Normality is direct product-closed

This article gives the statement, and possibly proof, of a subgroup property satisfying a subgroup metaproperty
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Statement

Property-theoretic statement

The subgroup property of being a normal subgroup is a direct product-closed subgroup property.

Symbolic statement

Suppose $I$ is a nonempty indexing set, and for each $i\in I$, we have a group-subgroup pair $H_i\le G_i$. Let $G$ be the external direct product of the $G_i$s, and $H$ the subgroup of $G$ obtained as the external direct product of the $H_i$s. Then $H$ is a normal subgroup of $G$.

Proof

Using notation from the symbolic statement.

Let $a\in G,b\in H$. It suffices to show that $ab{a}^{-1}\in H$.

Denote by $a_i,b_i$ the $G_i$-coordinates of $a$ and $b$. Then the $G_i$-coordinate of $ab{a}^{-1}$ is $a_i{b}_i{a}_i^{-1}$.

Since $H_i$ is normal in $G_i$, and $a_i\in G_i,b_i\in H_i$, $a_i{b}_i{a}_i^{-1}$ lies in $H_i$. Hence, the $i^{th}$ coordinate of $ab{a}^{-1}$ is in $H_i$ for each $i$, thus $ab{a}^{-1}\in H$.