Equal Numbers are Congruent

Theorem

$\forall x, y, z \in \R: x = y \implies x \equiv y\, \bmod \, z$

where $x \equiv y \, \bmod \, z$ denotes congruence modulo $z$.

Proof

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle x\)	\(=\)	\(\displaystyle y\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \implies\)	\(\displaystyle \)	\(\displaystyle x - y\)	\(=\)	\(\displaystyle 0\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \implies\)	\(\displaystyle \)	\(\displaystyle x - y\)	\(=\)	\(\displaystyle 0 \cdot z\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \implies\)	\(\displaystyle \)	\(\displaystyle x\)	\(\equiv\)	\(\displaystyle y \, \bmod \, z\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	by definition of congruence modulo $z$

$\blacksquare$

Sources

• George E. Andrews: Number Theory (1971): $\S 4.1$: Example $4.3$