Theorems · Theorem · commutative algebra
squarefree_mul_iff
∀ {R : Type u_1} [inst : CommMonoidWithZero R] [IsCancelMulZero R] {x y : R} [DecompositionMonoid R],
Squarefree (x * y) ↔ IsRelPrime x y ∧ Squarefree x ∧ Squarefree yx * y is square-free iff x and y have no common factors and are themselves square-free.
- Defined in
- Mathlib.Algebra.Squarefree.Basic
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 77 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites11
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- IsUnitproof · cited by 1,602
- CommMonoidWithZerostatement and proof · cited by 913
- IsCancelMulZerostatement and proof · cited by 177
- IsRelPrimestatement and proof · cited by 136
- Squarefreestatement and proof · cited by 112
- DecompositionMonoidstatement and proof · cited by 39
- Squarefree.of_mul_rightproof · cited by 3
- IsRelPrime.of_squarefree_mulproof · cited by 2
- Squarefree.dvd_of_squarefree_of_mul_dvd_mul_rightproof · cited by 2
- Squarefree.of_mul_leftproof · cited by 2
- Squarefree.dvd_of_squarefree_of_mul_dvd_mul_leftproof · cited by 1
Cited by4
Results whose statement or proof uses this declaration.
- Nat.squarefree_mul_iffproof · cited by 1
- Finset.squarefree_prod_of_pairwise_isCoprimeproof · cited by 1
- exists_squarefree_dvd_pow_of_ne_zeroproof · cited by 1
- IsRelPrime.moebius_mulproof · cited by 0