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.