# Product with Field Negative

## 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

$\blacksquare$