How to avoid plagiarism when including definitions and proofs from textbook?
It is very common for theses in the mathematical sciences to spend a significant amount of time and space repeating known or even standard definitions and results.  (In fact, up through the master's level, at least in many places one can write a perfectly acceptable thesis that only does this.)  In fact this is generally viewed as a positive feature of the thesis: the candidate has taken the time and effort to synthesize a presentation which is complete and self-contained up to a certain point.  It is also generally very helpful to do so in terms of readability: a math paper that repeated nothing that was already known would be well-nigh impenetrable except (perhaps) by a select coterie of insiders.   

Also the virtue of rewording is not as strong in this area.  If you are going to give the definition of, say, entropy, in a thesis, then I would say the best thing to do is to close all your textbooks and write down what you think is a good wording of the definition.  Once you've done that you check back with the sources to see that you've actually gotten the definition right, i.e., that it is mathematically equivalent to the one from the textbooks.  But if your language is similar or identical to what you found in the textbooks: okay, fine.  You don't need to change it for that purpose.   There are a lot of ways to list the axioms for a group, and if you wanted to you could pull out fifty textbooks and make sure that your wording is different from all fifty of them.  But this would be a big waste of time: it is not necessary to do so, and what do you bet that these fifty wordings capture most of those that are best in terms of efficiency, readability, and so forth?

When it comes to copying entire proofs word for word, I would pay close attention to how often you are doing this.  If you are simply copying multiple pages of proofs verbatim out of a single standard source, then you should start wondering about the value added in doing so (and, after a certain point, issues of copyright do emerge).  There is a key word that I used in the first paragraph: synthesis.  When you revisit old results, ideally you are synthesizing them: i.e., no one source has everything that you want, so you are combining multiple sources in a novel way.  Too much copying and too little synthesizing does not necessarily put you at risk for plagiarism and copyright violation -- it would have to be quite extreme for that to be an issue -- but it does not sound like the path to a strong thesis.  

I hope that everyone who is writing an academic thesis has a thesis advisor.  You should talk to her about this issue.  To a certain degree, the right answer is what she thinks is best.