Did Rush influence Prog Metal?

PT had metal influences in the 2000s. 

They were not the originators of metal, like I would not put them next to Black Sabbath, but they had an influence on progressive metal. By that I mean they influenced bands and allowed them to record music like Yes or Genesis or Pink Floyd but with a much harder attitude, essentially Rush music since they themselves were influenced by Yes for example.

In very fine well defined buckets, to me Black Sabbath influenced the creation of hard heavy metal like say Iron Maiden, Judas Priest. Rush helped bands like Tool, DT, Riverside create much harder rock, maybe call it metal-lite with prog attributes. I try not to mix those two buckets......And we all know Rush were influenced by Zeppelin, Yardbirds, Cream but also listened to Yes, Genesis and put those two together and you have the start of prog-metal and in early days, Rush.

Enchant is prog metal?  Not to me they weren't. Tiles would be a better example although they weren't really prog metal either.

In my opinion though they were definitely influenced by Rush especially Mike Portnoy(although he's no longer in DT). I admit there is a thin line between being inspired by someone and being influenced by them but Rush were heavy enough to make it plausible they were both.