-- Mathlib/Analysis/Calculus/UniformLimitsDeriv.lean
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

Proof state:
ι : Type u_1
l : Filter ι
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
𝕜 : Type u_3
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : IsRCLikeNormedField 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✝ : l.NeBot
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 : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
⊢ Tendsto (fun x_1 => (↑‖x_1.2 - x‖)⁻¹ • (g x_1.2 - g x - (g' x) (x_1.2 - x))) (l.curry (𝓝 x)) (𝓝 0)

Tactic:
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]
  abel

Result:
ι : Type u_1
l : Filter ι
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
𝕜 : Type u_3
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : IsRCLikeNormedField 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✝ : l.NeBot
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✝ : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
this :
  (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)
⊢ Tendsto (fun x_1 => (↑‖x_1.2 - x‖)⁻¹ • (g x_1.2 - g x - (g' x) (x_1.2 - x))) (l.curry (𝓝 x)) (𝓝 0)

Proof state:
ι : Type u_1
l : Filter ι
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
𝕜 : Type u_3
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : IsRCLikeNormedField 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✝ : l.NeBot
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 : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
⊢ (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)

Tactic:
ext

Result:
case h
ι : Type u_1
l : Filter ι
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
𝕜 : Type u_3
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : IsRCLikeNormedField 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✝ : l.NeBot
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 : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
x✝ : ι × E
⊢ (↑‖x✝.2 - x‖)⁻¹ • (g x✝.2 - g x - (g' x) (x✝.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))
      x✝

Proof state:
ι : Type u_1
l : Filter ι
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
𝕜 : Type u_3
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : IsRCLikeNormedField 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✝ : l.NeBot
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✝ : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
this : Tendsto (fun y => ‖y.2 - x‖⁻¹ * ‖g y.2 - g x - (g' x) (y.2 - x)‖) (l.curry (𝓝 x)) (𝓝 0)
⊢ Tendsto (fun x' => ‖x' - x‖⁻¹ * ‖g x' - g x - (g' x) (x' - x)‖) (𝓝 x) (𝓝 0)

Tactic:
rw [Metric.tendsto_nhds] at this ⊢

Result:
ι : Type u_1
l : Filter ι
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
𝕜 : Type u_3
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : IsRCLikeNormedField 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✝ : l.NeBot
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✝ : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
this : ∀ ε > 0, ∀ᶠ (x_1 : ι × E) in l.curry (𝓝 x), dist (‖x_1.2 - x‖⁻¹ * ‖g x_1.2 - g x - (g' x) (x_1.2 - x)‖) 0 < ε
⊢ ∀ ε > 0, ∀ᶠ (x_1 : E) in 𝓝 x, dist (‖x_1 - x‖⁻¹ * ‖g x_1 - g x - (g' x) (x_1 - x)‖) 0 < ε