I nominate the Chinese Remainder Theorem, in the form of an isomorphism of a ring of residues with a cartesian product ring. This isn't "profound" mathematics, but simply unpacking it (with construction of the underlying idempotents) should convince students that algebraic structure has "content". I recall a conversation about the analogue for polynomials in one variable over a finite field, in which my side was really stating that if you understand the original CRT in the correct way, this is no sweat at all.