Integer Coprime to Modulus iff Linear Congruence to 1 exists/Corollary

Theorem
Let $p$ be a prime number.

The linear congruence:
 * $a x \equiv 1 \pmod p$

has a solution $x$ $a \not \equiv 0 \pmod p$.

Proof
By definition of congruence:
 * $a \not \equiv 0 \pmod p \iff p \nmid a$

where $p \nmid a$ denotes that $p$ is not a divisor of $a$.

From Prime not Divisor implies Coprime:
 * $p \nmid a \iff p \perp a$

where $p \perp a$ denotes that $p$ and $a$ are coprime.

The result follows from Integer Coprime to Modulus iff Linear Congruence to 1 exists.