Theorem: isOpenMap_smul_of_sigmaCompact
File: Mathlib/Topology/Algebra/Group/OpenMapping.lean

tactic: simp_rw [isOpenMap_iff_nhds_le, Filter.le_map_iff]
state_before: G : Type u_1
X : Type u_2
inst✝⁹ : TopologicalSpace G
inst✝⁸ : TopologicalSpace X
inst✝⁷ : Group G
inst✝⁶ : TopologicalGroup G
inst✝⁵ : MulAction G X
inst✝⁴ : SigmaCompactSpace G
inst✝³ : BaireSpace X
inst✝² : T2Space X
inst✝¹ : ContinuousSMul G X
inst✝ : IsPretransitive G X
x : X
⊢ IsOpenMap fun g => g • x
state_after: G : Type u_1
X : Type u_2
inst✝⁹ : TopologicalSpace G
inst✝⁸ : TopologicalSpace X
inst✝⁷ : Group G
inst✝⁶ : TopologicalGroup G
inst✝⁵ : MulAction G X
inst✝⁴ : SigmaCompactSpace G
inst✝³ : BaireSpace X
inst✝² : T2Space X
inst✝¹ : ContinuousSMul G X
inst✝ : IsPretransitive G X
x : X
⊢ ∀ (x_1 : G), ∀ s ∈ 𝓝 x_1, (fun g => g • x) '' s ∈ 𝓝 (x_1 • x)

tactic: intro g U hU
state_before: G : Type u_1
X : Type u_2
inst✝⁹ : TopologicalSpace G
inst✝⁸ : TopologicalSpace X
inst✝⁷ : Group G
inst✝⁶ : TopologicalGroup G
inst✝⁵ : MulAction G X
inst✝⁴ : SigmaCompactSpace G
inst✝³ : BaireSpace X
inst✝² : T2Space X
inst✝¹ : ContinuousSMul G X
inst✝ : IsPretransitive G X
x : X
⊢ ∀ (x_1 : G), ∀ s ∈ 𝓝 x_1, (fun g => g • x) '' s ∈ 𝓝 (x_1 • x)
state_after: G : Type u_1
X : Type u_2
inst✝⁹ : TopologicalSpace G
inst✝⁸ : TopologicalSpace X
inst✝⁷ : Group G
inst✝⁶ : TopologicalGroup G
inst✝⁵ : MulAction G X
inst✝⁴ : SigmaCompactSpace G
inst✝³ : BaireSpace X
inst✝² : T2Space X
inst✝¹ : ContinuousSMul G X
inst✝ : IsPretransitive G X
x : X
g : G
U : Set G
hU : U ∈ 𝓝 g
⊢ (fun g => g • x) '' U ∈ 𝓝 (g • x)

tactic: rw [this, image_comp, ← smul_singleton]
state_before: G : Type u_1
X : Type u_2
inst✝⁹ : TopologicalSpace G
inst✝⁸ : TopologicalSpace X
inst✝⁷ : Group G
inst✝⁶ : TopologicalGroup G
inst✝⁵ : MulAction G X
inst✝⁴ : SigmaCompactSpace G
inst✝³ : BaireSpace X
inst✝² : T2Space X
inst✝¹ : ContinuousSMul G X
inst✝ : IsPretransitive G X
x : X
g : G
U : Set G
hU : U ∈ 𝓝 g
this : (fun x_1 => x_1 • x) = (fun x_1 => x_1 • g • x) ∘ fun x => x * g⁻¹
⊢ (fun g => g • x) '' U ∈ 𝓝 (g • x)
state_after: G : Type u_1
X : Type u_2
inst✝⁹ : TopologicalSpace G
inst✝⁸ : TopologicalSpace X
inst✝⁷ : Group G
inst✝⁶ : TopologicalGroup G
inst✝⁵ : MulAction G X
inst✝⁴ : SigmaCompactSpace G
inst✝³ : BaireSpace X
inst✝² : T2Space X
inst✝¹ : ContinuousSMul G X
inst✝ : IsPretransitive G X
x : X
g : G
U : Set G
hU : U ∈ 𝓝 g
this : (fun x_1 => x_1 • x) = (fun x_1 => x_1 • g • x) ∘ fun x => x * g⁻¹
⊢ (fun x => x * g⁻¹) '' U • {g • x} ∈ 𝓝 (g • x)

tactic: apply smul_singleton_mem_nhds_of_sigmaCompact
state_before: G : Type u_1
X : Type u_2
inst✝⁹ : TopologicalSpace G
inst✝⁸ : TopologicalSpace X
inst✝⁷ : Group G
inst✝⁶ : TopologicalGroup G
inst✝⁵ : MulAction G X
inst✝⁴ : SigmaCompactSpace G
inst✝³ : BaireSpace X
inst✝² : T2Space X
inst✝¹ : ContinuousSMul G X
inst✝ : IsPretransitive G X
x : X
g : G
U : Set G
hU : U ∈ 𝓝 g
this : (fun x_1 => x_1 • x) = (fun x_1 => x_1 • g • x) ∘ fun x => x * g⁻¹
⊢ (fun x => x * g⁻¹) '' U • {g • x} ∈ 𝓝 (g • x)
state_after: case hU
G : Type u_1
X : Type u_2
inst✝⁹ : TopologicalSpace G
inst✝⁸ : TopologicalSpace X
inst✝⁷ : Group G
inst✝⁶ : TopologicalGroup G
inst✝⁵ : MulAction G X
inst✝⁴ : SigmaCompactSpace G
inst✝³ : BaireSpace X
inst✝² : T2Space X
inst✝¹ : ContinuousSMul G X
inst✝ : IsPretransitive G X
x : X
g : G
U : Set G
hU : U ∈ 𝓝 g
this : (fun x_1 => x_1 • x) = (fun x_1 => x_1 • g • x) ∘ fun x => x * g⁻¹
⊢ (fun x => x * g⁻¹) '' U ∈ 𝓝 1

tactic: simpa using isOpenMap_mul_right g⁻¹ |>.image_mem_nhds hU
state_before: case hU
G : Type u_1
X : Type u_2
inst✝⁹ : TopologicalSpace G
inst✝⁸ : TopologicalSpace X
inst✝⁷ : Group G
inst✝⁶ : TopologicalGroup G
inst✝⁵ : MulAction G X
inst✝⁴ : SigmaCompactSpace G
inst✝³ : BaireSpace X
inst✝² : T2Space X
inst✝¹ : ContinuousSMul G X
inst✝ : IsPretransitive G X
x : X
g : G
U : Set G
hU : U ∈ 𝓝 g
this : (fun x_1 => x_1 • x) = (fun x_1 => x_1 • g • x) ∘ fun x => x * g⁻¹
⊢ (fun x => x * g⁻¹) '' U ∈ 𝓝 1
state_after: no goals

tactic: ext g
state_before: G : Type u_1
X : Type u_2
inst✝⁹ : TopologicalSpace G
inst✝⁸ : TopologicalSpace X
inst✝⁷ : Group G
inst✝⁶ : TopologicalGroup G
inst✝⁵ : MulAction G X
inst✝⁴ : SigmaCompactSpace G
inst✝³ : BaireSpace X
inst✝² : T2Space X
inst✝¹ : ContinuousSMul G X
inst✝ : IsPretransitive G X
x : X
g : G
U : Set G
hU : U ∈ 𝓝 g
⊢ (fun x_1 => x_1 • x) = (fun x_1 => x_1 • g • x) ∘ fun x => x * g⁻¹
state_after: case h
G : Type u_1
X : Type u_2
inst✝⁹ : TopologicalSpace G
inst✝⁸ : TopologicalSpace X
inst✝⁷ : Group G
inst✝⁶ : TopologicalGroup G
inst✝⁵ : MulAction G X
inst✝⁴ : SigmaCompactSpace G
inst✝³ : BaireSpace X
inst✝² : T2Space X
inst✝¹ : ContinuousSMul G X
inst✝ : IsPretransitive G X
x : X
g✝ : G
U : Set G
hU : U ∈ 𝓝 g✝
g : G
⊢ g • x = ((fun x_1 => x_1 • g✝ • x) ∘ fun x => x * g✝⁻¹) g

tactic: simp [smul_smul]
state_before: case h
G : Type u_1
X : Type u_2
inst✝⁹ : TopologicalSpace G
inst✝⁸ : TopologicalSpace X
inst✝⁷ : Group G
inst✝⁶ : TopologicalGroup G
inst✝⁵ : MulAction G X
inst✝⁴ : SigmaCompactSpace G
inst✝³ : BaireSpace X
inst✝² : T2Space X
inst✝¹ : ContinuousSMul G X
inst✝ : IsPretransitive G X
x : X
g✝ : G
U : Set G
hU : U ∈ 𝓝 g✝
g : G
⊢ g • x = ((fun x_1 => x_1 • g✝ • x) ∘ fun x => x * g✝⁻¹) g
state_after: no goals