instIsTopologicalSemiring — Mathlib

English

The semiring topology on MvPowerSeries σ R makes it a topological semiring when R is a topological semiring.

Русский

Полупрямой топологией полупрядомого MvPowerSeries σ R является топологическое полупространство полупринципа, если R — топологическое полупространство.

LaTeX

$$IsTopologicalSemiring (MvPowerSeries σ R)$$

Lean4

/-- The semiring topology on `MvPowerSeries` of a topological semiring -/
@[scoped instance]
theorem instIsTopologicalSemiring [Semiring R] [IsTopologicalSemiring R] : IsTopologicalSemiring (MvPowerSeries σ R)
    where
  continuous_add :=
    continuous_pi fun d => continuous_add.comp (((continuous_coeff R d).fst').prodMk (continuous_coeff R d).snd')
  continuous_mul :=
    continuous_pi fun _ =>
      continuous_finset_sum _ fun i _ =>
        continuous_mul.comp ((continuous_coeff R i.fst).fst'.prodMk (continuous_coeff R i.snd).snd')

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