Composition of Addition Mappings on Natural Numbers

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

$\blacksquare$