Axiom:Multiplicative Norm Axioms

Definition
Let $\struct {R, +, \circ}$ be a division ring whose zero is denoted $0_R$.

Let $\norm {\,\cdot\,}: R \to \R_{\ge 0}$ be a norm on $R$.

The norm axioms are the following conditions on $\norm {\,\cdot\,}$ which define $\norm {\,\cdot\,}$ as being a norm: