piEquiv — Mathlib

English

The map f ↦ (i ↦ f(i)) is continuous from A to ∀ i, X_i when each f(i) is continuous.

Русский

Отображение f ↦ (i ↦ f(i)) непрерывно от A в ∀_i X_i, если каждый f(i) непрерывен.

LaTeX

$$$\\pi: (\\forall i, C(A, X_i)) \\to C(A, \\forall i, X_i)$ with $(\pi f)(a)(i)=f(i)(a)$$$

Lean4

/-- Giving a continuous map out of a disjoint union is the same as giving a continuous map out of
each term
-/
@[simps]
def piEquiv : (∀ i, C(A, X i)) ≃ C(A, ∀ i, X i) where
  toFun := pi
  invFun f i := (eval i).comp f

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