Comment by ssivark

That is an idealized caricature, and far from the reality. One needs to validate the statement of the theorem and the boundary conditions needed to prove it (definitions, axioms, kernel soundness, etc), a la dependency injection. Mathlib is a common shared platform of vetted truths, but not all proofs restrict themselves to Mathlib AFAIK, and neither is Mathlib perfect -- particularly subtle mismatches in the definitions.