/-- `(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
    rw [← abs_norm, ← abs_inv, ← @IsROrC.norm_ofReal 𝕜 _ _, IsROrC.ofReal_inv, ← norm_smul]
  rw [← tendsto_zero_iff_norm_tendsto_zero]
  have :
    (fun a : ι × E => (‖a.2 - x‖⁻¹ : 𝕜) • (g a.2 - g x - (g' x) (a.2 - x))) =
      ((fun a : ι × E => (‖a.2 - x‖⁻¹ : 𝕜) • (g a.2 - g x - (f a.1 a.2 - f a.1 x))) +
          fun a : ι × E =>
          (‖a.2 - x‖⁻¹ : 𝕜) • (f a.1 a.2 - f a.1 x - ((f' a.1 x) a.2 - (f' a.1 x) x))) +
        fun a : ι × E => (‖a.2 - x‖⁻¹ : 𝕜) • (f' a.1 x - g' x) (a.2 - x) := by
    ext; simp only [Pi.add_apply]; rw [← smul_add, ← smul_add]; congr
    simp only [map_sub, sub_add_sub_cancel, ContinuousLinearMap.coe_sub', Pi.sub_apply]
    -- Porting note: added
    abel
  simp_rw [this]
  have : 𝓝 (0 : G) = 𝓝 (0 + 0 + 0); simp only [add_zero]
  rw [this]
  refine' Tendsto.add (Tendsto.add _ _) _
  simp only
  · have := difference_quotients_converge_uniformly hf' hf hfg
    rw [Metric.tendstoUniformlyOnFilter_iff] at this
    rw [Metric.tendsto_nhds]
    intro ε hε
    apply ((this ε hε).filter_mono curry_le_prod).mono
    intro n hn
    rw [dist_eq_norm] at hn ⊢
    rw [← smul_sub] at hn
    rwa [sub_zero]
  · -- (Almost) the definition of the derivatives
    rw [Metric.tendsto_nhds]
    intro ε hε
    rw [eventually_curry_iff]
    refine' hf.curry.mono fun n hn => _
    have := hn.self_of_nhds
    rw [hasFDerivAt_iff_tendsto, Metric.tendsto_nhds] at this
    refine' (this ε hε).mono fun y hy => _
    rw [dist_eq_norm] at hy ⊢
    simp only [sub_zero, map_sub, norm_mul, norm_inv, norm_norm] at hy ⊢
    

Proof state:
case refine'_2
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))
this✝¹ :
  (fun a => (↑‖a.2 - x‖)⁻¹ • (g a.2 - g x - (g' x) (a.2 - x))) =
    ((fun a => (↑‖a.2 - x‖)⁻¹ • (g a.2 - g x - (f a.1 a.2 - f a.1 x))) + fun a =>
        (↑‖a.2 - x‖)⁻¹ • (f a.1 a.2 - f a.1 x - ((f' a.1 x) a.2 - (f' a.1 x) x))) +
      fun a => (↑‖a.2 - x‖)⁻¹ • (f' a.1 x - g' x) (a.2 - x)
this✝ : 𝓝 0 = 𝓝 (0 + 0 + 0)
ε : ℝ
hε : ε > 0
n : ι
hn : ∀ᶠ (y : E) in 𝓝 x, HasFDerivAt (f (n, y).1) (f' (n, y).1 (n, y).2) (n, y).2
this :
  ∀ ε > 0,
    ∀ᶠ (x_1 : E) in 𝓝 (n, x).2,
      dist (‖x_1 - (n, x).2‖⁻¹ * ‖f (n, x).1 x_1 - f (n, x).1 (n, x).2 - (f' (n, x).1 (n, x).2) (x_1 - (n, x).2)‖) 0 < ε
y : E
hy : ‖y - x‖⁻¹ * ‖f n y - f n x - ((f' n x) y - (f' n x) x)‖ < ε
⊢ ‖(↑‖y - x‖)⁻¹ • (f n y - f n x - ((f' n x) y - (f' n x) x))‖ < ε

Next tactic:
rw [norm_smul, norm_inv, IsROrC.norm_coe_norm]