Mathematical proof

Summary

A mathematical proof is an argument that shows that a statement follows from axioms, definitions and statements that are already proved. Each step of a proof follows from earlier steps by a valid rule of inference, such as modus ponens. A proved statement is a theorem.

Remarks

Common methods are direct proof, proof by contrapositive, proof by contradiction and proof by cases. A formal proof is a sequence of formulas in a formal system. A computer can check it, for example in Lean. Examples, however many, do not prove a universal statement.

Explanations

L1 human

A proof is a chain of small steps, and each step is certainly correct. It starts from things that you accept or already know, and it ends at the statement. After a correct proof, the statement is true in every case, not only in the cases that you tested.