Gödel's Beta Function is Arithmetically Definable

Theorem
Let Gödel's $\beta$ function $\beta: \N^3 \to \N$ be defined as:
 * $\map \beta {x, y, z} = \map \rem {x, 1 + \paren {z + 1} \times y}$

Then there exists a $\Sigma_1$ WFF of $4$ free variables:
 * $\map \phi {r, x, y, z}$

such that:
 * $r = \map \beta {x, y, z} \iff \N \models \map \phi {\sqbrk r, \sqbrk x, \sqbrk y, \sqbrk z}$

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

Proof
Follows from:
 * Basic Primitive Recursive Functions are Arithmetically Definable
 * Addition is Arithmetically Definable
 * Multiplication is Arithmetically Definable
 * Remainder is Arithmetically Definable
 * Substitution of Arithmetically Definable Functions is Arithmetically Definable