Do we need to prove that $b-a = b + (-a)$, where $a,b$ are real numbers? A consequence of the axioms of $\mathbb{R}$ is that it is possible to define subtraction. Thus, $b-a$ is merely defined to be $b+(-a)$,  which is a number, say $x$, that satisfies $x+a = b$.
But in Apostol's Calculus (Vol. I), the possibility of subtraction is a theorem, and $b-a = b + (-a)$  is also a theorem. Here's the proof for the first theorem:
Let $y+a=0$, and $x=y+b$. Then $x+a=a+y+b=a+(-a)+b=b$. Since $y$ is unique, it follows that $x+a$ is unique, and is well defined. 
To prove the second theorem, let $x=b-a$ and $y=b+(-a)$. By the above theorem, $x+a=b$, and $y+a=b$, thus $x=y$, and we are done.
This brings us to my question: To what extent do definitions need to be rigorously proven? I'm only starting to delve into elementary proofs via elementary analysis, and it's not clear how rigorous our arguments and definitions need to be, and also what constitutes rigor. Couldn't we just accept the definition $b-a=b+(-a)$ and move on? Also, I find it difficult to get the gist of proving that we can subtract real numbers: it works, but it seems to be conjured out of thin air.
 A: There are essentially two possible approaches to laying rigorous foundations for analysis. 
The first would be to define exactly what a real number is, to define the operations of addition and multiplication on real numbers, and to prove all the usual properties of these operations. This approach would be quite lengthy.
The second, which is the one Apostol chooses, is to assume that there exist a set, whose elements are called real numbers, and operations of addition and multiplication on real numbers, such that certain assumptions are satisfied, called the axioms of real numbers. How this is done varies slightly from author to author. Then, starting from those assumptions, which are not proved, you prove all the remaining properties of real numbers that you need.
Apostol assumes as an axiom that for every number $x$ there is some $y$ such that $x + y = 0$. He doesn't immediately call this number $-x$; that will come later. It would in any case not be appropriate at this point to introduce the notation $-x$, because the uniqueness of the element $y$ has not yet been asserted.
Using that axiom and others, he then proves that for any numbers $a$ and $b$, there is a unique number $x$ satisfying $a + x = b$. This number he denotes $b - a$. Then $-a$ is used as an abbreviated form of $0 - a$.
He would not have used the notation $-a$ in the proof of the existence and uniqueness of $b - a$, since the notation is introduced only after this theorem has been proved. Likewise, the definition $b - a = b + (-a)$ would not have been possible because $-a$ was itself as yet undefined. Part of your confusion may stem from the fact that other authors define the notation $-a$ before they define $b - a$.
A: Definitions are often of the form 

Given a thing $X$ with property $\Phi(X)$,  the foobar of $X$ is the unique thing $Y$ with the property $\Psi(X,Y)$.

So, yes, such a definition requires a proof of a statement before. You don't need to proof the definition itself, but rather the implicit claim

For all things $X$ with property $\Phi(X)$ there exists one and only one thing $Y$ with the property $\Psi(x,y)$.

Without a proof of this, however, your definition is not useful. Remember, a definition is always only something like the introduction of an abbreviation that should always be possible to be eliminated. 

Idiomatically, you will therefore often find something like this: First a theorem that shows that (under certain conditions) something exists and ist unique, followed by a definition that this existing and unique thing gets a name.
Or another combination that you may encounter often: First a theorem that several properties are equivalent, followed by a definition that gives a name to an object having any (and hence all) of these properties.
A: My favorite answer to this question is, when you ask yourself "Do I need to prove this?", if you immediately know how you would prove it if you did, then perhaps you don't. If you dont immediately know how to prove it just from looking at it, you probably should. 
