One should distinguish between short, trivial and elementary proofs. Trivial proofs can be omitted with no harm. Omission of proof in the other two classes does seldom any good to the reader. In you shoes I would put full proof in the preprint on arXiv. It can be easily edited out in the published version.

One should distinguish between short, trivial and elementary proofs. Trivial proofs can be omitted with no harm. Omission of proof in the other two classes does seldom any good to the reader. In your shoes I would put full proof in the preprint on arXiv. It can be easily edited out in the published version.