# Composition of Product Mappings on Natural Numbers

## Theorem

Let $a \in \N$ be a natural number.

Let $\mu_a: \N \to \N$ be the mapping defined as:

$\forall x \in \N: \map {\mu_a} x = x a$

Then:

$\mu_{a b} = \mu_b \circ \mu_a$

## Proof

 $\displaystyle \mu_{a b}$ $=$ $\displaystyle x \paren {a b}$ Definition of $\mu$ $\displaystyle$ $=$ $\displaystyle \paren {x a} b$ $\displaystyle$ $=$ $\displaystyle \paren {\map {\mu_a} x} b$ Definition of $\mu$ $\displaystyle$ $=$ $\displaystyle \map {\mu_b} {\map {\mu_a} x}$ Definition of $\mu$ $\displaystyle$ $=$ $\displaystyle \map {\paren {\mu_b \circ \mu_a} } x$ Definition of Composition of Mappings

$\blacksquare$