Simplifying After Using the Bloch Ansatz in the Stationary Schrödinger Equation

Question

I am working on deriving the intensity equations for the dynamical diffraction of neutrons following along with a paper by Hartmut Lemmel (Hartmut Lemmel. Dynamical diffraction of neutrons and transition from beam splitter to phase shifter case. Phys. Rev. B, 76:144305, Oct 2007).

The first task is to get the following equation for an infinite system of coupled equations

$$ \Biggl(\frac{\hbar^2}{2m}|\vec{K}-\vec{H}|^2-E\Biggr)u_\vec{H}=-\sum_\vec{H^\prime}V_{\vec{H^\prime}-\vec{H}}u_\vec{H^\prime} $$

I am trying to work out all the math myself so I am plugging in both $$ \psi(\vec{r})=e^{i\vec{K}\vec{r}}\sum_{\vec{H}}u_{\vec{H}}e^{i\vec{H}\vec{r}} $$ and $$ V(\vec{r})=\sum_\vec{H}V_\vec{H}e^{i\vec{H}\vec{r}} $$ into the stationary Schrödinger equation $$ -\frac{\hbar^2}{2m}\nabla^2\psi(\vec{r})+V(\vec{r})\psi(\vec{r})=E\psi(\vec{r}) $$ and simplifying down to the first equation above. I have gotten through enough of the math to get to the following $$ e^{i\vec{K}\vec{r}}\sum_\vec{H}\Biggl(\frac{\hbar^2}{2m}|\vec{K}-\vec{H}|^2-E\Biggr)u_\vec{H}e^{i\vec{H\vec{r}}}=-\sum_\vec{H}V_\vec{H}e^{i\vec{H}\vec{r}}\Biggl(e^{i\vec{K}\vec{r}}\sum_{H^\prime}u_\vec{H^\prime}e^{i\vec{H^\prime}\vec{r}}\Biggr) $$ I assume from here I am able to cancel the $e^{i\vec{K}\vec{r}}$ and $e^{i\vec{H}\vec{r}}$ on both sides in order to leave me with $$ \sum_\vec{H}\Biggl(\frac{\hbar^2}{2m}|\vec{K}-\vec{H}|^2-E\Biggr)u_\vec{H}=-\sum_\vec{H}V_\vec{H}\Biggl(\sum_{H^\prime}u_\vec{H^\prime}e^{i\vec{H^\prime}\vec{r}}\Biggr) $$ I have never been that great at working with summations so if that is not allowed please do let me know. If it is, I am a little lost from here.

1. How does the sum over $\vec{H}$ on the left side go away?
2. How do we accomplish the change in the summation indices on the right side to get it to look like we want it to?
3. What happens to the $e^{i\vec{H^\prime}\vec{r}}$ in the second summation on the right side? How does that disappear?

I know this is probably such a trivial step, but it just isn't clear to me how this is accomplished or reasoned through. Any help would be greatly appreciated!