Proof of the properties of an ordered feild

[EV33]

Homework Statement

If F is an ordered field the the following property holds for any elements a and b of F.
If b<a, the -a<-b.

My task is to prove this property. My question is whether I need to use the definition of an ordered field. I used the basic axioms but I didn't use the definition of an ordered field.

Homework Equations

The basic axioms such... communitivity, addative inverse...
The definition of an ordered field.

The Attempt at a Solution

Assume b<a. Then add -a-b to each side, which gives us b-a-b<a-a-b. by using communitivity on the left we can rewrite it as b-b-a<a-a-b. By the use of the addative inverse we can simplify it to -a<-b.(QED)

So does this work without the definition of an ordered field?

 

[Pagan Harpoon]

Those properties, commutativity etc are part of the definition of a field, aren't they?

 

[Fredrik]

And the fact that you can add the same thing to both sides of an inequality is part of the definition of an ordered field.

 

[HallsofIvy]

The "definition of an ordered field" is that it is a set of objects, together with two operations and an order relation, that satisfy those axioms!