Product with Field Negative

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $\struct {F, +, \times}$ be a field whose zero is $0_F$ and whose unity is $1_F$.

Let $a, b \in F$.


Then:

$-\paren {a \times b} = a \times \paren {-b} = \paren {-a} \times b$


Corollary

$\paren {-1_F} \times a = \paren {-a}$


Proof

\(\ds a \times b + a \times \paren {-b}\) \(=\) \(\ds a \times \paren {b + \paren {-b} }\) Field Axiom $\text A1$: Associativity of Addition
\(\ds \) \(=\) \(\ds a \times 0_F\) Field Axiom $\text A4$: Inverses for Addition
\(\ds \) \(=\) \(\ds 0_F\) Field Product with Zero
\(\ds \leadsto \ \ \) \(\ds -\paren {a \times b}\) \(=\) \(\ds a \times \paren {-b}\) Definition of Ring Negative


Similarly:

\(\ds \paren {-a} \times b + a \times b\) \(=\) \(\ds \paren {\paren {-a} + a} \times b\) Field Axiom $\text A1$: Associativity of Addition
\(\ds \) \(=\) \(\ds 0_F \times a\) Field Axiom $\text A4$: Inverses for Addition
\(\ds \) \(=\) \(\ds 0_F\) Field Product with Zero
\(\ds \leadsto \ \ \) \(\ds -\paren {a \times b}\) \(=\) \(\ds \paren {-a} \times b\) Definition of Ring Negative

$\blacksquare$


Sources