Let $K$ be a number field, and write $\mathbb{A}_{K,f}^\times$ for the group of finite ideles of $K$. That is $$ \mathbb{A}_{K,f}^\times = \{(u_v)_v \in \prod_{v \nmid \infty} K_v^\times : v(u_v) = 0 \text{ for all but finitely many $v$}\}, $$ where the product is over all finite places of $K$. For each ideal $I$ of $K$, define $$ U_{K,I} = \{u \in \mathbb{A}_{K,f}^\times : v(u_v) = 0 \text{ and } v(u_v - 1) \geq v(I)\text{ for all finite places $v$}\}. $$ Then I think that the ray class group of $K$ modulo $I$ is isomorphic to the double quotient $$ K^\times \backslash \mathbb{A}_{K,f}^\times / U_{K,I}, $$ but I have not been able to find a reference. Of course, there are lots of references for the analogous statement without the finiteness (indeed many authors define ray class groups that way). It's not too hard to deduce the statement I want using the definitions, but ideally I'd like a quick reference without having to write out a proof myself.