To what extent do you explain a proof? What previous proofs are assumed and which are referenced? Is it based on the target audience of the proof (if that is a thing)? Level's of intuition may also vary drastically, a logical step for one person may need a paragraph of explanation for another. 
For example, I can't imagine quoting the definition of a derivative and the standard rules of differentiation, when finding the minimum of a function. In some cases would you even need to show the process, could you just write the minimum?
 A: Yes, all proofs are written for a particular audience, each with its own level of expertise and its own peculiar body of knowledge, which may be assumed without explanation. One has to tailor the mathematical writing to the audience. When I give a proof to a graduate seminar, it will look very different than when I give a proof to an undergraduate analysis class, or when I am speaking at a conference of experts.
When writing a mathematics paper for publication, one doesn't always know exactly the audience you will get, and so one should err on the side of extra explanation. A math talk amongst people whose background you know well will be very different. 
When I teach a math class and have my students write proofs, then I always tell them to write for the rest of the class as the audience, rather than writing to me. They should explain the things that they think the rest of the class would want explained, and can assume the things that the rest of the class would freely assume along with them. 
