Problem: Let $f,g \in \mathbb{C}[[\mathrm{x}]] = \mathbb{C}[[x_1,\dots,x_n]]$. Show that if $f \sim^{\mathcal{R}} g$ then $\text{ord } (f) = \text{ord } (g)$.
My attempt: Follow the definition, we have $f \sim^{\mathcal{R}} g$ if there exists a $\mathbb C$-algebra isomorphism $\varphi \colon \mathbb{C}[[\mathrm{x}]] \rightarrow \mathbb{C}[[\mathrm{x}]]$ such that $\varphi(f) = g$. Furthermore, suppose $\text{ord}(f) = b$. We have $g$ is $x_n$-general of order $b$, $\varphi$ is an isomorphism so $\text{ord} (g) = b$.
Is my proof correct? Thank all!