Composition of Product Mappings on Natural Numbers

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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

$\blacksquare$


Sources