I am currently studying valuation theory and came across the concept of composite valuation. I think, I proved that the composite valuation is always given by a lexicographic product of the two comparable valuations, but I am not sure whether I made a mistake. It seems too easy that this works out. So let's make the question precise:
Let $K$ be a field and $V$ a valuation ring of $K$ with valuation $v$ and value group $\Gamma$. Let $M' \subseteq V$ be a prime ideal, $V'=V_{M'}$ and $\Delta$ the unique isolated subgroup of $\Gamma$ such that we get a valuation $v':K\setminus \{0\} \to \Gamma / \Delta$ having $V'$ as a valuation ring. Moreover, Let $\bar{V} = V/M'$ the corresponding valuation ring of the residue field $\kappa'$ of $V'$ and let $\bar{v}$ be its valuation.
In this situation, $v$ is written as $v' \circ \bar{v}$ and called the composite of $v'$ and $\bar v$.
Now we can define a map $v_0: V'\setminus \{0\} \to \Gamma/\Delta \times \Delta$ by setting $v_0(a) = (v'(a),\bar v(a+M'))$. Extending multiplicatively to $K \setminus \{0\}$ and endowing $\Gamma / \Delta \times \Delta$ with lexicographic order (strong first component), I think, I proved that $v_0$ is a valuation on $K$ with valuation ring $V$, whence equivalent to $v$.
Is this true? Or where did I make the mistake?
Thank you for your help!