Among the important results of APStat that teachers should know how to prove mathematically are the following three:
1. The Central Limit Theorem (CLT) 2. The Chi-Square Test Stat Is Approximately Chi-Square Distributed 3. The T Test Stat Has A Student's t Distribution (exactly so when sampling from a normal population)
For those still in a proof-reading frame of mind, feel free to request either of the first two off-List.
Briefly, there are two versions of each: The first for the CLT is the usual version (sans the rigor of such --- thanks to JS for making me attentive of such) and is good for those who have not seen a proof of the CLT before. The second proof of the CLT uses two applications of L'Hospital's Rule to complete the proof.
Wrt 2 above, the first version is Fisherian in nature (especially good for those teaching BCCalc) and the second version is Neyman- Pearsonian in nature and good for those who were introduced to math-stat from books with such an approach.
Thus, feel free to let me know off-List if you'd like any of the proofs. (Suggestion: You should request one at a time and request another one if you were comfortable following the one requested.)
Wrt 3 above, in a way, such might seem like an entire math prob-and- stat course by itself, and so I'm holding it off for a while. (If anyone else wants to read and comment on it, then let me know off- List and I'll forward a copy to you --- btw, it runs six pages, even though the above theorem  and proof of such itself run only six lines...)
-- David Bee
PS: Please allow for some time for me to respond. (I should be able to do so within the same day.)