Theorems · Theorem · number theory
IsUnit.mul_left_dvd
∀ {α : Type u_1} [inst : CommMonoid α] {a b u : α}, IsUnit u → (u * a ∣ b ↔ a ∣ b)In a commutative monoid, an element a divides an element b iff all
left associates of a divide b.
- Defined in
- Mathlib.Algebra.Divisibility.Units
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 11 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.mul_left_dvdproof · cited by 1
Cited by3
Results whose statement or proof uses this declaration.
- Polynomial.Monic.irreducible_iff_degree_ltproof · cited by 1
- pow_dvd_of_mul_eq_powproof · cited by 1
- Irreducible.prime_of_isPrimalproof · cited by 1