-- Mathlib/MeasureTheory/Measure/Haar/NormedSpace.lean
theorem setIntegral_comp_smul (f : E → F) {R : ℝ} (s : Set E) (hR : R ≠ 0) :
    ∫ x in s, f (R • x) ∂μ = |(R ^ finrank ℝ E)⁻¹| • ∫ x in R • s, f x ∂μ

Proof state:
E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : MeasurableSpace E
inst✝⁴ : BorelSpace E
inst✝³ : FiniteDimensional ℝ E
μ : Measure E
inst✝² : μ.IsAddHaarMeasure
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
R : ℝ
s : Set E
hR : R ≠ 0
e : E ≃ᵐ E := (Homeomorph.smul (Units.mk0 R hR)).toMeasurableEquiv
⊢ ∫ (x : E) in s, f (R • x) ∂μ = |(R ^ finrank ℝ E)⁻¹| • ∫ (x : E) in R • s, f x ∂μ

Tactic:
calc
∫ x in s, f (R • x) ∂μ
  = ∫ x in e ⁻¹' (e.symm ⁻¹' s), f (e x) ∂μ := by simp [← preimage_comp]; rfl
_ = ∫ y in e.symm ⁻¹' s, f y ∂map (fun x ↦ R • x) μ := (setIntegral_map_equiv _ _ _).symm
_ = |(R ^ finrank ℝ E)⁻¹| • ∫ y in e.symm ⁻¹' s, f y ∂μ := by
  simp [map_addHaar_smul μ hR, integral_smul_measure, ENNReal.toReal_ofReal, abs_nonneg]
_ = |(R ^ finrank ℝ E)⁻¹| • ∫ x in R • s, f x ∂μ := by
  congr
  ext y
  rw [mem_smul_set_iff_inv_smul_mem₀ hR]
  rfl

Result:
no goals

Proof state:
E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : MeasurableSpace E
inst✝⁴ : BorelSpace E
inst✝³ : FiniteDimensional ℝ E
μ : Measure E
inst✝² : μ.IsAddHaarMeasure
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
R : ℝ
s : Set E
hR : R ≠ 0
e : E ≃ᵐ E := (Homeomorph.smul (Units.mk0 R hR)).toMeasurableEquiv
⊢ ∫ (y : E) in ⇑e.symm ⁻¹' s, f y ∂map (fun x => R • x) μ = |(R ^ finrank ℝ E)⁻¹| • ∫ (y : E) in ⇑e.symm ⁻¹' s, f y ∂μ

Tactic:
simp [map_addHaar_smul μ hR, integral_smul_measure, ENNReal.toReal_ofReal, abs_nonneg]

Result:
no goals

Proof state:
E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : MeasurableSpace E
inst✝⁴ : BorelSpace E
inst✝³ : FiniteDimensional ℝ E
μ : Measure E
inst✝² : μ.IsAddHaarMeasure
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
R : ℝ
s : Set E
hR : R ≠ 0
e : E ≃ᵐ E := (Homeomorph.smul (Units.mk0 R hR)).toMeasurableEquiv
⊢ ∫ (x : E) in s, f (R • x) ∂μ = ∫ (x : E) in s, f (e x) ∂μ

Tactic:
rfl

Result:
no goals

Proof state:
case e_a.e_μ.e_s.h
E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : MeasurableSpace E
inst✝⁴ : BorelSpace E
inst✝³ : FiniteDimensional ℝ E
μ : Measure E
inst✝² : μ.IsAddHaarMeasure
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
R : ℝ
s : Set E
hR : R ≠ 0
e : E ≃ᵐ E := (Homeomorph.smul (Units.mk0 R hR)).toMeasurableEquiv
y : E
⊢ y ∈ ⇑e.symm ⁻¹' s ↔ R⁻¹ • y ∈ s

Tactic:
rfl

Result:
no goals

Proof state:
E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : MeasurableSpace E
inst✝⁴ : BorelSpace E
inst✝³ : FiniteDimensional ℝ E
μ : Measure E
inst✝² : μ.IsAddHaarMeasure
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
R : ℝ
s : Set E
hR : R ≠ 0
e : E ≃ᵐ E := (Homeomorph.smul (Units.mk0 R hR)).toMeasurableEquiv
⊢ ∫ (x : E) in s, f (R • x) ∂μ = ∫ (x : E) in ⇑e ⁻¹' (⇑e.symm ⁻¹' s), f (e x) ∂μ

Tactic:
simp [← preimage_comp]

Result:
E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : MeasurableSpace E
inst✝⁴ : BorelSpace E
inst✝³ : FiniteDimensional ℝ E
μ : Measure E
inst✝² : μ.IsAddHaarMeasure
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
R : ℝ
s : Set E
hR : R ≠ 0
e : E ≃ᵐ E := (Homeomorph.smul (Units.mk0 R hR)).toMeasurableEquiv
⊢ ∫ (x : E) in s, f (R • x) ∂μ = ∫ (x : E) in s, f (e x) ∂μ

Proof state:
case e_a.e_μ.e_s
E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : MeasurableSpace E
inst✝⁴ : BorelSpace E
inst✝³ : FiniteDimensional ℝ E
μ : Measure E
inst✝² : μ.IsAddHaarMeasure
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
R : ℝ
s : Set E
hR : R ≠ 0
e : E ≃ᵐ E := (Homeomorph.smul (Units.mk0 R hR)).toMeasurableEquiv
⊢ ⇑e.symm ⁻¹' s = R • s

Tactic:
ext y

Result:
case e_a.e_μ.e_s.h
E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : MeasurableSpace E
inst✝⁴ : BorelSpace E
inst✝³ : FiniteDimensional ℝ E
μ : Measure E
inst✝² : μ.IsAddHaarMeasure
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
R : ℝ
s : Set E
hR : R ≠ 0
e : E ≃ᵐ E := (Homeomorph.smul (Units.mk0 R hR)).toMeasurableEquiv
y : E
⊢ y ∈ ⇑e.symm ⁻¹' s ↔ y ∈ R • s

Proof state:
E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : MeasurableSpace E
inst✝⁴ : BorelSpace E
inst✝³ : FiniteDimensional ℝ E
μ : Measure E
inst✝² : μ.IsAddHaarMeasure
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
R : ℝ
s : Set E
hR : R ≠ 0
⊢ ∫ (x : E) in s, f (R • x) ∂μ = |(R ^ finrank ℝ E)⁻¹| • ∫ (x : E) in R • s, f x ∂μ

Tactic:
let e : E ≃ᵐ E := (Homeomorph.smul (Units.mk0 R hR)).toMeasurableEquiv

Result:
E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : MeasurableSpace E
inst✝⁴ : BorelSpace E
inst✝³ : FiniteDimensional ℝ E
μ : Measure E
inst✝² : μ.IsAddHaarMeasure
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
R : ℝ
s : Set E
hR : R ≠ 0
e : E ≃ᵐ E := (Homeomorph.smul (Units.mk0 R hR)).toMeasurableEquiv
⊢ ∫ (x : E) in s, f (R • x) ∂μ = |(R ^ finrank ℝ E)⁻¹| • ∫ (x : E) in R • s, f x ∂μ

Proof state:
E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : MeasurableSpace E
inst✝⁴ : BorelSpace E
inst✝³ : FiniteDimensional ℝ E
μ : Measure E
inst✝² : μ.IsAddHaarMeasure
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
R : ℝ
s : Set E
hR : R ≠ 0
e : E ≃ᵐ E := (Homeomorph.smul (Units.mk0 R hR)).toMeasurableEquiv
⊢ |(R ^ finrank ℝ E)⁻¹| • ∫ (y : E) in ⇑e.symm ⁻¹' s, f y ∂μ = |(R ^ finrank ℝ E)⁻¹| • ∫ (x : E) in R • s, f x ∂μ

Tactic:
congr

Result:
case e_a.e_μ.e_s
E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : MeasurableSpace E
inst✝⁴ : BorelSpace E
inst✝³ : FiniteDimensional ℝ E
μ : Measure E
inst✝² : μ.IsAddHaarMeasure
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
R : ℝ
s : Set E
hR : R ≠ 0
e : E ≃ᵐ E := (Homeomorph.smul (Units.mk0 R hR)).toMeasurableEquiv
⊢ ⇑e.symm ⁻¹' s = R • s

Proof state:
case e_a.e_μ.e_s.h
E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : MeasurableSpace E
inst✝⁴ : BorelSpace E
inst✝³ : FiniteDimensional ℝ E
μ : Measure E
inst✝² : μ.IsAddHaarMeasure
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
R : ℝ
s : Set E
hR : R ≠ 0
e : E ≃ᵐ E := (Homeomorph.smul (Units.mk0 R hR)).toMeasurableEquiv
y : E
⊢ y ∈ ⇑e.symm ⁻¹' s ↔ y ∈ R • s

Tactic:
rw [mem_smul_set_iff_inv_smul_mem₀ hR]

Result:
case e_a.e_μ.e_s.h
E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : MeasurableSpace E
inst✝⁴ : BorelSpace E
inst✝³ : FiniteDimensional ℝ E
μ : Measure E
inst✝² : μ.IsAddHaarMeasure
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
R : ℝ
s : Set E
hR : R ≠ 0
e : E ≃ᵐ E := (Homeomorph.smul (Units.mk0 R hR)).toMeasurableEquiv
y : E
⊢ y ∈ ⇑e.symm ⁻¹' s ↔ R⁻¹ • y ∈ s