I am trying to prove the explicit, non-recursive formula for the subdivision of an n-simplex, $\operatorname{Sd}_{n}:S_n(\triangle^{n}_{top}) \to S_n(\triangle^{n}_{top})$. I found the exercise in Hatcher (section 2.1, exercise 25.) and tried to do it with the formula from our lecture.
I found out that for $\sigma = \operatorname{conv}(v_0,...,v_n):\:$ $\operatorname{Sd}_{n}(\sigma) = \sum \limits_{\pi \in \sum_{n+1}}sgn(\pi)\sigma^{\pi}$, where $\sigma^{\pi} = \operatorname{conv}(v_0^{\pi},...,v_n^{\pi})$ with $v_i^{\pi} = \frac{v_{\pi(i)}+...+v_{\pi(n)}}{n+1-i}$.
Now I already proved the start of the induction. Using our formula from the lecture, I calculated that: (For coordinates $t$ in an $(n+1)$-simplex):
$Sd_{n+1}(\sigma)(t_0,...,t_{n+1}) = t_0\frac{v_0+...+v_{n+1}}{n+2} + \\\\ \sum \limits_{i = 0}^{n+1}(-1)^{i} \sum \limits_{\pi \in \sum_{n+1}}sgn(\pi) \operatorname{conv}(v_0^{\pi},...,\hat{v}_i,...,v_{n+1}^{\pi})(t_1,...,t_{n+1})$.
Where the hat indicates the entry is left out.
But I don't know how to proceed.
Thanks in advance for any help.