The hdmi will display the full 1920x1080 through the pc? I see that they advertise the D-sub as the pc input, and im pretty sure they use wxga which is 1366x768 instead of wuxga which is 1920x1200/ 1920x1080
DVI connector on your 5870 to one of the HDMI connectors on your tv.
Now you have to listen to sound from the computer speakers or set up the computer to enable audio thru the HDMI-DVI. Or connect audio fron computer to your TV.
More info here: http://en.wikipedia.org/wiki/Digital_Visual_Interface