Minimum Function is Primitive Recursive

Theorem
The minimum function $\min: \N^2 \to \N$, defined as:
 * $\min \left({n, m}\right) = \begin{cases}

n: & n \le m \\ m: & m \le n \end{cases}$ is primitive recursive‎.

Proof
From Sum Less Maximum is Minimum we have that:
 * $\min \left({n, m}\right) = n + m - \max \left({n, m}\right)$.

As $n + m \ge \max \left({n, m}\right)$, we have that:
 * $\min \left({n, m}\right) = n + m \ \dot - \ \max \left({n, m}\right)$

Hence we see that $\min$ is obtained by substitution from:
 * the primitive recursive function $n \ \dot - \ m$
 * the primitive recursive function $\max \left({n, m}\right)$.

Hence the result.