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.

From Multiple Rule for Continuous Mappings of Topological Ring:
 * $x * \iota_R, \,\iota_R * x$ are continuous

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

Lemma 1

 * $\forall y \in R : \lambda_y = y * \iota_{_R} \land \rho_y = \iota_{_R} * y$

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 $x^{-1} * \iota_R$.

Now:

From Equality of Mappings, $\lambda^\circ_x = x * \iota_{_R}$.

Similarly, $\rho^\circ_x = \iota_{_R} * x$.

Hence