Theorems · Theorem · commutative algebra
lcm_dvd_iff
∀ {α : Type u_1} [inst : CommMonoidWithZero α] [inst_1 : GCDMonoid α] {a b c : α}, lcm a b ∣ c ↔ a ∣ c ∧ b ∣ c- Defined in
- Mathlib.Algebra.GCDMonoid.Basic
- Cited by
- 6 results in Mathlib
- Foundations
- Depth 18 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommMonoidWithZeroGCDMonoid
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites15
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- mul_commproof · cited by 2,262
- CommMonoidWithZerostatement and proof · cited by 913
- GCDMonoid.gcdproof · cited by 143
- GCDMonoidstatement and proof · cited by 96
- GCDMonoid.lcmstatement · cited by 78
- mul_dvd_mul_iff_leftproof · cited by 24
- Associated.dvd_iff_dvd_rightproof · cited by 11
- dvd_gcd_iffproof · cited by 8
- mul_dvd_mul_iff_rightproof · cited by 8
- Associated.dvd_iff_dvd_leftproof · cited by 6
- GCDMonoid.gcd_mul_lcmproof · cited by 3
- GCDMonoid.lcm_zero_leftproof · cited by 3
Cited by6
Results whose statement or proof uses this declaration.
- dvd_lcm_leftproof · cited by 17
- dvd_lcm_rightproof · cited by 17
- lcm_dvdproof · cited by 14
- Ideal.sup_mul_infproof · cited by 1
- isGCDMonoid_iff_exists_lcmproof · cited by 0
- span_singleton_inf_span_singletonproof · cited by 0