I was wondering if this proof I wrote for $0x = 0$ (using only the field axioms) is correct. The proof is as follows:
$$0x = (1-1)x = x-x = 0$$
2026-04-25 16:11:29.1777133489
Is this proof that $0x = 0$ correct?
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There are a lot of steps you skipped. First, -1 is defined as the inverse of 1, so a rigorous notation is (-1).
$0 x = (1 + (-1))x = x + (-1)x$
The first equality is the axiom of the inverse element, the second the distributive axiom. Now you should you prove $(-1) x = -x$, by definition, you'll have to prove that $x + (-1)x = 0$. So, after that $x + (-1)x = x + (-x) = 0$. The last equality is from the axiom of the inverse element.