In composing a proof that is reliant on proven theorems, does one simply assume the reader's familiarity with said theorems, or does one explicitly include their logic in the new logic?
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It depends on the context. My masters thesis, for example, began with saying that I expect the reader to be familiar with forcing. I am not expecting the reader to be familiar with other topics which are relatively common, though. These topics are fully explained in my thesis.
When writing something one can usually foresee who is going to read the text, and what the readers are expecting to see written.
Over-detailed writing is very hard to read; but under-detailed writing is very hard to understand. It takes time and experience to find the balance. Consult an advisor or referee regarding your actual text.
This may also be affected by where you are sending the text. When writing a very short article (say 3-4 pages) it's fine to just quote results, but when writing a book it's expected to prove them as well.
It's a judgment call, depending, in part, on the audience you are addressing, and your purpose for writing a proof, or a paper with proofs, etc. And so, as Asaf answers, the answer to your question: "depends on the context."
If you are writing proofs for classes (assigned or recommended), for example, it is best to error on the side of caution and be more explicit rather than less, and include more rather than less. One rarely loses marks for including more information than needed, whereas it is common to lose marks for not including enough. So when justifying a statement in such a proof: you can refer to proofs/theorems in your class text provided they have been covered in class. You can often do so by referencing the number/letter used in the text or in class, or by referring to it by using a commonly-used name of a particular theorem. E.g., "By the Fundamental Theorem of Arithmetic, it follows that...".
It is often helpful, when writing proofs for classes, to also justify an assertion by referring to definition(s) that your assertion relies upon: e.g. "by the definition of congruence modulo n, we know that...", without needing to restate the entire definition, unless you are introducing an unfamiliar term that you plan to use in your proof.
These suggestions are just as much for your (present and future) benefit as they are for your audience, and/or for demonstrating mastery of the material relevant to the statement you are asked to prove. Of course, for proofs required in coursework or in an exam, I'd highly recommend that you consult your instructor on this matter. If preparing for preliminary examinations, you can access and work out some problems from past prelims, and discuss with your advisor whether your proofs/solutions are adequate: (What should I have included? What could I have excluded? etc).
You might find the following exposition written by Dr. John M. Lee helpful:
If interested, you might want to explore the links available at
See also the previous post: What can a writer assume in a proof?.
With all proofs (in whatever format), you must first identify your intended audience. That will allow you to decide what you can / cannot assume, and how to include in the right amount of detail.
Are they students in your field that are interested in understanding the buildup of the theory? Are they 'professionals' in related fields who believe in the truth of your statements and only want to understand implications of your results? Are they high schoolers who want to know why quintics can't be 'solved' and only know the basics of group theory?
Having additional proofs is preferable. You can helps others along by labeling them as Theorem 1, Proof (or even placed in the Appendix), so if readers are familiar, they can skip it.