I want to proof that :
$\prod_D$GL$_n$($\Bbb F$)$\cong$GL$_n$($\prod_D$$\Bbb F$)
for any ultrafilter D and any field $\Bbb F$.
I was trying to crate an isomorphism between the two stractues as follows :
Let $\varphi$ : $\prod_D$GL$_n$($\Bbb F$)$\to$GL$_n$($\prod_D$$\Bbb F$)
where For any A $\in$ $\prod_D$GL$_n$($\Bbb F$): $\varphi$(A):= [$\varphi$(A$_i$)].
where [$\varphi$(A$_i$)] represents the equivalence class of the matrix $\varphi$(A$_i$) under the ultrapower construction. Here, $\varphi$(A$_i$) denotes the matrix obtained by applying the mapping $\varphi$ entry-wise to each matrix A$_i$.
To show that $\varphi$ is well-defined, we need to verify that the choice of representative {A$_i$} does not affect the equivalence class $\varphi$(A$_i$). Suppose we have another representative {B$_i$} for A, where B$_i$ $\in$ GL$_n$($\Bbb F$). Then, for every σ $\in$ D, we have B$_i$σ ≈ A$_i$σ (equivalent matrices). Since $\varphi$ is an entry-wise mapping, it follows that $\varphi$(B$_i$σ) = $\varphi$(A$_i$σ) for every σ $\in$ D. Therefore, [$\varphi$(A$_i$σ)] = [$\varphi$(B$_i$σ)], and $\varphi$(A) is well-defined.
Now I am going to show that $\varphi$ is a group homorphism. Let A,B $\in$ $\prod_D$GL$_n$($\Bbb F$). We have: $\varphi$(A·B) = [$\varphi$(A$_i$·B$_i$)] = [$\varphi$(A$_i$)·$\varphi$(B$_i$)] = [$\varphi$(A$_i$)]·[$\varphi$(B$_i$)] = $\varphi$(A)·$\varphi$(B), where the first equality follows from the definition of matrix multiplication, and the rest follows from the properties of ultrapowers and matrix multiplication.
To show that $\varphi$ is bijecntion I thought about definding an inverse map $\psi$ :GL$_n$($\prod_D$$\Bbb F$)$\to$$\prod_D$GL$_n$($\Bbb F$) as follows : $\psi$(B) = [$\psi$(B$_i$)], where [$\psi$(B$_i$)] represents the equivalence class of the matrix $\psi$(B$_i$) under the ultrafilter D. Here, $\psi$(B$_i$) denotes the matrix obtained by applying the inverse mapping $\psi$ entry-wise to each matrix B$_i$. Then I would show that $\psi$ is a group hompmorphism and $\varphi$ inverse hence $\varphi$ is an isomorphsim between the two structes hence: $\prod_D$GL$_n$($\Bbb F$)$\cong$GL$_n$($\prod_D$$\Bbb F$)
Is my proof valid? Thanks!