-- Mathlib/Analysis/Calculus/UniformLimitsDeriv.lean
theorem uniformCauchySeqOnFilter_of_fderiv (hf' : UniformCauchySeqOnFilter f' 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 : Cauchy (map (fun n => f n x) l)) : UniformCauchySeqOnFilter f l (𝓝 x)

Proof state:
case right.intro.intro
ι : Type u_1
l : Filter ι
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
𝕜 : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : IsRCLikeNormedField 𝕜
inst✝² : NormedSpace 𝕜 E
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : ι → E → G
f' : ι → E → E →L[𝕜] G
x : E
hf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (f' n.1 n.2) n.2
hfg : Cauchy (map (fun n => f n x) l)
this✝ : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
hf' : TendstoUniformlyOnFilter (fun n z => f' n.1 z - f' n.2 z) 0 (l ×ˢ l) (𝓝 x)
this : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E
ε : ℝ
hε : ε > 0
t : Set G
ht : t ∈ map (fun n => f n x) l
ht' : ∀ x ∈ t, ∀ y ∈ t, dist x y < ε
⊢ ∀ᶠ (n : (ι × ι) × E) in (l ×ˢ l) ×ˢ 𝓝 x, dist (0 n.2) (f n.1.1 x - f n.1.2 x) < ε

Tactic:
exact
  eventually_prod_iff.mpr
    ⟨fun n : ι × ι => f n.1 x ∈ t ∧ f n.2 x ∈ t,
      eventually_prod_iff.mpr ⟨_, ht, _, ht, fun {n} hn {n'} hn' => ⟨hn, hn'⟩⟩,
      fun _ => True,
      by simp,
      fun {n} hn {y} _ => by simpa [norm_sub_rev, dist_eq_norm] using ht' _ hn.1 _ hn.2⟩

Result:
no goals

Proof state:
ι : Type u_1
l : Filter ι
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
𝕜 : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : IsRCLikeNormedField 𝕜
inst✝² : NormedSpace 𝕜 E
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : ι → E → G
f' : ι → E → E →L[𝕜] G
x : E
hf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (f' n.1 n.2) n.2
hfg : Cauchy (map (fun n => f n x) l)
this✝ : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
hf' : TendstoUniformlyOnFilter (fun n z => f' n.1 z - f' n.2 z) 0 (l ×ˢ l) (𝓝 x)
this : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E
⊢ TendstoUniformlyOnFilter (fun n z => f n.1 z - f n.2 z) 0 (l ×ˢ l) (𝓝 x)

Tactic:
suffices
  TendstoUniformlyOnFilter (fun (n : ι × ι) (z : E) => f n.1 z - f n.2 z - (f n.1 x - f n.2 x)) 0
      (l ×ˢ l) (𝓝 x) ∧
    TendstoUniformlyOnFilter (fun (n : ι × ι) (_ : E) => f n.1 x - f n.2 x) 0 (l ×ˢ l) (𝓝 x) by
  have := this.1.add this.2
  rw [add_zero] at this
  exact this.congr (by simp)

Result:
ι : Type u_1
l : Filter ι
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
𝕜 : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : IsRCLikeNormedField 𝕜
inst✝² : NormedSpace 𝕜 E
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : ι → E → G
f' : ι → E → E →L[𝕜] G
x : E
hf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (f' n.1 n.2) n.2
hfg : Cauchy (map (fun n => f n x) l)
this✝ : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
hf' : TendstoUniformlyOnFilter (fun n z => f' n.1 z - f' n.2 z) 0 (l ×ˢ l) (𝓝 x)
this : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E
⊢ TendstoUniformlyOnFilter (fun n z => f n.1 z - f n.2 z - (f n.1 x - f n.2 x)) 0 (l ×ˢ l) (𝓝 x) ∧
    TendstoUniformlyOnFilter (fun n x_1 => f n.1 x - f n.2 x) 0 (l ×ˢ l) (𝓝 x)

Proof state:
case left.intro.intro.intro.intro.intro.intro
ι : Type u_1
l : Filter ι
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
𝕜 : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : IsRCLikeNormedField 𝕜
inst✝² : NormedSpace 𝕜 E
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : ι → E → G
f' : ι → E → E →L[𝕜] G
x : E
hf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (f' n.1 n.2) n.2
hfg : Cauchy (map (fun n => f n x) l)
this✝¹ : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
hf' : ∀ ε > 0, ∀ᶠ (n : (ι × ι) × E) in (l ×ˢ l) ×ˢ 𝓝 x, dist (0 n.2) (f' n.1.1 n.2 - f' n.1.2 n.2) < ε
this✝ : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E
ε : ℝ
hε : ε > 0
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
R : ℝ
hR : 0 < R
hR' : ∀ ⦃x_1 : E⦄, x_1 ∈ Metric.ball x R → c x_1
r : ℝ := 1 ⊓ R
hr : 0 < r
hr' : ∀ ⦃y : E⦄, y ∈ Metric.ball x r → c y
hxy : ∀ y ∈ Metric.ball x r, ‖y - x‖ < 1
hxyε : ∀ y ∈ Metric.ball x r, ε * ‖y - x‖ < ε
n : ι × ι
hn : a n
y : E
hy : (fun e => Metric.ball x r e) y
e :
  ∀ {x : ι × ι},
    a x →
      ∀ {y : E}, c y → ‖f' x.1 y - f' x.2 y‖ < ε ∧ HasFDerivAt (f x.1) (f' x.1 y) y ∧ HasFDerivAt (f x.2) (f' x.2 y) y
⊢ ‖f n.1 y - f n.2 y - (f n.1 x - f n.2 x)‖ < ε

Tactic:
refine lt_of_le_of_lt ?_ (hxyε y hy)

Result:
case left.intro.intro.intro.intro.intro.intro
ι : Type u_1
l : Filter ι
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
𝕜 : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : IsRCLikeNormedField 𝕜
inst✝² : NormedSpace 𝕜 E
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : ι → E → G
f' : ι → E → E →L[𝕜] G
x : E
hf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (f' n.1 n.2) n.2
hfg : Cauchy (map (fun n => f n x) l)
this✝¹ : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
hf' : ∀ ε > 0, ∀ᶠ (n : (ι × ι) × E) in (l ×ˢ l) ×ˢ 𝓝 x, dist (0 n.2) (f' n.1.1 n.2 - f' n.1.2 n.2) < ε
this✝ : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E
ε : ℝ
hε : ε > 0
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
R : ℝ
hR : 0 < R
hR' : ∀ ⦃x_1 : E⦄, x_1 ∈ Metric.ball x R → c x_1
r : ℝ := 1 ⊓ R
hr : 0 < r
hr' : ∀ ⦃y : E⦄, y ∈ Metric.ball x r → c y
hxy : ∀ y ∈ Metric.ball x r, ‖y - x‖ < 1
hxyε : ∀ y ∈ Metric.ball x r, ε * ‖y - x‖ < ε
n : ι × ι
hn : a n
y : E
hy : (fun e => Metric.ball x r e) y
e :
  ∀ {x : ι × ι},
    a x →
      ∀ {y : E}, c y → ‖f' x.1 y - f' x.2 y‖ < ε ∧ HasFDerivAt (f x.1) (f' x.1 y) y ∧ HasFDerivAt (f x.2) (f' x.2 y) y
⊢ ‖f n.1 y - f n.2 y - (f n.1 x - f n.2 x)‖ ≤ ε * ‖y - x‖