# Product of Reciprocals of Real Numbers

## Theorem

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

## Proof

 $\displaystyle \frac 1 {x \times y} \times \paren {x \times y}$ $=$ $\displaystyle 1$ Real Number Axioms: $\R \text M 4$: Inverses $\displaystyle \leadsto \ \$ $\displaystyle \frac 1 {x \times y} \times \paren {x \times y} \times \frac 1 y$ $=$ $\displaystyle 1 \times \frac 1 y$ as $y \ne 0$ $\displaystyle \leadsto \ \$ $\displaystyle \paren {\frac 1 {x \times y} \times x} \times \paren {y \times \frac 1 y}$ $=$ $\displaystyle 1 \times \frac 1 y$ Real Number Axioms: $\R \text M 1$: Associativity $\displaystyle \leadsto \ \$ $\displaystyle \paren {\frac 1 {x \times y} \times x} \times 1$ $=$ $\displaystyle 1 \times \frac 1 y$ Real Number Axioms: $\R \text M 4$: Inverse $\displaystyle \leadsto \ \$ $\displaystyle \frac 1 {x \times y} \times x$ $=$ $\displaystyle \frac 1 y$ Real Number Axioms: $\R \text M 3$: Identity $\displaystyle \leadsto \ \$ $\displaystyle \paren {\frac 1 {x \times y} \times x} \times \frac 1 x$ $=$ $\displaystyle \frac 1 y \times \frac 1 x$ as $x \ne 0$ $\displaystyle \leadsto \ \$ $\displaystyle \frac 1 {x \times y} \times \paren {x \times \frac 1 x}$ $=$ $\displaystyle \frac 1 y \times \frac 1 x$ Real Number Axioms: $\R \text M 1$: Associativity $\displaystyle \leadsto \ \$ $\displaystyle \frac 1 {x \times y} \times 1$ $=$ $\displaystyle \frac 1 y \times \frac 1 x$ Real Number Axioms: $\R \text M 4$: Inverse $\displaystyle \leadsto \ \$ $\displaystyle \frac 1 {x \times y}$ $=$ $\displaystyle \frac 1 y \times \frac 1 x$ Real Number Axioms: $\R \text M 3$: Identity $\displaystyle \leadsto \ \$ $\displaystyle \frac 1 {x \times y}$ $=$ $\displaystyle \frac 1 x \times \frac 1 y$ Real Number Axioms: $\R \text M 2$: Commutativity

$\blacksquare$