I'm stuck with the first exercise of chapter 9 from Jech and Hrbacek Introduction to set theory. It states:
If $J_i\,(i\in I)$ are mutually disjoint sets and $J=\bigcup_{i\in I}J_i$, and if $\kappa_j\,(j\in J)$ are cardinals, then $$\sum_{i\in I}\left(\sum_{j\in J_i}\kappa_j\right)=\sum_{j\in J}\kappa_j$$
Any help will be very appretiated.
Advanced greetings.
Set $A_{j}=\kappa_{j}\times\{j\}$ for $j\in J$.
Then one can write $$\sum_{i\in I}\left(\sum_{j\in J_{i}}\kappa_j\right)=\sum_{i\in I}\left|\bigcup_{j\in J_{i}}A_j\right|$$ And $$\sum_{i\in I}\left|\bigcup_{j\in J_{i}}A_j\right|=\left|\bigcup_{i\in I}\left(\bigcup_{j\in J_i}A_j\right)\right|$$ Using that $(\bigcup_{j\in J_i}A_j)\cap(\bigcup_{k\in J_k}A_k)=\bigcup_{j\in J_i}(A_i\cap(\bigcup_{j\in J_k}A_k))=\emptyset$ because $A_j\cap A_i=\emptyset$ if $i\neq j.$
Then $$\sum_{i\in I}\left(\sum_{j\in J_{i}}\kappa_j\right)=\left|\bigcup_{i\in I}\left(\bigcup_{j\in J_i}A_j\right)\right|=\left|\bigcup_{j\in(\bigcup_{i\in I}J_i)}A_j\right|=\left|\bigcup_{j\in J}A_j\right|=\sum_{j\in J}\kappa_j$$