I'm trying to prove commutativity of addition for vector spaces, using the axioms for vector spaces. Apparently commutativity can be proven! Im having trouble getting a good feel for what is allowed and what is not. Here's my work so far:
$u+v+u+v = 2(u+v) = 2u + 2v = u+u+v+v = u+(u+v)+v$
Here I just wanna claim that $u+(v+u)+v = u+(u+v)+v$
$\Rightarrow -u+u+(v+u)+v+(-v) = -u+u+(u+v)+v+(-v)$ : here im just adding -u to the right, and -v to the left. Question: is this "adding to both sides" really legit in this context? Why?
Quick help proof: $-v+v = (-1)v+(1)v = (-1+1)v = 0v = 0 = v-v$
And another: $ 0+v = v+(-v) + v = (1)v + (-1)v + v = (1-1)v + v = v = v+0$
We have $0 + (u+v) + 0 = 0+(v+u)+0 \Rightarrow u+v = v+u$
This feels ugly and not at all elegant, especially the great leap "add -u to both sides" feels completely out of place. Do I need more lemmas? Is there a more elegant way?
//not homework or anything, just for my own pleasure, feel free to provide theory, as it is more insightful than solutions. :) Thanks!
EDIT: corrected notation a little.