-- Mathlib/Analysis/Calculus/UniformLimitsDeriv.lean
theorem hasDerivAt_of_tendstoUniformlyOnFilter [NeBot l]
    (hf' : TendstoUniformlyOnFilter f' g' l (𝓝 x))
    (hf : ∀ᶠ n : ι × 𝕜 in l ×ˢ 𝓝 x, HasDerivAt (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))) : HasDerivAt g (g' x) x

Proof state:
case intro.intro
ι : Type u_1
l : Filter ι
𝕜 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
G : Type u_3
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
f : ι → 𝕜 → G
g : 𝕜 → G
f' : ι → 𝕜 → G
g' : 𝕜 → G
x : 𝕜
inst✝¹ : IsRCLikeNormedField 𝕜
inst✝ : l.NeBot
hf' : ∀ ε > 0, ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, dist (g' n.2) (f' n.1 n.2) < ε
hfg : ∀ᶠ (y : 𝕜) in 𝓝 x, Tendsto (fun n => f n y) l (𝓝 (g y))
F' : ι → 𝕜 → 𝕜 →L[𝕜] G := fun n z => ContinuousLinearMap.smulRight 1 (f' n z)
G' : 𝕜 → 𝕜 →L[𝕜] G := fun z => ContinuousLinearMap.smulRight 1 (g' z)
hf : ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (ContinuousLinearMap.smulRight 1 (f' n.1 n.2)) n.2
ε : ℝ
hε : ε > 0
q : ℝ
hq : 0 < q
hq' : q < ε
n : ι × 𝕜
hn : ‖g' n.2 - f' n.1 n.2‖ < q
⊢ ∀ (x : 𝕜), ‖(G' n.2 - F' n.1 n.2) x‖ ≤ q * ‖x‖

Tactic:
intro z

Result:
case intro.intro
ι : Type u_1
l : Filter ι
𝕜 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
G : Type u_3
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
f : ι → 𝕜 → G
g : 𝕜 → G
f' : ι → 𝕜 → G
g' : 𝕜 → G
x : 𝕜
inst✝¹ : IsRCLikeNormedField 𝕜
inst✝ : l.NeBot
hf' : ∀ ε > 0, ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, dist (g' n.2) (f' n.1 n.2) < ε
hfg : ∀ᶠ (y : 𝕜) in 𝓝 x, Tendsto (fun n => f n y) l (𝓝 (g y))
F' : ι → 𝕜 → 𝕜 →L[𝕜] G := fun n z => ContinuousLinearMap.smulRight 1 (f' n z)
G' : 𝕜 → 𝕜 →L[𝕜] G := fun z => ContinuousLinearMap.smulRight 1 (g' z)
hf : ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (ContinuousLinearMap.smulRight 1 (f' n.1 n.2)) n.2
ε : ℝ
hε : ε > 0
q : ℝ
hq : 0 < q
hq' : q < ε
n : ι × 𝕜
hn : ‖g' n.2 - f' n.1 n.2‖ < q
z : 𝕜
⊢ ‖(G' n.2 - F' n.1 n.2) z‖ ≤ q * ‖z‖

Proof state:
ι : Type u_1
l : Filter ι
𝕜 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
G : Type u_3
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
f : ι → 𝕜 → G
g : 𝕜 → G
f' : ι → 𝕜 → G
g' : 𝕜 → G
x : 𝕜
inst✝¹ : IsRCLikeNormedField 𝕜
inst✝ : l.NeBot
hf' : TendstoUniformlyOnFilter f' g' l (𝓝 x)
hf : ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, HasDerivAt (f n.1) (f' n.1 n.2) n.2
hfg : ∀ᶠ (y : 𝕜) in 𝓝 x, Tendsto (fun n => f n y) l (𝓝 (g y))
F' : ι → 𝕜 → 𝕜 →L[𝕜] G := fun n z => ContinuousLinearMap.smulRight 1 (f' n z)
G' : 𝕜 → 𝕜 →L[𝕜] G := fun z => ContinuousLinearMap.smulRight 1 (g' z)
⊢ HasDerivAt g (g' x) x

Tactic:
simp_rw [hasDerivAt_iff_hasFDerivAt] at hf ⊢

Result:
ι : Type u_1
l : Filter ι
𝕜 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
G : Type u_3
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
f : ι → 𝕜 → G
g : 𝕜 → G
f' : ι → 𝕜 → G
g' : 𝕜 → G
x : 𝕜
inst✝¹ : IsRCLikeNormedField 𝕜
inst✝ : l.NeBot
hf' : TendstoUniformlyOnFilter f' g' l (𝓝 x)
hfg : ∀ᶠ (y : 𝕜) in 𝓝 x, Tendsto (fun n => f n y) l (𝓝 (g y))
F' : ι → 𝕜 → 𝕜 →L[𝕜] G := fun n z => ContinuousLinearMap.smulRight 1 (f' n z)
G' : 𝕜 → 𝕜 →L[𝕜] G := fun z => ContinuousLinearMap.smulRight 1 (g' z)
hf : ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (ContinuousLinearMap.smulRight 1 (f' n.1 n.2)) n.2
⊢ HasFDerivAt g (ContinuousLinearMap.smulRight 1 (g' x)) x

Proof state:
case intro.intro
ι : Type u_1
l : Filter ι
𝕜 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
G : Type u_3
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
f : ι → 𝕜 → G
g : 𝕜 → G
f' : ι → 𝕜 → G
g' : 𝕜 → G
x : 𝕜
inst✝¹ : IsRCLikeNormedField 𝕜
inst✝ : l.NeBot
hf' : ∀ ε > 0, ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, dist (g' n.2) (f' n.1 n.2) < ε
hfg : ∀ᶠ (y : 𝕜) in 𝓝 x, Tendsto (fun n => f n y) l (𝓝 (g y))
F' : ι → 𝕜 → 𝕜 →L[𝕜] G := fun n z => ContinuousLinearMap.smulRight 1 (f' n z)
G' : 𝕜 → 𝕜 →L[𝕜] G := fun z => ContinuousLinearMap.smulRight 1 (g' z)
hf : ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (ContinuousLinearMap.smulRight 1 (f' n.1 n.2)) n.2
ε : ℝ
hε : ε > 0
q : ℝ
hq : 0 < q
hq' : q < ε
n : ι × 𝕜
hn : ‖g' n.2 - f' n.1 n.2‖ < q
⊢ ‖G' n.2 - F' n.1 n.2‖ ≤ q

Tactic:
refine ContinuousLinearMap.opNorm_le_bound _ hq.le ?_

Result:
case intro.intro
ι : Type u_1
l : Filter ι
𝕜 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
G : Type u_3
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
f : ι → 𝕜 → G
g : 𝕜 → G
f' : ι → 𝕜 → G
g' : 𝕜 → G
x : 𝕜
inst✝¹ : IsRCLikeNormedField 𝕜
inst✝ : l.NeBot
hf' : ∀ ε > 0, ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, dist (g' n.2) (f' n.1 n.2) < ε
hfg : ∀ᶠ (y : 𝕜) in 𝓝 x, Tendsto (fun n => f n y) l (𝓝 (g y))
F' : ι → 𝕜 → 𝕜 →L[𝕜] G := fun n z => ContinuousLinearMap.smulRight 1 (f' n z)
G' : 𝕜 → 𝕜 →L[𝕜] G := fun z => ContinuousLinearMap.smulRight 1 (g' z)
hf : ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (ContinuousLinearMap.smulRight 1 (f' n.1 n.2)) n.2
ε : ℝ
hε : ε > 0
q : ℝ
hq : 0 < q
hq' : q < ε
n : ι × 𝕜
hn : ‖g' n.2 - f' n.1 n.2‖ < q
⊢ ∀ (x : 𝕜), ‖(G' n.2 - F' n.1 n.2) x‖ ≤ q * ‖x‖

Proof state:
ι : Type u_1
l : Filter ι
𝕜 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
G : Type u_3
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
f : ι → 𝕜 → G
g : 𝕜 → G
f' : ι → 𝕜 → G
g' : 𝕜 → G
x : 𝕜
inst✝¹ : IsRCLikeNormedField 𝕜
inst✝ : l.NeBot
hf' : TendstoUniformlyOnFilter f' g' l (𝓝 x)
hf : ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, HasDerivAt (f n.1) (f' n.1 n.2) n.2
hfg : ∀ᶠ (y : 𝕜) in 𝓝 x, Tendsto (fun n => f n y) l (𝓝 (g y))
⊢ HasDerivAt g (g' x) x

Tactic:
let F' n z := (1 : 𝕜 →L[𝕜] 𝕜).smulRight (f' n z)

Result:
ι : Type u_1
l : Filter ι
𝕜 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
G : Type u_3
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
f : ι → 𝕜 → G
g : 𝕜 → G
f' : ι → 𝕜 → G
g' : 𝕜 → G
x : 𝕜
inst✝¹ : IsRCLikeNormedField 𝕜
inst✝ : l.NeBot
hf' : TendstoUniformlyOnFilter f' g' l (𝓝 x)
hf : ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, HasDerivAt (f n.1) (f' n.1 n.2) n.2
hfg : ∀ᶠ (y : 𝕜) in 𝓝 x, Tendsto (fun n => f n y) l (𝓝 (g y))
F' : ι → 𝕜 → 𝕜 →L[𝕜] G := fun n z => ContinuousLinearMap.smulRight 1 (f' n z)
⊢ HasDerivAt g (g' x) x

Proof state:
ι : Type u_1
l : Filter ι
𝕜 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
G : Type u_3
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
f : ι → 𝕜 → G
g : 𝕜 → G
f' : ι → 𝕜 → G
g' : 𝕜 → G
x : 𝕜
inst✝¹ : IsRCLikeNormedField 𝕜
inst✝ : l.NeBot
hf' : ∀ ε > 0, ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, dist (g' n.2) (f' n.1 n.2) < ε
hfg : ∀ᶠ (y : 𝕜) in 𝓝 x, Tendsto (fun n => f n y) l (𝓝 (g y))
F' : ι → 𝕜 → 𝕜 →L[𝕜] G := fun n z => ContinuousLinearMap.smulRight 1 (f' n z)
G' : 𝕜 → 𝕜 →L[𝕜] G := fun z => ContinuousLinearMap.smulRight 1 (g' z)
hf : ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (ContinuousLinearMap.smulRight 1 (f' n.1 n.2)) n.2
ε : ℝ
hε : ε > 0
⊢ ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, dist (G' n.2) (F' n.1 n.2) < ε

Tactic:
obtain ⟨q, hq, hq'⟩ := exists_between hε.lt

Result:
case intro.intro
ι : Type u_1
l : Filter ι
𝕜 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
G : Type u_3
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
f : ι → 𝕜 → G
g : 𝕜 → G
f' : ι → 𝕜 → G
g' : 𝕜 → G
x : 𝕜
inst✝¹ : IsRCLikeNormedField 𝕜
inst✝ : l.NeBot
hf' : ∀ ε > 0, ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, dist (g' n.2) (f' n.1 n.2) < ε
hfg : ∀ᶠ (y : 𝕜) in 𝓝 x, Tendsto (fun n => f n y) l (𝓝 (g y))
F' : ι → 𝕜 → 𝕜 →L[𝕜] G := fun n z => ContinuousLinearMap.smulRight 1 (f' n z)
G' : 𝕜 → 𝕜 →L[𝕜] G := fun z => ContinuousLinearMap.smulRight 1 (g' z)
hf : ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (ContinuousLinearMap.smulRight 1 (f' n.1 n.2)) n.2
ε : ℝ
hε : ε > 0
q : ℝ
hq : 0 < q
hq' : q < ε
⊢ ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, dist (G' n.2) (F' n.1 n.2) < ε