Hey I am m having trouble proving this OFC10:
If $x < y$ and $z < 0$, then $yz < xz$.
I started with $x+z < y+0$ by OFC8, but right now I'm stuck. Any suggestions?
Note: OFC10 means Ordered Field Consequence 10.
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1$\begingroup$ Only those using your textbook will know what OFC10 means. $\endgroup$ – GEdgar Sep 7 '13 at 19:18
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$\begingroup$ @GEdgar Yes, thanks for that and I have added a little note to my question. $\endgroup$ – user87274 Sep 7 '13 at 19:33
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2$\begingroup$ @Adib "Ordered field consequence 10" is still meaningless to those who aren't using the textbook. $\endgroup$ – user61527 Sep 7 '13 at 21:05
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$z<0$ means $0<-z$ and $x<y$ means $0<y+(-x)$.
You may want to prove $(-a)b = -(ab), \,(-a)(-b)= ab$ beforehand, but you may also prove them directly in your proof:
The product of two positive numbers is positive, thus
\begin{align} & 0<(-z)(y+(-x)) \\ & = (-z)y + (-z)(-x) \\ \end{align} You can jump to the last line if you've already proved the aforementioned statements, otherwise: \begin{align} & = (-z)y + (-z)(-x) + -(zy) + zy \\ & = y(z+(-z)) + (-z)(-x) + -(zy)\\ & = y\cdot0 + (-z)(-x) + -(zy) \\ & = y\cdot0 + (-z)(-x) + -(zy)+ y\cdot0 + -(y\cdot0) + x\cdot0 + -(x\cdot0) \\ & = y\cdot(0+0) -(y\cdot0)+ (-z)(-x) + -(zy) + x\cdot(0+0) + -(x\cdot0) \\ & = y\cdot0 -(y\cdot0)+ (-z)(-x) + -(zy) + x\cdot0 + x\cdot0 -(x\cdot0) \\ & = (-z)(-x) + -(zy) + x\cdot0\\ & = (-z)(-x) + -(zy) + x(z+(-z))\\ & = (-z)(-x) + -(zy) + (-z)x+zx \\ & = (-z)(-x + x) + -(zy) + zx \\ &= (-z)\cdot0 + (-z)\cdot0 + -((-z)\cdot0) + -(zy) + zx \\ &= (-z)\cdot(0+0)+ -((-z)\cdot0) + -(zy) + zx \\ & = -(zy) + zx \end{align}
which by definition means, $zy<zx$.
(Note that this proof can be shortened a lot if we already knew $a\cdot0=0$.)
Here, I've used these axioms of $\mathbb R$.
Note that $x\in \mathbb {R}^+\iff x>0$. Using the definition of $x>y$ as $x+(-y)\in \mathbb R^+$, this shouldn't be hard to prove.
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$\begingroup$ @CameronWilliams Yes, I've fixed it. Thank you. $\endgroup$ – Alraxite Sep 7 '13 at 19:20
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$\begingroup$ @Alraxite Thanks. The problem for me was to make the right assumptions to start the proof. $\endgroup$ – user87274 Sep 7 '13 at 19:43
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Hint: If $x-y < 0$ and $z<0$ then $(x-y)z>\ldots$?
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