Arithmetic in [0,∞]: Monoid Laws, Distributivity, Monotonicity and Cancellation

In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, let $[0,\infty]$, $\infty$ and the order $\le$ be as in The Extended Real Line, Its Order, and [0,∞] (ii) and The Extended Real Line, Its Order, and [0,∞] (iii), and the sum $+$ and the product $\cdot$ on $[0,\infty]$ as in Addition and Multiplication on [0,∞] (i) and Addition and Multiplication on [0,∞] (ii). Let $a,b,c\in[0,\infty]$.

(i) The sum $+$ on $[0,\infty]$ is associative and commutative, and $0$ is a neutral element of it.

(ii) The product $\cdot$ on $[0,\infty]$ is associative and commutative, and $1$ is a neutral element of it.

(iii) $a(b+c)=ab+ac$.

(iv) If $a\le b$, then $a+c\le b+c$ and $ac\le bc$.

(v) $a\le a+c$.

(vi) If $c\in\mathbb{R}$ and $a+c\le b+c$, then $a\le b$.