The axioms I use for formal number theory in first-order logic come from Mendelson's "Introduction to Mathematical Logic." In the first edition that I have, the axioms are on page 103, and the theorem that you want is on page 104. In this fourth edition the axioms are on page 155 and the desired theorem on page 156. However, in both editions the theorem with some others " are left as exercises for the reader." Note that $x \cdot 0 = 0$ is an axiom, but not $0 \cdot x = 0$.
However, it is easy to prove by induction (covered in axiom S9). By axiom S7, it is provable that $0 \cdot 0 = 0$. By axiom S8, $0 \cdot (x') = (0 \cdot x) + 0$, which by axiom S5 equals $0 \cdot x$, which by the induction hypothesis is $0$.
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