quantum mechanics - Show that Propagator satisfies Schrödinger equation

I want to show that $$K=K(x,x',t-t')=\sum_{\beta}\exp\left[\frac{-iE_{\beta}}{\hbar}(t-t')\right]$$ satisfies the Schrödinger equation $$H|\psi\rangle = i\hbar\partial_t|\psi\rangle$$ with respect to $x$ and $t$, where the $\beta$'s are the Eigenstates of the Hamiltonian and satisfy $$\sum |\beta\rangle\langle\beta|=1.$$ So I started calculating and got $$i\hbar\partial_t K =...= \langle x\vert\exp\left[\frac{-iE_{\beta}}{\hbar}(t-t')\right]\vert x'\rangle.$$ But here I am stuck since I am not that familiar with QM and bra-ket notation/relations. What is the next step or which fact do I have to use to show the claim?