-- Mathlib/NumberTheory/LSeries/ZMod.lean
theorem completedLFunction_one_sub_even (hΦ : Φ.Even) (s : ℂ)
    (hs₀ : s ≠ 0 ∨ ∑ j, Φ j = 0) (hs₁ : s ≠ 1 ∨ Φ 0 = 0) :
    completedLFunction Φ (1 - s) = N ^ (s - 1) * completedLFunction (𝓕 Φ) s

Proof state:
case h.e'_3.h
N : ℕ
inst✝ : NeZero N
Φ : ZMod N → ℂ
hΦ : Function.Even Φ
s : ℂ
hs₀ : s ≠ 0 ∨ ∑ j : ZMod N, Φ j = 0
hs₁ : s ≠ 1 ∨ Φ 0 = 0
F : ℂ → ℂ := fun t => completedLFunction Φ (1 - t)
G : ℂ → ℂ := fun t => ↑N ^ (t - 1) * completedLFunction (𝓕 Φ) t
U : Set ℂ := {t | (t ≠ 0 ∨ ∑ j : ZMod N, Φ j = 0) ∧ (t ≠ 1 ∨ Φ 0 = 0)}
hsU : s ∈ U
h2U : 2 ∈ U
hUo : IsOpen U
Uc : Set ℂ := (if ∑ j : ZMod N, Φ j = 0 then ∅ else {0}) ∪ if Φ 0 = 0 then ∅ else {1}
this : Uc.Countable
x : ℂ
⊢ x ∈ U ↔ x ∈ Ucᶜ

Tactic:
by_cases h : Φ 0 = 0 <;>
by_cases h' : ∑ j, Φ j = 0 <;>
simp [U, Uc, h, h', and_comm]

Result:
no goals

Proof state:
N : ℕ
inst✝ : NeZero N
Φ : ZMod N → ℂ
hΦ : Function.Even Φ
s : ℂ
hs₀ : s ≠ 0 ∨ ∑ j : ZMod N, Φ j = 0
hs₁ : s ≠ 1 ∨ Φ 0 = 0
F : ℂ → ℂ := fun t => completedLFunction Φ (1 - t)
G : ℂ → ℂ := fun t => ↑N ^ (t - 1) * completedLFunction (𝓕 Φ) t
U : Set ℂ := {t | (t ≠ 0 ∨ ∑ j : ZMod N, Φ j = 0) ∧ (t ≠ 1 ∨ Φ 0 = 0)}
⊢ completedLFunction Φ (1 - s) = ↑N ^ (s - 1) * completedLFunction (𝓕 Φ) s

Tactic:
have hsU : s ∈ U := ⟨hs₀, hs₁⟩

Result:
N : ℕ
inst✝ : NeZero N
Φ : ZMod N → ℂ
hΦ : Function.Even Φ
s : ℂ
hs₀ : s ≠ 0 ∨ ∑ j : ZMod N, Φ j = 0
hs₁ : s ≠ 1 ∨ Φ 0 = 0
F : ℂ → ℂ := fun t => completedLFunction Φ (1 - t)
G : ℂ → ℂ := fun t => ↑N ^ (t - 1) * completedLFunction (𝓕 Φ) t
U : Set ℂ := {t | (t ≠ 0 ∨ ∑ j : ZMod N, Φ j = 0) ∧ (t ≠ 1 ∨ Φ 0 = 0)}
hsU : s ∈ U
⊢ completedLFunction Φ (1 - s) = ↑N ^ (s - 1) * completedLFunction (𝓕 Φ) s

Proof state:
N : ℕ
inst✝ : NeZero N
Φ : ZMod N → ℂ
hΦ : Function.Even Φ
s : ℂ
hs₀ : s ≠ 0 ∨ ∑ j : ZMod N, Φ j = 0
hs₁ : s ≠ 1 ∨ Φ 0 = 0
F : ℂ → ℂ := fun t => completedLFunction Φ (1 - t)
G : ℂ → ℂ := fun t => ↑N ^ (t - 1) * completedLFunction (𝓕 Φ) t
U : Set ℂ := {t | (t ≠ 0 ∨ ∑ j : ZMod N, Φ j = 0) ∧ (t ≠ 1 ∨ Φ 0 = 0)}
hsU : s ∈ U
h2U : 2 ∈ U
hUo : IsOpen U
hUp : IsPreconnected U
hF : AnalyticOnNhd ℂ F U
hG : AnalyticOnNhd ℂ G U
hV : {z | 1 < z.re} ∈ 𝓝 2
hFG : F =ᶠ[𝓝 2] G
⊢ completedLFunction Φ (1 - s) = ↑N ^ (s - 1) * completedLFunction (𝓕 Φ) s

Tactic:
exact hF.eqOn_of_preconnected_of_eventuallyEq hG hUp h2U hFG hsU

Result:
no goals

Proof state:
N : ℕ
inst✝ : NeZero N
Φ : ZMod N → ℂ
hΦ : Function.Even Φ
s : ℂ
hs₀ : s ≠ 0 ∨ ∑ j : ZMod N, Φ j = 0
hs₁ : s ≠ 1 ∨ Φ 0 = 0
F : ℂ → ℂ := fun t => completedLFunction Φ (1 - t)
G : ℂ → ℂ := fun t => ↑N ^ (t - 1) * completedLFunction (𝓕 Φ) t
U : Set ℂ := {t | (t ≠ 0 ∨ ∑ j : ZMod N, Φ j = 0) ∧ (t ≠ 1 ∨ Φ 0 = 0)}
hsU : s ∈ U
h2U : 2 ∈ U
hUo : IsOpen U
Uc : Set ℂ := (if ∑ j : ZMod N, Φ j = 0 then ∅ else {0}) ∪ if Φ 0 = 0 then ∅ else {1}
⊢ IsPreconnected U

Tactic:
have : Uc.Countable := by
  apply Countable.union <;>
  split_ifs <;>
  simp only [countable_singleton, countable_empty]

Result:
N : ℕ
inst✝ : NeZero N
Φ : ZMod N → ℂ
hΦ : Function.Even Φ
s : ℂ
hs₀ : s ≠ 0 ∨ ∑ j : ZMod N, Φ j = 0
hs₁ : s ≠ 1 ∨ Φ 0 = 0
F : ℂ → ℂ := fun t => completedLFunction Φ (1 - t)
G : ℂ → ℂ := fun t => ↑N ^ (t - 1) * completedLFunction (𝓕 Φ) t
U : Set ℂ := {t | (t ≠ 0 ∨ ∑ j : ZMod N, Φ j = 0) ∧ (t ≠ 1 ∨ Φ 0 = 0)}
hsU : s ∈ U
h2U : 2 ∈ U
hUo : IsOpen U
Uc : Set ℂ := (if ∑ j : ZMod N, Φ j = 0 then ∅ else {0}) ∪ if Φ 0 = 0 then ∅ else {1}
this : Uc.Countable
⊢ IsPreconnected U

Proof state:
N : ℕ
inst✝ : NeZero N
Φ : ZMod N → ℂ
hΦ : Function.Even Φ
s : ℂ
hs₀ : s ≠ 0 ∨ ∑ j : ZMod N, Φ j = 0
hs₁ : s ≠ 1 ∨ Φ 0 = 0
F : ℂ → ℂ := fun t => completedLFunction Φ (1 - t)
G : ℂ → ℂ := fun t => ↑N ^ (t - 1) * completedLFunction (𝓕 Φ) t
U : Set ℂ := {t | (t ≠ 0 ∨ ∑ j : ZMod N, Φ j = 0) ∧ (t ≠ 1 ∨ Φ 0 = 0)}
hsU : s ∈ U
h2U : 2 ∈ U
hUo : IsOpen U
Uc : Set ℂ := (if ∑ j : ZMod N, Φ j = 0 then ∅ else {0}) ∪ if Φ 0 = 0 then ∅ else {1}
this : Uc.Countable
⊢ 1 < Module.rank ℝ ℂ

Tactic:
simp only [rank_real_complex, Nat.one_lt_ofNat]

Result:
no goals

Proof state:
N : ℕ
inst✝ : NeZero N
Φ : ZMod N → ℂ
hΦ : Function.Even Φ
s : ℂ
hs₀ : s ≠ 0 ∨ ∑ j : ZMod N, Φ j = 0
hs₁ : s ≠ 1 ∨ Φ 0 = 0
F : ℂ → ℂ := fun t => completedLFunction Φ (1 - t)
G : ℂ → ℂ := fun t => ↑N ^ (t - 1) * completedLFunction (𝓕 Φ) t
U : Set ℂ := {t | (t ≠ 0 ∨ ∑ j : ZMod N, Φ j = 0) ∧ (t ≠ 1 ∨ Φ 0 = 0)}
hsU : s ∈ U
h2U : 2 ∈ U
hUo : IsOpen U
hUp : IsPreconnected U
hF : AnalyticOnNhd ℂ F U
hG : AnalyticOnNhd ℂ G U
⊢ 2 ∈ re ⁻¹' Set.Ioi 1

Tactic:
simp

Result:
no goals

Proof state:
N : ℕ
inst✝ : NeZero N
Φ : ZMod N → ℂ
hΦ : Function.Even Φ
s : ℂ
hs₀ : s ≠ 0 ∨ ∑ j : ZMod N, Φ j = 0
hs₁ : s ≠ 1 ∨ Φ 0 = 0
F : ℂ → ℂ := fun t => completedLFunction Φ (1 - t)
G : ℂ → ℂ := fun t => ↑N ^ (t - 1) * completedLFunction (𝓕 Φ) t
U : Set ℂ := {t | (t ≠ 0 ∨ ∑ j : ZMod N, Φ j = 0) ∧ (t ≠ 1 ∨ Φ 0 = 0)}
hsU : s ∈ U
h2U : 2 ∈ U
hUo : IsOpen U
⊢ IsPreconnected U

Tactic:
let Uc : Set ℂ := (if ∑ j, Φ j = 0 then ∅ else {0}) ∪ (if Φ 0 = 0 then ∅ else {1})

Result:
N : ℕ
inst✝ : NeZero N
Φ : ZMod N → ℂ
hΦ : Function.Even Φ
s : ℂ
hs₀ : s ≠ 0 ∨ ∑ j : ZMod N, Φ j = 0
hs₁ : s ≠ 1 ∨ Φ 0 = 0
F : ℂ → ℂ := fun t => completedLFunction Φ (1 - t)
G : ℂ → ℂ := fun t => ↑N ^ (t - 1) * completedLFunction (𝓕 Φ) t
U : Set ℂ := {t | (t ≠ 0 ∨ ∑ j : ZMod N, Φ j = 0) ∧ (t ≠ 1 ∨ Φ 0 = 0)}
hsU : s ∈ U
h2U : 2 ∈ U
hUo : IsOpen U
Uc : Set ℂ := (if ∑ j : ZMod N, Φ j = 0 then ∅ else {0}) ∪ if Φ 0 = 0 then ∅ else {1}
⊢ IsPreconnected U

Proof state:
N : ℕ
inst✝ : NeZero N
Φ : ZMod N → ℂ
hΦ : Function.Even Φ
s : ℂ
hs₀ : s ≠ 0 ∨ ∑ j : ZMod N, Φ j = 0
hs₁ : s ≠ 1 ∨ Φ 0 = 0
⊢ completedLFunction Φ (1 - s) = ↑N ^ (s - 1) * completedLFunction (𝓕 Φ) s

Tactic:
let F (t) := completedLFunction Φ (1 - t)

Result:
N : ℕ
inst✝ : NeZero N
Φ : ZMod N → ℂ
hΦ : Function.Even Φ
s : ℂ
hs₀ : s ≠ 0 ∨ ∑ j : ZMod N, Φ j = 0
hs₁ : s ≠ 1 ∨ Φ 0 = 0
F : ℂ → ℂ := fun t => completedLFunction Φ (1 - t)
⊢ completedLFunction Φ (1 - s) = ↑N ^ (s - 1) * completedLFunction (𝓕 Φ) s

Proof state:
N : ℕ
inst✝ : NeZero N
Φ : ZMod N → ℂ
hΦ : Function.Even Φ
s : ℂ
hs₀ : s ≠ 0 ∨ ∑ j : ZMod N, Φ j = 0
hs₁ : s ≠ 1 ∨ Φ 0 = 0
F : ℂ → ℂ := fun t => completedLFunction Φ (1 - t)
G : ℂ → ℂ := fun t => ↑N ^ (t - 1) * completedLFunction (𝓕 Φ) t
U : Set ℂ := {t | (t ≠ 0 ∨ ∑ j : ZMod N, Φ j = 0) ∧ (t ≠ 1 ∨ Φ 0 = 0)}
hsU : s ∈ U
h2U : 2 ∈ U
hUo : IsOpen U
hUp : IsPreconnected U
hF : AnalyticOnNhd ℂ F U
t : ℂ
ht : t ∈ U
⊢ DifferentiableAt ℂ G t

Tactic:
apply ((differentiableAt_id.sub_const 1).const_cpow (.inl (NeZero.ne _))).mul

Result:
N : ℕ
inst✝ : NeZero N
Φ : ZMod N → ℂ
hΦ : Function.Even Φ
s : ℂ
hs₀ : s ≠ 0 ∨ ∑ j : ZMod N, Φ j = 0
hs₁ : s ≠ 1 ∨ Φ 0 = 0
F : ℂ → ℂ := fun t => completedLFunction Φ (1 - t)
G : ℂ → ℂ := fun t => ↑N ^ (t - 1) * completedLFunction (𝓕 Φ) t
U : Set ℂ := {t | (t ≠ 0 ∨ ∑ j : ZMod N, Φ j = 0) ∧ (t ≠ 1 ∨ Φ 0 = 0)}
hsU : s ∈ U
h2U : 2 ∈ U
hUo : IsOpen U
hUp : IsPreconnected U
hF : AnalyticOnNhd ℂ F U
t : ℂ
ht : t ∈ U
⊢ DifferentiableAt ℂ (completedLFunction (𝓕 Φ)) t

Proof state:
N : ℕ
inst✝ : NeZero N
Φ : ZMod N → ℂ
hΦ : Function.Even Φ
s : ℂ
hs₀ : s ≠ 0 ∨ ∑ j : ZMod N, Φ j = 0
hs₁ : s ≠ 1 ∨ Φ 0 = 0
F : ℂ → ℂ := fun t => completedLFunction Φ (1 - t)
⊢ completedLFunction Φ (1 - s) = ↑N ^ (s - 1) * completedLFunction (𝓕 Φ) s

Tactic:
let G (t) := ↑N ^ (t - 1) * completedLFunction (𝓕 Φ) t

Result:
N : ℕ
inst✝ : NeZero N
Φ : ZMod N → ℂ
hΦ : Function.Even Φ
s : ℂ
hs₀ : s ≠ 0 ∨ ∑ j : ZMod N, Φ j = 0
hs₁ : s ≠ 1 ∨ Φ 0 = 0
F : ℂ → ℂ := fun t => completedLFunction Φ (1 - t)
G : ℂ → ℂ := fun t => ↑N ^ (t - 1) * completedLFunction (𝓕 Φ) t
⊢ completedLFunction Φ (1 - s) = ↑N ^ (s - 1) * completedLFunction (𝓕 Φ) s