Multiplicative Regular Representations of Units of Topological Ring are Homeomorphisms

Theorem
Let $\struct{R, +, \circ, \tau}$ be a topological ring with unity $1_R$.

For all $y \in R$, let $\lambda_y$ and $\rho_y$ denote the left and right regular representations of $\struct{R, \circ}$ with respect to $y$.

Let $x \in R$ be a unit of $R$ with product inverse $x^{-1}$.

Then $\lambda_x, \, \rho_x: \struct{R, \tau} \to \struct{R, \tau}$ are homeomorphisms with inverse mappings $\lambda_{x^{-1} }, \, \rho_{x^{-1} }: \struct{R, \tau} \to \struct{R, \tau}$ respectively.

Proof
Let $I_{_R} : R \to R$ be the identity mapping on $R$.

For all $y \in R$, let $y * I_{_R} : R \to R$ be the mapping defined by:
 * $\forall z \in R: \map {\paren {y * I_{_R} } } z = y * \map {I_{_R}} z$

For all $y \in R$, let $I_{_R} * y : R \to R$ be the mapping defined by:
 * $\forall z \in R: \map {\paren {I_{_R} * y}} z = \map {I_{_R}} z * y$

Lemma 1
From Identity Mapping is Continuous, $I_{_R} : \struct{R, \tau} \to \struct{R, \tau}$ is continuous.

From Multiple Rule for Continuous Mappings into Topological Ring:
 * $x * I_{_R}, \, I_{_R} * x, x^{-1} * I_{_R}, \,I_{_R} * x^{-1}$ are continuous.

Lemma 2
By definition of homeomorphism, $\lambda_x, \,\rho_x : \struct{R, \tau} \to \struct{R, \tau}$ are homeomorphisms with inverse mappings $\lambda_{x^{-1} }, \,\rho_{x^{-1} }  : \struct{R, \tau} \to \struct{R, \tau}$ respectively.