similar_iff_exists_pairwise_dist_eq

English

Similarity is equivalent to existence of r > 0 with pairwise equalities dist(v1 i1, v1 i2) = r · dist(v2 i1, v2 i2).

Русский

Сходство существует равным r > 0 с попарными равенствами dist(v1 i1, v1 i2) = r · dist(v2 i1, v2 i2).

LaTeX

$$Similar v1 v2 ↔ ∃ r>0, Pairwise (i1,i2) ↦ (dist(v1 i1, v1 i2) = r · dist(v2 i1, v2 i2)).$$

Lean4

/-- Similarity holds if and only if all distances between points with different indices are
proportional. -/
theorem similar_iff_exists_pairwise_dist_eq :
    Similar v₁ v₂ ↔ (∃ r : ℝ≥0, r ≠ 0 ∧ Pairwise fun i₁ i₂ ↦ (dist (v₁ i₁) (v₁ i₂) = r * dist (v₂ i₁) (v₂ i₂))) :=
  by
  simp_rw [similar_iff_exists_pairwise_nndist_eq, dist_nndist]
  exact_mod_cast Iff.rfl

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