How detailed should proofs be in a mathematical PhD 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.