How do we know there is such a thing as the strong nuclear force?

We know there is a strong force because of how the nucleus stay compact holding together all the positively charged protons that they should have otherwise repelled one another. If you consider the nucleus of atoms (except hydrogen) you will see that it contain a number of proton, carrying positive charge, yet the nucleus instead of bursting as the result of the repulsive force it remains intact. So a clearly there must be a force "The Strong Force" or "Nuclear Force" that overcome this repulsion and keeps the nucleus together. This Force is created by the exchange of particles called mesons. As we all know, protons are located in the nucleus of the atom. All of them are positively charged. And as we know, same charges repel each other. But come to think of it. Why don't the protons in the nucleus repel each other if they all have the same charges. The strong acts at a very short distance and is manifested by the decay events that occur in elements with a lot of protons.