symm — Mathlib

English

Similarity is transitive: if v1 ∼ v2 and v2 ∼ v3, then v1 ∼ v3 with the product of scalars.

Русский

Сходство транзитивно: если v1 ∼ v2 и v2 ∼ v3, то v1 ∼ v3 с масштабом r1 r2.

LaTeX

$$If v1 ∼ v2 with scale r1 and v2 ∼ v3 with scale r2, then v1 ∼ v3 with scale r1 r2.$$

Lean4

@[symm]
protected theorem symm (h : v₁ ∼ v₂) : v₂ ∼ v₁ :=
  by
  rcases h with ⟨r, hr, h⟩
  refine ⟨r⁻¹, inv_ne_zero hr, fun _ _ => ?_⟩
  rw [ENNReal.coe_inv hr, ← ENNReal.div_eq_inv_mul, ENNReal.eq_div_iff _ ENNReal.coe_ne_top, h]
  norm_cast

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