External Direct Product Inverses/Necessary Condition

Theorem
Let $\struct {S \times T, \circ}$ be the external direct product of the two algebraic structures $\struct {S, \circ_1}$ and $\struct {T, \circ_2}$.

Let $\tuple {s^{-1}, t^{-1} }$ be an inverse of $\tuple {s, t} \in \struct {S \times T, \circ}$.

Then:
 * $s^{-1}$ is an inverse of $s \in \struct {S, \circ_1}$

and:
 * $t^{-1}$ is an inverse of $t \in \struct {T, \circ_2}$.

Proof
Let $\tuple {e_S, e_T}$ be the identity element of $\struct {S \times T, \circ}$.

Let $\tuple {s^{-1}, t^{-1} }$ be an inverse element of $\tuple {s, t} \in \struct {S \times T, \circ}$.

Then we have:

and:

Thus $s^{-1}$ and $t^{-1}$ are inverse elements of $s \in \struct {S, \circ_1}$ and $t \in \struct {T, \circ_2}$ respectively.

Also see

 * External Direct Product Associativity
 * External Direct Product Commutativity
 * External Direct Product Inverses