Question about proof that sub $*$-algebra of $B(H)$ is strongly dense in bicommutant.

Question

Consider following fragment from Murphy's "$C^*$-algebras and operator theory":

In the proof of lemma 4.1.4, why does $u(x) \in K$ follow from $pu = up?$

Answer 1

First, we will show that $u(K) \subseteq K$. Let $y \in K$, then $p(y) = y$ and $$u(y) = u(p(y)) = p(u(y)) \in K.$$

Now, as $id_H \in A$, we have $x = id_H (x) \in K$ from the definition of $A$. Hence, $u(x) \in K$.