Theorems · Theorem · commutative algebra
UniqueFactorizationMonoid.isRelPrime_iff_no_prime_factors
∀ {R : Type u_2} [inst : CommMonoidWithZero R] [UniqueFactorizationMonoid R] {a b : R},
a ≠ 0 → (IsRelPrime a b ↔ ∀ ⦃d : R⦄, d ∣ a → d ∣ b → ¬Prime d)- Cited by
- 2 results in Mathlib
- Foundations
- Depth 17 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites8
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- CommMonoidWithZerostatement and proof · cited by 913
- Irreducibleproof · cited by 496
- UniqueFactorizationMonoidstatement and proof · cited by 279
- Primestatement and proof · cited by 277
- IsRelPrimestatement and proof · cited by 136
- Prime.not_isUnitproof · cited by 29
- UniqueFactorizationMonoid.irreducible_iff_primeproof · cited by 8
- WfDvdMonoid.isRelPrime_of_no_irreducible_factorsproof · cited by 2
Cited by2
Results whose statement or proof uses this declaration.
- UniqueFactorizationMonoid.dvd_of_dvd_mul_left_of_no_prime_factorsproof · cited by 3
- UniqueFactorizationMonoid.prime_pow_coprime_prod_of_coprime_insertproof · cited by 2