Consider the stochastic process $\{Z_n - n\mu, n\geq 1\}$, where $Z_n \equiv \sum_{i=1}^n X_i$ and $X_1, X_2, \dots$ are independent random variables with mean $\mu$. As \begin{align}\mathbf{E}[Z_{n+1} - (n+1)\mu | X_1, X_2, \dots, X_n] &= \mathbf{E}[Z_{n+1} | X_1, X_2,\dots, X_n] - (n+1)\mu \\ &= Z_n +\mu -(n+1)\mu\\ &=Z_n - n\mu,\end{align} $\{Z_n - n\mu, n\geq 1\}$ is a martingale with respect to $\{I_n\}$, where $I_n \equiv \{X_1, X_2, \dots, X_n\}$.
This is an example in my textbook. Since I’m self-learning I want to be absolutely clear about the step between the first and the second line in the derivation.
I assume that from the linearity of the conditional expectation we have $$\mathbf{E}[Z_{n+1} - (n+1)\mu | X_1, X_2, \dots, X_n] = \mathbf{E}[Z_{n}| X_1, X_2, \dots, X_n] + \mathbf{E}[X_{n+1} | X_1, X_2, \dots, X_n] - (n+1)\mu$$ where we use that the second term $\mathbf{E}[X_{n+1}|X_1, X_2, \dots, X_n] = \mu$, because the random variables are independent (their results give no information). Is this correct?