The Complex Plane as $\mathbb{R}^2$: the Map $(x,y)\mapsto x+iy$ Is a Real-Linear Isometric Bijection

In the setting of Complex Analysis: Standing Notation and of Multivariable Calculus: Notation, with $n=2$, let $\Phi:\mathbb{R}^{2}\to\mathbb{C}$ be the function $\Phi(x,y)=x+iy$ (Values, Restriction and Equality of a Map, and the Map $u\mapsto t(u)$ as a Defined Set Symbol (v)).

(i) Bijection. $\Phi$ is a bijection of $\mathbb{R}^{2}$ onto $\mathbb{C}$, with inverse

$$\Phi^{-1}(z)=(\operatorname{Re}z,\operatorname{Im}z)\qquad(z\in\mathbb{C}).$$

(ii) Real linearity. For all $p,q\in\mathbb{R}^{2}$ and $t\in\mathbb{R}$, $\Phi(p+q)=\Phi(p)+\Phi(q)$ and $\Phi(tp)=t\,\Phi(p)$.

(iii) Isometry. For all $p,q\in\mathbb{R}^{2}$,

$$|\Phi(p)-\Phi(q)|=\lVert p-q\rVert=d_{E}(p,q).$$

(iv) Open sets. A set $W\subseteq\mathbb{C}$ is open in $(\mathbb{C},d_{\mathbb{C}})$ if and only if $\{p\in\mathbb{R}^{2}:\Phi(p)\in W\}$ is open in $(\mathbb{R}^{2},d_{E})$, the openness of Multivariable Calculus: Notation (ii).