P-Trap/Vent question for new washing machine in old Jacuzzi area

[George4678]

I just removed a jacuzzi and want to add a stackable washable dryer in the corner of the room (see pic). I have enough depth to build out another wall in front of the exterior wall for the plumbing. Can someone help me understand how to get from the stand pipe/ptrap which will be on the wall behind the green box in the pic across to the existing vent pipe and drain to code? I am having a professional come in but would like to know it should be done correctly. To the left of the W/D will be a dead corner, I was planning on framing it out almost vanity counter height so we can store the laundry soap on top. This would leave a huge void underneath, can the plumbing run through that void? see second pic.

Thank you in advance!