[GOAL]
G : Type u_1
inst✝ : Group G
V : Set G
⊢ index ∅ V = 0
[PROOFSTEP]
simp only [index, Nat.sInf_eq_zero]
[GOAL]
G : Type u_1
inst✝ : Group G
V : Set G
⊢ 0 ∈ Finset.card '' {t | ∅ ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h => g * h) ⁻¹' V} ∨
    Finset.card '' {t | ∅ ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h => g * h) ⁻¹' V} = ∅
[PROOFSTEP]
left
[GOAL]
case h
G : Type u_1
inst✝ : Group G
V : Set G
⊢ 0 ∈ Finset.card '' {t | ∅ ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h => g * h) ⁻¹' V}
[PROOFSTEP]
use∅
[GOAL]
case h
G : Type u_1
inst✝ : Group G
V : Set G
⊢ ∅ ∈ {t | ∅ ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h => g * h) ⁻¹' V} ∧ Finset.card ∅ = 0
[PROOFSTEP]
simp only [Finset.card_empty, empty_subset, mem_setOf_eq, eq_self_iff_true, and_self_iff]
[GOAL]
G : Type u_1
inst✝¹ : Group G
inst✝ : TopologicalSpace G
K₀ : PositiveCompacts G
U : Set G
⊢ prehaar (↑K₀) U ⊥ = 0
[PROOFSTEP]
rw [prehaar, Compacts.coe_bot, index_empty, Nat.cast_zero, zero_div]
[GOAL]
G : Type u_1
inst✝¹ : Group G
inst✝ : TopologicalSpace G
K₀ : PositiveCompacts G
U : Set G
K : Compacts G
⊢ 0 ≤ prehaar (↑K₀) U K
[PROOFSTEP]
apply div_nonneg
[GOAL]
case ha
G : Type u_1
inst✝¹ : Group G
inst✝ : TopologicalSpace G
K₀ : PositiveCompacts G
U : Set G
K : Compacts G
⊢ 0 ≤ ↑(index (↑K) U)
[PROOFSTEP]
norm_cast
[GOAL]
case hb
G : Type u_1
inst✝¹ : Group G
inst✝ : TopologicalSpace G
K₀ : PositiveCompacts G
U : Set G
K : Compacts G
⊢ 0 ≤ ↑(index (↑K₀) U)
[PROOFSTEP]
norm_cast
[GOAL]
case ha
G : Type u_1
inst✝¹ : Group G
inst✝ : TopologicalSpace G
K₀ : PositiveCompacts G
U : Set G
K : Compacts G
⊢ 0 ≤ index (↑K) U
[PROOFSTEP]
apply zero_le
[GOAL]
case hb
G : Type u_1
inst✝¹ : Group G
inst✝ : TopologicalSpace G
K₀ : PositiveCompacts G
U : Set G
K : Compacts G
⊢ 0 ≤ index (↑K₀) U
[PROOFSTEP]
apply zero_le
[GOAL]
G : Type u_1
inst✝¹ : Group G
inst✝ : TopologicalSpace G
K₀ : Set G
f : Compacts G → ℝ
⊢ f ∈ haarProduct K₀ ↔ ∀ (K : Compacts G), f K ∈ Icc 0 ↑(index (↑K) K₀)
[PROOFSTEP]
simp only [haarProduct, Set.pi, forall_prop_of_true, mem_univ, mem_setOf_eq]
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K V : Set G
hK : IsCompact K
hV : Set.Nonempty (interior V)
⊢ ∃ n, n ∈ Finset.card '' {t | K ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h => g * h) ⁻¹' V}
[PROOFSTEP]
rcases compact_covered_by_mul_left_translates hK hV with ⟨t, ht⟩
[GOAL]
case intro
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K V : Set G
hK : IsCompact K
hV : Set.Nonempty (interior V)
t : Finset G
ht : K ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h => g * h) ⁻¹' V
⊢ ∃ n, n ∈ Finset.card '' {t | K ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h => g * h) ⁻¹' V}
[PROOFSTEP]
exact ⟨t.card, t, ht, rfl⟩
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K V : Set G
hK : IsCompact K
hV : Set.Nonempty (interior V)
⊢ ∃ t, K ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h => g * h) ⁻¹' V ∧ Finset.card t = index K V
[PROOFSTEP]
have := Nat.sInf_mem (index_defined hK hV)
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K V : Set G
hK : IsCompact K
hV : Set.Nonempty (interior V)
this :
  sInf (Finset.card '' {t | K ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h => g * h) ⁻¹' V}) ∈
    Finset.card '' {t | K ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h => g * h) ⁻¹' V}
⊢ ∃ t, K ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h => g * h) ⁻¹' V ∧ Finset.card t = index K V
[PROOFSTEP]
rwa [mem_image] at this
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
K : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
⊢ index (↑K) V ≤ index ↑K ↑K₀ * index (↑K₀) V
[PROOFSTEP]
obtain ⟨s, h1s, h2s⟩ := index_elim K.isCompact K₀.interior_nonempty
[GOAL]
case intro.intro
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
K : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
s : Finset G
h1s : ↑K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' ↑K₀
h2s : Finset.card s = index ↑K ↑K₀
⊢ index (↑K) V ≤ index ↑K ↑K₀ * index (↑K₀) V
[PROOFSTEP]
obtain ⟨t, h1t, h2t⟩ := index_elim K₀.isCompact hV
[GOAL]
case intro.intro.intro.intro
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
K : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
s : Finset G
h1s : ↑K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' ↑K₀
h2s : Finset.card s = index ↑K ↑K₀
t : Finset G
h1t : ↑K₀ ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h => g * h) ⁻¹' V
h2t : Finset.card t = index (↑K₀) V
⊢ index (↑K) V ≤ index ↑K ↑K₀ * index (↑K₀) V
[PROOFSTEP]
rw [← h2s, ← h2t, mul_comm]
[GOAL]
case intro.intro.intro.intro
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
K : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
s : Finset G
h1s : ↑K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' ↑K₀
h2s : Finset.card s = index ↑K ↑K₀
t : Finset G
h1t : ↑K₀ ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h => g * h) ⁻¹' V
h2t : Finset.card t = index (↑K₀) V
⊢ index (↑K) V ≤ Finset.card t * Finset.card s
[PROOFSTEP]
refine' le_trans _ Finset.card_mul_le
[GOAL]
case intro.intro.intro.intro
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
K : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
s : Finset G
h1s : ↑K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' ↑K₀
h2s : Finset.card s = index ↑K ↑K₀
t : Finset G
h1t : ↑K₀ ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h => g * h) ⁻¹' V
h2t : Finset.card t = index (↑K₀) V
⊢ index (↑K) V ≤ Finset.card (t * s)
[PROOFSTEP]
apply Nat.sInf_le
[GOAL]
case intro.intro.intro.intro.hm
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
K : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
s : Finset G
h1s : ↑K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' ↑K₀
h2s : Finset.card s = index ↑K ↑K₀
t : Finset G
h1t : ↑K₀ ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h => g * h) ⁻¹' V
h2t : Finset.card t = index (↑K₀) V
⊢ Finset.card (t * s) ∈ Finset.card '' {t | ↑K ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h => g * h) ⁻¹' V}
[PROOFSTEP]
refine' ⟨_, _, rfl⟩
[GOAL]
case intro.intro.intro.intro.hm
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
K : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
s : Finset G
h1s : ↑K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' ↑K₀
h2s : Finset.card s = index ↑K ↑K₀
t : Finset G
h1t : ↑K₀ ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h => g * h) ⁻¹' V
h2t : Finset.card t = index (↑K₀) V
⊢ t * s ∈ {t | ↑K ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h => g * h) ⁻¹' V}
[PROOFSTEP]
rw [mem_setOf_eq]
[GOAL]
case intro.intro.intro.intro.hm
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
K : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
s : Finset G
h1s : ↑K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' ↑K₀
h2s : Finset.card s = index ↑K ↑K₀
t : Finset G
h1t : ↑K₀ ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h => g * h) ⁻¹' V
h2t : Finset.card t = index (↑K₀) V
⊢ ↑K ⊆ ⋃ (g : G) (_ : g ∈ t * s), (fun h => g * h) ⁻¹' V
[PROOFSTEP]
refine' Subset.trans h1s _
[GOAL]
case intro.intro.intro.intro.hm
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
K : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
s : Finset G
h1s : ↑K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' ↑K₀
h2s : Finset.card s = index ↑K ↑K₀
t : Finset G
h1t : ↑K₀ ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h => g * h) ⁻¹' V
h2t : Finset.card t = index (↑K₀) V
⊢ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' ↑K₀ ⊆ ⋃ (g : G) (_ : g ∈ t * s), (fun h => g * h) ⁻¹' V
[PROOFSTEP]
apply iUnion₂_subset
[GOAL]
case intro.intro.intro.intro.hm.h
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
K : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
s : Finset G
h1s : ↑K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' ↑K₀
h2s : Finset.card s = index ↑K ↑K₀
t : Finset G
h1t : ↑K₀ ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h => g * h) ⁻¹' V
h2t : Finset.card t = index (↑K₀) V
⊢ ∀ (i : G), i ∈ s → (fun h => i * h) ⁻¹' ↑K₀ ⊆ ⋃ (g : G) (_ : g ∈ t * s), (fun h => g * h) ⁻¹' V
[PROOFSTEP]
intro g₁ hg₁
[GOAL]
case intro.intro.intro.intro.hm.h
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
K : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
s : Finset G
h1s : ↑K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' ↑K₀
h2s : Finset.card s = index ↑K ↑K₀
t : Finset G
h1t : ↑K₀ ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h => g * h) ⁻¹' V
h2t : Finset.card t = index (↑K₀) V
g₁ : G
hg₁ : g₁ ∈ s
⊢ (fun h => g₁ * h) ⁻¹' ↑K₀ ⊆ ⋃ (g : G) (_ : g ∈ t * s), (fun h => g * h) ⁻¹' V
[PROOFSTEP]
rw [preimage_subset_iff]
[GOAL]
case intro.intro.intro.intro.hm.h
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
K : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
s : Finset G
h1s : ↑K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' ↑K₀
h2s : Finset.card s = index ↑K ↑K₀
t : Finset G
h1t : ↑K₀ ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h => g * h) ⁻¹' V
h2t : Finset.card t = index (↑K₀) V
g₁ : G
hg₁ : g₁ ∈ s
⊢ ∀ (a : G), g₁ * a ∈ ↑K₀ → a ∈ ⋃ (g : G) (_ : g ∈ t * s), (fun h => g * h) ⁻¹' V
[PROOFSTEP]
intro g₂ hg₂
[GOAL]
case intro.intro.intro.intro.hm.h
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
K : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
s : Finset G
h1s : ↑K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' ↑K₀
h2s : Finset.card s = index ↑K ↑K₀
t : Finset G
h1t : ↑K₀ ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h => g * h) ⁻¹' V
h2t : Finset.card t = index (↑K₀) V
g₁ : G
hg₁ : g₁ ∈ s
g₂ : G
hg₂ : g₁ * g₂ ∈ ↑K₀
⊢ g₂ ∈ ⋃ (g : G) (_ : g ∈ t * s), (fun h => g * h) ⁻¹' V
[PROOFSTEP]
have := h1t hg₂
[GOAL]
case intro.intro.intro.intro.hm.h
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
K : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
s : Finset G
h1s : ↑K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' ↑K₀
h2s : Finset.card s = index ↑K ↑K₀
t : Finset G
h1t : ↑K₀ ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h => g * h) ⁻¹' V
h2t : Finset.card t = index (↑K₀) V
g₁ : G
hg₁ : g₁ ∈ s
g₂ : G
hg₂ : g₁ * g₂ ∈ ↑K₀
this : g₁ * g₂ ∈ ⋃ (g : G) (_ : g ∈ t), (fun h => g * h) ⁻¹' V
⊢ g₂ ∈ ⋃ (g : G) (_ : g ∈ t * s), (fun h => g * h) ⁻¹' V
[PROOFSTEP]
rcases this with ⟨_, ⟨g₃, rfl⟩, A, ⟨hg₃, rfl⟩, h2V⟩
[GOAL]
case intro.intro.intro.intro.hm.h.intro.intro.intro.intro.intro.intro
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
K : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
s : Finset G
h1s : ↑K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' ↑K₀
h2s : Finset.card s = index ↑K ↑K₀
t : Finset G
h1t : ↑K₀ ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h => g * h) ⁻¹' V
h2t : Finset.card t = index (↑K₀) V
g₁ : G
hg₁ : g₁ ∈ s
g₂ : G
hg₂ : g₁ * g₂ ∈ ↑K₀
g₃ : G
hg₃ : g₃ ∈ t
h2V : g₁ * g₂ ∈ (fun h => (fun h => g₃ * h) ⁻¹' V) hg₃
⊢ g₂ ∈ ⋃ (g : G) (_ : g ∈ t * s), (fun h => g * h) ⁻¹' V
[PROOFSTEP]
rw [mem_preimage, ← mul_assoc] at h2V
[GOAL]
case intro.intro.intro.intro.hm.h.intro.intro.intro.intro.intro.intro
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
K : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
s : Finset G
h1s : ↑K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' ↑K₀
h2s : Finset.card s = index ↑K ↑K₀
t : Finset G
h1t : ↑K₀ ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h => g * h) ⁻¹' V
h2t : Finset.card t = index (↑K₀) V
g₁ : G
hg₁ : g₁ ∈ s
g₂ : G
hg₂ : g₁ * g₂ ∈ ↑K₀
g₃ : G
hg₃ : g₃ ∈ t
h2V : g₃ * g₁ * g₂ ∈ V
⊢ g₂ ∈ ⋃ (g : G) (_ : g ∈ t * s), (fun h => g * h) ⁻¹' V
[PROOFSTEP]
exact mem_biUnion (Finset.mul_mem_mul hg₃ hg₁) h2V
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K : PositiveCompacts G
V : Set G
hV : Set.Nonempty (interior V)
⊢ 0 < index (↑K) V
[PROOFSTEP]
unfold index
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K : PositiveCompacts G
V : Set G
hV : Set.Nonempty (interior V)
⊢ 0 < sInf (Finset.card '' {t | ↑K ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h => g * h) ⁻¹' V})
[PROOFSTEP]
rw [Nat.sInf_def, Nat.find_pos, mem_image]
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K : PositiveCompacts G
V : Set G
hV : Set.Nonempty (interior V)
⊢ ¬∃ x, x ∈ {t | ↑K ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h => g * h) ⁻¹' V} ∧ Finset.card x = 0
[PROOFSTEP]
rintro ⟨t, h1t, h2t⟩
[GOAL]
case intro.intro
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K : PositiveCompacts G
V : Set G
hV : Set.Nonempty (interior V)
t : Finset G
h1t : t ∈ {t | ↑K ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h => g * h) ⁻¹' V}
h2t : Finset.card t = 0
⊢ False
[PROOFSTEP]
rw [Finset.card_eq_zero] at h2t
[GOAL]
case intro.intro
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K : PositiveCompacts G
V : Set G
hV : Set.Nonempty (interior V)
t : Finset G
h1t : t ∈ {t | ↑K ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h => g * h) ⁻¹' V}
h2t : t = ∅
⊢ False
[PROOFSTEP]
subst h2t
[GOAL]
case intro.intro
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K : PositiveCompacts G
V : Set G
hV : Set.Nonempty (interior V)
h1t : ∅ ∈ {t | ↑K ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h => g * h) ⁻¹' V}
⊢ False
[PROOFSTEP]
obtain ⟨g, hg⟩ := K.interior_nonempty
[GOAL]
case intro.intro.intro
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K : PositiveCompacts G
V : Set G
hV : Set.Nonempty (interior V)
h1t : ∅ ∈ {t | ↑K ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h => g * h) ⁻¹' V}
g : G
hg : g ∈ interior ↑K
⊢ False
[PROOFSTEP]
show g ∈ (∅ : Set G)
[GOAL]
case intro.intro.intro
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K : PositiveCompacts G
V : Set G
hV : Set.Nonempty (interior V)
h1t : ∅ ∈ {t | ↑K ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h => g * h) ⁻¹' V}
g : G
hg : g ∈ interior ↑K
⊢ g ∈ ∅
[PROOFSTEP]
convert h1t (interior_subset hg)
[GOAL]
case h.e'_5
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K : PositiveCompacts G
V : Set G
hV : Set.Nonempty (interior V)
h1t : ∅ ∈ {t | ↑K ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h => g * h) ⁻¹' V}
g : G
hg : g ∈ interior ↑K
⊢ ∅ = ⋃ (g : G) (_ : g ∈ ∅), (fun h => g * h) ⁻¹' V
[PROOFSTEP]
symm
[GOAL]
case h.e'_5
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K : PositiveCompacts G
V : Set G
hV : Set.Nonempty (interior V)
h1t : ∅ ∈ {t | ↑K ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h => g * h) ⁻¹' V}
g : G
hg : g ∈ interior ↑K
⊢ ⋃ (g : G) (_ : g ∈ ∅), (fun h => g * h) ⁻¹' V = ∅
[PROOFSTEP]
simp only [Finset.not_mem_empty, iUnion_of_empty, iUnion_empty]
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K : PositiveCompacts G
V : Set G
hV : Set.Nonempty (interior V)
⊢ ∃ n, n ∈ Finset.card '' {t | ↑K ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h => g * h) ⁻¹' V}
[PROOFSTEP]
exact index_defined K.isCompact hV
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K K' V : Set G
hK' : IsCompact K'
h : K ⊆ K'
hV : Set.Nonempty (interior V)
⊢ index K V ≤ index K' V
[PROOFSTEP]
rcases index_elim hK' hV with ⟨s, h1s, h2s⟩
[GOAL]
case intro.intro
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K K' V : Set G
hK' : IsCompact K'
h : K ⊆ K'
hV : Set.Nonempty (interior V)
s : Finset G
h1s : K' ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
h2s : Finset.card s = index K' V
⊢ index K V ≤ index K' V
[PROOFSTEP]
apply Nat.sInf_le
[GOAL]
case intro.intro.hm
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K K' V : Set G
hK' : IsCompact K'
h : K ⊆ K'
hV : Set.Nonempty (interior V)
s : Finset G
h1s : K' ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
h2s : Finset.card s = index K' V
⊢ index K' V ∈ Finset.card '' {t | K ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h => g * h) ⁻¹' V}
[PROOFSTEP]
rw [mem_image]
[GOAL]
case intro.intro.hm
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K K' V : Set G
hK' : IsCompact K'
h : K ⊆ K'
hV : Set.Nonempty (interior V)
s : Finset G
h1s : K' ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
h2s : Finset.card s = index K' V
⊢ ∃ x, x ∈ {t | K ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h => g * h) ⁻¹' V} ∧ Finset.card x = index K' V
[PROOFSTEP]
refine' ⟨s, Subset.trans h h1s, h2s⟩
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
⊢ index (K₁.carrier ∪ K₂.carrier) V ≤ index K₁.carrier V + index K₂.carrier V
[PROOFSTEP]
rcases index_elim K₁.2 hV with ⟨s, h1s, h2s⟩
[GOAL]
case intro.intro
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
s : Finset G
h1s : K₁.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
h2s : Finset.card s = index K₁.carrier V
⊢ index (K₁.carrier ∪ K₂.carrier) V ≤ index K₁.carrier V + index K₂.carrier V
[PROOFSTEP]
rcases index_elim K₂.2 hV with ⟨t, h1t, h2t⟩
[GOAL]
case intro.intro.intro.intro
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
s : Finset G
h1s : K₁.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
h2s : Finset.card s = index K₁.carrier V
t : Finset G
h1t : K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h => g * h) ⁻¹' V
h2t : Finset.card t = index K₂.carrier V
⊢ index (K₁.carrier ∪ K₂.carrier) V ≤ index K₁.carrier V + index K₂.carrier V
[PROOFSTEP]
rw [← h2s, ← h2t]
[GOAL]
case intro.intro.intro.intro
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
s : Finset G
h1s : K₁.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
h2s : Finset.card s = index K₁.carrier V
t : Finset G
h1t : K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h => g * h) ⁻¹' V
h2t : Finset.card t = index K₂.carrier V
⊢ index (K₁.carrier ∪ K₂.carrier) V ≤ Finset.card s + Finset.card t
[PROOFSTEP]
refine' le_trans _ (Finset.card_union_le _ _)
[GOAL]
case intro.intro.intro.intro
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
s : Finset G
h1s : K₁.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
h2s : Finset.card s = index K₁.carrier V
t : Finset G
h1t : K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h => g * h) ⁻¹' V
h2t : Finset.card t = index K₂.carrier V
⊢ index (K₁.carrier ∪ K₂.carrier) V ≤ Finset.card (s ∪ t)
[PROOFSTEP]
apply Nat.sInf_le
[GOAL]
case intro.intro.intro.intro.hm
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
s : Finset G
h1s : K₁.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
h2s : Finset.card s = index K₁.carrier V
t : Finset G
h1t : K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h => g * h) ⁻¹' V
h2t : Finset.card t = index K₂.carrier V
⊢ Finset.card (s ∪ t) ∈ Finset.card '' {t | K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h => g * h) ⁻¹' V}
[PROOFSTEP]
refine' ⟨_, _, rfl⟩
[GOAL]
case intro.intro.intro.intro.hm
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
s : Finset G
h1s : K₁.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
h2s : Finset.card s = index K₁.carrier V
t : Finset G
h1t : K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h => g * h) ⁻¹' V
h2t : Finset.card t = index K₂.carrier V
⊢ s ∪ t ∈ {t | K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h => g * h) ⁻¹' V}
[PROOFSTEP]
rw [mem_setOf_eq]
[GOAL]
case intro.intro.intro.intro.hm
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
s : Finset G
h1s : K₁.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
h2s : Finset.card s = index K₁.carrier V
t : Finset G
h1t : K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h => g * h) ⁻¹' V
h2t : Finset.card t = index K₂.carrier V
⊢ K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s ∪ t), (fun h => g * h) ⁻¹' V
[PROOFSTEP]
apply union_subset
[GOAL]
case intro.intro.intro.intro.hm.sr
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
s : Finset G
h1s : K₁.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
h2s : Finset.card s = index K₁.carrier V
t : Finset G
h1t : K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h => g * h) ⁻¹' V
h2t : Finset.card t = index K₂.carrier V
⊢ K₁.carrier ⊆ ⋃ (g : G) (_ : g ∈ s ∪ t), (fun h => g * h) ⁻¹' V
[PROOFSTEP]
refine' Subset.trans (by assumption) _
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
s : Finset G
h1s : K₁.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
h2s : Finset.card s = index K₁.carrier V
t : Finset G
h1t : K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h => g * h) ⁻¹' V
h2t : Finset.card t = index K₂.carrier V
⊢ K₁.carrier ⊆ ?m.46641
[PROOFSTEP]
assumption
[GOAL]
case intro.intro.intro.intro.hm.tr
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
s : Finset G
h1s : K₁.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
h2s : Finset.card s = index K₁.carrier V
t : Finset G
h1t : K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h => g * h) ⁻¹' V
h2t : Finset.card t = index K₂.carrier V
⊢ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s ∪ t), (fun h => g * h) ⁻¹' V
[PROOFSTEP]
refine' Subset.trans (by assumption) _
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
s : Finset G
h1s : K₁.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
h2s : Finset.card s = index K₁.carrier V
t : Finset G
h1t : K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h => g * h) ⁻¹' V
h2t : Finset.card t = index K₂.carrier V
⊢ K₂.carrier ⊆ ?m.46676
[PROOFSTEP]
assumption
[GOAL]
case intro.intro.intro.intro.hm.sr
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
s : Finset G
h1s : K₁.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
h2s : Finset.card s = index K₁.carrier V
t : Finset G
h1t : K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h => g * h) ⁻¹' V
h2t : Finset.card t = index K₂.carrier V
⊢ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V ⊆ ⋃ (g : G) (_ : g ∈ s ∪ t), (fun h => g * h) ⁻¹' V
[PROOFSTEP]
apply biUnion_subset_biUnion_left
[GOAL]
case intro.intro.intro.intro.hm.tr
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
s : Finset G
h1s : K₁.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
h2s : Finset.card s = index K₁.carrier V
t : Finset G
h1t : K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h => g * h) ⁻¹' V
h2t : Finset.card t = index K₂.carrier V
⊢ ⋃ (g : G) (_ : g ∈ t), (fun h => g * h) ⁻¹' V ⊆ ⋃ (g : G) (_ : g ∈ s ∪ t), (fun h => g * h) ⁻¹' V
[PROOFSTEP]
apply biUnion_subset_biUnion_left
[GOAL]
case intro.intro.intro.intro.hm.sr.h
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
s : Finset G
h1s : K₁.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
h2s : Finset.card s = index K₁.carrier V
t : Finset G
h1t : K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h => g * h) ⁻¹' V
h2t : Finset.card t = index K₂.carrier V
⊢ (fun x => x ∈ s.val) ⊆ fun x => x ∈ (s ∪ t).val
[PROOFSTEP]
intro g hg
[GOAL]
case intro.intro.intro.intro.hm.tr.h
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
s : Finset G
h1s : K₁.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
h2s : Finset.card s = index K₁.carrier V
t : Finset G
h1t : K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h => g * h) ⁻¹' V
h2t : Finset.card t = index K₂.carrier V
⊢ (fun x => x ∈ t.val) ⊆ fun x => x ∈ (s ∪ t).val
[PROOFSTEP]
intro g hg
[GOAL]
case intro.intro.intro.intro.hm.sr.h
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
s : Finset G
h1s : K₁.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
h2s : Finset.card s = index K₁.carrier V
t : Finset G
h1t : K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h => g * h) ⁻¹' V
h2t : Finset.card t = index K₂.carrier V
g : G
hg : g ∈ fun x => x ∈ s.val
⊢ g ∈ fun x => x ∈ (s ∪ t).val
[PROOFSTEP]
simp only [mem_def] at hg
[GOAL]
case intro.intro.intro.intro.hm.tr.h
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
s : Finset G
h1s : K₁.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
h2s : Finset.card s = index K₁.carrier V
t : Finset G
h1t : K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h => g * h) ⁻¹' V
h2t : Finset.card t = index K₂.carrier V
g : G
hg : g ∈ fun x => x ∈ t.val
⊢ g ∈ fun x => x ∈ (s ∪ t).val
[PROOFSTEP]
simp only [mem_def] at hg
[GOAL]
case intro.intro.intro.intro.hm.sr.h
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
s : Finset G
h1s : K₁.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
h2s : Finset.card s = index K₁.carrier V
t : Finset G
h1t : K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h => g * h) ⁻¹' V
h2t : Finset.card t = index K₂.carrier V
g : G
hg : g ∈ s.val
⊢ g ∈ fun x => x ∈ (s ∪ t).val
[PROOFSTEP]
simp only [mem_def, Multiset.mem_union, Finset.union_val, hg, or_true_iff, true_or_iff]
[GOAL]
case intro.intro.intro.intro.hm.tr.h
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
s : Finset G
h1s : K₁.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
h2s : Finset.card s = index K₁.carrier V
t : Finset G
h1t : K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h => g * h) ⁻¹' V
h2t : Finset.card t = index K₂.carrier V
g : G
hg : g ∈ t.val
⊢ g ∈ fun x => x ∈ (s ∪ t).val
[PROOFSTEP]
simp only [mem_def, Multiset.mem_union, Finset.union_val, hg, or_true_iff, true_or_iff]
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
h : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)
⊢ index (K₁.carrier ∪ K₂.carrier) V = index K₁.carrier V + index K₂.carrier V
[PROOFSTEP]
apply le_antisymm (index_union_le K₁ K₂ hV)
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
h : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)
⊢ index K₁.carrier V + index K₂.carrier V ≤ index (K₁.carrier ∪ K₂.carrier) V
[PROOFSTEP]
rcases index_elim (K₁.2.union K₂.2) hV with ⟨s, h1s, h2s⟩
[GOAL]
case intro.intro
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
h : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)
s : Finset G
h1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
h2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V
⊢ index K₁.carrier V + index K₂.carrier V ≤ index (K₁.carrier ∪ K₂.carrier) V
[PROOFSTEP]
rw [← h2s]
[GOAL]
case intro.intro
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
h : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)
s : Finset G
h1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
h2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V
⊢ index K₁.carrier V + index K₂.carrier V ≤ Finset.card s
[PROOFSTEP]
have :
  ∀ K : Set G,
    (K ⊆ ⋃ g ∈ s, (fun h => g * h) ⁻¹' V) →
      index K V ≤ (s.filter fun g => ((fun h : G => g * h) ⁻¹' V ∩ K).Nonempty).card :=
  by
  intro K hK; apply Nat.sInf_le; refine' ⟨_, _, rfl⟩; rw [mem_setOf_eq]
  intro g hg; rcases hK hg with ⟨_, ⟨g₀, rfl⟩, _, ⟨h1g₀, rfl⟩, h2g₀⟩
  simp only [mem_preimage] at h2g₀
  simp only [mem_iUnion]; use g₀; constructor; swap
  · simp only [Finset.mem_filter, h1g₀, true_and_iff]; use g
    simp only [hg, h2g₀, mem_inter_iff, mem_preimage, and_self_iff]
  exact h2g₀
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
h : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)
s : Finset G
h1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
h2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V
⊢ ∀ (K : Set G),
    K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V →
      index K V ≤ Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s)
[PROOFSTEP]
intro K hK
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
h : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)
s : Finset G
h1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
h2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V
K : Set G
hK : K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
⊢ index K V ≤ Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s)
[PROOFSTEP]
apply Nat.sInf_le
[GOAL]
case hm
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
h : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)
s : Finset G
h1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
h2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V
K : Set G
hK : K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
⊢ Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s) ∈
    Finset.card '' {t | K ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h => g * h) ⁻¹' V}
[PROOFSTEP]
refine' ⟨_, _, rfl⟩
[GOAL]
case hm
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
h : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)
s : Finset G
h1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
h2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V
K : Set G
hK : K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
⊢ Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s ∈
    {t | K ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h => g * h) ⁻¹' V}
[PROOFSTEP]
rw [mem_setOf_eq]
[GOAL]
case hm
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
h : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)
s : Finset G
h1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
h2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V
K : Set G
hK : K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
⊢ K ⊆ ⋃ (g : G) (_ : g ∈ Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s), (fun h => g * h) ⁻¹' V
[PROOFSTEP]
intro g hg
[GOAL]
case hm
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
h : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)
s : Finset G
h1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
h2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V
K : Set G
hK : K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
g : G
hg : g ∈ K
⊢ g ∈ ⋃ (g : G) (_ : g ∈ Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s), (fun h => g * h) ⁻¹' V
[PROOFSTEP]
rcases hK hg with ⟨_, ⟨g₀, rfl⟩, _, ⟨h1g₀, rfl⟩, h2g₀⟩
[GOAL]
case hm.intro.intro.intro.intro.intro.intro
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
h : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)
s : Finset G
h1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
h2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V
K : Set G
hK : K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
g : G
hg : g ∈ K
g₀ : G
h1g₀ : g₀ ∈ s
h2g₀ : g ∈ (fun h => (fun h => g₀ * h) ⁻¹' V) h1g₀
⊢ g ∈ ⋃ (g : G) (_ : g ∈ Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s), (fun h => g * h) ⁻¹' V
[PROOFSTEP]
simp only [mem_preimage] at h2g₀
[GOAL]
case hm.intro.intro.intro.intro.intro.intro
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
h : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)
s : Finset G
h1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
h2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V
K : Set G
hK : K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
g : G
hg : g ∈ K
g₀ : G
h1g₀ : g₀ ∈ s
h2g₀ : g₀ * g ∈ V
⊢ g ∈ ⋃ (g : G) (_ : g ∈ Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s), (fun h => g * h) ⁻¹' V
[PROOFSTEP]
simp only [mem_iUnion]
[GOAL]
case hm.intro.intro.intro.intro.intro.intro
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
h : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)
s : Finset G
h1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
h2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V
K : Set G
hK : K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
g : G
hg : g ∈ K
g₀ : G
h1g₀ : g₀ ∈ s
h2g₀ : g₀ * g ∈ V
⊢ ∃ i i_1, g ∈ (fun h => i * h) ⁻¹' V
[PROOFSTEP]
use g₀
[GOAL]
case h
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
h : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)
s : Finset G
h1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
h2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V
K : Set G
hK : K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
g : G
hg : g ∈ K
g₀ : G
h1g₀ : g₀ ∈ s
h2g₀ : g₀ * g ∈ V
⊢ ∃ i, g ∈ (fun h => g₀ * h) ⁻¹' V
[PROOFSTEP]
constructor
[GOAL]
case h.h
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
h : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)
s : Finset G
h1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
h2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V
K : Set G
hK : K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
g : G
hg : g ∈ K
g₀ : G
h1g₀ : g₀ ∈ s
h2g₀ : g₀ * g ∈ V
⊢ g ∈ (fun h => g₀ * h) ⁻¹' V
case h.w
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
h : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)
s : Finset G
h1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
h2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V
K : Set G
hK : K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
g : G
hg : g ∈ K
g₀ : G
h1g₀ : g₀ ∈ s
h2g₀ : g₀ * g ∈ V
⊢ g₀ ∈ Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s
[PROOFSTEP]
swap
[GOAL]
case h.w
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
h : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)
s : Finset G
h1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
h2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V
K : Set G
hK : K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
g : G
hg : g ∈ K
g₀ : G
h1g₀ : g₀ ∈ s
h2g₀ : g₀ * g ∈ V
⊢ g₀ ∈ Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s
[PROOFSTEP]
simp only [Finset.mem_filter, h1g₀, true_and_iff]
[GOAL]
case h.w
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
h : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)
s : Finset G
h1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
h2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V
K : Set G
hK : K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
g : G
hg : g ∈ K
g₀ : G
h1g₀ : g₀ ∈ s
h2g₀ : g₀ * g ∈ V
⊢ Set.Nonempty ((fun h => g₀ * h) ⁻¹' V ∩ K)
[PROOFSTEP]
use g
[GOAL]
case h
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
h : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)
s : Finset G
h1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
h2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V
K : Set G
hK : K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
g : G
hg : g ∈ K
g₀ : G
h1g₀ : g₀ ∈ s
h2g₀ : g₀ * g ∈ V
⊢ g ∈ (fun h => g₀ * h) ⁻¹' V ∩ K
[PROOFSTEP]
simp only [hg, h2g₀, mem_inter_iff, mem_preimage, and_self_iff]
[GOAL]
case h.h
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
h : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)
s : Finset G
h1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
h2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V
K : Set G
hK : K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
g : G
hg : g ∈ K
g₀ : G
h1g₀ : g₀ ∈ s
h2g₀ : g₀ * g ∈ V
⊢ g ∈ (fun h => g₀ * h) ⁻¹' V
[PROOFSTEP]
exact h2g₀
[GOAL]
case intro.intro
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
h : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)
s : Finset G
h1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
h2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V
this :
  ∀ (K : Set G),
    K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V →
      index K V ≤ Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s)
⊢ index K₁.carrier V + index K₂.carrier V ≤ Finset.card s
[PROOFSTEP]
refine'
  le_trans
    (add_le_add (this K₁.1 <| Subset.trans (subset_union_left _ _) h1s)
      (this K₂.1 <| Subset.trans (subset_union_right _ _) h1s))
    _
[GOAL]
case intro.intro
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
h : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)
s : Finset G
h1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
h2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V
this :
  ∀ (K : Set G),
    K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V →
      index K V ≤ Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s)
⊢ Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K₁.carrier)) s) +
      Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K₂.carrier)) s) ≤
    Finset.card s
[PROOFSTEP]
rw [← Finset.card_union_eq, Finset.filter_union_right]
[GOAL]
case intro.intro
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
h : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)
s : Finset G
h1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
h2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V
this :
  ∀ (K : Set G),
    K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V →
      index K V ≤ Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s)
⊢ Finset.card
      (Finset.filter
        (fun x =>
          Set.Nonempty ((fun h => x * h) ⁻¹' V ∩ K₁.carrier) ∨ Set.Nonempty ((fun h => x * h) ⁻¹' V ∩ K₂.carrier))
        s) ≤
    Finset.card s
case intro.intro
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
h : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)
s : Finset G
h1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
h2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V
this :
  ∀ (K : Set G),
    K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V →
      index K V ≤ Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s)
⊢ Disjoint (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K₁.carrier)) s)
    (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K₂.carrier)) s)
[PROOFSTEP]
exact s.card_filter_le _
[GOAL]
case intro.intro
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
h : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)
s : Finset G
h1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
h2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V
this :
  ∀ (K : Set G),
    K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V →
      index K V ≤ Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s)
⊢ Disjoint (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K₁.carrier)) s)
    (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K₂.carrier)) s)
[PROOFSTEP]
apply Finset.disjoint_filter.mpr
[GOAL]
case intro.intro
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
h : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)
s : Finset G
h1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
h2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V
this :
  ∀ (K : Set G),
    K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V →
      index K V ≤ Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s)
⊢ ∀ (x : G),
    x ∈ s → Set.Nonempty ((fun h => x * h) ⁻¹' V ∩ K₁.carrier) → ¬Set.Nonempty ((fun h => x * h) ⁻¹' V ∩ K₂.carrier)
[PROOFSTEP]
rintro g₁ _ ⟨g₂, h1g₂, h2g₂⟩ ⟨g₃, h1g₃, h2g₃⟩
[GOAL]
case intro.intro.intro.intro.intro.intro
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
h : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)
s : Finset G
h1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
h2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V
this :
  ∀ (K : Set G),
    K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V →
      index K V ≤ Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s)
g₁ : G
a✝ : g₁ ∈ s
g₂ : G
h1g₂ : g₂ ∈ (fun h => g₁ * h) ⁻¹' V
h2g₂ : g₂ ∈ K₁.carrier
g₃ : G
h1g₃ : g₃ ∈ (fun h => g₁ * h) ⁻¹' V
h2g₃ : g₃ ∈ K₂.carrier
⊢ False
[PROOFSTEP]
simp only [mem_preimage] at h1g₃ h1g₂
[GOAL]
case intro.intro.intro.intro.intro.intro
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
h : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)
s : Finset G
h1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
h2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V
this :
  ∀ (K : Set G),
    K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V →
      index K V ≤ Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s)
g₁ : G
a✝ : g₁ ∈ s
g₂ : G
h2g₂ : g₂ ∈ K₁.carrier
g₃ : G
h2g₃ : g₃ ∈ K₂.carrier
h1g₃ : g₁ * g₃ ∈ V
h1g₂ : g₁ * g₂ ∈ V
⊢ False
[PROOFSTEP]
refine' h.le_bot (_ : g₁⁻¹ ∈ _)
[GOAL]
case intro.intro.intro.intro.intro.intro
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
h : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)
s : Finset G
h1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
h2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V
this :
  ∀ (K : Set G),
    K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V →
      index K V ≤ Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s)
g₁ : G
a✝ : g₁ ∈ s
g₂ : G
h2g₂ : g₂ ∈ K₁.carrier
g₃ : G
h2g₃ : g₃ ∈ K₂.carrier
h1g₃ : g₁ * g₃ ∈ V
h1g₂ : g₁ * g₂ ∈ V
⊢ g₁⁻¹ ∈ K₁.carrier * V⁻¹ ⊓ K₂.carrier * V⁻¹
[PROOFSTEP]
constructor
[GOAL]
case intro.intro.intro.intro.intro.intro.left
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
h : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)
s : Finset G
h1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
h2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V
this :
  ∀ (K : Set G),
    K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V →
      index K V ≤ Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s)
g₁ : G
a✝ : g₁ ∈ s
g₂ : G
h2g₂ : g₂ ∈ K₁.carrier
g₃ : G
h2g₃ : g₃ ∈ K₂.carrier
h1g₃ : g₁ * g₃ ∈ V
h1g₂ : g₁ * g₂ ∈ V
⊢ g₁⁻¹ ∈ K₁.carrier * V⁻¹
[PROOFSTEP]
simp only [Set.mem_inv, Set.mem_mul, exists_exists_and_eq_and, exists_and_left]
[GOAL]
case intro.intro.intro.intro.intro.intro.right
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
h : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)
s : Finset G
h1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
h2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V
this :
  ∀ (K : Set G),
    K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V →
      index K V ≤ Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s)
g₁ : G
a✝ : g₁ ∈ s
g₂ : G
h2g₂ : g₂ ∈ K₁.carrier
g₃ : G
h2g₃ : g₃ ∈ K₂.carrier
h1g₃ : g₁ * g₃ ∈ V
h1g₂ : g₁ * g₂ ∈ V
⊢ g₁⁻¹ ∈ K₂.carrier * V⁻¹
[PROOFSTEP]
simp only [Set.mem_inv, Set.mem_mul, exists_exists_and_eq_and, exists_and_left]
[GOAL]
case intro.intro.intro.intro.intro.intro.left
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
h : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)
s : Finset G
h1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
h2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V
this :
  ∀ (K : Set G),
    K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V →
      index K V ≤ Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s)
g₁ : G
a✝ : g₁ ∈ s
g₂ : G
h2g₂ : g₂ ∈ K₁.carrier
g₃ : G
h2g₃ : g₃ ∈ K₂.carrier
h1g₃ : g₁ * g₃ ∈ V
h1g₂ : g₁ * g₂ ∈ V
⊢ ∃ x, x ∈ K₁.carrier ∧ ∃ x_1, x_1⁻¹ ∈ V ∧ x * x_1 = g₁⁻¹
[PROOFSTEP]
refine' ⟨_, h2g₂, (g₁ * g₂)⁻¹, _, _⟩
[GOAL]
case intro.intro.intro.intro.intro.intro.left.refine'_1
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
h : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)
s : Finset G
h1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
h2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V
this :
  ∀ (K : Set G),
    K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V →
      index K V ≤ Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s)
g₁ : G
a✝ : g₁ ∈ s
g₂ : G
h2g₂ : g₂ ∈ K₁.carrier
g₃ : G
h2g₃ : g₃ ∈ K₂.carrier
h1g₃ : g₁ * g₃ ∈ V
h1g₂ : g₁ * g₂ ∈ V
⊢ (g₁ * g₂)⁻¹⁻¹ ∈ V
case intro.intro.intro.intro.intro.intro.left.refine'_2
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
h : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)
s : Finset G
h1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
h2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V
this :
  ∀ (K : Set G),
    K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V →
      index K V ≤ Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s)
g₁ : G
a✝ : g₁ ∈ s
g₂ : G
h2g₂ : g₂ ∈ K₁.carrier
g₃ : G
h2g₃ : g₃ ∈ K₂.carrier
h1g₃ : g₁ * g₃ ∈ V
h1g₂ : g₁ * g₂ ∈ V
⊢ g₂ * (g₁ * g₂)⁻¹ = g₁⁻¹
[PROOFSTEP]
simp only [inv_inv, h1g₂]
[GOAL]
case intro.intro.intro.intro.intro.intro.left.refine'_2
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
h : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)
s : Finset G
h1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
h2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V
this :
  ∀ (K : Set G),
    K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V →
      index K V ≤ Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s)
g₁ : G
a✝ : g₁ ∈ s
g₂ : G
h2g₂ : g₂ ∈ K₁.carrier
g₃ : G
h2g₃ : g₃ ∈ K₂.carrier
h1g₃ : g₁ * g₃ ∈ V
h1g₂ : g₁ * g₂ ∈ V
⊢ g₂ * (g₁ * g₂)⁻¹ = g₁⁻¹
[PROOFSTEP]
simp only [mul_inv_rev, mul_inv_cancel_left]
[GOAL]
case intro.intro.intro.intro.intro.intro.right
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
h : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)
s : Finset G
h1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
h2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V
this :
  ∀ (K : Set G),
    K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V →
      index K V ≤ Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s)
g₁ : G
a✝ : g₁ ∈ s
g₂ : G
h2g₂ : g₂ ∈ K₁.carrier
g₃ : G
h2g₃ : g₃ ∈ K₂.carrier
h1g₃ : g₁ * g₃ ∈ V
h1g₂ : g₁ * g₂ ∈ V
⊢ ∃ x, x ∈ K₂.carrier ∧ ∃ x_1, x_1⁻¹ ∈ V ∧ x * x_1 = g₁⁻¹
[PROOFSTEP]
refine' ⟨_, h2g₃, (g₁ * g₃)⁻¹, _, _⟩
[GOAL]
case intro.intro.intro.intro.intro.intro.right.refine'_1
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
h : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)
s : Finset G
h1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
h2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V
this :
  ∀ (K : Set G),
    K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V →
      index K V ≤ Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s)
g₁ : G
a✝ : g₁ ∈ s
g₂ : G
h2g₂ : g₂ ∈ K₁.carrier
g₃ : G
h2g₃ : g₃ ∈ K₂.carrier
h1g₃ : g₁ * g₃ ∈ V
h1g₂ : g₁ * g₂ ∈ V
⊢ (g₁ * g₃)⁻¹⁻¹ ∈ V
case intro.intro.intro.intro.intro.intro.right.refine'_2
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
h : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)
s : Finset G
h1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
h2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V
this :
  ∀ (K : Set G),
    K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V →
      index K V ≤ Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s)
g₁ : G
a✝ : g₁ ∈ s
g₂ : G
h2g₂ : g₂ ∈ K₁.carrier
g₃ : G
h2g₃ : g₃ ∈ K₂.carrier
h1g₃ : g₁ * g₃ ∈ V
h1g₂ : g₁ * g₂ ∈ V
⊢ g₃ * (g₁ * g₃)⁻¹ = g₁⁻¹
[PROOFSTEP]
simp only [inv_inv, h1g₃]
[GOAL]
case intro.intro.intro.intro.intro.intro.right.refine'_2
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : Set.Nonempty (interior V)
h : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)
s : Finset G
h1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
h2s : Finset.card s = index (K₁.carrier ∪ K₂.carrier) V
this :
  ∀ (K : Set G),
    K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V →
      index K V ≤ Finset.card (Finset.filter (fun g => Set.Nonempty ((fun h => g * h) ⁻¹' V ∩ K)) s)
g₁ : G
a✝ : g₁ ∈ s
g₂ : G
h2g₂ : g₂ ∈ K₁.carrier
g₃ : G
h2g₃ : g₃ ∈ K₂.carrier
h1g₃ : g₁ * g₃ ∈ V
h1g₂ : g₁ * g₂ ∈ V
⊢ g₃ * (g₁ * g₃)⁻¹ = g₁⁻¹
[PROOFSTEP]
simp only [mul_inv_rev, mul_inv_cancel_left]
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K : Set G
hK : IsCompact K
V : Set G
hV : Set.Nonempty (interior V)
g : G
⊢ index ((fun h => g * h) '' K) V ≤ index K V
[PROOFSTEP]
rcases index_elim hK hV with ⟨s, h1s, h2s⟩
[GOAL]
case intro.intro
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K : Set G
hK : IsCompact K
V : Set G
hV : Set.Nonempty (interior V)
g : G
s : Finset G
h1s : K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
h2s : Finset.card s = index K V
⊢ index ((fun h => g * h) '' K) V ≤ index K V
[PROOFSTEP]
rw [← h2s]
[GOAL]
case intro.intro
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K : Set G
hK : IsCompact K
V : Set G
hV : Set.Nonempty (interior V)
g : G
s : Finset G
h1s : K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
h2s : Finset.card s = index K V
⊢ index ((fun h => g * h) '' K) V ≤ Finset.card s
[PROOFSTEP]
apply Nat.sInf_le
[GOAL]
case intro.intro.hm
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K : Set G
hK : IsCompact K
V : Set G
hV : Set.Nonempty (interior V)
g : G
s : Finset G
h1s : K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
h2s : Finset.card s = index K V
⊢ Finset.card s ∈ Finset.card '' {t | (fun h => g * h) '' K ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h => g * h) ⁻¹' V}
[PROOFSTEP]
rw [mem_image]
[GOAL]
case intro.intro.hm
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K : Set G
hK : IsCompact K
V : Set G
hV : Set.Nonempty (interior V)
g : G
s : Finset G
h1s : K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
h2s : Finset.card s = index K V
⊢ ∃ x, x ∈ {t | (fun h => g * h) '' K ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h => g * h) ⁻¹' V} ∧ Finset.card x = Finset.card s
[PROOFSTEP]
refine' ⟨s.map (Equiv.mulRight g⁻¹).toEmbedding, _, Finset.card_map _⟩
[GOAL]
case intro.intro.hm
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K : Set G
hK : IsCompact K
V : Set G
hV : Set.Nonempty (interior V)
g : G
s : Finset G
h1s : K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
h2s : Finset.card s = index K V
⊢ Finset.map (Equiv.toEmbedding (Equiv.mulRight g⁻¹)) s ∈
    {t | (fun h => g * h) '' K ⊆ ⋃ (g : G) (_ : g ∈ t), (fun h => g * h) ⁻¹' V}
[PROOFSTEP]
simp only [mem_setOf_eq]
[GOAL]
case intro.intro.hm
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K : Set G
hK : IsCompact K
V : Set G
hV : Set.Nonempty (interior V)
g : G
s : Finset G
h1s : K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
h2s : Finset.card s = index K V
⊢ (fun h => g * h) '' K ⊆
    ⋃ (g_1 : G) (_ : g_1 ∈ Finset.map (Equiv.toEmbedding (Equiv.mulRight g⁻¹)) s), (fun h => g_1 * h) ⁻¹' V
[PROOFSTEP]
refine' Subset.trans (image_subset _ h1s) _
[GOAL]
case intro.intro.hm
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K : Set G
hK : IsCompact K
V : Set G
hV : Set.Nonempty (interior V)
g : G
s : Finset G
h1s : K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
h2s : Finset.card s = index K V
⊢ (fun h => g * h) '' ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V ⊆
    ⋃ (g_1 : G) (_ : g_1 ∈ Finset.map (Equiv.toEmbedding (Equiv.mulRight g⁻¹)) s), (fun h => g_1 * h) ⁻¹' V
[PROOFSTEP]
rintro _ ⟨g₁, ⟨_, ⟨g₂, rfl⟩, ⟨_, ⟨hg₂, rfl⟩, hg₁⟩⟩, rfl⟩
[GOAL]
case intro.intro.hm.intro.intro.intro.intro.intro.intro.intro.intro
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K : Set G
hK : IsCompact K
V : Set G
hV : Set.Nonempty (interior V)
g : G
s : Finset G
h1s : K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
h2s : Finset.card s = index K V
g₁ g₂ : G
hg₂ : g₂ ∈ s
hg₁ : g₁ ∈ (fun h => (fun h => g₂ * h) ⁻¹' V) hg₂
⊢ (fun h => g * h) g₁ ∈
    ⋃ (g_1 : G) (_ : g_1 ∈ Finset.map (Equiv.toEmbedding (Equiv.mulRight g⁻¹)) s), (fun h => g_1 * h) ⁻¹' V
[PROOFSTEP]
simp only [mem_preimage] at hg₁
[GOAL]
case intro.intro.hm.intro.intro.intro.intro.intro.intro.intro.intro
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K : Set G
hK : IsCompact K
V : Set G
hV : Set.Nonempty (interior V)
g : G
s : Finset G
h1s : K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
h2s : Finset.card s = index K V
g₁ g₂ : G
hg₂ : g₂ ∈ s
hg₁ : g₂ * g₁ ∈ V
⊢ (fun h => g * h) g₁ ∈
    ⋃ (g_1 : G) (_ : g_1 ∈ Finset.map (Equiv.toEmbedding (Equiv.mulRight g⁻¹)) s), (fun h => g_1 * h) ⁻¹' V
[PROOFSTEP]
simp only [exists_prop, mem_iUnion, Finset.mem_map, Equiv.coe_mulRight, exists_exists_and_eq_and, mem_preimage,
  Equiv.toEmbedding_apply]
[GOAL]
case intro.intro.hm.intro.intro.intro.intro.intro.intro.intro.intro
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K : Set G
hK : IsCompact K
V : Set G
hV : Set.Nonempty (interior V)
g : G
s : Finset G
h1s : K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
h2s : Finset.card s = index K V
g₁ g₂ : G
hg₂ : g₂ ∈ s
hg₁ : g₂ * g₁ ∈ V
⊢ ∃ a, a ∈ s ∧ a * g⁻¹ * (g * g₁) ∈ V
[PROOFSTEP]
refine' ⟨_, hg₂, _⟩
[GOAL]
case intro.intro.hm.intro.intro.intro.intro.intro.intro.intro.intro
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K : Set G
hK : IsCompact K
V : Set G
hV : Set.Nonempty (interior V)
g : G
s : Finset G
h1s : K ⊆ ⋃ (g : G) (_ : g ∈ s), (fun h => g * h) ⁻¹' V
h2s : Finset.card s = index K V
g₁ g₂ : G
hg₂ : g₂ ∈ s
hg₁ : g₂ * g₁ ∈ V
⊢ g₂ * g⁻¹ * (g * g₁) ∈ V
[PROOFSTEP]
simp only [mul_assoc, hg₁, inv_mul_cancel_left]
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K : Set G
hK : IsCompact K
g : G
V : Set G
hV : Set.Nonempty (interior V)
⊢ index ((fun h => g * h) '' K) V = index K V
[PROOFSTEP]
refine' le_antisymm (mul_left_index_le hK hV g) _
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K : Set G
hK : IsCompact K
g : G
V : Set G
hV : Set.Nonempty (interior V)
⊢ index K V ≤ index ((fun h => g * h) '' K) V
[PROOFSTEP]
convert mul_left_index_le (hK.image <| continuous_mul_left g) hV g⁻¹
[GOAL]
case h.e'_3.h.e'_3
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K : Set G
hK : IsCompact K
g : G
V : Set G
hV : Set.Nonempty (interior V)
⊢ K = (fun h => g⁻¹ * h) '' ((fun b => g * b) '' K)
[PROOFSTEP]
rw [image_image]
[GOAL]
case h.e'_3.h.e'_3
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K : Set G
hK : IsCompact K
g : G
V : Set G
hV : Set.Nonempty (interior V)
⊢ K = (fun x => g⁻¹ * (g * x)) '' K
[PROOFSTEP]
symm
[GOAL]
case h.e'_3.h.e'_3
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K : Set G
hK : IsCompact K
g : G
V : Set G
hV : Set.Nonempty (interior V)
⊢ (fun x => g⁻¹ * (g * x)) '' K = K
[PROOFSTEP]
convert image_id' _ with h
[GOAL]
case h.e'_2.h.e'_3.h
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K : Set G
hK : IsCompact K
g : G
V : Set G
hV : Set.Nonempty (interior V)
h : G
⊢ g⁻¹ * (g * h) = h
[PROOFSTEP]
apply inv_mul_cancel_left
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
U : Set G
K : Compacts G
hU : Set.Nonempty (interior U)
⊢ prehaar (↑K₀) U K ≤ ↑(index ↑K ↑K₀)
[PROOFSTEP]
unfold prehaar
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
U : Set G
K : Compacts G
hU : Set.Nonempty (interior U)
⊢ ↑(index (↑K) U) / ↑(index (↑K₀) U) ≤ ↑(index ↑K ↑K₀)
[PROOFSTEP]
rw [div_le_iff]
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
U : Set G
K : Compacts G
hU : Set.Nonempty (interior U)
⊢ ↑(index (↑K) U) ≤ ↑(index ↑K ↑K₀) * ↑(index (↑K₀) U)
[PROOFSTEP]
norm_cast
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
U : Set G
K : Compacts G
hU : Set.Nonempty (interior U)
⊢ 0 < ↑(index (↑K₀) U)
[PROOFSTEP]
norm_cast
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
U : Set G
K : Compacts G
hU : Set.Nonempty (interior U)
⊢ index (↑K) U ≤ index ↑K ↑K₀ * index (↑K₀) U
[PROOFSTEP]
apply le_index_mul K₀ K hU
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
U : Set G
K : Compacts G
hU : Set.Nonempty (interior U)
⊢ 0 < index (↑K₀) U
[PROOFSTEP]
exact index_pos K₀ hU
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
U : Set G
hU : Set.Nonempty (interior U)
K : Set G
h1K : IsCompact K
h2K : Set.Nonempty (interior K)
⊢ 0 < prehaar (↑K₀) U { carrier := K, isCompact' := h1K }
[PROOFSTEP]
apply div_pos
[GOAL]
case ha
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
U : Set G
hU : Set.Nonempty (interior U)
K : Set G
h1K : IsCompact K
h2K : Set.Nonempty (interior K)
⊢ 0 < ↑(index (↑{ carrier := K, isCompact' := h1K }) U)
[PROOFSTEP]
norm_cast
[GOAL]
case hb
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
U : Set G
hU : Set.Nonempty (interior U)
K : Set G
h1K : IsCompact K
h2K : Set.Nonempty (interior K)
⊢ 0 < ↑(index (↑K₀) U)
[PROOFSTEP]
norm_cast
[GOAL]
case ha
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
U : Set G
hU : Set.Nonempty (interior U)
K : Set G
h1K : IsCompact K
h2K : Set.Nonempty (interior K)
⊢ 0 < index (↑{ carrier := K, isCompact' := h1K }) U
case hb
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
U : Set G
hU : Set.Nonempty (interior U)
K : Set G
h1K : IsCompact K
h2K : Set.Nonempty (interior K)
⊢ 0 < index (↑K₀) U
[PROOFSTEP]
apply index_pos ⟨⟨K, h1K⟩, h2K⟩ hU
[GOAL]
case hb
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
U : Set G
hU : Set.Nonempty (interior U)
K : Set G
h1K : IsCompact K
h2K : Set.Nonempty (interior K)
⊢ 0 < index (↑K₀) U
[PROOFSTEP]
exact index_pos K₀ hU
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
U : Set G
hU : Set.Nonempty (interior U)
K₁ K₂ : Compacts G
h : ↑K₁ ⊆ K₂.carrier
⊢ prehaar (↑K₀) U K₁ ≤ prehaar (↑K₀) U K₂
[PROOFSTEP]
simp only [prehaar]
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
U : Set G
hU : Set.Nonempty (interior U)
K₁ K₂ : Compacts G
h : ↑K₁ ⊆ K₂.carrier
⊢ ↑(index (↑K₁) U) / ↑(index (↑K₀) U) ≤ ↑(index (↑K₂) U) / ↑(index (↑K₀) U)
[PROOFSTEP]
rw [div_le_div_right]
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
U : Set G
hU : Set.Nonempty (interior U)
K₁ K₂ : Compacts G
h : ↑K₁ ⊆ K₂.carrier
⊢ ↑(index (↑K₁) U) ≤ ↑(index (↑K₂) U)
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
U : Set G
hU : Set.Nonempty (interior U)
K₁ K₂ : Compacts G
h : ↑K₁ ⊆ K₂.carrier
⊢ 0 < ↑(index (↑K₀) U)
[PROOFSTEP]
exact_mod_cast index_mono K₂.2 h hU
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
U : Set G
hU : Set.Nonempty (interior U)
K₁ K₂ : Compacts G
h : ↑K₁ ⊆ K₂.carrier
⊢ 0 < ↑(index (↑K₀) U)
[PROOFSTEP]
exact_mod_cast index_pos K₀ hU
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
U : Set G
hU : Set.Nonempty (interior U)
⊢ 0 < ↑(index (↑K₀.toCompacts) U)
[PROOFSTEP]
exact_mod_cast index_pos K₀ hU
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
U : Set G
K₁ K₂ : Compacts G
hU : Set.Nonempty (interior U)
⊢ prehaar (↑K₀) U (K₁ ⊔ K₂) ≤ prehaar (↑K₀) U K₁ + prehaar (↑K₀) U K₂
[PROOFSTEP]
simp only [prehaar]
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
U : Set G
K₁ K₂ : Compacts G
hU : Set.Nonempty (interior U)
⊢ ↑(index (↑(K₁ ⊔ K₂)) U) / ↑(index (↑K₀) U) ≤ ↑(index (↑K₁) U) / ↑(index (↑K₀) U) + ↑(index (↑K₂) U) / ↑(index (↑K₀) U)
[PROOFSTEP]
rw [div_add_div_same, div_le_div_right]
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
U : Set G
K₁ K₂ : Compacts G
hU : Set.Nonempty (interior U)
⊢ ↑(index (↑(K₁ ⊔ K₂)) U) ≤ ↑(index (↑K₁) U) + ↑(index (↑K₂) U)
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
U : Set G
K₁ K₂ : Compacts G
hU : Set.Nonempty (interior U)
⊢ 0 < ↑(index (↑K₀) U)
[PROOFSTEP]
exact_mod_cast index_union_le K₁ K₂ hU
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
U : Set G
K₁ K₂ : Compacts G
hU : Set.Nonempty (interior U)
⊢ 0 < ↑(index (↑K₀) U)
[PROOFSTEP]
exact_mod_cast index_pos K₀ hU
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
U : Set G
K₁ K₂ : Compacts G
hU : Set.Nonempty (interior U)
h : Disjoint (K₁.carrier * U⁻¹) (K₂.carrier * U⁻¹)
⊢ prehaar (↑K₀) U (K₁ ⊔ K₂) = prehaar (↑K₀) U K₁ + prehaar (↑K₀) U K₂
[PROOFSTEP]
simp only [prehaar]
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
U : Set G
K₁ K₂ : Compacts G
hU : Set.Nonempty (interior U)
h : Disjoint (K₁.carrier * U⁻¹) (K₂.carrier * U⁻¹)
⊢ ↑(index (↑(K₁ ⊔ K₂)) U) / ↑(index (↑K₀) U) = ↑(index (↑K₁) U) / ↑(index (↑K₀) U) + ↑(index (↑K₂) U) / ↑(index (↑K₀) U)
[PROOFSTEP]
rw [div_add_div_same]
  -- Porting note: Here was `congr`, but `to_additive` failed to generate a theorem.
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
U : Set G
K₁ K₂ : Compacts G
hU : Set.Nonempty (interior U)
h : Disjoint (K₁.carrier * U⁻¹) (K₂.carrier * U⁻¹)
⊢ ↑(index (↑(K₁ ⊔ K₂)) U) / ↑(index (↑K₀) U) = (↑(index (↑K₁) U) + ↑(index (↑K₂) U)) / ↑(index (↑K₀) U)
[PROOFSTEP]
refine congr_arg (fun x : ℝ => x / index K₀ U) ?_
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
U : Set G
K₁ K₂ : Compacts G
hU : Set.Nonempty (interior U)
h : Disjoint (K₁.carrier * U⁻¹) (K₂.carrier * U⁻¹)
⊢ ↑(index (↑(K₁ ⊔ K₂)) U) = ↑(index (↑K₁) U) + ↑(index (↑K₂) U)
[PROOFSTEP]
exact_mod_cast index_union_eq K₁ K₂ hU h
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
U : Set G
hU : Set.Nonempty (interior U)
g : G
K : Compacts G
⊢ prehaar (↑K₀) U (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) = prehaar (↑K₀) U K
[PROOFSTEP]
simp only [prehaar, Compacts.coe_map, is_left_invariant_index K.isCompact _ hU]
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
U : Set G
hU : Set.Nonempty (interior U)
⊢ prehaar (↑K₀) U ∈ haarProduct ↑K₀
[PROOFSTEP]
rintro ⟨K, hK⟩ _
[GOAL]
case mk
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
U : Set G
hU : Set.Nonempty (interior U)
K : Set G
hK : IsCompact K
a✝ : { carrier := K, isCompact' := hK } ∈ univ
⊢ prehaar (↑K₀) U { carrier := K, isCompact' := hK } ∈
    (fun K => Icc 0 ↑(index ↑K ↑K₀)) { carrier := K, isCompact' := hK }
[PROOFSTEP]
rw [mem_Icc]
[GOAL]
case mk
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
U : Set G
hU : Set.Nonempty (interior U)
K : Set G
hK : IsCompact K
a✝ : { carrier := K, isCompact' := hK } ∈ univ
⊢ 0 ≤ prehaar (↑K₀) U { carrier := K, isCompact' := hK } ∧
    prehaar (↑K₀) U { carrier := K, isCompact' := hK } ≤ ↑(index ↑{ carrier := K, isCompact' := hK } ↑K₀)
[PROOFSTEP]
exact ⟨prehaar_nonneg K₀ _, prehaar_le_index K₀ _ hU⟩
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
⊢ Set.Nonempty (haarProduct ↑K₀ ∩ ⋂ (V : OpenNhdsOf 1), clPrehaar (↑K₀) V)
[PROOFSTEP]
have : IsCompact (haarProduct (K₀ : Set G)) := by apply isCompact_univ_pi; intro K; apply isCompact_Icc
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
⊢ IsCompact (haarProduct ↑K₀)
[PROOFSTEP]
apply isCompact_univ_pi
[GOAL]
case h
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
⊢ ∀ (i : Compacts G), IsCompact (Icc 0 ↑(index ↑i ↑K₀))
[PROOFSTEP]
intro K
[GOAL]
case h
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
K : Compacts G
⊢ IsCompact (Icc 0 ↑(index ↑K ↑K₀))
[PROOFSTEP]
apply isCompact_Icc
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
this : IsCompact (haarProduct ↑K₀)
⊢ Set.Nonempty (haarProduct ↑K₀ ∩ ⋂ (V : OpenNhdsOf 1), clPrehaar (↑K₀) V)
[PROOFSTEP]
refine' this.inter_iInter_nonempty (clPrehaar K₀) (fun s => isClosed_closure) fun t => _
[GOAL]
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)
⊢ Set.Nonempty (haarProduct ↑K₀ ∩ ⋂ (i : OpenNhdsOf 1) (_ : i ∈ t), clPrehaar (↑K₀) i)
[PROOFSTEP]
let V₀ := ⋂ V ∈ t, (V : OpenNhdsOf (1 : G)).carrier
[GOAL]
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 : OpenNhdsOf 1) (_ : V ∈ t), V.carrier
⊢ Set.Nonempty (haarProduct ↑K₀ ∩ ⋂ (i : OpenNhdsOf 1) (_ : i ∈ t), clPrehaar (↑K₀) i)
[PROOFSTEP]
have h1V₀ : IsOpen V₀ := by apply isOpen_biInter; apply Finset.finite_toSet; rintro ⟨⟨V, hV₁⟩, hV₂⟩ _; exact hV₁
[GOAL]
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 : OpenNhdsOf 1) (_ : V ∈ t), V.carrier
⊢ IsOpen V₀
[PROOFSTEP]
apply isOpen_biInter
[GOAL]
case hs
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 : OpenNhdsOf 1) (_ : V ∈ t), V.carrier
⊢ Set.Finite fun i => i ∈ t.val
case h
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 : OpenNhdsOf 1) (_ : V ∈ t), V.carrier
⊢ ∀ (i : OpenNhdsOf 1), (i ∈ fun i => i ∈ t.val) → IsOpen i.carrier
[PROOFSTEP]
apply Finset.finite_toSet
[GOAL]
case h
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 : OpenNhdsOf 1) (_ : V ∈ t), V.carrier
⊢ ∀ (i : OpenNhdsOf 1), (i ∈ fun i => i ∈ t.val) → IsOpen i.carrier
[PROOFSTEP]
rintro ⟨⟨V, hV₁⟩, hV₂⟩ _
[GOAL]
case h.mk.mk
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 : OpenNhdsOf 1) (_ : V ∈ t), V.carrier
V : Set G
hV₁ : IsOpen V
hV₂ : 1 ∈ { carrier := V, is_open' := hV₁ }.carrier
a✝ : { toOpens := { carrier := V, is_open' := hV₁ }, mem' := hV₂ } ∈ fun i => i ∈ t.val
⊢ IsOpen { toOpens := { carrier := V, is_open' := hV₁ }, mem' := hV₂ }.toOpens.carrier
[PROOFSTEP]
exact hV₁
[GOAL]
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 : OpenNhdsOf 1) (_ : V ∈ t), V.carrier
h1V₀ : IsOpen V₀
⊢ Set.Nonempty (haarProduct ↑K₀ ∩ ⋂ (i : OpenNhdsOf 1) (_ : i ∈ t), clPrehaar (↑K₀) i)
[PROOFSTEP]
have h2V₀ : (1 : G) ∈ V₀ := by simp only [mem_iInter]; rintro ⟨⟨V, hV₁⟩, hV₂⟩ _; exact hV₂
[GOAL]
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 : OpenNhdsOf 1) (_ : V ∈ t), V.carrier
h1V₀ : IsOpen V₀
⊢ 1 ∈ V₀
[PROOFSTEP]
simp only [mem_iInter]
[GOAL]
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 : OpenNhdsOf 1) (_ : V ∈ t), V.carrier
h1V₀ : IsOpen V₀
⊢ ∀ (i : OpenNhdsOf 1), i ∈ t → 1 ∈ i.carrier
[PROOFSTEP]
rintro ⟨⟨V, hV₁⟩, hV₂⟩ _
[GOAL]
case mk.mk
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 : OpenNhdsOf 1) (_ : V ∈ t), V.carrier
h1V₀ : IsOpen V₀
V : Set G
hV₁ : IsOpen V
hV₂ : 1 ∈ { carrier := V, is_open' := hV₁ }.carrier
i✝ : { toOpens := { carrier := V, is_open' := hV₁ }, mem' := hV₂ } ∈ t
⊢ 1 ∈ { toOpens := { carrier := V, is_open' := hV₁ }, mem' := hV₂ }.toOpens.carrier
[PROOFSTEP]
exact hV₂
[GOAL]
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 : OpenNhdsOf 1) (_ : V ∈ t), V.carrier
h1V₀ : IsOpen V₀
h2V₀ : 1 ∈ V₀
⊢ Set.Nonempty (haarProduct ↑K₀ ∩ ⋂ (i : OpenNhdsOf 1) (_ : i ∈ t), clPrehaar (↑K₀) i)
[PROOFSTEP]
refine' ⟨prehaar K₀ V₀, _⟩
[GOAL]
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 : OpenNhdsOf 1) (_ : V ∈ t), V.carrier
h1V₀ : IsOpen V₀
h2V₀ : 1 ∈ V₀
⊢ prehaar (↑K₀) V₀ ∈ haarProduct ↑K₀ ∩ ⋂ (i : OpenNhdsOf 1) (_ : i ∈ t), clPrehaar (↑K₀) i
[PROOFSTEP]
constructor
[GOAL]
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 : OpenNhdsOf 1) (_ : V ∈ t), V.carrier
h1V₀ : IsOpen V₀
h2V₀ : 1 ∈ V₀
⊢ prehaar (↑K₀) V₀ ∈ haarProduct ↑K₀
[PROOFSTEP]
apply prehaar_mem_haarProduct K₀
[GOAL]
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 : OpenNhdsOf 1) (_ : V ∈ t), V.carrier
h1V₀ : IsOpen V₀
h2V₀ : 1 ∈ V₀
⊢ Set.Nonempty (interior V₀)
[PROOFSTEP]
use 1
[GOAL]
case h
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 : OpenNhdsOf 1) (_ : V ∈ t), V.carrier
h1V₀ : IsOpen V₀
h2V₀ : 1 ∈ V₀
⊢ 1 ∈ interior V₀
[PROOFSTEP]
rwa [h1V₀.interior_eq]
[GOAL]
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 : OpenNhdsOf 1) (_ : V ∈ t), V.carrier
h1V₀ : IsOpen V₀
h2V₀ : 1 ∈ V₀
⊢ prehaar (↑K₀) V₀ ∈ ⋂ (i : OpenNhdsOf 1) (_ : i ∈ t), clPrehaar (↑K₀) i
[PROOFSTEP]
simp only [mem_iInter]
[GOAL]
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 : OpenNhdsOf 1) (_ : V ∈ t), V.carrier
h1V₀ : IsOpen V₀
h2V₀ : 1 ∈ V₀
⊢ ∀ (i : OpenNhdsOf 1), i ∈ t → prehaar (↑K₀) (⋂ (V : OpenNhdsOf 1) (_ : V ∈ t), V.carrier) ∈ clPrehaar (↑K₀) i
[PROOFSTEP]
rintro ⟨V, hV⟩ h2V
[GOAL]
case right.mk
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 : OpenNhdsOf 1) (_ : V ∈ t), V.carrier
h1V₀ : IsOpen V₀
h2V₀ : 1 ∈ V₀
V : Opens G
hV : 1 ∈ V.carrier
h2V : { toOpens := V, mem' := hV } ∈ t
⊢ prehaar (↑K₀) (⋂ (V : OpenNhdsOf 1) (_ : V ∈ t), V.carrier) ∈ clPrehaar ↑K₀ { toOpens := V, mem' := hV }
[PROOFSTEP]
apply subset_closure
[GOAL]
case right.mk.a
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 : OpenNhdsOf 1) (_ : V ∈ t), V.carrier
h1V₀ : IsOpen V₀
h2V₀ : 1 ∈ V₀
V : Opens G
hV : 1 ∈ V.carrier
h2V : { toOpens := V, mem' := hV } ∈ t
⊢ prehaar (↑K₀) (⋂ (V : OpenNhdsOf 1) (_ : V ∈ t), V.carrier) ∈
    prehaar ↑K₀ '' {U | U ⊆ ↑{ toOpens := V, mem' := hV }.toOpens ∧ IsOpen U ∧ 1 ∈ U}
[PROOFSTEP]
apply mem_image_of_mem
[GOAL]
case right.mk.a.h
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 : OpenNhdsOf 1) (_ : V ∈ t), V.carrier
h1V₀ : IsOpen V₀
h2V₀ : 1 ∈ V₀
V : Opens G
hV : 1 ∈ V.carrier
h2V : { toOpens := V, mem' := hV } ∈ t
⊢ ⋂ (V : OpenNhdsOf 1) (_ : V ∈ t), V.carrier ∈ {U | U ⊆ ↑{ toOpens := V, mem' := hV }.toOpens ∧ IsOpen U ∧ 1 ∈ U}
[PROOFSTEP]
rw [mem_setOf_eq]
[GOAL]
case right.mk.a.h
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 : OpenNhdsOf 1) (_ : V ∈ t), V.carrier
h1V₀ : IsOpen V₀
h2V₀ : 1 ∈ V₀
V : Opens G
hV : 1 ∈ V.carrier
h2V : { toOpens := V, mem' := hV } ∈ t
⊢ ⋂ (V : OpenNhdsOf 1) (_ : V ∈ t), V.carrier ⊆ ↑{ toOpens := V, mem' := hV }.toOpens ∧
    IsOpen (⋂ (V : OpenNhdsOf 1) (_ : V ∈ t), V.carrier) ∧ 1 ∈ ⋂ (V : OpenNhdsOf 1) (_ : V ∈ t), V.carrier
[PROOFSTEP]
exact ⟨Subset.trans (iInter_subset _ ⟨V, hV⟩) (iInter_subset _ h2V), h1V₀, h2V₀⟩
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
V : OpenNhdsOf 1
⊢ chaar K₀ ∈ clPrehaar (↑K₀) V
[PROOFSTEP]
have := (Classical.choose_spec (nonempty_iInter_clPrehaar K₀)).2
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
V : OpenNhdsOf 1
this :
  Classical.choose (_ : Set.Nonempty (haarProduct ↑K₀ ∩ ⋂ (V : OpenNhdsOf 1), clPrehaar (↑K₀) V)) ∈
    ⋂ (V : OpenNhdsOf 1), clPrehaar (↑K₀) V
⊢ chaar K₀ ∈ clPrehaar (↑K₀) V
[PROOFSTEP]
rw [mem_iInter] at this
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
V : OpenNhdsOf 1
this :
  ∀ (i : OpenNhdsOf 1),
    Classical.choose (_ : Set.Nonempty (haarProduct ↑K₀ ∩ ⋂ (V : OpenNhdsOf 1), clPrehaar (↑K₀) V)) ∈ clPrehaar (↑K₀) i
⊢ chaar K₀ ∈ clPrehaar (↑K₀) V
[PROOFSTEP]
exact this V
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
K : Compacts G
⊢ 0 ≤ chaar K₀ K
[PROOFSTEP]
have := chaar_mem_haarProduct K₀ K (mem_univ _)
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
K : Compacts G
this : chaar K₀ K ∈ (fun K => Icc 0 ↑(index ↑K ↑K₀)) K
⊢ 0 ≤ chaar K₀ K
[PROOFSTEP]
rw [mem_Icc] at this
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
K : Compacts G
this : 0 ≤ chaar K₀ K ∧ chaar K₀ K ≤ ↑(index ↑K ↑K₀)
⊢ 0 ≤ chaar K₀ K
[PROOFSTEP]
exact this.1
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
⊢ chaar K₀ ⊥ = 0
[PROOFSTEP]
let eval : (Compacts G → ℝ) → ℝ := fun f => f ⊥
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
eval : (Compacts G → ℝ) → ℝ := fun f => f ⊥
⊢ chaar K₀ ⊥ = 0
[PROOFSTEP]
have : Continuous eval := continuous_apply ⊥
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
eval : (Compacts G → ℝ) → ℝ := fun f => f ⊥
this : Continuous eval
⊢ chaar K₀ ⊥ = 0
[PROOFSTEP]
show chaar K₀ ∈ eval ⁻¹' {(0 : ℝ)}
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
eval : (Compacts G → ℝ) → ℝ := fun f => f ⊥
this : Continuous eval
⊢ chaar K₀ ∈ eval ⁻¹' {0}
[PROOFSTEP]
apply mem_of_subset_of_mem _ (chaar_mem_clPrehaar K₀ ⊤)
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
eval : (Compacts G → ℝ) → ℝ := fun f => f ⊥
this : Continuous eval
⊢ clPrehaar ↑K₀ ⊤ ⊆ eval ⁻¹' {0}
[PROOFSTEP]
unfold clPrehaar
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
eval : (Compacts G → ℝ) → ℝ := fun f => f ⊥
this : Continuous eval
⊢ closure (prehaar ↑K₀ '' {U | U ⊆ ↑⊤.toOpens ∧ IsOpen U ∧ 1 ∈ U}) ⊆ eval ⁻¹' {0}
[PROOFSTEP]
rw [IsClosed.closure_subset_iff]
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
eval : (Compacts G → ℝ) → ℝ := fun f => f ⊥
this : Continuous eval
⊢ prehaar ↑K₀ '' {U | U ⊆ ↑⊤.toOpens ∧ IsOpen U ∧ 1 ∈ U} ⊆ eval ⁻¹' {0}
[PROOFSTEP]
rintro _ ⟨U, _, rfl⟩
[GOAL]
case intro.intro
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
eval : (Compacts G → ℝ) → ℝ := fun f => f ⊥
this : Continuous eval
U : Set G
left✝ : U ∈ {U | U ⊆ ↑⊤.toOpens ∧ IsOpen U ∧ 1 ∈ U}
⊢ prehaar (↑K₀) U ∈ eval ⁻¹' {0}
[PROOFSTEP]
apply prehaar_empty
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
eval : (Compacts G → ℝ) → ℝ := fun f => f ⊥
this : Continuous eval
⊢ IsClosed (eval ⁻¹' {0})
[PROOFSTEP]
apply continuous_iff_isClosed.mp this
[GOAL]
case a
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
eval : (Compacts G → ℝ) → ℝ := fun f => f ⊥
this : Continuous eval
⊢ IsClosed {0}
[PROOFSTEP]
exact isClosed_singleton
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
⊢ chaar K₀ K₀.toCompacts = 1
[PROOFSTEP]
let eval : (Compacts G → ℝ) → ℝ := fun f => f K₀.toCompacts
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
eval : (Compacts G → ℝ) → ℝ := fun f => f K₀.toCompacts
⊢ chaar K₀ K₀.toCompacts = 1
[PROOFSTEP]
have : Continuous eval := continuous_apply _
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
eval : (Compacts G → ℝ) → ℝ := fun f => f K₀.toCompacts
this : Continuous eval
⊢ chaar K₀ K₀.toCompacts = 1
[PROOFSTEP]
show chaar K₀ ∈ eval ⁻¹' {(1 : ℝ)}
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
eval : (Compacts G → ℝ) → ℝ := fun f => f K₀.toCompacts
this : Continuous eval
⊢ chaar K₀ ∈ eval ⁻¹' {1}
[PROOFSTEP]
apply mem_of_subset_of_mem _ (chaar_mem_clPrehaar K₀ ⊤)
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
eval : (Compacts G → ℝ) → ℝ := fun f => f K₀.toCompacts
this : Continuous eval
⊢ clPrehaar ↑K₀ ⊤ ⊆ eval ⁻¹' {1}
[PROOFSTEP]
unfold clPrehaar
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
eval : (Compacts G → ℝ) → ℝ := fun f => f K₀.toCompacts
this : Continuous eval
⊢ closure (prehaar ↑K₀ '' {U | U ⊆ ↑⊤.toOpens ∧ IsOpen U ∧ 1 ∈ U}) ⊆ eval ⁻¹' {1}
[PROOFSTEP]
rw [IsClosed.closure_subset_iff]
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
eval : (Compacts G → ℝ) → ℝ := fun f => f K₀.toCompacts
this : Continuous eval
⊢ prehaar ↑K₀ '' {U | U ⊆ ↑⊤.toOpens ∧ IsOpen U ∧ 1 ∈ U} ⊆ eval ⁻¹' {1}
[PROOFSTEP]
rintro _ ⟨U, ⟨_, h2U, h3U⟩, rfl⟩
[GOAL]
case intro.intro.intro.intro
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
eval : (Compacts G → ℝ) → ℝ := fun f => f K₀.toCompacts
this : Continuous eval
U : Set G
left✝ : U ⊆ ↑⊤.toOpens
h2U : IsOpen U
h3U : 1 ∈ U
⊢ prehaar (↑K₀) U ∈ eval ⁻¹' {1}
[PROOFSTEP]
apply prehaar_self
[GOAL]
case intro.intro.intro.intro.hU
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
eval : (Compacts G → ℝ) → ℝ := fun f => f K₀.toCompacts
this : Continuous eval
U : Set G
left✝ : U ⊆ ↑⊤.toOpens
h2U : IsOpen U
h3U : 1 ∈ U
⊢ Set.Nonempty (interior U)
[PROOFSTEP]
rw [h2U.interior_eq]
[GOAL]
case intro.intro.intro.intro.hU
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
eval : (Compacts G → ℝ) → ℝ := fun f => f K₀.toCompacts
this : Continuous eval
U : Set G
left✝ : U ⊆ ↑⊤.toOpens
h2U : IsOpen U
h3U : 1 ∈ U
⊢ Set.Nonempty U
[PROOFSTEP]
exact ⟨1, h3U⟩
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
eval : (Compacts G → ℝ) → ℝ := fun f => f K₀.toCompacts
this : Continuous eval
⊢ IsClosed (eval ⁻¹' {1})
[PROOFSTEP]
apply continuous_iff_isClosed.mp this
[GOAL]
case a
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
eval : (Compacts G → ℝ) → ℝ := fun f => f K₀.toCompacts
this : Continuous eval
⊢ IsClosed {1}
[PROOFSTEP]
exact isClosed_singleton
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : ↑K₁ ⊆ ↑K₂
⊢ chaar K₀ K₁ ≤ chaar K₀ K₂
[PROOFSTEP]
let eval : (Compacts G → ℝ) → ℝ := fun f => f K₂ - f K₁
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : ↑K₁ ⊆ ↑K₂
eval : (Compacts G → ℝ) → ℝ := fun f => f K₂ - f K₁
⊢ chaar K₀ K₁ ≤ chaar K₀ K₂
[PROOFSTEP]
have : Continuous eval := (continuous_apply K₂).sub (continuous_apply K₁)
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : ↑K₁ ⊆ ↑K₂
eval : (Compacts G → ℝ) → ℝ := fun f => f K₂ - f K₁
this : Continuous eval
⊢ chaar K₀ K₁ ≤ chaar K₀ K₂
[PROOFSTEP]
rw [← sub_nonneg]
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : ↑K₁ ⊆ ↑K₂
eval : (Compacts G → ℝ) → ℝ := fun f => f K₂ - f K₁
this : Continuous eval
⊢ 0 ≤ chaar K₀ K₂ - chaar K₀ K₁
[PROOFSTEP]
show chaar K₀ ∈ eval ⁻¹' Ici (0 : ℝ)
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : ↑K₁ ⊆ ↑K₂
eval : (Compacts G → ℝ) → ℝ := fun f => f K₂ - f K₁
this : Continuous eval
⊢ chaar K₀ ∈ eval ⁻¹' Ici 0
[PROOFSTEP]
apply mem_of_subset_of_mem _ (chaar_mem_clPrehaar K₀ ⊤)
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : ↑K₁ ⊆ ↑K₂
eval : (Compacts G → ℝ) → ℝ := fun f => f K₂ - f K₁
this : Continuous eval
⊢ clPrehaar ↑K₀ ⊤ ⊆ eval ⁻¹' Ici 0
[PROOFSTEP]
unfold clPrehaar
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : ↑K₁ ⊆ ↑K₂
eval : (Compacts G → ℝ) → ℝ := fun f => f K₂ - f K₁
this : Continuous eval
⊢ closure (prehaar ↑K₀ '' {U | U ⊆ ↑⊤.toOpens ∧ IsOpen U ∧ 1 ∈ U}) ⊆ eval ⁻¹' Ici 0
[PROOFSTEP]
rw [IsClosed.closure_subset_iff]
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : ↑K₁ ⊆ ↑K₂
eval : (Compacts G → ℝ) → ℝ := fun f => f K₂ - f K₁
this : Continuous eval
⊢ prehaar ↑K₀ '' {U | U ⊆ ↑⊤.toOpens ∧ IsOpen U ∧ 1 ∈ U} ⊆ eval ⁻¹' Ici 0
[PROOFSTEP]
rintro _ ⟨U, ⟨_, h2U, h3U⟩, rfl⟩
[GOAL]
case intro.intro.intro.intro
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : ↑K₁ ⊆ ↑K₂
eval : (Compacts G → ℝ) → ℝ := fun f => f K₂ - f K₁
this : Continuous eval
U : Set G
left✝ : U ⊆ ↑⊤.toOpens
h2U : IsOpen U
h3U : 1 ∈ U
⊢ prehaar (↑K₀) U ∈ eval ⁻¹' Ici 0
[PROOFSTEP]
simp only [mem_preimage, mem_Ici, sub_nonneg]
[GOAL]
case intro.intro.intro.intro
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : ↑K₁ ⊆ ↑K₂
eval : (Compacts G → ℝ) → ℝ := fun f => f K₂ - f K₁
this : Continuous eval
U : Set G
left✝ : U ⊆ ↑⊤.toOpens
h2U : IsOpen U
h3U : 1 ∈ U
⊢ prehaar (↑K₀) U K₁ ≤ prehaar (↑K₀) U K₂
[PROOFSTEP]
apply prehaar_mono _ h
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : ↑K₁ ⊆ ↑K₂
eval : (Compacts G → ℝ) → ℝ := fun f => f K₂ - f K₁
this : Continuous eval
U : Set G
left✝ : U ⊆ ↑⊤.toOpens
h2U : IsOpen U
h3U : 1 ∈ U
⊢ Set.Nonempty (interior U)
[PROOFSTEP]
rw [h2U.interior_eq]
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : ↑K₁ ⊆ ↑K₂
eval : (Compacts G → ℝ) → ℝ := fun f => f K₂ - f K₁
this : Continuous eval
U : Set G
left✝ : U ⊆ ↑⊤.toOpens
h2U : IsOpen U
h3U : 1 ∈ U
⊢ Set.Nonempty U
[PROOFSTEP]
exact ⟨1, h3U⟩
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : ↑K₁ ⊆ ↑K₂
eval : (Compacts G → ℝ) → ℝ := fun f => f K₂ - f K₁
this : Continuous eval
⊢ IsClosed (eval ⁻¹' Ici 0)
[PROOFSTEP]
apply continuous_iff_isClosed.mp this
[GOAL]
case a
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : ↑K₁ ⊆ ↑K₂
eval : (Compacts G → ℝ) → ℝ := fun f => f K₂ - f K₁
this : Continuous eval
⊢ IsClosed (Ici 0)
[PROOFSTEP]
exact isClosed_Ici
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
⊢ chaar K₀ (K₁ ⊔ K₂) ≤ chaar K₀ K₁ + chaar K₀ K₂
[PROOFSTEP]
let eval : (Compacts G → ℝ) → ℝ := fun f => f K₁ + f K₂ - f (K₁ ⊔ K₂)
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
eval : (Compacts G → ℝ) → ℝ := fun f => f K₁ + f K₂ - f (K₁ ⊔ K₂)
⊢ chaar K₀ (K₁ ⊔ K₂) ≤ chaar K₀ K₁ + chaar K₀ K₂
[PROOFSTEP]
have : Continuous eval := by exact ((continuous_apply K₁).add (continuous_apply K₂)).sub (continuous_apply (K₁ ⊔ K₂))
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
eval : (Compacts G → ℝ) → ℝ := fun f => f K₁ + f K₂ - f (K₁ ⊔ K₂)
⊢ Continuous eval
[PROOFSTEP]
exact ((continuous_apply K₁).add (continuous_apply K₂)).sub (continuous_apply (K₁ ⊔ K₂))
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
eval : (Compacts G → ℝ) → ℝ := fun f => f K₁ + f K₂ - f (K₁ ⊔ K₂)
this : Continuous eval
⊢ chaar K₀ (K₁ ⊔ K₂) ≤ chaar K₀ K₁ + chaar K₀ K₂
[PROOFSTEP]
rw [← sub_nonneg]
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
eval : (Compacts G → ℝ) → ℝ := fun f => f K₁ + f K₂ - f (K₁ ⊔ K₂)
this : Continuous eval
⊢ 0 ≤ chaar K₀ K₁ + chaar K₀ K₂ - chaar K₀ (K₁ ⊔ K₂)
[PROOFSTEP]
show chaar K₀ ∈ eval ⁻¹' Ici (0 : ℝ)
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
eval : (Compacts G → ℝ) → ℝ := fun f => f K₁ + f K₂ - f (K₁ ⊔ K₂)
this : Continuous eval
⊢ chaar K₀ ∈ eval ⁻¹' Ici 0
[PROOFSTEP]
apply mem_of_subset_of_mem _ (chaar_mem_clPrehaar K₀ ⊤)
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
eval : (Compacts G → ℝ) → ℝ := fun f => f K₁ + f K₂ - f (K₁ ⊔ K₂)
this : Continuous eval
⊢ clPrehaar ↑K₀ ⊤ ⊆ eval ⁻¹' Ici 0
[PROOFSTEP]
unfold clPrehaar
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
eval : (Compacts G → ℝ) → ℝ := fun f => f K₁ + f K₂ - f (K₁ ⊔ K₂)
this : Continuous eval
⊢ closure (prehaar ↑K₀ '' {U | U ⊆ ↑⊤.toOpens ∧ IsOpen U ∧ 1 ∈ U}) ⊆ eval ⁻¹' Ici 0
[PROOFSTEP]
rw [IsClosed.closure_subset_iff]
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
eval : (Compacts G → ℝ) → ℝ := fun f => f K₁ + f K₂ - f (K₁ ⊔ K₂)
this : Continuous eval
⊢ prehaar ↑K₀ '' {U | U ⊆ ↑⊤.toOpens ∧ IsOpen U ∧ 1 ∈ U} ⊆ eval ⁻¹' Ici 0
[PROOFSTEP]
rintro _ ⟨U, ⟨_, h2U, h3U⟩, rfl⟩
[GOAL]
case intro.intro.intro.intro
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
eval : (Compacts G → ℝ) → ℝ := fun f => f K₁ + f K₂ - f (K₁ ⊔ K₂)
this : Continuous eval
U : Set G
left✝ : U ⊆ ↑⊤.toOpens
h2U : IsOpen U
h3U : 1 ∈ U
⊢ prehaar (↑K₀) U ∈ eval ⁻¹' Ici 0
[PROOFSTEP]
simp only [mem_preimage, mem_Ici, sub_nonneg]
[GOAL]
case intro.intro.intro.intro
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
eval : (Compacts G → ℝ) → ℝ := fun f => f K₁ + f K₂ - f (K₁ ⊔ K₂)
this : Continuous eval
U : Set G
left✝ : U ⊆ ↑⊤.toOpens
h2U : IsOpen U
h3U : 1 ∈ U
⊢ prehaar (↑K₀) U (K₁ ⊔ K₂) ≤ prehaar (↑K₀) U K₁ + prehaar (↑K₀) U K₂
[PROOFSTEP]
apply prehaar_sup_le
[GOAL]
case intro.intro.intro.intro.hU
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
eval : (Compacts G → ℝ) → ℝ := fun f => f K₁ + f K₂ - f (K₁ ⊔ K₂)
this : Continuous eval
U : Set G
left✝ : U ⊆ ↑⊤.toOpens
h2U : IsOpen U
h3U : 1 ∈ U
⊢ Set.Nonempty (interior U)
[PROOFSTEP]
rw [h2U.interior_eq]
[GOAL]
case intro.intro.intro.intro.hU
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
eval : (Compacts G → ℝ) → ℝ := fun f => f K₁ + f K₂ - f (K₁ ⊔ K₂)
this : Continuous eval
U : Set G
left✝ : U ⊆ ↑⊤.toOpens
h2U : IsOpen U
h3U : 1 ∈ U
⊢ Set.Nonempty U
[PROOFSTEP]
exact ⟨1, h3U⟩
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
eval : (Compacts G → ℝ) → ℝ := fun f => f K₁ + f K₂ - f (K₁ ⊔ K₂)
this : Continuous eval
⊢ IsClosed (eval ⁻¹' Ici 0)
[PROOFSTEP]
apply continuous_iff_isClosed.mp this
[GOAL]
case a
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
eval : (Compacts G → ℝ) → ℝ := fun f => f K₁ + f K₂ - f (K₁ ⊔ K₂)
this : Continuous eval
⊢ IsClosed (Ici 0)
[PROOFSTEP]
exact isClosed_Ici
[GOAL]
G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : Disjoint K₁.carrier K₂.carrier
⊢ chaar K₀ (K₁ ⊔ K₂) = chaar K₀ K₁ + chaar K₀ K₂
[PROOFSTEP]
rcases isCompact_isCompact_separated K₁.2 K₂.2 h with ⟨U₁, U₂, h1U₁, h1U₂, h2U₁, h2U₂, hU⟩
[GOAL]
case intro.intro.intro.intro.intro.intro
G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : Disjoint K₁.carrier K₂.carrier
U₁ U₂ : Set G
h1U₁ : IsOpen U₁
h1U₂ : IsOpen U₂
h2U₁ : K₁.carrier ⊆ U₁
h2U₂ : K₂.carrier ⊆ U₂
hU : Disjoint U₁ U₂
⊢ chaar K₀ (K₁ ⊔ K₂) = chaar K₀ K₁ + chaar K₀ K₂
[PROOFSTEP]
rcases compact_open_separated_mul_right K₁.2 h1U₁ h2U₁ with ⟨L₁, h1L₁, h2L₁⟩
[GOAL]
case intro.intro.intro.intro.intro.intro.intro.intro
G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : Disjoint K₁.carrier K₂.carrier
U₁ U₂ : Set G
h1U₁ : IsOpen U₁
h1U₂ : IsOpen U₂
h2U₁ : K₁.carrier ⊆ U₁
h2U₂ : K₂.carrier ⊆ U₂
hU : Disjoint U₁ U₂
L₁ : Set G
h1L₁ : L₁ ∈ 𝓝 1
h2L₁ : K₁.carrier * L₁ ⊆ U₁
⊢ chaar K₀ (K₁ ⊔ K₂) = chaar K₀ K₁ + chaar K₀ K₂
[PROOFSTEP]
rcases mem_nhds_iff.mp h1L₁ with ⟨V₁, h1V₁, h2V₁, h3V₁⟩
[GOAL]
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : Disjoint K₁.carrier K₂.carrier
U₁ U₂ : Set G
h1U₁ : IsOpen U₁
h1U₂ : IsOpen U₂
h2U₁ : K₁.carrier ⊆ U₁
h2U₂ : K₂.carrier ⊆ U₂
hU : Disjoint U₁ U₂
L₁ : Set G
h1L₁ : L₁ ∈ 𝓝 1
h2L₁ : K₁.carrier * L₁ ⊆ U₁
V₁ : Set G
h1V₁ : V₁ ⊆ L₁
h2V₁ : IsOpen V₁
h3V₁ : 1 ∈ V₁
⊢ chaar K₀ (K₁ ⊔ K₂) = chaar K₀ K₁ + chaar K₀ K₂
[PROOFSTEP]
replace h2L₁ := Subset.trans (mul_subset_mul_left h1V₁) h2L₁
[GOAL]
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : Disjoint K₁.carrier K₂.carrier
U₁ U₂ : Set G
h1U₁ : IsOpen U₁
h1U₂ : IsOpen U₂
h2U₁ : K₁.carrier ⊆ U₁
h2U₂ : K₂.carrier ⊆ U₂
hU : Disjoint U₁ U₂
L₁ : Set G
h1L₁ : L₁ ∈ 𝓝 1
V₁ : Set G
h1V₁ : V₁ ⊆ L₁
h2V₁ : IsOpen V₁
h3V₁ : 1 ∈ V₁
h2L₁ : K₁.carrier * V₁ ⊆ U₁
⊢ chaar K₀ (K₁ ⊔ K₂) = chaar K₀ K₁ + chaar K₀ K₂
[PROOFSTEP]
rcases compact_open_separated_mul_right K₂.2 h1U₂ h2U₂ with ⟨L₂, h1L₂, h2L₂⟩
[GOAL]
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : Disjoint K₁.carrier K₂.carrier
U₁ U₂ : Set G
h1U₁ : IsOpen U₁
h1U₂ : IsOpen U₂
h2U₁ : K₁.carrier ⊆ U₁
h2U₂ : K₂.carrier ⊆ U₂
hU : Disjoint U₁ U₂
L₁ : Set G
h1L₁ : L₁ ∈ 𝓝 1
V₁ : Set G
h1V₁ : V₁ ⊆ L₁
h2V₁ : IsOpen V₁
h3V₁ : 1 ∈ V₁
h2L₁ : K₁.carrier * V₁ ⊆ U₁
L₂ : Set G
h1L₂ : L₂ ∈ 𝓝 1
h2L₂ : K₂.carrier * L₂ ⊆ U₂
⊢ chaar K₀ (K₁ ⊔ K₂) = chaar K₀ K₁ + chaar K₀ K₂
[PROOFSTEP]
rcases mem_nhds_iff.mp h1L₂ with ⟨V₂, h1V₂, h2V₂, h3V₂⟩
[GOAL]
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : Disjoint K₁.carrier K₂.carrier
U₁ U₂ : Set G
h1U₁ : IsOpen U₁
h1U₂ : IsOpen U₂
h2U₁ : K₁.carrier ⊆ U₁
h2U₂ : K₂.carrier ⊆ U₂
hU : Disjoint U₁ U₂
L₁ : Set G
h1L₁ : L₁ ∈ 𝓝 1
V₁ : Set G
h1V₁ : V₁ ⊆ L₁
h2V₁ : IsOpen V₁
h3V₁ : 1 ∈ V₁
h2L₁ : K₁.carrier * V₁ ⊆ U₁
L₂ : Set G
h1L₂ : L₂ ∈ 𝓝 1
h2L₂ : K₂.carrier * L₂ ⊆ U₂
V₂ : Set G
h1V₂ : V₂ ⊆ L₂
h2V₂ : IsOpen V₂
h3V₂ : 1 ∈ V₂
⊢ chaar K₀ (K₁ ⊔ K₂) = chaar K₀ K₁ + chaar K₀ K₂
[PROOFSTEP]
replace h2L₂ := Subset.trans (mul_subset_mul_left h1V₂) h2L₂
[GOAL]
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : Disjoint K₁.carrier K₂.carrier
U₁ U₂ : Set G
h1U₁ : IsOpen U₁
h1U₂ : IsOpen U₂
h2U₁ : K₁.carrier ⊆ U₁
h2U₂ : K₂.carrier ⊆ U₂
hU : Disjoint U₁ U₂
L₁ : Set G
h1L₁ : L₁ ∈ 𝓝 1
V₁ : Set G
h1V₁ : V₁ ⊆ L₁
h2V₁ : IsOpen V₁
h3V₁ : 1 ∈ V₁
h2L₁ : K₁.carrier * V₁ ⊆ U₁
L₂ : Set G
h1L₂ : L₂ ∈ 𝓝 1
V₂ : Set G
h1V₂ : V₂ ⊆ L₂
h2V₂ : IsOpen V₂
h3V₂ : 1 ∈ V₂
h2L₂ : K₂.carrier * V₂ ⊆ U₂
⊢ chaar K₀ (K₁ ⊔ K₂) = chaar K₀ K₁ + chaar K₀ K₂
[PROOFSTEP]
let eval : (Compacts G → ℝ) → ℝ := fun f => f K₁ + f K₂ - f (K₁ ⊔ K₂)
[GOAL]
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : Disjoint K₁.carrier K₂.carrier
U₁ U₂ : Set G
h1U₁ : IsOpen U₁
h1U₂ : IsOpen U₂
h2U₁ : K₁.carrier ⊆ U₁
h2U₂ : K₂.carrier ⊆ U₂
hU : Disjoint U₁ U₂
L₁ : Set G
h1L₁ : L₁ ∈ 𝓝 1
V₁ : Set G
h1V₁ : V₁ ⊆ L₁
h2V₁ : IsOpen V₁
h3V₁ : 1 ∈ V₁
h2L₁ : K₁.carrier * V₁ ⊆ U₁
L₂ : Set G
h1L₂ : L₂ ∈ 𝓝 1
V₂ : Set G
h1V₂ : V₂ ⊆ L₂
h2V₂ : IsOpen V₂
h3V₂ : 1 ∈ V₂
h2L₂ : K₂.carrier * V₂ ⊆ U₂
eval : (Compacts G → ℝ) → ℝ := fun f => f K₁ + f K₂ - f (K₁ ⊔ K₂)
⊢ chaar K₀ (K₁ ⊔ K₂) = chaar K₀ K₁ + chaar K₀ K₂
[PROOFSTEP]
have : Continuous eval := ((continuous_apply K₁).add (continuous_apply K₂)).sub (continuous_apply (K₁ ⊔ K₂))
[GOAL]
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : Disjoint K₁.carrier K₂.carrier
U₁ U₂ : Set G
h1U₁ : IsOpen U₁
h1U₂ : IsOpen U₂
h2U₁ : K₁.carrier ⊆ U₁
h2U₂ : K₂.carrier ⊆ U₂
hU : Disjoint U₁ U₂
L₁ : Set G
h1L₁ : L₁ ∈ 𝓝 1
V₁ : Set G
h1V₁ : V₁ ⊆ L₁
h2V₁ : IsOpen V₁
h3V₁ : 1 ∈ V₁
h2L₁ : K₁.carrier * V₁ ⊆ U₁
L₂ : Set G
h1L₂ : L₂ ∈ 𝓝 1
V₂ : Set G
h1V₂ : V₂ ⊆ L₂
h2V₂ : IsOpen V₂
h3V₂ : 1 ∈ V₂
h2L₂ : K₂.carrier * V₂ ⊆ U₂
eval : (Compacts G → ℝ) → ℝ := fun f => f K₁ + f K₂ - f (K₁ ⊔ K₂)
this : Continuous eval
⊢ chaar K₀ (K₁ ⊔ K₂) = chaar K₀ K₁ + chaar K₀ K₂
[PROOFSTEP]
rw [eq_comm, ← sub_eq_zero]
[GOAL]
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : Disjoint K₁.carrier K₂.carrier
U₁ U₂ : Set G
h1U₁ : IsOpen U₁
h1U₂ : IsOpen U₂
h2U₁ : K₁.carrier ⊆ U₁
h2U₂ : K₂.carrier ⊆ U₂
hU : Disjoint U₁ U₂
L₁ : Set G
h1L₁ : L₁ ∈ 𝓝 1
V₁ : Set G
h1V₁ : V₁ ⊆ L₁
h2V₁ : IsOpen V₁
h3V₁ : 1 ∈ V₁
h2L₁ : K₁.carrier * V₁ ⊆ U₁
L₂ : Set G
h1L₂ : L₂ ∈ 𝓝 1
V₂ : Set G
h1V₂ : V₂ ⊆ L₂
h2V₂ : IsOpen V₂
h3V₂ : 1 ∈ V₂
h2L₂ : K₂.carrier * V₂ ⊆ U₂
eval : (Compacts G → ℝ) → ℝ := fun f => f K₁ + f K₂ - f (K₁ ⊔ K₂)
this : Continuous eval
⊢ chaar K₀ K₁ + chaar K₀ K₂ - chaar K₀ (K₁ ⊔ K₂) = 0
[PROOFSTEP]
show chaar K₀ ∈ eval ⁻¹' {(0 : ℝ)}
[GOAL]
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : Disjoint K₁.carrier K₂.carrier
U₁ U₂ : Set G
h1U₁ : IsOpen U₁
h1U₂ : IsOpen U₂
h2U₁ : K₁.carrier ⊆ U₁
h2U₂ : K₂.carrier ⊆ U₂
hU : Disjoint U₁ U₂
L₁ : Set G
h1L₁ : L₁ ∈ 𝓝 1
V₁ : Set G
h1V₁ : V₁ ⊆ L₁
h2V₁ : IsOpen V₁
h3V₁ : 1 ∈ V₁
h2L₁ : K₁.carrier * V₁ ⊆ U₁
L₂ : Set G
h1L₂ : L₂ ∈ 𝓝 1
V₂ : Set G
h1V₂ : V₂ ⊆ L₂
h2V₂ : IsOpen V₂
h3V₂ : 1 ∈ V₂
h2L₂ : K₂.carrier * V₂ ⊆ U₂
eval : (Compacts G → ℝ) → ℝ := fun f => f K₁ + f K₂ - f (K₁ ⊔ K₂)
this : Continuous eval
⊢ chaar K₀ ∈ eval ⁻¹' {0}
[PROOFSTEP]
let V := V₁ ∩ V₂
[GOAL]
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : Disjoint K₁.carrier K₂.carrier
U₁ U₂ : Set G
h1U₁ : IsOpen U₁
h1U₂ : IsOpen U₂
h2U₁ : K₁.carrier ⊆ U₁
h2U₂ : K₂.carrier ⊆ U₂
hU : Disjoint U₁ U₂
L₁ : Set G
h1L₁ : L₁ ∈ 𝓝 1
V₁ : Set G
h1V₁ : V₁ ⊆ L₁
h2V₁ : IsOpen V₁
h3V₁ : 1 ∈ V₁
h2L₁ : K₁.carrier * V₁ ⊆ U₁
L₂ : Set G
h1L₂ : L₂ ∈ 𝓝 1
V₂ : Set G
h1V₂ : V₂ ⊆ L₂
h2V₂ : IsOpen V₂
h3V₂ : 1 ∈ V₂
h2L₂ : K₂.carrier * V₂ ⊆ U₂
eval : (Compacts G → ℝ) → ℝ := fun f => f K₁ + f K₂ - f (K₁ ⊔ K₂)
this : Continuous eval
V : Set G := V₁ ∩ V₂
⊢ chaar K₀ ∈ eval ⁻¹' {0}
[PROOFSTEP]
apply
  mem_of_subset_of_mem _
    (chaar_mem_clPrehaar K₀
      ⟨⟨V⁻¹, (h2V₁.inter h2V₂).preimage continuous_inv⟩, by
        simp only [mem_inv, inv_one, h3V₁, h3V₂, mem_inter_iff, true_and_iff]⟩)
[GOAL]
G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : Disjoint K₁.carrier K₂.carrier
U₁ U₂ : Set G
h1U₁ : IsOpen U₁
h1U₂ : IsOpen U₂
h2U₁ : K₁.carrier ⊆ U₁
h2U₂ : K₂.carrier ⊆ U₂
hU : Disjoint U₁ U₂
L₁ : Set G
h1L₁ : L₁ ∈ 𝓝 1
V₁ : Set G
h1V₁ : V₁ ⊆ L₁
h2V₁ : IsOpen V₁
h3V₁ : 1 ∈ V₁
h2L₁ : K₁.carrier * V₁ ⊆ U₁
L₂ : Set G
h1L₂ : L₂ ∈ 𝓝 1
V₂ : Set G
h1V₂ : V₂ ⊆ L₂
h2V₂ : IsOpen V₂
h3V₂ : 1 ∈ V₂
h2L₂ : K₂.carrier * V₂ ⊆ U₂
eval : (Compacts G → ℝ) → ℝ := fun f => f K₁ + f K₂ - f (K₁ ⊔ K₂)
this : Continuous eval
V : Set G := V₁ ∩ V₂
⊢ 1 ∈ { carrier := V⁻¹, is_open' := (_ : IsOpen ((fun a => a⁻¹) ⁻¹' (V₁ ∩ V₂))) }.carrier
[PROOFSTEP]
simp only [mem_inv, inv_one, h3V₁, h3V₂, mem_inter_iff, true_and_iff]
[GOAL]
G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : Disjoint K₁.carrier K₂.carrier
U₁ U₂ : Set G
h1U₁ : IsOpen U₁
h1U₂ : IsOpen U₂
h2U₁ : K₁.carrier ⊆ U₁
h2U₂ : K₂.carrier ⊆ U₂
hU : Disjoint U₁ U₂
L₁ : Set G
h1L₁ : L₁ ∈ 𝓝 1
V₁ : Set G
h1V₁ : V₁ ⊆ L₁
h2V₁ : IsOpen V₁
h3V₁ : 1 ∈ V₁
h2L₁ : K₁.carrier * V₁ ⊆ U₁
L₂ : Set G
h1L₂ : L₂ ∈ 𝓝 1
V₂ : Set G
h1V₂ : V₂ ⊆ L₂
h2V₂ : IsOpen V₂
h3V₂ : 1 ∈ V₂
h2L₂ : K₂.carrier * V₂ ⊆ U₂
eval : (Compacts G → ℝ) → ℝ := fun f => f K₁ + f K₂ - f (K₁ ⊔ K₂)
this : Continuous eval
V : Set G := V₁ ∩ V₂
⊢ clPrehaar ↑K₀
      { toOpens := { carrier := V⁻¹, is_open' := (_ : IsOpen ((fun a => a⁻¹) ⁻¹' (V₁ ∩ V₂))) },
        mem' := (_ : 1 ∈ (V₁ ∩ V₂)⁻¹) } ⊆
    eval ⁻¹' {0}
[PROOFSTEP]
unfold clPrehaar
[GOAL]
G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : Disjoint K₁.carrier K₂.carrier
U₁ U₂ : Set G
h1U₁ : IsOpen U₁
h1U₂ : IsOpen U₂
h2U₁ : K₁.carrier ⊆ U₁
h2U₂ : K₂.carrier ⊆ U₂
hU : Disjoint U₁ U₂
L₁ : Set G
h1L₁ : L₁ ∈ 𝓝 1
V₁ : Set G
h1V₁ : V₁ ⊆ L₁
h2V₁ : IsOpen V₁
h3V₁ : 1 ∈ V₁
h2L₁ : K₁.carrier * V₁ ⊆ U₁
L₂ : Set G
h1L₂ : L₂ ∈ 𝓝 1
V₂ : Set G
h1V₂ : V₂ ⊆ L₂
h2V₂ : IsOpen V₂
h3V₂ : 1 ∈ V₂
h2L₂ : K₂.carrier * V₂ ⊆ U₂
eval : (Compacts G → ℝ) → ℝ := fun f => f K₁ + f K₂ - f (K₁ ⊔ K₂)
this : Continuous eval
V : Set G := V₁ ∩ V₂
⊢ closure
      (prehaar ↑K₀ ''
        {U |
          U ⊆
              ↑{ toOpens := { carrier := V⁻¹, is_open' := (_ : IsOpen ((fun a => a⁻¹) ⁻¹' (V₁ ∩ V₂))) },
                    mem' := (_ : 1 ∈ (V₁ ∩ V₂)⁻¹) }.toOpens ∧
            IsOpen U ∧ 1 ∈ U}) ⊆
    eval ⁻¹' {0}
[PROOFSTEP]
rw [IsClosed.closure_subset_iff]
[GOAL]
G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : Disjoint K₁.carrier K₂.carrier
U₁ U₂ : Set G
h1U₁ : IsOpen U₁
h1U₂ : IsOpen U₂
h2U₁ : K₁.carrier ⊆ U₁
h2U₂ : K₂.carrier ⊆ U₂
hU : Disjoint U₁ U₂
L₁ : Set G
h1L₁ : L₁ ∈ 𝓝 1
V₁ : Set G
h1V₁ : V₁ ⊆ L₁
h2V₁ : IsOpen V₁
h3V₁ : 1 ∈ V₁
h2L₁ : K₁.carrier * V₁ ⊆ U₁
L₂ : Set G
h1L₂ : L₂ ∈ 𝓝 1
V₂ : Set G
h1V₂ : V₂ ⊆ L₂
h2V₂ : IsOpen V₂
h3V₂ : 1 ∈ V₂
h2L₂ : K₂.carrier * V₂ ⊆ U₂
eval : (Compacts G → ℝ) → ℝ := fun f => f K₁ + f K₂ - f (K₁ ⊔ K₂)
this : Continuous eval
V : Set G := V₁ ∩ V₂
⊢ prehaar ↑K₀ ''
      {U |
        U ⊆
            ↑{ toOpens := { carrier := V⁻¹, is_open' := (_ : IsOpen ((fun a => a⁻¹) ⁻¹' (V₁ ∩ V₂))) },
                  mem' := (_ : 1 ∈ (V₁ ∩ V₂)⁻¹) }.toOpens ∧
          IsOpen U ∧ 1 ∈ U} ⊆
    eval ⁻¹' {0}
[PROOFSTEP]
rintro _ ⟨U, ⟨h1U, h2U, h3U⟩, rfl⟩
[GOAL]
case intro.intro.intro.intro
G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : Disjoint K₁.carrier K₂.carrier
U₁ U₂ : Set G
h1U₁ : IsOpen U₁
h1U₂ : IsOpen U₂
h2U₁ : K₁.carrier ⊆ U₁
h2U₂ : K₂.carrier ⊆ U₂
hU : Disjoint U₁ U₂
L₁ : Set G
h1L₁ : L₁ ∈ 𝓝 1
V₁ : Set G
h1V₁ : V₁ ⊆ L₁
h2V₁ : IsOpen V₁
h3V₁ : 1 ∈ V₁
h2L₁ : K₁.carrier * V₁ ⊆ U₁
L₂ : Set G
h1L₂ : L₂ ∈ 𝓝 1
V₂ : Set G
h1V₂ : V₂ ⊆ L₂
h2V₂ : IsOpen V₂
h3V₂ : 1 ∈ V₂
h2L₂ : K₂.carrier * V₂ ⊆ U₂
eval : (Compacts G → ℝ) → ℝ := fun f => f K₁ + f K₂ - f (K₁ ⊔ K₂)
this : Continuous eval
V : Set G := V₁ ∩ V₂
U : Set G
h1U :
  U ⊆
    ↑{ toOpens := { carrier := V⁻¹, is_open' := (_ : IsOpen ((fun a => a⁻¹) ⁻¹' (V₁ ∩ V₂))) },
          mem' := (_ : 1 ∈ (V₁ ∩ V₂)⁻¹) }.toOpens
h2U : IsOpen U
h3U : 1 ∈ U
⊢ prehaar (↑K₀) U ∈ eval ⁻¹' {0}
[PROOFSTEP]
simp only [mem_preimage, sub_eq_zero, mem_singleton_iff]
[GOAL]
case intro.intro.intro.intro
G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : Disjoint K₁.carrier K₂.carrier
U₁ U₂ : Set G
h1U₁ : IsOpen U₁
h1U₂ : IsOpen U₂
h2U₁ : K₁.carrier ⊆ U₁
h2U₂ : K₂.carrier ⊆ U₂
hU : Disjoint U₁ U₂
L₁ : Set G
h1L₁ : L₁ ∈ 𝓝 1
V₁ : Set G
h1V₁ : V₁ ⊆ L₁
h2V₁ : IsOpen V₁
h3V₁ : 1 ∈ V₁
h2L₁ : K₁.carrier * V₁ ⊆ U₁
L₂ : Set G
h1L₂ : L₂ ∈ 𝓝 1
V₂ : Set G
h1V₂ : V₂ ⊆ L₂
h2V₂ : IsOpen V₂
h3V₂ : 1 ∈ V₂
h2L₂ : K₂.carrier * V₂ ⊆ U₂
eval : (Compacts G → ℝ) → ℝ := fun f => f K₁ + f K₂ - f (K₁ ⊔ K₂)
this : Continuous eval
V : Set G := V₁ ∩ V₂
U : Set G
h1U :
  U ⊆
    ↑{ toOpens := { carrier := V⁻¹, is_open' := (_ : IsOpen ((fun a => a⁻¹) ⁻¹' (V₁ ∩ V₂))) },
          mem' := (_ : 1 ∈ (V₁ ∩ V₂)⁻¹) }.toOpens
h2U : IsOpen U
h3U : 1 ∈ U
⊢ prehaar (↑K₀) U K₁ + prehaar (↑K₀) U K₂ = prehaar (↑K₀) U (K₁ ⊔ K₂)
[PROOFSTEP]
rw [eq_comm]
[GOAL]
case intro.intro.intro.intro
G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : Disjoint K₁.carrier K₂.carrier
U₁ U₂ : Set G
h1U₁ : IsOpen U₁
h1U₂ : IsOpen U₂
h2U₁ : K₁.carrier ⊆ U₁
h2U₂ : K₂.carrier ⊆ U₂
hU : Disjoint U₁ U₂
L₁ : Set G
h1L₁ : L₁ ∈ 𝓝 1
V₁ : Set G
h1V₁ : V₁ ⊆ L₁
h2V₁ : IsOpen V₁
h3V₁ : 1 ∈ V₁
h2L₁ : K₁.carrier * V₁ ⊆ U₁
L₂ : Set G
h1L₂ : L₂ ∈ 𝓝 1
V₂ : Set G
h1V₂ : V₂ ⊆ L₂
h2V₂ : IsOpen V₂
h3V₂ : 1 ∈ V₂
h2L₂ : K₂.carrier * V₂ ⊆ U₂
eval : (Compacts G → ℝ) → ℝ := fun f => f K₁ + f K₂ - f (K₁ ⊔ K₂)
this : Continuous eval
V : Set G := V₁ ∩ V₂
U : Set G
h1U :
  U ⊆
    ↑{ toOpens := { carrier := V⁻¹, is_open' := (_ : IsOpen ((fun a => a⁻¹) ⁻¹' (V₁ ∩ V₂))) },
          mem' := (_ : 1 ∈ (V₁ ∩ V₂)⁻¹) }.toOpens
h2U : IsOpen U
h3U : 1 ∈ U
⊢ prehaar (↑K₀) U (K₁ ⊔ K₂) = prehaar (↑K₀) U K₁ + prehaar (↑K₀) U K₂
[PROOFSTEP]
apply prehaar_sup_eq
[GOAL]
case intro.intro.intro.intro.hU
G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : Disjoint K₁.carrier K₂.carrier
U₁ U₂ : Set G
h1U₁ : IsOpen U₁
h1U₂ : IsOpen U₂
h2U₁ : K₁.carrier ⊆ U₁
h2U₂ : K₂.carrier ⊆ U₂
hU : Disjoint U₁ U₂
L₁ : Set G
h1L₁ : L₁ ∈ 𝓝 1
V₁ : Set G
h1V₁ : V₁ ⊆ L₁
h2V₁ : IsOpen V₁
h3V₁ : 1 ∈ V₁
h2L₁ : K₁.carrier * V₁ ⊆ U₁
L₂ : Set G
h1L₂ : L₂ ∈ 𝓝 1
V₂ : Set G
h1V₂ : V₂ ⊆ L₂
h2V₂ : IsOpen V₂
h3V₂ : 1 ∈ V₂
h2L₂ : K₂.carrier * V₂ ⊆ U₂
eval : (Compacts G → ℝ) → ℝ := fun f => f K₁ + f K₂ - f (K₁ ⊔ K₂)
this : Continuous eval
V : Set G := V₁ ∩ V₂
U : Set G
h1U :
  U ⊆
    ↑{ toOpens := { carrier := V⁻¹, is_open' := (_ : IsOpen ((fun a => a⁻¹) ⁻¹' (V₁ ∩ V₂))) },
          mem' := (_ : 1 ∈ (V₁ ∩ V₂)⁻¹) }.toOpens
h2U : IsOpen U
h3U : 1 ∈ U
⊢ Set.Nonempty (interior U)
[PROOFSTEP]
rw [h2U.interior_eq]
[GOAL]
case intro.intro.intro.intro.hU
G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : Disjoint K₁.carrier K₂.carrier
U₁ U₂ : Set G
h1U₁ : IsOpen U₁
h1U₂ : IsOpen U₂
h2U₁ : K₁.carrier ⊆ U₁
h2U₂ : K₂.carrier ⊆ U₂
hU : Disjoint U₁ U₂
L₁ : Set G
h1L₁ : L₁ ∈ 𝓝 1
V₁ : Set G
h1V₁ : V₁ ⊆ L₁
h2V₁ : IsOpen V₁
h3V₁ : 1 ∈ V₁
h2L₁ : K₁.carrier * V₁ ⊆ U₁
L₂ : Set G
h1L₂ : L₂ ∈ 𝓝 1
V₂ : Set G
h1V₂ : V₂ ⊆ L₂
h2V₂ : IsOpen V₂
h3V₂ : 1 ∈ V₂
h2L₂ : K₂.carrier * V₂ ⊆ U₂
eval : (Compacts G → ℝ) → ℝ := fun f => f K₁ + f K₂ - f (K₁ ⊔ K₂)
this : Continuous eval
V : Set G := V₁ ∩ V₂
U : Set G
h1U :
  U ⊆
    ↑{ toOpens := { carrier := V⁻¹, is_open' := (_ : IsOpen ((fun a => a⁻¹) ⁻¹' (V₁ ∩ V₂))) },
          mem' := (_ : 1 ∈ (V₁ ∩ V₂)⁻¹) }.toOpens
h2U : IsOpen U
h3U : 1 ∈ U
⊢ Set.Nonempty U
[PROOFSTEP]
exact ⟨1, h3U⟩
[GOAL]
case intro.intro.intro.intro.h
G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : Disjoint K₁.carrier K₂.carrier
U₁ U₂ : Set G
h1U₁ : IsOpen U₁
h1U₂ : IsOpen U₂
h2U₁ : K₁.carrier ⊆ U₁
h2U₂ : K₂.carrier ⊆ U₂
hU : Disjoint U₁ U₂
L₁ : Set G
h1L₁ : L₁ ∈ 𝓝 1
V₁ : Set G
h1V₁ : V₁ ⊆ L₁
h2V₁ : IsOpen V₁
h3V₁ : 1 ∈ V₁
h2L₁ : K₁.carrier * V₁ ⊆ U₁
L₂ : Set G
h1L₂ : L₂ ∈ 𝓝 1
V₂ : Set G
h1V₂ : V₂ ⊆ L₂
h2V₂ : IsOpen V₂
h3V₂ : 1 ∈ V₂
h2L₂ : K₂.carrier * V₂ ⊆ U₂
eval : (Compacts G → ℝ) → ℝ := fun f => f K₁ + f K₂ - f (K₁ ⊔ K₂)
this : Continuous eval
V : Set G := V₁ ∩ V₂
U : Set G
h1U :
  U ⊆
    ↑{ toOpens := { carrier := V⁻¹, is_open' := (_ : IsOpen ((fun a => a⁻¹) ⁻¹' (V₁ ∩ V₂))) },
          mem' := (_ : 1 ∈ (V₁ ∩ V₂)⁻¹) }.toOpens
h2U : IsOpen U
h3U : 1 ∈ U
⊢ Disjoint (K₁.carrier * U⁻¹) (K₂.carrier * U⁻¹)
[PROOFSTEP]
refine' disjoint_of_subset _ _ hU
[GOAL]
case intro.intro.intro.intro.h.refine'_1
G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : Disjoint K₁.carrier K₂.carrier
U₁ U₂ : Set G
h1U₁ : IsOpen U₁
h1U₂ : IsOpen U₂
h2U₁ : K₁.carrier ⊆ U₁
h2U₂ : K₂.carrier ⊆ U₂
hU : Disjoint U₁ U₂
L₁ : Set G
h1L₁ : L₁ ∈ 𝓝 1
V₁ : Set G
h1V₁ : V₁ ⊆ L₁
h2V₁ : IsOpen V₁
h3V₁ : 1 ∈ V₁
h2L₁ : K₁.carrier * V₁ ⊆ U₁
L₂ : Set G
h1L₂ : L₂ ∈ 𝓝 1
V₂ : Set G
h1V₂ : V₂ ⊆ L₂
h2V₂ : IsOpen V₂
h3V₂ : 1 ∈ V₂
h2L₂ : K₂.carrier * V₂ ⊆ U₂
eval : (Compacts G → ℝ) → ℝ := fun f => f K₁ + f K₂ - f (K₁ ⊔ K₂)
this : Continuous eval
V : Set G := V₁ ∩ V₂
U : Set G
h1U :
  U ⊆
    ↑{ toOpens := { carrier := V⁻¹, is_open' := (_ : IsOpen ((fun a => a⁻¹) ⁻¹' (V₁ ∩ V₂))) },
          mem' := (_ : 1 ∈ (V₁ ∩ V₂)⁻¹) }.toOpens
h2U : IsOpen U
h3U : 1 ∈ U
⊢ K₁.carrier * U⁻¹ ⊆ U₁
[PROOFSTEP]
refine' Subset.trans (mul_subset_mul Subset.rfl _) h2L₁
[GOAL]
case intro.intro.intro.intro.h.refine'_1
G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : Disjoint K₁.carrier K₂.carrier
U₁ U₂ : Set G
h1U₁ : IsOpen U₁
h1U₂ : IsOpen U₂
h2U₁ : K₁.carrier ⊆ U₁
h2U₂ : K₂.carrier ⊆ U₂
hU : Disjoint U₁ U₂
L₁ : Set G
h1L₁ : L₁ ∈ 𝓝 1
V₁ : Set G
h1V₁ : V₁ ⊆ L₁
h2V₁ : IsOpen V₁
h3V₁ : 1 ∈ V₁
h2L₁ : K₁.carrier * V₁ ⊆ U₁
L₂ : Set G
h1L₂ : L₂ ∈ 𝓝 1
V₂ : Set G
h1V₂ : V₂ ⊆ L₂
h2V₂ : IsOpen V₂
h3V₂ : 1 ∈ V₂
h2L₂ : K₂.carrier * V₂ ⊆ U₂
eval : (Compacts G → ℝ) → ℝ := fun f => f K₁ + f K₂ - f (K₁ ⊔ K₂)
this : Continuous eval
V : Set G := V₁ ∩ V₂
U : Set G
h1U :
  U ⊆
    ↑{ toOpens := { carrier := V⁻¹, is_open' := (_ : IsOpen ((fun a => a⁻¹) ⁻¹' (V₁ ∩ V₂))) },
          mem' := (_ : 1 ∈ (V₁ ∩ V₂)⁻¹) }.toOpens
h2U : IsOpen U
h3U : 1 ∈ U
⊢ U⁻¹ ⊆ V₁
[PROOFSTEP]
exact Subset.trans (inv_subset.mpr h1U) (inter_subset_left _ _)
[GOAL]
case intro.intro.intro.intro.h.refine'_2
G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : Disjoint K₁.carrier K₂.carrier
U₁ U₂ : Set G
h1U₁ : IsOpen U₁
h1U₂ : IsOpen U₂
h2U₁ : K₁.carrier ⊆ U₁
h2U₂ : K₂.carrier ⊆ U₂
hU : Disjoint U₁ U₂
L₁ : Set G
h1L₁ : L₁ ∈ 𝓝 1
V₁ : Set G
h1V₁ : V₁ ⊆ L₁
h2V₁ : IsOpen V₁
h3V₁ : 1 ∈ V₁
h2L₁ : K₁.carrier * V₁ ⊆ U₁
L₂ : Set G
h1L₂ : L₂ ∈ 𝓝 1
V₂ : Set G
h1V₂ : V₂ ⊆ L₂
h2V₂ : IsOpen V₂
h3V₂ : 1 ∈ V₂
h2L₂ : K₂.carrier * V₂ ⊆ U₂
eval : (Compacts G → ℝ) → ℝ := fun f => f K₁ + f K₂ - f (K₁ ⊔ K₂)
this : Continuous eval
V : Set G := V₁ ∩ V₂
U : Set G
h1U :
  U ⊆
    ↑{ toOpens := { carrier := V⁻¹, is_open' := (_ : IsOpen ((fun a => a⁻¹) ⁻¹' (V₁ ∩ V₂))) },
          mem' := (_ : 1 ∈ (V₁ ∩ V₂)⁻¹) }.toOpens
h2U : IsOpen U
h3U : 1 ∈ U
⊢ K₂.carrier * U⁻¹ ⊆ U₂
[PROOFSTEP]
refine' Subset.trans (mul_subset_mul Subset.rfl _) h2L₂
[GOAL]
case intro.intro.intro.intro.h.refine'_2
G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : Disjoint K₁.carrier K₂.carrier
U₁ U₂ : Set G
h1U₁ : IsOpen U₁
h1U₂ : IsOpen U₂
h2U₁ : K₁.carrier ⊆ U₁
h2U₂ : K₂.carrier ⊆ U₂
hU : Disjoint U₁ U₂
L₁ : Set G
h1L₁ : L₁ ∈ 𝓝 1
V₁ : Set G
h1V₁ : V₁ ⊆ L₁
h2V₁ : IsOpen V₁
h3V₁ : 1 ∈ V₁
h2L₁ : K₁.carrier * V₁ ⊆ U₁
L₂ : Set G
h1L₂ : L₂ ∈ 𝓝 1
V₂ : Set G
h1V₂ : V₂ ⊆ L₂
h2V₂ : IsOpen V₂
h3V₂ : 1 ∈ V₂
h2L₂ : K₂.carrier * V₂ ⊆ U₂
eval : (Compacts G → ℝ) → ℝ := fun f => f K₁ + f K₂ - f (K₁ ⊔ K₂)
this : Continuous eval
V : Set G := V₁ ∩ V₂
U : Set G
h1U :
  U ⊆
    ↑{ toOpens := { carrier := V⁻¹, is_open' := (_ : IsOpen ((fun a => a⁻¹) ⁻¹' (V₁ ∩ V₂))) },
          mem' := (_ : 1 ∈ (V₁ ∩ V₂)⁻¹) }.toOpens
h2U : IsOpen U
h3U : 1 ∈ U
⊢ U⁻¹ ⊆ V₂
[PROOFSTEP]
exact Subset.trans (inv_subset.mpr h1U) (inter_subset_right _ _)
[GOAL]
G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : Disjoint K₁.carrier K₂.carrier
U₁ U₂ : Set G
h1U₁ : IsOpen U₁
h1U₂ : IsOpen U₂
h2U₁ : K₁.carrier ⊆ U₁
h2U₂ : K₂.carrier ⊆ U₂
hU : Disjoint U₁ U₂
L₁ : Set G
h1L₁ : L₁ ∈ 𝓝 1
V₁ : Set G
h1V₁ : V₁ ⊆ L₁
h2V₁ : IsOpen V₁
h3V₁ : 1 ∈ V₁
h2L₁ : K₁.carrier * V₁ ⊆ U₁
L₂ : Set G
h1L₂ : L₂ ∈ 𝓝 1
V₂ : Set G
h1V₂ : V₂ ⊆ L₂
h2V₂ : IsOpen V₂
h3V₂ : 1 ∈ V₂
h2L₂ : K₂.carrier * V₂ ⊆ U₂
eval : (Compacts G → ℝ) → ℝ := fun f => f K₁ + f K₂ - f (K₁ ⊔ K₂)
this : Continuous eval
V : Set G := V₁ ∩ V₂
⊢ IsClosed (eval ⁻¹' {0})
[PROOFSTEP]
apply continuous_iff_isClosed.mp this
[GOAL]
case a
G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : Disjoint K₁.carrier K₂.carrier
U₁ U₂ : Set G
h1U₁ : IsOpen U₁
h1U₂ : IsOpen U₂
h2U₁ : K₁.carrier ⊆ U₁
h2U₂ : K₂.carrier ⊆ U₂
hU : Disjoint U₁ U₂
L₁ : Set G
h1L₁ : L₁ ∈ 𝓝 1
V₁ : Set G
h1V₁ : V₁ ⊆ L₁
h2V₁ : IsOpen V₁
h3V₁ : 1 ∈ V₁
h2L₁ : K₁.carrier * V₁ ⊆ U₁
L₂ : Set G
h1L₂ : L₂ ∈ 𝓝 1
V₂ : Set G
h1V₂ : V₂ ⊆ L₂
h2V₂ : IsOpen V₂
h3V₂ : 1 ∈ V₂
h2L₂ : K₂.carrier * V₂ ⊆ U₂
eval : (Compacts G → ℝ) → ℝ := fun f => f K₁ + f K₂ - f (K₁ ⊔ K₂)
this : Continuous eval
V : Set G := V₁ ∩ V₂
⊢ IsClosed {0}
[PROOFSTEP]
exact isClosed_singleton
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
g : G
K : Compacts G
⊢ chaar K₀ (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) = chaar K₀ K
[PROOFSTEP]
let eval : (Compacts G → ℝ) → ℝ := fun f => f (K.map _ <| continuous_mul_left g) - f K
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
g : G
K : Compacts G
eval : (Compacts G → ℝ) → ℝ := fun f => f (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) - f K
⊢ chaar K₀ (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) = chaar K₀ K
[PROOFSTEP]
have : Continuous eval := (continuous_apply (K.map _ _)).sub (continuous_apply K)
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
g : G
K : Compacts G
eval : (Compacts G → ℝ) → ℝ := fun f => f (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) - f K
this : Continuous eval
⊢ chaar K₀ (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) = chaar K₀ K
[PROOFSTEP]
rw [← sub_eq_zero]
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
g : G
K : Compacts G
eval : (Compacts G → ℝ) → ℝ := fun f => f (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) - f K
this : Continuous eval
⊢ chaar K₀ (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) - chaar K₀ K = 0
[PROOFSTEP]
show chaar K₀ ∈ eval ⁻¹' {(0 : ℝ)}
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
g : G
K : Compacts G
eval : (Compacts G → ℝ) → ℝ := fun f => f (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) - f K
this : Continuous eval
⊢ chaar K₀ ∈ eval ⁻¹' {0}
[PROOFSTEP]
apply mem_of_subset_of_mem _ (chaar_mem_clPrehaar K₀ ⊤)
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
g : G
K : Compacts G
eval : (Compacts G → ℝ) → ℝ := fun f => f (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) - f K
this : Continuous eval
⊢ clPrehaar ↑K₀ ⊤ ⊆ eval ⁻¹' {0}
[PROOFSTEP]
unfold clPrehaar
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
g : G
K : Compacts G
eval : (Compacts G → ℝ) → ℝ := fun f => f (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) - f K
this : Continuous eval
⊢ closure (prehaar ↑K₀ '' {U | U ⊆ ↑⊤.toOpens ∧ IsOpen U ∧ 1 ∈ U}) ⊆ eval ⁻¹' {0}
[PROOFSTEP]
rw [IsClosed.closure_subset_iff]
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
g : G
K : Compacts G
eval : (Compacts G → ℝ) → ℝ := fun f => f (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) - f K
this : Continuous eval
⊢ prehaar ↑K₀ '' {U | U ⊆ ↑⊤.toOpens ∧ IsOpen U ∧ 1 ∈ U} ⊆ eval ⁻¹' {0}
[PROOFSTEP]
rintro _ ⟨U, ⟨_, h2U, h3U⟩, rfl⟩
[GOAL]
case intro.intro.intro.intro
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
g : G
K : Compacts G
eval : (Compacts G → ℝ) → ℝ := fun f => f (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) - f K
this : Continuous eval
U : Set G
left✝ : U ⊆ ↑⊤.toOpens
h2U : IsOpen U
h3U : 1 ∈ U
⊢ prehaar (↑K₀) U ∈ eval ⁻¹' {0}
[PROOFSTEP]
simp only [mem_singleton_iff, mem_preimage, sub_eq_zero]
[GOAL]
case intro.intro.intro.intro
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
g : G
K : Compacts G
eval : (Compacts G → ℝ) → ℝ := fun f => f (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) - f K
this : Continuous eval
U : Set G
left✝ : U ⊆ ↑⊤.toOpens
h2U : IsOpen U
h3U : 1 ∈ U
⊢ prehaar (↑K₀) U (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) = prehaar (↑K₀) U K
[PROOFSTEP]
apply is_left_invariant_prehaar
[GOAL]
case intro.intro.intro.intro.hU
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
g : G
K : Compacts G
eval : (Compacts G → ℝ) → ℝ := fun f => f (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) - f K
this : Continuous eval
U : Set G
left✝ : U ⊆ ↑⊤.toOpens
h2U : IsOpen U
h3U : 1 ∈ U
⊢ Set.Nonempty (interior U)
[PROOFSTEP]
rw [h2U.interior_eq]
[GOAL]
case intro.intro.intro.intro.hU
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
g : G
K : Compacts G
eval : (Compacts G → ℝ) → ℝ := fun f => f (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) - f K
this : Continuous eval
U : Set G
left✝ : U ⊆ ↑⊤.toOpens
h2U : IsOpen U
h3U : 1 ∈ U
⊢ Set.Nonempty U
[PROOFSTEP]
exact ⟨1, h3U⟩
[GOAL]
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
g : G
K : Compacts G
eval : (Compacts G → ℝ) → ℝ := fun f => f (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) - f K
this : Continuous eval
⊢ IsClosed (eval ⁻¹' {0})
[PROOFSTEP]
apply continuous_iff_isClosed.mp this
[GOAL]
case a
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : TopologicalGroup G
K₀ : PositiveCompacts G
g : G
K : Compacts G
eval : (Compacts G → ℝ) → ℝ := fun f => f (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) - f K
this : Continuous eval
⊢ IsClosed {0}
[PROOFSTEP]
exact isClosed_singleton
[GOAL]
G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : ↑K₁ ⊆ ↑K₂
⊢ (fun K => { val := chaar K₀ K, property := (_ : 0 ≤ chaar K₀ K) }) K₁ ≤
    (fun K => { val := chaar K₀ K, property := (_ : 0 ≤ chaar K₀ K) }) K₂
[PROOFSTEP]
simp only [← NNReal.coe_le_coe, NNReal.toReal, chaar_mono, h]
[GOAL]
G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : Disjoint ↑K₁ ↑K₂
⊢ (fun K => { val := chaar K₀ K, property := (_ : 0 ≤ chaar K₀ K) }) (K₁ ⊔ K₂) =
    (fun K => { val := chaar K₀ K, property := (_ : 0 ≤ chaar K₀ K) }) K₁ +
      (fun K => { val := chaar K₀ K, property := (_ : 0 ≤ chaar K₀ K) }) K₂
[PROOFSTEP]
simp only [chaar_sup_eq h]
[GOAL]
G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
h : Disjoint ↑K₁ ↑K₂
⊢ { val := chaar K₀ K₁ + chaar K₀ K₂, property := (_ : (fun r => 0 ≤ r) (chaar K₀ K₁ + chaar K₀ K₂)) } =
    { val := chaar K₀ K₁, property := (_ : 0 ≤ chaar K₀ K₁) } +
      { val := chaar K₀ K₂, property := (_ : 0 ≤ chaar K₀ K₂) }
[PROOFSTEP]
rfl
[GOAL]
G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
⊢ (fun K => { val := chaar K₀ K, property := (_ : 0 ≤ chaar K₀ K) }) (K₁ ⊔ K₂) ≤
    (fun K => { val := chaar K₀ K, property := (_ : 0 ≤ chaar K₀ K) }) K₁ +
      (fun K => { val := chaar K₀ K, property := (_ : 0 ≤ chaar K₀ K) }) K₂
[PROOFSTEP]
simp only [← NNReal.coe_le_coe, NNReal.coe_add]
[GOAL]
G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K₀ : PositiveCompacts G
K₁ K₂ : Compacts G
⊢ ↑{ val := chaar K₀ (K₁ ⊔ K₂), property := (_ : 0 ≤ chaar K₀ (K₁ ⊔ K₂)) } ≤
    ↑{ val := chaar K₀ K₁, property := (_ : 0 ≤ chaar K₀ K₁) } +
      ↑{ val := chaar K₀ K₂, property := (_ : 0 ≤ chaar K₀ K₂) }
[PROOFSTEP]
simp only [NNReal.toReal, chaar_sup_le]
[GOAL]
G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K₀ : PositiveCompacts G
⊢ (fun s => ↑(Content.toFun (haarContent K₀) s)) K₀.toCompacts = 1
[PROOFSTEP]
simp_rw [← ENNReal.coe_one, haarContent_apply, ENNReal.coe_eq_coe, chaar_self]
[GOAL]
G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K₀ : PositiveCompacts G
⊢ { val := 1, property := (_ : (fun r => 0 ≤ r) 1) } = 1
[PROOFSTEP]
rfl
[GOAL]
G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K₀ : PositiveCompacts G
g : G
K : Compacts G
⊢ (fun s => ↑(Content.toFun (haarContent K₀) s)) (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) =
    (fun s => ↑(Content.toFun (haarContent K₀) s)) K
[PROOFSTEP]
simpa only [ENNReal.coe_eq_coe, ← NNReal.coe_eq, haarContent_apply] using is_left_invariant_chaar g K
[GOAL]
G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K₀ : PositiveCompacts G
⊢ 0 < ↑(Content.outerMeasure (haarContent K₀)) ↑K₀
[PROOFSTEP]
refine' zero_lt_one.trans_le _
[GOAL]
G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K₀ : PositiveCompacts G
⊢ 1 ≤ ↑(Content.outerMeasure (haarContent K₀)) ↑K₀
[PROOFSTEP]
rw [Content.outerMeasure_eq_iInf]
[GOAL]
G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K₀ : PositiveCompacts G
⊢ 1 ≤
    ⨅ (U : Set G) (hU : IsOpen U) (_ : ↑K₀ ⊆ U), Content.innerContent (haarContent K₀) { carrier := U, is_open' := hU }
[PROOFSTEP]
refine' le_iInf₂ fun U hU => le_iInf fun hK₀ => le_trans _ <| le_iSup₂ K₀.toCompacts hK₀
[GOAL]
G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K₀ : PositiveCompacts G
U : Set G
hU : IsOpen U
hK₀ : ↑K₀ ⊆ U
⊢ 1 ≤ (fun s => ↑(Content.toFun (haarContent K₀) s)) K₀.toCompacts
[PROOFSTEP]
exact haarContent_self.ge
[GOAL]
G : Type u_1
inst✝⁵ : Group G
inst✝⁴ : TopologicalSpace G
inst✝³ : T2Space G
inst✝² : TopologicalGroup G
inst✝¹ : MeasurableSpace G
inst✝ : BorelSpace G
K₀ : PositiveCompacts G
s : Set G
hs : MeasurableSet s
⊢ ↑↑(haarMeasure K₀) s = ↑(Content.outerMeasure (haarContent K₀)) s / ↑(Content.outerMeasure (haarContent K₀)) ↑K₀
[PROOFSTEP]
change ((haarContent K₀).outerMeasure K₀)⁻¹ * (haarContent K₀).measure s = _
[GOAL]
G : Type u_1
inst✝⁵ : Group G
inst✝⁴ : TopologicalSpace G
inst✝³ : T2Space G
inst✝² : TopologicalGroup G
inst✝¹ : MeasurableSpace G
inst✝ : BorelSpace G
K₀ : PositiveCompacts G
s : Set G
hs : MeasurableSet s
⊢ (↑(Content.outerMeasure (haarContent K₀)) ↑K₀)⁻¹ * ↑↑(Content.measure (haarContent K₀)) s =
    ↑(Content.outerMeasure (haarContent K₀)) s / ↑(Content.outerMeasure (haarContent K₀)) ↑K₀
[PROOFSTEP]
simp only [hs, div_eq_mul_inv, mul_comm, Content.measure_apply]
[GOAL]
G : Type u_1
inst✝⁵ : Group G
inst✝⁴ : TopologicalSpace G
inst✝³ : T2Space G
inst✝² : TopologicalGroup G
inst✝¹ : MeasurableSpace G
inst✝ : BorelSpace G
K₀ : PositiveCompacts G
⊢ IsMulLeftInvariant (haarMeasure K₀)
[PROOFSTEP]
rw [← forall_measure_preimage_mul_iff]
[GOAL]
G : Type u_1
inst✝⁵ : Group G
inst✝⁴ : TopologicalSpace G
inst✝³ : T2Space G
inst✝² : TopologicalGroup G
inst✝¹ : MeasurableSpace G
inst✝ : BorelSpace G
K₀ : PositiveCompacts G
⊢ ∀ (g : G) (A : Set G), MeasurableSet A → ↑↑(haarMeasure K₀) ((fun h => g * h) ⁻¹' A) = ↑↑(haarMeasure K₀) A
[PROOFSTEP]
intro g A hA
[GOAL]
G : Type u_1
inst✝⁵ : Group G
inst✝⁴ : TopologicalSpace G
inst✝³ : T2Space G
inst✝² : TopologicalGroup G
inst✝¹ : MeasurableSpace G
inst✝ : BorelSpace G
K₀ : PositiveCompacts G
g : G
A : Set G
hA : MeasurableSet A
⊢ ↑↑(haarMeasure K₀) ((fun h => g * h) ⁻¹' A) = ↑↑(haarMeasure K₀) A
[PROOFSTEP]
rw [haarMeasure_apply hA, haarMeasure_apply (measurable_const_mul g hA)]
  -- Porting note: Here was `congr 1`, but `to_additive` failed to generate a theorem.
[GOAL]
G : Type u_1
inst✝⁵ : Group G
inst✝⁴ : TopologicalSpace G
inst✝³ : T2Space G
inst✝² : TopologicalGroup G
inst✝¹ : MeasurableSpace G
inst✝ : BorelSpace G
K₀ : PositiveCompacts G
g : G
A : Set G
hA : MeasurableSet A
⊢ ↑(Content.outerMeasure (haarContent K₀)) ((fun x => g * x) ⁻¹' A) / ↑(Content.outerMeasure (haarContent K₀)) ↑K₀ =
    ↑(Content.outerMeasure (haarContent K₀)) A / ↑(Content.outerMeasure (haarContent K₀)) ↑K₀
[PROOFSTEP]
refine congr_arg (fun x : ℝ≥0∞ => x / (haarContent K₀).outerMeasure K₀) ?_
[GOAL]
G : Type u_1
inst✝⁵ : Group G
inst✝⁴ : TopologicalSpace G
inst✝³ : T2Space G
inst✝² : TopologicalGroup G
inst✝¹ : MeasurableSpace G
inst✝ : BorelSpace G
K₀ : PositiveCompacts G
g : G
A : Set G
hA : MeasurableSet A
⊢ ↑(Content.outerMeasure (haarContent K₀)) ((fun x => g * x) ⁻¹' A) = ↑(Content.outerMeasure (haarContent K₀)) A
[PROOFSTEP]
apply Content.is_mul_left_invariant_outerMeasure
[GOAL]
case h
G : Type u_1
inst✝⁵ : Group G
inst✝⁴ : TopologicalSpace G
inst✝³ : T2Space G
inst✝² : TopologicalGroup G
inst✝¹ : MeasurableSpace G
inst✝ : BorelSpace G
K₀ : PositiveCompacts G
g : G
A : Set G
hA : MeasurableSet A
⊢ ∀ (g : G) {K : Compacts G},
    (fun s => ↑(Content.toFun (haarContent K₀) s)) (Compacts.map (fun b => g * b) (_ : Continuous fun b => g * b) K) =
      (fun s => ↑(Content.toFun (haarContent K₀) s)) K
[PROOFSTEP]
apply is_left_invariant_haarContent
[GOAL]
G : Type u_1
inst✝⁵ : Group G
inst✝⁴ : TopologicalSpace G
inst✝³ : T2Space G
inst✝² : TopologicalGroup G
inst✝¹ : MeasurableSpace G
inst✝ : BorelSpace G
K₀ : PositiveCompacts G
⊢ ↑↑(haarMeasure K₀) ↑K₀ = 1
[PROOFSTEP]
haveI : LocallyCompactSpace G := K₀.locallyCompactSpace_of_group
[GOAL]
G : Type u_1
inst✝⁵ : Group G
inst✝⁴ : TopologicalSpace G
inst✝³ : T2Space G
inst✝² : TopologicalGroup G
inst✝¹ : MeasurableSpace G
inst✝ : BorelSpace G
K₀ : PositiveCompacts G
this : LocallyCompactSpace G
⊢ ↑↑(haarMeasure K₀) ↑K₀ = 1
[PROOFSTEP]
rw [haarMeasure_apply K₀.isCompact.measurableSet, ENNReal.div_self]
[GOAL]
case h0
G : Type u_1
inst✝⁵ : Group G
inst✝⁴ : TopologicalSpace G
inst✝³ : T2Space G
inst✝² : TopologicalGroup G
inst✝¹ : MeasurableSpace G
inst✝ : BorelSpace G
K₀ : PositiveCompacts G
this : LocallyCompactSpace G
⊢ ↑(Content.outerMeasure (haarContent K₀)) ↑K₀ ≠ 0
[PROOFSTEP]
rw [← pos_iff_ne_zero]
[GOAL]
case h0
G : Type u_1
inst✝⁵ : Group G
inst✝⁴ : TopologicalSpace G
inst✝³ : T2Space G
inst✝² : TopologicalGroup G
inst✝¹ : MeasurableSpace G
inst✝ : BorelSpace G
K₀ : PositiveCompacts G
this : LocallyCompactSpace G
⊢ 0 < ↑(Content.outerMeasure (haarContent K₀)) ↑K₀
[PROOFSTEP]
exact haarContent_outerMeasure_self_pos
[GOAL]
case hI
G : Type u_1
inst✝⁵ : Group G
inst✝⁴ : TopologicalSpace G
inst✝³ : T2Space G
inst✝² : TopologicalGroup G
inst✝¹ : MeasurableSpace G
inst✝ : BorelSpace G
K₀ : PositiveCompacts G
this : LocallyCompactSpace G
⊢ ↑(Content.outerMeasure (haarContent K₀)) ↑K₀ ≠ ⊤
[PROOFSTEP]
exact (Content.outerMeasure_lt_top_of_isCompact _ K₀.isCompact).ne
[GOAL]
G : Type u_1
inst✝⁵ : Group G
inst✝⁴ : TopologicalSpace G
inst✝³ : T2Space G
inst✝² : TopologicalGroup G
inst✝¹ : MeasurableSpace G
inst✝ : BorelSpace G
K₀ : PositiveCompacts G
⊢ Regular (haarMeasure K₀)
[PROOFSTEP]
haveI : LocallyCompactSpace G := K₀.locallyCompactSpace_of_group
[GOAL]
G : Type u_1
inst✝⁵ : Group G
inst✝⁴ : TopologicalSpace G
inst✝³ : T2Space G
inst✝² : TopologicalGroup G
inst✝¹ : MeasurableSpace G
inst✝ : BorelSpace G
K₀ : PositiveCompacts G
this : LocallyCompactSpace G
⊢ Regular (haarMeasure K₀)
[PROOFSTEP]
apply Regular.smul
[GOAL]
case hx
G : Type u_1
inst✝⁵ : Group G
inst✝⁴ : TopologicalSpace G
inst✝³ : T2Space G
inst✝² : TopologicalGroup G
inst✝¹ : MeasurableSpace G
inst✝ : BorelSpace G
K₀ : PositiveCompacts G
this : LocallyCompactSpace G
⊢ (↑(Content.outerMeasure (haarContent K₀)) ↑K₀)⁻¹ ≠ ⊤
[PROOFSTEP]
rw [ENNReal.inv_ne_top]
[GOAL]
case hx
G : Type u_1
inst✝⁵ : Group G
inst✝⁴ : TopologicalSpace G
inst✝³ : T2Space G
inst✝² : TopologicalGroup G
inst✝¹ : MeasurableSpace G
inst✝ : BorelSpace G
K₀ : PositiveCompacts G
this : LocallyCompactSpace G
⊢ ↑(Content.outerMeasure (haarContent K₀)) ↑K₀ ≠ 0
[PROOFSTEP]
exact haarContent_outerMeasure_self_pos.ne'
[GOAL]
G : Type u_1
inst✝⁶ : Group G
inst✝⁵ : TopologicalSpace G
inst✝⁴ : T2Space G
inst✝³ : TopologicalGroup G
inst✝² : MeasurableSpace G
inst✝¹ : BorelSpace G
inst✝ : SecondCountableTopology G
K₀ : PositiveCompacts G
⊢ SigmaFinite (haarMeasure K₀)
[PROOFSTEP]
haveI : LocallyCompactSpace G := K₀.locallyCompactSpace_of_group
[GOAL]
G : Type u_1
inst✝⁶ : Group G
inst✝⁵ : TopologicalSpace G
inst✝⁴ : T2Space G
inst✝³ : TopologicalGroup G
inst✝² : MeasurableSpace G
inst✝¹ : BorelSpace G
inst✝ : SecondCountableTopology G
K₀ : PositiveCompacts G
this : LocallyCompactSpace G
⊢ SigmaFinite (haarMeasure K₀)
[PROOFSTEP]
infer_instance
[GOAL]
G : Type u_1
inst✝⁵ : Group G
inst✝⁴ : TopologicalSpace G
inst✝³ : T2Space G
inst✝² : TopologicalGroup G
inst✝¹ : MeasurableSpace G
inst✝ : BorelSpace G
K₀ : PositiveCompacts G
⊢ IsHaarMeasure (haarMeasure K₀)
[PROOFSTEP]
apply isHaarMeasure_of_isCompact_nonempty_interior (haarMeasure K₀) K₀ K₀.isCompact K₀.interior_nonempty
[GOAL]
case h
G : Type u_1
inst✝⁵ : Group G
inst✝⁴ : TopologicalSpace G
inst✝³ : T2Space G
inst✝² : TopologicalGroup G
inst✝¹ : MeasurableSpace G
inst✝ : BorelSpace G
K₀ : PositiveCompacts G
⊢ ↑↑(haarMeasure K₀) ↑K₀ ≠ 0
[PROOFSTEP]
simp only [haarMeasure_self]
[GOAL]
case h
G : Type u_1
inst✝⁵ : Group G
inst✝⁴ : TopologicalSpace G
inst✝³ : T2Space G
inst✝² : TopologicalGroup G
inst✝¹ : MeasurableSpace G
inst✝ : BorelSpace G
K₀ : PositiveCompacts G
⊢ 1 ≠ 0
[PROOFSTEP]
exact one_ne_zero
[GOAL]
case h'
G : Type u_1
inst✝⁵ : Group G
inst✝⁴ : TopologicalSpace G
inst✝³ : T2Space G
inst✝² : TopologicalGroup G
inst✝¹ : MeasurableSpace G
inst✝ : BorelSpace G
K₀ : PositiveCompacts G
⊢ ↑↑(haarMeasure K₀) ↑K₀ ≠ ⊤
[PROOFSTEP]
simp only [haarMeasure_self]
[GOAL]
G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : T2Space G
inst✝⁵ : TopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
inst✝² : SecondCountableTopology G
μ : Measure G
inst✝¹ : SigmaFinite μ
inst✝ : IsMulLeftInvariant μ
K₀ : PositiveCompacts G
⊢ (↑↑μ ↑K₀ / ↑↑(haarMeasure K₀) ↑K₀) • haarMeasure K₀ = ↑↑μ ↑K₀ • haarMeasure K₀
[PROOFSTEP]
rw [haarMeasure_self, div_one]
[GOAL]
G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : T2Space G
inst✝⁵ : TopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
inst✝² : SecondCountableTopology G
μ : Measure G
inst✝¹ : SigmaFinite μ
inst✝ : IsMulLeftInvariant μ
K : Set G
hK : IsCompact K
h2K : Set.Nonempty (interior K)
hμK : ↑↑μ K ≠ ⊤
⊢ Regular μ
[PROOFSTEP]
rw [haarMeasure_unique μ ⟨⟨K, hK⟩, h2K⟩]
[GOAL]
G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : T2Space G
inst✝⁵ : TopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
inst✝² : SecondCountableTopology G
μ : Measure G
inst✝¹ : SigmaFinite μ
inst✝ : IsMulLeftInvariant μ
K : Set G
hK : IsCompact K
h2K : Set.Nonempty (interior K)
hμK : ↑↑μ K ≠ ⊤
⊢ Regular
    (↑↑μ ↑{ toCompacts := { carrier := K, isCompact' := hK }, interior_nonempty' := h2K } •
      haarMeasure { toCompacts := { carrier := K, isCompact' := hK }, interior_nonempty' := h2K })
[PROOFSTEP]
exact Regular.smul hμK
[GOAL]
G : Type u_1
inst✝⁹ : Group G
inst✝⁸ : TopologicalSpace G
inst✝⁷ : T2Space G
inst✝⁶ : TopologicalGroup G
inst✝⁵ : MeasurableSpace G
inst✝⁴ : BorelSpace G
inst✝³ : SecondCountableTopology G
inst✝² : LocallyCompactSpace G
μ ν : Measure G
inst✝¹ : IsHaarMeasure μ
inst✝ : IsHaarMeasure ν
⊢ ∃ c, c ≠ 0 ∧ c ≠ ⊤ ∧ μ = c • ν
[PROOFSTEP]
have K : PositiveCompacts G := Classical.arbitrary _
[GOAL]
G : Type u_1
inst✝⁹ : Group G
inst✝⁸ : TopologicalSpace G
inst✝⁷ : T2Space G
inst✝⁶ : TopologicalGroup G
inst✝⁵ : MeasurableSpace G
inst✝⁴ : BorelSpace G
inst✝³ : SecondCountableTopology G
inst✝² : LocallyCompactSpace G
μ ν : Measure G
inst✝¹ : IsHaarMeasure μ
inst✝ : IsHaarMeasure ν
K : PositiveCompacts G
⊢ ∃ c, c ≠ 0 ∧ c ≠ ⊤ ∧ μ = c • ν
[PROOFSTEP]
have νpos : 0 < ν K := measure_pos_of_nonempty_interior _ K.interior_nonempty
[GOAL]
G : Type u_1
inst✝⁹ : Group G
inst✝⁸ : TopologicalSpace G
inst✝⁷ : T2Space G
inst✝⁶ : TopologicalGroup G
inst✝⁵ : MeasurableSpace G
inst✝⁴ : BorelSpace G
inst✝³ : SecondCountableTopology G
inst✝² : LocallyCompactSpace G
μ ν : Measure G
inst✝¹ : IsHaarMeasure μ
inst✝ : IsHaarMeasure ν
K : PositiveCompacts G
νpos : 0 < ↑↑ν ↑K
⊢ ∃ c, c ≠ 0 ∧ c ≠ ⊤ ∧ μ = c • ν
[PROOFSTEP]
have νne : ν K ≠ ∞ := K.isCompact.measure_lt_top.ne
[GOAL]
G : Type u_1
inst✝⁹ : Group G
inst✝⁸ : TopologicalSpace G
inst✝⁷ : T2Space G
inst✝⁶ : TopologicalGroup G
inst✝⁵ : MeasurableSpace G
inst✝⁴ : BorelSpace G
inst✝³ : SecondCountableTopology G
inst✝² : LocallyCompactSpace G
μ ν : Measure G
inst✝¹ : IsHaarMeasure μ
inst✝ : IsHaarMeasure ν
K : PositiveCompacts G
νpos : 0 < ↑↑ν ↑K
νne : ↑↑ν ↑K ≠ ⊤
⊢ ∃ c, c ≠ 0 ∧ c ≠ ⊤ ∧ μ = c • ν
[PROOFSTEP]
refine' ⟨μ K / ν K, _, _, _⟩
[GOAL]
case refine'_1
G : Type u_1
inst✝⁹ : Group G
inst✝⁸ : TopologicalSpace G
inst✝⁷ : T2Space G
inst✝⁶ : TopologicalGroup G
inst✝⁵ : MeasurableSpace G
inst✝⁴ : BorelSpace G
inst✝³ : SecondCountableTopology G
inst✝² : LocallyCompactSpace G
μ ν : Measure G
inst✝¹ : IsHaarMeasure μ
inst✝ : IsHaarMeasure ν
K : PositiveCompacts G
νpos : 0 < ↑↑ν ↑K
νne : ↑↑ν ↑K ≠ ⊤
⊢ ↑↑μ ↑K / ↑↑ν ↑K ≠ 0
[PROOFSTEP]
simp only [νne, (μ.measure_pos_of_nonempty_interior K.interior_nonempty).ne', Ne.def, ENNReal.div_eq_zero_iff,
  not_false_iff, or_self_iff]
[GOAL]
case refine'_2
G : Type u_1
inst✝⁹ : Group G
inst✝⁸ : TopologicalSpace G
inst✝⁷ : T2Space G
inst✝⁶ : TopologicalGroup G
inst✝⁵ : MeasurableSpace G
inst✝⁴ : BorelSpace G
inst✝³ : SecondCountableTopology G
inst✝² : LocallyCompactSpace G
μ ν : Measure G
inst✝¹ : IsHaarMeasure μ
inst✝ : IsHaarMeasure ν
K : PositiveCompacts G
νpos : 0 < ↑↑ν ↑K
νne : ↑↑ν ↑K ≠ ⊤
⊢ ↑↑μ ↑K / ↑↑ν ↑K ≠ ⊤
[PROOFSTEP]
simp only [div_eq_mul_inv, νpos.ne', (K.isCompact.measure_lt_top (μ := μ)).ne, or_self_iff, ENNReal.inv_eq_top,
  ENNReal.mul_eq_top, Ne.def, not_false_iff, and_false_iff, false_and_iff]
[GOAL]
case refine'_3
G : Type u_1
inst✝⁹ : Group G
inst✝⁸ : TopologicalSpace G
inst✝⁷ : T2Space G
inst✝⁶ : TopologicalGroup G
inst✝⁵ : MeasurableSpace G
inst✝⁴ : BorelSpace G
inst✝³ : SecondCountableTopology G
inst✝² : LocallyCompactSpace G
μ ν : Measure G
inst✝¹ : IsHaarMeasure μ
inst✝ : IsHaarMeasure ν
K : PositiveCompacts G
νpos : 0 < ↑↑ν ↑K
νne : ↑↑ν ↑K ≠ ⊤
⊢ μ = (↑↑μ ↑K / ↑↑ν ↑K) • ν
[PROOFSTEP]
calc
  μ = μ K • haarMeasure K := haarMeasure_unique μ K
  _ = (μ K / ν K) • ν K • haarMeasure K := by
    rw [smul_smul, div_eq_mul_inv, mul_assoc, ENNReal.inv_mul_cancel νpos.ne' νne, mul_one]
  _ = (μ K / ν K) • ν := by rw [← haarMeasure_unique ν K]
[GOAL]
G : Type u_1
inst✝⁹ : Group G
inst✝⁸ : TopologicalSpace G
inst✝⁷ : T2Space G
inst✝⁶ : TopologicalGroup G
inst✝⁵ : MeasurableSpace G
inst✝⁴ : BorelSpace G
inst✝³ : SecondCountableTopology G
inst✝² : LocallyCompactSpace G
μ ν : Measure G
inst✝¹ : IsHaarMeasure μ
inst✝ : IsHaarMeasure ν
K : PositiveCompacts G
νpos : 0 < ↑↑ν ↑K
νne : ↑↑ν ↑K ≠ ⊤
⊢ ↑↑μ ↑K • haarMeasure K = (↑↑μ ↑K / ↑↑ν ↑K) • ↑↑ν ↑K • haarMeasure K
[PROOFSTEP]
rw [smul_smul, div_eq_mul_inv, mul_assoc, ENNReal.inv_mul_cancel νpos.ne' νne, mul_one]
[GOAL]
G : Type u_1
inst✝⁹ : Group G
inst✝⁸ : TopologicalSpace G
inst✝⁷ : T2Space G
inst✝⁶ : TopologicalGroup G
inst✝⁵ : MeasurableSpace G
inst✝⁴ : BorelSpace G
inst✝³ : SecondCountableTopology G
inst✝² : LocallyCompactSpace G
μ ν : Measure G
inst✝¹ : IsHaarMeasure μ
inst✝ : IsHaarMeasure ν
K : PositiveCompacts G
νpos : 0 < ↑↑ν ↑K
νne : ↑↑ν ↑K ≠ ⊤
⊢ (↑↑μ ↑K / ↑↑ν ↑K) • ↑↑ν ↑K • haarMeasure K = (↑↑μ ↑K / ↑↑ν ↑K) • ν
[PROOFSTEP]
rw [← haarMeasure_unique ν K]
[GOAL]
G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : T2Space G
inst✝⁵ : TopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
inst✝² : SecondCountableTopology G
inst✝¹ : LocallyCompactSpace G
μ : Measure G
inst✝ : IsHaarMeasure μ
⊢ Regular μ
[PROOFSTEP]
have K : PositiveCompacts G := Classical.arbitrary _
[GOAL]
G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : T2Space G
inst✝⁵ : TopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
inst✝² : SecondCountableTopology G
inst✝¹ : LocallyCompactSpace G
μ : Measure G
inst✝ : IsHaarMeasure μ
K : PositiveCompacts G
⊢ Regular μ
[PROOFSTEP]
obtain ⟨c, _, ctop, hμ⟩ : ∃ c : ℝ≥0∞, c ≠ 0 ∧ c ≠ ∞ ∧ μ = c • haarMeasure K := isHaarMeasure_eq_smul_isHaarMeasure μ _
[GOAL]
case intro.intro.intro
G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : T2Space G
inst✝⁵ : TopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
inst✝² : SecondCountableTopology G
inst✝¹ : LocallyCompactSpace G
μ : Measure G
inst✝ : IsHaarMeasure μ
K : PositiveCompacts G
c : ℝ≥0∞
left✝ : c ≠ 0
ctop : c ≠ ⊤
hμ : μ = c • haarMeasure K
⊢ Regular μ
[PROOFSTEP]
rw [hμ]
[GOAL]
case intro.intro.intro
G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : T2Space G
inst✝⁵ : TopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
inst✝² : SecondCountableTopology G
inst✝¹ : LocallyCompactSpace G
μ : Measure G
inst✝ : IsHaarMeasure μ
K : PositiveCompacts G
c : ℝ≥0∞
left✝ : c ≠ 0
ctop : c ≠ ⊤
hμ : μ = c • haarMeasure K
⊢ Regular (c • haarMeasure K)
[PROOFSTEP]
exact Regular.smul ctop
[GOAL]
G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : T2Space G
inst✝⁵ : TopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
inst✝² : SecondCountableTopology G
μ : Measure G
inst✝¹ : IsHaarMeasure μ
inst✝ : LocallyCompactSpace G
E : Set G
hE : MeasurableSet E
hEpos : 0 < ↑↑μ E
⊢ E / E ∈ 𝓝 1
[PROOFSTEP]
obtain ⟨L, hL, hLE, hLpos, hLtop⟩ : ∃ L : Set G, MeasurableSet L ∧ L ⊆ E ∧ 0 < μ L ∧ μ L < ∞ :=
  exists_subset_measure_lt_top hE hEpos
[GOAL]
case intro.intro.intro.intro
G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : T2Space G
inst✝⁵ : TopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
inst✝² : SecondCountableTopology G
μ : Measure G
inst✝¹ : IsHaarMeasure μ
inst✝ : LocallyCompactSpace G
E : Set G
hE : MeasurableSet E
hEpos : 0 < ↑↑μ E
L : Set G
hL : MeasurableSet L
hLE : L ⊆ E
hLpos : 0 < ↑↑μ L
hLtop : ↑↑μ L < ⊤
⊢ E / E ∈ 𝓝 1
[PROOFSTEP]
obtain ⟨K, hKL, hK, hKpos⟩ : ∃ (K : Set G), K ⊆ L ∧ IsCompact K ∧ 0 < μ K :=
  MeasurableSet.exists_lt_isCompact_of_ne_top hL (ne_of_lt hLtop) hLpos
[GOAL]
case intro.intro.intro.intro.intro.intro.intro
G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : T2Space G
inst✝⁵ : TopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
inst✝² : SecondCountableTopology G
μ : Measure G
inst✝¹ : IsHaarMeasure μ
inst✝ : LocallyCompactSpace G
E : Set G
hE : MeasurableSet E
hEpos : 0 < ↑↑μ E
L : Set G
hL : MeasurableSet L
hLE : L ⊆ E
hLpos : 0 < ↑↑μ L
hLtop : ↑↑μ L < ⊤
K : Set G
hKL : K ⊆ L
hK : IsCompact K
hKpos : 0 < ↑↑μ K
⊢ E / E ∈ 𝓝 1
[PROOFSTEP]
have hKtop : μ K ≠ ∞ := by
  apply ne_top_of_le_ne_top (ne_of_lt hLtop)
  apply measure_mono hKL
[GOAL]
G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : T2Space G
inst✝⁵ : TopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
inst✝² : SecondCountableTopology G
μ : Measure G
inst✝¹ : IsHaarMeasure μ
inst✝ : LocallyCompactSpace G
E : Set G
hE : MeasurableSet E
hEpos : 0 < ↑↑μ E
L : Set G
hL : MeasurableSet L
hLE : L ⊆ E
hLpos : 0 < ↑↑μ L
hLtop : ↑↑μ L < ⊤
K : Set G
hKL : K ⊆ L
hK : IsCompact K
hKpos : 0 < ↑↑μ K
⊢ ↑↑μ K ≠ ⊤
[PROOFSTEP]
apply ne_top_of_le_ne_top (ne_of_lt hLtop)
[GOAL]
G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : T2Space G
inst✝⁵ : TopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
inst✝² : SecondCountableTopology G
μ : Measure G
inst✝¹ : IsHaarMeasure μ
inst✝ : LocallyCompactSpace G
E : Set G
hE : MeasurableSet E
hEpos : 0 < ↑↑μ E
L : Set G
hL : MeasurableSet L
hLE : L ⊆ E
hLpos : 0 < ↑↑μ L
hLtop : ↑↑μ L < ⊤
K : Set G
hKL : K ⊆ L
hK : IsCompact K
hKpos : 0 < ↑↑μ K
⊢ ↑↑μ K ≤ ↑↑μ L
[PROOFSTEP]
apply measure_mono hKL
[GOAL]
case intro.intro.intro.intro.intro.intro.intro
G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : T2Space G
inst✝⁵ : TopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
inst✝² : SecondCountableTopology G
μ : Measure G
inst✝¹ : IsHaarMeasure μ
inst✝ : LocallyCompactSpace G
E : Set G
hE : MeasurableSet E
hEpos : 0 < ↑↑μ E
L : Set G
hL : MeasurableSet L
hLE : L ⊆ E
hLpos : 0 < ↑↑μ L
hLtop : ↑↑μ L < ⊤
K : Set G
hKL : K ⊆ L
hK : IsCompact K
hKpos : 0 < ↑↑μ K
hKtop : ↑↑μ K ≠ ⊤
⊢ E / E ∈ 𝓝 1
[PROOFSTEP]
obtain ⟨U, hUK, hU, hμUK⟩ : ∃ (U : Set G), U ⊇ K ∧ IsOpen U ∧ μ U < μ K + μ K :=
  Set.exists_isOpen_lt_add K hKtop hKpos.ne'
[GOAL]
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : T2Space G
inst✝⁵ : TopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
inst✝² : SecondCountableTopology G
μ : Measure G
inst✝¹ : IsHaarMeasure μ
inst✝ : LocallyCompactSpace G
E : Set G
hE : MeasurableSet E
hEpos : 0 < ↑↑μ E
L : Set G
hL : MeasurableSet L
hLE : L ⊆ E
hLpos : 0 < ↑↑μ L
hLtop : ↑↑μ L < ⊤
K : Set G
hKL : K ⊆ L
hK : IsCompact K
hKpos : 0 < ↑↑μ K
hKtop : ↑↑μ K ≠ ⊤
U : Set G
hUK : U ⊇ K
hU : IsOpen U
hμUK : ↑↑μ U < ↑↑μ K + ↑↑μ K
⊢ E / E ∈ 𝓝 1
[PROOFSTEP]
obtain ⟨V, hV1, hVKU⟩ : ∃ V ∈ 𝓝 (1 : G), V * K ⊆ U := compact_open_separated_mul_left hK hU hUK
[GOAL]
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : T2Space G
inst✝⁵ : TopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
inst✝² : SecondCountableTopology G
μ : Measure G
inst✝¹ : IsHaarMeasure μ
inst✝ : LocallyCompactSpace G
E : Set G
hE : MeasurableSet E
hEpos : 0 < ↑↑μ E
L : Set G
hL : MeasurableSet L
hLE : L ⊆ E
hLpos : 0 < ↑↑μ L
hLtop : ↑↑μ L < ⊤
K : Set G
hKL : K ⊆ L
hK : IsCompact K
hKpos : 0 < ↑↑μ K
hKtop : ↑↑μ K ≠ ⊤
U : Set G
hUK : U ⊇ K
hU : IsOpen U
hμUK : ↑↑μ U < ↑↑μ K + ↑↑μ K
V : Set G
hV1 : V ∈ 𝓝 1
hVKU : V * K ⊆ U
⊢ E / E ∈ 𝓝 1
[PROOFSTEP]
have hv : ∀ v : G, v ∈ V → ¬Disjoint ({ v } * K) K :=
  by
  intro v hv hKv
  have hKvsub : { v } * K ∪ K ⊆ U := by
    apply Set.union_subset _ hUK
    apply _root_.subset_trans _ hVKU
    apply Set.mul_subset_mul _ (Set.Subset.refl K)
    simp only [Set.singleton_subset_iff, hv]
  replace hKvsub := @measure_mono _ _ μ _ _ hKvsub
  have hcontr := lt_of_le_of_lt hKvsub hμUK
  rw [measure_union hKv (IsCompact.measurableSet hK)] at hcontr
  have hKtranslate : μ ({ v } * K) = μ K := by simp only [singleton_mul, image_mul_left, measure_preimage_mul]
  rw [hKtranslate, lt_self_iff_false] at hcontr
  assumption
[GOAL]
G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : T2Space G
inst✝⁵ : TopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
inst✝² : SecondCountableTopology G
μ : Measure G
inst✝¹ : IsHaarMeasure μ
inst✝ : LocallyCompactSpace G
E : Set G
hE : MeasurableSet E
hEpos : 0 < ↑↑μ E
L : Set G
hL : MeasurableSet L
hLE : L ⊆ E
hLpos : 0 < ↑↑μ L
hLtop : ↑↑μ L < ⊤
K : Set G
hKL : K ⊆ L
hK : IsCompact K
hKpos : 0 < ↑↑μ K
hKtop : ↑↑μ K ≠ ⊤
U : Set G
hUK : U ⊇ K
hU : IsOpen U
hμUK : ↑↑μ U < ↑↑μ K + ↑↑μ K
V : Set G
hV1 : V ∈ 𝓝 1
hVKU : V * K ⊆ U
⊢ ∀ (v : G), v ∈ V → ¬Disjoint ({v} * K) K
[PROOFSTEP]
intro v hv hKv
[GOAL]
G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : T2Space G
inst✝⁵ : TopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
inst✝² : SecondCountableTopology G
μ : Measure G
inst✝¹ : IsHaarMeasure μ
inst✝ : LocallyCompactSpace G
E : Set G
hE : MeasurableSet E
hEpos : 0 < ↑↑μ E
L : Set G
hL : MeasurableSet L
hLE : L ⊆ E
hLpos : 0 < ↑↑μ L
hLtop : ↑↑μ L < ⊤
K : Set G
hKL : K ⊆ L
hK : IsCompact K
hKpos : 0 < ↑↑μ K
hKtop : ↑↑μ K ≠ ⊤
U : Set G
hUK : U ⊇ K
hU : IsOpen U
hμUK : ↑↑μ U < ↑↑μ K + ↑↑μ K
V : Set G
hV1 : V ∈ 𝓝 1
hVKU : V * K ⊆ U
v : G
hv : v ∈ V
hKv : Disjoint ({v} * K) K
⊢ False
[PROOFSTEP]
have hKvsub : { v } * K ∪ K ⊆ U := by
  apply Set.union_subset _ hUK
  apply _root_.subset_trans _ hVKU
  apply Set.mul_subset_mul _ (Set.Subset.refl K)
  simp only [Set.singleton_subset_iff, hv]
[GOAL]
G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : T2Space G
inst✝⁵ : TopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
inst✝² : SecondCountableTopology G
μ : Measure G
inst✝¹ : IsHaarMeasure μ
inst✝ : LocallyCompactSpace G
E : Set G
hE : MeasurableSet E
hEpos : 0 < ↑↑μ E
L : Set G
hL : MeasurableSet L
hLE : L ⊆ E
hLpos : 0 < ↑↑μ L
hLtop : ↑↑μ L < ⊤
K : Set G
hKL : K ⊆ L
hK : IsCompact K
hKpos : 0 < ↑↑μ K
hKtop : ↑↑μ K ≠ ⊤
U : Set G
hUK : U ⊇ K
hU : IsOpen U
hμUK : ↑↑μ U < ↑↑μ K + ↑↑μ K
V : Set G
hV1 : V ∈ 𝓝 1
hVKU : V * K ⊆ U
v : G
hv : v ∈ V
hKv : Disjoint ({v} * K) K
⊢ {v} * K ∪ K ⊆ U
[PROOFSTEP]
apply Set.union_subset _ hUK
[GOAL]
G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : T2Space G
inst✝⁵ : TopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
inst✝² : SecondCountableTopology G
μ : Measure G
inst✝¹ : IsHaarMeasure μ
inst✝ : LocallyCompactSpace G
E : Set G
hE : MeasurableSet E
hEpos : 0 < ↑↑μ E
L : Set G
hL : MeasurableSet L
hLE : L ⊆ E
hLpos : 0 < ↑↑μ L
hLtop : ↑↑μ L < ⊤
K : Set G
hKL : K ⊆ L
hK : IsCompact K
hKpos : 0 < ↑↑μ K
hKtop : ↑↑μ K ≠ ⊤
U : Set G
hUK : U ⊇ K
hU : IsOpen U
hμUK : ↑↑μ U < ↑↑μ K + ↑↑μ K
V : Set G
hV1 : V ∈ 𝓝 1
hVKU : V * K ⊆ U
v : G
hv : v ∈ V
hKv : Disjoint ({v} * K) K
⊢ {v} * K ⊆ U
[PROOFSTEP]
apply _root_.subset_trans _ hVKU
[GOAL]
G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : T2Space G
inst✝⁵ : TopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
inst✝² : SecondCountableTopology G
μ : Measure G
inst✝¹ : IsHaarMeasure μ
inst✝ : LocallyCompactSpace G
E : Set G
hE : MeasurableSet E
hEpos : 0 < ↑↑μ E
L : Set G
hL : MeasurableSet L
hLE : L ⊆ E
hLpos : 0 < ↑↑μ L
hLtop : ↑↑μ L < ⊤
K : Set G
hKL : K ⊆ L
hK : IsCompact K
hKpos : 0 < ↑↑μ K
hKtop : ↑↑μ K ≠ ⊤
U : Set G
hUK : U ⊇ K
hU : IsOpen U
hμUK : ↑↑μ U < ↑↑μ K + ↑↑μ K
V : Set G
hV1 : V ∈ 𝓝 1
hVKU : V * K ⊆ U
v : G
hv : v ∈ V
hKv : Disjoint ({v} * K) K
⊢ {v} * K ⊆ V * K
[PROOFSTEP]
apply Set.mul_subset_mul _ (Set.Subset.refl K)
[GOAL]
G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : T2Space G
inst✝⁵ : TopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
inst✝² : SecondCountableTopology G
μ : Measure G
inst✝¹ : IsHaarMeasure μ
inst✝ : LocallyCompactSpace G
E : Set G
hE : MeasurableSet E
hEpos : 0 < ↑↑μ E
L : Set G
hL : MeasurableSet L
hLE : L ⊆ E
hLpos : 0 < ↑↑μ L
hLtop : ↑↑μ L < ⊤
K : Set G
hKL : K ⊆ L
hK : IsCompact K
hKpos : 0 < ↑↑μ K
hKtop : ↑↑μ K ≠ ⊤
U : Set G
hUK : U ⊇ K
hU : IsOpen U
hμUK : ↑↑μ U < ↑↑μ K + ↑↑μ K
V : Set G
hV1 : V ∈ 𝓝 1
hVKU : V * K ⊆ U
v : G
hv : v ∈ V
hKv : Disjoint ({v} * K) K
⊢ {v} ⊆ V
[PROOFSTEP]
simp only [Set.singleton_subset_iff, hv]
[GOAL]
G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : T2Space G
inst✝⁵ : TopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
inst✝² : SecondCountableTopology G
μ : Measure G
inst✝¹ : IsHaarMeasure μ
inst✝ : LocallyCompactSpace G
E : Set G
hE : MeasurableSet E
hEpos : 0 < ↑↑μ E
L : Set G
hL : MeasurableSet L
hLE : L ⊆ E
hLpos : 0 < ↑↑μ L
hLtop : ↑↑μ L < ⊤
K : Set G
hKL : K ⊆ L
hK : IsCompact K
hKpos : 0 < ↑↑μ K
hKtop : ↑↑μ K ≠ ⊤
U : Set G
hUK : U ⊇ K
hU : IsOpen U
hμUK : ↑↑μ U < ↑↑μ K + ↑↑μ K
V : Set G
hV1 : V ∈ 𝓝 1
hVKU : V * K ⊆ U
v : G
hv : v ∈ V
hKv : Disjoint ({v} * K) K
hKvsub : {v} * K ∪ K ⊆ U
⊢ False
[PROOFSTEP]
replace hKvsub := @measure_mono _ _ μ _ _ hKvsub
[GOAL]
G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : T2Space G
inst✝⁵ : TopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
inst✝² : SecondCountableTopology G
μ : Measure G
inst✝¹ : IsHaarMeasure μ
inst✝ : LocallyCompactSpace G
E : Set G
hE : MeasurableSet E
hEpos : 0 < ↑↑μ E
L : Set G
hL : MeasurableSet L
hLE : L ⊆ E
hLpos : 0 < ↑↑μ L
hLtop : ↑↑μ L < ⊤
K : Set G
hKL : K ⊆ L
hK : IsCompact K
hKpos : 0 < ↑↑μ K
hKtop : ↑↑μ K ≠ ⊤
U : Set G
hUK : U ⊇ K
hU : IsOpen U
hμUK : ↑↑μ U < ↑↑μ K + ↑↑μ K
V : Set G
hV1 : V ∈ 𝓝 1
hVKU : V * K ⊆ U
v : G
hv : v ∈ V
hKv : Disjoint ({v} * K) K
hKvsub : ↑↑μ ({v} * K ∪ K) ≤ ↑↑μ U
⊢ False
[PROOFSTEP]
have hcontr := lt_of_le_of_lt hKvsub hμUK
[GOAL]
G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : T2Space G
inst✝⁵ : TopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
inst✝² : SecondCountableTopology G
μ : Measure G
inst✝¹ : IsHaarMeasure μ
inst✝ : LocallyCompactSpace G
E : Set G
hE : MeasurableSet E
hEpos : 0 < ↑↑μ E
L : Set G
hL : MeasurableSet L
hLE : L ⊆ E
hLpos : 0 < ↑↑μ L
hLtop : ↑↑μ L < ⊤
K : Set G
hKL : K ⊆ L
hK : IsCompact K
hKpos : 0 < ↑↑μ K
hKtop : ↑↑μ K ≠ ⊤
U : Set G
hUK : U ⊇ K
hU : IsOpen U
hμUK : ↑↑μ U < ↑↑μ K + ↑↑μ K
V : Set G
hV1 : V ∈ 𝓝 1
hVKU : V * K ⊆ U
v : G
hv : v ∈ V
hKv : Disjoint ({v} * K) K
hKvsub : ↑↑μ ({v} * K ∪ K) ≤ ↑↑μ U
hcontr : ↑↑μ ({v} * K ∪ K) < ↑↑μ K + ↑↑μ K
⊢ False
[PROOFSTEP]
rw [measure_union hKv (IsCompact.measurableSet hK)] at hcontr
[GOAL]
G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : T2Space G
inst✝⁵ : TopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
inst✝² : SecondCountableTopology G
μ : Measure G
inst✝¹ : IsHaarMeasure μ
inst✝ : LocallyCompactSpace G
E : Set G
hE : MeasurableSet E
hEpos : 0 < ↑↑μ E
L : Set G
hL : MeasurableSet L
hLE : L ⊆ E
hLpos : 0 < ↑↑μ L
hLtop : ↑↑μ L < ⊤
K : Set G
hKL : K ⊆ L
hK : IsCompact K
hKpos : 0 < ↑↑μ K
hKtop : ↑↑μ K ≠ ⊤
U : Set G
hUK : U ⊇ K
hU : IsOpen U
hμUK : ↑↑μ U < ↑↑μ K + ↑↑μ K
V : Set G
hV1 : V ∈ 𝓝 1
hVKU : V * K ⊆ U
v : G
hv : v ∈ V
hKv : Disjoint ({v} * K) K
hKvsub : ↑↑μ ({v} * K ∪ K) ≤ ↑↑μ U
hcontr : ↑↑μ ({v} * K) + ↑↑μ K < ↑↑μ K + ↑↑μ K
⊢ False
[PROOFSTEP]
have hKtranslate : μ ({ v } * K) = μ K := by simp only [singleton_mul, image_mul_left, measure_preimage_mul]
[GOAL]
G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : T2Space G
inst✝⁵ : TopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
inst✝² : SecondCountableTopology G
μ : Measure G
inst✝¹ : IsHaarMeasure μ
inst✝ : LocallyCompactSpace G
E : Set G
hE : MeasurableSet E
hEpos : 0 < ↑↑μ E
L : Set G
hL : MeasurableSet L
hLE : L ⊆ E
hLpos : 0 < ↑↑μ L
hLtop : ↑↑μ L < ⊤
K : Set G
hKL : K ⊆ L
hK : IsCompact K
hKpos : 0 < ↑↑μ K
hKtop : ↑↑μ K ≠ ⊤
U : Set G
hUK : U ⊇ K
hU : IsOpen U
hμUK : ↑↑μ U < ↑↑μ K + ↑↑μ K
V : Set G
hV1 : V ∈ 𝓝 1
hVKU : V * K ⊆ U
v : G
hv : v ∈ V
hKv : Disjoint ({v} * K) K
hKvsub : ↑↑μ ({v} * K ∪ K) ≤ ↑↑μ U
hcontr : ↑↑μ ({v} * K) + ↑↑μ K < ↑↑μ K + ↑↑μ K
⊢ ↑↑μ ({v} * K) = ↑↑μ K
[PROOFSTEP]
simp only [singleton_mul, image_mul_left, measure_preimage_mul]
[GOAL]
G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : T2Space G
inst✝⁵ : TopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
inst✝² : SecondCountableTopology G
μ : Measure G
inst✝¹ : IsHaarMeasure μ
inst✝ : LocallyCompactSpace G
E : Set G
hE : MeasurableSet E
hEpos : 0 < ↑↑μ E
L : Set G
hL : MeasurableSet L
hLE : L ⊆ E
hLpos : 0 < ↑↑μ L
hLtop : ↑↑μ L < ⊤
K : Set G
hKL : K ⊆ L
hK : IsCompact K
hKpos : 0 < ↑↑μ K
hKtop : ↑↑μ K ≠ ⊤
U : Set G
hUK : U ⊇ K
hU : IsOpen U
hμUK : ↑↑μ U < ↑↑μ K + ↑↑μ K
V : Set G
hV1 : V ∈ 𝓝 1
hVKU : V * K ⊆ U
v : G
hv : v ∈ V
hKv : Disjoint ({v} * K) K
hKvsub : ↑↑μ ({v} * K ∪ K) ≤ ↑↑μ U
hcontr : ↑↑μ ({v} * K) + ↑↑μ K < ↑↑μ K + ↑↑μ K
hKtranslate : ↑↑μ ({v} * K) = ↑↑μ K
⊢ False
[PROOFSTEP]
rw [hKtranslate, lt_self_iff_false] at hcontr
[GOAL]
G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : T2Space G
inst✝⁵ : TopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
inst✝² : SecondCountableTopology G
μ : Measure G
inst✝¹ : IsHaarMeasure μ
inst✝ : LocallyCompactSpace G
E : Set G
hE : MeasurableSet E
hEpos : 0 < ↑↑μ E
L : Set G
hL : MeasurableSet L
hLE : L ⊆ E
hLpos : 0 < ↑↑μ L
hLtop : ↑↑μ L < ⊤
K : Set G
hKL : K ⊆ L
hK : IsCompact K
hKpos : 0 < ↑↑μ K
hKtop : ↑↑μ K ≠ ⊤
U : Set G
hUK : U ⊇ K
hU : IsOpen U
hμUK : ↑↑μ U < ↑↑μ K + ↑↑μ K
V : Set G
hV1 : V ∈ 𝓝 1
hVKU : V * K ⊆ U
v : G
hv : v ∈ V
hKv : Disjoint ({v} * K) K
hKvsub : ↑↑μ ({v} * K ∪ K) ≤ ↑↑μ U
hcontr : False
hKtranslate : ↑↑μ ({v} * K) = ↑↑μ K
⊢ False
[PROOFSTEP]
assumption
[GOAL]
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : T2Space G
inst✝⁵ : TopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
inst✝² : SecondCountableTopology G
μ : Measure G
inst✝¹ : IsHaarMeasure μ
inst✝ : LocallyCompactSpace G
E : Set G
hE : MeasurableSet E
hEpos : 0 < ↑↑μ E
L : Set G
hL : MeasurableSet L
hLE : L ⊆ E
hLpos : 0 < ↑↑μ L
hLtop : ↑↑μ L < ⊤
K : Set G
hKL : K ⊆ L
hK : IsCompact K
hKpos : 0 < ↑↑μ K
hKtop : ↑↑μ K ≠ ⊤
U : Set G
hUK : U ⊇ K
hU : IsOpen U
hμUK : ↑↑μ U < ↑↑μ K + ↑↑μ K
V : Set G
hV1 : V ∈ 𝓝 1
hVKU : V * K ⊆ U
hv : ∀ (v : G), v ∈ V → ¬Disjoint ({v} * K) K
⊢ E / E ∈ 𝓝 1
[PROOFSTEP]
suffices V ⊆ E / E from Filter.mem_of_superset hV1 this
[GOAL]
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : T2Space G
inst✝⁵ : TopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
inst✝² : SecondCountableTopology G
μ : Measure G
inst✝¹ : IsHaarMeasure μ
inst✝ : LocallyCompactSpace G
E : Set G
hE : MeasurableSet E
hEpos : 0 < ↑↑μ E
L : Set G
hL : MeasurableSet L
hLE : L ⊆ E
hLpos : 0 < ↑↑μ L
hLtop : ↑↑μ L < ⊤
K : Set G
hKL : K ⊆ L
hK : IsCompact K
hKpos : 0 < ↑↑μ K
hKtop : ↑↑μ K ≠ ⊤
U : Set G
hUK : U ⊇ K
hU : IsOpen U
hμUK : ↑↑μ U < ↑↑μ K + ↑↑μ K
V : Set G
hV1 : V ∈ 𝓝 1
hVKU : V * K ⊆ U
hv : ∀ (v : G), v ∈ V → ¬Disjoint ({v} * K) K
⊢ V ⊆ E / E
[PROOFSTEP]
intro v hvV
[GOAL]
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : T2Space G
inst✝⁵ : TopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
inst✝² : SecondCountableTopology G
μ : Measure G
inst✝¹ : IsHaarMeasure μ
inst✝ : LocallyCompactSpace G
E : Set G
hE : MeasurableSet E
hEpos : 0 < ↑↑μ E
L : Set G
hL : MeasurableSet L
hLE : L ⊆ E
hLpos : 0 < ↑↑μ L
hLtop : ↑↑μ L < ⊤
K : Set G
hKL : K ⊆ L
hK : IsCompact K
hKpos : 0 < ↑↑μ K
hKtop : ↑↑μ K ≠ ⊤
U : Set G
hUK : U ⊇ K
hU : IsOpen U
hμUK : ↑↑μ U < ↑↑μ K + ↑↑μ K
V : Set G
hV1 : V ∈ 𝓝 1
hVKU : V * K ⊆ U
hv : ∀ (v : G), v ∈ V → ¬Disjoint ({v} * K) K
v : G
hvV : v ∈ V
⊢ v ∈ E / E
[PROOFSTEP]
obtain ⟨x, hxK, hxvK⟩ : ∃ x : G, x ∈ { v } * K ∧ x ∈ K := Set.not_disjoint_iff.1 (hv v hvV)
[GOAL]
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : T2Space G
inst✝⁵ : TopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
inst✝² : SecondCountableTopology G
μ : Measure G
inst✝¹ : IsHaarMeasure μ
inst✝ : LocallyCompactSpace G
E : Set G
hE : MeasurableSet E
hEpos : 0 < ↑↑μ E
L : Set G
hL : MeasurableSet L
hLE : L ⊆ E
hLpos : 0 < ↑↑μ L
hLtop : ↑↑μ L < ⊤
K : Set G
hKL : K ⊆ L
hK : IsCompact K
hKpos : 0 < ↑↑μ K
hKtop : ↑↑μ K ≠ ⊤
U : Set G
hUK : U ⊇ K
hU : IsOpen U
hμUK : ↑↑μ U < ↑↑μ K + ↑↑μ K
V : Set G
hV1 : V ∈ 𝓝 1
hVKU : V * K ⊆ U
hv : ∀ (v : G), v ∈ V → ¬Disjoint ({v} * K) K
v : G
hvV : v ∈ V
x : G
hxK : x ∈ {v} * K
hxvK : x ∈ K
⊢ v ∈ E / E
[PROOFSTEP]
refine' ⟨x, v⁻¹ * x, hLE (hKL hxvK), _, _⟩
[GOAL]
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1
G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : T2Space G
inst✝⁵ : TopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
inst✝² : SecondCountableTopology G
μ : Measure G
inst✝¹ : IsHaarMeasure μ
inst✝ : LocallyCompactSpace G
E : Set G
hE : MeasurableSet E
hEpos : 0 < ↑↑μ E
L : Set G
hL : MeasurableSet L
hLE : L ⊆ E
hLpos : 0 < ↑↑μ L
hLtop : ↑↑μ L < ⊤
K : Set G
hKL : K ⊆ L
hK : IsCompact K
hKpos : 0 < ↑↑μ K
hKtop : ↑↑μ K ≠ ⊤
U : Set G
hUK : U ⊇ K
hU : IsOpen U
hμUK : ↑↑μ U < ↑↑μ K + ↑↑μ K
V : Set G
hV1 : V ∈ 𝓝 1
hVKU : V * K ⊆ U
hv : ∀ (v : G), v ∈ V → ¬Disjoint ({v} * K) K
v : G
hvV : v ∈ V
x : G
hxK : x ∈ {v} * K
hxvK : x ∈ K
⊢ v⁻¹ * x ∈ E
[PROOFSTEP]
apply hKL.trans hLE
[GOAL]
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_1.a
G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : T2Space G
inst✝⁵ : TopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
inst✝² : SecondCountableTopology G
μ : Measure G
inst✝¹ : IsHaarMeasure μ
inst✝ : LocallyCompactSpace G
E : Set G
hE : MeasurableSet E
hEpos : 0 < ↑↑μ E
L : Set G
hL : MeasurableSet L
hLE : L ⊆ E
hLpos : 0 < ↑↑μ L
hLtop : ↑↑μ L < ⊤
K : Set G
hKL : K ⊆ L
hK : IsCompact K
hKpos : 0 < ↑↑μ K
hKtop : ↑↑μ K ≠ ⊤
U : Set G
hUK : U ⊇ K
hU : IsOpen U
hμUK : ↑↑μ U < ↑↑μ K + ↑↑μ K
V : Set G
hV1 : V ∈ 𝓝 1
hVKU : V * K ⊆ U
hv : ∀ (v : G), v ∈ V → ¬Disjoint ({v} * K) K
v : G
hvV : v ∈ V
x : G
hxK : x ∈ {v} * K
hxvK : x ∈ K
⊢ v⁻¹ * x ∈ K
[PROOFSTEP]
simpa only [singleton_mul, image_mul_left, mem_preimage] using hxK
[GOAL]
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine'_2
G : Type u_1
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : T2Space G
inst✝⁵ : TopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
inst✝² : SecondCountableTopology G
μ : Measure G
inst✝¹ : IsHaarMeasure μ
inst✝ : LocallyCompactSpace G
E : Set G
hE : MeasurableSet E
hEpos : 0 < ↑↑μ E
L : Set G
hL : MeasurableSet L
hLE : L ⊆ E
hLpos : 0 < ↑↑μ L
hLtop : ↑↑μ L < ⊤
K : Set G
hKL : K ⊆ L
hK : IsCompact K
hKpos : 0 < ↑↑μ K
hKtop : ↑↑μ K ≠ ⊤
U : Set G
hUK : U ⊇ K
hU : IsOpen U
hμUK : ↑↑μ U < ↑↑μ K + ↑↑μ K
V : Set G
hV1 : V ∈ 𝓝 1
hVKU : V * K ⊆ U
hv : ∀ (v : G), v ∈ V → ¬Disjoint ({v} * K) K
v : G
hvV : v ∈ V
x : G
hxK : x ∈ {v} * K
hxvK : x ∈ K
⊢ (fun x x_1 => x / x_1) x (v⁻¹ * x) = v
[PROOFSTEP]
simp only [div_eq_iff_eq_mul, ← mul_assoc, mul_right_inv, one_mul]
[GOAL]
G : Type u_1
inst✝⁸ : CommGroup G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : TopologicalGroup G
inst✝⁵ : T2Space G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
inst✝² : SecondCountableTopology G
μ : Measure G
inst✝¹ : IsHaarMeasure μ
inst✝ : LocallyCompactSpace G
⊢ IsInvInvariant μ
[PROOFSTEP]
constructor
[GOAL]
case inv_eq_self
G : Type u_1
inst✝⁸ : CommGroup G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : TopologicalGroup G
inst✝⁵ : T2Space G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
inst✝² : SecondCountableTopology G
μ : Measure G
inst✝¹ : IsHaarMeasure μ
inst✝ : LocallyCompactSpace G
⊢ Measure.inv μ = μ
[PROOFSTEP]
haveI : IsHaarMeasure (Measure.map Inv.inv μ) := (MulEquiv.inv G).isHaarMeasure_map μ continuous_inv continuous_inv
[GOAL]
case inv_eq_self
G : Type u_1
inst✝⁸ : CommGroup G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : TopologicalGroup G
inst✝⁵ : T2Space G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
inst✝² : SecondCountableTopology G
μ : Measure G
inst✝¹ : IsHaarMeasure μ
inst✝ : LocallyCompactSpace G
this : IsHaarMeasure (map inv μ)
⊢ Measure.inv μ = μ
[PROOFSTEP]
obtain ⟨c, _, _, hc⟩ : ∃ c : ℝ≥0∞, c ≠ 0 ∧ c ≠ ∞ ∧ Measure.map Inv.inv μ = c • μ :=
  isHaarMeasure_eq_smul_isHaarMeasure _ _
[GOAL]
case inv_eq_self.intro.intro.intro
G : Type u_1
inst✝⁸ : CommGroup G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : TopologicalGroup G
inst✝⁵ : T2Space G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
inst✝² : SecondCountableTopology G
μ : Measure G
inst✝¹ : IsHaarMeasure μ
inst✝ : LocallyCompactSpace G
this : IsHaarMeasure (map inv μ)
c : ℝ≥0∞
left✝¹ : c ≠ 0
left✝ : c ≠ ⊤
hc : map inv μ = c • μ
⊢ Measure.inv μ = μ
[PROOFSTEP]
have : map Inv.inv (map Inv.inv μ) = c ^ 2 • μ := by simp only [hc, smul_smul, pow_two, Measure.map_smul]
[GOAL]
G : Type u_1
inst✝⁸ : CommGroup G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : TopologicalGroup G
inst✝⁵ : T2Space G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
inst✝² : SecondCountableTopology G
μ : Measure G
inst✝¹ : IsHaarMeasure μ
inst✝ : LocallyCompactSpace G
this : IsHaarMeasure (map inv μ)
c : ℝ≥0∞
left✝¹ : c ≠ 0
left✝ : c ≠ ⊤
hc : map inv μ = c • μ
⊢ map inv (map inv μ) = c ^ 2 • μ
[PROOFSTEP]
simp only [hc, smul_smul, pow_two, Measure.map_smul]
[GOAL]
case inv_eq_self.intro.intro.intro
G : Type u_1
inst✝⁸ : CommGroup G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : TopologicalGroup G
inst✝⁵ : T2Space G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
inst✝² : SecondCountableTopology G
μ : Measure G
inst✝¹ : IsHaarMeasure μ
inst✝ : LocallyCompactSpace G
this✝ : IsHaarMeasure (map inv μ)
c : ℝ≥0∞
left✝¹ : c ≠ 0
left✝ : c ≠ ⊤
hc : map inv μ = c • μ
this : map inv (map inv μ) = c ^ 2 • μ
⊢ Measure.inv μ = μ
[PROOFSTEP]
have μeq : μ = c ^ 2 • μ :=
  by
  rw [map_map continuous_inv.measurable continuous_inv.measurable] at this
  simpa only [inv_involutive, Involutive.comp_self, map_id]
[GOAL]
G : Type u_1
inst✝⁸ : CommGroup G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : TopologicalGroup G
inst✝⁵ : T2Space G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
inst✝² : SecondCountableTopology G
μ : Measure G
inst✝¹ : IsHaarMeasure μ
inst✝ : LocallyCompactSpace G
this✝ : IsHaarMeasure (map inv μ)
c : ℝ≥0∞
left✝¹ : c ≠ 0
left✝ : c ≠ ⊤
hc : map inv μ = c • μ
this : map inv (map inv μ) = c ^ 2 • μ
⊢ μ = c ^ 2 • μ
[PROOFSTEP]
rw [map_map continuous_inv.measurable continuous_inv.measurable] at this
[GOAL]
G : Type u_1
inst✝⁸ : CommGroup G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : TopologicalGroup G
inst✝⁵ : T2Space G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
inst✝² : SecondCountableTopology G
μ : Measure G
inst✝¹ : IsHaarMeasure μ
inst✝ : LocallyCompactSpace G
this✝ : IsHaarMeasure (map inv μ)
c : ℝ≥0∞
left✝¹ : c ≠ 0
left✝ : c ≠ ⊤
hc : map inv μ = c • μ
this : map ((fun a => a⁻¹) ∘ fun a => a⁻¹) μ = c ^ 2 • μ
⊢ μ = c ^ 2 • μ
[PROOFSTEP]
simpa only [inv_involutive, Involutive.comp_self, map_id]
[GOAL]
case inv_eq_self.intro.intro.intro
G : Type u_1
inst✝⁸ : CommGroup G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : TopologicalGroup G
inst✝⁵ : T2Space G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
inst✝² : SecondCountableTopology G
μ : Measure G
inst✝¹ : IsHaarMeasure μ
inst✝ : LocallyCompactSpace G
this✝ : IsHaarMeasure (map inv μ)
c : ℝ≥0∞
left✝¹ : c ≠ 0
left✝ : c ≠ ⊤
hc : map inv μ = c • μ
this : map inv (map inv μ) = c ^ 2 • μ
μeq : μ = c ^ 2 • μ
⊢ Measure.inv μ = μ
[PROOFSTEP]
have K : PositiveCompacts G := Classical.arbitrary _
[GOAL]
case inv_eq_self.intro.intro.intro
G : Type u_1
inst✝⁸ : CommGroup G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : TopologicalGroup G
inst✝⁵ : T2Space G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
inst✝² : SecondCountableTopology G
μ : Measure G
inst✝¹ : IsHaarMeasure μ
inst✝ : LocallyCompactSpace G
this✝ : IsHaarMeasure (map inv μ)
c : ℝ≥0∞
left✝¹ : c ≠ 0
left✝ : c ≠ ⊤
hc : map inv μ = c • μ
this : map inv (map inv μ) = c ^ 2 • μ
μeq : μ = c ^ 2 • μ
K : PositiveCompacts G
⊢ Measure.inv μ = μ
[PROOFSTEP]
have : c ^ 2 * μ K = 1 ^ 2 * μ K := by
  conv_rhs => rw [μeq]
  simp
[GOAL]
G : Type u_1
inst✝⁸ : CommGroup G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : TopologicalGroup G
inst✝⁵ : T2Space G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
inst✝² : SecondCountableTopology G
μ : Measure G
inst✝¹ : IsHaarMeasure μ
inst✝ : LocallyCompactSpace G
this✝ : IsHaarMeasure (map inv μ)
c : ℝ≥0∞
left✝¹ : c ≠ 0
left✝ : c ≠ ⊤
hc : map inv μ = c • μ
this : map inv (map inv μ) = c ^ 2 • μ
μeq : μ = c ^ 2 • μ
K : PositiveCompacts G
⊢ c ^ 2 * ↑↑μ ↑K = 1 ^ 2 * ↑↑μ ↑K
[PROOFSTEP]
conv_rhs => rw [μeq]
[GOAL]
G : Type u_1
inst✝⁸ : CommGroup G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : TopologicalGroup G
inst✝⁵ : T2Space G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
inst✝² : SecondCountableTopology G
μ : Measure G
inst✝¹ : IsHaarMeasure μ
inst✝ : LocallyCompactSpace G
this✝ : IsHaarMeasure (map inv μ)
c : ℝ≥0∞
left✝¹ : c ≠ 0
left✝ : c ≠ ⊤
hc : map inv μ = c • μ
this : map inv (map inv μ) = c ^ 2 • μ
μeq : μ = c ^ 2 • μ
K : PositiveCompacts G
| 1 ^ 2 * ↑↑μ ↑K
[PROOFSTEP]
rw [μeq]
[GOAL]
G : Type u_1
inst✝⁸ : CommGroup G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : TopologicalGroup G
inst✝⁵ : T2Space G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
inst✝² : SecondCountableTopology G
μ : Measure G
inst✝¹ : IsHaarMeasure μ
inst✝ : LocallyCompactSpace G
this✝ : IsHaarMeasure (map inv μ)
c : ℝ≥0∞
left✝¹ : c ≠ 0
left✝ : c ≠ ⊤
hc : map inv μ = c • μ
this : map inv (map inv μ) = c ^ 2 • μ
μeq : μ = c ^ 2 • μ
K : PositiveCompacts G
| 1 ^ 2 * ↑↑μ ↑K
[PROOFSTEP]
rw [μeq]
[GOAL]
G : Type u_1
inst✝⁸ : CommGroup G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : TopologicalGroup G
inst✝⁵ : T2Space G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
inst✝² : SecondCountableTopology G
μ : Measure G
inst✝¹ : IsHaarMeasure μ
inst✝ : LocallyCompactSpace G
this✝ : IsHaarMeasure (map inv μ)
c : ℝ≥0∞
left✝¹ : c ≠ 0
left✝ : c ≠ ⊤
hc : map inv μ = c • μ
this : map inv (map inv μ) = c ^ 2 • μ
μeq : μ = c ^ 2 • μ
K : PositiveCompacts G
| 1 ^ 2 * ↑↑μ ↑K
[PROOFSTEP]
rw [μeq]
[GOAL]
G : Type u_1
inst✝⁸ : CommGroup G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : TopologicalGroup G
inst✝⁵ : T2Space G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
inst✝² : SecondCountableTopology G
μ : Measure G
inst✝¹ : IsHaarMeasure μ
inst✝ : LocallyCompactSpace G
this✝ : IsHaarMeasure (map inv μ)
c : ℝ≥0∞
left✝¹ : c ≠ 0
left✝ : c ≠ ⊤
hc : map inv μ = c • μ
this : map inv (map inv μ) = c ^ 2 • μ
μeq : μ = c ^ 2 • μ
K : PositiveCompacts G
⊢ c ^ 2 * ↑↑μ ↑K = 1 ^ 2 * ↑↑(c ^ 2 • μ) ↑K
[PROOFSTEP]
simp
[GOAL]
case inv_eq_self.intro.intro.intro
G : Type u_1
inst✝⁸ : CommGroup G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : TopologicalGroup G
inst✝⁵ : T2Space G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
inst✝² : SecondCountableTopology G
μ : Measure G
inst✝¹ : IsHaarMeasure μ
inst✝ : LocallyCompactSpace G
this✝¹ : IsHaarMeasure (map inv μ)
c : ℝ≥0∞
left✝¹ : c ≠ 0
left✝ : c ≠ ⊤
hc : map inv μ = c • μ
this✝ : map inv (map inv μ) = c ^ 2 • μ
μeq : μ = c ^ 2 • μ
K : PositiveCompacts G
this : c ^ 2 * ↑↑μ ↑K = 1 ^ 2 * ↑↑μ ↑K
⊢ Measure.inv μ = μ
[PROOFSTEP]
have : c ^ 2 = 1 ^ 2 :=
  (ENNReal.mul_eq_mul_right (measure_pos_of_nonempty_interior _ K.interior_nonempty).ne'
        K.isCompact.measure_lt_top.ne).1
    this
[GOAL]
case inv_eq_self.intro.intro.intro
G : Type u_1
inst✝⁸ : CommGroup G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : TopologicalGroup G
inst✝⁵ : T2Space G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
inst✝² : SecondCountableTopology G
μ : Measure G
inst✝¹ : IsHaarMeasure μ
inst✝ : LocallyCompactSpace G
this✝² : IsHaarMeasure (map inv μ)
c : ℝ≥0∞
left✝¹ : c ≠ 0
left✝ : c ≠ ⊤
hc : map inv μ = c • μ
this✝¹ : map inv (map inv μ) = c ^ 2 • μ
μeq : μ = c ^ 2 • μ
K : PositiveCompacts G
this✝ : c ^ 2 * ↑↑μ ↑K = 1 ^ 2 * ↑↑μ ↑K
this : c ^ 2 = 1 ^ 2
⊢ Measure.inv μ = μ
[PROOFSTEP]
have : c = 1 := (ENNReal.pow_strictMono two_ne_zero).injective this
[GOAL]
case inv_eq_self.intro.intro.intro
G : Type u_1
inst✝⁸ : CommGroup G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : TopologicalGroup G
inst✝⁵ : T2Space G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
inst✝² : SecondCountableTopology G
μ : Measure G
inst✝¹ : IsHaarMeasure μ
inst✝ : LocallyCompactSpace G
this✝³ : IsHaarMeasure (map inv μ)
c : ℝ≥0∞
left✝¹ : c ≠ 0
left✝ : c ≠ ⊤
hc : map inv μ = c • μ
this✝² : map inv (map inv μ) = c ^ 2 • μ
μeq : μ = c ^ 2 • μ
K : PositiveCompacts G
this✝¹ : c ^ 2 * ↑↑μ ↑K = 1 ^ 2 * ↑↑μ ↑K
this✝ : c ^ 2 = 1 ^ 2
this : c = 1
⊢ Measure.inv μ = μ
[PROOFSTEP]
rw [Measure.inv, hc, this, one_smul]
[GOAL]
G : Type u_1
inst✝⁹ : CommGroup G
inst✝⁸ : TopologicalSpace G
inst✝⁷ : TopologicalGroup G
inst✝⁶ : T2Space G
inst✝⁵ : MeasurableSpace G
inst✝⁴ : BorelSpace G
inst✝³ : SecondCountableTopology G
μ : Measure G
inst✝² : IsHaarMeasure μ
inst✝¹ : CompactSpace G
inst✝ : RootableBy G ℤ
n : ℤ
hn : n ≠ 0
⊢ map (fun g => g ^ n) μ = μ
[PROOFSTEP]
let f := @zpowGroupHom G _ n
[GOAL]
G : Type u_1
inst✝⁹ : CommGroup G
inst✝⁸ : TopologicalSpace G
inst✝⁷ : TopologicalGroup G
inst✝⁶ : T2Space G
inst✝⁵ : MeasurableSpace G
inst✝⁴ : BorelSpace G
inst✝³ : SecondCountableTopology G
μ : Measure G
inst✝² : IsHaarMeasure μ
inst✝¹ : CompactSpace G
inst✝ : RootableBy G ℤ
n : ℤ
hn : n ≠ 0
f : G →* G := zpowGroupHom n
⊢ map (fun g => g ^ n) μ = μ
[PROOFSTEP]
have hf : Continuous f := continuous_zpow n
[GOAL]
G : Type u_1
inst✝⁹ : CommGroup G
inst✝⁸ : TopologicalSpace G
inst✝⁷ : TopologicalGroup G
inst✝⁶ : T2Space G
inst✝⁵ : MeasurableSpace G
inst✝⁴ : BorelSpace G
inst✝³ : SecondCountableTopology G
μ : Measure G
inst✝² : IsHaarMeasure μ
inst✝¹ : CompactSpace G
inst✝ : RootableBy G ℤ
n : ℤ
hn : n ≠ 0
f : G →* G := zpowGroupHom n
hf : Continuous ↑f
⊢ map (fun g => g ^ n) μ = μ
[PROOFSTEP]
haveI : (μ.map f).IsHaarMeasure := isHaarMeasure_map μ f hf (RootableBy.surjective_pow G ℤ hn) (by simp)
[GOAL]
G : Type u_1
inst✝⁹ : CommGroup G
inst✝⁸ : TopologicalSpace G
inst✝⁷ : TopologicalGroup G
inst✝⁶ : T2Space G
inst✝⁵ : MeasurableSpace G
inst✝⁴ : BorelSpace G
inst✝³ : SecondCountableTopology G
μ : Measure G
inst✝² : IsHaarMeasure μ
inst✝¹ : CompactSpace G
inst✝ : RootableBy G ℤ
n : ℤ
hn : n ≠ 0
f : G →* G := zpowGroupHom n
hf : Continuous ↑f
⊢ Filter.Tendsto (↑f) (Filter.cocompact G) (Filter.cocompact G)
[PROOFSTEP]
simp
[GOAL]
G : Type u_1
inst✝⁹ : CommGroup G
inst✝⁸ : TopologicalSpace G
inst✝⁷ : TopologicalGroup G
inst✝⁶ : T2Space G
inst✝⁵ : MeasurableSpace G
inst✝⁴ : BorelSpace G
inst✝³ : SecondCountableTopology G
μ : Measure G
inst✝² : IsHaarMeasure μ
inst✝¹ : CompactSpace G
inst✝ : RootableBy G ℤ
n : ℤ
hn : n ≠ 0
f : G →* G := zpowGroupHom n
hf : Continuous ↑f
this : IsHaarMeasure (map (↑f) μ)
⊢ map (fun g => g ^ n) μ = μ
[PROOFSTEP]
obtain ⟨C, -, -, hC⟩ := isHaarMeasure_eq_smul_isHaarMeasure (μ.map f) μ
[GOAL]
case intro.intro.intro
G : Type u_1
inst✝⁹ : CommGroup G
inst✝⁸ : TopologicalSpace G
inst✝⁷ : TopologicalGroup G
inst✝⁶ : T2Space G
inst✝⁵ : MeasurableSpace G
inst✝⁴ : BorelSpace G
inst✝³ : SecondCountableTopology G
μ : Measure G
inst✝² : IsHaarMeasure μ
inst✝¹ : CompactSpace G
inst✝ : RootableBy G ℤ
n : ℤ
hn : n ≠ 0
f : G →* G := zpowGroupHom n
hf : Continuous ↑f
this : IsHaarMeasure (map (↑f) μ)
C : ℝ≥0∞
hC : map (↑f) μ = C • μ
⊢ map (fun g => g ^ n) μ = μ
[PROOFSTEP]
suffices C = 1 by rwa [this, one_smul] at hC
[GOAL]
G : Type u_1
inst✝⁹ : CommGroup G
inst✝⁸ : TopologicalSpace G
inst✝⁷ : TopologicalGroup G
inst✝⁶ : T2Space G
inst✝⁵ : MeasurableSpace G
inst✝⁴ : BorelSpace G
inst✝³ : SecondCountableTopology G
μ : Measure G
inst✝² : IsHaarMeasure μ
inst✝¹ : CompactSpace G
inst✝ : RootableBy G ℤ
n : ℤ
hn : n ≠ 0
f : G →* G := zpowGroupHom n
hf : Continuous ↑f
this✝ : IsHaarMeasure (map (↑f) μ)
C : ℝ≥0∞
hC : map (↑f) μ = C • μ
this : C = 1
⊢ map (fun g => g ^ n) μ = μ
[PROOFSTEP]
rwa [this, one_smul] at hC
[GOAL]
case intro.intro.intro
G : Type u_1
inst✝⁹ : CommGroup G
inst✝⁸ : TopologicalSpace G
inst✝⁷ : TopologicalGroup G
inst✝⁶ : T2Space G
inst✝⁵ : MeasurableSpace G
inst✝⁴ : BorelSpace G
inst✝³ : SecondCountableTopology G
μ : Measure G
inst✝² : IsHaarMeasure μ
inst✝¹ : CompactSpace G
inst✝ : RootableBy G ℤ
n : ℤ
hn : n ≠ 0
f : G →* G := zpowGroupHom n
hf : Continuous ↑f
this : IsHaarMeasure (map (↑f) μ)
C : ℝ≥0∞
hC : map (↑f) μ = C • μ
⊢ C = 1
[PROOFSTEP]
have h_univ : (μ.map f) univ = μ univ := by
  rw [map_apply_of_aemeasurable hf.measurable.aemeasurable MeasurableSet.univ, preimage_univ]
[GOAL]
G : Type u_1
inst✝⁹ : CommGroup G
inst✝⁸ : TopologicalSpace G
inst✝⁷ : TopologicalGroup G
inst✝⁶ : T2Space G
inst✝⁵ : MeasurableSpace G
inst✝⁴ : BorelSpace G
inst✝³ : SecondCountableTopology G
μ : Measure G
inst✝² : IsHaarMeasure μ
inst✝¹ : CompactSpace G
inst✝ : RootableBy G ℤ
n : ℤ
hn : n ≠ 0
f : G →* G := zpowGroupHom n
hf : Continuous ↑f
this : IsHaarMeasure (map (↑f) μ)
C : ℝ≥0∞
hC : map (↑f) μ = C • μ
⊢ ↑↑(map (↑f) μ) univ = ↑↑μ univ
[PROOFSTEP]
rw [map_apply_of_aemeasurable hf.measurable.aemeasurable MeasurableSet.univ, preimage_univ]
[GOAL]
case intro.intro.intro
G : Type u_1
inst✝⁹ : CommGroup G
inst✝⁸ : TopologicalSpace G
inst✝⁷ : TopologicalGroup G
inst✝⁶ : T2Space G
inst✝⁵ : MeasurableSpace G
inst✝⁴ : BorelSpace G
inst✝³ : SecondCountableTopology G
μ : Measure G
inst✝² : IsHaarMeasure μ
inst✝¹ : CompactSpace G
inst✝ : RootableBy G ℤ
n : ℤ
hn : n ≠ 0
f : G →* G := zpowGroupHom n
hf : Continuous ↑f
this : IsHaarMeasure (map (↑f) μ)
C : ℝ≥0∞
hC : map (↑f) μ = C • μ
h_univ : ↑↑(map (↑f) μ) univ = ↑↑μ univ
⊢ C = 1
[PROOFSTEP]
have hμ₀ : μ univ ≠ 0 := IsOpenPosMeasure.open_pos univ isOpen_univ univ_nonempty
[GOAL]
case intro.intro.intro
G : Type u_1
inst✝⁹ : CommGroup G
inst✝⁸ : TopologicalSpace G
inst✝⁷ : TopologicalGroup G
inst✝⁶ : T2Space G
inst✝⁵ : MeasurableSpace G
inst✝⁴ : BorelSpace G
inst✝³ : SecondCountableTopology G
μ : Measure G
inst✝² : IsHaarMeasure μ
inst✝¹ : CompactSpace G
inst✝ : RootableBy G ℤ
n : ℤ
hn : n ≠ 0
f : G →* G := zpowGroupHom n
hf : Continuous ↑f
this : IsHaarMeasure (map (↑f) μ)
C : ℝ≥0∞
hC : map (↑f) μ = C • μ
h_univ : ↑↑(map (↑f) μ) univ = ↑↑μ univ
hμ₀ : ↑↑μ univ ≠ 0
⊢ C = 1
[PROOFSTEP]
have hμ₁ : μ univ ≠ ∞ := CompactSpace.isFiniteMeasure.measure_univ_lt_top.ne
[GOAL]
case intro.intro.intro
G : Type u_1
inst✝⁹ : CommGroup G
inst✝⁸ : TopologicalSpace G
inst✝⁷ : TopologicalGroup G
inst✝⁶ : T2Space G
inst✝⁵ : MeasurableSpace G
inst✝⁴ : BorelSpace G
inst✝³ : SecondCountableTopology G
μ : Measure G
inst✝² : IsHaarMeasure μ
inst✝¹ : CompactSpace G
inst✝ : RootableBy G ℤ
n : ℤ
hn : n ≠ 0
f : G →* G := zpowGroupHom n
hf : Continuous ↑f
this : IsHaarMeasure (map (↑f) μ)
C : ℝ≥0∞
hC : map (↑f) μ = C • μ
h_univ : ↑↑(map (↑f) μ) univ = ↑↑μ univ
hμ₀ : ↑↑μ univ ≠ 0
hμ₁ : ↑↑μ univ ≠ ⊤
⊢ C = 1
[PROOFSTEP]
rwa [hC, smul_apply, Algebra.id.smul_eq_mul, mul_comm, ← ENNReal.eq_div_iff hμ₀ hμ₁, ENNReal.div_self hμ₀ hμ₁] at h_univ