---
tag: 0B94TKFQJX
kind: concept
name: Mathematical proof
aliases: [proof]
msc: [00A35]
uses: [0Y0MQYST0M]
sources:
  - url: https://en.wikipedia.org/wiki/Mathematical_proof
    locator: lead section
same_as:
  wikidata: Q11538
  wikipedia: 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|0Y0MQYST0M]].
A proved statement is a theorem.

## Remarks

Common methods are [[direct proof|0NTF6DMVAP]], [[proof by contrapositive|0BB5TPGQ1N]], [[proof by contradiction|0J0XBKG64S]] and [[proof by cases|0WCP0PVET6]].
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.