I am reading Tu's book "An introduction to Manifolds". Specifically, I am reading about the the exterior algebra of multicovectors (subchapter 1.3) and I am following a proof stating that the wedge product is anticommutative (i.e. proposition 3.21). So, I will state the steps followed by the book and then I will explain my concerns.
If $f\in A_k(V)$ and $g\in A_l(V)$, then we define $\tau\in S_{k+l}$ to be the permutation $$\tau=\begin{bmatrix} 1 & ... & l & l+1 & ... & l+k\\ k+1 & ... & k+l & 1 & ... & k \end{bmatrix}$$
And we will try to prove that $A(f\otimes g)(v_1,...,v_{k+l})=\text{sgn}(\tau)A(g\otimes f)(v_1,...,v_{k+l})$ hence showing that the wedge product is commutative as well. We start with the definition of the alternating operator $$A(f\otimes g)(v_1,...,v_{k+l})=\sum_{\sigma\in S_{k+l}} \text{sgn}(\sigma)f(v_{\sigma(1)},...v_{\sigma(l)})g(v_{\sigma(k+1)},...v_{\sigma(k+l)}) \\= \sum_{\sigma\in S_{k+l}} \text{sgn}(\sigma) f(v_{\sigma\tau(l+1)},...v_{\sigma\tau(l+k)})g(v_{\sigma\tau(1)},...v_{\sigma\tau(l)}) \\=\text{sgn}(\tau) \sum_{\sigma\in S_{k+l}} \text{sgn}(\sigma\tau) g(v_{\sigma\tau(1)},...v_{\sigma\tau(l)})f(v_{\sigma\tau(l+1)},...v_{\sigma\tau(l+k)}) \\= \text{sgn}(\tau)A(g\otimes f)(v_1,...,v_{k+l})$$ where in the second step the definition of $\tau$ was used and then from the second to third step lies my question: why is the author allowed to swap the order of the functions $f$ and $g$? I take it that the wedge product is not commutative, otherwise it would be stated somewhere in the book, right? So, is he allowed to change the order of the two functions because they are inside the sum? Some help would be appreciated
The remaining proof is okay, since the author simply identifies the result with the alternating operator, acting on the tensor product of $g$ and $f$, according to the definition...
Thanks in advance.