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Math Help - reciprocal in ordered field

  1. #1
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    reciprocal in ordered field

    Let  \mathbb{F} be an ordered field and  a \in \mathbb{F} . If  a > 0 , show that  a^{-1} > 0 . If  a < 0 , show that  a^{-1} < 0 .

    So let  a,b,c \in \mathbb{F} . If  ab = c and any two of  a,b, \ \text{or} \ c is positive, then so is the third.

    Take  b = a^{-1} and the Corollary is shown?

    Is this correct? Generally, with corollaries, you invoke the theorem?
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  2. #2
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    Quote Originally Posted by particlejohn View Post
    Let  \mathbb{F} be an ordered field and  a \in \mathbb{F} . If  a > 0 , show that  a^{-1} > 0 . If  a < 0 , show that  a^{-1} < 0 .

    So let  a,b,c \in \mathbb{F} . If  ab = c and any two of  a,b, \ \text{or} \ c is positive, then so is the third.

    Take  b = a^{-1} and the Corollary is shown?

    Is this correct? Generally, with corollaries, you invoke the theorem?
    Well thats correct (since it should be easy to establish 1 > 0 by your axioms)
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