so I was learning some abstract algebra and group theory, when they went over the proof of the cancellation law
$$ ab = ac\implies a^{-1}(ab) = a^{-1}(ac)\implies (a^{-1}a)b = (a^{-1}a)c\implies eb=ec \implies b=c $$
But the first step in which you add the additional term seems jarring to me, especially since I felt we were proving ever trivial thing from the ground up. Obviously I'm familiar with middle school pre-algebra, so I know that it is true that if we perform an operation on both sides it preserves equality, but I didn't know how we know this is the case always. Is it an axiom or is it proven?
Here is my attempt at a proof, let me know if I am going in the correct direction.
Assume via axiom that $x=x$ and if $a=b$ and $b=c$ then $a=c$, and prove that $a=b\implies ka=kb$
We know that $a=b$, we define $x\mid x=a \implies a=x$. Since $a=b$ and $a=x$, then $b=x$ which we can rewrite as $x=x$. Now we perform the operation on both sides $kx=kx$, which is true via our axiom. Then we re-substitute $x=a$ and $x=b$ to get $ka=kb$. Q.E.D
That was my original idea but I don't know if that's watertight. Thank you!