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

$\ds \frac 1 {x \times y} \times \paren {x \times y}$

$=$

$\ds 1$

Real Number Axiom $\R \text M4$: Inverses for Multiplication

$\ds \leadsto \ \ $

$\ds \frac 1 {x \times y} \times \paren {x \times y} \times \frac 1 y$

$=$

$\ds 1 \times \frac 1 y$

as $y \ne 0$

$\ds \leadsto \ \ $

$\ds \paren {\frac 1 {x \times y} \times x} \times \paren {y \times \frac 1 y}$

$=$

$\ds 1 \times \frac 1 y$

Real Number Axiom $\R \text M1$: Associativity of Multiplication

$\ds \leadsto \ \ $

$\ds \paren {\frac 1 {x \times y} \times x} \times 1$

$=$

$\ds 1 \times \frac 1 y$

Real Number Axiom $\R \text M4$: Inverses for Multiplication

$\ds \leadsto \ \ $

$\ds \frac 1 {x \times y} \times x$

$=$

$\ds \frac 1 y$

Real Number Axiom $\R \text M3$: Identity Element for Multiplication

$\ds \leadsto \ \ $

$\ds \paren {\frac 1 {x \times y} \times x} \times \frac 1 x$

$=$

$\ds \frac 1 y \times \frac 1 x$

as $x \ne 0$

$\ds \leadsto \ \ $

$\ds \frac 1 {x \times y} \times \paren {x \times \frac 1 x}$

$=$

$\ds \frac 1 y \times \frac 1 x$

Real Number Axiom $\R \text M1$: Associativity of Multiplication

$\ds \leadsto \ \ $

$\ds \frac 1 {x \times y} \times 1$

$=$

$\ds \frac 1 y \times \frac 1 x$

Real Number Axiom $\R \text M4$: Inverses for Multiplication

$\ds \leadsto \ \ $

$\ds \frac 1 {x \times y}$

$=$

$\ds \frac 1 y \times \frac 1 x$

Real Number Axiom $\R \text M3$: Identity Element for Multiplication

$\ds \leadsto \ \ $

$\ds \frac 1 {x \times y}$

$=$

$\ds \frac 1 x \times \frac 1 y$

Real Number Axiom $\R \text M2$: Commutativity of Multiplication

$\blacksquare$

Sources

2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 4$: The Integers and the Real Numbers: Exercise $1 \ \text{(m)}$