Modulo Addition is Well-Defined/Real Modulus

Theorem

Let $z \in \R$ be a real number.

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Let:

$a \equiv b \pmod z$

and:

$x \equiv y \pmod z$

where $a, b, x, y \in \R$.

Then:

$a + x \equiv b + y \pmod z$

Proof

$\ds a$

$\equiv$

$\ds b$

$\ds \pmod z$

$\ds c$

$\equiv$

$\ds d$

$\ds \pmod z$

$\ds \leadsto \ \ $

$\ds a \bmod z$

$=$

$\ds b \bmod z$

Definition of Congruence

$\ds x \bmod z$

$=$

$\ds y \bmod z$

$\ds \leadsto \ \ $

$\ds \exists k_1 \in \Z: \, $

$\ds a - b$

$=$

$\ds k_1 z$

$\ds \exists k_2 \in \Z: \, $

$\ds x - y$

$=$

$\ds k_2 z$

$\ds \leadsto \ \ $

$\ds \paren {a + x} - \paren {b + y}$

$=$

$\ds \paren {k_1 + k_2} z$

Definition of Integer Addition

$\ds \leadsto \ \ $

$\ds a + x$

$\equiv$

$\ds b + y$

$\ds \pmod z$

Definition of Congruence

$\blacksquare$

Sources

1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.4$: Integer Functions and Elementary Number Theory: Exercise $24$