TITLE: How do we know whether certain mathematical theorems are circular?
QUESTION [14 upvotes]: There are countless mathematical theorems and lemmata, some of which, obviously, depend on others.
My question is: how do we know that, say, Theorem $A_1$- which uses a result proved in Theorem $A_2$ which uses a result proved in ... which uses a result proved in Theorem $A_n$ which, in fact, relies on the fact that Theorem $A_1$ is true - doesn't make the proof of Theorem $A_1$ circular?
Essentially, what I'm saying is that, since there's no comprehensive list of all mathematical theorems, lemmata and corollaries (and what statements they rely on), how can we assume that no two theorems will be circular (not directly- but down a long chain of theorems)?
If I'm not articulating myself properly, please ask me to elaborate.
Thanks

REPLY [5 votes]: Extending on @Josefs answer and your phrase "comprehensive list of all mathematical theorems": I asked about this sort of quest here: What is an efficient nesting of mathematical theorems?. In the last 2.5 years, I researched this a bit.
The biggest math library, starting from set theory axioms as initial "theorems" is Mizar. You find a page where they publish they theorems in pdf-format too, see here. The two languages Agda and Coq have a more active community and I know the first one has a library too. And there are more such projects, in particular in the Computer Science community. And there is, in fact, a subfield now, called Mathematical knowledge management which tried to forms tools to do that sort of thing in a less ad hoc fashion. 
A year ago I started a similar project for myself, where I mostly sort sets by the inclusion relation, AxiomsOfChoice.org. (It has a graph rendered with the graphviz software, but now that there are over 200 entries it got completely messy. Anyone with serious interest in a coding challenge please write me!) I encountered the algebraic stacks project Joseph links too before I also know of some guys wiki on groups, which thousands of pages.
I also want to say that circular arguments are bad as far as the "math game" goes, but one should keep in mind that for many it's important to foster the things that work. I could point you to the logic project Reverse Mathematics, where people explore interdependencies by "starting at the good stuff".
I could post more links in that spirit, but my message is that people are thinking alot about this kind of stuff :)