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

tactic: let eval : (Compacts G → ℝ) → ℝ := fun f => f (K.map _ <| continuous_mul_left g) - f K
state_before: 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

tactic: have : Continuous eval := (continuous_apply (K.map _ _)).sub (continuous_apply K)
state_before: 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

tactic: rw [← sub_eq_zero]
state_before: 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

tactic: show chaar K₀ ∈ eval ⁻¹' {(0 : ℝ)}
state_before: 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

tactic: apply mem_of_subset_of_mem _ (chaar_mem_clPrehaar K₀ ⊤)
state_before: 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}

tactic: unfold clPrehaar
state_before: 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}

tactic: rw [IsClosed.closure_subset_iff]
state_before: 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}

tactic: rintro _ ⟨U, ⟨_, h2U, h3U⟩, rfl⟩
state_before: 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}

tactic: simp only [mem_singleton_iff, mem_preimage, sub_eq_zero]
state_before: 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}

tactic: apply is_left_invariant_prehaar
state_before: 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

tactic: rw [h2U.interior_eq]
state_before: 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)

tactic: exact ⟨1, h3U⟩
state_before: 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

tactic: apply continuous_iff_isClosed.mp this
state_before: 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})

tactic: exact isClosed_singleton
state_before: 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}