I want to cite a theorem from a book written by two computer scientist that I consider to be influential and renowned. If you want the specifics, I want to cite Theorem 11.8 from the book Computational Complexity: A Modern Approach, by Sanjeev Arora & Boaz Barak. If it affects in any way, I am planning to submit this paper for a conference.
This theorem is stated on the book, however the proof is omitted since it is similar to another proof that is presented in the book. I keep wondering: should I include my own proof of this theorem or is it unnecessary? I could write the proof but I feel that it could be distracting and take space in a paper that will already have many short proofs (at least at this point in my research it seems that way). Additionally, I don't think someone would contest the validity of the theorem. On the other side, the result is vital to the content of the paper I am writing.
UPDATE: I should have mentioned before that in the book, three sources are cited for the theorem. However, when I read those sources, none of them state directly the theorem, even if they give the "building blocks" for proving it. I also tried searching for the same theorem in other sources and I found one in which it is proved, but it is stated in a way that could be confusing to read.