prod — Mathlib

English

The cartesian product of analytic within-at functions is analytic within-at.

Русский

Карточная пара аналитичных внутри-от функций аналитична внутри-ат.

LaTeX

$$$\text{AnalyticWithinAt}(f, s, x) \land \text{AnalyticWithinAt}(g, s, x) \Rightarrow \text{AnalyticWithinAt}(x \mapsto (f(x), g(x)), s, x)$$$

Lean4

/-- The Cartesian product of analytic functions is analytic. -/
theorem prod {e : E} {f : E → F} {g : E → G} {s : Set E} (hf : AnalyticWithinAt 𝕜 f s e)
    (hg : AnalyticWithinAt 𝕜 g s e) : AnalyticWithinAt 𝕜 (fun x ↦ (f x, g x)) s e :=
  by
  rcases hf with ⟨_, hf⟩
  rcases hg with ⟨_, hg⟩
  exact ⟨_, hf.prod hg⟩

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