/-- If `f_n → g` pointwise and the derivatives `(f_n)' → h` _uniformly_ converge, then
in fact for a fixed `y`, the difference quotients `‖z - y‖⁻¹ • (f_n z - f_n y)` converge
_uniformly_ to `‖z - y‖⁻¹ • (g z - g y)` -/
theorem difference_quotients_converge_uniformly (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))) :
    TendstoUniformlyOnFilter (fun n : ι => fun y : E => (‖y - x‖⁻¹ : 𝕜) • (f n y - f n x))
      (fun y : E => (‖y - x‖⁻¹ : 𝕜) • (g y - g x)) l (𝓝 x) := by
  let : NormedSpace ℝ E; exact NormedSpace.restrictScalars ℝ 𝕜 _
  rcases eq_or_ne l ⊥ with (hl | hl)
  · simp only [hl, TendstoUniformlyOnFilter, bot_prod, eventually_bot, imp_true_iff]
  haveI : NeBot l := ⟨hl⟩
  refine'
    UniformCauchySeqOnFilter.tendstoUniformlyOnFilter_of_tendsto _
      ((hfg.and (eventually_const.mpr hfg.self_of_nhds)).mono fun y hy =>
        (hy.1.sub hy.2).const_smul _)
  rw [SeminormedAddGroup.uniformCauchySeqOnFilter_iff_tendstoUniformlyOnFilter_zero]
  rw [Metric.tendstoUniformlyOnFilter_iff]
  have hfg' := hf'.uniformCauchySeqOnFilter
  rw [SeminormedAddGroup.uniformCauchySeqOnFilter_iff_tendstoUniformlyOnFilter_zero] at hfg'
  rw [Metric.tendstoUniformlyOnFilter_iff] at hfg'
  intro ε hε
  obtain ⟨q, hqpos, hqε⟩ := exists_pos_rat_lt hε
  specialize hfg' (q : ℝ) (by simp [hqpos])
  have := (tendsto_swap4_prod.eventually (hf.prod_mk hf)).diag_of_prod_right
  obtain ⟨a, b, c, d, e⟩ := eventually_prod_iff.1 (hfg'.and this)
  obtain ⟨r, hr, hr'⟩ := Metric.nhds_basis_ball.eventually_iff.mp d
  rw [eventually_prod_iff]
  refine'
    ⟨_, b, fun e : E => Metric.ball x r e,
      eventually_mem_set.mpr (Metric.nhds_basis_ball.mem_of_mem hr), fun {n} hn {y} hy => _⟩
  simp only [Pi.zero_apply, dist_zero_left]
  

Proof state:
case inr.intro.intro.intro.intro.intro.intro.intro.intro
ι : Type u_1
l : Filter ι
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
𝕜 : Type u_3
inst✝³ : IsROrC 𝕜
inst✝² : NormedSpace 𝕜 E
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : 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
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 : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E
hl : l ≠ ⊥
this✝ : NeBot l
ε : ℝ
hε : ε > 0
q : ℚ
hqpos : 0 < q
hqε : ↑q < ε
hfg' : ∀ᶠ (n : (ι × ι) × E) in (l ×ˢ l) ×ˢ 𝓝 x, dist (OfNat.ofNat 0 n.2) (f' n.1.1 n.2 - f' n.1.2 n.2) < ↑q
this :
  ∀ᶠ (x : (ι × ι) × E) in (l ×ˢ l) ×ˢ 𝓝 x,
    HasFDerivAt (f (((x.1, x.2, x.2).1.1, (x.1, x.2, x.2).2.1), (x.1, x.2, x.2).1.2, (x.1, x.2, x.2).2.2).1.1)
        (f' (((x.1, x.2, x.2).1.1, (x.1, x.2, x.2).2.1), (x.1, x.2, x.2).1.2, (x.1, x.2, x.2).2.2).1.1
          (((x.1, x.2, x.2).1.1, (x.1, x.2, x.2).2.1), (x.1, x.2, x.2).1.2, (x.1, x.2, x.2).2.2).1.2)
        (((x.1, x.2, x.2).1.1, (x.1, x.2, x.2).2.1), (x.1, x.2, x.2).1.2, (x.1, x.2, x.2).2.2).1.2 ∧
      HasFDerivAt (f (((x.1, x.2, x.2).1.1, (x.1, x.2, x.2).2.1), (x.1, x.2, x.2).1.2, (x.1, x.2, x.2).2.2).2.1)
        (f' (((x.1, x.2, x.2).1.1, (x.1, x.2, x.2).2.1), (x.1, x.2, x.2).1.2, (x.1, x.2, x.2).2.2).2.1
          (((x.1, x.2, x.2).1.1, (x.1, x.2, x.2).2.1), (x.1, x.2, x.2).1.2, (x.1, x.2, x.2).2.2).2.2)
        (((x.1, x.2, x.2).1.1, (x.1, x.2, x.2).2.1), (x.1, x.2, x.2).1.2, (x.1, x.2, x.2).2.2).2.2
a : ι × ι → Prop
b : ∀ᶠ (x : ι × ι) in l ×ˢ l, a x
c : E → Prop
d : ∀ᶠ (y : E) in 𝓝 x, c y
e :
  ∀ {x : ι × ι},
    a x →
      ∀ {y : E},
        c y →
          dist (OfNat.ofNat 0 (x, y).2) (f' (x, y).1.1 (x, y).2 - f' (x, y).1.2 (x, y).2) < ↑q ∧
            HasFDerivAt
                (f
                  ((((x, y).1, (x, y).2, (x, y).2).1.1, ((x, y).1, (x, y).2, (x, y).2).2.1),
                        ((x, y).1, (x, y).2, (x, y).2).1.2, ((x, y).1, (x, y).2, (x, y).2).2.2).1.1)
                (f'
                  ((((x, y).1, (x, y).2, (x, y).2).1.1, ((x, y).1, (x, y).2, (x, y).2).2.1),
                        ((x, y).1, (x, y).2, (x, y).2).1.2, ((x, y).1, (x, y).2, (x, y).2).2.2).1.1
                  ((((x, y).1, (x, y).2, (x, y).2).1.1, ((x, y).1, (x, y).2, (x, y).2).2.1),
                        ((x, y).1, (x, y).2, (x, y).2).1.2, ((x, y).1, (x, y).2, (x, y).2).2.2).1.2)
                ((((x, y).1, (x, y).2, (x, y).2).1.1, ((x, y).1, (x, y).2, (x, y).2).2.1),
                      ((x, y).1, (x, y).2, (x, y).2).1.2, ((x, y).1, (x, y).2, (x, y).2).2.2).1.2 ∧
              HasFDerivAt
                (f
                  ((((x, y).1, (x, y).2, (x, y).2).1.1, ((x, y).1, (x, y).2, (x, y).2).2.1),
                        ((x, y).1, (x, y).2, (x, y).2).1.2, ((x, y).1, (x, y).2, (x, y).2).2.2).2.1)
                (f'
                  ((((x, y).1, (x, y).2, (x, y).2).1.1, ((x, y).1, (x, y).2, (x, y).2).2.1),
                        ((x, y).1, (x, y).2, (x, y).2).1.2, ((x, y).1, (x, y).2, (x, y).2).2.2).2.1
                  ((((x, y).1, (x, y).2, (x, y).2).1.1, ((x, y).1, (x, y).2, (x, y).2).2.1),
                        ((x, y).1, (x, y).2, (x, y).2).1.2, ((x, y).1, (x, y).2, (x, y).2).2.2).2.2)
                ((((x, y).1, (x, y).2, (x, y).2).1.1, ((x, y).1, (x, y).2, (x, y).2).2.1),
                      ((x, y).1, (x, y).2, (x, y).2).1.2, ((x, y).1, (x, y).2, (x, y).2).2.2).2.2
r : ℝ
hr : 0 < r
hr' : ∀ ⦃x_1 : E⦄, x_1 ∈ Metric.ball x r → c x_1
n : ι × ι
hn : a n
y : E
hy : (fun e => Metric.ball x r e) y
⊢ ‖(↑‖y - x‖)⁻¹ • (f n.1 y - f n.1 x) - (↑‖y - x‖)⁻¹ • (f n.2 y - f n.2 x)‖ < ε

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