15’ should not be a problem for a decent HDMI cable.
I’ve used a 30’ Monoprice a few times to directly connect our Internet modem to the cable box (usually run WiFi, but it craps out from time to time) and it worked fine.
This said, our previous and current TV cap out @ 1080P, and like I said I’m not picky about TV (it just needs to look/sound OK).
DeKay