I have to solev next task: Proove that the operation of multiplication is definable operation in $(\mathbb{N}, + , |)$.
If I define mapping $f: \mathbb{N}^{2} \to \mathbb{N}$, as $f(x,y) = x \cdot y$, I have to show that its graph $$ G_{f} = \{ (x,y,f(x,y)) : (x,y) \in \mathbb{N}^{2} \} $$ is definable set. So, I need formule which define $G_{f}$.
I tried something but I am not sure that is correct: Product $x \cdot y$ is natural number, for example $x \cdot y = z$. From here it follows that $x | z$ and $y | z$ and I stopped here. My problem is how to use operations $+$ and $|$ to define formule which I need.
Thanks a lot.