That's true, but in practice mathematicians rarely check a proof to that level of detail. In fact, they rarely write a proof at that level of detail. There just

That's true, but in practice mathematicians rarely check a proof to that level of detail. In fact, they rarely write a proof at that level of detail. There just isn't enough time to do that for every result/review, so people take shortcuts. Most of the time it's fine because trained mathematicians take good shortcuts, but sometimes things slip through.