Theorems · Theorem · commutative algebra
Prime.dvd_lcm
∀ {α : Type u_1} [inst : CommMonoidWithZero α] [inst_1 : GCDMonoid α] {a b p : α},
Prime p → (p ∣ lcm a b ↔ p ∣ a ∨ p ∣ b)- Defined in
- Mathlib.Algebra.GCDMonoid.Basic
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 22 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.
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
- Primestatement and proof · cited by 277
- GCDMonoidstatement and proof · cited by 96
- GCDMonoid.lcmstatement · cited by 78
- dvd_lcm_of_dvd_leftproof · cited by 2
- dvd_lcm_of_dvd_rightproof · cited by 2
- Prime.dvd_or_dvd_of_dvd_lcmproof · cited by 1
Cited by1
Results whose statement or proof uses this declaration.
- Prime.not_dvd_lcmproof · cited by 0