Theorems · Theorem · commutative algebra
dvd_neg
∀ {α : Type u_1} [inst : Semigroup α] [inst_1 : HasDistribNeg α] {a b : α}, a ∣ -b ↔ a ∣ bAn element a of a semigroup with a distributive negation divides the negation of an element
b iff a divides b.
- Defined in
- Mathlib.Algebra.Ring.Divisibility.Basic
- Cited by
- 19 results in Mathlib
- Foundations
- Depth 17 from the axioms · uses propext, Quot.sound
- Assumes
- SemigroupHasDistribNeg
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites6
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- mul_negproof · cited by 590
- Semigroupstatement and proof · cited by 202
- HasDistribNegstatement and proof · cited by 114
- Equiv.negproof · cited by 53
- Equiv.neg_applyproof · cited by 34
- Equiv.exists_congr_leftproof · cited by 33
Cited by19
Results whose statement or proof uses this declaration.
- IsAlgebraic.exists_nonzero_dvdproof · cited by 5
- Dvd.dvd.neg_rightproof · cited by 5
- addOrderOf_dvd_iff_zsmul_eq_zeroproof · cited by 3
- IsRelPrime.neg_leftproof · cited by 3
- orderOf_dvd_iff_zpow_eq_oneproof · cited by 2
- PadicInt.zmod_congr_of_sub_mem_span_auxproof · cited by 2
- Equiv.Perm.IsCycleOn.zpow_apply_eqproof · cited by 2
- Polynomial.irreducible_of_mirrorproof · cited by 1
- divRadical_dvd_wronskian_rightproof · cited by 1
- sub_one_dvd_natCast_of_pow_eq_oneproof · cited by 1
- IsPrimitiveRoot.zpow_eq_one_iff_dvdproof · cited by 1
- Irreducible.natDegree_dvd_of_dvd_X_pow_card_pow_sub_Xproof · cited by 1