Upstairs bathroom; drain questions

I just bought a 1950's era slab house and I am remodeling the full bath on the first floor. While I am doing so, I figured it would be a good time to cut in the piping for a second floor bathroom. I know I am shooting for 1/4" per foot slope, but I am unsure of the way to handle the vertical drop(s). Ideally I would like to use the existing 3" cast iron vent in the downstairs bathroom as my tie-in to the house sewer. I have no idea if this is even remotely reasonable, but it seems like a good solution with a manageable amount of work.

I have included a quick (and bad, forgive me) sketch of what I have going on, and would appreciate any suggestions/correctins/warnings.

Thanks in advance for all you help.

P.S. Once I am upstairs, are 45's on their side (horizontal) acceptable to get me out from between the floor/ceiling joists as long as my slope is ok?

Sorry about that, the picture fell by the wayside...

drain

Your description implied you were going to something that is not permitted. Your drawing verifies it. You can connect the upstairs plumbing into the downstairs piping but it has to be done below the floor after, or below, the existing connections. The exact location will depend on how the existing piping is arranged and we cannot tell you how, or where, that would be.