Cardinal Product Distributes over Cardinal Sum

Theorem
Let $\mathbf a$, $\mathbf b$ and $\mathbf c$ be cardinals.

Then:
 * $\mathbf a \paren {\mathbf b + \mathbf c} = \mathbf a \mathbf b + \mathbf a \mathbf c$

where:
 * $\mathbf a + \mathbf b$ denotes the sum of $\mathbf a$ and $\mathbf b$.
 * $\mathbf a \mathbf b$ denotes the product of $\mathbf a$ and $\mathbf b$.

Proof
Let $\mathbf a = \card A$, $\mathbf b = \card B$ and $\mathbf c = \card C$ for some sets $A$, $B$ and $C$.

Let $B$ and $C$ be pairwise disjoint, that is:
 * $B \cap C = \O$

Then we can define:
 * $B \sqcup C := B \cup C$

where $B \sqcup C$ denotes the disjoint union of $B$ and $C$.

Then we have:
 * $\mathbf b + \mathbf c = \card {B \sqcup C} = \card {B \cup C}$

Then:

Then:

Then: