A remark on projections in von Neumann algebras

Question

Let $M$ be a von Neumann algebra and $e,f$ be projections in $M$. For a given central projection $z\in M$, is the following true?

$$z(e\vee f)=ze\vee zf$$

Answer 1

Assume $M\subset B(H)$.

I think this is pretty straightforward. Looking at the range subspaces, $$ z(e\vee f)H=z(\overline{\text{span}}\{eH\cup fH\})=\overline{\text{span}}\{zeH\cup zfH\}=(ze\vee zf)H. $$