Multiplicative Regular Representations of Units of Topological Ring are Homeomorphisms/Lemma 1

Theorem
Let $\struct{R, +, \circ}$ be a ring.

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

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:
 * $\forall z \in R: \map {\paren{\iota_{_R} * y}} z = \map {\iota_{_R}} z * y$

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

Proof
Let $y \in R$.

From Equality of Mappings, $\lambda_y = y * \iota_{_R}$.

From Equality of Mappings, $\rho_y = \iota_{_R} * y$.