Let $K$ be a number field, $\Bbb I_K$ its idèles, and $C_K := \Bbb I_K/K^\times$ the idèle class group of $K$. For an extension $L/K$ of number fields define the Artin map $\theta_{L/K} : \Bbb I_K \to \operatorname{Gal}(L/K)$ by sending an idèle $(a_v)$ of $K$ to $\prod_v \theta_{L^v/K_v}$, whereby $\theta_{L^v/K_v}$ denotes the local Artin map (and $L^v$ denoting the unique extension of $K_v$ corresponding to primes of $L$ lying over $v$, these all being equivalent since $L/K$ is Galois).
It's my understanding that one first proves that $\theta_{L/K}$ induces an isomorphism $C_K/N_{L/K} C_L \cong \operatorname{Gal}(L/K)$ ($N_{L/K}$ being the norm) in the case where $L/K$ is a cyclic extension and uses this to extend to any abelian extension of number fields by using that Artin reciprocity for linearly disjoint extensions implies Artin reciprocity for their compositum.
My question is: Is the final claim (Reciprocity for linearly disjoint extensions implies reciprocity for composita) easily proven? If not, can somebody point me to a text in which it is proven, since most seem to just skim over this fact!