Module: Mathlib.MeasureTheory.Measure.Haar.Unique

Goal state before:
pp: case pos
G : Type u_1
inst✝⁶ : TopologicalSpace G
inst✝⁵ : Group G
inst✝⁴ : IsTopologicalGroup G
inst✝³ : MeasurableSpace G
inst✝² : BorelSpace G
μ' μ : Measure G
inst✝¹ : μ.IsHaarMeasure
inst✝ : μ'.IsHaarMeasure
φ : G ≃ₜ* G
hG : LocallyCompactSpace G
f : G → ℝ
f_cont : Continuous[inst✝⁶, _] f
hf :
  HasCompactSupport ⇑{ toFun := f, continuous_toFun := f_cont } ∧
    0 ≤ ⇑{ toFun := f, continuous_toFun := f_cont } ∧ { toFun := f, continuous_toFun := f_cont } 1 ≠ 0
int_f_ne_zero : ∫ (x : G), f x ∂map (⇑φ) μ ≠ 0
⊢ (∫ (x : G), { toFun := f, continuous_toFun := f_cont } (φ x) ∂μ') /
      ∫ (x : G), { toFun := f, continuous_toFun := f_cont } (φ x) ∂μ =
    (∫ (x : G), (f ∘ ⇑φ.toHomeomorph) x ∂μ') / ∫ (x : G), (f ∘ ⇑φ.toHomeomorph) x ∂μ

pp: case pos.h'f
G : Type u_1
inst✝⁶ : TopologicalSpace G
inst✝⁵ : Group G
inst✝⁴ : IsTopologicalGroup G
inst✝³ : MeasurableSpace G
inst✝² : BorelSpace G
μ' μ : Measure G
inst✝¹ : μ.IsHaarMeasure
inst✝ : μ'.IsHaarMeasure
φ : G ≃ₜ* G
hG : LocallyCompactSpace G
f : G → ℝ
f_cont : Continuous[inst✝⁶, _] f
hf :
  HasCompactSupport ⇑{ toFun := f, continuous_toFun := f_cont } ∧
    0 ≤ ⇑{ toFun := f, continuous_toFun := f_cont } ∧ { toFun := f, continuous_toFun := f_cont } 1 ≠ 0
int_f_ne_zero : ∫ (x : G), f x ∂map (⇑φ) μ ≠ 0
⊢ HasCompactSupport (f ∘ ⇑φ.toHomeomorph)

pp: case pos.int_nonzero
G : Type u_1
inst✝⁶ : TopologicalSpace G
inst✝⁵ : Group G
inst✝⁴ : IsTopologicalGroup G
inst✝³ : MeasurableSpace G
inst✝² : BorelSpace G
μ' μ : Measure G
inst✝¹ : μ.IsHaarMeasure
inst✝ : μ'.IsHaarMeasure
φ : G ≃ₜ* G
hG : LocallyCompactSpace G
f : G → ℝ
f_cont : Continuous[inst✝⁶, _] f
hf :
  HasCompactSupport ⇑{ toFun := f, continuous_toFun := f_cont } ∧
    0 ≤ ⇑{ toFun := f, continuous_toFun := f_cont } ∧ { toFun := f, continuous_toFun := f_cont } 1 ≠ 0
int_f_ne_zero : ∫ (x : G), f x ∂map (⇑φ) μ ≠ 0
⊢ ∫ (x : G), (f ∘ ⇑φ.toHomeomorph) x ∂μ ≠ 0

Tactic:
]

Goal state after:
case pos
G : Type u_1
inst✝⁶ : TopologicalSpace G
inst✝⁵ : Group G
inst✝⁴ : IsTopologicalGroup G
inst✝³ : MeasurableSpace G
inst✝² : BorelSpace G
μ' μ : Measure G
inst✝¹ : μ.IsHaarMeasure
inst✝ : μ'.IsHaarMeasure
φ : G ≃ₜ* G
hG : LocallyCompactSpace G
f : G → ℝ
f_cont : Continuous[inst✝⁶, _] f
hf :
  HasCompactSupport ⇑{ toFun := f, continuous_toFun := f_cont } ∧
    0 ≤ ⇑{ toFun := f, continuous_toFun := f_cont } ∧ { toFun := f, continuous_toFun := f_cont } 1 ≠ 0
int_f_ne_zero : ∫ (x : G), f x ∂map (⇑φ) μ ≠ 0
⊢ (∫ (x : G), { toFun := f, continuous_toFun := f_cont } (φ x) ∂μ') /
      ∫ (x : G), { toFun := f, continuous_toFun := f_cont } (φ x) ∂μ =
    (∫ (x : G), (f ∘ ⇑φ.toHomeomorph) x ∂μ') / ∫ (x : G), (f ∘ ⇑φ.toHomeomorph) x ∂μ

case pos.h'f
G : Type u_1
inst✝⁶ : TopologicalSpace G
inst✝⁵ : Group G
inst✝⁴ : IsTopologicalGroup G
inst✝³ : MeasurableSpace G
inst✝² : BorelSpace G
μ' μ : Measure G
inst✝¹ : μ.IsHaarMeasure
inst✝ : μ'.IsHaarMeasure
φ : G ≃ₜ* G
hG : LocallyCompactSpace G
f : G → ℝ
f_cont : Continuous[inst✝⁶, _] f
hf :
  HasCompactSupport ⇑{ toFun := f, continuous_toFun := f_cont } ∧
    0 ≤ ⇑{ toFun := f, continuous_toFun := f_cont } ∧ { toFun := f, continuous_toFun := f_cont } 1 ≠ 0
int_f_ne_zero : ∫ (x : G), f x ∂map (⇑φ) μ ≠ 0
⊢ HasCompactSupport (f ∘ ⇑φ.toHomeomorph)

case pos.int_nonzero
G : Type u_1
inst✝⁶ : TopologicalSpace G
inst✝⁵ : Group G
inst✝⁴ : IsTopologicalGroup G
inst✝³ : MeasurableSpace G
inst✝² : BorelSpace G
μ' μ : Measure G
inst✝¹ : μ.IsHaarMeasure
inst✝ : μ'.IsHaarMeasure
φ : G ≃ₜ* G
hG : LocallyCompactSpace G
f : G → ℝ
f_cont : Continuous[inst✝⁶, _] f
hf :
  HasCompactSupport ⇑{ toFun := f, continuous_toFun := f_cont } ∧
    0 ≤ ⇑{ toFun := f, continuous_toFun := f_cont } ∧ { toFun := f, continuous_toFun := f_cont } 1 ≠ 0
int_f_ne_zero : ∫ (x : G), f x ∂map (⇑φ) μ ≠ 0
⊢ ∫ (x : G), (f ∘ ⇑φ.toHomeomorph) x ∂μ ≠ 0