Are Shimano 105 R7000 STI levers compatible with the older R5800 deraillers? My bike currently has Shimano 105 5800 components (shifters, rim brakes, derailleurs), and I've recently discovered that both my shifters are wearing out. 

The derailleurs seem to be in good shape, though.  I'd like to just upgrade the shifters to the newer R7000. Will they work with the older derailleurs and rim brakes?
Short and easy answer - yes.

Shimano compatibility charts show ST-R7000 shifters are compatible with FD-5800 front derailleur and RD-5800 rear derailleur. All Shimano road rim caliper brakes are compatible with each other.

http://productinfo.shimano.com/#/com/3.7?cid=C-453&amp;acid=C-455