Why proving the solution of a problem is polynomial time is sufficient enough to say that it is a NP prolbem? [duplicate]

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• What is the definition of P, NP, NP-complete and NP-hard? 7 answers

Why proving that we can verify the solution of a problem is polynomial time is sufficient enough to say that the problem is nondeterministic polynomial time? Please note: this is not a question on how to prove a questions is NP, but instead asking why, we can just do so? If there is more step we need to do, what are missing here?

I am not sure why proving that we can verify the solution of a problem is polynomial time is sufficient enough to say that the problem in NP because seem to me that we can also verify a solution of a problem is polynomial time while it is actually linear time, can't we?

Proving that a problem is in NP seem requires one additional step? it is really necessary ?