dist_inf_sup_pi — Mathlib

English

The map y ↦ dist(a, y) is monotone in y on the set { y ≥ y } coordinatewise, i.e., monotone in each coordinate when y is raised.

Русский

Отображение y ↦ dist(a, y) монотонно по каждому индексу при возрастании y.

LaTeX

$$$\text{MonotoneOn } (\mathrm{dist}(\cdot, y)) \; (\, \mathrm Ici \; y\,)$$$

Lean4

theorem dist_inf_sup_pi (x y : ι → ℝ) : dist (x ⊓ y) (x ⊔ y) = dist x y :=
  by
  refine congr_arg NNReal.toReal (Finset.sup_congr rfl fun i _ ↦ ?_)
  simp only [Real.nndist_eq', max_sub_min_eq_abs, Pi.inf_apply, Pi.sup_apply, Real.nnabs_of_nonneg, abs_nonneg,
    Real.toNNReal_abs]

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