Theorems · Theorem · general topology
dist_nonneg
∀ {α : Type u} [inst : PseudoMetricSpace α] {x y : α}, 0 ≤ dist x y- Defined in
- Mathlib.Topology.MetricSpace.Pseudo.Defs
- Cited by
- 126 results in Mathlib
- Foundations
- Depth 111 from the axioms, rests on 2,419 definitions · uses propext, Classical.choice, Quot.sound
- Assumes
- PseudoMetricSpace
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.
- Realstatement · cited by 25,697
- PseudoMetricSpacestatement and proof · cited by 1,550
- Dist.diststatement and proof · cited by 1,539
- dist_commproof · cited by 188
- dist_selfproof · cited by 116
- dist_triangleproof · cited by 44
Cited by126
Results whose statement or proof uses this declaration.
- norm_nonnegproof · cited by 725
- norm_nonneg'proof · cited by 21
- dist_edistproof · cited by 21
- Metric.nonempty_closedBallproof · cited by 13
- abs_distproof · cited by 10
- Metric.pos_of_mem_ballproof · cited by 9
- Set.Nontrivial.le_infsep_iffproof · cited by 6
- UpperHalfPlane.cmp_dist_eq_cmp_dist_coe_centerproof · cited by 5
- Metric.eball_ofRealproof · cited by 5
- Seminorm.uniformContinuous_of_continuousAt_zeroproof · cited by 5
- EuclideanGeometry.dist_orthogonalProjection_eq_infDistproof · cited by 5
- aux_hasSum_of_le_geometricproof · cited by 4