Here is the layout and I am questioning the need to have the purple segment
the green blue selected segment is already in place, the green segment between the shower vent and the sink is what I am building (still dry fitting the pieces but I did not drill the holes for the studs to the left of the vertical green segment
As mentioned the distance between the two green verticals is 4'
I am also considering an air admittance valve
Update: Here is why I need to go down and not horizontal with the drain
Best Answer
If the slope of the drain is 1/4"-per-foot you can do without the pink vent. Be sure to measure your slope with a level. Don't just measure from the floor.
You don't need an AAV to delete the pink vent.
Instead of the green branch drain having a vertical drop, just run it horizontally (with a 1/4"-per-foot slope) over to the corner. That is within the max length before siphoning could occur.![edited CAD drawing with red line for branch drain](https://i.stack.imgur.com/jR5pO.png)