I am not sure I fully understand the question. Is it the case that the theorem itself gives you a huge mileage while its proof is extremely difficult, (Characterization of finite simple group is an ultimate example; the Atiyah-Singer index theorem and the BBD(G)-decomposition theorem are other examples; or is it a case that understanding the proof (which is feasible) gives you a lot of mileage and a feeling that you got grip with the subject.

Anyway, a theorem which, to some extent, has both these features is Adams's theorem asserting that d-dimensional vectors form an algebra (even non-associative) in which division (except by 0) is always possible only for , 2, 4, and 8. (In these cases there are examles: the Complex, Quaternions and Cayley algebras.)

I am not sure I fully understand the question. Is it the case that the theorem itself gives you a huge mileage while its proof is extremely difficult, (Characterization of finite simple group is an ultimate example; the Atiyah-Singer index theorem and the BBD(G)-decomposition theorem are other examples; or is it a case that understanding the proof (which is feasible) gives you a lot of mileage and a feeling that you got grip with the subject.

Anyway, a theorem which, to some extent, has both these features is Adams's theorem asserting that d-dimensional vectors form an algebra (even non-associative) in which division (except by 0) is always possible only for , 2, 4, and 8. (In these cases there are examles: the Complex, Quaternions and Cayley algebras.)