My water closet empties into a 3" horizontal drain (using a combo wye for the change from vertical to horizontal). I plan to add a vent, connected with a wye, about 7' downstream from that point. What is the correct pipe size for this vent? NC code says the vent must be at least 1/2 the diameter of the drain pipe so I figured 1 1/2" would work and meet code. Also, am I correct that the distance to the vent on a 3" drain cannot be more than 10'? The slope of the horizontal drain is 1/4" per foot.

Thanks for your help. Just want to be sure.