External Direct Product Inverses/General Result

Theorem
Let $\displaystyle \left({S, \circ}\right) = \prod_{k \mathop = 1}^n S_k$ be the external direct product of the algebraic structures $\left({S_1, \circ_1}\right), \left({S_2, \circ_2}\right), \ldots, \left({S_n, \circ_n}\right)$.

Let $\left({x_1, x_2, \ldots, x_n}\right) \in S$.

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

Then $\left({y_1, y_2, \ldots, y_n}\right)$ is the inverse of $\left({x_1, x_2, \ldots, x_n}\right) \in S$ in $\left({S, \circ}\right)$.

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

Let $x := \left({x_1, x_2, \ldots, x_n}\right)$.

Let $y := \left({x_1, x_2, \ldots, x_n}\right)$.

From External Direct Product Identity, $e := \left({e_1, e_2, \ldots, e_n}\right)$ is the identity element of $S^n$.

Then:

and: