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)
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
simp only [prehaar]
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)
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)
rw [div_add_div_same]
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)
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)
refine congr_arg (fun x : ℝ => x / index K₀ U) ?_
no goals
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)
exact_mod_cast index_union_eq K₁ K₂ hU h