Theorem: Polynomial.cyclotomic_eq_X_pow_sub_one_div
File: Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean

tactic: nontriviality R
state_before: R : Type u_1
inst✝ : CommRing R
n : ℕ
hpos : 0 < n
⊢ cyclotomic n R = (X ^ n - 1) /ₘ ∏ i in Nat.properDivisors n, cyclotomic i R

tactic: rw [← prod_cyclotomic_eq_X_pow_sub_one hpos, ← Nat.cons_self_properDivisors hpos.ne',
  Finset.prod_cons]
state_before: R : Type u_1
inst✝ : CommRing R
n : ℕ
hpos : 0 < n
✝ : Nontrivial R
⊢ cyclotomic n R = (X ^ n - 1) /ₘ ∏ i in Nat.properDivisors n, cyclotomic i R

tactic: have prod_monic : (∏ i in Nat.properDivisors n, cyclotomic i R).Monic := by
  apply monic_prod_of_monic
  intro i _
  exact cyclotomic.monic i R
state_before: R : Type u_1
inst✝ : CommRing R
n : ℕ
hpos : 0 < n
✝ : Nontrivial R
⊢ cyclotomic n R =
    (cyclotomic n R * ∏ x in Nat.properDivisors n, cyclotomic x R) /ₘ ∏ i in Nat.properDivisors n, cyclotomic i R

tactic: rw [(div_modByMonic_unique (cyclotomic n R) 0 prod_monic _).1]
state_before: R : Type u_1
inst✝ : CommRing R
n : ℕ
hpos : 0 < n
✝ : Nontrivial R
prod_monic : Monic (∏ i in Nat.properDivisors n, cyclotomic i R)
⊢ cyclotomic n R =
    (cyclotomic n R * ∏ x in Nat.properDivisors n, cyclotomic x R) /ₘ ∏ i in Nat.properDivisors n, cyclotomic i R

tactic: simp only [degree_zero, zero_add]
state_before: R : Type u_1
inst✝ : CommRing R
n : ℕ
hpos : 0 < n
✝ : Nontrivial R
prod_monic : Monic (∏ i in Nat.properDivisors n, cyclotomic i R)
⊢ 0 + (∏ i in Nat.properDivisors n, cyclotomic i R) * cyclotomic n R =
      cyclotomic n R * ∏ x in Nat.properDivisors n, cyclotomic x R ∧
    degree 0 < degree (∏ i in Nat.properDivisors n, cyclotomic i R)

tactic: constructor
state_before: R : Type u_1
inst✝ : CommRing R
n : ℕ
hpos : 0 < n
✝ : Nontrivial R
prod_monic : Monic (∏ i in Nat.properDivisors n, cyclotomic i R)
⊢ (∏ i in Nat.properDivisors n, cyclotomic i R) * cyclotomic n R =
      cyclotomic n R * ∏ i in Nat.properDivisors n, cyclotomic i R ∧
    ⊥ < degree (∏ i in Nat.properDivisors n, cyclotomic i R)

tactic: rw [bot_lt_iff_ne_bot]
state_before: case right
R : Type u_1
inst✝ : CommRing R
n : ℕ
hpos : 0 < n
✝ : Nontrivial R
prod_monic : Monic (∏ i in Nat.properDivisors n, cyclotomic i R)
⊢ ⊥ < degree (∏ i in Nat.properDivisors n, cyclotomic i R)

tactic: intro h
state_before: case right
R : Type u_1
inst✝ : CommRing R
n : ℕ
hpos : 0 < n
✝ : Nontrivial R
prod_monic : Monic (∏ i in Nat.properDivisors n, cyclotomic i R)
⊢ degree (∏ i in Nat.properDivisors n, cyclotomic i R) ≠ ⊥

tactic: exact Monic.ne_zero prod_monic (degree_eq_bot.1 h)
state_before: case right
R : Type u_1
inst✝ : CommRing R
n : ℕ
hpos : 0 < n
✝ : Nontrivial R
prod_monic : Monic (∏ i in Nat.properDivisors n, cyclotomic i R)
h : degree (∏ i in Nat.properDivisors n, cyclotomic i R) = ⊥
⊢ False

tactic: apply monic_prod_of_monic
state_before: R : Type u_1
inst✝ : CommRing R
n : ℕ
hpos : 0 < n
✝ : Nontrivial R
⊢ Monic (∏ i in Nat.properDivisors n, cyclotomic i R)

tactic: intro i _
state_before: case hs
R : Type u_1
inst✝ : CommRing R
n : ℕ
hpos : 0 < n
✝ : Nontrivial R
⊢ ∀ (i : ℕ), i ∈ Nat.properDivisors n → Monic (cyclotomic i R)

tactic: exact cyclotomic.monic i R
state_before: case hs
R : Type u_1
inst✝ : CommRing R
n : ℕ
hpos : 0 < n
✝ : Nontrivial R
i : ℕ
a✝ : i ∈ Nat.properDivisors n
⊢ Monic (cyclotomic i R)

tactic: rw [mul_comm]
state_before: case left
R : Type u_1
inst✝ : CommRing R
n : ℕ
hpos : 0 < n
✝ : Nontrivial R
prod_monic : Monic (∏ i in Nat.properDivisors n, cyclotomic i R)
⊢ (∏ i in Nat.properDivisors n, cyclotomic i R) * cyclotomic n R =
    cyclotomic n R * ∏ i in Nat.properDivisors n, cyclotomic i R