Math doesn't have a property that you can verify everything you're doing is correct with little cost. Humans simply tend to prefer theorems and proofs that are simpler.
You can, in principle, formalize any correct mathematical proof and verify its validity procedurally with a "simple" algorithm, that actually exists (See Coq, Lean...). Coming up with the proof is much harder, and deciding what to attempt to prove even harder, though.
You can verify it with a simple algorithm, but that verification won't always be cheap. If it was, curry-howard would be incredibly uninteresting. It only seems cheap because we usually have little interest in exploring the expensive theorems. Sometimes we do though and get things like the 4 color theorem whose proof verification amounts to combinatorial search.