# Composite of Evaluation Mapping and Projection

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## Theorem

Let $X$ be a topological space.

Let $\family {Y_i}_{i \mathop \in I}$ be an indexed family of topological spaces for some indexing set $I$.

Let $\family {f_i : X \to Y_i}_{i \mathop \in I}$ be an indexed family of continuous mappings.

Let $\ds Y = \prod_{i \mathop \in I} Y_i$ be the product space of $\family {Y_i}_{i \mathop \in I}$.

Let $f : X \to Y$ be the evaluation mapping induced by $\family{f_i}_{i \mathop \in I}$.

Let:

- $pr_i : Y \to Y_i$ denote the $i$th projection on $Y$

Then:

- $\forall i \in I: pr_i \circ f = f_i$

## Proof

By definition of projection:

\(\ds \forall x \in X, i \in I: \, \) | \(\ds \map {\paren{pr_i \circ f} } x\) | \(=\) | \(\ds \map {pr_i} {\map f x}\) | Definition of Composite Mapping | ||||||||||

\(\ds \) | \(=\) | \(\ds \map {pr_i} {\family{\map {f_i} x} }\) | Definition of Evaluation Mapping | |||||||||||

\(\ds \) | \(=\) | \(\ds \map {f_i} x\) | Definition of Projection |

From Equality of Mappings:

- $\forall i \in I : pr_i \circ f = f_i$

$\blacksquare$