Theorem: HurwitzZeta.differentiableAt_one_completedHurwitzZetaEven_sub_completedHurwitzZetaEven
File: Mathlib/NumberTheory/LSeries/HurwitzZetaEven.lean

tactic: have (s) : completedHurwitzZetaEven a s - completedHurwitzZetaEven b s =
    completedHurwitzZetaEven₀ a s - completedHurwitzZetaEven₀ b s -
    ((if a = 0 then 1 else 0) - (if b = 0 then 1 else 0)) / s := by
  simp_rw [completedHurwitzZetaEven_eq, sub_div]
  abel
state_before: a b : UnitAddCircle
⊢ DifferentiableAt ℂ (fun s => completedHurwitzZetaEven a s - completedHurwitzZetaEven b s) 1
state_after: a b : UnitAddCircle
this :
  ∀ (s : ℂ),
    completedHurwitzZetaEven a s - completedHurwitzZetaEven b s =
      completedHurwitzZetaEven₀ a s - completedHurwitzZetaEven₀ b s -
        ((if a = 0 then 1 else 0) - if b = 0 then 1 else 0) / s
⊢ DifferentiableAt ℂ (fun s => completedHurwitzZetaEven a s - completedHurwitzZetaEven b s) 1

tactic: rw [funext this]
state_before: a b : UnitAddCircle
this :
  ∀ (s : ℂ),
    completedHurwitzZetaEven a s - completedHurwitzZetaEven b s =
      completedHurwitzZetaEven₀ a s - completedHurwitzZetaEven₀ b s -
        ((if a = 0 then 1 else 0) - if b = 0 then 1 else 0) / s
⊢ DifferentiableAt ℂ (fun s => completedHurwitzZetaEven a s - completedHurwitzZetaEven b s) 1
state_after: a b : UnitAddCircle
this :
  ∀ (s : ℂ),
    completedHurwitzZetaEven a s - completedHurwitzZetaEven b s =
      completedHurwitzZetaEven₀ a s - completedHurwitzZetaEven₀ b s -
        ((if a = 0 then 1 else 0) - if b = 0 then 1 else 0) / s
⊢ DifferentiableAt ℂ
    (fun x =>
      completedHurwitzZetaEven₀ a x - completedHurwitzZetaEven₀ b x -
        ((if a = 0 then 1 else 0) - if b = 0 then 1 else 0) / x)
    1

tactic: refine .sub ?_ <| (differentiable_const _ _).div (differentiable_id _) one_ne_zero
state_before: a b : UnitAddCircle
this :
  ∀ (s : ℂ),
    completedHurwitzZetaEven a s - completedHurwitzZetaEven b s =
      completedHurwitzZetaEven₀ a s - completedHurwitzZetaEven₀ b s -
        ((if a = 0 then 1 else 0) - if b = 0 then 1 else 0) / s
⊢ DifferentiableAt ℂ
    (fun x =>
      completedHurwitzZetaEven₀ a x - completedHurwitzZetaEven₀ b x -
        ((if a = 0 then 1 else 0) - if b = 0 then 1 else 0) / x)
    1
state_after: a b : UnitAddCircle
this :
  ∀ (s : ℂ),
    completedHurwitzZetaEven a s - completedHurwitzZetaEven b s =
      completedHurwitzZetaEven₀ a s - completedHurwitzZetaEven₀ b s -
        ((if a = 0 then 1 else 0) - if b = 0 then 1 else 0) / s
⊢ DifferentiableAt ℂ (fun x => completedHurwitzZetaEven₀ a x - completedHurwitzZetaEven₀ b x) 1

tactic: apply DifferentiableAt.sub <;> apply differentiable_completedHurwitzZetaEven₀
state_before: a b : UnitAddCircle
this :
  ∀ (s : ℂ),
    completedHurwitzZetaEven a s - completedHurwitzZetaEven b s =
      completedHurwitzZetaEven₀ a s - completedHurwitzZetaEven₀ b s -
        ((if a = 0 then 1 else 0) - if b = 0 then 1 else 0) / s
⊢ DifferentiableAt ℂ (fun x => completedHurwitzZetaEven₀ a x - completedHurwitzZetaEven₀ b x) 1
state_after: no goals

tactic: simp_rw [completedHurwitzZetaEven_eq, sub_div]
state_before: a b : UnitAddCircle
s : ℂ
⊢ completedHurwitzZetaEven a s - completedHurwitzZetaEven b s =
    completedHurwitzZetaEven₀ a s - completedHurwitzZetaEven₀ b s -
      ((if a = 0 then 1 else 0) - if b = 0 then 1 else 0) / s
state_after: a b : UnitAddCircle
s : ℂ
⊢ completedHurwitzZetaEven₀ a s - (if a = 0 then 1 else 0) / s - 1 / (1 - s) -
      (completedHurwitzZetaEven₀ b s - (if b = 0 then 1 else 0) / s - 1 / (1 - s)) =
    completedHurwitzZetaEven₀ a s - completedHurwitzZetaEven₀ b s -
      ((if a = 0 then 1 else 0) / s - (if b = 0 then 1 else 0) / s)

tactic: abel
state_before: a b : UnitAddCircle
s : ℂ
⊢ completedHurwitzZetaEven₀ a s - (if a = 0 then 1 else 0) / s - 1 / (1 - s) -
      (completedHurwitzZetaEven₀ b s - (if b = 0 then 1 else 0) / s - 1 / (1 - s)) =
    completedHurwitzZetaEven₀ a s - completedHurwitzZetaEven₀ b s -
      ((if a = 0 then 1 else 0) / s - (if b = 0 then 1 else 0) / s)
state_after: no goals