Milne defines the conductor of an abelian extension $L/K$ to be the smallest modulus $\mathfrak{m}$ s.t. the Artin map factors as
$\psi_{L/K}:I_K^{\mathfrak{m}}\to \textrm{Cl}_\mathfrak{m}(K)\to \textrm{Gal}(L/K)$
I'm trying to understand the class field theoretic proof of the Kronecker-Weber theorem. For the proof to work, one needs to show that given some conductor $\mathfrak{m}$ and its ray class field $L_\mathfrak{m}$, then the abelian extensions of $K$ contained in $L_\mathfrak{m}$ are precisely the fields s.t. their conductor divides $\mathfrak{m}$. Hence, one needs to prove (denoting the conductor of $L/K$ by $\mathfrak{f}(L/K)$) that
$\mathfrak{f}(L/K)\mid \mathfrak{m}\Rightarrow L\subset L_\mathfrak{m}$
Anyone know how to prove this? I've been looking at a few other books, but they seem to define things differently, so the proofs did not translate to Milne's definition, which is what I'm currently trying to understand.
EDIT: This question is about trying to fill in the details in Remark 3.8 on page 155 in his notes: http://www.jmilne.org/math/CourseNotes/CFT.pdf