hasSum_eval₂ — Mathlib

English

The domain of evaluation forms an ideal again.

Русский

Область оценки образует идеал снова.

LaTeX

$$$\text{hasEvalIdeal is an ideal in } S$$$

Lean4

theorem hasSum_eval₂ (hφ : Continuous φ) (ha : HasEval a) (f : PowerSeries R) :
    HasSum (fun (d : ℕ) ↦ φ (coeff d f) * a ^ d) (f.eval₂ φ a) :=
  by
  have := MvPowerSeries.hasSum_eval₂ hφ (hasEval ha) f
  simp only [PowerSeries.eval₂]
  rw [← (Finsupp.single_injective ()).hasSum_iff] at this
  · convert this; simp; congr
  · intro d hd
    exact False.elim (hd ⟨d (), by ext; simp⟩)

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