Proposition 5.16.158EFDC4A2C19E

Proof. By the divisorial multiplier-ideal estimate, for every positive integer $k$, \[k\nu(\varphi,E)-A_X(E)<\operatorname{ord}_E\mathcal I(k\varphi)\leq k\nu(\varphi,E).\] The log discrepancy $A_X(E)$ is a fixed finite number on a fixed smooth model carrying $E$. Divide by $k$ and let $k\to\infty$. Both bounds tend to $\nu(\varphi,E)$, giving the formula. The order is that of the pulled-back ideal on the model; it is independent of its chosen generators and of a further model, as proved in the same estimate.