Inversion Mapping is Isomorphism from Ordered Abelian Group to its Dual

Theorem
Let $\struct {G, \circ, \preccurlyeq}$ be an ordered Abelian group.

Let $\struct {G, \circ, \succcurlyeq}$ be the dual of $\struct {G, \circ, \preccurlyeq}$.

Let $\phi: \struct {G, \circ, \preccurlyeq} \to \struct {G, \circ, \succcurlyeq}$ be the inversion mapping from $\struct {G, \circ, \preccurlyeq}$ to $\struct {G, \circ, \succcurlyeq}$ defined as:
 * $\forall x \in G: \map \phi x = x^{-1}$

where $x^{-1}$ denotes the inverse of $x$ in $\struct {G, \circ$.

Then $\phi$ is an isomorphism.