Is it better to be more long-winded than concise in writing a mathematical Ph.D. 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?