Product of Real Number with Quotient

Theorem

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

Proof

\(\ds \frac {a \times x} y\)

\(=\)

\(\ds \paren {a \times x} \times \frac 1 y\)

Definition of Real Division

\(\ds \)

\(=\)

\(\ds a \times \paren {x \times \frac 1 y}\)

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

\(\ds \)

\(=\)

\(\ds a \times \frac x y\)

Definition of Real Division

$\blacksquare$

Sources

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