I am reading Algebraic Number Theory by Cassel, Frohlich. I have a question about the proof that norm residue map maps uniformizing elements to Frobenius elements in unramified extension. This is in the chapter of local class field theory by Serre. More formally, the proposition is
Let $L/K$ be an unramified extension of local fields with degree $n$ and let $f$ be the Frobenius element of $G(L/K)$ the Galois group. Then for every $\alpha\in K^\times$, we have $$(\alpha, L/K)=f^{v(\alpha)}$$ where $v$ is the additive valuation of $K$ and $p$ is a uniformizing element of $K$, that is, $v(\pi)=1$.
The proof proceeds as below. First, it suffices that for every $\chi\in \text{Hom}(G(L/K),\mathbb{Q}/\mathbb{Z})\cong H^1(G(L/K),\mathbb{Q}/\mathbb{Z})$ we have $\chi((\alpha,L/K))=\chi(f^{v(\alpha)})$. By a proposition proven before, we have $\chi((\alpha,L/K))=\text{inv}(\bar{\alpha}\smile\delta_\chi)$.
Let me explain all the symbols here. $\text{inv}:H^2(G(K_s/K),K_s^\times)\to \mathbb{Q}/\mathbb{Z}$ is the invariant isomorphism where $K_s$ is the separable closure of $K$. $\delta_\chi$ is the image of $\chi$ under the connecting homomorphism $$H^1(G(L/K),\mathbb{Q}/\mathbb{Z})\to H^2(G(L/K),\mathbb{Z})$$ induced by the short exact sequence $0\to \mathbb{Z}\to \mathbb{Q}\to \mathbb{Q}/\mathbb{Z}\to 0$. Finally, $\hat{\alpha}$ denotes the image of $\alpha$ in $\hat{H}^0(G(L/K),L^\times)\cong K^\times/N(L^\times)$ and $\smile$ denotes the cup product $$\hat{H}^0(G(L/K),L^\times)\otimes H^2(G(L/K),\mathbb{Z})\to H^2(G(L/K),L^\times).$$
Then he claimed that $v(\bar{\alpha}\smile \delta_\chi)=v(\alpha)\smile \delta_\chi$, but I don't see why this is true. I tried to check it explicitly following the construction of cup products and connecting homomorphisms but I still don't see why this holds. In Serre's book Local Fields, he made the same proof and he said this claim is clear, so I guess maybe I just miss something silly.
My question looks a bit technical and I will really appreciate any help :)