Ordering is Preserved on Integers by Addition

Theorem

The usual ordering on the integers is preserved by the operation of addition:

$\forall a, b, c, d, \in \Z: a \le b, c \le d \implies a + c \le b + d$

$\forall x, y, z \in \Z: x \le y \implies \paren {x + z} \le \paren {y + z}$

$\forall x, y, z \in \Z: x \le y \implies \paren {z + x} \le \paren {z + y}$

So:

$\ds a$

$\le$

$\ds b$

$\ds \leadsto \ \ $

$\ds a + c$

$\le$

$\ds b + c$

$\ds c$

$\le$

$\ds d$

$\ds \leadsto \ \ $

$\ds b + c$

$\le$

$\ds b + d$

Finally:

$\ds a + c$

$\le$

$\ds b + c$

$\le$

$\ds b + d$

$\ds \leadsto \ \ $

$\ds a + c$

$\le$

$\ds b + d$

Definition of Ordering

$\blacksquare$

1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $2$: Integers and natural numbers: $\S 2.1$: The integers: $\mathbf Z. \, 7$
1994: H.E. Rose: A Course in Number Theory (2nd ed.) ... (previous) ... (next): $1$ Divisibility: $1.1$ The Euclidean algorithm and unique factorization