Equality Relation is Primitive Recursive

Theorem
The relation $\operatorname{eq} \subseteq \N^2$, defined as:
 * $\operatorname{eq} \left({n, m}\right) \iff n = m$

is primitive recursive.

Proof
We note that:
 * $n = m \iff \left|{n - m}\right| = 0$
 * $n \ne m \iff \left|{n - m}\right| > 0$

So it can be seen that the characteristic function of $\operatorname{eq}$ is given by:
 * $\chi_{\operatorname{eq}} \left({n, m}\right) = \overline{\operatorname{sgn}} \left({\operatorname{adf}\left({n, m}\right)}\right)$.

So $\chi_{\operatorname{eq}} \left({n, m}\right)$ is defined by substitution from:
 * the primitive recursive function $\overline{\operatorname{sgn}}$
 * the primitive recursive function $\operatorname{adf}$.

Thus $\chi_{\operatorname{eq}}$ is primitive recursive.

Hence the result.