analyticAt_fun_prod — Mathlib

English

A finite product of analyticAt functions is analyticAt.

Русский

Конечное произведение аналитических функций в точке аналитично в точке.

LaTeX

$$$\forall N\,\text{Finset},\ (\forall n\in N,\ AnalyticAt_{\mathfrak k}(f(n),c))\Rightarrow AnalyticAt_{\mathfrak k}\Bigl(\prod_{n\in N} f(n)\Bigr)(c).$$$

Lean4

/-- Finite products of analytic functions are analytic -/
@[fun_prop]
theorem analyticAt_fun_prod {A : Type*} [NormedCommRing A] [NormedAlgebra 𝕜 A] {f : α → E → A} {c : E} (N : Finset α)
    (h : ∀ n ∈ N, AnalyticAt 𝕜 (f n) c) : AnalyticAt 𝕜 (fun z ↦ ∏ n ∈ N, f n z) c :=
  by
  simp_rw [← analyticWithinAt_univ] at h ⊢
  exact N.analyticWithinAt_fun_prod h

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