Homomorphism from Reals to Circle Group

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Theorem

Let $\left({\R, +}\right)$ be the Additive Group of Real Numbers.

Let $\left({K, \times}\right)$ be the circle group.

Let $\phi: \left({\R, +}\right) \to \left({K, \times}\right)$ be the mapping defined as $\phi \left ({x}\right) = e^{i x}$.

Let $x, y \in \R$. Then:

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle \phi \left ({x}\right) \times \phi \left ({y}\right)$	$=$	$\displaystyle e^{i x} e^{i y}$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle e^{i \left({x + y}\right)}$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \phi \left({x + y}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $

$\blacksquare$

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \phi \left ({x}\right) \times \phi \left ({y}\right)\)	\(=\)	\(\displaystyle e^{i x} e^{i y}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle e^{i \left({x + y}\right)}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \phi \left({x + y}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)