Under the UPC the waste arm from the toilet to the stack may be up to 8ft long without adding additional venting.

In the IRC there is an exception in the code that says if a toilet is the only fixture served by the drain line it may run an indefinite length from the toilet to the stack.

You proposed configuration at the stack is completely legal...however I would be interested to know how your venting the tub because code prohibits any horizontal vents until the vent reaches an elevation 6" higher than the tallest fixture served by that vent.

When I referred to the tub's vent, I was meaning the design built-in to the tub which also serves as overflow control. There is a pipe which connects the hole located just below the faucet to the drain in the bottom of the tub. Those two combine and then enter the P-trap which is located within inches of the stack. Does this not serve as a functioning vent for the tub?

A vent must be on the discharge side of a trap. The overflow is on the input side and it has absolutely nothing to do with venting.

If the distance from the trap weir to the vertical stack is within the maximum allowable length of the fixture arm there is no additional venting required.

Under the IRC an 1-1/2" tub fixture arm with a 1/4" per foot pitch is allowed to be a maximum of 6'.

Under the UPC an 1-1/2" fixture arm is limited to 3'6".

The minimum allowable distance from the trap weir to the vent or vertical stack is 2x the pipe diameter.