Prove that if .

*Proof.*Since we know there exist elements such that and . So,

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Stumbling Robot

A Fraction of a Dot
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Prove that the inverse of a product is the product of the inverses

* Proof. * Since we know there exist elements such that and . So,
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Prove that if .

My solution:

Since a ≠ 0 and b ≠ 0, by Theorem I.11, it follows that ab ≠ 0. Hence, the multiplicative inverse of ab exists. To show that this multiplicative inverse is nothing more than the product of the multiplicative inverses of a and b, we can notice the following:

(ab)(a^{-1}b^{-1}) = (ba)(a^{-1}b^{-1}) = b(aa^{-1})b^{-1} = b(1)b^{-1} = bb^{-1} = 1.

Hence, the multiplicative inverse of ab is a^{-1}b^{-1}; that is, (ab)^{-1} = a^{-1}b^{-1}.