Let $A$ be an integer matrix, $z \in \mathbb{Q}^n$, $a$ an integer vector and $z_0 \in \mathbb{Q}$.
Consider the statement:
The inequality $a\cdot x \leq z_0$ holds for all $x \in \mathbb{Q}^n$ if and only if exists a row vector $c$ with nonnegative entries such that $cA = a$ and $cz \leq z_0$
Is this statement a first order logic statement over $\mathbb{Q}$?
I'm very confused on distinguish first order logic statements. It is not clear to me in particular because I'm considering integers. Thanks in advance.