z : ℂ
⊢ HasSum (fun n => (z * I) ^ (2 * n) / ↑(2 * n)!) ((_root_.exp ℂ (z * I) + _root_.exp ℂ (-z * I)) / 2)
z : ℂ
⊢ HasSum (fun n => (z * I) ^ (2 * n) / ↑(2 * n)!) (cos z)
rw [Complex.cos, Complex.exp_eq_exp_ℂ]
z : ℂ
this :
  HasSum (fun i => ((z * I) ^ i / ↑i ! + (-z * I) ^ i / ↑i !) / 2) ((_root_.exp ℂ (z * I) + _root_.exp ℂ (-z * I)) / 2)
⊢ HasSum (fun n => (z * I) ^ (2 * n) / ↑(2 * n)!) ((_root_.exp ℂ (z * I) + _root_.exp ℂ (-z * I)) / 2)
z : ℂ
⊢ HasSum (fun n => (z * I) ^ (2 * n) / ↑(2 * n)!) ((_root_.exp ℂ (z * I) + _root_.exp ℂ (-z * I)) / 2)
have := ((expSeries_div_hasSum_exp ℂ (z * Complex.I)).add
  (expSeries_div_hasSum_exp ℂ (-z * Complex.I))).div_const 2
z : ℂ
this :
  HasSum ((fun i => ((z * I) ^ i / ↑i ! + (-z * I) ^ i / ↑i !) / 2) ∘ ↑(Nat.divModEquiv 2).symm)
    ((_root_.exp ℂ (z * I) + _root_.exp ℂ (-z * I)) / 2)
⊢ HasSum (fun n => (z * I) ^ (2 * n) / ↑(2 * n)!) ((_root_.exp ℂ (z * I) + _root_.exp ℂ (-z * I)) / 2)
z : ℂ
this :
  HasSum (fun i => ((z * I) ^ i / ↑i ! + (-z * I) ^ i / ↑i !) / 2) ((_root_.exp ℂ (z * I) + _root_.exp ℂ (-z * I)) / 2)
⊢ HasSum (fun n => (z * I) ^ (2 * n) / ↑(2 * n)!) ((_root_.exp ℂ (z * I) + _root_.exp ℂ (-z * I)) / 2)
replace := (Nat.divModEquiv 2).symm.hasSum_iff.mpr this
z : ℂ
this :
  HasSum
    (fun x =>
      ((z * I) ^ (x.fst * 2 + ↑x.snd) / ↑(x.fst * 2 + ↑x.snd)! +
          (-z * I) ^ (x.fst * 2 + ↑x.snd) / ↑(x.fst * 2 + ↑x.snd)!) /
        2)
    ((_root_.exp ℂ (z * I) + _root_.exp ℂ (-z * I)) / 2)
⊢ HasSum (fun n => (z * I) ^ (2 * n) / ↑(2 * n)!) ((_root_.exp ℂ (z * I) + _root_.exp ℂ (-z * I)) / 2)
z : ℂ
this :
  HasSum ((fun i => ((z * I) ^ i / ↑i ! + (-z * I) ^ i / ↑i !) / 2) ∘ ↑(Nat.divModEquiv 2).symm)
    ((_root_.exp ℂ (z * I) + _root_.exp ℂ (-z * I)) / 2)
⊢ HasSum (fun n => (z * I) ^ (2 * n) / ↑(2 * n)!) ((_root_.exp ℂ (z * I) + _root_.exp ℂ (-z * I)) / 2)
dsimp [Function.comp] at this
z : ℂ
this :
  HasSum
    (fun x =>
      ((z * I) ^ (2 * x.fst + ↑x.snd) / ↑(2 * x.fst + ↑x.snd)! +
          (-z * I) ^ (2 * x.fst + ↑x.snd) / ↑(2 * x.fst + ↑x.snd)!) /
        2)
    ((_root_.exp ℂ (z * I) + _root_.exp ℂ (-z * I)) / 2)
⊢ HasSum (fun n => (z * I) ^ (2 * n) / ↑(2 * n)!) ((_root_.exp ℂ (z * I) + _root_.exp ℂ (-z * I)) / 2)
z : ℂ
this :
  HasSum
    (fun x =>
      ((z * I) ^ (x.fst * 2 + ↑x.snd) / ↑(x.fst * 2 + ↑x.snd)! +
          (-z * I) ^ (x.fst * 2 + ↑x.snd) / ↑(x.fst * 2 + ↑x.snd)!) /
        2)
    ((_root_.exp ℂ (z * I) + _root_.exp ℂ (-z * I)) / 2)
⊢ HasSum (fun n => (z * I) ^ (2 * n) / ↑(2 * n)!) ((_root_.exp ℂ (z * I) + _root_.exp ℂ (-z * I)) / 2)
simp_rw [← mul_comm 2 _] at this
z : ℂ
this :
  HasSum
    (fun x =>
      ((z * I) ^ (2 * x.fst + ↑x.snd) / ↑(2 * x.fst + ↑x.snd)! +
          (-z * I) ^ (2 * x.fst + ↑x.snd) / ↑(2 * x.fst + ↑x.snd)!) /
        2)
    ((_root_.exp ℂ (z * I) + _root_.exp ℂ (-z * I)) / 2)
k : ℕ
⊢ HasSum
    (fun c =>
      ((z * I) ^ (2 * (k, c).fst + ↑(k, c).snd) / ↑(2 * (k, c).fst + ↑(k, c).snd)! +
          (-z * I) ^ (2 * (k, c).fst + ↑(k, c).snd) / ↑(2 * (k, c).fst + ↑(k, c).snd)!) /
        2)
    ((z * I) ^ (2 * k) / ↑(2 * k)!)
z : ℂ
this :
  HasSum
    (fun x =>
      ((z * I) ^ (2 * x.fst + ↑x.snd) / ↑(2 * x.fst + ↑x.snd)! +
          (-z * I) ^ (2 * x.fst + ↑x.snd) / ↑(2 * x.fst + ↑x.snd)!) /
        2)
    ((_root_.exp ℂ (z * I) + _root_.exp ℂ (-z * I)) / 2)
⊢ HasSum (fun n => (z * I) ^ (2 * n) / ↑(2 * n)!) ((_root_.exp ℂ (z * I) + _root_.exp ℂ (-z * I)) / 2)
refine' this.prod_fiberwise fun k => _
z : ℂ
this :
  HasSum
    (fun x =>
      ((z * I) ^ (2 * x.fst + ↑x.snd) / ↑(2 * x.fst + ↑x.snd)! +
          (-z * I) ^ (2 * x.fst + ↑x.snd) / ↑(2 * x.fst + ↑x.snd)!) /
        2)
    ((_root_.exp ℂ (z * I) + _root_.exp ℂ (-z * I)) / 2)
k : ℕ
⊢ HasSum (fun c => ((z * I) ^ (2 * k + ↑c) / ↑(2 * k + ↑c)! + (-z * I) ^ (2 * k + ↑c) / ↑(2 * k + ↑c)!) / 2)
    ((z * I) ^ (2 * k) / ↑(2 * k)!)
z : ℂ
this :
  HasSum
    (fun x =>
      ((z * I) ^ (2 * x.fst + ↑x.snd) / ↑(2 * x.fst + ↑x.snd)! +
          (-z * I) ^ (2 * x.fst + ↑x.snd) / ↑(2 * x.fst + ↑x.snd)!) /
        2)
    ((_root_.exp ℂ (z * I) + _root_.exp ℂ (-z * I)) / 2)
k : ℕ
⊢ HasSum
    (fun c =>
      ((z * I) ^ (2 * (k, c).fst + ↑(k, c).snd) / ↑(2 * (k, c).fst + ↑(k, c).snd)! +
          (-z * I) ^ (2 * (k, c).fst + ↑(k, c).snd) / ↑(2 * (k, c).fst + ↑(k, c).snd)!) /
        2)
    ((z * I) ^ (2 * k) / ↑(2 * k)!)
dsimp only
case h.e'_6
z : ℂ
this :
  HasSum
    (fun x =>
      ((z * I) ^ (2 * x.fst + ↑x.snd) / ↑(2 * x.fst + ↑x.snd)! +
          (-z * I) ^ (2 * x.fst + ↑x.snd) / ↑(2 * x.fst + ↑x.snd)!) /
        2)
    ((_root_.exp ℂ (z * I) + _root_.exp ℂ (-z * I)) / 2)
k : ℕ
⊢ (z * I) ^ (2 * k) / ↑(2 * k)! =
    Finset.sum Finset.univ fun b =>
      ((z * I) ^ (2 * k + ↑b) / ↑(2 * k + ↑b)! + (-z * I) ^ (2 * k + ↑b) / ↑(2 * k + ↑b)!) / 2
z : ℂ
this :
  HasSum
    (fun x =>
      ((z * I) ^ (2 * x.fst + ↑x.snd) / ↑(2 * x.fst + ↑x.snd)! +
          (-z * I) ^ (2 * x.fst + ↑x.snd) / ↑(2 * x.fst + ↑x.snd)!) /
        2)
    ((_root_.exp ℂ (z * I) + _root_.exp ℂ (-z * I)) / 2)
k : ℕ
⊢ HasSum (fun c => ((z * I) ^ (2 * k + ↑c) / ↑(2 * k + ↑c)! + (-z * I) ^ (2 * k + ↑c) / ↑(2 * k + ↑c)!) / 2)
    ((z * I) ^ (2 * k) / ↑(2 * k)!)
convert hasSum_fintype (_ : Fin 2 → ℂ) using 1
case h.e'_6
z : ℂ
this :
  HasSum
    (fun x =>
      ((z * I) ^ (2 * x.fst + ↑x.snd) / ↑(2 * x.fst + ↑x.snd)! +
          (-z * I) ^ (2 * x.fst + ↑x.snd) / ↑(2 * x.fst + ↑x.snd)!) /
        2)
    ((_root_.exp ℂ (z * I) + _root_.exp ℂ (-z * I)) / 2)
k : ℕ
⊢ (z * I) ^ (2 * k) / ↑(2 * k)! =
    ((z * I) ^ (2 * k + ↑0) / ↑(2 * k + ↑0)! + (-z * I) ^ (2 * k + ↑0) / ↑(2 * k + ↑0)!) / 2 +
      ((z * I) ^ (2 * k + ↑1) / ↑(2 * k + ↑1)! + (-z * I) ^ (2 * k + ↑1) / ↑(2 * k + ↑1)!) / 2
case h.e'_6
z : ℂ
this :
  HasSum
    (fun x =>
      ((z * I) ^ (2 * x.fst + ↑x.snd) / ↑(2 * x.fst + ↑x.snd)! +
          (-z * I) ^ (2 * x.fst + ↑x.snd) / ↑(2 * x.fst + ↑x.snd)!) /
        2)
    ((_root_.exp ℂ (z * I) + _root_.exp ℂ (-z * I)) / 2)
k : ℕ
⊢ (z * I) ^ (2 * k) / ↑(2 * k)! =
    Finset.sum Finset.univ fun b =>
      ((z * I) ^ (2 * k + ↑b) / ↑(2 * k + ↑b)! + (-z * I) ^ (2 * k + ↑b) / ↑(2 * k + ↑b)!) / 2
rw [Fin.sum_univ_two]
no goals
case h.e'_6
z : ℂ
this :
  HasSum
    (fun x =>
      ((z * I) ^ (2 * x.fst + ↑x.snd) / ↑(2 * x.fst + ↑x.snd)! +
          (-z * I) ^ (2 * x.fst + ↑x.snd) / ↑(2 * x.fst + ↑x.snd)!) /
        2)
    ((_root_.exp ℂ (z * I) + _root_.exp ℂ (-z * I)) / 2)
k : ℕ
⊢ (z * I) ^ (2 * k) / ↑(2 * k)! =
    ((z * I) ^ (2 * k + ↑0) / ↑(2 * k + ↑0)! + (-z * I) ^ (2 * k + ↑0) / ↑(2 * k + ↑0)!) / 2 +
      ((z * I) ^ (2 * k + ↑1) / ↑(2 * k + ↑1)! + (-z * I) ^ (2 * k + ↑1) / ↑(2 * k + ↑1)!) / 2
simp_rw [Fin.val_zero, Fin.val_one, add_zero, pow_succ', pow_mul, mul_pow, neg_sq, ← two_mul,
  neg_mul, mul_neg, neg_div, add_right_neg, zero_div, add_zero,
  mul_div_cancel_left _ (two_ne_zero : (2 : ℂ) ≠ 0)]