Hey Matt. First it seems like you know how you want your math to be formatted but don't know how to do it on the forums so I would suggest checking out this thread.
I think what you're asking was how to get from your first probability statement to the second one in a rigorous way. I guess my question for you is what definitions you know for a t-statistic? Do you have something in terms of the ratio of a normal r.v. and and a chi-square r.v.? Or does you book use something else?