The only problem that I can think of would be the width of the cage. The 7-speed chain might not have as much clearance with the narrower cage. Actually I believe it would work fine but I would go for a 7700 instead of a 7800 if possible. You could also try a narrower chain. I'm pretty sure you could use an 8-speed or 9-speed chain with your 7-speed drive train. I have see a 9-speed Shimano chain work perfectly on a 7-speed Shimano drive train with a triple crankset.