Proving that addition on both sides preserves equality I'm doing some very low level proofs in abstract algebra and I have derived the fact that (on a particular ring) $0 = -1$ where $0$ is the additive identity of the ring in question and $-1$ is the inverse of its multiplicative identity.
I wish to prove that this implies $0 = 1$ (which I can use to prove that my ring is the trivial ring) but I am struggling to do so.  Intuitively we can add $1$ to both sides and perform the following derivation
\begin{align*}
0 = -1 &\implies 0 + 1 = (-1) + 1 \tag{???} \\
&\implies 1 = (-1) + 1 \tag{Identity} \\
&\implies 1 = 0 \tag{Inverse} \\
\end{align*}
But I'm not really sure how to justify adding $1$ to both sides.  I believe the definition of equality on a ring has been inherited from set theory, however I can't figure out how one might prove this using the axioms of set theory.
How might I go about proving that $a = b \implies a+x = b+x$?  Is this something that is proven or is it derived from some definition somewhere? 
 A: It's just because addition (and all binary operations, for that matter) are functions.
If $a=b$, then the ordered pair $(a,x)=(b,x)\in R\times R$.
By the definition of a function, $a+x=+(a, x)=+(b, x)=b+x$
A: Most traditional set theories imply the basic concepts of function and relation: a set is in most of the axiomatizations simply a relation. In particular 2 relations are usually considered fundamental: "is an element of" and "is equal to".
Ring axioms take for granted a certain set theory axiomatization with which they define the ring set and add to that 2 total binary functions (or 2 tertiary relations +(x,y,z) and *(x,y,z) whose meaning is self-explainatory I guess). But this isn't really relevant if I understand what you're doing.
Being the 2 ring actions just standard functions (i.e. non multi-valued) then:
(a + x = b &&  a + x = c) implies (b = c)
(a + x = b &&  b = c) implies (a + x = c)
(a + b = x &&  a + c = x) implies (b = c)
Etc... including of course your "both sides" theorem given the proper substitutions.
