Equality of Rational Numbers

Theorem
Let $a, b, c, d$ be integers, with $b$ and $d$ nonzero.


 * $(1): \quad$ The rational numbers $\dfrac a b$ and $\dfrac c d$ are equal.
 * $(2): \quad$ The integers $a d$ and $b c$ are equal.

Proof
Note that by definition, $\Q$ is the field of quotients of $\Z$.

1 implies 2
Let $\dfrac a b = \dfrac c d$ in $\Q$.

Then $b c = a d$ in $\Q$.

By Canonical Mapping to Field of Quotients is Injective, $b c = a d$ in $\Z$.

2 implies 1
Let $bd = ac$ in $\Z$.

By definition of ring homomorphism, $b c = a d$ in $\Q$.

Thus Let $\dfrac a b = \dfrac c d$ in $\Q$.