Normality is direct product-closed

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 $$aba^{-1} \in H$$.

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

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