[GOAL]
ι : Type u_1
X : Type u_2
E : Type u_3
inst✝⁷ : TopologicalSpace X
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module ℝ E
inst✝⁴ : NormalSpace X
inst✝³ : ParacompactSpace X
inst✝² : TopologicalSpace E
inst✝¹ : ContinuousAdd E
inst✝ : ContinuousSMul ℝ E
t : X → Set E
ht : ∀ (x : X), Convex ℝ (t x)
H : ∀ (x : X), ∃ U, U ∈ 𝓝 x ∧ ∃ g, ContinuousOn g U ∧ ∀ (y : X), y ∈ U → g y ∈ t y
⊢ ∃ g, ∀ (x : X), ↑g x ∈ t x
[PROOFSTEP]
choose U hU g hgc hgt using H
[GOAL]
ι : Type u_1
X : Type u_2
E : Type u_3
inst✝⁷ : TopologicalSpace X
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module ℝ E
inst✝⁴ : NormalSpace X
inst✝³ : ParacompactSpace X
inst✝² : TopologicalSpace E
inst✝¹ : ContinuousAdd E
inst✝ : ContinuousSMul ℝ E
t : X → Set E
ht : ∀ (x : X), Convex ℝ (t x)
U : X → Set X
hU : ∀ (x : X), U x ∈ 𝓝 x
g : X → X → E
hgc : ∀ (x : X), ContinuousOn (g x) (U x)
hgt : ∀ (x y : X), y ∈ U x → g x y ∈ t y
⊢ ∃ g, ∀ (x : X), ↑g x ∈ t x
[PROOFSTEP]
obtain ⟨f, hf⟩ :=
  PartitionOfUnity.exists_isSubordinate isClosed_univ (fun x => interior (U x)) (fun x => isOpen_interior) fun x _ =>
    mem_iUnion.2 ⟨x, mem_interior_iff_mem_nhds.2 (hU x)⟩
[GOAL]
case intro
ι : Type u_1
X : Type u_2
E : Type u_3
inst✝⁷ : TopologicalSpace X
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module ℝ E
inst✝⁴ : NormalSpace X
inst✝³ : ParacompactSpace X
inst✝² : TopologicalSpace E
inst✝¹ : ContinuousAdd E
inst✝ : ContinuousSMul ℝ E
t : X → Set E
ht : ∀ (x : X), Convex ℝ (t x)
U : X → Set X
hU : ∀ (x : X), U x ∈ 𝓝 x
g : X → X → E
hgc : ∀ (x : X), ContinuousOn (g x) (U x)
hgt : ∀ (x y : X), y ∈ U x → g x y ∈ t y
f : PartitionOfUnity X X univ
hf : PartitionOfUnity.IsSubordinate f fun x => interior (U x)
⊢ ∃ g, ∀ (x : X), ↑g x ∈ t x
[PROOFSTEP]
refine'
  ⟨⟨fun x => ∑ᶠ i, f i x • g i x,
      hf.continuous_finsum_smul (fun i => isOpen_interior) fun i => (hgc i).mono interior_subset⟩,
    fun x => f.finsum_smul_mem_convex (mem_univ x) (fun i hi => hgt _ _ _) (ht _)⟩
[GOAL]
case intro
ι : Type u_1
X : Type u_2
E : Type u_3
inst✝⁷ : TopologicalSpace X
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module ℝ E
inst✝⁴ : NormalSpace X
inst✝³ : ParacompactSpace X
inst✝² : TopologicalSpace E
inst✝¹ : ContinuousAdd E
inst✝ : ContinuousSMul ℝ E
t : X → Set E
ht : ∀ (x : X), Convex ℝ (t x)
U : X → Set X
hU : ∀ (x : X), U x ∈ 𝓝 x
g : X → X → E
hgc : ∀ (x : X), ContinuousOn (g x) (U x)
hgt : ∀ (x y : X), y ∈ U x → g x y ∈ t y
f : PartitionOfUnity X X univ
hf : PartitionOfUnity.IsSubordinate f fun x => interior (U x)
x i : X
hi : ↑(↑f i) x ≠ 0
⊢ x ∈ U i
[PROOFSTEP]
exact interior_subset (hf _ <| subset_closure hi)
