Theorems · Theorem · commutative algebra
Irreducible.gcd_eq_one_iff
∀ {α : Type u_1} [inst : CommMonoidWithZero α] [inst_1 : NormalizedGCDMonoid α] {x y : α},
Irreducible x → (gcd x y = 1 ↔ ¬x ∣ y)- Defined in
- Mathlib.Algebra.GCDMonoid.Basic
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 20 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.
- CommMonoidWithZerostatement and proof · cited by 913
- Irreduciblestatement and proof · cited by 496
- NormalizedGCDMonoidstatement and proof · cited by 159
- GCDMonoid.gcdstatement and proof · cited by 143
- normalize_eq_oneproof · cited by 10
- NormalizedGCDMonoid.normalize_gcdproof · cited by 2
- Irreducible.isUnit_gcd_iffproof · cited by 1
Cited by2
Results whose statement or proof uses this declaration.
- dvd_differentIdeal_of_not_isSeparableproof · cited by 1
- IsCyclotomicExtension.Rat.isCoprime_of_not_zeta_sub_one_dvdproof · cited by 0