Inverse of Field Product with Inverse/Proof 2

Theorem

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

Let $a, b \in F$ such that $a \ne 0$.

Then:

$\paren {a \times b^{-1} }^{-1} = b \times a^{-1}$

This can also be expressed using the notation of division as:

$1 / \paren {a / b} = b / a$

Proof

\(\ds \paren {a \times b^{-1} }^{-1}\)

\(=\)

\(\ds \paren {b^{-1} }^{-1} \times a^{-1}\)

Inverse of Field Product

\(\ds \)

\(=\)

\(\ds b \times a^{-1}\)

Inverse of Multiplicative Inverse

$\blacksquare$