dist_eq — Mathlib

English

For x,y ∈ α, dist (x : Completion α) y = dist x y. Thus the distance on the completion extends the original distance.

Русский

Для x,y ∈ α верно dist( x в Completion α, y ) = dist x y. Таким образом, расстояние на завершении extension оригинального расстояния.

LaTeX

$$$\operatorname{dist}(x:\mathrm{Completion}\;\alpha, y)=\operatorname{dist}(x,y)$$$

Lean4

/-- The new distance is an extension of the original distance. -/
@[simp]
protected theorem dist_eq (x y : α) : dist (x : Completion α) y = dist x y :=
  Completion.extension₂_coe_coe uniformContinuous_dist _
    _
      /- Let us check that the new distance satisfies the axioms of a distance, by starting from the
      properties on α and extending them to `Completion α` by continuity. -/

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