new shower, pipe diameters sufficient?

[shouldknowbetter]

I am planning to install a shower into a new extension. The supplies will be mains CW, gravity HW. I have read other posts about using single pumps and pressure equalising valves to balance the supplies. The HW feed to the pump will be as a branch from the supply to the kitchen. It is 15mm copper pipe, and after the branch I will fit a check valve to prevent siphonage from the kitchen sink. Is 15mm sufficient to feed to the pump,
and from the pump should I then run 15mm or 22mm to the mixer,
and then 22mm from mixer to shower head. (Yet to buy pump, valves, mixer and shower head. )

Any advice greatly appreciated