C*-algebra theory has a number of good examples of this.
- Voiculescu's theorem: an ample representation of a C*-algebra essentially absorbs any nondegenerate representation
- Kasparov's technical theorem: if anybody really cares, I'll repeat the statement. The point is that anybody who works with bivariant K-theory uses this result ALL THE TIME, e.g. for excision or the existence of Kasparov products.
- Stinespring's theorem: any completely positive map into $B(H)$ dilates to a representation
I have been using Voicalescu's theorem and KTT for a about a year or so longer than I knew the proofs. I probably still wouldn't know the proofs if it hadn't become necessary. Stinespring's theorem is probably better known among the people who use it because it's not so difficult, but it could be tempting to use it as a black box.