Product of Reciprocals of Real Numbers

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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 Axioms: $\R \text M 4$: Inverses
\(\ds \leadsto \ \ \) \(\ds \frac 1 {x \times y} \times \paren {x \times y} \times \frac 1 y\) \(=\) \(\ds 1 \times \frac 1 y\) 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 Axioms: $\R \text M 1$: Associativity
\(\ds \leadsto \ \ \) \(\ds \paren {\frac 1 {x \times y} \times x} \times 1\) \(=\) \(\ds 1 \times \frac 1 y\) Real Number Axioms: $\R \text M 4$: Inverse
\(\ds \leadsto \ \ \) \(\ds \frac 1 {x \times y} \times x\) \(=\) \(\ds \frac 1 y\) Real Number Axioms: $\R \text M 3$: Identity
\(\ds \leadsto \ \ \) \(\ds \paren {\frac 1 {x \times y} \times x} \times \frac 1 x\) \(=\) \(\ds \frac 1 y \times \frac 1 x\) 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 Axioms: $\R \text M 1$: Associativity
\(\ds \leadsto \ \ \) \(\ds \frac 1 {x \times y} \times 1\) \(=\) \(\ds \frac 1 y \times \frac 1 x\) Real Number Axioms: $\R \text M 4$: Inverse
\(\ds \leadsto \ \ \) \(\ds \frac 1 {x \times y}\) \(=\) \(\ds \frac 1 y \times \frac 1 x\) Real Number Axioms: $\R \text M 3$: Identity
\(\ds \leadsto \ \ \) \(\ds \frac 1 {x \times y}\) \(=\) \(\ds \frac 1 x \times \frac 1 y\) Real Number Axioms: $\R \text M 2$: Commutativity

$\blacksquare$


Sources