Synopsis:
The concept of a mathematical proof involves writing a sequence of unbroken logic statements that leads to a definitive conclusion about a theorem/conjecture.
This article will lay out why using previous proofs/theorems/corollaries is inherently a necessary step of any solid proof in Abstract Algebra, Topology, Complex Analysis, and so on.
Use previously proven theorems and conjectures for your own proof:
A famous example of this is the Fermat's Last Theorem[1] by Andrew Wiles, per FLT, the proof involves the Taniyama-Shimura[2] conjecture, that links number theory to geometry.
This essential framework provides the foundation of a contradiction proof of Fermat's Last Theorem. The lesson here is to use the previous literature to avoid circularity and logical gaps in your proof.
A Meta-Framework for such proofs:
Here is such framework that one may use as a general framework for proof-writing:
- [ THEOREM 0...n ]
- [ PROOF STRUCTURE ]
- [ INVOCATION OF THEOREM 0...n ]
- [ LINK BETWEEN THEOREM AND PROOF ]
- [ CONCLUSION OF PROOF ]