# A very simple question

$vx = z$

$zb = y$

Which means $vxb = y$

Does this build on an axiom (and which)? I have to prove a statement using only some specific axioms. But I don't know if I'm allowed to do that.

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It would be very useful to the question to say which axioms you are allowed to use, and what is the context of the multiplication. –  Asaf Karagila Sep 20 '11 at 16:25
Also, is this homework? If this is homework please add [homework] tag to the question. –  Asaf Karagila Sep 20 '11 at 16:25

2. This binary function is associative (so that $(ab)c = a(bc)$ for all $a$, $b$, and $c$;
then, yes. If $vx=z$ and $zb=y$, then $y = zb = (vx)b = vxb$. You are using the fact that the product is a function, so evaluating at $z$ and $b$ is the same as evaluating at $vx$ and $b$ (since $vx=z$; this is sometimes called the "Principle of Substitution", which is an axiom of the underlying logic). So $zb= (vx)b$. And because the operation is assumed to be associative, then the two possible meanings of "$vxb$" (namely, $(vx)b$ and $v(xb)$) have the same value, so we do not need to distinguish between them.