Adding washing machine and tray to bathroom DWV

[artistsnature]

Hi guys,

I'm trying to tie a new laundry room into an existing bathroom on a raised slab. The copper supply lines (as well as cast iron drains) are all under concrete, so I'm going to cut them off and re-route them through the walls. My concern is, is it okay to patch the washer and tray into the lav drain line? Currently there are no vents, so I am adding them to the utility sink, bathroom sink and at the end of the 3 inch main line above the shower and toilet. Will that work? Is it to code?

Thanks!

PS. I intend to plumb the washer and tray the Terry Love way! I'm not sure why the program (Quick Plumb) put the extra detour - there's no obstacle (except for a cinder block wall ;-)