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Mathematics is rife with the fruit of abstraction. Many problems which first are solved via "direct" methods (long and difficult calculations, tricky estimates, and gritty technical theorems) later turn out to follow beautifully from basic properties of simple devices, though it often takes some work to set up the new machinery. I would like to hear about some examples of problems which were originally solved using arduous direct techniques, but were later found to be corollaries of more sophisticated results.

I am not as interested in problems which motivated the development of complex machinery that eventually solved them, such as the Poincare conjecture in dimension five or higher (which motivated the development of surgery theory) or the Weil conjectures (which motivated the development of l-adic and other cohomology theories). I would also prefer results which really did have difficult solutions before the quick proofs were found. Finally, I insist that the proofs really be quick (it should be possible to explain it in a few sentences granting the machinery on which it depends) but certainly not necessarily easy (i.e. it is fine if the machinery is extremely difficult to construct).

In summary, I'm looking for results that everyone thought was really hard but which turned out to be almost trivial (or at least natural) when looked at in the right way. I'll post an answer which gives what I would consider to be an example.

I decided to make this a community wiki, and I think the usual "one example per answer" guideline makes sense here.

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Power series. Both conceptually and computationally, in the 17th century they replaced a multitude of ad-hoc methods that had been used for millennia.

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Most of the problems tackled in introductory calculus courses (tangent lines of and areas under basic curves, volumes and areas of solids of revolution, etc) had to be solved on a case-by-case basis, with some pretty complicated and ingenious proofs; now any undergraduate can solve them in a few lines by rote methodology.

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Hindman's theorem : if I remember correctly, the original proof was elementary but very long and complicated; whereas a proof using an idempotent ultrafilter can be explained in less than a page.

Still with ultrafilters, there's also Tychonoff's theorem : the original proofs, either Tychonoff's with complete accumulation points, or the one with the Alexander subbase theorem are somewhat technical and require some imagination.

The proof using ultrafilters is extremely straightforward and anyone who has learned about ultrafilters can find it with little to no imagination. It also has the advantage of showing that if you work with Hausdorff spaces, you don't need the full axiom of choice, whereas the Alexander subbase theorem uses Zorn's lemma indistinctly.

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