norm_def — Mathlib

English

The map expMapBasis is continuous as a function from realSpace K to realSpace K.

Русский

Функция expMapBasis непрерывна.

LaTeX

$$$\text{Continuous}(\expMapBasis : \mathbb{R}^{r} \to \mathbb{R}^{r})$$$

Lean4

/-- The norm of the image after the embedding associated to `v` is equal to the `v`-adic absolute
value. -/
theorem norm_def (x : WithVal (v.valuation K)) : ‖embedding v x‖ = adicAbv v x := by
  simp [NormedField.toNorm, instNormedFieldValuedAdicCompletion, Valued.toNormedField, Valued.norm,
    Valuation.RankOne.hom, embedding_apply, ← toNNReal_valued_eq_adicAbv]

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