In four sentences or less, explain what a proof is in mathematics.
A proof is a logical argument that uses a set of axioms or previously proven statements to demonstrate the truth of a mathematical statement. It works by deriving conclusions from a series of premises or by contradiction, showing that the statement must be true in all possible cases. Proofs provide a rigorous way to establish and verify theorems, which are general facts or principles that can be applied to solve complex problems. A proof must be complete, logically sound, and free from any ambiguity or contradiction to be considered valid in mathematics.
Complete the following task based on the instruction and optional input provided.### Instruction:
In four sentences or less, explain what a proof is in mathematics.

### Output:
A proof is a logical argument that uses a set of axioms or previously proven statements to demonstrate the truth of a mathematical statement. It works by deriving conclusions from a series of premises or by contradiction, showing that the statement must be true in all possible cases. Proofs provide a rigorous way to establish and verify theorems, which are general facts or principles that can be applied to solve complex problems. A proof must be complete, logically sound, and free from any ambiguity or contradiction to be considered valid in mathematics.