Lemma 5.16.17C85C2748AE0B

Proof. For a coherent ideal $\mathfrak a$ near $x$, the order of $\mathfrak a\mathcal O_Y$ along the exceptional divisor of the blowup at $x$ is its ordinary vanishing order at $x$. Indeed, in a blowup chart substitute $z=(t,tw_2,\ldots,tw_n)$ into each Taylor series. Its lowest nonzero homogeneous term factors as $t^d$ times a polynomial in $w$ which is not identically zero on the exceptional divisor, so its generic order there is $d$; taking the minimum over generators proves the ideal assertion.

Apply this to $\mathfrak a=\mathcal I(k\varphi)$. The pointwise order estimate gives $k^{-1}\operatorname{ord}_x\mathcal I(k\varphi)\to\nu(\varphi,x)$, while its divisorial counterpart gives $k^{-1}\operatorname{ord}_E\mathcal I(k\varphi)\to\nu(\varphi,E)$. The two sequences are identical, proving the equality. In dimension one the blowup of the point ideal is an isomorphism and the assertion reduces to the point itself.