I'm reading chapter 4 of Gerard Murphy's C*-algebras book and am confused by a statement in his proof of theorem 4.1.10. In his proof, he says, "$A'$ is the linear span of its unitaries" (where $A'$ denotes the commutant of $A$). I can't find any reference to why this is true anywhere earlier in the chapter. Am I missing something, or is this referring to some other result?

For context, 4.1.10 states:

Let $A$ be a von Neumann algebra on a Hilbert space $H$ and $v$ an element of $A$ with polar decomposition $v = u|v|$. Then $u \in A$.