l3lab/ntp-mathlib huggingface dataset:
/-- Consider a sigma-compact group acting continuously and transitively on a Baire space. Then
the orbit map is open around the identity. It follows in `isOpenMap_smul_of_sigmaCompact` that it
is open around any point. -/
@[to_additive "Consider a sigma-compact additive group acting continuously and transitively on a
Baire space. Then the orbit map is open around zero. It follows in
`isOpenMap_vadd_of_sigmaCompact` that it is open around any point."]
theorem smul_singleton_mem_nhds_of_sigmaCompact
    {U : Set G} (hU : U ∈ 𝓝 1) (x : X) : U • {x} ∈ 𝓝 x := by
  /- Consider a small closed neighborhood `V` of the identity. Then the group is covered by
  countably many translates of `V`, say `gᵢ V`. Let also `Kₙ` be a sequence of compact sets covering
  the space. Then the image of `Kₙ ∩ gᵢ V` in the orbit is compact, and their unions covers the
  space. By Baire, one of them has nonempty interior. Then `gᵢ V • x` has nonempty interior, and
  so does `V • x`. Its interior contains a point `g' x` with `g' ∈ V`. Then `g'⁻¹ • V • x` contains
  a neighborhood of `x`, and it is included in `V⁻¹ • V • x`, which is itself contained in `U • x`
  if `V` is small enough. -/
  obtain ⟨V, V_mem, V_closed, V_symm, VU⟩ : ∃ V ∈ 𝓝 (1 : G), IsClosed V ∧ V⁻¹ = V ∧ V * V ⊆ U :=
    exists_closed_nhds_one_inv_eq_mul_subset hU
  obtain ⟨s, s_count, hs⟩ : ∃ (s : Set G), s.Countable ∧ ⋃ g ∈ s, g • V = univ := by
    apply countable_cover_nhds_of_sigma_compact (fun g ↦ ?_)
    convert smul_mem_nhds g V_mem
    simp only [smul_eq_mul, mul_one]
  let K : ℕ → Set G := compactCovering G
  let F : ℕ × s → Set X := fun p ↦ (K p.1 ∩ (p.2 : G) • V) • ({x} : Set X)
  obtain ⟨⟨n, ⟨g, hg⟩⟩, hi⟩ : ∃ i, (interior (F i)).Nonempty := by
    have : Nonempty X := ⟨x⟩
    

Proof state:
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
U : Set G
hU : U ∈ 𝓝 1
x : X
V : Set G
V_mem : V ∈ 𝓝 1
V_closed : IsClosed V
V_symm : V⁻¹ = V
VU : V * V ⊆ U
s : Set G
s_count : Set.Countable s
hs : ⋃ g ∈ s, g • V = univ
K : ℕ → Set G := compactCovering G
F : ℕ × ↑s → Set X := fun p => (K p.1 ∩ ↑p.2 • V) • {x}
this : Nonempty X
⊢ ∃ i, Set.Nonempty (interior (F i))

Next tactic:
have : Encodable s := Countable.toEncodable s_count
/-- Consider a sigma-compact group acting continuously and transitively on a Baire space. Then
the orbit map is open around the identity. It follows in `isOpenMap_smul_of_sigmaCompact` that it
is open around any point. -/
@[to_additive "Consider a sigma-compact additive group acting continuously and transitively on a
Baire space. Then the orbit map is open around zero. It follows in
`isOpenMap_vadd_of_sigmaCompact` that it is open around any point."]
theorem smul_singleton_mem_nhds_of_sigmaCompact
    {U : Set G} (hU : U ∈ 𝓝 1) (x : X) : U • {x} ∈ 𝓝 x := by
  /- Consider a small closed neighborhood `V` of the identity. Then the group is covered by
  countably many translates of `V`, say `gᵢ V`. Let also `Kₙ` be a sequence of compact sets covering
  the space. Then the image of `Kₙ ∩ gᵢ V` in the orbit is compact, and their unions covers the
  space. By Baire, one of them has nonempty interior. Then `gᵢ V • x` has nonempty interior, and
  so does `V • x`. Its interior contains a point `g' x` with `g' ∈ V`. Then `g'⁻¹ • V • x` contains
  a neighborhood of `x`, and it is included in `V⁻¹ • V • x`, which is itself contained in `U • x`
  if `V` is small enough. -/
  obtain ⟨V, V_mem, V_closed, V_symm, VU⟩ : ∃ V ∈ 𝓝 (1 : G), IsClosed V ∧ V⁻¹ = V ∧ V * V ⊆ U :=
    exists_closed_nhds_one_inv_eq_mul_subset hU
  obtain ⟨s, s_count, hs⟩ : ∃ (s : Set G), s.Countable ∧ ⋃ g ∈ s, g • V = univ := by
    apply countable_cover_nhds_of_sigma_compact (fun g ↦ ?_)
    convert smul_mem_nhds g V_mem
    simp only [smul_eq_mul, mul_one]
  let K : ℕ → Set G := compactCovering G
  let F : ℕ × s → Set X := fun p ↦ (K p.1 ∩ (p.2 : G) • V) • ({x} : Set X)
  obtain ⟨⟨n, ⟨g, hg⟩⟩, hi⟩ : ∃ i, (interior (F i)).Nonempty := by
    have : Nonempty X := ⟨x⟩
    have : Encodable s := Countable.toEncodable s_count
    

Proof state:
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
U : Set G
hU : U ∈ 𝓝 1
x : X
V : Set G
V_mem : V ∈ 𝓝 1
V_closed : IsClosed V
V_symm : V⁻¹ = V
VU : V * V ⊆ U
s : Set G
s_count : Set.Countable s
hs : ⋃ g ∈ s, g • V = univ
K : ℕ → Set G := compactCovering G
F : ℕ × ↑s → Set X := fun p => (K p.1 ∩ ↑p.2 • V) • {x}
this✝ : Nonempty X
this : Encodable ↑s
⊢ ∃ i, Set.Nonempty (interior (F i))

Next tactic:
apply nonempty_interior_of_iUnion_of_closed
/-- Consider a sigma-compact group acting continuously and transitively on a Baire space. Then
the orbit map is open around the identity. It follows in `isOpenMap_smul_of_sigmaCompact` that it
is open around any point. -/
@[to_additive "Consider a sigma-compact additive group acting continuously and transitively on a
Baire space. Then the orbit map is open around zero. It follows in
`isOpenMap_vadd_of_sigmaCompact` that it is open around any point."]
theorem smul_singleton_mem_nhds_of_sigmaCompact
    {U : Set G} (hU : U ∈ 𝓝 1) (x : X) : U • {x} ∈ 𝓝 x := by
  /- Consider a small closed neighborhood `V` of the identity. Then the group is covered by
  countably many translates of `V`, say `gᵢ V`. Let also `Kₙ` be a sequence of compact sets covering
  the space. Then the image of `Kₙ ∩ gᵢ V` in the orbit is compact, and their unions covers the
  space. By Baire, one of them has nonempty interior. Then `gᵢ V • x` has nonempty interior, and
  so does `V • x`. Its interior contains a point `g' x` with `g' ∈ V`. Then `g'⁻¹ • V • x` contains
  a neighborhood of `x`, and it is included in `V⁻¹ • V • x`, which is itself contained in `U • x`
  if `V` is small enough. -/
  obtain ⟨V, V_mem, V_closed, V_symm, VU⟩ : ∃ V ∈ 𝓝 (1 : G), IsClosed V ∧ V⁻¹ = V ∧ V * V ⊆ U :=
    exists_closed_nhds_one_inv_eq_mul_subset hU
  obtain ⟨s, s_count, hs⟩ : ∃ (s : Set G), s.Countable ∧ ⋃ g ∈ s, g • V = univ := by
    apply countable_cover_nhds_of_sigma_compact (fun g ↦ ?_)
    convert smul_mem_nhds g V_mem
    simp only [smul_eq_mul, mul_one]
  let K : ℕ → Set G := compactCovering G
  let F : ℕ × s → Set X := fun p ↦ (K p.1 ∩ (p.2 : G) • V) • ({x} : Set X)
  obtain ⟨⟨n, ⟨g, hg⟩⟩, hi⟩ : ∃ i, (interior (F i)).Nonempty := by
    have : Nonempty X := ⟨x⟩
    have : Encodable s := Countable.toEncodable s_count
    apply nonempty_interior_of_iUnion_of_closed
    · 

Proof state:
case hc
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
U : Set G
hU : U ∈ 𝓝 1
x : X
V : Set G
V_mem : V ∈ 𝓝 1
V_closed : IsClosed V
V_symm : V⁻¹ = V
VU : V * V ⊆ U
s : Set G
s_count : Set.Countable s
hs : ⋃ g ∈ s, g • V = univ
K : ℕ → Set G := compactCovering G
F : ℕ × ↑s → Set X := fun p => (K p.1 ∩ ↑p.2 • V) • {x}
this✝ : Nonempty X
this : Encodable ↑s
⊢ ∀ (s : ℕ × ↑s), IsClosed (F s)

Next tactic:
rintro ⟨n, ⟨g, hg⟩⟩
/-- Consider a sigma-compact group acting continuously and transitively on a Baire space. Then
the orbit map is open around the identity. It follows in `isOpenMap_smul_of_sigmaCompact` that it
is open around any point. -/
@[to_additive "Consider a sigma-compact additive group acting continuously and transitively on a
Baire space. Then the orbit map is open around zero. It follows in
`isOpenMap_vadd_of_sigmaCompact` that it is open around any point."]
theorem smul_singleton_mem_nhds_of_sigmaCompact
    {U : Set G} (hU : U ∈ 𝓝 1) (x : X) : U • {x} ∈ 𝓝 x := by
  /- Consider a small closed neighborhood `V` of the identity. Then the group is covered by
  countably many translates of `V`, say `gᵢ V`. Let also `Kₙ` be a sequence of compact sets covering
  the space. Then the image of `Kₙ ∩ gᵢ V` in the orbit is compact, and their unions covers the
  space. By Baire, one of them has nonempty interior. Then `gᵢ V • x` has nonempty interior, and
  so does `V • x`. Its interior contains a point `g' x` with `g' ∈ V`. Then `g'⁻¹ • V • x` contains
  a neighborhood of `x`, and it is included in `V⁻¹ • V • x`, which is itself contained in `U • x`
  if `V` is small enough. -/
  obtain ⟨V, V_mem, V_closed, V_symm, VU⟩ : ∃ V ∈ 𝓝 (1 : G), IsClosed V ∧ V⁻¹ = V ∧ V * V ⊆ U :=
    exists_closed_nhds_one_inv_eq_mul_subset hU
  obtain ⟨s, s_count, hs⟩ : ∃ (s : Set G), s.Countable ∧ ⋃ g ∈ s, g • V = univ := by
    apply countable_cover_nhds_of_sigma_compact (fun g ↦ ?_)
    convert smul_mem_nhds g V_mem
    simp only [smul_eq_mul, mul_one]
  let K : ℕ → Set G := compactCovering G
  let F : ℕ × s → Set X := fun p ↦ (K p.1 ∩ (p.2 : G) • V) • ({x} : Set X)
  obtain ⟨⟨n, ⟨g, hg⟩⟩, hi⟩ : ∃ i, (interior (F i)).Nonempty := by
    have : Nonempty X := ⟨x⟩
    have : Encodable s := Countable.toEncodable s_count
    apply nonempty_interior_of_iUnion_of_closed
    · rintro ⟨n, ⟨g, hg⟩⟩
      

Proof state:
case hc.mk.mk
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
U : Set G
hU : U ∈ 𝓝 1
x : X
V : Set G
V_mem : V ∈ 𝓝 1
V_closed : IsClosed V
V_symm : V⁻¹ = V
VU : V * V ⊆ U
s : Set G
s_count : Set.Countable s
hs : ⋃ g ∈ s, g • V = univ
K : ℕ → Set G := compactCovering G
F : ℕ × ↑s → Set X := fun p => (K p.1 ∩ ↑p.2 • V) • {x}
this✝ : Nonempty X
this : Encodable ↑s
n : ℕ
g : G
hg : g ∈ s
⊢ IsClosed (F (n, { val := g, property := hg }))

Next tactic:
apply IsCompact.isClosed
/-- Consider a sigma-compact group acting continuously and transitively on a Baire space. Then
the orbit map is open around the identity. It follows in `isOpenMap_smul_of_sigmaCompact` that it
is open around any point. -/
@[to_additive "Consider a sigma-compact additive group acting continuously and transitively on a
Baire space. Then the orbit map is open around zero. It follows in
`isOpenMap_vadd_of_sigmaCompact` that it is open around any point."]
theorem smul_singleton_mem_nhds_of_sigmaCompact
    {U : Set G} (hU : U ∈ 𝓝 1) (x : X) : U • {x} ∈ 𝓝 x := by
  /- Consider a small closed neighborhood `V` of the identity. Then the group is covered by
  countably many translates of `V`, say `gᵢ V`. Let also `Kₙ` be a sequence of compact sets covering
  the space. Then the image of `Kₙ ∩ gᵢ V` in the orbit is compact, and their unions covers the
  space. By Baire, one of them has nonempty interior. Then `gᵢ V • x` has nonempty interior, and
  so does `V • x`. Its interior contains a point `g' x` with `g' ∈ V`. Then `g'⁻¹ • V • x` contains
  a neighborhood of `x`, and it is included in `V⁻¹ • V • x`, which is itself contained in `U • x`
  if `V` is small enough. -/
  obtain ⟨V, V_mem, V_closed, V_symm, VU⟩ : ∃ V ∈ 𝓝 (1 : G), IsClosed V ∧ V⁻¹ = V ∧ V * V ⊆ U :=
    exists_closed_nhds_one_inv_eq_mul_subset hU
  obtain ⟨s, s_count, hs⟩ : ∃ (s : Set G), s.Countable ∧ ⋃ g ∈ s, g • V = univ := by
    apply countable_cover_nhds_of_sigma_compact (fun g ↦ ?_)
    convert smul_mem_nhds g V_mem
    simp only [smul_eq_mul, mul_one]
  let K : ℕ → Set G := compactCovering G
  let F : ℕ × s → Set X := fun p ↦ (K p.1 ∩ (p.2 : G) • V) • ({x} : Set X)
  obtain ⟨⟨n, ⟨g, hg⟩⟩, hi⟩ : ∃ i, (interior (F i)).Nonempty := by
    have : Nonempty X := ⟨x⟩
    have : Encodable s := Countable.toEncodable s_count
    apply nonempty_interior_of_iUnion_of_closed
    · rintro ⟨n, ⟨g, hg⟩⟩
      apply IsCompact.isClosed
      

Proof state:
case hc.mk.mk.hs
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
U : Set G
hU : U ∈ 𝓝 1
x : X
V : Set G
V_mem : V ∈ 𝓝 1
V_closed : IsClosed V
V_symm : V⁻¹ = V
VU : V * V ⊆ U
s : Set G
s_count : Set.Countable s
hs : ⋃ g ∈ s, g • V = univ
K : ℕ → Set G := compactCovering G
F : ℕ × ↑s → Set X := fun p => (K p.1 ∩ ↑p.2 • V) • {x}
this✝ : Nonempty X
this : Encodable ↑s
n : ℕ
g : G
hg : g ∈ s
⊢ IsCompact (F (n, { val := g, property := hg }))

Next tactic:
suffices H : IsCompact ((fun (g : G) ↦ g • x) '' (K n ∩ g • V)) by
        simpa only [smul_singleton] using H
/-- Consider a sigma-compact group acting continuously and transitively on a Baire space. Then
the orbit map is open around the identity. It follows in `isOpenMap_smul_of_sigmaCompact` that it
is open around any point. -/
@[to_additive "Consider a sigma-compact additive group acting continuously and transitively on a
Baire space. Then the orbit map is open around zero. It follows in
`isOpenMap_vadd_of_sigmaCompact` that it is open around any point."]
theorem smul_singleton_mem_nhds_of_sigmaCompact
    {U : Set G} (hU : U ∈ 𝓝 1) (x : X) : U • {x} ∈ 𝓝 x := by
  /- Consider a small closed neighborhood `V` of the identity. Then the group is covered by
  countably many translates of `V`, say `gᵢ V`. Let also `Kₙ` be a sequence of compact sets covering
  the space. Then the image of `Kₙ ∩ gᵢ V` in the orbit is compact, and their unions covers the
  space. By Baire, one of them has nonempty interior. Then `gᵢ V • x` has nonempty interior, and
  so does `V • x`. Its interior contains a point `g' x` with `g' ∈ V`. Then `g'⁻¹ • V • x` contains
  a neighborhood of `x`, and it is included in `V⁻¹ • V • x`, which is itself contained in `U • x`
  if `V` is small enough. -/
  obtain ⟨V, V_mem, V_closed, V_symm, VU⟩ : ∃ V ∈ 𝓝 (1 : G), IsClosed V ∧ V⁻¹ = V ∧ V * V ⊆ U :=
    exists_closed_nhds_one_inv_eq_mul_subset hU
  obtain ⟨s, s_count, hs⟩ : ∃ (s : Set G), s.Countable ∧ ⋃ g ∈ s, g • V = univ := by
    apply countable_cover_nhds_of_sigma_compact (fun g ↦ ?_)
    convert smul_mem_nhds g V_mem
    simp only [smul_eq_mul, mul_one]
  let K : ℕ → Set G := compactCovering G
  let F : ℕ × s → Set X := fun p ↦ (K p.1 ∩ (p.2 : G) • V) • ({x} : Set X)
  obtain ⟨⟨n, ⟨g, hg⟩⟩, hi⟩ : ∃ i, (interior (F i)).Nonempty := by
    have : Nonempty X := ⟨x⟩
    have : Encodable s := Countable.toEncodable s_count
    apply nonempty_interior_of_iUnion_of_closed
    · rintro ⟨n, ⟨g, hg⟩⟩
      apply IsCompact.isClosed
      suffices H : IsCompact ((fun (g : G) ↦ g • x) '' (K n ∩ g • V)) by
        

Proof state:
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
U : Set G
hU : U ∈ 𝓝 1
x : X
V : Set G
V_mem : V ∈ 𝓝 1
V_closed : IsClosed V
V_symm : V⁻¹ = V
VU : V * V ⊆ U
s : Set G
s_count : Set.Countable s
hs : ⋃ g ∈ s, g • V = univ
K : ℕ → Set G := compactCovering G
F : ℕ × ↑s → Set X := fun p => (K p.1 ∩ ↑p.2 • V) • {x}
this✝ : Nonempty X
this : Encodable ↑s
n : ℕ
g : G
hg : g ∈ s
H : IsCompact ((fun g => g • x) '' (K n ∩ g • V))
⊢ IsCompact (F (n, { val := g, property := hg }))

Next tactic:
simpa only [smul_singleton] using H
/-- Consider a sigma-compact group acting continuously and transitively on a Baire space. Then
the orbit map is open around the identity. It follows in `isOpenMap_smul_of_sigmaCompact` that it
is open around any point. -/
@[to_additive "Consider a sigma-compact additive group acting continuously and transitively on a
Baire space. Then the orbit map is open around zero. It follows in
`isOpenMap_vadd_of_sigmaCompact` that it is open around any point."]
theorem smul_singleton_mem_nhds_of_sigmaCompact
    {U : Set G} (hU : U ∈ 𝓝 1) (x : X) : U • {x} ∈ 𝓝 x := by
  /- Consider a small closed neighborhood `V` of the identity. Then the group is covered by
  countably many translates of `V`, say `gᵢ V`. Let also `Kₙ` be a sequence of compact sets covering
  the space. Then the image of `Kₙ ∩ gᵢ V` in the orbit is compact, and their unions covers the
  space. By Baire, one of them has nonempty interior. Then `gᵢ V • x` has nonempty interior, and
  so does `V • x`. Its interior contains a point `g' x` with `g' ∈ V`. Then `g'⁻¹ • V • x` contains
  a neighborhood of `x`, and it is included in `V⁻¹ • V • x`, which is itself contained in `U • x`
  if `V` is small enough. -/
  obtain ⟨V, V_mem, V_closed, V_symm, VU⟩ : ∃ V ∈ 𝓝 (1 : G), IsClosed V ∧ V⁻¹ = V ∧ V * V ⊆ U :=
    exists_closed_nhds_one_inv_eq_mul_subset hU
  obtain ⟨s, s_count, hs⟩ : ∃ (s : Set G), s.Countable ∧ ⋃ g ∈ s, g • V = univ := by
    apply countable_cover_nhds_of_sigma_compact (fun g ↦ ?_)
    convert smul_mem_nhds g V_mem
    simp only [smul_eq_mul, mul_one]
  let K : ℕ → Set G := compactCovering G
  let F : ℕ × s → Set X := fun p ↦ (K p.1 ∩ (p.2 : G) • V) • ({x} : Set X)
  obtain ⟨⟨n, ⟨g, hg⟩⟩, hi⟩ : ∃ i, (interior (F i)).Nonempty := by
    have : Nonempty X := ⟨x⟩
    have : Encodable s := Countable.toEncodable s_count
    apply nonempty_interior_of_iUnion_of_closed
    · rintro ⟨n, ⟨g, hg⟩⟩
      apply IsCompact.isClosed
      suffices H : IsCompact ((fun (g : G) ↦ g • x) '' (K n ∩ g • V)) by
        simpa only [smul_singleton] using H
      

Proof state:
case hc.mk.mk.hs
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
U : Set G
hU : U ∈ 𝓝 1
x : X
V : Set G
V_mem : V ∈ 𝓝 1
V_closed : IsClosed V
V_symm : V⁻¹ = V
VU : V * V ⊆ U
s : Set G
s_count : Set.Countable s
hs : ⋃ g ∈ s, g • V = univ
K : ℕ → Set G := compactCovering G
F : ℕ × ↑s → Set X := fun p => (K p.1 ∩ ↑p.2 • V) • {x}
this✝ : Nonempty X
this : Encodable ↑s
n : ℕ
g : G
hg : g ∈ s
⊢ IsCompact ((fun g => g • x) '' (K n ∩ g • V))

Next tactic:
apply IsCompact.image
/-- Consider a sigma-compact group acting continuously and transitively on a Baire space. Then
the orbit map is open around the identity. It follows in `isOpenMap_smul_of_sigmaCompact` that it
is open around any point. -/
@[to_additive "Consider a sigma-compact additive group acting continuously and transitively on a
Baire space. Then the orbit map is open around zero. It follows in
`isOpenMap_vadd_of_sigmaCompact` that it is open around any point."]
theorem smul_singleton_mem_nhds_of_sigmaCompact
    {U : Set G} (hU : U ∈ 𝓝 1) (x : X) : U • {x} ∈ 𝓝 x := by
  /- Consider a small closed neighborhood `V` of the identity. Then the group is covered by
  countably many translates of `V`, say `gᵢ V`. Let also `Kₙ` be a sequence of compact sets covering
  the space. Then the image of `Kₙ ∩ gᵢ V` in the orbit is compact, and their unions covers the
  space. By Baire, one of them has nonempty interior. Then `gᵢ V • x` has nonempty interior, and
  so does `V • x`. Its interior contains a point `g' x` with `g' ∈ V`. Then `g'⁻¹ • V • x` contains
  a neighborhood of `x`, and it is included in `V⁻¹ • V • x`, which is itself contained in `U • x`
  if `V` is small enough. -/
  obtain ⟨V, V_mem, V_closed, V_symm, VU⟩ : ∃ V ∈ 𝓝 (1 : G), IsClosed V ∧ V⁻¹ = V ∧ V * V ⊆ U :=
    exists_closed_nhds_one_inv_eq_mul_subset hU
  obtain ⟨s, s_count, hs⟩ : ∃ (s : Set G), s.Countable ∧ ⋃ g ∈ s, g • V = univ := by
    apply countable_cover_nhds_of_sigma_compact (fun g ↦ ?_)
    convert smul_mem_nhds g V_mem
    simp only [smul_eq_mul, mul_one]
  let K : ℕ → Set G := compactCovering G
  let F : ℕ × s → Set X := fun p ↦ (K p.1 ∩ (p.2 : G) • V) • ({x} : Set X)
  obtain ⟨⟨n, ⟨g, hg⟩⟩, hi⟩ : ∃ i, (interior (F i)).Nonempty := by
    have : Nonempty X := ⟨x⟩
    have : Encodable s := Countable.toEncodable s_count
    apply nonempty_interior_of_iUnion_of_closed
    · rintro ⟨n, ⟨g, hg⟩⟩
      apply IsCompact.isClosed
      suffices H : IsCompact ((fun (g : G) ↦ g • x) '' (K n ∩ g • V)) by
        simpa only [smul_singleton] using H
      apply IsCompact.image
      · 

Proof state:
case hc.mk.mk.hs.hs
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
U : Set G
hU : U ∈ 𝓝 1
x : X
V : Set G
V_mem : V ∈ 𝓝 1
V_closed : IsClosed V
V_symm : V⁻¹ = V
VU : V * V ⊆ U
s : Set G
s_count : Set.Countable s
hs : ⋃ g ∈ s, g • V = univ
K : ℕ → Set G := compactCovering G
F : ℕ × ↑s → Set X := fun p => (K p.1 ∩ ↑p.2 • V) • {x}
this✝ : Nonempty X
this : Encodable ↑s
n : ℕ
g : G
hg : g ∈ s
⊢ IsCompact (K n ∩ g • V)

Next tactic:
exact (isCompact_compactCovering G n).inter_right (V_closed.smul g)
/-- Consider a sigma-compact group acting continuously and transitively on a Baire space. Then
the orbit map is open around the identity. It follows in `isOpenMap_smul_of_sigmaCompact` that it
is open around any point. -/
@[to_additive "Consider a sigma-compact additive group acting continuously and transitively on a
Baire space. Then the orbit map is open around zero. It follows in
`isOpenMap_vadd_of_sigmaCompact` that it is open around any point."]
theorem smul_singleton_mem_nhds_of_sigmaCompact
    {U : Set G} (hU : U ∈ 𝓝 1) (x : X) : U • {x} ∈ 𝓝 x := by
  /- Consider a small closed neighborhood `V` of the identity. Then the group is covered by
  countably many translates of `V`, say `gᵢ V`. Let also `Kₙ` be a sequence of compact sets covering
  the space. Then the image of `Kₙ ∩ gᵢ V` in the orbit is compact, and their unions covers the
  space. By Baire, one of them has nonempty interior. Then `gᵢ V • x` has nonempty interior, and
  so does `V • x`. Its interior contains a point `g' x` with `g' ∈ V`. Then `g'⁻¹ • V • x` contains
  a neighborhood of `x`, and it is included in `V⁻¹ • V • x`, which is itself contained in `U • x`
  if `V` is small enough. -/
  obtain ⟨V, V_mem, V_closed, V_symm, VU⟩ : ∃ V ∈ 𝓝 (1 : G), IsClosed V ∧ V⁻¹ = V ∧ V * V ⊆ U :=
    exists_closed_nhds_one_inv_eq_mul_subset hU
  obtain ⟨s, s_count, hs⟩ : ∃ (s : Set G), s.Countable ∧ ⋃ g ∈ s, g • V = univ := by
    apply countable_cover_nhds_of_sigma_compact (fun g ↦ ?_)
    convert smul_mem_nhds g V_mem
    simp only [smul_eq_mul, mul_one]
  let K : ℕ → Set G := compactCovering G
  let F : ℕ × s → Set X := fun p ↦ (K p.1 ∩ (p.2 : G) • V) • ({x} : Set X)
  obtain ⟨⟨n, ⟨g, hg⟩⟩, hi⟩ : ∃ i, (interior (F i)).Nonempty := by
    have : Nonempty X := ⟨x⟩
    have : Encodable s := Countable.toEncodable s_count
    apply nonempty_interior_of_iUnion_of_closed
    · rintro ⟨n, ⟨g, hg⟩⟩
      apply IsCompact.isClosed
      suffices H : IsCompact ((fun (g : G) ↦ g • x) '' (K n ∩ g • V)) by
        simpa only [smul_singleton] using H
      apply IsCompact.image
      · exact (isCompact_compactCovering G n).inter_right (V_closed.smul g)
      · 

Proof state:
case hc.mk.mk.hs.hf
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
U : Set G
hU : U ∈ 𝓝 1
x : X
V : Set G
V_mem : V ∈ 𝓝 1
V_closed : IsClosed V
V_symm : V⁻¹ = V
VU : V * V ⊆ U
s : Set G
s_count : Set.Countable s
hs : ⋃ g ∈ s, g • V = univ
K : ℕ → Set G := compactCovering G
F : ℕ × ↑s → Set X := fun p => (K p.1 ∩ ↑p.2 • V) • {x}
this✝ : Nonempty X
this : Encodable ↑s
n : ℕ
g : G
hg : g ∈ s
⊢ Continuous fun g => g • x

Next tactic:
exact continuous_id.smul continuous_const