norm_def_int — Mathlib

English

The expMapBasis applied to x equals expMap of the inverse basis applied to x.

Русский

expMapBasis применимо к x через expMap обратного базиса.

LaTeX

$$$\text{expMapBasis}(x) = \exp( (\text{completeBasis}(K).\text{equivFun}).\text{symm}(x) )$$$

Lean4

/-- The norm of the image after the embedding associated to `v` is equal to the norm of `v` raised
to the power of the `v`-adic valuation for integers. -/
theorem norm_def_int (x : 𝓞 (WithVal (v.valuation K))) :
    ‖embedding v x‖ = toNNReal (absNorm_ne_zero v) (v.intValuation x) := by
  rw [norm_def, adicAbv_def, valuation_of_algebraMap]

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