Filling in details from other papers, is it good or bad? Say I find a Math Proposition in another paper, which is just stated, and no proof (nor even sketch of proof) is given.

If I am writing a paper referencing that paper, is it good or bad (from reviewer's point of view) for me to fill in the proof of that proposition?

If I do so, what is a good way for me to indicate it in the paper? (that the proof written is by me, though the proposition is by the other paper)

The proof is not trivial, though it is not particularly difficult either.

Or should I just privately verify if the proposition is correct? And just cite the proposition (without proof) in my paper?

Thanks.

I am concerned about this issue since I followed the news story in the case of Grigori Perelman's proof of Poincare conjecture; the authors who filled in "gaps" in his proof did not go well with the reviewers and the general public. (https://www.newyorker.com/magazine/2006/08/28/manifold-destiny)
Let's call the other paper [A].

First of all, you might like to check whether a proof already appears in some other paper or book, [B].  You can search MathSciNet or other databases for papers which cite [A].  If so then you can cite both ("The following proposition was stated in [A]; see [B] for a proof.")

Otherwise, whether to give a proof is at your discretion.  If you think the reader would find it helpful to see the proof, and it doesn't distract from the main purpose of your paper, and it isn't excessively long, then sure, you can include it.  People often use phrasing like:


  We make use of the following proposition, which is stated in [A].  Since [A] does not include a proof, we give one here.


Another option is to put the proof in an appendix, or to write it up as a separate note which you post on arXiv or something.

The Perelman case is about something different - people disagreeing over whether the original paper actually solved the problem, and how much of the credit for the result is due to those who filled in whatever gaps there may have been.  In this case, you describe the proof of the proposition as "not particularly difficult", so it's likely that the authors of [A] did in fact know how to prove it, and so would most of their readers who took the time to do it.  It's reasonable to give all the credit for this proposition to the authors of [A].  There's nothing wrong with you including a proof, and as long as you don't try to take credit for the result, no controversy will result.