Code Number for Non-Positive Integer is Primitive Recursive

Theorem
Let $c : \N \to \N$ be defined as:
 * $\map c n = m$

where $m$ is the code number for the integer $-n : \Z$.

Then $c$ is a primitive recursive function.

Proof
Let $c : \N \to \N$ be defined as:
 * $\map c n = n + n$

which is primitive recursive by:
 * Addition is Primitive Recursive

For every $n \in \N$, we have:
 * $-n \le 0$

Thus:
 * $m = -2 \paren {-n} = 2 n$

Therefore:
 * $\map c n = m$