Why $u = v \Rightarrow u + w = v + w$ for all $u,v,w$ in a vector space I think the question is simple, but I'm a bit confused with proving the following rule for any $u,v,w$ in a vector space $V$ (with the usual 8 axioms for vector spaces, as in it's Wikipedia's page):
$$
u = v \Rightarrow u+w = v+w
$$
I think it's related to $"="$ being a binary relation, but I don't know where to start to prove it, any help would be appreciated.
 A: This is purely because $+$ is a function and because $u=v$ implies $(u,w)=(v,w)$. Let $f:X\to Z$ be a function and $x=y$. Then $f(x)=f(y)$.
The property is also known as $+$ being well defined. Does this help?
A: You're right that this is about the meaning of "$=$". The assertion $u=v$ means essentially that $u$ and $v$ are different names for the same object. So of course $u+w = v+w$.
This will be true whenever "$+$" makes sense, and has nothing to do with vector spaces.
A: A fundamental property of equality is that whenever $x=y$ holds, and you have some true statement involving $x$, then you can replace some or all of the occurrences of $x$ with $y$, and get another true statement. This doesn't depend on whether you look at statements in set theory, group theory, field theory, linear algebra, analysis or whatever other theory you can or cannot think of.
So in your case, if you know that $u=v$, then you can take the fact that $u+w=u+w$ (this is another fundamental property of equality: Anything is equivalent to itself) and replace the second $u$ with $v$ to get $u+w=v+w$.
Note that to be able to do this replacement, you do not need to know that $u$, $v$ and $w$ are vectors, or what $+$ does (other than that it is a well-defined operation), or whatever else. All you need to know is that expressions of the form $x+y$ make sense and describe an object in whatever theory you are looking at.
