# Product with Field Negative/Corollary

## Theorem

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

Let $a \in F$.

Then:

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

## Proof

 $\ds a + \paren {-1_F} \times a$ $=$ $\ds 1_F \times a + \paren {-1_F} \times a$ Field Axiom $\text M3$: Identity for Product $\ds$ $=$ $\ds \paren {1_F + \paren {-1_F} } \times a$ Field Axiom $\text D$: Distributivity of Product over Addition $\ds$ $=$ $\ds 0_F \times a$ Field Axiom $\text A4$: Inverses for Addition $\ds$ $=$ $\ds 0_F$ Field Product with Zero $\ds \leadsto \ \$ $\ds \paren {-a}$ $=$ $\ds \paren {-1_F} \times a$ Definition of Ring Negative

$\blacksquare$