Product of Real Number with Quotient
Jump to navigation
Jump to search
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 $1 \ \text{(s)}$