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The HDMI output won't work if both DVI connections are used and assuming that your HD5770 is the original design with 2x DVI and 1x HDMI and 1x Displayport, then you have no other choice but to use the Displayport output.
Since your HDTV uses a HDMI connector you will need to use a converter from Displayport to HDMI, however, I'm unsure if there are any certified companies that sell Displayport to HDMI that will work and not screw up. The 5770 should have come with a DVI to HDMI and you can use that to connect the HDTV to that by HDMI. You'll need to purchase a Displayport to DVI adapter and use that to connect to one of your monitors from DVI.