continuous_uncurry — Mathlib

English

The uncurried form of a continuous map X → C(Y,Z) is continuous Y→Z with locally compact Y.

Русский

Неприкоснованная форма отображения X → C(Y,Z) непрерывна при локальной компактности Y.

LaTeX

$$$\text{continuous_uncurry}:[X,Y,Z] \; (f : C(X, C(Y,Z))) : C(X\times Y, Z)$ is continuous$$

Lean4

/-- The uncurrying process is a continuous map between function spaces. -/
theorem continuous_uncurry [LocallyCompactSpace X] [LocallyCompactSpace Y] :
    Continuous (uncurry : C(X, C(Y, Z)) → C(X × Y, Z)) :=
  by
  apply continuous_of_continuous_uncurry
  rw [← (Homeomorph.prodAssoc _ _ _).comp_continuous_iff']
  dsimp [Function.comp_def]
  exact (continuous_fst.fst.eval continuous_fst.snd).eval continuous_snd

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