l3lab/ntp-mathlib huggingface dataset:
/-- Consider a series of functions `∑' n, f n x`. If the series converges at a
point, and all functions in the series are differentiable with a summable bound on the derivatives,
then the series is differentiable and its derivative is the sum of the derivatives. -/
theorem hasFDerivAt_tsum (hu : Summable u) (hf : ∀ n x, HasFDerivAt (f n) (f' n x) x)
    (hf' : ∀ n x, ‖f' n x‖ ≤ u n) (hf0 : Summable fun n => f n x₀) (x : E) :
    HasFDerivAt (fun y => ∑' n, f n y) (∑' n, f' n x) x := by
  let : NormedSpace ℝ E; 

Proof state:
case this
α : Type u_1
β : Type u_2
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁵ : IsROrC 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : CompleteSpace F
u : α → ℝ
inst✝ : NormedSpace 𝕜 F
f : α → E → F
f' : α → E → E →L[𝕜] F
v : ℕ → α → ℝ
s : Set E
x₀ x✝ : E
N : ℕ∞
hu : Summable u
hf : ∀ (n : α) (x : E), HasFDerivAt (f n) (f' n x) x
hf' : ∀ (n : α) (x : E), ‖f' n x‖ ≤ u n
hf0 : Summable fun n => f n x₀
x : E
⊢ NormedSpace ℝ E
α : Type u_1
β : Type u_2
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁵ : IsROrC 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : CompleteSpace F
u : α → ℝ
inst✝ : NormedSpace 𝕜 F
f : α → E → F
f' : α → E → E →L[𝕜] F
v : ℕ → α → ℝ
s : Set E
x₀ x✝ : E
N : ℕ∞
hu : Summable u
hf : ∀ (n : α) (x : E), HasFDerivAt (f n) (f' n x) x
hf' : ∀ (n : α) (x : E), ‖f' n x‖ ≤ u n
hf0 : Summable fun n => f n x₀
x : E
this : NormedSpace ℝ E := ?this
⊢ HasFDerivAt (fun y => ∑' (n : α), f n y) (∑' (n : α), f' n x) x

Next tactic:
exact NormedSpace.restrictScalars ℝ 𝕜 _
/-- Consider a series of functions `∑' n, f n x`. If the series converges at a
point, and all functions in the series are differentiable with a summable bound on the derivatives,
then the series is differentiable and its derivative is the sum of the derivatives. -/
theorem hasFDerivAt_tsum (hu : Summable u) (hf : ∀ n x, HasFDerivAt (f n) (f' n x) x)
    (hf' : ∀ n x, ‖f' n x‖ ≤ u n) (hf0 : Summable fun n => f n x₀) (x : E) :
    HasFDerivAt (fun y => ∑' n, f n y) (∑' n, f' n x) x := by
  let : NormedSpace ℝ E; exact NormedSpace.restrictScalars ℝ 𝕜 _
  

Proof state:
α : Type u_1
β : Type u_2
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁵ : IsROrC 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : CompleteSpace F
u : α → ℝ
inst✝ : NormedSpace 𝕜 F
f : α → E → F
f' : α → E → E →L[𝕜] F
v : ℕ → α → ℝ
s : Set E
x₀ x✝ : E
N : ℕ∞
hu : Summable u
hf : ∀ (n : α) (x : E), HasFDerivAt (f n) (f' n x) x
hf' : ∀ (n : α) (x : E), ‖f' n x‖ ≤ u n
hf0 : Summable fun n => f n x₀
x : E
this : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E
⊢ HasFDerivAt (fun y => ∑' (n : α), f n y) (∑' (n : α), f' n x) x

Next tactic:
exact hasFDerivAt_tsum_of_isPreconnected hu isOpen_univ isPreconnected_univ
    (fun n x _ => hf n x) (fun n x _ => hf' n x) (mem_univ _) hf0 (mem_univ _)
/-- Consider a series of functions `∑' n, f n x`. If all functions in the series are differentiable
with a summable bound on the derivatives, then the series is differentiable.
Note that our assumptions do not ensure the pointwise convergence, but if there is no pointwise
convergence then the series is zero everywhere so the result still holds. -/
theorem differentiable_tsum (hu : Summable u) (hf : ∀ n x, HasFDerivAt (f n) (f' n x) x)
    (hf' : ∀ n x, ‖f' n x‖ ≤ u n) : Differentiable 𝕜 fun y => ∑' n, f n y := by
  

Proof state:
α : Type u_1
β : Type u_2
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁵ : IsROrC 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : CompleteSpace F
u : α → ℝ
inst✝ : NormedSpace 𝕜 F
f : α → E → F
f' : α → E → E →L[𝕜] F
v : ℕ → α → ℝ
s : Set E
x₀ x : E
N : ℕ∞
hu : Summable u
hf : ∀ (n : α) (x : E), HasFDerivAt (f n) (f' n x) x
hf' : ∀ (n : α) (x : E), ‖f' n x‖ ≤ u n
⊢ Differentiable 𝕜 fun y => ∑' (n : α), f n y

Next tactic:
by_cases h : ∃ x₀, Summable fun n => f n x₀
/-- Consider a series of functions `∑' n, f n x`. If all functions in the series are differentiable
with a summable bound on the derivatives, then the series is differentiable.
Note that our assumptions do not ensure the pointwise convergence, but if there is no pointwise
convergence then the series is zero everywhere so the result still holds. -/
theorem differentiable_tsum (hu : Summable u) (hf : ∀ n x, HasFDerivAt (f n) (f' n x) x)
    (hf' : ∀ n x, ‖f' n x‖ ≤ u n) : Differentiable 𝕜 fun y => ∑' n, f n y := by
  by_cases h : ∃ x₀, Summable fun n => f n x₀
  · 

Proof state:
case pos
α : Type u_1
β : Type u_2
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁵ : IsROrC 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : CompleteSpace F
u : α → ℝ
inst✝ : NormedSpace 𝕜 F
f : α → E → F
f' : α → E → E →L[𝕜] F
v : ℕ → α → ℝ
s : Set E
x₀ x : E
N : ℕ∞
hu : Summable u
hf : ∀ (n : α) (x : E), HasFDerivAt (f n) (f' n x) x
hf' : ∀ (n : α) (x : E), ‖f' n x‖ ≤ u n
h : ∃ x₀, Summable fun n => f n x₀
⊢ Differentiable 𝕜 fun y => ∑' (n : α), f n y

Next tactic:
rcases h with ⟨x₀, hf0⟩
/-- Consider a series of functions `∑' n, f n x`. If all functions in the series are differentiable
with a summable bound on the derivatives, then the series is differentiable.
Note that our assumptions do not ensure the pointwise convergence, but if there is no pointwise
convergence then the series is zero everywhere so the result still holds. -/
theorem differentiable_tsum (hu : Summable u) (hf : ∀ n x, HasFDerivAt (f n) (f' n x) x)
    (hf' : ∀ n x, ‖f' n x‖ ≤ u n) : Differentiable 𝕜 fun y => ∑' n, f n y := by
  by_cases h : ∃ x₀, Summable fun n => f n x₀
  · rcases h with ⟨x₀, hf0⟩
    

Proof state:
case pos.intro
α : Type u_1
β : Type u_2
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁵ : IsROrC 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : CompleteSpace F
u : α → ℝ
inst✝ : NormedSpace 𝕜 F
f : α → E → F
f' : α → E → E →L[𝕜] F
v : ℕ → α → ℝ
s : Set E
x₀✝ x : E
N : ℕ∞
hu : Summable u
hf : ∀ (n : α) (x : E), HasFDerivAt (f n) (f' n x) x
hf' : ∀ (n : α) (x : E), ‖f' n x‖ ≤ u n
x₀ : E
hf0 : Summable fun n => f n x₀
⊢ Differentiable 𝕜 fun y => ∑' (n : α), f n y

Next tactic:
intro x
/-- Consider a series of functions `∑' n, f n x`. If all functions in the series are differentiable
with a summable bound on the derivatives, then the series is differentiable.
Note that our assumptions do not ensure the pointwise convergence, but if there is no pointwise
convergence then the series is zero everywhere so the result still holds. -/
theorem differentiable_tsum (hu : Summable u) (hf : ∀ n x, HasFDerivAt (f n) (f' n x) x)
    (hf' : ∀ n x, ‖f' n x‖ ≤ u n) : Differentiable 𝕜 fun y => ∑' n, f n y := by
  by_cases h : ∃ x₀, Summable fun n => f n x₀
  · rcases h with ⟨x₀, hf0⟩
    intro x
    

Proof state:
case pos.intro
α : Type u_1
β : Type u_2
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁵ : IsROrC 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : CompleteSpace F
u : α → ℝ
inst✝ : NormedSpace 𝕜 F
f : α → E → F
f' : α → E → E →L[𝕜] F
v : ℕ → α → ℝ
s : Set E
x₀✝ x✝ : E
N : ℕ∞
hu : Summable u
hf : ∀ (n : α) (x : E), HasFDerivAt (f n) (f' n x) x
hf' : ∀ (n : α) (x : E), ‖f' n x‖ ≤ u n
x₀ : E
hf0 : Summable fun n => f n x₀
x : E
⊢ DifferentiableAt 𝕜 (fun y => ∑' (n : α), f n y) x

Next tactic:
exact (hasFDerivAt_tsum hu hf hf' hf0 x).differentiableAt
/-- Consider a series of functions `∑' n, f n x`. If all functions in the series are differentiable
with a summable bound on the derivatives, then the series is differentiable.
Note that our assumptions do not ensure the pointwise convergence, but if there is no pointwise
convergence then the series is zero everywhere so the result still holds. -/
theorem differentiable_tsum (hu : Summable u) (hf : ∀ n x, HasFDerivAt (f n) (f' n x) x)
    (hf' : ∀ n x, ‖f' n x‖ ≤ u n) : Differentiable 𝕜 fun y => ∑' n, f n y := by
  by_cases h : ∃ x₀, Summable fun n => f n x₀
  · rcases h with ⟨x₀, hf0⟩
    intro x
    exact (hasFDerivAt_tsum hu hf hf' hf0 x).differentiableAt
  · 

Proof state:
case neg
α : Type u_1
β : Type u_2
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁵ : IsROrC 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : CompleteSpace F
u : α → ℝ
inst✝ : NormedSpace 𝕜 F
f : α → E → F
f' : α → E → E →L[𝕜] F
v : ℕ → α → ℝ
s : Set E
x₀ x : E
N : ℕ∞
hu : Summable u
hf : ∀ (n : α) (x : E), HasFDerivAt (f n) (f' n x) x
hf' : ∀ (n : α) (x : E), ‖f' n x‖ ≤ u n
h : ¬∃ x₀, Summable fun n => f n x₀
⊢ Differentiable 𝕜 fun y => ∑' (n : α), f n y

Next tactic:
push_neg at h
/-- Consider a series of functions `∑' n, f n x`. If all functions in the series are differentiable
with a summable bound on the derivatives, then the series is differentiable.
Note that our assumptions do not ensure the pointwise convergence, but if there is no pointwise
convergence then the series is zero everywhere so the result still holds. -/
theorem differentiable_tsum (hu : Summable u) (hf : ∀ n x, HasFDerivAt (f n) (f' n x) x)
    (hf' : ∀ n x, ‖f' n x‖ ≤ u n) : Differentiable 𝕜 fun y => ∑' n, f n y := by
  by_cases h : ∃ x₀, Summable fun n => f n x₀
  · rcases h with ⟨x₀, hf0⟩
    intro x
    exact (hasFDerivAt_tsum hu hf hf' hf0 x).differentiableAt
  · push_neg at h
    

Proof state:
case neg
α : Type u_1
β : Type u_2
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁵ : IsROrC 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : CompleteSpace F
u : α → ℝ
inst✝ : NormedSpace 𝕜 F
f : α → E → F
f' : α → E → E →L[𝕜] F
v : ℕ → α → ℝ
s : Set E
x₀ x : E
N : ℕ∞
hu : Summable u
hf : ∀ (n : α) (x : E), HasFDerivAt (f n) (f' n x) x
hf' : ∀ (n : α) (x : E), ‖f' n x‖ ≤ u n
h : ∀ (x₀ : E), ¬Summable fun n => f n x₀
⊢ Differentiable 𝕜 fun y => ∑' (n : α), f n y

Next tactic:
have : (fun x => ∑' n, f n x) = 0 := by ext1 x; exact tsum_eq_zero_of_not_summable (h x)
/-- Consider a series of functions `∑' n, f n x`. If all functions in the series are differentiable
with a summable bound on the derivatives, then the series is differentiable.
Note that our assumptions do not ensure the pointwise convergence, but if there is no pointwise
convergence then the series is zero everywhere so the result still holds. -/
theorem differentiable_tsum (hu : Summable u) (hf : ∀ n x, HasFDerivAt (f n) (f' n x) x)
    (hf' : ∀ n x, ‖f' n x‖ ≤ u n) : Differentiable 𝕜 fun y => ∑' n, f n y := by
  by_cases h : ∃ x₀, Summable fun n => f n x₀
  · rcases h with ⟨x₀, hf0⟩
    intro x
    exact (hasFDerivAt_tsum hu hf hf' hf0 x).differentiableAt
  · push_neg at h
    have : (fun x => ∑' n, f n x) = 0 := by 

Proof state:
α : Type u_1
β : Type u_2
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁵ : IsROrC 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : CompleteSpace F
u : α → ℝ
inst✝ : NormedSpace 𝕜 F
f : α → E → F
f' : α → E → E →L[𝕜] F
v : ℕ → α → ℝ
s : Set E
x₀ x : E
N : ℕ∞
hu : Summable u
hf : ∀ (n : α) (x : E), HasFDerivAt (f n) (f' n x) x
hf' : ∀ (n : α) (x : E), ‖f' n x‖ ≤ u n
h : ∀ (x₀ : E), ¬Summable fun n => f n x₀
⊢ (fun x => ∑' (n : α), f n x) = 0

Next tactic:
ext1 x
/-- Consider a series of functions `∑' n, f n x`. If all functions in the series are differentiable
with a summable bound on the derivatives, then the series is differentiable.
Note that our assumptions do not ensure the pointwise convergence, but if there is no pointwise
convergence then the series is zero everywhere so the result still holds. -/
theorem differentiable_tsum (hu : Summable u) (hf : ∀ n x, HasFDerivAt (f n) (f' n x) x)
    (hf' : ∀ n x, ‖f' n x‖ ≤ u n) : Differentiable 𝕜 fun y => ∑' n, f n y := by
  by_cases h : ∃ x₀, Summable fun n => f n x₀
  · rcases h with ⟨x₀, hf0⟩
    intro x
    exact (hasFDerivAt_tsum hu hf hf' hf0 x).differentiableAt
  · push_neg at h
    have : (fun x => ∑' n, f n x) = 0 := by ext1 x; 

Proof state:
case h
α : Type u_1
β : Type u_2
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁵ : IsROrC 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : CompleteSpace F
u : α → ℝ
inst✝ : NormedSpace 𝕜 F
f : α → E → F
f' : α → E → E →L[𝕜] F
v : ℕ → α → ℝ
s : Set E
x₀ x✝ : E
N : ℕ∞
hu : Summable u
hf : ∀ (n : α) (x : E), HasFDerivAt (f n) (f' n x) x
hf' : ∀ (n : α) (x : E), ‖f' n x‖ ≤ u n
h : ∀ (x₀ : E), ¬Summable fun n => f n x₀
x : E
⊢ ∑' (n : α), f n x = OfNat.ofNat 0 x

Next tactic:
exact tsum_eq_zero_of_not_summable (h x)
/-- Consider a series of functions `∑' n, f n x`. If all functions in the series are differentiable
with a summable bound on the derivatives, then the series is differentiable.
Note that our assumptions do not ensure the pointwise convergence, but if there is no pointwise
convergence then the series is zero everywhere so the result still holds. -/
theorem differentiable_tsum (hu : Summable u) (hf : ∀ n x, HasFDerivAt (f n) (f' n x) x)
    (hf' : ∀ n x, ‖f' n x‖ ≤ u n) : Differentiable 𝕜 fun y => ∑' n, f n y := by
  by_cases h : ∃ x₀, Summable fun n => f n x₀
  · rcases h with ⟨x₀, hf0⟩
    intro x
    exact (hasFDerivAt_tsum hu hf hf' hf0 x).differentiableAt
  · push_neg at h
    have : (fun x => ∑' n, f n x) = 0 := by ext1 x; exact tsum_eq_zero_of_not_summable (h x)
    

Proof state:
case neg
α : Type u_1
β : Type u_2
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁵ : IsROrC 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : CompleteSpace F
u : α → ℝ
inst✝ : NormedSpace 𝕜 F
f : α → E → F
f' : α → E → E →L[𝕜] F
v : ℕ → α → ℝ
s : Set E
x₀ x : E
N : ℕ∞
hu : Summable u
hf : ∀ (n : α) (x : E), HasFDerivAt (f n) (f' n x) x
hf' : ∀ (n : α) (x : E), ‖f' n x‖ ≤ u n
h : ∀ (x₀ : E), ¬Summable fun n => f n x₀
this : (fun x => ∑' (n : α), f n x) = 0
⊢ Differentiable 𝕜 fun y => ∑' (n : α), f n y

Next tactic:
rw [this]
/-- Consider a series of functions `∑' n, f n x`. If all functions in the series are differentiable
with a summable bound on the derivatives, then the series is differentiable.
Note that our assumptions do not ensure the pointwise convergence, but if there is no pointwise
convergence then the series is zero everywhere so the result still holds. -/
theorem differentiable_tsum (hu : Summable u) (hf : ∀ n x, HasFDerivAt (f n) (f' n x) x)
    (hf' : ∀ n x, ‖f' n x‖ ≤ u n) : Differentiable 𝕜 fun y => ∑' n, f n y := by
  by_cases h : ∃ x₀, Summable fun n => f n x₀
  · rcases h with ⟨x₀, hf0⟩
    intro x
    exact (hasFDerivAt_tsum hu hf hf' hf0 x).differentiableAt
  · push_neg at h
    have : (fun x => ∑' n, f n x) = 0 := by ext1 x; exact tsum_eq_zero_of_not_summable (h x)
    rw [this]
    

Proof state:
case neg
α : Type u_1
β : Type u_2
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁵ : IsROrC 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : CompleteSpace F
u : α → ℝ
inst✝ : NormedSpace 𝕜 F
f : α → E → F
f' : α → E → E →L[𝕜] F
v : ℕ → α → ℝ
s : Set E
x₀ x : E
N : ℕ∞
hu : Summable u
hf : ∀ (n : α) (x : E), HasFDerivAt (f n) (f' n x) x
hf' : ∀ (n : α) (x : E), ‖f' n x‖ ≤ u n
h : ∀ (x₀ : E), ¬Summable fun n => f n x₀
this : (fun x => ∑' (n : α), f n x) = 0
⊢ Differentiable 𝕜 0

Next tactic:
exact differentiable_const 0
theorem fderiv_tsum (hu : Summable u) (hf : ∀ n, Differentiable 𝕜 (f n))
    (hf' : ∀ n x, ‖fderiv 𝕜 (f n) x‖ ≤ u n) {x₀ : E} (hf0 : Summable fun n => f n x₀) :
    (fderiv 𝕜 fun y => ∑' n, f n y) = fun x => ∑' n, fderiv 𝕜 (f n) x := by
  

Proof state:
α : Type u_1
β : Type u_2
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁵ : IsROrC 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : CompleteSpace F
u : α → ℝ
inst✝ : NormedSpace 𝕜 F
f : α → E → F
f' : α → E → E →L[𝕜] F
v : ℕ → α → ℝ
s : Set E
x₀✝ x : E
N : ℕ∞
hu : Summable u
hf : ∀ (n : α), Differentiable 𝕜 (f n)
hf' : ∀ (n : α) (x : E), ‖fderiv 𝕜 (f n) x‖ ≤ u n
x₀ : E
hf0 : Summable fun n => f n x₀
⊢ (fderiv 𝕜 fun y => ∑' (n : α), f n y) = fun x => ∑' (n : α), fderiv 𝕜 (f n) x

Next tactic:
ext1 x
theorem fderiv_tsum (hu : Summable u) (hf : ∀ n, Differentiable 𝕜 (f n))
    (hf' : ∀ n x, ‖fderiv 𝕜 (f n) x‖ ≤ u n) {x₀ : E} (hf0 : Summable fun n => f n x₀) :
    (fderiv 𝕜 fun y => ∑' n, f n y) = fun x => ∑' n, fderiv 𝕜 (f n) x := by
  ext1 x
  

Proof state:
case h
α : Type u_1
β : Type u_2
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁵ : IsROrC 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : CompleteSpace F
u : α → ℝ
inst✝ : NormedSpace 𝕜 F
f : α → E → F
f' : α → E → E →L[𝕜] F
v : ℕ → α → ℝ
s : Set E
x₀✝ x✝ : E
N : ℕ∞
hu : Summable u
hf : ∀ (n : α), Differentiable 𝕜 (f n)
hf' : ∀ (n : α) (x : E), ‖fderiv 𝕜 (f n) x‖ ≤ u n
x₀ : E
hf0 : Summable fun n => f n x₀
x : E
⊢ fderiv 𝕜 (fun y => ∑' (n : α), f n y) x = ∑' (n : α), fderiv 𝕜 (f n) x

Next tactic:
exact fderiv_tsum_apply hu hf hf' hf0 x
/-- Consider a series of smooth functions, with summable uniform bounds on the successive
derivatives. Then the iterated derivative of the sum is the sum of the iterated derivative. -/
theorem iteratedFDeriv_tsum (hf : ∀ i, ContDiff 𝕜 N (f i))
    (hv : ∀ k : ℕ, (k : ℕ∞) ≤ N → Summable (v k))
    (h'f : ∀ (k : ℕ) (i : α) (x : E), (k : ℕ∞) ≤ N → ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i) {k : ℕ}
    (hk : (k : ℕ∞) ≤ N) :
    (iteratedFDeriv 𝕜 k fun y => ∑' n, f n y) = fun x => ∑' n, iteratedFDeriv 𝕜 k (f n) x := by
  

Proof state:
α : Type u_1
β : Type u_2
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁵ : IsROrC 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : CompleteSpace F
u : α → ℝ
inst✝ : NormedSpace 𝕜 F
f : α → E → F
f' : α → E → E →L[𝕜] F
v : ℕ → α → ℝ
s : Set E
x₀ x : E
N : ℕ∞
hf : ∀ (i : α), ContDiff 𝕜 N (f i)
hv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)
h'f : ∀ (k : ℕ) (i : α) (x : E), ↑k ≤ N → ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i
k : ℕ
hk : ↑k ≤ N
⊢ (iteratedFDeriv 𝕜 k fun y => ∑' (n : α), f n y) = fun x => ∑' (n : α), iteratedFDeriv 𝕜 k (f n) x

Next tactic:
induction' k with k IH
/-- Consider a series of smooth functions, with summable uniform bounds on the successive
derivatives. Then the iterated derivative of the sum is the sum of the iterated derivative. -/
theorem iteratedFDeriv_tsum (hf : ∀ i, ContDiff 𝕜 N (f i))
    (hv : ∀ k : ℕ, (k : ℕ∞) ≤ N → Summable (v k))
    (h'f : ∀ (k : ℕ) (i : α) (x : E), (k : ℕ∞) ≤ N → ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i) {k : ℕ}
    (hk : (k : ℕ∞) ≤ N) :
    (iteratedFDeriv 𝕜 k fun y => ∑' n, f n y) = fun x => ∑' n, iteratedFDeriv 𝕜 k (f n) x := by
  induction' k with k IH
  · 

Proof state:
case zero
α : Type u_1
β : Type u_2
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁵ : IsROrC 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : CompleteSpace F
u : α → ℝ
inst✝ : NormedSpace 𝕜 F
f : α → E → F
f' : α → E → E →L[𝕜] F
v : ℕ → α → ℝ
s : Set E
x₀ x : E
N : ℕ∞
hf : ∀ (i : α), ContDiff 𝕜 N (f i)
hv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)
h'f : ∀ (k : ℕ) (i : α) (x : E), ↑k ≤ N → ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i
hk : ↑Nat.zero ≤ N
⊢ (iteratedFDeriv 𝕜 Nat.zero fun y => ∑' (n : α), f n y) = fun x => ∑' (n : α), iteratedFDeriv 𝕜 Nat.zero (f n) x

Next tactic:
ext1 x
/-- Consider a series of smooth functions, with summable uniform bounds on the successive
derivatives. Then the iterated derivative of the sum is the sum of the iterated derivative. -/
theorem iteratedFDeriv_tsum (hf : ∀ i, ContDiff 𝕜 N (f i))
    (hv : ∀ k : ℕ, (k : ℕ∞) ≤ N → Summable (v k))
    (h'f : ∀ (k : ℕ) (i : α) (x : E), (k : ℕ∞) ≤ N → ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i) {k : ℕ}
    (hk : (k : ℕ∞) ≤ N) :
    (iteratedFDeriv 𝕜 k fun y => ∑' n, f n y) = fun x => ∑' n, iteratedFDeriv 𝕜 k (f n) x := by
  induction' k with k IH
  · ext1 x
    

Proof state:
case zero.h
α : Type u_1
β : Type u_2
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁵ : IsROrC 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : CompleteSpace F
u : α → ℝ
inst✝ : NormedSpace 𝕜 F
f : α → E → F
f' : α → E → E →L[𝕜] F
v : ℕ → α → ℝ
s : Set E
x₀ x✝ : E
N : ℕ∞
hf : ∀ (i : α), ContDiff 𝕜 N (f i)
hv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)
h'f : ∀ (k : ℕ) (i : α) (x : E), ↑k ≤ N → ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i
hk : ↑Nat.zero ≤ N
x : E
⊢ iteratedFDeriv 𝕜 Nat.zero (fun y => ∑' (n : α), f n y) x = ∑' (n : α), iteratedFDeriv 𝕜 Nat.zero (f n) x

Next tactic:
simp_rw [iteratedFDeriv_zero_eq_comp]
/-- Consider a series of smooth functions, with summable uniform bounds on the successive
derivatives. Then the iterated derivative of the sum is the sum of the iterated derivative. -/
theorem iteratedFDeriv_tsum (hf : ∀ i, ContDiff 𝕜 N (f i))
    (hv : ∀ k : ℕ, (k : ℕ∞) ≤ N → Summable (v k))
    (h'f : ∀ (k : ℕ) (i : α) (x : E), (k : ℕ∞) ≤ N → ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i) {k : ℕ}
    (hk : (k : ℕ∞) ≤ N) :
    (iteratedFDeriv 𝕜 k fun y => ∑' n, f n y) = fun x => ∑' n, iteratedFDeriv 𝕜 k (f n) x := by
  induction' k with k IH
  · ext1 x
    simp_rw [iteratedFDeriv_zero_eq_comp]
    

Proof state:
case zero.h
α : Type u_1
β : Type u_2
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁵ : IsROrC 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : CompleteSpace F
u : α → ℝ
inst✝ : NormedSpace 𝕜 F
f : α → E → F
f' : α → E → E →L[𝕜] F
v : ℕ → α → ℝ
s : Set E
x₀ x✝ : E
N : ℕ∞
hf : ∀ (i : α), ContDiff 𝕜 N (f i)
hv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)
h'f : ∀ (k : ℕ) (i : α) (x : E), ↑k ≤ N → ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i
hk : ↑Nat.zero ≤ N
x : E
⊢ (⇑(LinearIsometryEquiv.symm (continuousMultilinearCurryFin0 𝕜 E F)) ∘ fun y => ∑' (n : α), f n y) x =
    ∑' (n : α), (⇑(LinearIsometryEquiv.symm (continuousMultilinearCurryFin0 𝕜 E F)) ∘ f n) x

Next tactic:
exact (continuousMultilinearCurryFin0 𝕜 E F).symm.toContinuousLinearEquiv.map_tsum
/-- Consider a series of smooth functions, with summable uniform bounds on the successive
derivatives. Then the iterated derivative of the sum is the sum of the iterated derivative. -/
theorem iteratedFDeriv_tsum (hf : ∀ i, ContDiff 𝕜 N (f i))
    (hv : ∀ k : ℕ, (k : ℕ∞) ≤ N → Summable (v k))
    (h'f : ∀ (k : ℕ) (i : α) (x : E), (k : ℕ∞) ≤ N → ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i) {k : ℕ}
    (hk : (k : ℕ∞) ≤ N) :
    (iteratedFDeriv 𝕜 k fun y => ∑' n, f n y) = fun x => ∑' n, iteratedFDeriv 𝕜 k (f n) x := by
  induction' k with k IH
  · ext1 x
    simp_rw [iteratedFDeriv_zero_eq_comp]
    exact (continuousMultilinearCurryFin0 𝕜 E F).symm.toContinuousLinearEquiv.map_tsum
  · 

Proof state:
case succ
α : Type u_1
β : Type u_2
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁵ : IsROrC 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : CompleteSpace F
u : α → ℝ
inst✝ : NormedSpace 𝕜 F
f : α → E → F
f' : α → E → E →L[𝕜] F
v : ℕ → α → ℝ
s : Set E
x₀ x : E
N : ℕ∞
hf : ∀ (i : α), ContDiff 𝕜 N (f i)
hv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)
h'f : ∀ (k : ℕ) (i : α) (x : E), ↑k ≤ N → ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i
k : ℕ
IH : ↑k ≤ N → (iteratedFDeriv 𝕜 k fun y => ∑' (n : α), f n y) = fun x => ∑' (n : α), iteratedFDeriv 𝕜 k (f n) x
hk : ↑(Nat.succ k) ≤ N
⊢ (iteratedFDeriv 𝕜 (Nat.succ k) fun y => ∑' (n : α), f n y) = fun x =>
    ∑' (n : α), iteratedFDeriv 𝕜 (Nat.succ k) (f n) x

Next tactic:
have h'k : (k : ℕ∞) < N := lt_of_lt_of_le (WithTop.coe_lt_coe.2 (Nat.lt_succ_self _)) hk