-- Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean
theorem cyclotomic_coeff_zero (R : Type*) [CommRing R] {n : ℕ} (hn : 1 < n) :
    (cyclotomic n R).coeff 0 = 1

Proof state:
case h
R : Type u_1
inst✝ : CommRing R
n : ℕ
hi : ∀ m < n, 1 < m → (cyclotomic m R).coeff 0 = 1
hn : 1 < n
hprod : ∏ i ∈ n.properDivisors, (cyclotomic i R).coeff 0 = -1
⊢ (cyclotomic n R).coeff 0 = 1

Tactic:
have heq : (X ^ n - 1 : R[X]).coeff 0 = -(cyclotomic n R).coeff 0 := by
  rw [← prod_cyclotomic_eq_X_pow_sub_one (zero_le_one.trans_lt hn), ←
    Nat.cons_self_properDivisors hn.ne_bot, Finset.prod_cons, mul_coeff_zero, coeff_zero_prod,
    hprod, mul_neg, mul_one]

Result:
case h
R : Type u_1
inst✝ : CommRing R
n : ℕ
hi : ∀ m < n, 1 < m → (cyclotomic m R).coeff 0 = 1
hn : 1 < n
hprod : ∏ i ∈ n.properDivisors, (cyclotomic i R).coeff 0 = -1
heq : (X ^ n - 1).coeff 0 = -(cyclotomic n R).coeff 0
⊢ (cyclotomic n R).coeff 0 = 1

Proof state:
R : Type u_1
inst✝ : CommRing R
n : ℕ
hi : ∀ m < n, 1 < m → (cyclotomic m R).coeff 0 = 1
hn : 1 < n
hleq : ∀ j ∈ n.properDivisors.erase 1, 2 ≤ j
hcongr : ∀ j ∈ n.properDivisors.erase 1, (cyclotomic j R).coeff 0 = 1
⊢ (X - 1).coeff 0 * ∏ x ∈ n.properDivisors.erase 1, (cyclotomic x R).coeff 0 = -1

Tactic:
have hrw : (∏ x ∈ n.properDivisors.erase 1, (cyclotomic x R).coeff 0) = 1 := by
  rw [Finset.prod_congr (refl (n.properDivisors.erase 1)) hcongr]
  simp only [Finset.prod_const_one]

Result:
R : Type u_1
inst✝ : CommRing R
n : ℕ
hi : ∀ m < n, 1 < m → (cyclotomic m R).coeff 0 = 1
hn : 1 < n
hleq : ∀ j ∈ n.properDivisors.erase 1, 2 ≤ j
hcongr : ∀ j ∈ n.properDivisors.erase 1, (cyclotomic j R).coeff 0 = 1
hrw : ∏ x ∈ n.properDivisors.erase 1, (cyclotomic x R).coeff 0 = 1
⊢ (X - 1).coeff 0 * ∏ x ∈ n.properDivisors.erase 1, (cyclotomic x R).coeff 0 = -1

Proof state:
R : Type u_1
inst✝ : CommRing R
n : ℕ
hi : ∀ m < n, 1 < m → (cyclotomic m R).coeff 0 = 1
hn : 1 < n
⊢ (X - 1).coeff 0 * ∏ x ∈ n.properDivisors.erase 1, (cyclotomic x R).coeff 0 = -1

Tactic:
have hleq : ∀ j ∈ n.properDivisors.erase 1, 2 ≤ j := by
  intro j hj
  apply Nat.succ_le_of_lt
  exact (Ne.le_iff_lt (Finset.mem_erase.1 hj).1.symm).mp
    (Nat.succ_le_of_lt (Nat.pos_of_mem_properDivisors (Finset.mem_erase.1 hj).2))

Result:
R : Type u_1
inst✝ : CommRing R
n : ℕ
hi : ∀ m < n, 1 < m → (cyclotomic m R).coeff 0 = 1
hn : 1 < n
hleq : ∀ j ∈ n.properDivisors.erase 1, 2 ≤ j
⊢ (X - 1).coeff 0 * ∏ x ∈ n.properDivisors.erase 1, (cyclotomic x R).coeff 0 = -1

Proof state:
R : Type u_1
inst✝ : CommRing R
n : ℕ
hi : ∀ m < n, 1 < m → (cyclotomic m R).coeff 0 = 1
hn : 1 < n
⊢ ∀ j ∈ n.properDivisors.erase 1, 2 ≤ j

Tactic:
intro j hj

Result:
R : Type u_1
inst✝ : CommRing R
n : ℕ
hi : ∀ m < n, 1 < m → (cyclotomic m R).coeff 0 = 1
hn : 1 < n
j : ℕ
hj : j ∈ n.properDivisors.erase 1
⊢ 2 ≤ j

Proof state:
R : Type u_1
inst✝ : CommRing R
n : ℕ
hi : ∀ m < n, 1 < m → (cyclotomic m R).coeff 0 = 1
hn : 1 < n
⊢ ∏ i ∈ n.properDivisors, (cyclotomic i R).coeff 0 = -1

Tactic:
rw [← Finset.insert_erase (Nat.one_mem_properDivisors_iff_one_lt.2
  (lt_of_lt_of_le one_lt_two hn)), Finset.prod_insert (Finset.not_mem_erase 1 _),
  cyclotomic_one R]

Result:
R : Type u_1
inst✝ : CommRing R
n : ℕ
hi : ∀ m < n, 1 < m → (cyclotomic m R).coeff 0 = 1
hn : 1 < n
⊢ (X - 1).coeff 0 * ∏ x ∈ n.properDivisors.erase 1, (cyclotomic x R).coeff 0 = -1

Proof state:
R : Type u_1
inst✝ : CommRing R
n : ℕ
hi : ∀ m < n, 1 < m → (cyclotomic m R).coeff 0 = 1
hn : 1 < n
hleq : ∀ j ∈ n.properDivisors.erase 1, 2 ≤ j
⊢ (X - 1).coeff 0 * ∏ x ∈ n.properDivisors.erase 1, (cyclotomic x R).coeff 0 = -1

Tactic:
have hcongr : ∀ j ∈ n.properDivisors.erase 1, (cyclotomic j R).coeff 0 = 1 := by
  intro j hj
  exact hi j (Nat.mem_properDivisors.1 (Finset.mem_erase.1 hj).2).2 (hleq j hj)

Result:
R : Type u_1
inst✝ : CommRing R
n : ℕ
hi : ∀ m < n, 1 < m → (cyclotomic m R).coeff 0 = 1
hn : 1 < n
hleq : ∀ j ∈ n.properDivisors.erase 1, 2 ≤ j
hcongr : ∀ j ∈ n.properDivisors.erase 1, (cyclotomic j R).coeff 0 = 1
⊢ (X - 1).coeff 0 * ∏ x ∈ n.properDivisors.erase 1, (cyclotomic x R).coeff 0 = -1

Proof state:
R : Type u_1
inst✝ : CommRing R
n : ℕ
hi : ∀ m < n, 1 < m → (cyclotomic m R).coeff 0 = 1
hn : 1 < n
hleq : ∀ j ∈ n.properDivisors.erase 1, 2 ≤ j
j : ℕ
hj : j ∈ n.properDivisors.erase 1
⊢ (cyclotomic j R).coeff 0 = 1

Tactic:
exact hi j (Nat.mem_properDivisors.1 (Finset.mem_erase.1 hj).2).2 (hleq j hj)

Result:
no goals

Proof state:
R : Type u_1
inst✝ : CommRing R
n : ℕ
hi : ∀ m < n, 1 < m → (cyclotomic m R).coeff 0 = 1
hn : 1 < n
hleq : ∀ j ∈ n.properDivisors.erase 1, 2 ≤ j
⊢ ∀ j ∈ n.properDivisors.erase 1, (cyclotomic j R).coeff 0 = 1

Tactic:
intro j hj

Result:
R : Type u_1
inst✝ : CommRing R
n : ℕ
hi : ∀ m < n, 1 < m → (cyclotomic m R).coeff 0 = 1
hn : 1 < n
hleq : ∀ j ∈ n.properDivisors.erase 1, 2 ≤ j
j : ℕ
hj : j ∈ n.properDivisors.erase 1
⊢ (cyclotomic j R).coeff 0 = 1