Field/Basic properties/Fact/Proof2

Proof

1. We have $a0=a(0+0)=a0+a0$. Subtracting $a0$ (meaning addition with the negative of $a0$) on both sides gives the claim.
2. See exercise.
3. See exercise.
4. See exercise.
5. We prove this by contradiction, so we assume that $a$ and $b$ are both not $0$. Then there exist inverse elements $a^{-1}$ and $b^{-1}$ and hence $(ab)\left(b^{-1}{a}^{-1}\right)=1$. On the other hand, we have $ab=0$ by the premise and so the annulation rule gives

   $(ab)\left(b^{-1}{a}^{-1}\right)=0\left(b^{-1}{a}^{-1}\right)=0,$

   hence $0=1$, which contradicts the field properties.

6. This follows with a double induction, see exercise.

To fact