Is this a mathematical law?

Suppose A = CB and A, C, and B are positive real numbers and C < 1. It must be so that A < B. Suppose instead that C > 1; then A > B. The inequalities are reversed (so that A > B and A < B, respectively) if we instead say that A and B are negative real numbers (C still being positive).

If what I am saying is true, what is this axiom or rule called?

I don’t think it has a special name; it’s just the statement that CB < B implies C > 1 if B > 0 and C < 1 if B < 0. This follows from the usual division rules for inequalities: divide the equation CB < B by B on both sides to see this.

The simplest formulation is that the sum and product of two positive numbers is positive. So if A > B and C > 0, then (A - B)C > 0, from which it follows that AC - BC > 0. Also BC > 0, so adding BC to both sides gives AC > BC. All the other rules about inequalities follow in similar ways, including the fact that a product of a positive and a negative number is negative and that a product of two negative numbers is positive.