similar_iff_exists_pos_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],
Similar v₁ v₂ ↔ ∃ r, 0 < r ∧ Pairwise fun i₁ i₂ => dist (v₁ i₁) (v₁ i₂) = r * dist (v₂ i₁) (v₂ i₂)Similarity holds iff pairwise distances are proportional with a positive ratio.
- Defined in
- Mathlib.Topology.MetricSpace.Similarity
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 155 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites8
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Realstatement and proof · cited by 25,697
- NNRealproof · cited by 4,310
- PseudoMetricSpacestatement and proof · cited by 1,550
- Dist.diststatement and proof · cited by 1,539
- NNReal.toRealproof · cited by 1,260
- Pairwisestatement and proof · cited by 516
- NNReal.mkproof · cited by 94
- Similarstatement · cited by 49
Cited by3
Results whose statement or proof uses this declaration.
- Similar.of_exists_pos_pairwise_dist_eqproof · cited by 1
- Similar.exists_pos_pairwise_dist_eqproof · cited by 0
- EuclideanGeometry.similar_of_side_angle_sideproof · cited by 0