mem_integers_of_powerSeries — Mathlib

English

The image of a power series under LaurentSeriesRingEquiv lies in the integers of the adic completion subring.

Русский

Образ ряда мощности через LaurentSeriesRingEquiv лежит в целых внутри адикционного завершения.

LaTeX

$$$\\mathrm{LaurentSeriesRingEquiv}\\;K\\;F \\in (\\mathrm{idealX}\\;K)^{\\mathrm{adicCompletionIntegers}}(\\mathrm{RatFunc}\\,K)$$$

Lean4

theorem mem_integers_of_powerSeries (F : K⟦X⟧) :
    (LaurentSeriesRingEquiv K) F ∈ (idealX K).adicCompletionIntegers (RatFunc K) :=
  by
  simp only [mem_adicCompletionIntegers, LaurentSeriesRingEquiv_def, valuation_compare, val_le_one_iff_eq_coe]
  exact
    ⟨F, rfl⟩
      /- Conversely, all elements in the unit ball inside the completion of `RatFunc K` come from a power
      series through the isomorphism `LaurentSeriesRingEquiv`. -/

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