How detailed should proofs be in a mathematical PhD thesis? I heard this suggestion from a mathematics professor:

In writing a mathematical Ph.D. thesis, it is far more tolerable to be tediously-lengthy than having a gap in the proofs.

I think what he means is that whenever in doubt, adding more details to make the argument clearer is always better, even if sometimes doing this may make the proof too wordy.

Now if I really follow his advice literally, it seems there are too many details for me to write. For example, I don't even feel safe to write "the case n=1 is trivial" when proving by induction, or "by a direct calculation we have the following result". Another example is, whenever writing a commutative diagram (some of my diagrams are 3-D), I doubt if I need to prove that every small triangle or rectangle is commutative. (Actually I have asked this question but have received no answer so far.)

Thus my question is: How detailed should be the proofs be in a mathematical Ph.D. thesis?
I usual give the advice 


  If something is clear, just write it down clearly.


Of course there is a caveat: What is "clear" clearly depends on the reader, and also what is considered a "clear explanation" certainly depends.

But the message is: If you are tempted to write "The case $n=0$ is obvious." think a minute about the explanation and how to write it down. If you then think that writing down the explanation does not add anything, leave it out.

But in general, I would say, being more verbose is the better option.

Another test you could apply is: Write down "obviously…" but note down a longer explanation somewhere else. After a week or two reread that part and try to recover the result again. If you don't have any problem than you are probably fine without a longer explanation.

Some resource that may be helpful for your proof-writing skills is the talk given by Leslie Lamport on proof-writing last year at the Heidelberg Laureate Forum: Here is a blog post about it and the whole talk is here.