Equal value groups preserve valuation rank

A nonzero $z\in K$ outside $V$ has $z^{-1}\in {\mathfrak{m}}_V$: otherwise $z^{-1}$ would be a unit of $V$. Localness sends this inverse into ${\mathfrak{m}}_W$, so $z$ cannot belong to $W$. Hence $W\cap K=V$. The map $\varphi$ is well defined since units of $V$ remain units of $W$, and it is injective since $V\cap W^{\times }=V^{\times }$. The equality $W\cap K=V$ shows that $\varphi (v_V(z))\ge 0$ if and only if $v_V(z)\ge 0$, so $\varphi$ preserves and reflects order. Its assumed surjectivity makes it an ordered-group isomorphism. Every nonzero $w\in W$ now has the same value as some nonzero $a\in V$: surjectivity first gives $a\in K^{\times }$, and order reflection puts $a$ in $V$. Thus $w\mathrm{/}a\in W^{\times }$. Consequently every ideal $J$ of $W$ satisfies $J=(J\cap V)W$, since each nonzero $w\in J$ is a unit times such an $a\in J\cap V$. Here $IW$ denotes finite sums of products $aw$ with $a\in I,w\in W$, and $J\cap V$ denotes contraction.

For a prime ideal $\mathfrak{p}$ of $V$, show $(\mathfrak{p}W)\cap V=\mathfrak{p}$. If $0\mathrm{\ne }x\in (\mathfrak{p}W)\cap V$, write $x={\sum }_{i=1}^n{a}_i{w}_i$ with $a_i\in \mathfrak{p},w_i\in W$. Some $a_i$ is nonzero; choose a nonzero $a_j$ with least value among this finite list. Then each $a_i\mathrm{/}{a}_j\in V$ (including zero coefficients), so $x\mathrm{/}{a}_j\in W\cap K=V$, and $x\in \mathfrak{p}$. The reverse containment is immediate. In particular $\mathfrak{p}W$ is proper. If $w{w}^{\mathrm{\prime }}\in \mathfrak{p}W$ with $w,w^{\mathrm{\prime }}\mathrm{\ne }0$, step 1.1 writes them as units times $a,a^{\mathrm{\prime }}\in V$. Thus $a{a}^{\mathrm{\prime }}\in \mathfrak{p}W\cap V=\mathfrak{p}$, so $a\in \mathfrak{p}$ or $a^{\mathrm{\prime }}\in \mathfrak{p}$, and $w\in \mathfrak{p}W$ or $w^{\mathrm{\prime }}\in \mathfrak{p}W$. A zero factor already lies in this ideal. Hence $\mathfrak{p}W$ is prime. Contraction of a prime of $W$ is prime by the product test and preservation of $1$. Step 1.1 and the contraction equality give inverse maps, each preserving inclusion; they therefore preserve strict chains and their lengths. Taking suprema proves equality of ranks, also for fields and for infinite rank. Only finitely many witnesses at a time were chosen.