Theorems · Theorem · order theory
AbsoluteValue.nonneg
∀ {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), 0 ≤ abv x- Cited by
- 16 results in Mathlib
- Foundations
- Depth 17 from the axioms · uses no axioms
- Assumes
- SemiringSemiringPartialOrder
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.
- DFunLike.coestatement · cited by 62,936
- Semiringstatement and proof · cited by 13,802
- PartialOrderstatement and proof · cited by 6,410
- AbsoluteValuestatement and proof · cited by 363
- AbsoluteValue.nonneg'proof · cited by 1
Cited by16
Results whose statement or proof uses this declaration.
- AbsoluteValue.pos_iffproof · cited by 6
- Height.mulHeight_smul_eq_mulHeightproof · cited by 6
- AbsoluteValue.isEquiv_iff_exists_rpow_eqproof · cited by 4
- Height.mulHeight_powproof · cited by 3
- NumberField.Units.dirichletUnitTheorem.seq_nextproof · cited by 3
- NumberField.mixedEmbedding.exists_ne_zero_mem_ideal_of_norm_leproof · cited by 2
- ClassGroup.norm_ltproof · cited by 1
- ClassGroup.exists_mem_finsetApproxproof · cited by 1
- ClassGroup.exists_mem_finset_approx'proof · cited by 1
- ClassGroup.exists_minproof · cited by 1
- Matrix.det_sum_smul_leproof · cited by 1
- Polynomial.cardPowDegree_anti_archimedeanproof · cited by 1