/-- `(d/dx) lim_{n → ∞} f n x = lim_{n → ∞} f' n x` when the `f' n` converge
_uniformly_ to their limit at `x`.

In words the assumptions mean the following:
  * `hf'`: The `f'` converge "uniformly at" `x` to `g'`. This does not mean that the `f' n` even
    converge away from `x`!
  * `hf`: For all `(y, n)` with `y` sufficiently close to `x` and `n` sufficiently large, `f' n` is
    the derivative of `f n`
  * `hfg`: The `f n` converge pointwise to `g` on a neighborhood of `x` -/
theorem hasFDerivAt_of_tendstoUniformlyOnFilter [NeBot l]
    (hf' : TendstoUniformlyOnFilter f' g' l (𝓝 x))
    (hf : ∀ᶠ n : ι × E in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (f' n.1 n.2) n.2)
    (hfg : ∀ᶠ y in 𝓝 x, Tendsto (fun n => f n y) l (𝓝 (g y))) : HasFDerivAt g (g' x) x := by
  -- The proof strategy follows several steps:
  --   1. The quantifiers in the definition of the derivative are
  --      `∀ ε > 0, ∃δ > 0, ∀y ∈ B_δ(x)`. We will introduce a quantifier in the middle:
  --      `∀ ε > 0, ∃N, ∀n ≥ N, ∃δ > 0, ∀y ∈ B_δ(x)` which will allow us to introduce the `f(') n`
  --   2. The order of the quantifiers `hfg` are opposite to what we need. We will be able to swap
  --      the quantifiers using the uniform convergence assumption
  rw [hasFDerivAt_iff_tendsto]
  -- Introduce extra quantifier via curried filters
  suffices
    Tendsto (fun y : ι × E => ‖y.2 - x‖⁻¹ * ‖g y.2 - g x - (g' x) (y.2 - x)‖)
      (l.curry (𝓝 x)) (𝓝 0) by
    rw [Metric.tendsto_nhds] at this ⊢
    intro ε hε
    specialize this ε hε
    rw [eventually_curry_iff] at this
    simp only at this
    exact (eventually_const.mp this).mono (by simp only [imp_self, forall_const])
  -- With the new quantifier in hand, we can perform the famous `ε/3` proof. Specifically,
  -- we will break up the limit (the difference functions minus the derivative go to 0) into 3:
  --   * The difference functions of the `f n` converge *uniformly* to the difference functions
  --     of the `g n`
  --   * The `f' n` are the derivatives of the `f n`
  --   * The `f' n` converge to `g'` at `x`
  conv =>
    congr
    ext
    

Proof state:
case f.h
ι : Type u_1
l : Filter ι
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
𝕜 : Type u_3
inst✝⁴ : IsROrC 𝕜
inst✝³ : NormedSpace 𝕜 E
G : Type u_4
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
f : ι → E → G
g : E → G
f' : ι → E → E →L[𝕜] G
g' : E → E →L[𝕜] G
x : E
inst✝ : NeBot l
hf' : TendstoUniformlyOnFilter f' g' l (𝓝 x)
hf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (f' n.1 n.2) n.2
hfg : ∀ᶠ (y : E) in 𝓝 x, Tendsto (fun n => f n y) l (𝓝 (g y))
x✝ : ι × E
| ‖x✝.2 - x‖⁻¹ * ‖g x✝.2 - g x - (g' x) (x✝.2 - x)‖
case l₁
ι : Type u_1
l : Filter ι
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
𝕜 : Type u_3
inst✝⁴ : IsROrC 𝕜
inst✝³ : NormedSpace 𝕜 E
G : Type u_4
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
f : ι → E → G
g : E → G
f' : ι → E → E →L[𝕜] G
g' : E → E →L[𝕜] G
x : E
inst✝ : NeBot l
hf' : TendstoUniformlyOnFilter f' g' l (𝓝 x)
hf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (f' n.1 n.2) n.2
hfg : ∀ᶠ (y : E) in 𝓝 x, Tendsto (fun n => f n y) l (𝓝 (g y))
| curry l (𝓝 x)
case l₂
ι : Type u_1
l : Filter ι
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
𝕜 : Type u_3
inst✝⁴ : IsROrC 𝕜
inst✝³ : NormedSpace 𝕜 E
G : Type u_4
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
f : ι → E → G
g : E → G
f' : ι → E → E →L[𝕜] G
g' : E → E →L[𝕜] G
x : E
inst✝ : NeBot l
hf' : TendstoUniformlyOnFilter f' g' l (𝓝 x)
hf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (f' n.1 n.2) n.2
hfg : ∀ᶠ (y : E) in 𝓝 x, Tendsto (fun n => f n y) l (𝓝 (g y))
| 𝓝 0

Next tactic:
rw [← abs_norm, ← abs_inv, ← @IsROrC.norm_ofReal 𝕜 _ _, IsROrC.ofReal_inv, ← norm_smul]