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 $\iota_{_R} : R \to R$ be the identity mapping on $R$.

$\iota_{_R} : \struct{R, \tau} \to \struct{R, \tau}$ is continuous by Identity Mapping is Continuous.

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

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

Lemma 1
From Multiple Rule for Continuous Mappings of Topological Ring:
 * for all $y \in R : y * \iota_R, \,\iota_R * y$ are continuous

Hence:
 * $x * \iota_R, \,\iota_R * x, x^{-1} * \iota_R, \,\iota_R * x^{-1}$ are continuous.

Lemma 2

 * $x^{-1} * \iota_{_R}$ is the inverse mapping of $x * \iota_{_R}$
 * $\iota_{_R} * x^{-1}$ is the inverse mapping of $\iota_{_R} * x$

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.

Consider the composite of $x * \iota_R$ with $x^{-1} * \iota_R$.