Proving that $0\cdot x=0$ using field axioms

You can't.

Let $F=\Bbb Q$, define addition as usual and $$x\cdot y =\begin{cases}xy&\text{if }x\ne 0\\y&\text{if }x=0\end{cases}$$ Then

• $(F,+)$ is an abelian group because $\Bbb Q$ really is a field;
• $(F\setminus\{0\},\cdot)$ is an abelian group because $\Bbb Q$ really is a field and $\cdot $ conincides with standard multiplication here
• Left distribution holds for $a\ne 0$ because it holds in the field $\Bbb Q$
• left distribution holds for $a=0$ by direct verification

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In other words: your collection of axioms is "wrong".