Theorem: Associates.count_of_coprime
File: Mathlib/RingTheory/UniqueFactorizationDomain.lean

tactic: rw [or_iff_not_imp_left, ← Ne]
state_before: α : Type u_1
inst✝⁴ : CancelCommMonoidWithZero α
inst✝³ : UniqueFactorizationMonoid α
inst✝² : DecidableEq (Associates α)
inst✝¹ : (p : Associates α) → Decidable (Irreducible p)
inst✝ : DecidableEq (Associates α)
a : Associates α
ha : a ≠ 0
b : Associates α
hb : b ≠ 0
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
p : Associates α
hp : Irreducible p
⊢ p.count a.factors = 0 ∨ p.count b.factors = 0
state_after: α : Type u_1
inst✝⁴ : CancelCommMonoidWithZero α
inst✝³ : UniqueFactorizationMonoid α
inst✝² : DecidableEq (Associates α)
inst✝¹ : (p : Associates α) → Decidable (Irreducible p)
inst✝ : DecidableEq (Associates α)
a : Associates α
ha : a ≠ 0
b : Associates α
hb : b ≠ 0
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
p : Associates α
hp : Irreducible p
⊢ p.count a.factors ≠ 0 → p.count b.factors = 0

tactic: intro hca
state_before: α : Type u_1
inst✝⁴ : CancelCommMonoidWithZero α
inst✝³ : UniqueFactorizationMonoid α
inst✝² : DecidableEq (Associates α)
inst✝¹ : (p : Associates α) → Decidable (Irreducible p)
inst✝ : DecidableEq (Associates α)
a : Associates α
ha : a ≠ 0
b : Associates α
hb : b ≠ 0
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
p : Associates α
hp : Irreducible p
⊢ p.count a.factors ≠ 0 → p.count b.factors = 0
state_after: α : Type u_1
inst✝⁴ : CancelCommMonoidWithZero α
inst✝³ : UniqueFactorizationMonoid α
inst✝² : DecidableEq (Associates α)
inst✝¹ : (p : Associates α) → Decidable (Irreducible p)
inst✝ : DecidableEq (Associates α)
a : Associates α
ha : a ≠ 0
b : Associates α
hb : b ≠ 0
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
p : Associates α
hp : Irreducible p
hca : p.count a.factors ≠ 0
⊢ p.count b.factors = 0

tactic: contrapose! hab with hcb
state_before: α : Type u_1
inst✝⁴ : CancelCommMonoidWithZero α
inst✝³ : UniqueFactorizationMonoid α
inst✝² : DecidableEq (Associates α)
inst✝¹ : (p : Associates α) → Decidable (Irreducible p)
inst✝ : DecidableEq (Associates α)
a : Associates α
ha : a ≠ 0
b : Associates α
hb : b ≠ 0
hab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d
p : Associates α
hp : Irreducible p
hca : p.count a.factors ≠ 0
⊢ p.count b.factors = 0
state_after: α : Type u_1
inst✝⁴ : CancelCommMonoidWithZero α
inst✝³ : UniqueFactorizationMonoid α
inst✝² : DecidableEq (Associates α)
inst✝¹ : (p : Associates α) → Decidable (Irreducible p)
inst✝ : DecidableEq (Associates α)
a : Associates α
ha : a ≠ 0
b : Associates α
hb : b ≠ 0
p : Associates α
hp : Irreducible p
hca : p.count a.factors ≠ 0
hcb : p.count b.factors ≠ 0
⊢ ∃ d, d ∣ a ∧ d ∣ b ∧ Prime d

tactic: exact ⟨p, le_of_count_ne_zero ha hp hca, le_of_count_ne_zero hb hp hcb,
  UniqueFactorizationMonoid.irreducible_iff_prime.mp hp⟩
state_before: α : Type u_1
inst✝⁴ : CancelCommMonoidWithZero α
inst✝³ : UniqueFactorizationMonoid α
inst✝² : DecidableEq (Associates α)
inst✝¹ : (p : Associates α) → Decidable (Irreducible p)
inst✝ : DecidableEq (Associates α)
a : Associates α
ha : a ≠ 0
b : Associates α
hb : b ≠ 0
p : Associates α
hp : Irreducible p
hca : p.count a.factors ≠ 0
hcb : p.count b.factors ≠ 0
⊢ ∃ d, d ∣ a ∧ d ∣ b ∧ Prime d
state_after: no goals