-- Mathlib/Analysis/Calculus/FDeriv/Measurable.lean
theorem D_subset_differentiable_set {K : Set (E →L[𝕜] F)} (hK : IsComplete K) :
    D f K ⊆ { x | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K }

Proof state:
case neg
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
K : Set (E →L[𝕜] F)
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
c : 𝕜
hc : 1 < ‖c‖
x : E
hx : x ∈ D f K
n : ℕ → ℕ
L : ℕ → ℕ → ℕ → E →L[𝕜] F
hn :
  ∀ (e p q : ℕ),
    n e ≤ p →
      n e ≤ q → L e p q ∈ K ∧ x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f (L e p q) ((1 / 2) ^ q) ((1 / 2) ^ e)
M :
  ∀ (e p q e' p' q' : ℕ),
    n e ≤ p → n e ≤ q → n e' ≤ p' → n e' ≤ q' → e ≤ e' → ‖L e p q - L e' p' q'‖ ≤ 12 * ‖c‖ * (1 / 2) ^ e
L0 : ℕ → E →L[𝕜] F := fun e => L e (n e) (n e)
this : CauchySeq L0
f' : E →L[𝕜] F
f'K : f' ∈ K
hf' : Tendsto L0 atTop (𝓝 f')
Lf' : ∀ (e p : ℕ), n e ≤ p → ‖L e (n e) p - f'‖ ≤ 12 * ‖c‖ * (1 / 2) ^ e
ε : ℝ
εpos : 0 < ε
pos : 0 < 4 + 12 * ‖c‖
e : ℕ
he : (1 / 2) ^ e < ε / (4 + 12 * ‖c‖)
y : E
hy : y ∈ ball 0 ((1 / 2) ^ (n e + 1))
y_pos : ¬y = 0
yzero : 0 < ‖y‖
y_lt : ‖y‖ < (1 / 2) ^ (n e + 1)
⊢ ‖f (x + y) - f x - f' y‖ ≤ ε * ‖y‖

Tactic:
have yone : ‖y‖ ≤ 1 := le_trans y_lt.le (pow_le_one₀ (by norm_num) (by norm_num))

Result:
case neg
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
K : Set (E →L[𝕜] F)
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
c : 𝕜
hc : 1 < ‖c‖
x : E
hx : x ∈ D f K
n : ℕ → ℕ
L : ℕ → ℕ → ℕ → E →L[𝕜] F
hn :
  ∀ (e p q : ℕ),
    n e ≤ p →
      n e ≤ q → L e p q ∈ K ∧ x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f (L e p q) ((1 / 2) ^ q) ((1 / 2) ^ e)
M :
  ∀ (e p q e' p' q' : ℕ),
    n e ≤ p → n e ≤ q → n e' ≤ p' → n e' ≤ q' → e ≤ e' → ‖L e p q - L e' p' q'‖ ≤ 12 * ‖c‖ * (1 / 2) ^ e
L0 : ℕ → E →L[𝕜] F := fun e => L e (n e) (n e)
this : CauchySeq L0
f' : E →L[𝕜] F
f'K : f' ∈ K
hf' : Tendsto L0 atTop (𝓝 f')
Lf' : ∀ (e p : ℕ), n e ≤ p → ‖L e (n e) p - f'‖ ≤ 12 * ‖c‖ * (1 / 2) ^ e
ε : ℝ
εpos : 0 < ε
pos : 0 < 4 + 12 * ‖c‖
e : ℕ
he : (1 / 2) ^ e < ε / (4 + 12 * ‖c‖)
y : E
hy : y ∈ ball 0 ((1 / 2) ^ (n e + 1))
y_pos : ¬y = 0
yzero : 0 < ‖y‖
y_lt : ‖y‖ < (1 / 2) ^ (n e + 1)
yone : ‖y‖ ≤ 1
⊢ ‖f (x + y) - f x - f' y‖ ≤ ε * ‖y‖

Proof state:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
K : Set (E →L[𝕜] F)
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
⊢ D f K ⊆ {x | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K}

Tactic:
rcases NormedField.exists_one_lt_norm 𝕜 with ⟨c, hc⟩

Result:
case intro
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
K : Set (E →L[𝕜] F)
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
c : 𝕜
hc : 1 < ‖c‖
⊢ D f K ⊆ {x | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K}

Proof state:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
K : Set (E →L[𝕜] F)
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
c : 𝕜
hc : 1 < ‖c‖
x : E
hx : x ∈ D f K
n : ℕ → ℕ
L : ℕ → ℕ → ℕ → E →L[𝕜] F
hn :
  ∀ (e p q : ℕ),
    n e ≤ p →
      n e ≤ q → L e p q ∈ K ∧ x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f (L e p q) ((1 / 2) ^ q) ((1 / 2) ^ e)
M :
  ∀ (e p q e' p' q' : ℕ),
    n e ≤ p → n e ≤ q → n e' ≤ p' → n e' ≤ q' → e ≤ e' → ‖L e p q - L e' p' q'‖ ≤ 12 * ‖c‖ * (1 / 2) ^ e
L0 : ℕ → E →L[𝕜] F := fun e => L e (n e) (n e)
this : CauchySeq L0
f' : E →L[𝕜] F
f'K : f' ∈ K
hf' : Tendsto L0 atTop (𝓝 f')
Lf' : ∀ (e p : ℕ), n e ≤ p → ‖L e (n e) p - f'‖ ≤ 12 * ‖c‖ * (1 / 2) ^ e
ε : ℝ
εpos : 0 < ε
pos : 0 < 4 + 12 * ‖c‖
e : ℕ
he : (1 / 2) ^ e < ε / (4 + 12 * ‖c‖)
y : E
hy : y ∈ ball 0 ((1 / 2) ^ (n e + 1))
y_pos : ¬y = 0
yzero : 0 < ‖y‖
⊢ ‖y‖ < (1 / 2) ^ (n e + 1)

Tactic:
simpa using mem_ball_iff_norm.1 hy

Result:
no goals

Proof state:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
K : Set (E →L[𝕜] F)
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
c : 𝕜
hc : 1 < ‖c‖
x : E
hx : x ∈ D f K
n : ℕ → ℕ
L : ℕ → ℕ → ℕ → E →L[𝕜] F
hn :
  ∀ (e p q : ℕ),
    n e ≤ p →
      n e ≤ q → L e p q ∈ K ∧ x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f (L e p q) ((1 / 2) ^ q) ((1 / 2) ^ e)
e p q e' p' q' : ℕ
hp : n e ≤ p
hq : n e ≤ q
hp' : n e' ≤ p'
hq' : n e' ≤ q'
he' : e ≤ e'
r : ℕ := n e ⊔ n e'
I : (1 / 2) ^ e' ≤ (1 / 2) ^ e
⊢ ‖L e p q - L e' p' q'‖ ≤ 12 * ‖c‖ * (1 / 2) ^ e

Tactic:
have J1 : ‖L e p q - L e p r‖ ≤ 4 * ‖c‖ * (1 / 2) ^ e := by
  have I1 : x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / 2) ^ e) := (hn e p q hp hq).2.1
  have I2 : x ∈ A f (L e p r) ((1 / 2) ^ p) ((1 / 2) ^ e) := (hn e p r hp (le_max_left _ _)).2.1
  exact norm_sub_le_of_mem_A hc P P I1 I2

Result:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
K : Set (E →L[𝕜] F)
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
c : 𝕜
hc : 1 < ‖c‖
x : E
hx : x ∈ D f K
n : ℕ → ℕ
L : ℕ → ℕ → ℕ → E →L[𝕜] F
hn :
  ∀ (e p q : ℕ),
    n e ≤ p →
      n e ≤ q → L e p q ∈ K ∧ x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f (L e p q) ((1 / 2) ^ q) ((1 / 2) ^ e)
e p q e' p' q' : ℕ
hp : n e ≤ p
hq : n e ≤ q
hp' : n e' ≤ p'
hq' : n e' ≤ q'
he' : e ≤ e'
r : ℕ := n e ⊔ n e'
I : (1 / 2) ^ e' ≤ (1 / 2) ^ e
J1 : ‖L e p q - L e p r‖ ≤ 4 * ‖c‖ * (1 / 2) ^ e
⊢ ‖L e p q - L e' p' q'‖ ≤ 12 * ‖c‖ * (1 / 2) ^ e

Proof state:
case intro
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
K : Set (E →L[𝕜] F)
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
c : 𝕜
hc : 1 < ‖c‖
x : E
hx : x ∈ D f K
⊢ x ∈ {x | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K}

Tactic:
have :
  ∀ e : ℕ, ∃ n : ℕ, ∀ p q, n ≤ p → n ≤ q →
    ∃ L ∈ K, x ∈ A f L ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f L ((1 / 2) ^ q) ((1 / 2) ^ e) := by
  intro e
  have := mem_iInter.1 hx e
  rcases mem_iUnion.1 this with ⟨n, hn⟩
  refine ⟨n, fun p q hp hq => ?_⟩
  simp only [mem_iInter] at hn
  rcases mem_iUnion.1 (hn p hp q hq) with ⟨L, hL⟩
  exact ⟨L, exists_prop.mp <| mem_iUnion.1 hL⟩

Result:
case intro
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
K : Set (E →L[𝕜] F)
hK : IsComplete K
P : ∀ {n : ℕ}, 0 < (1 / 2) ^ n
c : 𝕜
hc : 1 < ‖c‖
x : E
hx : x ∈ D f K
this :
  ∀ (e : ℕ),
    ∃ n, ∀ (p q : ℕ), n ≤ p → n ≤ q → ∃ L ∈ K, x ∈ A f L ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A f L ((1 / 2) ^ q) ((1 / 2) ^ e)
⊢ x ∈ {x | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K}