Suppose there's a metal in which electrons interact with themselves and with the phonons. The hamiltonian might look like this
\begin{equation} H= \sum_{k}\epsilon_k c^\dagger_k c_k + \sum_{k}\omega_ka^\dagger_ka_k+H_{e-e}+H_{e-p} \end{equation}
where $\epsilon_k$ is the energy of the electrons, $\omega_k$ is the energy of the phonons and the other two terms are interaction terms. Without going into the details, let's say that I calculate the resistivity only due to e-e scattering and only due to e-p scattering and I get
\begin{equation} \rho_{ee}=a T^2 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \rho_{ep}=b T \end{equation}
where $a$ and $b$ are just constants. The question is: How should I add these resistivities? Is it reasonable to just say $\rho_{total}=\rho_{ee}+\rho_{ep}$? I'm trying to avoid a calculation where I need to treat both scattering processes simultaneously, of course.
Some extra information: The e-e resistivity is calculated in the context of Umklapp scattering at low temperatures when the Fermi surface has a similar size as the Brillouin zone. The e-p resistivity is the tipical calculation at high temperatures when the phonons become classical. There's a temperature range where both effects coincide and that's where I'm trying to get the total resistivity.
P.S.: I read about Matthiesen rule and also, in a different place, about the "parallel resistor" formula but I'm not sure how to apply those concepts here. (Just naming those terms in case they ring a bell).