Theorems · Theorem · general topology
congruent_iff_pairwise_dist_eq
∀ {ι : Type u_1} {P₁ : Type u_3} {P₂ : Type u_4} {v₁ : ι → P₁} {v₂ : ι → P₂} [inst : PseudoMetricSpace P₁]
[inst_1 : PseudoMetricSpace P₂], Congruent v₁ v₂ ↔ Pairwise fun i₁ i₂ => dist (v₁ i₁) (v₁ i₂) = dist (v₂ i₁) (v₂ i₂)Congruence holds if and only if all non-negative distances between points with different indices are the same.
- Defined in
- Mathlib.Topology.MetricSpace.Congruence
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 152 from the axioms · uses propext, Classical.choice, Quot.sound
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
- Pairwisestatement and proof · cited by 516
- NNDist.nndistproof · cited by 235
- Congruentstatement · cited by 37
Cited by2
Results whose statement or proof uses this declaration.
- Congruent.of_pairwise_dist_eqproof · cited by 0
- Congruent.pairwise_dist_eqproof · cited by 0