Product with Matrix Exponential of Commutative Matrices

Theorem
Let $\mathbf A$ and $\mathbf B$ be square matrices of order $m$ for some $m \in \Z_{\ge 1}$.

Let $t \in \R$ be a real number.

Let $e^{\mathbf A t}$ denote the matrix exponential of $\mathbf A$.

Let $\mathbf A \mathbf B = \mathbf B \mathbf A$.

Then:
 * $e^{\mathbf A t} \mathbf B = \mathbf B e^{\mathbf A t}$

Proof
Let:
 * $\map \Phi t = e^{\mathbf A t} \mathbf B - \mathbf B e^{\mathbf A t}$