Inversion Mapping is Mapping

Theorem

Let $\struct {G, \circ}$ be a group.

Let $\iota: G \to G$ be the inversion mapping on $G$.

Then $\iota$ is indeed a mapping.

To show that $\iota$ is a mapping, it is sufficient to show that:

$\map \iota a \ne \map \iota b \implies a \ne b$:

$\ds \map \iota a$

$\ne$

$\ds \map \iota b$

$\ds \leadsto \ \ $

$\ds a^{-1}$

$\ne$

$\ds b^{-1}$

Definition of $\iota$

$\ds \leadsto \ \ $

$\ds a \circ a^{-1} \circ b$

$\ne$

$\ds a \circ b^{-1} \circ b$

$\ds \leadsto \ \ $

$\ds e \circ b$

$\ne$

$\ds a \circ e$

$\ds \leadsto \ \ $

$\ds b$

$\ne$

$\ds a$

$\blacksquare$