Odd Number Coprime to Number is also Coprime to its Double

Theorem
Let $a, b \in \Z$ be integers.

Let $a$ be odd.

Let:
 * $a \perp b$

where $\perp$ denotes coprimality.

Then:
 * $a \perp 2 b$

Proof
By definition of odd number:
 * $a \perp 2$

The result follows from Integer Coprime to Factors is Coprime to Whole.