Theorem: MeasureTheory.Measure.div_mem_nhds_one_of_haar_pos
File: Mathlib/MeasureTheory/Measure/Haar/Basic.lean

Proof steps:
state_before: case intro.intro.intro.intro
G : Type u_1
inst✝⁷ : Group G
inst✝⁶ : TopologicalSpace G
inst✝⁵ : TopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
μ : Measure G
inst✝² : μ.IsHaarMeasure
inst✝¹ : LocallyCompactSpace G
inst✝ : μ.InnerRegular
E : Set G
hE : MeasurableSet E
hEpos : 0 < μ E
K : Set G
hKE : K ⊆ E
hK : IsCompact K
K_closed : IsClosed K
hKpos : 0 < μ K
⊢ E / E ∈ 𝓝 1
state_after: case h
G : Type u_1
inst✝⁷ : Group G
inst✝⁶ : TopologicalSpace G
inst✝⁵ : TopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
μ : Measure G
inst✝² : μ.IsHaarMeasure
inst✝¹ : LocallyCompactSpace G
inst✝ : μ.InnerRegular
E : Set G
hE : MeasurableSet E
hEpos : 0 < μ E
K : Set G
hKE : K ⊆ E
hK : IsCompact K
K_closed : IsClosed K
hKpos : 0 < μ K
g : G
hg : μ (g • K \ K) < μ K
⊢ g ∈ E / E
tactic: filter_upwards [eventually_nhds_one_measure_smul_diff_lt hK K_closed hKpos.ne' (μ := μ)] with g hg
state_before: case h
G : Type u_1
inst✝⁷ : Group G
inst✝⁶ : TopologicalSpace G
inst✝⁵ : TopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
μ : Measure G
inst✝² : μ.IsHaarMeasure
inst✝¹ : LocallyCompactSpace G
inst✝ : μ.InnerRegular
E : Set G
hE : MeasurableSet E
hEpos : 0 < μ E
K : Set G
hKE : K ⊆ E
hK : IsCompact K
K_closed : IsClosed K
hKpos : 0 < μ K
g : G
hg : μ (g • K \ K) < μ K
⊢ g ∈ E / E
state_after: case h
G : Type u_1
inst✝⁷ : Group G
inst✝⁶ : TopologicalSpace G
inst✝⁵ : TopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
μ : Measure G
inst✝² : μ.IsHaarMeasure
inst✝¹ : LocallyCompactSpace G
inst✝ : μ.InnerRegular
E : Set G
hE : MeasurableSet E
hEpos : 0 < μ E
K : Set G
hKE : K ⊆ E
hK : IsCompact K
K_closed : IsClosed K
hKpos : 0 < μ K
g : G
hg : μ (g • K \ K) < μ K
this : ¬Disjoint (g • K) K
⊢ g ∈ E / E
tactic: have : ¬Disjoint (g • K) K := fun hd ↦ by
    rw [hd.symm.sdiff_eq_right, measure_smul] at hg
    exact hg.false
state_before: case h
G : Type u_1
inst✝⁷ : Group G
inst✝⁶ : TopologicalSpace G
inst✝⁵ : TopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
μ : Measure G
inst✝² : μ.IsHaarMeasure
inst✝¹ : LocallyCompactSpace G
inst✝ : μ.InnerRegular
E : Set G
hE : MeasurableSet E
hEpos : 0 < μ E
K : Set G
hKE : K ⊆ E
hK : IsCompact K
K_closed : IsClosed K
hKpos : 0 < μ K
g : G
hg : μ (g • K \ K) < μ K
this : ¬Disjoint (g • K) K
⊢ g ∈ E / E
state_after: case h.intro.intro.intro.intro
G : Type u_1
inst✝⁷ : Group G
inst✝⁶ : TopologicalSpace G
inst✝⁵ : TopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
μ : Measure G
inst✝² : μ.IsHaarMeasure
inst✝¹ : LocallyCompactSpace G
inst✝ : μ.InnerRegular
E : Set G
hE : MeasurableSet E
hEpos : 0 < μ E
K : Set G
hKE : K ⊆ E
hK : IsCompact K
K_closed : IsClosed K
hKpos : 0 < μ K
g : G
hg : μ (g • K \ K) < μ K
this : ¬Disjoint (g • K) K
x : G
hxK : x ∈ K
hgxK : (fun x => g • x) x ∈ K
⊢ g ∈ E / E
tactic: rcases Set.not_disjoint_iff.1 this with ⟨_, ⟨x, hxK, rfl⟩, hgxK⟩
state_before: case h.intro.intro.intro.intro
G : Type u_1
inst✝⁷ : Group G
inst✝⁶ : TopologicalSpace G
inst✝⁵ : TopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
μ : Measure G
inst✝² : μ.IsHaarMeasure
inst✝¹ : LocallyCompactSpace G
inst✝ : μ.InnerRegular
E : Set G
hE : MeasurableSet E
hEpos : 0 < μ E
K : Set G
hKE : K ⊆ E
hK : IsCompact K
K_closed : IsClosed K
hKpos : 0 < μ K
g : G
hg : μ (g • K \ K) < μ K
this : ¬Disjoint (g • K) K
x : G
hxK : x ∈ K
hgxK : (fun x => g • x) x ∈ K
⊢ g ∈ E / E
state_after: no goals
tactic: simpa using div_mem_div (hKE hgxK) (hKE hxK)