by_cases h : NeBot p
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hr : 0 < r
F_cont : ∀ᶠ (n : ι) in p, ContinuousOn (F n) (sphere z₀ r)
F_conv : TendstoUniformlyOn F f p (sphere z₀ r)
⊢ Tendsto (fun i => ∮ (z : ℂ) in C(z₀, r), F i z) p (𝓝 (∮ (z : ℂ) in C(z₀, r), f z))
case pos
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hr : 0 < r
F_cont : ∀ᶠ (n : ι) in p, ContinuousOn (F n) (sphere z₀ r)
F_conv : TendstoUniformlyOn F f p (sphere z₀ r)
h : NeBot p
⊢ Tendsto (fun i => ∮ (z : ℂ) in C(z₀, r), F i z) p (𝓝 (∮ (z : ℂ) in C(z₀, r), f z))

case neg
ι : Type u_1
F : ι → ℂ → ℂ
f : ℂ → ℂ
z₀ : ℂ
p : Filter ι
r : ℝ
U : Set ℂ
hr : 0 < r
F_cont : ∀ᶠ (n : ι) in p, ContinuousOn (F n) (sphere z₀ r)
F_conv : TendstoUniformlyOn F f p (sphere z₀ r)
h : ¬NeBot p
⊢ Tendsto (fun i => ∮ (z : ℂ) in C(z₀, r), F i z) p (𝓝 (∮ (z : ℂ) in C(z₀, r), f z))