prod_nndist_eq_add — Mathlib

English

There is a product Bornology on WithLp p (α × β) given by Prod.instBornology.

Русский

Существует произведение борологии на WithLp p (α × β), задаваемое Prod.instBornology.

LaTeX

$$$ \\operatorname{Bornology}(\\mathrm{WithLp}_{p}(\\alpha \\times \\beta)) $$$

Lean4

theorem prod_nndist_eq_add [PseudoMetricSpace α] [PseudoMetricSpace β] (hp : p ≠ ∞) (x y : WithLp p (α × β)) :
    nndist x y = (nndist x.fst y.fst ^ p.toReal + nndist x.snd y.snd ^ p.toReal) ^ (1 / p.toReal) :=
  NNReal.eq <| by
    push_cast
    exact prod_dist_eq_add (p.toReal_pos_iff_ne_top.mpr hp) _ _

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