Ring Product preserves Inequalities on Positive Elements
Jump to navigation
Jump to search
Theorem
Let $\struct {R, +, \circ, \le}$ be an ordered ring.
Let $x, y, z, w \in R$.
Let $0 < x < y$ and $0 < z < w$.
Then:
- $0 < z \circ x < w \circ y$
Proof
By Properties of Ordered Ring $(6)$:
- $z \circ x < z \circ y$
- $z \circ y < w \circ y$
Then by transitivity of $\circ$:
- $z \circ x < w \circ y$
Also by Properties of Ordered Ring $(6)$:
- $z \circ 0 < z \circ x$
Hence by Ring Product with Zero:
- $0 < z \circ x$
$\blacksquare$