> Lemmas are simple, obvious, and yet they completely change how you think.
speak only for yourself :o) , they are the reason we have:
lemma now, dilemma laterback
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Yeah, I can't say lemmas are (generally, or even often) simple and obvious. To me, they often seem arbitrary: what do you mean before we prove this grand theorem we have to prove these completely unrelated lemmas? Okay, proved the lemmas. Now the proof of the theorem has "according to such and such lemmas..." sprinkled around, but I've already forgotten what the lemmas were and why they're applicable. I also can't name any lemmas that changed how I think.
My first ever paper was a pretty good entry at this. https://dspace.library.uvic.ca/server/api/core/bitstreams/66... 16 lemmas, one theorem. The theorem makes all of the lemmas obvious. Though, in retrospect, I should have separated out the characteristic 0 case into a second theorem.
But the fact that most lemmas are like this, does not mean that all are. Whether we call it a lemma, or something else, the more important ideas are the ones that result in thinking differently. And something like Zorn's lemma, makes us think differently.