Zero of Cardinal Product is Zero

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Theorem

Let $\mathbf a$ be a cardinal.


Then:

$\mathbf 0 \mathbf a = \mathbf 0$

where $\mathbf 0 \mathbf a$ denotes the product of the (cardinal) zero and $\mathbf a$.


That is, $\mathbf 0$ is the zero element of the product operation on cardinals.


Proof

Let $\mathbf a = \operatorname{Card} \left({A}\right)$ for some set $A$.

From the definition of (cardinal) zero, $\mathbf 0$ is the cardinal associated with the empty set $\varnothing$.

We have by definition of product of cardinals that $\mathbf 0 \mathbf a$ is the cardinal associated with $\varnothing \times A$.

But from Cartesian Product is Empty iff Factor is Empty:

$\varnothing \times A = \varnothing$

Hence the result.

$\blacksquare$


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