Negative of Sum of Real Numbers
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Theorem
- $\forall x, y \in \R: -\paren {x + y} = -x - y$
Proof
\(\ds -\paren {x + y}\) | \(=\) | \(\ds \paren {-1} \times \paren {x + y}\) | Multiplication by Negative Real Number: Corollary | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\paren {-1} \times x} + \paren {\paren {-1} \times y}\) | Real Number Axiom $\R \text D$: Distributivity of Multiplication over Addition | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {-x} + \paren {-y}\) | Multiplication by Negative Real Number: Corollary | |||||||||||
\(\ds \) | \(=\) | \(\ds -x - y\) | Definition of Real Subtraction |
$\blacksquare$
Sources
- 2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 4$: The Integers and the Real Numbers: Exercise $1 \ \text{(h)}$