coe_algebraMap_mem — Mathlib

English

The valuation on the adic completion is surjective onto the value group.

Русский

Оценка на адик завершении сюръективна на группе значений.

LaTeX

$$valuedAdicCompletion_surjective$$

Lean4

theorem coe_algebraMap_mem (r : R) : ↑((algebraMap R K) r) ∈ adicCompletionIntegers K v :=
  by
  rw [mem_adicCompletionIntegers]
  letI : Valued K ℤᵐ⁰ := adicValued v
  dsimp only [adicCompletion]
  rw [Valued.valuedCompletion_apply]
  exact v.valuation_le_one _

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