The complexity of proving the theorem in a particular state and in general

Question

Is it possible to give an example of a theorem that is difficult to prove in a particular case, but easy to prove in general because of the symmetry and order of the theorem?

Answer 1

1. I would not describe the direct proofs of Fermat's Little Theorem as difficult, but that theorem is an immediate consequence of Lagrange's theorem that the order of a subgroup divides the order of the group.

   Of course that short conceptual proof isn't free. It depends on developing the abstract concept of a group.

2. Hadamard and de la Vallée Poussin first proved the prime number theorem using analytic number theory. Erdős and Selberg's subsequent "elementary proof" is arguably more complex.

3. Allen R. Bernstein and Abraham Robinson used nonstandard analysis to prove that polynomially compact operators have an invariant subspace. Halmos then rewrote their proof using only standard constructions.

4. Noether's theorem guarantees the existence of invariants in physical systems that result from symmetries.