Exercise (Lee Smooth Manifolds Exercise 14.22): Let $M$ be a smooth manifold and $X$ be a smooth vector field on $M$, $\omega$ a smooth differential $k$-form. $\omega \in \Omega^k(M)$. Show, that $i_X\omega$ (interior product) is smooth.
Please check if my solution is correct. This is my first time dealing with differential forms and I'm very unsure if I'm using them correctly.
My solution: Let $(U,\varphi)$ be a chart in $M$. In these coordinates $$X = \sum X^i\frac{\partial}{\partial \varphi^i}$$ $$\omega = \sum_I ' \omega^I d\varphi^I$$ To show that $i_X\omega\in \Omega^{k-1}(M)$ is smooth, it is enough to show that each coefficient $a^I$ of the summands in $i_X\omega = \sum_I ' a^I d\varphi^I$ vary smoothly. For $J=(j_2,j_3,\dots, j_k)$ We have $$a^J(p) = i_X\omega \left(\frac{\partial}{\partial \varphi^{j_2}}\Bigg|_p, \dots , \frac{\partial}{\partial \varphi^{j_k}}\Bigg|_p \right)=$$ $$=\sum_I ' \omega^I(p) d\varphi^I \left(X_p, \frac{\partial}{\partial \varphi^{j_2}}\Bigg|_p, \dots , \frac{\partial}{\partial \varphi^{j_k}}\Bigg|_p \right)=$$ $$=\sum_I ' \omega^I(p) \det \begin{pmatrix} d\varphi^{i_1}(X_p) & d\varphi^{i_1}(\frac{\partial}{\partial \varphi^{j_2}}) & \cdots & d\varphi^{i_1}(\frac{\partial}{\partial \varphi^{j_k}}) \\ d\varphi^{i_2}(X_p) & d\varphi^{i_2}(\frac{\partial}{\partial \varphi^{j_2}}) & \cdots & d\varphi^{i_2}(\frac{\partial}{\partial \varphi^{j_k}}) \\ \vdots & \vdots & \ddots & \vdots \\ d\varphi^{i_k}(X_p) & d\varphi^{i_k}(\frac{\partial}{\partial \varphi^{j_2}}) & \cdots & d\varphi^{i_k}(\frac{\partial}{\partial \varphi^{j_k}}) \end{pmatrix}$$
Every entry of the determinant is smooth, hence we have shown $i_X\omega$ is smooth.