Theorems · Theorem · order theory
nonpos_iff_eq_zero
∀ {α : Type u_1} {a : α} [inst : PartialOrder α] [inst_1 : Zero α] [IsBotZeroClass α], a ≤ 0 ↔ a = 0- Defined in
- Mathlib.Algebra.Order.IsBotOne
- Cited by
- 100 results in Mathlib
- Foundations
- Depth 8 from the axioms, rests on 20 definitions · uses no axioms
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites4
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- PartialOrderstatement and proof · cited by 6,410
- zero_leproof · cited by 382
- IsBotZeroClassstatement and proof · cited by 71
- LE.le.ge_iff_eq'proof · cited by 50
Cited by100
Results whose statement or proof uses this declaration.
- tsub_eq_zero_iff_leproof · cited by 32
- ENNReal.le_div_iff_mul_leproof · cited by 15
- le_zero_iffproof · cited by 15
- MeasureTheory.Measure.quasiMeasurePreserving_sndproof · cited by 15
- MeasureTheory.Measure.absolutelyContinuous_of_leproof · cited by 14
- MeasureTheory.ae_of_ae_restrict_of_ae_restrict_complproof · cited by 14
- Polynomial.natDegree_eq_zero_iff_degree_le_zeroproof · cited by 12
- MeasureTheory.Measure.quasiMeasurePreserving_fstproof · cited by 12
- MeasureTheory.Measure.dirac_applyproof · cited by 9
- Finset.sum_eq_sum_Ico_succ_botproof · cited by 8
- Metric.infEDist_zero_of_memproof · cited by 7
- MeasureTheory.Measure.eq_singularPartproof · cited by 6