Natural Number Addition Commutes with Zero

Theorem
Let $\N$ be the natural numbers.

Then:
 * $\forall n \in \N: 0 + n = n = n + 0$

Proof
Proof by induction:

From definition of addition:

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

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

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, where $k \ge 0$, then it logically follows that $\map P {k^+}$ is true.

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

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

Induction Step
This is our induction step:

By definition:
 * $k^+ + 0 = k^+$

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

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