Proof in field theory

Question

Im having trouble with this question:
$\mathbb F$ is an ordered field
Prove that for all $x,y\in\mathbb F$ , $xy > 0 \iff ((x<0 \land y<0)\lor(x>0 \land y>0))$

I managed to prove that right side implies left side, but I couldnt manage to prove vice versa

Thanks.

Answer 1

By Trichotomy Law, for any $x\in F$ $$ x>0\lor x=0\lor x<0 $$ If $x=0$ or $y=0$, then $xy=0$. So $$ xy>0\implies x\ne0 \land y\ne 0 $$ If $x>0$ and $y<0$, then $-y>0$ and $$ -xy=x(-y)>0\quad \text{and so}\quad xy<0 $$ Likewise if $x<0$ and $y>0$, then $xy<0$. Thus $$ xy > 0 \implies (x<0 \land y<0)\lor(x>0 \land y>0) $$

Answer 2

Hint: Try proving the two statements:

1. $(x<0\wedge y<0)\vee(x>0\wedge y>0)\ \implies\ xy>0$
2. $(x<0\wedge y>0)\vee(x>0\wedge y<0)\ \implies\ xy<0$