Deep Purple + Iron Maiden + Pink Floyd + Metallica + Rush = DT? I think we are onto something.
FTFY.
Besides, it might look like almost too many variables but I think those 5 has DTs styles pretty much covered. Of course I generalise the characteristics here but we've got the heavy riffing covered with Metallica, we've got our ambient more atmospheric soundscapes covered with Pink Floyd, Iron Maiden kinda represents the other aspect of their 'heavy' side and in
some ways kinda reflects some of the vocal styles. And obviously Deep Purple has the classic rock and "keyboard" driven sound covered. Out of those 5 I feel that Rush more represents the proggy (and to a similar extent, the progressive) side of the band. It's possible some of their other influences have driven the 'prog' such as; Yes, Genesis, Kansas etc.
On a side note, I'm not much of a Yes listener (tbh, listen to Genesis more than Yes). So excuse my ignorance as far as this band's music goes, but when I heard DTs cover of Heart of the Sunrise, I could definitely hear multiple influences from that one song alone. For example; tell me the bit with the keyboard synth/strings and the bassline doesn't remind you of the similar section near the start of Learning to Live?