Equal Numbers are Congruent

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Theorem

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

where $x \equiv y \pmod z$ denotes congruence modulo $z$.


Proof

\(\ds x\) \(=\) \(\ds y\)
\(\ds \leadsto \ \ \) \(\ds x - y\) \(=\) \(\ds 0\)
\(\ds \leadsto \ \ \) \(\ds x - y\) \(=\) \(\ds 0 \cdot z\)
\(\ds \leadsto \ \ \) \(\ds x\) \(\equiv\) \(\ds y\) \(\ds \pmod z\) Definition of Congruence Modulo $z$

$\blacksquare$


Also see


Sources