Reciprocal of Quotient of Real Numbers

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Theorem

$\forall x, y \in \R_{\ne 0}: \dfrac 1 {x / y} = \dfrac y x$


Proof

\(\displaystyle \dfrac 1 {x / y}\) \(=\) \(\displaystyle \frac 1 {x \times \dfrac 1 y}\) Definition of Real Division
\(\displaystyle \) \(=\) \(\displaystyle 1 \times \frac 1 {x \times \dfrac 1 y}\) Real Number Axioms: $\R M3$: Identity
\(\displaystyle \) \(=\) \(\displaystyle \paren {y \times \frac 1 y} \times \frac 1 {x \times \dfrac 1 y}\) Real Number Axioms: $\R M4$: Inverses
\(\displaystyle \) \(=\) \(\displaystyle y \times \paren {\frac 1 y \times \frac 1 {x \times \dfrac 1 y} }\) Real Number Axioms: $\R M1$: Associativity
\(\displaystyle \) \(=\) \(\displaystyle y \times \frac 1 {y \times \paren {x \times \dfrac 1 y} }\) Product of Reciprocals of Real Numbers
\(\displaystyle \) \(=\) \(\displaystyle y \times \frac 1 {x \times \paren {y \times \dfrac 1 y} }\) Real Number Axioms: $\R M1$: Associativity and $\R M2$: Commutativity
\(\displaystyle \) \(=\) \(\displaystyle y \times \frac 1 {x \times 1}\) Real Number Axioms: $\R M4$: Inverses
\(\displaystyle \) \(=\) \(\displaystyle y \times \frac 1 x\) Real Number Axioms: $\R M3$: Identity
\(\displaystyle \) \(=\) \(\displaystyle \frac y x\) Definition of Real Division

$\blacksquare$


Sources