Proposition 18.12.284E9AB1A05B4

Proof. Set $q_ F=P_{\theta ,F}[u]_{\mathcal I}$. Translation of competitors gives

so $q_ F\sim q$. The identity $q_ F-F=P_{\theta +\mathrm{dd}^{\mathrm{c}}F}[u-F]_{\mathcal I}$ and theorem 18.4.2 give $q_ F\sim _{\mathcal I}u$. Since $q_ F\leq F$ and $q_ F\sim q$, it is a competitor for $P_{\theta ,F}[q]$. Conversely, the maximum of a competitor for $P_{\theta ,F}[q]$ with a sufficiently low translate of $q$ is a competitor for $P_{\theta ,F}[q]_{\mathcal I}$, and this ideal envelope equals $q_ F$ because $q\sim _{\mathcal I}u$. These bounds give

The last assertion follows from proposition 16.28.65, which gives $P_{\theta ,F}[u]=P_{\theta ,F}[q]$. $\square$