𝕜 : Type u_1
E : Type u_2
inst✝⁶ : LinearOrderedField 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : TopologicalSpace E
inst✝² : TopologicalAddGroup E
inst✝¹ : ContinuousConstSMul 𝕜 E
inst✝ : LocallyConvexSpace 𝕜 E
s t : Set E
disj : Disjoint s t
hs₁ : Convex 𝕜 s
hs₂ : IsCompact s
ht₁ : Convex 𝕜 t
ht₂ : IsClosed t
this : UniformSpace E := TopologicalAddGroup.toUniformSpace E
⊢ ∃ u v, IsOpen u ∧ IsOpen v ∧ Convex 𝕜 u ∧ Convex 𝕜 v ∧ s ⊆ u ∧ t ⊆ v ∧ Disjoint u v
𝕜 : Type u_1
E : Type u_2
inst✝⁶ : LinearOrderedField 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : TopologicalSpace E
inst✝² : TopologicalAddGroup E
inst✝¹ : ContinuousConstSMul 𝕜 E
inst✝ : LocallyConvexSpace 𝕜 E
s t : Set E
disj : Disjoint s t
hs₁ : Convex 𝕜 s
hs₂ : IsCompact s
ht₁ : Convex 𝕜 t
ht₂ : IsClosed t
⊢ ∃ u v, IsOpen u ∧ IsOpen v ∧ Convex 𝕜 u ∧ Convex 𝕜 v ∧ s ⊆ u ∧ t ⊆ v ∧ Disjoint u v
letI : UniformSpace E := TopologicalAddGroup.toUniformSpace E
𝕜 : Type u_1
E : Type u_2
inst✝⁶ : LinearOrderedField 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : TopologicalSpace E
inst✝² : TopologicalAddGroup E
inst✝¹ : ContinuousConstSMul 𝕜 E
inst✝ : LocallyConvexSpace 𝕜 E
s t : Set E
disj : Disjoint s t
hs₁ : Convex 𝕜 s
hs₂ : IsCompact s
ht₁ : Convex 𝕜 t
ht₂ : IsClosed t
this✝ : UniformSpace E := TopologicalAddGroup.toUniformSpace E
this : UniformAddGroup E
⊢ ∃ u v, IsOpen u ∧ IsOpen v ∧ Convex 𝕜 u ∧ Convex 𝕜 v ∧ s ⊆ u ∧ t ⊆ v ∧ Disjoint u v
𝕜 : Type u_1
E : Type u_2
inst✝⁶ : LinearOrderedField 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : TopologicalSpace E
inst✝² : TopologicalAddGroup E
inst✝¹ : ContinuousConstSMul 𝕜 E
inst✝ : LocallyConvexSpace 𝕜 E
s t : Set E
disj : Disjoint s t
hs₁ : Convex 𝕜 s
hs₂ : IsCompact s
ht₁ : Convex 𝕜 t
ht₂ : IsClosed t
this : UniformSpace E := TopologicalAddGroup.toUniformSpace E
⊢ ∃ u v, IsOpen u ∧ IsOpen v ∧ Convex 𝕜 u ∧ Convex 𝕜 v ∧ s ⊆ u ∧ t ⊆ v ∧ Disjoint u v
haveI : UniformAddGroup E := comm_topologicalAddGroup_is_uniform
𝕜 : Type u_1
E : Type u_2
inst✝⁶ : LinearOrderedField 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : TopologicalSpace E
inst✝² : TopologicalAddGroup E
inst✝¹ : ContinuousConstSMul 𝕜 E
inst✝ : LocallyConvexSpace 𝕜 E
s t : Set E
disj : Disjoint s t
hs₁ : Convex 𝕜 s
hs₂ : IsCompact s
ht₁ : Convex 𝕜 t
ht₂ : IsClosed t
this✝¹ : UniformSpace E := TopologicalAddGroup.toUniformSpace E
this✝ : UniformAddGroup E
this :
  HasBasis (comap (fun x => x.snd - x.fst) (𝓝 0)) (fun s => 0 ∈ s ∧ IsOpen s ∧ Convex 𝕜 s) fun i =>
    (fun x => x.snd - x.fst) ⁻¹' id i
⊢ ∃ u v, IsOpen u ∧ IsOpen v ∧ Convex 𝕜 u ∧ Convex 𝕜 v ∧ s ⊆ u ∧ t ⊆ v ∧ Disjoint u v
𝕜 : Type u_1
E : Type u_2
inst✝⁶ : LinearOrderedField 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : TopologicalSpace E
inst✝² : TopologicalAddGroup E
inst✝¹ : ContinuousConstSMul 𝕜 E
inst✝ : LocallyConvexSpace 𝕜 E
s t : Set E
disj : Disjoint s t
hs₁ : Convex 𝕜 s
hs₂ : IsCompact s
ht₁ : Convex 𝕜 t
ht₂ : IsClosed t
this✝ : UniformSpace E := TopologicalAddGroup.toUniformSpace E
this : UniformAddGroup E
⊢ ∃ u v, IsOpen u ∧ IsOpen v ∧ Convex 𝕜 u ∧ Convex 𝕜 v ∧ s ⊆ u ∧ t ⊆ v ∧ Disjoint u v
have := (LocallyConvexSpace.convex_open_basis_zero 𝕜 E).comap fun x : E × E => x.2 - x.1
𝕜 : Type u_1
E : Type u_2
inst✝⁶ : LinearOrderedField 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : TopologicalSpace E
inst✝² : TopologicalAddGroup E
inst✝¹ : ContinuousConstSMul 𝕜 E
inst✝ : LocallyConvexSpace 𝕜 E
s t : Set E
disj : Disjoint s t
hs₁ : Convex 𝕜 s
hs₂ : IsCompact s
ht₁ : Convex 𝕜 t
ht₂ : IsClosed t
this✝¹ : UniformSpace E := TopologicalAddGroup.toUniformSpace E
this✝ : UniformAddGroup E
this : HasBasis (uniformity E) (fun s => 0 ∈ s ∧ IsOpen s ∧ Convex 𝕜 s) fun i => (fun x => x.snd - x.fst) ⁻¹' id i
⊢ ∃ u v, IsOpen u ∧ IsOpen v ∧ Convex 𝕜 u ∧ Convex 𝕜 v ∧ s ⊆ u ∧ t ⊆ v ∧ Disjoint u v
𝕜 : Type u_1
E : Type u_2
inst✝⁶ : LinearOrderedField 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : TopologicalSpace E
inst✝² : TopologicalAddGroup E
inst✝¹ : ContinuousConstSMul 𝕜 E
inst✝ : LocallyConvexSpace 𝕜 E
s t : Set E
disj : Disjoint s t
hs₁ : Convex 𝕜 s
hs₂ : IsCompact s
ht₁ : Convex 𝕜 t
ht₂ : IsClosed t
this✝¹ : UniformSpace E := TopologicalAddGroup.toUniformSpace E
this✝ : UniformAddGroup E
this :
  HasBasis (comap (fun x => x.snd - x.fst) (𝓝 0)) (fun s => 0 ∈ s ∧ IsOpen s ∧ Convex 𝕜 s) fun i =>
    (fun x => x.snd - x.fst) ⁻¹' id i
⊢ ∃ u v, IsOpen u ∧ IsOpen v ∧ Convex 𝕜 u ∧ Convex 𝕜 v ∧ s ⊆ u ∧ t ⊆ v ∧ Disjoint u v
rw [← uniformity_eq_comap_nhds_zero] at this
case intro.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝⁶ : LinearOrderedField 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : TopologicalSpace E
inst✝² : TopologicalAddGroup E
inst✝¹ : ContinuousConstSMul 𝕜 E
inst✝ : LocallyConvexSpace 𝕜 E
s t : Set E
disj : Disjoint s t
hs₁ : Convex 𝕜 s
hs₂ : IsCompact s
ht₁ : Convex 𝕜 t
ht₂ : IsClosed t
this✝¹ : UniformSpace E := TopologicalAddGroup.toUniformSpace E
this✝ : UniformAddGroup E
this : HasBasis (uniformity E) (fun s => 0 ∈ s ∧ IsOpen s ∧ Convex 𝕜 s) fun i => (fun x => x.snd - x.fst) ⁻¹' id i
V : Set E
hV :
  Disjoint (⋃ (x : E) (_ : x ∈ s), UniformSpace.ball x ((fun x => x.snd - x.fst) ⁻¹' id V))
    (⋃ (x : E) (_ : x ∈ t), UniformSpace.ball x ((fun x => x.snd - x.fst) ⁻¹' id V))
hV0 : 0 ∈ V
hVopen : IsOpen V
hVconvex : Convex 𝕜 V
⊢ ∃ u v, IsOpen u ∧ IsOpen v ∧ Convex 𝕜 u ∧ Convex 𝕜 v ∧ s ⊆ u ∧ t ⊆ v ∧ Disjoint u v
𝕜 : Type u_1
E : Type u_2
inst✝⁶ : LinearOrderedField 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : TopologicalSpace E
inst✝² : TopologicalAddGroup E
inst✝¹ : ContinuousConstSMul 𝕜 E
inst✝ : LocallyConvexSpace 𝕜 E
s t : Set E
disj : Disjoint s t
hs₁ : Convex 𝕜 s
hs₂ : IsCompact s
ht₁ : Convex 𝕜 t
ht₂ : IsClosed t
this✝¹ : UniformSpace E := TopologicalAddGroup.toUniformSpace E
this✝ : UniformAddGroup E
this : HasBasis (uniformity E) (fun s => 0 ∈ s ∧ IsOpen s ∧ Convex 𝕜 s) fun i => (fun x => x.snd - x.fst) ⁻¹' id i
⊢ ∃ u v, IsOpen u ∧ IsOpen v ∧ Convex 𝕜 u ∧ Convex 𝕜 v ∧ s ⊆ u ∧ t ⊆ v ∧ Disjoint u v
rcases disj.exists_uniform_thickening_of_basis this hs₂ ht₂ with ⟨V, ⟨hV0, hVopen, hVconvex⟩, hV⟩
case intro.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝⁶ : LinearOrderedField 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : TopologicalSpace E
inst✝² : TopologicalAddGroup E
inst✝¹ : ContinuousConstSMul 𝕜 E
inst✝ : LocallyConvexSpace 𝕜 E
s t : Set E
disj : Disjoint s t
hs₁ : Convex 𝕜 s
hs₂ : IsCompact s
ht₁ : Convex 𝕜 t
ht₂ : IsClosed t
this✝¹ : UniformSpace E := TopologicalAddGroup.toUniformSpace E
this✝ : UniformAddGroup E
this : HasBasis (uniformity E) (fun s => 0 ∈ s ∧ IsOpen s ∧ Convex 𝕜 s) fun i => (fun x => x.snd - x.fst) ⁻¹' id i
V : Set E
hV :
  Disjoint (⋃ (x : E) (_ : x ∈ s), UniformSpace.ball x ((fun x => x.snd - x.fst) ⁻¹' id V))
    (⋃ (x : E) (_ : x ∈ t), UniformSpace.ball x ((fun x => x.snd - x.fst) ⁻¹' id V))
hV0 : 0 ∈ V
hVopen : IsOpen V
hVconvex : Convex 𝕜 V
⊢ Disjoint (s + V) (t + V)
case intro.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝⁶ : LinearOrderedField 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : TopologicalSpace E
inst✝² : TopologicalAddGroup E
inst✝¹ : ContinuousConstSMul 𝕜 E
inst✝ : LocallyConvexSpace 𝕜 E
s t : Set E
disj : Disjoint s t
hs₁ : Convex 𝕜 s
hs₂ : IsCompact s
ht₁ : Convex 𝕜 t
ht₂ : IsClosed t
this✝¹ : UniformSpace E := TopologicalAddGroup.toUniformSpace E
this✝ : UniformAddGroup E
this : HasBasis (uniformity E) (fun s => 0 ∈ s ∧ IsOpen s ∧ Convex 𝕜 s) fun i => (fun x => x.snd - x.fst) ⁻¹' id i
V : Set E
hV :
  Disjoint (⋃ (x : E) (_ : x ∈ s), UniformSpace.ball x ((fun x => x.snd - x.fst) ⁻¹' id V))
    (⋃ (x : E) (_ : x ∈ t), UniformSpace.ball x ((fun x => x.snd - x.fst) ⁻¹' id V))
hV0 : 0 ∈ V
hVopen : IsOpen V
hVconvex : Convex 𝕜 V
⊢ ∃ u v, IsOpen u ∧ IsOpen v ∧ Convex 𝕜 u ∧ Convex 𝕜 v ∧ s ⊆ u ∧ t ⊆ v ∧ Disjoint u v
refine' ⟨s + V, t + V, hVopen.add_left, hVopen.add_left, hs₁.add hVconvex, ht₁.add hVconvex,
  subset_add_left _ hV0, subset_add_left _ hV0, _⟩
case intro.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝⁶ : LinearOrderedField 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : TopologicalSpace E
inst✝² : TopologicalAddGroup E
inst✝¹ : ContinuousConstSMul 𝕜 E
inst✝ : LocallyConvexSpace 𝕜 E
s t : Set E
disj : Disjoint s t
hs₁ : Convex 𝕜 s
hs₂ : IsCompact s
ht₁ : Convex 𝕜 t
ht₂ : IsClosed t
this✝¹ : UniformSpace E := TopologicalAddGroup.toUniformSpace E
this✝ : UniformAddGroup E
this : HasBasis (uniformity E) (fun s => 0 ∈ s ∧ IsOpen s ∧ Convex 𝕜 s) fun i => (fun x => x.snd - x.fst) ⁻¹' id i
V : Set E
hV :
  Disjoint (⋃ (x : E) (_ : x ∈ s), UniformSpace.ball x ((fun x => x.snd - x.fst) ⁻¹' id V))
    (⋃ (x : E) (_ : x ∈ t), UniformSpace.ball x ((fun x => x.snd - x.fst) ⁻¹' id V))
hV0 : 0 ∈ V
hVopen : IsOpen V
hVconvex : Convex 𝕜 V
⊢ Disjoint (⋃ (a : E) (_ : a ∈ s), (fun x => -a + x) ⁻¹' V) (⋃ (a : E) (_ : a ∈ t), (fun x => -a + x) ⁻¹' V)
case intro.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝⁶ : LinearOrderedField 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : TopologicalSpace E
inst✝² : TopologicalAddGroup E
inst✝¹ : ContinuousConstSMul 𝕜 E
inst✝ : LocallyConvexSpace 𝕜 E
s t : Set E
disj : Disjoint s t
hs₁ : Convex 𝕜 s
hs₂ : IsCompact s
ht₁ : Convex 𝕜 t
ht₂ : IsClosed t
this✝¹ : UniformSpace E := TopologicalAddGroup.toUniformSpace E
this✝ : UniformAddGroup E
this : HasBasis (uniformity E) (fun s => 0 ∈ s ∧ IsOpen s ∧ Convex 𝕜 s) fun i => (fun x => x.snd - x.fst) ⁻¹' id i
V : Set E
hV :
  Disjoint (⋃ (x : E) (_ : x ∈ s), UniformSpace.ball x ((fun x => x.snd - x.fst) ⁻¹' id V))
    (⋃ (x : E) (_ : x ∈ t), UniformSpace.ball x ((fun x => x.snd - x.fst) ⁻¹' id V))
hV0 : 0 ∈ V
hVopen : IsOpen V
hVconvex : Convex 𝕜 V
⊢ Disjoint (s + V) (t + V)
simp_rw [← iUnion_add_left_image, image_add_left]
case intro.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝⁶ : LinearOrderedField 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : TopologicalSpace E
inst✝² : TopologicalAddGroup E
inst✝¹ : ContinuousConstSMul 𝕜 E
inst✝ : LocallyConvexSpace 𝕜 E
s t : Set E
disj : Disjoint s t
hs₁ : Convex 𝕜 s
hs₂ : IsCompact s
ht₁ : Convex 𝕜 t
ht₂ : IsClosed t
this✝¹ : UniformSpace E := TopologicalAddGroup.toUniformSpace E
this✝ : UniformAddGroup E
this : HasBasis (uniformity E) (fun s => 0 ∈ s ∧ IsOpen s ∧ Convex 𝕜 s) fun i => (fun x => x.snd - x.fst) ⁻¹' id i
V : Set E
hV0 : 0 ∈ V
hVopen : IsOpen V
hVconvex : Convex 𝕜 V
hV :
  Disjoint (⋃ (x : E) (_ : x ∈ s), (fun x => -x.fst + x.snd) ∘ Prod.mk x ⁻¹' id V)
    (⋃ (x : E) (_ : x ∈ t), (fun x => -x.fst + x.snd) ∘ Prod.mk x ⁻¹' id V)
⊢ Disjoint (⋃ (a : E) (_ : a ∈ s), (fun x => -a + x) ⁻¹' V) (⋃ (a : E) (_ : a ∈ t), (fun x => -a + x) ⁻¹' V)
case intro.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝⁶ : LinearOrderedField 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : TopologicalSpace E
inst✝² : TopologicalAddGroup E
inst✝¹ : ContinuousConstSMul 𝕜 E
inst✝ : LocallyConvexSpace 𝕜 E
s t : Set E
disj : Disjoint s t
hs₁ : Convex 𝕜 s
hs₂ : IsCompact s
ht₁ : Convex 𝕜 t
ht₂ : IsClosed t
this✝¹ : UniformSpace E := TopologicalAddGroup.toUniformSpace E
this✝ : UniformAddGroup E
this : HasBasis (uniformity E) (fun s => 0 ∈ s ∧ IsOpen s ∧ Convex 𝕜 s) fun i => (fun x => x.snd - x.fst) ⁻¹' id i
V : Set E
hV :
  Disjoint (⋃ (x : E) (_ : x ∈ s), UniformSpace.ball x ((fun x => x.snd - x.fst) ⁻¹' id V))
    (⋃ (x : E) (_ : x ∈ t), UniformSpace.ball x ((fun x => x.snd - x.fst) ⁻¹' id V))
hV0 : 0 ∈ V
hVopen : IsOpen V
hVconvex : Convex 𝕜 V
⊢ Disjoint (⋃ (a : E) (_ : a ∈ s), (fun x => -a + x) ⁻¹' V) (⋃ (a : E) (_ : a ∈ t), (fun x => -a + x) ⁻¹' V)
simp_rw [UniformSpace.ball, ← preimage_comp, sub_eq_neg_add] at hV
no goals
case intro.intro.intro.intro
𝕜 : Type u_1
E : Type u_2
inst✝⁶ : LinearOrderedField 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : TopologicalSpace E
inst✝² : TopologicalAddGroup E
inst✝¹ : ContinuousConstSMul 𝕜 E
inst✝ : LocallyConvexSpace 𝕜 E
s t : Set E
disj : Disjoint s t
hs₁ : Convex 𝕜 s
hs₂ : IsCompact s
ht₁ : Convex 𝕜 t
ht₂ : IsClosed t
this✝¹ : UniformSpace E := TopologicalAddGroup.toUniformSpace E
this✝ : UniformAddGroup E
this : HasBasis (uniformity E) (fun s => 0 ∈ s ∧ IsOpen s ∧ Convex 𝕜 s) fun i => (fun x => x.snd - x.fst) ⁻¹' id i
V : Set E
hV0 : 0 ∈ V
hVopen : IsOpen V
hVconvex : Convex 𝕜 V
hV :
  Disjoint (⋃ (x : E) (_ : x ∈ s), (fun x => -x.fst + x.snd) ∘ Prod.mk x ⁻¹' id V)
    (⋃ (x : E) (_ : x ∈ t), (fun x => -x.fst + x.snd) ∘ Prod.mk x ⁻¹' id V)
⊢ Disjoint (⋃ (a : E) (_ : a ∈ s), (fun x => -a + x) ⁻¹' V) (⋃ (a : E) (_ : a ∈ t), (fun x => -a + x) ⁻¹' V)
exact hV