Zero is Zero Element for Natural Number Multiplication

Theorem
Let $\N$ be the natural numbers.

Then $0$ is a zero element for multiplication:
 * $\forall n \in \N: 0 \times n = 0 = n \times 0$

Proof
Proof by induction.

For all $n \in \N$, let $\map P n$ be the proposition:
 * $0 \times n = 0 = n \times 0$

Basis for the Induction
By definition, we have:
 * $0 \times 0 = 0 = 0 \times 0$

Thus $\map P 0$ is seen to be true.

This is our basis for the induction.

Induction Hypothesis
Now we need to show that, if $\map P k$ is true, then it logically follows that $\map P {k + 1}$ is true.

So this is our induction hypothesis $\map P k$:
 * $0 \times k = 0 = k \times 0$

Then we need to show that $\map P {k + 1}$ follows directly from $\map P k$:
 * $0 \times \paren {k + 1} = 0 = \paren {k + 1} \times 0$

Induction Step
This is our induction step:

By definition:
 * $\paren {k + 1} \times 0 = 0$

So $\map P k \implies \map P {k + 1}$, and the result follows by the Principle of Mathematical Induction.

Therefore:
 * $\forall n \in \N: 0 \times n = 0 = n \times 0$