norm_extends — Mathlib

English

The norm on PadicAlgCl p extends the p-adic norm from ℚ_p to its algebraic closure, i.e., for x ∈ ℚ_p, ||x|| = ||x||_p.

Русский

Норма на PadicAlgCl p extendирует p-адическую норму ℚ_p; для x ∈ ℚ_p выполняется ||x|| = ||x||_p.

LaTeX

$$$$\|x\|_{\operatorname{PadicAlgCl}(p)} = \|x\|_p \quad(x\in \mathbb{Q}_p).$$$$

Lean4

/-- The norm on `PadicAlgCl p` extends the `p`-adic norm on `ℚ_[p]`. -/
@[simp]
theorem norm_extends (x : ℚ_[p]) : ‖(x : PadicAlgCl p)‖ = ‖x‖ :=
  spectralAlgNorm_extends (K := ℚ_[p]) (L := PadicAlgCl p) _

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