Let be $\mathfrak X$ a indexed collection over a set $I$: so it is a well know result that if $*_i$ is for each $i$ in $I$ an operation on $X_i$ in $\mathfrak X$ then the equality $$ (a*b)(i):=a(i)*_i b(i) $$ with $a,b\in\prod_{i\in I}X_i$ defines an operation in $\prod_{i\in I}X_i$; moreover, if $Y$ is a set with an operation $+$ then it is well know that the equality $$ U\oplus V:=\big\{y\in Y:y=u+v\,\text{with}\,(u,v)\in U\times V\big\} $$ defines an operation on $Y$ so thatlet be $\star$ and $\star_i$ with $i\in I$ respectively the corresponding operation on $\prod_{i\in I}X_i$ and on $X_i$ induced by $*$ and by $*_i$.
So If $A_i$ and $B_i$ are subsets of $X_i$ for each $i\in I$ then I would like to prove or disprove if the equality \begin{equation} \tag{0}\label{0}\left(\prod_{i\in I}A_i\right)\star\left(\prod_{i\in I}B_i\right)=\prod_{i\in I}(A_i\star_i B_i) \end{equation} holds.
So if $x$ is in $\left(\prod_{i\in I}A_i\right)\star\left(\prod_{i\in I}B_i\right)$ then there exists $a$ in $\prod_{i\in I}A_i$ and $b$ in $\prod_{i\in I}B_i$ such that $$ x=a*b $$ so that if $\big(a(i)\big)*_i\big(b(i)\big)$ is in $A_i\star_i B_i$ for each $i$ in $I$ then by the equality $$ x(i)=\big(a(i)\big)*_i\big(b(i)\big) $$ we conclude that $x$ is in $\prod_{i\in I}(A_i\star_i B_i)$ for each $i$ in $I$ and so the inclusion \begin{equation} \tag{1}\label{1}\left(\prod_{i\in I}A_i\right)\star\left(\prod_{i\in I}B_i\right)\subseteq\prod_{i\in I}(A_i\star_i B_i) \end{equation} holds. However if $x$ is in $\prod_{i\in I}(A_i\star_i B_i)$ then $x(i)$ is in $A_i\star B_i$ for each $i$ in $I$ and so there exist $a_i$ in $A_i$ and $b_i$ in $B_i$ such that the equality $$ x(i)=a_i*_i b_i $$ holds and so putting $$ a(i):=a_i\quad\text{and}\quad b(i):=b_i $$ with $i$ in $I$ we define an element $a$ of $\prod_{i\in I}A_i$ and an element $b$ of $\prod_{i\in I}B_i$ such that $$ x=a*b $$ and this proves the inclusion \begin{equation} \tag{2}\label{2}\prod_{i\in I}(A_i\star_i B_i)\subseteq\left(\prod_{i\in I}A_i\right)\star\left(\prod_{i\in I}B_i\right) \end{equation} holds. Finally by incl. \eqref{1} and incl. \eqref{2} we conclue that eq. \eqref{0} is true.
Moreover if $Y$ is a subset of $\prod_{i\in I}X_i$ then it seem to me that only the inclusion \begin{equation} \tag{3}\label{3}Y\star\prod_{i\in I}A_i\subseteq\prod_{i\in I}\big(\pi_i[Y]\star_i A_i\big) \end{equation} generally holds since generally only the inclusion \begin{equation} \tag{4}\label{4}Y\subseteq\prod_{i\in I}\pi_i[Y] \end{equation} holds.
Well unfortunately I found worth of eq. \eqref{0} by myself so that I would like to know if it is actually true and so if I well proved it; moreover I would like to understand if eq. \eqref{3} is true or false: so could someone help me, please?