Theorem: TopologicalSpace.PositiveCompacts.locallyCompactSpace_of_group
File: Mathlib/Topology/Algebra/Group/Compact.lean

tactic: refine' locally_compact_of_compact_nhds fun x => _
state_before: α : Type u
β : Type v
G : Type w
H : Type x
inst✝³ : TopologicalSpace G
inst✝² : Group G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K : PositiveCompacts G
⊢ LocallyCompactSpace G

tactic: obtain ⟨y, hy⟩ := K.interior_nonempty
state_before: α : Type u
β : Type v
G : Type w
H : Type x
inst✝³ : TopologicalSpace G
inst✝² : Group G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K : PositiveCompacts G
x : G
⊢ ∃ s, s ∈ 𝓝 x ∧ IsCompact s

tactic: let F := Homeomorph.mulLeft (x * y⁻¹)
state_before: case intro
α : Type u
β : Type v
G : Type w
H : Type x
inst✝³ : TopologicalSpace G
inst✝² : Group G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K : PositiveCompacts G
x y : G
hy : y ∈ interior ↑K
⊢ ∃ s, s ∈ 𝓝 x ∧ IsCompact s

tactic: refine' ⟨F '' K, _, K.isCompact.image F.continuous⟩
state_before: case intro
α : Type u
β : Type v
G : Type w
H : Type x
inst✝³ : TopologicalSpace G
inst✝² : Group G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K : PositiveCompacts G
x y : G
hy : y ∈ interior ↑K
F : G ≃ₜ G := Homeomorph.mulLeft (x * y⁻¹)
⊢ ∃ s, s ∈ 𝓝 x ∧ IsCompact s

tactic: suffices F.symm ⁻¹' K ∈ 𝓝 x by
  convert this using 1
  apply Equiv.image_eq_preimage
state_before: case intro
α : Type u
β : Type v
G : Type w
H : Type x
inst✝³ : TopologicalSpace G
inst✝² : Group G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K : PositiveCompacts G
x y : G
hy : y ∈ interior ↑K
F : G ≃ₜ G := Homeomorph.mulLeft (x * y⁻¹)
⊢ ↑F '' ↑K ∈ 𝓝 x

tactic: apply ContinuousAt.preimage_mem_nhds F.symm.continuous.continuousAt
state_before: case intro
α : Type u
β : Type v
G : Type w
H : Type x
inst✝³ : TopologicalSpace G
inst✝² : Group G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K : PositiveCompacts G
x y : G
hy : y ∈ interior ↑K
F : G ≃ₜ G := Homeomorph.mulLeft (x * y⁻¹)
⊢ ↑(Homeomorph.symm F) ⁻¹' ↑K ∈ 𝓝 x

tactic: have : F.symm x = y := by simp only [Homeomorph.mulLeft_symm, mul_inv_rev,
    inv_inv, Homeomorph.coe_mulLeft, inv_mul_cancel_right]
state_before: case intro
α : Type u
β : Type v
G : Type w
H : Type x
inst✝³ : TopologicalSpace G
inst✝² : Group G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K : PositiveCompacts G
x y : G
hy : y ∈ interior ↑K
F : G ≃ₜ G := Homeomorph.mulLeft (x * y⁻¹)
⊢ ↑K ∈ 𝓝 (↑(Homeomorph.symm F) x)

tactic: rw [this]
state_before: case intro
α : Type u
β : Type v
G : Type w
H : Type x
inst✝³ : TopologicalSpace G
inst✝² : Group G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K : PositiveCompacts G
x y : G
hy : y ∈ interior ↑K
F : G ≃ₜ G := Homeomorph.mulLeft (x * y⁻¹)
this : ↑(Homeomorph.symm F) x = y
⊢ ↑K ∈ 𝓝 (↑(Homeomorph.symm F) x)

tactic: exact mem_interior_iff_mem_nhds.1 hy
state_before: case intro
α : Type u
β : Type v
G : Type w
H : Type x
inst✝³ : TopologicalSpace G
inst✝² : Group G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K : PositiveCompacts G
x y : G
hy : y ∈ interior ↑K
F : G ≃ₜ G := Homeomorph.mulLeft (x * y⁻¹)
this : ↑(Homeomorph.symm F) x = y
⊢ ↑K ∈ 𝓝 y

tactic: convert this using 1
state_before: α : Type u
β : Type v
G : Type w
H : Type x
inst✝³ : TopologicalSpace G
inst✝² : Group G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K : PositiveCompacts G
x y : G
hy : y ∈ interior ↑K
F : G ≃ₜ G := Homeomorph.mulLeft (x * y⁻¹)
this : ↑(Homeomorph.symm F) ⁻¹' ↑K ∈ 𝓝 x
⊢ ↑F '' ↑K ∈ 𝓝 x

tactic: apply Equiv.image_eq_preimage
state_before: case h.e'_4
α : Type u
β : Type v
G : Type w
H : Type x
inst✝³ : TopologicalSpace G
inst✝² : Group G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K : PositiveCompacts G
x y : G
hy : y ∈ interior ↑K
F : G ≃ₜ G := Homeomorph.mulLeft (x * y⁻¹)
this : ↑(Homeomorph.symm F) ⁻¹' ↑K ∈ 𝓝 x
⊢ ↑F '' ↑K = ↑(Homeomorph.symm F) ⁻¹' ↑K

tactic: simp only [Homeomorph.mulLeft_symm, mul_inv_rev,
inv_inv, Homeomorph.coe_mulLeft, inv_mul_cancel_right]
state_before: α : Type u
β : Type v
G : Type w
H : Type x
inst✝³ : TopologicalSpace G
inst✝² : Group G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K : PositiveCompacts G
x y : G
hy : y ∈ interior ↑K
F : G ≃ₜ G := Homeomorph.mulLeft (x * y⁻¹)
⊢ ↑(Homeomorph.symm F) x = y