[Math] My proof of $m \cdot 0 = 0 = 0 \cdot m$ for all $m \in \mathbb{Z}$

elementary-number-theoryproof-verification

I have the following proposition to prove:

For all $m \in\mathbb\ Z$, $m \cdot 0 = 0 = 0 \cdot m$

I can use the following axioms:

  1. commutativity
  2. associativity
  3. distributivity
  4. identity for addition ($0$)
  5. identity for multiplication ($1$)
  6. additive inverse
  7. cancellation: Let $m,n,p$ be integers. If $m \cdot n = m \cdot p$ and $m \ne 0$, then $n = p$.

Here is my proof:

\begin{align*}
m \cdot 0 &= m \cdot (m + (-m))\\
m \cdot 0 &= (m \cdot m) + (m \cdot (-m))\\
m \cdot 0 &= (m \cdot m) +(m \cdot -1 \cdot m) \\
m \cdot 0 &= (m \cdot m) +-1 \cdot (m \cdot m) \\
m \cdot 0 &= (m \cdot m) – (m \cdot m) \\
m \cdot 0 &= 0
\end{align*}

However, I am not sure, given a simple set of axioms, that this solution is correct. More specifically, is factoring $-m$ as $-1 \cdot m$ acceptable? Or is another proposition that I should prove beforehand?

Best Answer

It depends. What are your axioms? What is your definition of the notation $-m$?

If $-m$ is defined as the additive inverse of $m$, then no, you cannot factor $-m = -1\cdot m$ until you prove that this is true.

EDIT: For the particular axioms you have listed, your proof in fact may well be circular, since the most straightforward way of proving that $-1\cdot m = -m$ is to add $m$ to $-1\cdot m$ and show that this sum is zero. Here's a hint to get you started for your original problem:

$$m\cdot 0 = m\cdot (0+0)$$ by the additive identity axiom. Can you see how to take it from here?

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