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Theorems · Theorem · order theory

AbsoluteValue.ne_zero_iff

∀ {R : Type u_5} {S : Type u_6} [inst : Semiring R] [inst_1 : Semiring S] [inst_2 : PartialOrder S]
  (abv : AbsoluteValue R S) {x : R}, abv x ≠ 0 ↔ x ≠ 0
Defined in
Mathlib.Algebra.Order.AbsoluteValue.Basic
Cited by
4 results in Mathlib
Foundations
Depth 18 from the axioms · uses no axioms
Assumes
SemiringSemiringPartialOrder
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Around this declaration

Dashed lines are statement dependencies; solid lines are citations in proofs.

Height.one_le_mulHeight · cited by 8Height.one_le_mulHeightAbsoluteValue.pos_iff · cited by 6AbsoluteValue.pos_iffAbsoluteValue.ne_zero · cited by 2AbsoluteValue.ne_zeroAbsoluteValue.IsNontrivial.exists_abv_lt_one · cited by 1IsNontrivial.exists_abv_l…DFunLike.coe · cited by 62936DFunLike.coeSemiring · cited by 13802SemiringPartialOrder · cited by 6410PartialOrderIff.not · cited by 489Iff.notAbsoluteValue · cited by 363AbsoluteValueAbsoluteValue.eq_zero · cited by 5AbsoluteValue.eq_zeroAbsoluteValue.ne_zero_iffCITED BYCITES

Cites6

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

Cited by4

Results whose statement or proof uses this declaration.