Prove or Proof

I don't fully understand the difference between prove and proof as it pertains to mathematics. Explain.

 

The word proof generally means evidence that's used to justify an argument.
Sometimes, it is a synonym for evidence. An effort, process, or operation designed to establish or discover a fact or truth; an act of testing; a test; a trial. (countable)
Proof is the act of validating; finding or testing the truth of something


The word prove means to validate the presence of something by evidence.
Prove is much simpler. It is a verb, where it usually means to verify something as true or to demonstrate one’s competence.
To demonstrate that something is true or viable; to give proof for. (transitive)
example:
‘I will prove that my method is more effective than yours.’;

 

You are saying this:

Prove when x = 4, 2x + 2x becomes 16.

Doing the math work for this is the actual proof. Yes?

Proof

2x + 2x

Let x = 4

2(4) + 2(4)

8 + 8 = 16