Remainder is Arithmetically Definable

Theorem
Let $\operatorname{rem}: \N^2 \to \N$ be defined as:
 * $\map \rem {n, m} = \begin{cases}

\text{the remainder when } n \text{ is divided by } m & : m \ne 0 \\ 0 & : m = 0 \end{cases}$ where the $\text{remainder}$ is as defined in the Division Theorem:
 * If $n = m q + r$, where $0 \le r < m$, then $r$ is the remainder.

Then there exists a $\Sigma_1$ WFF of $3$ free variables:
 * $\map \phi {r, n, m}$

such that:
 * $r = \map \rem {n, m} \iff \N \models \map \phi {\sqbrk r, \sqbrk n, \sqbrk m}$

where $\sqbrk a$ denotes the unary representation of $a \in \N$.

Proof
Define:
 * $\map \phi {r, n, m} := \paren {m = 0 \land r = 0} \lor \paren {m \ne 0 \land r < m \land \exists q: n = \paren {m \times q} + r}$

Suppose $m = 0$.

Then, only the first case:
 * $m = 0 \land r = 0$

can possibly hold.

Therefore, the formula holds $r = 0$, which matches the definition.

Suppose $m \ne 0$.

Then, only the second case:
 * $m \ne 0 \land r < m \land \exists q: n = \paren {m \times q} + r$

can possibly hold.

But:
 * $r < m \land \exists q: n = \paren {m \times q} + r$

is the definition of $r$.

Therefore, in all cases:
 * $r = \map \rem {n, m} \iff \N \models \map \rem {\sqbrk r, \sqbrk n, \sqbrk m}$

Finally, that $\phi$ is $\Sigma_1$ follows from:
 * Conjunction of Existential Quantifier
 * Existential Quantifier Distributes over Disjunction