External Direct Product Inverses/General Result

Theorem
Let $\ds \struct {\SS, \circ} = \prod_{k \mathop = 1}^n S_k$ be the external direct product of the algebraic structures $\struct {S_1, \circ_1}, \struct {S_2, \circ_2}, \ldots, \struct {S_n, \circ_n}$.

Let $\tuple {x_1, x_2, \ldots, x_n} \in S$.

Let $y_k$ be an inverse of $x_k$ in $\struct {S_k, \circ_k}$ for each of $k \in \N^*_n$.

Then $\tuple {y_1, y_2, \ldots, y_n}$ is the inverse of $\tuple {x_1, x_2, \ldots, x_n} \in \SS$ in $\struct {\SS, \circ}$.

Proof
Let $e_1, e_2, \ldots, e_n$ be the identity elements of $\struct {S_1, \circ_1}, \struct {S_2, \circ_2}, \ldots, \struct {S_n, \circ_n}$ respectively.

Let $x := \tuple {x_1, x_2, \ldots, x_n}$.

Let $y := \tuple {x_1, x_2, \ldots, x_n}$.

From External Direct Product Identity, $e := \tuple {e_1, e_2, \ldots, e_n}$ is the identity element of $\SS$.

Then:

and: