Large-scale molecular dynamics simulations are used to determine the phase diagram of a Lennard-Jones system with and without prewetting. The simulations show that the normal condensation in the homogeneous bulk system, in the case of an attractive solid surface, is extended with a single first-order prewetting phase transition to a fluid layer of a thickness that varies from only one fluid layer for a strong attractive surface to a film of three layers of particles in the case of a weaker prewetting surface. A particle-structured surface (face-centered cubic (111)) has only marginal impact on the prewetting transition. The new triple-point temperature is significantly lower than the triple-point temperature for the bulk system.