When writing a PhD thesis on mathematics one needs to quote many results by others, such as
The following theorem was proved by ABC in [1].
Theorem. Bla, bla, bla...
My question is when should one include a proof of the result in his/her PhD thesis, if it is basically the same as in the reference? I do not quite see the point of copy-pasting the proof by others. Of course if one has a completely different proof of the same result, it is probably suitable to include it.
As a general rule, you can cite other people's theorems without explaining their proofs, and omitting a proof is a good idea if it would be a lengthy distraction. However, there are several reasons why including such a proof could be helpful:
Including it may be convenient for the reader if the proof is short. It's annoying to look up another paper and discover that you only needed a short argument that could easily have been explained in the original paper. Extracting information from a reference can be cumbersome (you have to locate exactly what you're looking for, figure out what it depends on, sort out the notation, etc.), while giving your own explanation can help readers avoid some of these difficulties.
Even if the proof is not particularly short, it may serve as a warm-up for new applications of the same techniques. Reminding the reader how they work may make your paper much easier to read than if you just dive into the newest and most complicated case.
Ph.D. dissertations are something of a special case, because your advisor may encourage you to include extra details in the background sections (beyond what you might include in a published paper). This is partly a matter of demonstrating your mastery of the area and partly a matter of writing a useful survey for others. Advisors differ in how they approach this: some think it's a waste of time and it's best just to focus on writing a published paper, while others think writing a more extensive dissertation is a valuable learning exercise. This is an issue you should discuss with your advisor.
This is a good question, and as Anonymous Mathematician indicates, it is well worth discussing with your advisor.
Essentially what you are asking is whether and when to include exposition in your PhD thesis. The answer is that it is rarely strictly required, but it is often expected, in many cases encouraged, and in some cases not necessary. There are a lot of nuances here and I don't foresee a comprehensive general answer being possible. (Anonymous Mathematician's answer is excellent, and I am essentially corroborating it.)
Mathematics has a proud tradition of PhD theses having significant expository content. (In my thesis, Chapter 0 is expository. It occupies about half of the thesis. This is a bit on the lengthy side, but not so unusual.)
One reason that this is done is because a PhD thesis is usually the last chance that your mentors get to lean on you and require that you show your mastery of highly difficult, technical concepts. When I am a committee member on a math PhD thesis, I generally want to see at least enough exposition to convince me that the writer has mastered the concepts, definitions and objects used in the thesis. Especially, I want to see key definitions in a lot of detail, even if they are long and taken from other sources.
Another reason this is done is that the cultural standard in mathematics is that PhD theses can be significantly more discursive than published papers. When a PhD thesis gets converted to a paper, often there is a compression of 2:1 or more in terms of the page count, and often the results that appear in the paper are stronger than what appear in the thesis. (In mathematics, I gather unlike some other fields, one most often publishes the lion's share of one's thesis work after completing the thesis, not before.) Something's gotta give, and often math papers published in the strongest journals are written so that every single page contains an important new idea or truly difficult calculation. This density of content is a point of pride of the top journals, but it can make the papers awfully difficult to read. A lot of theses are famous for being the best sources of exposition for the topics they contain.
Having said all this, it seems clear that little value is added by "copy-pasting". Taken literally: copying lengthy proofs verbatim from other sources would be plagiarism if carried too far. Most exposition in a PhD thesis is filling a gap in the literature, not reproducing it. Good exposition synthesizes several sources, offers new perspectives (including a bridge to the novel results, as AM mentions), chooses notation and hypotheses in a globally appropriate way, and so forth.
Finally: formal proof is often the least important part of good mathematical exposition. Getting the definitions and statements just right and putting them in context is more important. Most contemporary math PhD theses build on significant technical foundations, not all of which the student is expected to be personally conversant with. A PhD thesis is not supposed to be "logically self-contained" in any formal sense, only to demonstrate mastery in the eyes of the committee members and to be a useful document for the reader in the eyes of the advisor and (most importantly) the writer. If you are thinking of more or less copying a proof "for completeness", that may not be the way to go.
Generally, you should only need to reproduce the proof verbatim in cases where you need to dissect it, call out one part of the proof in particular, or you intend to extend it directly using similar arguments, otherwise, stating the theorem and citing a work where it is proved should be sufficient.
I don't see any reason to copy a proof found somewhere else. I think its fine to say that this result and its proof can be found on this article. In my opinion, the proofs in a dissertation should be your own.
