Hadronic matrix elements of proton decay are essential ingredients to bridge the grand unification theory to low energy observables like proton lifetime. In this paper we non-perturbatively calculate the matrix elements, relevant for the process of a nucleon decaying into a pseudoscalar meson and an anti-lepton through generic baryon number violating four-fermi operators. Lattice QCD with 2+1 flavor dynamical domain-wall fermions with the {it direct} method, which is direct measurement of matrix element from three-point function without chiral perturbation theory, are used for this study to have good control over the lattice discretization error, operator renormalization, and chiral extrapolation. The relevant form factors for possible transition process from an initial proton or neutron to a final pion or kaon induced by all types of three quark operators are obtained through three-point functions of (nucleon)-(three-quark operator)-(meson) with physical kinematics. In this study all the relevant systematic uncertainties of the form factors are taken into account for the first time, and the total error is found to be the range 30%-40% for $pi$ and 20%-40% for $K$ final states.
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