Composition of Addition Mappings on Natural Numbers

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Theorem

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

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

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


Then:

$\alpha_{a + b} = \alpha_b \circ \alpha_a$


Proof

\(\displaystyle \alpha_{a + b}\) \(=\) \(\displaystyle x + \paren {a + b}\) Definition of $\alpha$
\(\displaystyle \) \(=\) \(\displaystyle \paren {x + a} + b\)
\(\displaystyle \) \(=\) \(\displaystyle \paren {\map {\alpha_a} x} + b\) Definition of $\alpha$
\(\displaystyle \) \(=\) \(\displaystyle \map {\alpha_b} {\map {\alpha_a} x}\) Definition of $\alpha$
\(\displaystyle \) \(=\) \(\displaystyle \map {\paren {\alpha_b \circ \alpha_a} } x\) Definition of Composition of Mappings

$\blacksquare$


Sources