dist_le_dist_of_le_pi — Mathlib

English

A direct corollary of the left-right monotonicity: dist(a₁,b₁) ≤ dist(a₂,b₂) under the same coordinatewise order assumptions as before.

Русский

Следствие монотонности: dist(a₁,b₁) ≤ dist(a₂,b₂) при тех же предпосылках.

LaTeX

$$$\text{If } a_{2} \le a_{1} \text{ and } a_{1} \le b_{1} \text{ and } b_{1} \le b_{2}, \; \operatorname{dist}(a_{1}, b_{1}) \le \operatorname{dist}(a_{2}, b_{2})$$$

Lean4

theorem dist_le_dist_of_le_pi (ha : a₂ ≤ a₁) (h₁ : a₁ ≤ b₁) (hb : b₁ ≤ b₂) : dist a₁ b₁ ≤ dist a₂ b₂ :=
  (dist_mono_right_pi h₁ (h₁.trans hb) hb).trans <| dist_anti_left_pi (ha.trans <| h₁.trans hb) (h₁.trans hb) ha

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