Negative of Field Negative
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Theorem
Let $\struct {F, +, \times}$ be a field whose zero is $0_F$.
Let $a \in F$ and let $-a$ be the field negative of $a$.
Then:
- $-\paren {-a} = a$
Proof
\(\ds \paren {-a} + a\) | \(=\) | \(\ds a + \paren {-a}\) | Field Axiom $\text A2$: Commutativity of Addition | |||||||||||
\(\ds \) | \(=\) | \(\ds 0_F\) | Field Axiom $\text A4$: Inverses for Addition | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds a\) | \(=\) | \(\ds -\paren {-a}\) | Definition of Field Negative |
$\blacksquare$
Sources
- 1973: C.R.J. Clapham: Introduction to Mathematical Analysis ... (previous) ... (next): Chapter $1$: Axioms for the Real Numbers: $2$. Fields: Theorem $2 \ \text {(i)}$