How do I cite a result that I've only seen stated without a proof, and I prove myself?
If it's a textbook-level result, you could just cite a textbook. Otherwise, if you derived it yourself or with colleagues, no citation is required unless you published that proof earlier (not likely if it's a minor one). But you should definitely thank the people who helped in acknowledgments.

Finally, note that there is a threshold for citation: if a result is trivial, and as such is likely to have been independently demonstrated by many people over time, with no clear historical/seminal work, then you do not need to cite it.