Theorems · Theorem · commutative algebra
isCoprime_of_irreducible_dvd
∀ {R : Type u} [inst : CommRing R] [IsDomain R] [IsPrincipalIdealRing R] {x y : R},
¬(x = 0 ∧ y = 0) → (∀ (z : R), Irreducible z → z ∣ x → ¬z ∣ y) → IsCoprime x y- Defined in
- Mathlib.RingTheory.PrincipalIdealDomain
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 86 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites7
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- CommRingstatement and proof · cited by 17,173
- IsDomainstatement and proof · cited by 2,196
- Irreduciblestatement and proof · cited by 496
- IsCoprimestatement · cited by 321
- IsPrincipalIdealRingstatement and proof · cited by 131
- IsRelPrime.isCoprimeproof · cited by 5
- WfDvdMonoid.isRelPrime_of_no_irreducible_factorsproof · cited by 2
Cited by3
Results whose statement or proof uses this declaration.
- PerfectField.separable_iff_squarefreeproof · cited by 2
- isCoprime_of_prime_dvdproof · cited by 1
- Polynomial.natSepDegree_mul_eq_iffproof · cited by 1