Theorems · Theorem · commutative algebra
dvd_gcd_iff
∀ {α : Type u_1} [inst : CommMonoidWithZero α] [inst_1 : GCDMonoid α] (a b c : α), a ∣ gcd b c ↔ a ∣ b ∧ a ∣ c- Defined in
- Mathlib.Algebra.GCDMonoid.Basic
- Cited by
- 8 results in Mathlib
- Foundations
- Depth 8 from the axioms · uses no axioms
- Assumes
- CommMonoidWithZeroGCDMonoid
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
- Dvd.dvd.transproof · cited by 148
- GCDMonoid.gcdstatement and proof · cited by 143
- GCDMonoidstatement and proof · cited by 96
- GCDMonoid.gcd_dvd_leftproof · cited by 36
- GCDMonoid.gcd_dvd_rightproof · cited by 34
- GCDMonoid.dvd_gcdproof · cited by 23
Cited by8
Results whose statement or proof uses this declaration.
- lcm_dvd_iffproof · cited by 6
- gcd_eq_of_dvd_sub_rightproof · cited by 2
- Ideal.sup_mul_infproof · cited by 1
- Ideal.absNorm_span_insertproof · cited by 0
- isGCDMonoid_iff_exists_gcdproof · cited by 0
- nonempty_strongNormalizedGCDMonoid_iffproof · cited by 0
- nonempty_normalizedGCDMonoid_iff_isGCDMonoidproof · cited by 0
- dvd_mul_sub_mul_mul_gcd_of_dvdproof · cited by 0