# Expectation conditional on a sum of random variables

The setting is a simple OLS regression where the true model has regressor $$x$$ and error term $$u$$, but we can only measure $$\bar{x}=x+v$$ where $$v$$ is iid with mean 0.

According to the textbook:

$$\mathbb{E}(u|x)=0$$ is satisfied, but $$\mathbb{E}(\bar{u}|\bar{x})=\mathbb{E}(u-\beta x|x+v)=-\beta v$$

I can see that $$x$$ and $$u$$ are correlated, but why am I allowed to just take $$\beta x$$ out of the expectation when I "only" condition on the sum $$x+v$$, and not explicitly on $$v$$?