watts = amber * volts, 28 amber in 12 V trail means it can deliver up 336 watts in the 12 Volts trail only an HD5770 working at 99% 3D (furmark 1.8 test) needs 22a * 12 Volts = 264 watts leaving less than 72 watts to the rest of your devices in your system, a hard disk for example needs 8.5 watts clearly the answer you need “yes it will work” is not that simple. If you planning to overclock the HD5770 the watts consumption will raised.
This is why they recommend bigger watts PSU. The PSU’s are ratted: 80 silver, 80 bronze and 80 gold (energy efficiency above 90%) this is why 80 gold is move expensive. Finally you may need to upgrade your PSU, I recommend for you to find an 80 Bronze Psu.