How do we know Mitzraim was Egypt?

Maybe it’s a stupid question. But it just doesn’t make sense to me that it would take so long to get from there to Israel. To say nothing of how many kingdoms we run into along the way. I also keep on hearing about how the Mitzrim worshipped lambs. That’s one of the only animals that wasn’t an Egyptian god. The only connection between Mitzraim and Egypt seems to be Yeor (the Nile). But I can’t find how we prove that those are the same either. It could be the Euphrates for all we know!