.mk c • MeasureTheory.indicatorConstLp p hs hμs b = MeasureTheory.indicatorConstLp p ⋯ ⋯ b

full_name: DomAddAct.mk_vadd_indicatorConstLp
code: @[to_additive]
theorem mk_smul_indicatorConstLp (c : M)
    {s : Set α} (hs : MeasurableSet s) (hμs : μ s ≠ ∞) (b : E) :
    mk c • indicatorConstLp p hs hμs b =
      indicatorConstLp p (hs.preimage <| measurable_const_smul c)
        (by rwa [SMulInvariantMeasure.measure_preimage_smul c hs]) b 
goal: DomAddAct.mk c +ᵥ MeasureTheory.indicatorConstLp p hs hμs b = MeasureTheory.indicatorConstLp p ⋯ ⋯ b

full_name: DomMulAct.instSMulCommClassSubtypeAEEqFunMemAddSubgroupLp
code: instance [SMul N α] [SMulCommClass M N α] [SMulInvariantMeasure N α μ] [MeasurableSMul N α] :
    SMulCommClass Mᵈᵐᵃ Nᵈᵐᵃ (Lp E p μ) 
goal: SMulCommClass Mᵈᵐᵃ Nᵈᵐᵃ ↥(MeasureTheory.Lp E p μ)

full_name: DomMulAct.instSMulCommClassSubtypeAEEqFunMemAddSubgroupLp_1
code: instance {𝕜 : Type*} [NormedRing 𝕜] [Module 𝕜 E] [BoundedSMul 𝕜 E] :
    SMulCommClass Mᵈᵐᵃ 𝕜 (Lp E p μ) 
goal: SMulCommClass Mᵈᵐᵃ 𝕜 ↥(MeasureTheory.Lp E p μ)

full_name: DomMulAct.instSMulCommClassSubtypeAEEqFunMemAddSubgroupLp_2
code: instance {𝕜 : Type*} [NormedRing 𝕜] [Module 𝕜 E] [BoundedSMul 𝕜 E] :
    SMulCommClass 𝕜 Mᵈᵐᵃ (Lp E p μ) 
goal: SMulCommClass 𝕜 Mᵈᵐᵃ ↥(MeasureTheory.Lp E p μ)

full_name: DomAddAct.vadd_Lp_add
code: @[to_additive]
theorem smul_Lp_add (c : Mᵈᵐᵃ) : ∀ f g : Lp E p μ, c • (f + g) = c • f + c • g 
goal: c +ᵥ (f + g) = c +ᵥ f + (c +ᵥ g)

full_name: DomMulAct.smul_Lp_add
code: @[to_additive]
theorem smul_Lp_add (c : Mᵈᵐᵃ) : ∀ f g : Lp E p μ, c • (f + g) = c • f + c • g 
goal: c • (f + g) = c • f + c • g

full_name: DomMulAct.smul_Lp_zero
code: @[to_additive (attr := simp 1001)]
theorem smul_Lp_zero (c : Mᵈᵐᵃ) : c • (0 : Lp E p μ) = 0 
goal: c • 0 = 0

full_name: DomAddAct.vadd_Lp_zero
code: @[to_additive (attr := simp 1001)]
theorem smul_Lp_zero (c : Mᵈᵐᵃ) : c • (0 : Lp E p μ) = 0 
goal: c +ᵥ 0 = 0

full_name: DomAddAct.vadd_Lp_neg
code: @[to_additive]
theorem smul_Lp_neg (c : Mᵈᵐᵃ) (f : Lp E p μ) : c • (-f) = -(c • f) 
goal: c +ᵥ -f = -(c +ᵥ f)

full_name: DomMulAct.smul_Lp_neg
code: @[to_additive]
theorem smul_Lp_neg (c : Mᵈᵐᵃ) (f : Lp E p μ) : c • (-f) = -(c • f) 
goal: c • -f = -(c • f)

full_name: DomAddAct.vadd_Lp_sub
code: @[to_additive]
theorem smul_Lp_sub (c : Mᵈᵐᵃ) : ∀ f g : Lp E p μ, c • (f - g) = c • f - c • g 
goal: c +ᵥ (f - g) = c +ᵥ f - (c +ᵥ g)

full_name: DomMulAct.smul_Lp_sub
code: @[to_additive]
theorem smul_Lp_sub (c : Mᵈᵐᵃ) : ∀ f g : Lp E p μ, c • (f - g) = c • f - c • g 
goal: c • (f - g) = c • f - c • g

full_name: DomMulAct.instDistribSMulSubtypeAEEqFunMemAddSubgroupLp
code: instance : DistribSMul Mᵈᵐᵃ (Lp E p μ) 
goal: DistribSMul Mᵈᵐᵃ ↥(MeasureTheory.Lp E p μ)

full_name: DomAddAct.norm_vadd_Lp
code: @[to_additive (attr := simp)]
theorem norm_smul_Lp (c : Mᵈᵐᵃ) (f : Lp E p μ) : ‖c • f‖ = ‖f‖ 
goal: ‖c +ᵥ f‖ = ‖f‖

full_name: DomMulAct.norm_smul_Lp
code: @[to_additive (attr := simp)]
theorem norm_smul_Lp (c : Mᵈᵐᵃ) (f : Lp E p μ) : ‖c • f‖ = ‖f‖ 
goal: ‖c • f‖ = ‖f‖

full_name: DomMulAct.nnnorm_smul_Lp
code: @[to_additive (attr := simp)]
theorem nnnorm_smul_Lp (c : Mᵈᵐᵃ) (f : Lp E p μ) : ‖c • f‖₊ = ‖f‖₊ 
goal: ‖c • f‖₊ = ‖f‖₊

full_name: DomAddAct.nnnorm_vadd_Lp
code: @[to_additive (attr := simp)]
theorem nnnorm_smul_Lp (c : Mᵈᵐᵃ) (f : Lp E p μ) : ‖c • f‖₊ = ‖f‖₊ 
goal: ‖c +ᵥ f‖₊ = ‖f‖₊

full_name: DomMulAct.dist_smul_Lp
code: @[to_additive (attr := simp)]
theorem dist_smul_Lp (c : Mᵈᵐᵃ) (f g : Lp E p μ) : dist (c • f) (c • g) = dist f g 
goal: dist (c • f) (c • g) = dist f g

full_name: DomAddAct.dist_vadd_Lp
code: @[to_additive (attr := simp)]
theorem dist_smul_Lp (c : Mᵈᵐᵃ) (f g : Lp E p μ) : dist (c • f) (c • g) = dist f g 
goal: dist (c +ᵥ f) (c +ᵥ g) = dist f g

full_name: DomAddAct.edist_vadd_Lp
code: @[to_additive (attr := simp)]
theorem edist_smul_Lp (c : Mᵈᵐᵃ) (f g : Lp E p μ) : edist (c • f) (c • g) = edist f g 
goal: edist (c +ᵥ f) (c +ᵥ g) = edist f g

full_name: DomMulAct.edist_smul_Lp
code: @[to_additive (attr := simp)]
theorem edist_smul_Lp (c : Mᵈᵐᵃ) (f g : Lp E p μ) : edist (c • f) (c • g) = edist f g 
goal: edist (c • f) (c • g) = edist f g

full_name: DomAddAct.instIsometricVAddSubtypeAEEqFunMemAddAddSubgroupLp
code: @[to_additive]
instance : IsometricSMul Mᵈᵐᵃ (Lp E p μ) 
goal: IsometricVAdd Mᵈᵃᵃ ↥(MeasureTheory.Lp E p μ)

full_name: DomMulAct.instIsometricSMulSubtypeAEEqFunMemAddSubgroupLp
code: @[to_additive]
instance : IsometricSMul Mᵈᵐᵃ (Lp E p μ) 
goal: IsometricSMul Mᵈᵐᵃ ↥(MeasureTheory.Lp E p μ)

full_name: DomMulAct.instMulActionSubtypeAEEqFunMemAddSubgroupLp
code: @[to_additive]
instance : MulAction Mᵈᵐᵃ (Lp E p μ) 
goal: MulAction Mᵈᵐᵃ ↥(MeasureTheory.Lp E p μ)

full_name: DomAddAct.instAddActionSubtypeAEEqFunMemAddAddSubgroupLp
code: @[to_additive]
instance : MulAction Mᵈᵐᵃ (Lp E p μ) 
goal: AddAction Mᵈᵃᵃ ↥(MeasureTheory.Lp E p μ)

full_name: DomMulAct.instDistribMulActionSubtypeAEEqFunMemAddSubgroupLp
code: instance : DistribMulAction Mᵈᵐᵃ (Lp E p μ) 
goal: DistribMulAction Mᵈᵐᵃ ↥(MeasureTheory.Lp E p μ)