Theorem: MeasureTheory.Measure.haar.nonempty_iInter_clAddPrehaar
File: Mathlib/MeasureTheory/Measure/Haar/Basic.lean

Proof steps:
state_before: G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
⊢ (haarProduct ↑K₀ ∩ ⋂ V, clPrehaar (↑K₀) V).Nonempty
state_after: G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
this : IsCompact (haarProduct ↑K₀)
⊢ (haarProduct ↑K₀ ∩ ⋂ V, clPrehaar (↑K₀) V).Nonempty
tactic: have : IsCompact (haarProduct (K₀ : Set G)) := by
    apply isCompact_univ_pi; intro K; apply isCompact_Icc
state_before: G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
this : IsCompact (haarProduct ↑K₀)
⊢ (haarProduct ↑K₀ ∩ ⋂ V, clPrehaar (↑K₀) V).Nonempty
state_after: G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
this : IsCompact (haarProduct ↑K₀)
t : Finset (OpenNhdsOf 1)
⊢ (haarProduct ↑K₀ ∩ ⋂ i ∈ t, clPrehaar (↑K₀) i).Nonempty
tactic: refine this.inter_iInter_nonempty (clPrehaar K₀) (fun s => isClosed_closure) fun t => ?_
state_before: G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
this : IsCompact (haarProduct ↑K₀)
t : Finset (OpenNhdsOf 1)
⊢ (haarProduct ↑K₀ ∩ ⋂ i ∈ t, clPrehaar (↑K₀) i).Nonempty
state_after: G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
this : IsCompact (haarProduct ↑K₀)
t : Finset (OpenNhdsOf 1)
V₀ : Set G := ⋂ V ∈ t, V.carrier
⊢ (haarProduct ↑K₀ ∩ ⋂ i ∈ t, clPrehaar (↑K₀) i).Nonempty
tactic: let V₀ := ⋂ V ∈ t, (V : OpenNhdsOf (1 : G)).carrier
state_before: G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
this : IsCompact (haarProduct ↑K₀)
t : Finset (OpenNhdsOf 1)
V₀ : Set G := ⋂ V ∈ t, V.carrier
⊢ (haarProduct ↑K₀ ∩ ⋂ i ∈ t, clPrehaar (↑K₀) i).Nonempty
state_after: G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
this : IsCompact (haarProduct ↑K₀)
t : Finset (OpenNhdsOf 1)
V₀ : Set G := ⋂ V ∈ t, V.carrier
h1V₀ : IsOpen V₀
⊢ (haarProduct ↑K₀ ∩ ⋂ i ∈ t, clPrehaar (↑K₀) i).Nonempty
tactic: have h1V₀ : IsOpen V₀ := isOpen_biInter_finset <| by rintro ⟨⟨V, hV₁⟩, hV₂⟩ _; exact hV₁
state_before: G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
this : IsCompact (haarProduct ↑K₀)
t : Finset (OpenNhdsOf 1)
V₀ : Set G := ⋂ V ∈ t, V.carrier
h1V₀ : IsOpen V₀
⊢ (haarProduct ↑K₀ ∩ ⋂ i ∈ t, clPrehaar (↑K₀) i).Nonempty
state_after: G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
this : IsCompact (haarProduct ↑K₀)
t : Finset (OpenNhdsOf 1)
V₀ : Set G := ⋂ V ∈ t, V.carrier
h1V₀ : IsOpen V₀
h2V₀ : 1 ∈ V₀
⊢ (haarProduct ↑K₀ ∩ ⋂ i ∈ t, clPrehaar (↑K₀) i).Nonempty
tactic: have h2V₀ : (1 : G) ∈ V₀ := by simp only [V₀, mem_iInter]; rintro ⟨⟨V, hV₁⟩, hV₂⟩ _; exact hV₂
state_before: G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
this : IsCompact (haarProduct ↑K₀)
t : Finset (OpenNhdsOf 1)
V₀ : Set G := ⋂ V ∈ t, V.carrier
h1V₀ : IsOpen V₀
h2V₀ : 1 ∈ V₀
⊢ (haarProduct ↑K₀ ∩ ⋂ i ∈ t, clPrehaar (↑K₀) i).Nonempty
state_after: G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
this : IsCompact (haarProduct ↑K₀)
t : Finset (OpenNhdsOf 1)
V₀ : Set G := ⋂ V ∈ t, V.carrier
h1V₀ : IsOpen V₀
h2V₀ : 1 ∈ V₀
⊢ prehaar (↑K₀) V₀ ∈ haarProduct ↑K₀ ∩ ⋂ i ∈ t, clPrehaar (↑K₀) i
tactic: refine ⟨prehaar K₀ V₀, ?_⟩
state_before: G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
this : IsCompact (haarProduct ↑K₀)
t : Finset (OpenNhdsOf 1)
V₀ : Set G := ⋂ V ∈ t, V.carrier
h1V₀ : IsOpen V₀
h2V₀ : 1 ∈ V₀
⊢ prehaar (↑K₀) V₀ ∈ haarProduct ↑K₀ ∩ ⋂ i ∈ t, clPrehaar (↑K₀) i
state_after: case left
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
this : IsCompact (haarProduct ↑K₀)
t : Finset (OpenNhdsOf 1)
V₀ : Set G := ⋂ V ∈ t, V.carrier
h1V₀ : IsOpen V₀
h2V₀ : 1 ∈ V₀
⊢ prehaar (↑K₀) V₀ ∈ haarProduct ↑K₀

case right
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
this : IsCompact (haarProduct ↑K₀)
t : Finset (OpenNhdsOf 1)
V₀ : Set G := ⋂ V ∈ t, V.carrier
h1V₀ : IsOpen V₀
h2V₀ : 1 ∈ V₀
⊢ prehaar (↑K₀) V₀ ∈ ⋂ i ∈ t, clPrehaar (↑K₀) i
tactic: constructor