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There are reliefs of Ramses II fighting on a chariot and he's from the 1200-1300 BC, so the wheel must have been used much earlier than that by Egyptians.

I'm not sure if we're taking about different time periods, but weren't they at the forefront of chariot warfare during the bronze age?

And I vaguely recall that inventing bellows to melt iron took a long time too, so steam machinery seems pretty far off if all you have is bronze and stone.

The Egyptians had chariots, they definitely used wheels.