Theorems · Theorem · number theory
IsUnit.dvd_mul_left
∀ {α : Type u_1} [inst : CommMonoid α] {a b u : α}, IsUnit u → (a ∣ u * b ↔ a ∣ b)In a commutative monoid, an element a divides an element b iff a divides all left
associates of b.
- Defined in
- Mathlib.Algebra.Divisibility.Units
- Cited by
- 5 results in Mathlib
- Foundations
- Depth 12 from the axioms · uses no axioms
- Assumes
- CommMonoid
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites5
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Unitsproof · cited by 2,804
- CommMonoidstatement and proof · cited by 2,264
- Units.valproof · cited by 1,966
- IsUnitstatement and proof · cited by 1,602
- Units.dvd_mul_leftproof · cited by 3
Cited by5
Results whose statement or proof uses this declaration.
- isRelPrime_mul_unit_left_leftproof · cited by 3
- Submonoid.LocalizationMap.map_primeproof · cited by 1
- Submonoid.LocalizationMap.map_dvd_mapproof · cited by 1
- Polynomial.scaleRoots_dvd'proof · cited by 1
- IsPrimitiveRoot.prime_dvd_of_dvd_norm_sub_oneproof · cited by 1