a cardinal notation

Question

How it follows here from $\aleph_\delta^{\operatorname {cf}(\delta)}<\aleph_ {(|\delta|^{\operatorname {cf}(\delta)})^+}$ that $2^{\aleph_\omega}<\aleph_{(2^{\aleph_0})^+}$ and how was strong limiteness of $\aleph_\omega$ used ? I've just tried to plug into the first formula $\omega$ for $\delta$ but the result is still unclear to me.