We construct one-dimensional nonlinear lattices having the special property such that the umklapp process vanishes and only the normal processes are included in the potential functions. These lattices have long-range quartic nonlinear and nearest-neighbor harmonic interactions with/without harmonic onsite potential. We study heat transport in two cases of the lattices with and without harmonic onsite potential by nonequilibrium molecular dynamics simulation. It is shown that the ballistic heat transport occurs in both cases, i.e., the scaling law κ∝N holds between the thermal conductivity κ and the lattice size N. This result directly validates Peierls's hypothesis that only the umklapp processes can cause the thermal resistance while the normal ones do not.