properties of cardinal numbers

Theorem.Let ${\left(c_{\gamma }\right)}_{\gamma \in \mathrm{\Gamma }}$ be an indexed family of cardinal numbers indexed by a nonempty index set $\mathrm{\Gamma }$. Also, let ${\left({\mathrm{\Gamma }}_{\delta }\right)}_{\delta \in \mathrm{\Delta }}$ be an arbitrary indexed partition of the index set. Then we have the following properties:

1. Associative Laws.

$$\sum _{\gamma \in \mathrm{\Gamma }}{c}_{\gamma }=\sum _{\delta \in \mathrm{\Delta }}\sum _{\gamma \in {\mathrm{\Gamma }}_{\delta }}{c}_{\gamma }$$

and

$$\prod _{\gamma \in \mathrm{\Gamma }}{c}_{\gamma }=\prod _{\delta \in \mathrm{\Delta }}\prod _{\gamma \in {\mathrm{\Gamma }}_{\delta }}{c}_{\gamma }.$$

2. Commutative Laws. Let $\pi :\mathrm{\Gamma }\to \mathrm{\Gamma }$ be a partition. Then

$$\sum _{\gamma \in \mathrm{\Gamma }}{c}_{\gamma }=\sum _{\gamma \in \mathrm{\Gamma }}{c}_{\pi (\gamma )}$$

and

$$\prod _{\gamma \in \mathrm{\Gamma }}{c}_{\gamma }=\prod _{\gamma \in \mathrm{\Gamma }}{c}_{\pi (\gamma )}.$$

3. Distributive Laws. Let $a$ be any arbitrary infinite cardinal number. Then

$$a\left(\sum _{\gamma \in \mathrm{\Gamma }}{c}_{\gamma }\right)=\sum _{\gamma \in \mathrm{\Gamma }}a{c}_{\gamma }$$