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]