G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : ContinuousMul G
H : Subgroup G
g : G
hg : ↑H ∈ 𝓝 g
x : G
hx : x ∈ ↑H
⊢ ↑H ∈ 𝓝 x
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : ContinuousMul G
H : Subgroup G
g : G
hg : ↑H ∈ 𝓝 g
⊢ IsOpen ↑H
refine' isOpen_iff_mem_nhds.2 fun x hx => _
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : ContinuousMul G
H : Subgroup G
g : G
hg : ↑H ∈ 𝓝 g
x : G
hx : x ∈ ↑H
hg' : g ∈ H
⊢ ↑H ∈ 𝓝 x
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : ContinuousMul G
H : Subgroup G
g : G
hg : ↑H ∈ 𝓝 g
x : G
hx : x ∈ ↑H
⊢ ↑H ∈ 𝓝 x
have hg' : g ∈ H := SetLike.mem_coe.1 (mem_of_mem_nhds hg)
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : ContinuousMul G
H : Subgroup G
g : G
hg : ↑H ∈ 𝓝 g
x : G
hx : x ∈ ↑H
hg' : g ∈ H
this : Filter.Tendsto (fun y => y * (x⁻¹ * g)) (𝓝 x) (𝓝 g)
⊢ ↑H ∈ 𝓝 x
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : ContinuousMul G
H : Subgroup G
g : G
hg : ↑H ∈ 𝓝 g
x : G
hx : x ∈ ↑H
hg' : g ∈ H
⊢ ↑H ∈ 𝓝 x
have : Filter.Tendsto (fun y => y * (x⁻¹ * g)) (𝓝 x) (𝓝 g) :=
  (continuous_id.mul continuous_const).tendsto' _ _ (mul_inv_cancel_left _ _)
no goals
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : ContinuousMul G
H : Subgroup G
g : G
hg : ↑H ∈ 𝓝 g
x : G
hx : x ∈ ↑H
hg' : g ∈ H
this : Filter.Tendsto (fun y => y * (x⁻¹ * g)) (𝓝 x) (𝓝 g)
⊢ ↑H ∈ 𝓝 x
simpa only [SetLike.mem_coe, Filter.mem_map',
  H.mul_mem_cancel_right (H.mul_mem (H.inv_mem hx) hg')] using this hg