Theorem: uniformCauchySeqOnFilter_of_fderiv
File: Mathlib/Analysis/Calculus/UniformLimitsDeriv.lean

tactic: letI : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 _
state_before: ι : Type u_3
l : Filter ι
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
𝕜 : Type u_4
inst✝³ : IsROrC 𝕜
inst✝² : NormedSpace 𝕜 E
G : Type u_2
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' : UniformCauchySeqOnFilter f' l (𝓝 x)
hf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.fst) (f' n.fst n.snd) n.snd
hfg : Cauchy (map (fun n => f n x) l)
⊢ UniformCauchySeqOnFilter f l (𝓝 x)

tactic: rw [SeminormedAddGroup.uniformCauchySeqOnFilter_iff_tendstoUniformlyOnFilter_zero] at hf' ⊢
state_before: ι : Type u_3
l : Filter ι
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
𝕜 : Type u_4
inst✝³ : IsROrC 𝕜
inst✝² : NormedSpace 𝕜 E
G : Type u_2
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' : UniformCauchySeqOnFilter f' l (𝓝 x)
hf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.fst) (f' n.fst n.snd) n.snd
hfg : Cauchy (map (fun n => f n x) l)
this : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E
⊢ UniformCauchySeqOnFilter f 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)
state_before: ι : Type u_3
l : Filter ι
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
𝕜 : Type u_4
inst✝³ : IsROrC 𝕜
inst✝² : NormedSpace 𝕜 E
G : Type u_2
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 (fun n z => f' n.fst z - f' n.snd z) 0 (l ×ˢ l) (𝓝 x)
hf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.fst) (f' n.fst n.snd) n.snd
hfg : Cauchy (map (fun n => f n x) l)
this : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E
⊢ TendstoUniformlyOnFilter (fun n z => f n.fst z - f n.snd z) 0 (l ×ˢ l) (𝓝 x)

tactic: constructor
state_before: ι : Type u_3
l : Filter ι
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
𝕜 : Type u_4
inst✝³ : IsROrC 𝕜
inst✝² : NormedSpace 𝕜 E
G : Type u_2
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 (fun n z => f' n.fst z - f' n.snd z) 0 (l ×ˢ l) (𝓝 x)
hf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.fst) (f' n.fst n.snd) n.snd
hfg : Cauchy (map (fun n => f n x) l)
this : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E
⊢ TendstoUniformlyOnFilter (fun n z => f n.fst z - f n.snd z - (f n.fst x - f n.snd x)) 0 (l ×ˢ l) (𝓝 x) ∧
    TendstoUniformlyOnFilter (fun n x_1 => f n.fst x - f n.snd x) 0 (l ×ˢ l) (𝓝 x)

tactic: have := this.1.add this.2
state_before: ι : Type u_3
l : Filter ι
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
𝕜 : Type u_4
inst✝³ : IsROrC 𝕜
inst✝² : NormedSpace 𝕜 E
G : Type u_2
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 (fun n z => f' n.fst z - f' n.snd z) 0 (l ×ˢ l) (𝓝 x)
hf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.fst) (f' n.fst n.snd) n.snd
hfg : Cauchy (map (fun n => f n x) l)
this✝ : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E
this :
  TendstoUniformlyOnFilter (fun n z => f n.fst z - f n.snd z - (f n.fst x - f n.snd x)) 0 (l ×ˢ l) (𝓝 x) ∧
    TendstoUniformlyOnFilter (fun n x_1 => f n.fst x - f n.snd x) 0 (l ×ˢ l) (𝓝 x)
⊢ TendstoUniformlyOnFilter (fun n z => f n.fst z - f n.snd z) 0 (l ×ˢ l) (𝓝 x)

tactic: rw [add_zero] at this
state_before: ι : Type u_3
l : Filter ι
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
𝕜 : Type u_4
inst✝³ : IsROrC 𝕜
inst✝² : NormedSpace 𝕜 E
G : Type u_2
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 (fun n z => f' n.fst z - f' n.snd z) 0 (l ×ˢ l) (𝓝 x)
hf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.fst) (f' n.fst n.snd) n.snd
hfg : Cauchy (map (fun n => f n x) l)
this✝¹ : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E
this✝ :
  TendstoUniformlyOnFilter (fun n z => f n.fst z - f n.snd z - (f n.fst x - f n.snd x)) 0 (l ×ˢ l) (𝓝 x) ∧
    TendstoUniformlyOnFilter (fun n x_1 => f n.fst x - f n.snd x) 0 (l ×ˢ l) (𝓝 x)
this :
  TendstoUniformlyOnFilter
    ((fun n z => f n.fst z - f n.snd z - (f n.fst x - f n.snd x)) + fun n x_1 => f n.fst x - f n.snd x) (0 + 0) (l ×ˢ l)
    (𝓝 x)
⊢ TendstoUniformlyOnFilter (fun n z => f n.fst z - f n.snd z) 0 (l ×ˢ l) (𝓝 x)

tactic: exact this.congr (by simp)
state_before: ι : Type u_3
l : Filter ι
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
𝕜 : Type u_4
inst✝³ : IsROrC 𝕜
inst✝² : NormedSpace 𝕜 E
G : Type u_2
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 (fun n z => f' n.fst z - f' n.snd z) 0 (l ×ˢ l) (𝓝 x)
hf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.fst) (f' n.fst n.snd) n.snd
hfg : Cauchy (map (fun n => f n x) l)
this✝¹ : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E
this✝ :
  TendstoUniformlyOnFilter (fun n z => f n.fst z - f n.snd z - (f n.fst x - f n.snd x)) 0 (l ×ˢ l) (𝓝 x) ∧
    TendstoUniformlyOnFilter (fun n x_1 => f n.fst x - f n.snd x) 0 (l ×ˢ l) (𝓝 x)
this :
  TendstoUniformlyOnFilter
    ((fun n z => f n.fst z - f n.snd z - (f n.fst x - f n.snd x)) + fun n x_1 => f n.fst x - f n.snd x) 0 (l ×ˢ l) (𝓝 x)
⊢ TendstoUniformlyOnFilter (fun n z => f n.fst z - f n.snd z) 0 (l ×ˢ l) (𝓝 x)

tactic: simp
state_before: ι : Type u_3
l : Filter ι
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
𝕜 : Type u_4
inst✝³ : IsROrC 𝕜
inst✝² : NormedSpace 𝕜 E
G : Type u_2
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 (fun n z => f' n.fst z - f' n.snd z) 0 (l ×ˢ l) (𝓝 x)
hf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.fst) (f' n.fst n.snd) n.snd
hfg : Cauchy (map (fun n => f n x) l)
this✝¹ : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E
this✝ :
  TendstoUniformlyOnFilter (fun n z => f n.fst z - f n.snd z - (f n.fst x - f n.snd x)) 0 (l ×ˢ l) (𝓝 x) ∧
    TendstoUniformlyOnFilter (fun n x_1 => f n.fst x - f n.snd x) 0 (l ×ˢ l) (𝓝 x)
this :
  TendstoUniformlyOnFilter
    ((fun n z => f n.fst z - f n.snd z - (f n.fst x - f n.snd x)) + fun n x_1 => f n.fst x - f n.snd x) 0 (l ×ˢ l) (𝓝 x)
⊢ ∀ᶠ (n : (ι × ι) × E) in (l ×ˢ l) ×ˢ 𝓝 x,
    ((fun n z => f n.fst z - f n.snd z - (f n.fst x - f n.snd x)) + fun n x_1 => f n.fst x - f n.snd x) n.fst n.snd =
      f n.fst.fst n.snd - f n.fst.snd n.snd

tactic: rw [Metric.tendstoUniformlyOnFilter_iff] at hf' ⊢
state_before: case left
ι : Type u_3
l : Filter ι
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
𝕜 : Type u_4
inst✝³ : IsROrC 𝕜
inst✝² : NormedSpace 𝕜 E
G : Type u_2
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 (fun n z => f' n.fst z - f' n.snd z) 0 (l ×ˢ l) (𝓝 x)
hf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.fst) (f' n.fst n.snd) n.snd
hfg : Cauchy (map (fun n => f n x) l)
this : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E
⊢ TendstoUniformlyOnFilter (fun n z => f n.fst z - f n.snd z - (f n.fst x - f n.snd x)) 0 (l ×ˢ l) (𝓝 x)

tactic: intro ε hε
state_before: case left
ι : Type u_3
l : Filter ι
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
𝕜 : Type u_4
inst✝³ : IsROrC 𝕜
inst✝² : NormedSpace 𝕜 E
G : Type u_2
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' :
  ∀ (ε : ℝ),
    ε > 0 →
      ∀ᶠ (n : (ι × ι) × E) in (l ×ˢ l) ×ˢ 𝓝 x, dist (OfNat.ofNat 0 n.snd) (f' n.fst.fst n.snd - f' n.fst.snd n.snd) < ε
hf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.fst) (f' n.fst n.snd) n.snd
hfg : Cauchy (map (fun n => f n x) l)
this : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E
⊢ ∀ (ε : ℝ),
    ε > 0 →
      ∀ᶠ (n : (ι × ι) × E) in (l ×ˢ l) ×ˢ 𝓝 x,
        dist (OfNat.ofNat 0 n.snd) (f n.fst.fst n.snd - f n.fst.snd n.snd - (f n.fst.fst x - f n.fst.snd x)) < ε

tactic: have := (tendsto_swap4_prod.eventually (hf.prod_mk hf)).diag_of_prod_right
state_before: case left
ι : Type u_3
l : Filter ι
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
𝕜 : Type u_4
inst✝³ : IsROrC 𝕜
inst✝² : NormedSpace 𝕜 E
G : Type u_2
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' :
  ∀ (ε : ℝ),
    ε > 0 →
      ∀ᶠ (n : (ι × ι) × E) in (l ×ˢ l) ×ˢ 𝓝 x, dist (OfNat.ofNat 0 n.snd) (f' n.fst.fst n.snd - f' n.fst.snd n.snd) < ε
hf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.fst) (f' n.fst n.snd) n.snd
hfg : Cauchy (map (fun n => f n x) l)
this : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E
ε : ℝ
hε : ε > 0
⊢ ∀ᶠ (n : (ι × ι) × E) in (l ×ˢ l) ×ˢ 𝓝 x,
    dist (OfNat.ofNat 0 n.snd) (f n.fst.fst n.snd - f n.fst.snd n.snd - (f n.fst.fst x - f n.fst.snd x)) < ε

tactic: obtain ⟨a, b, c, d, e⟩ := eventually_prod_iff.1 ((hf' ε hε).and this)
state_before: case left
ι : Type u_3
l : Filter ι
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
𝕜 : Type u_4
inst✝³ : IsROrC 𝕜
inst✝² : NormedSpace 𝕜 E
G : Type u_2
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' :
  ∀ (ε : ℝ),
    ε > 0 →
      ∀ᶠ (n : (ι × ι) × E) in (l ×ˢ l) ×ˢ 𝓝 x, dist (OfNat.ofNat 0 n.snd) (f' n.fst.fst n.snd - f' n.fst.snd n.snd) < ε
hf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.fst) (f' n.fst n.snd) n.snd
hfg : Cauchy (map (fun n => f n x) l)
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.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).fst.fst)
        (f'
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).fst.fst
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).fst.snd)
        (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
              (x.fst, x.snd, x.snd).snd.snd).fst.snd ∧
      HasFDerivAt
        (f
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).snd.fst)
        (f'
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).snd.fst
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).snd.snd)
        (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
              (x.fst, x.snd, x.snd).snd.snd).snd.snd
⊢ ∀ᶠ (n : (ι × ι) × E) in (l ×ˢ l) ×ˢ 𝓝 x,
    dist (OfNat.ofNat 0 n.snd) (f n.fst.fst n.snd - f n.fst.snd n.snd - (f n.fst.fst x - f n.fst.snd x)) < ε

tactic: obtain ⟨R, hR, hR'⟩ := Metric.nhds_basis_ball.eventually_iff.mp d
state_before: case left.intro.intro.intro.intro
ι : Type u_3
l : Filter ι
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
𝕜 : Type u_4
inst✝³ : IsROrC 𝕜
inst✝² : NormedSpace 𝕜 E
G : Type u_2
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' :
  ∀ (ε : ℝ),
    ε > 0 →
      ∀ᶠ (n : (ι × ι) × E) in (l ×ˢ l) ×ˢ 𝓝 x, dist (OfNat.ofNat 0 n.snd) (f' n.fst.fst n.snd - f' n.fst.snd n.snd) < ε
hf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.fst) (f' n.fst n.snd) n.snd
hfg : Cauchy (map (fun n => f n x) l)
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.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).fst.fst)
        (f'
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).fst.fst
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).fst.snd)
        (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
              (x.fst, x.snd, x.snd).snd.snd).fst.snd ∧
      HasFDerivAt
        (f
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).snd.fst)
        (f'
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).snd.fst
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).snd.snd)
        (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
              (x.fst, x.snd, x.snd).snd.snd).snd.snd
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).snd) (f' (x, y).fst.fst (x, y).snd - f' (x, y).fst.snd (x, y).snd) < ε ∧
            HasFDerivAt
                (f
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).fst.fst)
                (f'
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).fst.fst
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).fst.snd)
                ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                      ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                      ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).fst.snd ∧
              HasFDerivAt
                (f
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).snd.fst)
                (f'
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).snd.fst
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).snd.snd)
                ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                      ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                      ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).snd.snd
⊢ ∀ᶠ (n : (ι × ι) × E) in (l ×ˢ l) ×ˢ 𝓝 x,
    dist (OfNat.ofNat 0 n.snd) (f n.fst.fst n.snd - f n.fst.snd n.snd - (f n.fst.fst x - f n.fst.snd x)) < ε

tactic: let r := min 1 R
state_before: case left.intro.intro.intro.intro.intro.intro
ι : Type u_3
l : Filter ι
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
𝕜 : Type u_4
inst✝³ : IsROrC 𝕜
inst✝² : NormedSpace 𝕜 E
G : Type u_2
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' :
  ∀ (ε : ℝ),
    ε > 0 →
      ∀ᶠ (n : (ι × ι) × E) in (l ×ˢ l) ×ˢ 𝓝 x, dist (OfNat.ofNat 0 n.snd) (f' n.fst.fst n.snd - f' n.fst.snd n.snd) < ε
hf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.fst) (f' n.fst n.snd) n.snd
hfg : Cauchy (map (fun n => f n x) l)
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.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).fst.fst)
        (f'
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).fst.fst
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).fst.snd)
        (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
              (x.fst, x.snd, x.snd).snd.snd).fst.snd ∧
      HasFDerivAt
        (f
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).snd.fst)
        (f'
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).snd.fst
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).snd.snd)
        (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
              (x.fst, x.snd, x.snd).snd.snd).snd.snd
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).snd) (f' (x, y).fst.fst (x, y).snd - f' (x, y).fst.snd (x, y).snd) < ε ∧
            HasFDerivAt
                (f
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).fst.fst)
                (f'
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).fst.fst
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).fst.snd)
                ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                      ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                      ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).fst.snd ∧
              HasFDerivAt
                (f
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).snd.fst)
                (f'
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).snd.fst
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).snd.snd)
                ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                      ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                      ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).snd.snd
R : ℝ
hR : 0 < R
hR' : ∀ ⦃x_1 : E⦄, x_1 ∈ Metric.ball x R → c x_1
⊢ ∀ᶠ (n : (ι × ι) × E) in (l ×ˢ l) ×ˢ 𝓝 x,
    dist (OfNat.ofNat 0 n.snd) (f n.fst.fst n.snd - f n.fst.snd n.snd - (f n.fst.fst x - f n.fst.snd x)) < ε

tactic: have hr : 0 < r := by simp [hR]
state_before: case left.intro.intro.intro.intro.intro.intro
ι : Type u_3
l : Filter ι
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
𝕜 : Type u_4
inst✝³ : IsROrC 𝕜
inst✝² : NormedSpace 𝕜 E
G : Type u_2
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' :
  ∀ (ε : ℝ),
    ε > 0 →
      ∀ᶠ (n : (ι × ι) × E) in (l ×ˢ l) ×ˢ 𝓝 x, dist (OfNat.ofNat 0 n.snd) (f' n.fst.fst n.snd - f' n.fst.snd n.snd) < ε
hf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.fst) (f' n.fst n.snd) n.snd
hfg : Cauchy (map (fun n => f n x) l)
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.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).fst.fst)
        (f'
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).fst.fst
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).fst.snd)
        (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
              (x.fst, x.snd, x.snd).snd.snd).fst.snd ∧
      HasFDerivAt
        (f
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).snd.fst)
        (f'
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).snd.fst
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).snd.snd)
        (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
              (x.fst, x.snd, x.snd).snd.snd).snd.snd
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).snd) (f' (x, y).fst.fst (x, y).snd - f' (x, y).fst.snd (x, y).snd) < ε ∧
            HasFDerivAt
                (f
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).fst.fst)
                (f'
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).fst.fst
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).fst.snd)
                ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                      ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                      ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).fst.snd ∧
              HasFDerivAt
                (f
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).snd.fst)
                (f'
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).snd.fst
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).snd.snd)
                ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                      ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                      ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).snd.snd
R : ℝ
hR : 0 < R
hR' : ∀ ⦃x_1 : E⦄, x_1 ∈ Metric.ball x R → c x_1
r : ℝ := min 1 R
⊢ ∀ᶠ (n : (ι × ι) × E) in (l ×ˢ l) ×ˢ 𝓝 x,
    dist (OfNat.ofNat 0 n.snd) (f n.fst.fst n.snd - f n.fst.snd n.snd - (f n.fst.fst x - f n.fst.snd x)) < ε

tactic: have hr' : ∀ ⦃y : E⦄, y ∈ Metric.ball x r → c y := fun y hy =>
  hR' (lt_of_lt_of_le (Metric.mem_ball.mp hy) (min_le_right _ _))
state_before: case left.intro.intro.intro.intro.intro.intro
ι : Type u_3
l : Filter ι
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
𝕜 : Type u_4
inst✝³ : IsROrC 𝕜
inst✝² : NormedSpace 𝕜 E
G : Type u_2
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' :
  ∀ (ε : ℝ),
    ε > 0 →
      ∀ᶠ (n : (ι × ι) × E) in (l ×ˢ l) ×ˢ 𝓝 x, dist (OfNat.ofNat 0 n.snd) (f' n.fst.fst n.snd - f' n.fst.snd n.snd) < ε
hf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.fst) (f' n.fst n.snd) n.snd
hfg : Cauchy (map (fun n => f n x) l)
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.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).fst.fst)
        (f'
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).fst.fst
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).fst.snd)
        (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
              (x.fst, x.snd, x.snd).snd.snd).fst.snd ∧
      HasFDerivAt
        (f
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).snd.fst)
        (f'
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).snd.fst
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).snd.snd)
        (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
              (x.fst, x.snd, x.snd).snd.snd).snd.snd
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).snd) (f' (x, y).fst.fst (x, y).snd - f' (x, y).fst.snd (x, y).snd) < ε ∧
            HasFDerivAt
                (f
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).fst.fst)
                (f'
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).fst.fst
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).fst.snd)
                ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                      ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                      ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).fst.snd ∧
              HasFDerivAt
                (f
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).snd.fst)
                (f'
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).snd.fst
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).snd.snd)
                ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                      ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                      ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).snd.snd
R : ℝ
hR : 0 < R
hR' : ∀ ⦃x_1 : E⦄, x_1 ∈ Metric.ball x R → c x_1
r : ℝ := min 1 R
hr : 0 < r
⊢ ∀ᶠ (n : (ι × ι) × E) in (l ×ˢ l) ×ˢ 𝓝 x,
    dist (OfNat.ofNat 0 n.snd) (f n.fst.fst n.snd - f n.fst.snd n.snd - (f n.fst.fst x - f n.fst.snd x)) < ε

tactic: have hxy : ∀ y : E, y ∈ Metric.ball x r → ‖y - x‖ < 1 := by
  intro y hy
  rw [Metric.mem_ball, dist_eq_norm] at hy
  exact lt_of_lt_of_le hy (min_le_left _ _)
state_before: case left.intro.intro.intro.intro.intro.intro
ι : Type u_3
l : Filter ι
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
𝕜 : Type u_4
inst✝³ : IsROrC 𝕜
inst✝² : NormedSpace 𝕜 E
G : Type u_2
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' :
  ∀ (ε : ℝ),
    ε > 0 →
      ∀ᶠ (n : (ι × ι) × E) in (l ×ˢ l) ×ˢ 𝓝 x, dist (OfNat.ofNat 0 n.snd) (f' n.fst.fst n.snd - f' n.fst.snd n.snd) < ε
hf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.fst) (f' n.fst n.snd) n.snd
hfg : Cauchy (map (fun n => f n x) l)
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.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).fst.fst)
        (f'
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).fst.fst
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).fst.snd)
        (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
              (x.fst, x.snd, x.snd).snd.snd).fst.snd ∧
      HasFDerivAt
        (f
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).snd.fst)
        (f'
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).snd.fst
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).snd.snd)
        (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
              (x.fst, x.snd, x.snd).snd.snd).snd.snd
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).snd) (f' (x, y).fst.fst (x, y).snd - f' (x, y).fst.snd (x, y).snd) < ε ∧
            HasFDerivAt
                (f
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).fst.fst)
                (f'
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).fst.fst
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).fst.snd)
                ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                      ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                      ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).fst.snd ∧
              HasFDerivAt
                (f
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).snd.fst)
                (f'
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).snd.fst
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).snd.snd)
                ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                      ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                      ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).snd.snd
R : ℝ
hR : 0 < R
hR' : ∀ ⦃x_1 : E⦄, x_1 ∈ Metric.ball x R → c x_1
r : ℝ := min 1 R
hr : 0 < r
hr' : ∀ ⦃y : E⦄, y ∈ Metric.ball x r → c y
⊢ ∀ᶠ (n : (ι × ι) × E) in (l ×ˢ l) ×ˢ 𝓝 x,
    dist (OfNat.ofNat 0 n.snd) (f n.fst.fst n.snd - f n.fst.snd n.snd - (f n.fst.fst x - f n.fst.snd x)) < ε

tactic: have hxyε : ∀ y : E, y ∈ Metric.ball x r → ε * ‖y - x‖ < ε := by
  intro y hy
  exact (mul_lt_iff_lt_one_right hε.lt).mpr (hxy y hy)
state_before: case left.intro.intro.intro.intro.intro.intro
ι : Type u_3
l : Filter ι
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
𝕜 : Type u_4
inst✝³ : IsROrC 𝕜
inst✝² : NormedSpace 𝕜 E
G : Type u_2
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' :
  ∀ (ε : ℝ),
    ε > 0 →
      ∀ᶠ (n : (ι × ι) × E) in (l ×ˢ l) ×ˢ 𝓝 x, dist (OfNat.ofNat 0 n.snd) (f' n.fst.fst n.snd - f' n.fst.snd n.snd) < ε
hf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.fst) (f' n.fst n.snd) n.snd
hfg : Cauchy (map (fun n => f n x) l)
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.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).fst.fst)
        (f'
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).fst.fst
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).fst.snd)
        (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
              (x.fst, x.snd, x.snd).snd.snd).fst.snd ∧
      HasFDerivAt
        (f
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).snd.fst)
        (f'
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).snd.fst
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).snd.snd)
        (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
              (x.fst, x.snd, x.snd).snd.snd).snd.snd
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).snd) (f' (x, y).fst.fst (x, y).snd - f' (x, y).fst.snd (x, y).snd) < ε ∧
            HasFDerivAt
                (f
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).fst.fst)
                (f'
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).fst.fst
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).fst.snd)
                ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                      ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                      ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).fst.snd ∧
              HasFDerivAt
                (f
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).snd.fst)
                (f'
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).snd.fst
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).snd.snd)
                ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                      ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                      ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).snd.snd
R : ℝ
hR : 0 < R
hR' : ∀ ⦃x_1 : E⦄, x_1 ∈ Metric.ball x R → c x_1
r : ℝ := min 1 R
hr : 0 < r
hr' : ∀ ⦃y : E⦄, y ∈ Metric.ball x r → c y
hxy : ∀ (y : E), y ∈ Metric.ball x r → ‖y - x‖ < 1
⊢ ∀ᶠ (n : (ι × ι) × E) in (l ×ˢ l) ×ˢ 𝓝 x,
    dist (OfNat.ofNat 0 n.snd) (f n.fst.fst n.snd - f n.fst.snd n.snd - (f n.fst.fst x - f n.fst.snd x)) < ε

tactic: refine'
  eventually_prod_iff.mpr
    ⟨_, 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 => _⟩
state_before: case left.intro.intro.intro.intro.intro.intro
ι : Type u_3
l : Filter ι
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
𝕜 : Type u_4
inst✝³ : IsROrC 𝕜
inst✝² : NormedSpace 𝕜 E
G : Type u_2
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' :
  ∀ (ε : ℝ),
    ε > 0 →
      ∀ᶠ (n : (ι × ι) × E) in (l ×ˢ l) ×ˢ 𝓝 x, dist (OfNat.ofNat 0 n.snd) (f' n.fst.fst n.snd - f' n.fst.snd n.snd) < ε
hf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.fst) (f' n.fst n.snd) n.snd
hfg : Cauchy (map (fun n => f n x) l)
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.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).fst.fst)
        (f'
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).fst.fst
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).fst.snd)
        (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
              (x.fst, x.snd, x.snd).snd.snd).fst.snd ∧
      HasFDerivAt
        (f
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).snd.fst)
        (f'
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).snd.fst
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).snd.snd)
        (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
              (x.fst, x.snd, x.snd).snd.snd).snd.snd
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).snd) (f' (x, y).fst.fst (x, y).snd - f' (x, y).fst.snd (x, y).snd) < ε ∧
            HasFDerivAt
                (f
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).fst.fst)
                (f'
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).fst.fst
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).fst.snd)
                ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                      ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                      ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).fst.snd ∧
              HasFDerivAt
                (f
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).snd.fst)
                (f'
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).snd.fst
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).snd.snd)
                ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                      ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                      ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).snd.snd
R : ℝ
hR : 0 < R
hR' : ∀ ⦃x_1 : E⦄, x_1 ∈ Metric.ball x R → c x_1
r : ℝ := min 1 R
hr : 0 < r
hr' : ∀ ⦃y : E⦄, y ∈ Metric.ball x r → c y
hxy : ∀ (y : E), y ∈ Metric.ball x r → ‖y - x‖ < 1
hxyε : ∀ (y : E), y ∈ Metric.ball x r → ε * ‖y - x‖ < ε
⊢ ∀ᶠ (n : (ι × ι) × E) in (l ×ˢ l) ×ˢ 𝓝 x,
    dist (OfNat.ofNat 0 n.snd) (f n.fst.fst n.snd - f n.fst.snd n.snd - (f n.fst.fst x - f n.fst.snd x)) < ε

tactic: simp only [Pi.zero_apply, dist_zero_left] at e ⊢
state_before: case left.intro.intro.intro.intro.intro.intro
ι : Type u_3
l : Filter ι
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
𝕜 : Type u_4
inst✝³ : IsROrC 𝕜
inst✝² : NormedSpace 𝕜 E
G : Type u_2
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' :
  ∀ (ε : ℝ),
    ε > 0 →
      ∀ᶠ (n : (ι × ι) × E) in (l ×ˢ l) ×ˢ 𝓝 x, dist (OfNat.ofNat 0 n.snd) (f' n.fst.fst n.snd - f' n.fst.snd n.snd) < ε
hf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.fst) (f' n.fst n.snd) n.snd
hfg : Cauchy (map (fun n => f n x) l)
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.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).fst.fst)
        (f'
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).fst.fst
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).fst.snd)
        (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
              (x.fst, x.snd, x.snd).snd.snd).fst.snd ∧
      HasFDerivAt
        (f
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).snd.fst)
        (f'
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).snd.fst
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).snd.snd)
        (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
              (x.fst, x.snd, x.snd).snd.snd).snd.snd
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).snd) (f' (x, y).fst.fst (x, y).snd - f' (x, y).fst.snd (x, y).snd) < ε ∧
            HasFDerivAt
                (f
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).fst.fst)
                (f'
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).fst.fst
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).fst.snd)
                ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                      ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                      ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).fst.snd ∧
              HasFDerivAt
                (f
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).snd.fst)
                (f'
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).snd.fst
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).snd.snd)
                ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                      ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                      ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).snd.snd
R : ℝ
hR : 0 < R
hR' : ∀ ⦃x_1 : E⦄, x_1 ∈ Metric.ball x R → c x_1
r : ℝ := min 1 R
hr : 0 < r
hr' : ∀ ⦃y : E⦄, y ∈ Metric.ball x r → c y
hxy : ∀ (y : E), y ∈ Metric.ball x r → ‖y - x‖ < 1
hxyε : ∀ (y : E), 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
⊢ dist (OfNat.ofNat 0 (n, y).snd)
      (f (n, y).fst.fst (n, y).snd - f (n, y).fst.snd (n, y).snd - (f (n, y).fst.fst x - f (n, y).fst.snd x)) <
    ε

tactic: refine' lt_of_le_of_lt _ (hxyε y hy)
state_before: case left.intro.intro.intro.intro.intro.intro
ι : Type u_3
l : Filter ι
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
𝕜 : Type u_4
inst✝³ : IsROrC 𝕜
inst✝² : NormedSpace 𝕜 E
G : Type u_2
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' :
  ∀ (ε : ℝ),
    ε > 0 →
      ∀ᶠ (n : (ι × ι) × E) in (l ×ˢ l) ×ˢ 𝓝 x, dist (OfNat.ofNat 0 n.snd) (f' n.fst.fst n.snd - f' n.fst.snd n.snd) < ε
hf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.fst) (f' n.fst n.snd) n.snd
hfg : Cauchy (map (fun n => f n x) l)
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.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).fst.fst)
        (f'
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).fst.fst
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).fst.snd)
        (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
              (x.fst, x.snd, x.snd).snd.snd).fst.snd ∧
      HasFDerivAt
        (f
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).snd.fst)
        (f'
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).snd.fst
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).snd.snd)
        (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
              (x.fst, x.snd, x.snd).snd.snd).snd.snd
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 := min 1 R
hr : 0 < r
hr' : ∀ ⦃y : E⦄, y ∈ Metric.ball x r → c y
hxy : ∀ (y : E), y ∈ Metric.ball x r → ‖y - x‖ < 1
hxyε : ∀ (y : E), 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.fst y - f' x.snd y‖ < ε ∧ HasFDerivAt (f x.fst) (f' x.fst y) y ∧ HasFDerivAt (f x.snd) (f' x.snd y) y
⊢ ‖f n.fst y - f n.snd y - (f n.fst x - f n.snd x)‖ < ε

tactic: exact
  Convex.norm_image_sub_le_of_norm_hasFDerivWithin_le
    (fun y hy => ((e hn (hr' hy)).2.1.sub (e hn (hr' hy)).2.2).hasFDerivWithinAt)
    (fun y hy => (e hn (hr' hy)).1.le) (convex_ball x r) (Metric.mem_ball_self hr) hy
state_before: case left.intro.intro.intro.intro.intro.intro
ι : Type u_3
l : Filter ι
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
𝕜 : Type u_4
inst✝³ : IsROrC 𝕜
inst✝² : NormedSpace 𝕜 E
G : Type u_2
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' :
  ∀ (ε : ℝ),
    ε > 0 →
      ∀ᶠ (n : (ι × ι) × E) in (l ×ˢ l) ×ˢ 𝓝 x, dist (OfNat.ofNat 0 n.snd) (f' n.fst.fst n.snd - f' n.fst.snd n.snd) < ε
hf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.fst) (f' n.fst n.snd) n.snd
hfg : Cauchy (map (fun n => f n x) l)
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.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).fst.fst)
        (f'
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).fst.fst
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).fst.snd)
        (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
              (x.fst, x.snd, x.snd).snd.snd).fst.snd ∧
      HasFDerivAt
        (f
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).snd.fst)
        (f'
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).snd.fst
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).snd.snd)
        (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
              (x.fst, x.snd, x.snd).snd.snd).snd.snd
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 := min 1 R
hr : 0 < r
hr' : ∀ ⦃y : E⦄, y ∈ Metric.ball x r → c y
hxy : ∀ (y : E), y ∈ Metric.ball x r → ‖y - x‖ < 1
hxyε : ∀ (y : E), 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.fst y - f' x.snd y‖ < ε ∧ HasFDerivAt (f x.fst) (f' x.fst y) y ∧ HasFDerivAt (f x.snd) (f' x.snd y) y
⊢ ‖f n.fst y - f n.snd y - (f n.fst x - f n.snd x)‖ ≤ ε * ‖y - x‖

tactic: simp [hR]
state_before: ι : Type u_3
l : Filter ι
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
𝕜 : Type u_4
inst✝³ : IsROrC 𝕜
inst✝² : NormedSpace 𝕜 E
G : Type u_2
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' :
  ∀ (ε : ℝ),
    ε > 0 →
      ∀ᶠ (n : (ι × ι) × E) in (l ×ˢ l) ×ˢ 𝓝 x, dist (OfNat.ofNat 0 n.snd) (f' n.fst.fst n.snd - f' n.fst.snd n.snd) < ε
hf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.fst) (f' n.fst n.snd) n.snd
hfg : Cauchy (map (fun n => f n x) l)
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.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).fst.fst)
        (f'
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).fst.fst
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).fst.snd)
        (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
              (x.fst, x.snd, x.snd).snd.snd).fst.snd ∧
      HasFDerivAt
        (f
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).snd.fst)
        (f'
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).snd.fst
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).snd.snd)
        (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
              (x.fst, x.snd, x.snd).snd.snd).snd.snd
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).snd) (f' (x, y).fst.fst (x, y).snd - f' (x, y).fst.snd (x, y).snd) < ε ∧
            HasFDerivAt
                (f
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).fst.fst)
                (f'
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).fst.fst
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).fst.snd)
                ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                      ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                      ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).fst.snd ∧
              HasFDerivAt
                (f
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).snd.fst)
                (f'
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).snd.fst
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).snd.snd)
                ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                      ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                      ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).snd.snd
R : ℝ
hR : 0 < R
hR' : ∀ ⦃x_1 : E⦄, x_1 ∈ Metric.ball x R → c x_1
r : ℝ := min 1 R
⊢ 0 < r

tactic: intro y hy
state_before: ι : Type u_3
l : Filter ι
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
𝕜 : Type u_4
inst✝³ : IsROrC 𝕜
inst✝² : NormedSpace 𝕜 E
G : Type u_2
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' :
  ∀ (ε : ℝ),
    ε > 0 →
      ∀ᶠ (n : (ι × ι) × E) in (l ×ˢ l) ×ˢ 𝓝 x, dist (OfNat.ofNat 0 n.snd) (f' n.fst.fst n.snd - f' n.fst.snd n.snd) < ε
hf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.fst) (f' n.fst n.snd) n.snd
hfg : Cauchy (map (fun n => f n x) l)
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.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).fst.fst)
        (f'
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).fst.fst
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).fst.snd)
        (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
              (x.fst, x.snd, x.snd).snd.snd).fst.snd ∧
      HasFDerivAt
        (f
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).snd.fst)
        (f'
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).snd.fst
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).snd.snd)
        (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
              (x.fst, x.snd, x.snd).snd.snd).snd.snd
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).snd) (f' (x, y).fst.fst (x, y).snd - f' (x, y).fst.snd (x, y).snd) < ε ∧
            HasFDerivAt
                (f
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).fst.fst)
                (f'
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).fst.fst
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).fst.snd)
                ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                      ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                      ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).fst.snd ∧
              HasFDerivAt
                (f
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).snd.fst)
                (f'
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).snd.fst
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).snd.snd)
                ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                      ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                      ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).snd.snd
R : ℝ
hR : 0 < R
hR' : ∀ ⦃x_1 : E⦄, x_1 ∈ Metric.ball x R → c x_1
r : ℝ := min 1 R
hr : 0 < r
hr' : ∀ ⦃y : E⦄, y ∈ Metric.ball x r → c y
⊢ ∀ (y : E), y ∈ Metric.ball x r → ‖y - x‖ < 1

tactic: rw [Metric.mem_ball, dist_eq_norm] at hy
state_before: ι : Type u_3
l : Filter ι
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
𝕜 : Type u_4
inst✝³ : IsROrC 𝕜
inst✝² : NormedSpace 𝕜 E
G : Type u_2
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' :
  ∀ (ε : ℝ),
    ε > 0 →
      ∀ᶠ (n : (ι × ι) × E) in (l ×ˢ l) ×ˢ 𝓝 x, dist (OfNat.ofNat 0 n.snd) (f' n.fst.fst n.snd - f' n.fst.snd n.snd) < ε
hf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.fst) (f' n.fst n.snd) n.snd
hfg : Cauchy (map (fun n => f n x) l)
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.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).fst.fst)
        (f'
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).fst.fst
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).fst.snd)
        (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
              (x.fst, x.snd, x.snd).snd.snd).fst.snd ∧
      HasFDerivAt
        (f
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).snd.fst)
        (f'
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).snd.fst
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).snd.snd)
        (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
              (x.fst, x.snd, x.snd).snd.snd).snd.snd
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).snd) (f' (x, y).fst.fst (x, y).snd - f' (x, y).fst.snd (x, y).snd) < ε ∧
            HasFDerivAt
                (f
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).fst.fst)
                (f'
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).fst.fst
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).fst.snd)
                ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                      ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                      ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).fst.snd ∧
              HasFDerivAt
                (f
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).snd.fst)
                (f'
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).snd.fst
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).snd.snd)
                ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                      ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                      ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).snd.snd
R : ℝ
hR : 0 < R
hR' : ∀ ⦃x_1 : E⦄, x_1 ∈ Metric.ball x R → c x_1
r : ℝ := min 1 R
hr : 0 < r
hr' : ∀ ⦃y : E⦄, y ∈ Metric.ball x r → c y
y : E
hy : y ∈ Metric.ball x r
⊢ ‖y - x‖ < 1

tactic: exact lt_of_lt_of_le hy (min_le_left _ _)
state_before: ι : Type u_3
l : Filter ι
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
𝕜 : Type u_4
inst✝³ : IsROrC 𝕜
inst✝² : NormedSpace 𝕜 E
G : Type u_2
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' :
  ∀ (ε : ℝ),
    ε > 0 →
      ∀ᶠ (n : (ι × ι) × E) in (l ×ˢ l) ×ˢ 𝓝 x, dist (OfNat.ofNat 0 n.snd) (f' n.fst.fst n.snd - f' n.fst.snd n.snd) < ε
hf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.fst) (f' n.fst n.snd) n.snd
hfg : Cauchy (map (fun n => f n x) l)
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.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).fst.fst)
        (f'
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).fst.fst
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).fst.snd)
        (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
              (x.fst, x.snd, x.snd).snd.snd).fst.snd ∧
      HasFDerivAt
        (f
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).snd.fst)
        (f'
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).snd.fst
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).snd.snd)
        (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
              (x.fst, x.snd, x.snd).snd.snd).snd.snd
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).snd) (f' (x, y).fst.fst (x, y).snd - f' (x, y).fst.snd (x, y).snd) < ε ∧
            HasFDerivAt
                (f
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).fst.fst)
                (f'
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).fst.fst
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).fst.snd)
                ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                      ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                      ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).fst.snd ∧
              HasFDerivAt
                (f
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).snd.fst)
                (f'
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).snd.fst
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).snd.snd)
                ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                      ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                      ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).snd.snd
R : ℝ
hR : 0 < R
hR' : ∀ ⦃x_1 : E⦄, x_1 ∈ Metric.ball x R → c x_1
r : ℝ := min 1 R
hr : 0 < r
hr' : ∀ ⦃y : E⦄, y ∈ Metric.ball x r → c y
y : E
hy : ‖y - x‖ < r
⊢ ‖y - x‖ < 1

tactic: intro y hy
state_before: ι : Type u_3
l : Filter ι
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
𝕜 : Type u_4
inst✝³ : IsROrC 𝕜
inst✝² : NormedSpace 𝕜 E
G : Type u_2
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' :
  ∀ (ε : ℝ),
    ε > 0 →
      ∀ᶠ (n : (ι × ι) × E) in (l ×ˢ l) ×ˢ 𝓝 x, dist (OfNat.ofNat 0 n.snd) (f' n.fst.fst n.snd - f' n.fst.snd n.snd) < ε
hf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.fst) (f' n.fst n.snd) n.snd
hfg : Cauchy (map (fun n => f n x) l)
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.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).fst.fst)
        (f'
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).fst.fst
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).fst.snd)
        (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
              (x.fst, x.snd, x.snd).snd.snd).fst.snd ∧
      HasFDerivAt
        (f
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).snd.fst)
        (f'
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).snd.fst
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).snd.snd)
        (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
              (x.fst, x.snd, x.snd).snd.snd).snd.snd
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).snd) (f' (x, y).fst.fst (x, y).snd - f' (x, y).fst.snd (x, y).snd) < ε ∧
            HasFDerivAt
                (f
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).fst.fst)
                (f'
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).fst.fst
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).fst.snd)
                ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                      ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                      ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).fst.snd ∧
              HasFDerivAt
                (f
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).snd.fst)
                (f'
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).snd.fst
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).snd.snd)
                ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                      ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                      ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).snd.snd
R : ℝ
hR : 0 < R
hR' : ∀ ⦃x_1 : E⦄, x_1 ∈ Metric.ball x R → c x_1
r : ℝ := min 1 R
hr : 0 < r
hr' : ∀ ⦃y : E⦄, y ∈ Metric.ball x r → c y
hxy : ∀ (y : E), y ∈ Metric.ball x r → ‖y - x‖ < 1
⊢ ∀ (y : E), y ∈ Metric.ball x r → ε * ‖y - x‖ < ε

tactic: exact (mul_lt_iff_lt_one_right hε.lt).mpr (hxy y hy)
state_before: ι : Type u_3
l : Filter ι
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
𝕜 : Type u_4
inst✝³ : IsROrC 𝕜
inst✝² : NormedSpace 𝕜 E
G : Type u_2
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' :
  ∀ (ε : ℝ),
    ε > 0 →
      ∀ᶠ (n : (ι × ι) × E) in (l ×ˢ l) ×ˢ 𝓝 x, dist (OfNat.ofNat 0 n.snd) (f' n.fst.fst n.snd - f' n.fst.snd n.snd) < ε
hf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.fst) (f' n.fst n.snd) n.snd
hfg : Cauchy (map (fun n => f n x) l)
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.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).fst.fst)
        (f'
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).fst.fst
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).fst.snd)
        (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
              (x.fst, x.snd, x.snd).snd.snd).fst.snd ∧
      HasFDerivAt
        (f
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).snd.fst)
        (f'
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).snd.fst
          (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
                (x.fst, x.snd, x.snd).snd.snd).snd.snd)
        (((x.fst, x.snd, x.snd).fst.fst, (x.fst, x.snd, x.snd).snd.fst), (x.fst, x.snd, x.snd).fst.snd,
              (x.fst, x.snd, x.snd).snd.snd).snd.snd
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).snd) (f' (x, y).fst.fst (x, y).snd - f' (x, y).fst.snd (x, y).snd) < ε ∧
            HasFDerivAt
                (f
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).fst.fst)
                (f'
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).fst.fst
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).fst.snd)
                ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                      ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                      ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).fst.snd ∧
              HasFDerivAt
                (f
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).snd.fst)
                (f'
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).snd.fst
                  ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                        ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                        ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).snd.snd)
                ((((x, y).fst, (x, y).snd, (x, y).snd).fst.fst, ((x, y).fst, (x, y).snd, (x, y).snd).snd.fst),
                      ((x, y).fst, (x, y).snd, (x, y).snd).fst.snd,
                      ((x, y).fst, (x, y).snd, (x, y).snd).snd.snd).snd.snd
R : ℝ
hR : 0 < R
hR' : ∀ ⦃x_1 : E⦄, x_1 ∈ Metric.ball x R → c x_1
r : ℝ := min 1 R
hr : 0 < r
hr' : ∀ ⦃y : E⦄, y ∈ Metric.ball x r → c y
hxy : ∀ (y : E), y ∈ Metric.ball x r → ‖y - x‖ < 1
y : E
hy : y ∈ Metric.ball x r
⊢ ε * ‖y - x‖ < ε

tactic: refine' Metric.tendstoUniformlyOnFilter_iff.mpr fun ε hε => _
state_before: case right
ι : Type u_3
l : Filter ι
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
𝕜 : Type u_4
inst✝³ : IsROrC 𝕜
inst✝² : NormedSpace 𝕜 E
G : Type u_2
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 (fun n z => f' n.fst z - f' n.snd z) 0 (l ×ˢ l) (𝓝 x)
hf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.fst) (f' n.fst n.snd) n.snd
hfg : Cauchy (map (fun n => f n x) l)
this : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E
⊢ TendstoUniformlyOnFilter (fun n x_1 => f n.fst x - f n.snd x) 0 (l ×ˢ l) (𝓝 x)

tactic: obtain ⟨t, ht, ht'⟩ := (Metric.cauchy_iff.mp hfg).2 ε hε
state_before: case right
ι : Type u_3
l : Filter ι
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
𝕜 : Type u_4
inst✝³ : IsROrC 𝕜
inst✝² : NormedSpace 𝕜 E
G : Type u_2
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 (fun n z => f' n.fst z - f' n.snd z) 0 (l ×ˢ l) (𝓝 x)
hf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.fst) (f' n.fst n.snd) n.snd
hfg : Cauchy (map (fun n => f n x) l)
this : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E
ε : ℝ
hε : ε > 0
⊢ ∀ᶠ (n : (ι × ι) × E) in (l ×ˢ l) ×ˢ 𝓝 x, dist (OfNat.ofNat 0 n.snd) (f n.fst.fst x - f n.fst.snd 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⟩
state_before: case right.intro.intro
ι : Type u_3
l : Filter ι
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
𝕜 : Type u_4
inst✝³ : IsROrC 𝕜
inst✝² : NormedSpace 𝕜 E
G : Type u_2
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 (fun n z => f' n.fst z - f' n.snd z) 0 (l ×ˢ l) (𝓝 x)
hf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.fst) (f' n.fst n.snd) n.snd
hfg : Cauchy (map (fun n => f n x) l)
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 : G), x ∈ t → ∀ (y : G), y ∈ t → dist x y < ε
⊢ ∀ᶠ (n : (ι × ι) × E) in (l ×ˢ l) ×ˢ 𝓝 x, dist (OfNat.ofNat 0 n.snd) (f n.fst.fst x - f n.fst.snd x) < ε

tactic: simp
state_before: ι : Type u_3
l : Filter ι
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
𝕜 : Type u_4
inst✝³ : IsROrC 𝕜
inst✝² : NormedSpace 𝕜 E
G : Type u_2
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 (fun n z => f' n.fst z - f' n.snd z) 0 (l ×ˢ l) (𝓝 x)
hf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.fst) (f' n.fst n.snd) n.snd
hfg : Cauchy (map (fun n => f n x) l)
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 : G), x ∈ t → ∀ (y : G), y ∈ t → dist x y < ε
⊢ ∀ᶠ (y : E) in 𝓝 x, (fun x => True) y

tactic: simpa [norm_sub_rev, dist_eq_norm] using ht' _ hn.1 _ hn.2
state_before: ι : Type u_3
l : Filter ι
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
𝕜 : Type u_4
inst✝³ : IsROrC 𝕜
inst✝² : NormedSpace 𝕜 E
G : Type u_2
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 (fun n z => f' n.fst z - f' n.snd z) 0 (l ×ˢ l) (𝓝 x)
hf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.fst) (f' n.fst n.snd) n.snd
hfg : Cauchy (map (fun n => f n x) l)
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 : G), x ∈ t → ∀ (y : G), y ∈ t → dist x y < ε
n : ι × ι
hn : (fun n => f n.fst x ∈ t ∧ f n.snd x ∈ t) n
y : E
x✝ : (fun x => True) y
⊢ dist (OfNat.ofNat 0 (n, y).snd) (f (n, y).fst.fst x - f (n, y).fst.snd x) < ε