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

Preliminaries. By The Extended Real Line, Its Order, and [0,∞] (ii) and The Extended Real Line, Its Order, and [0,∞] (i), every element of $[0,\infty]$ is either a real number $x$ with $0\le x$ or $\infty$, and not both, as $\infty\notin\mathbb{R}$. By Order and Suprema in the Extended Real Line (i), Order and Suprema in the Extended Real Line (ii) and Order and Suprema in the Extended Real Line (viii), $\le$ is a total order on $[-\infty,\infty]$ which agrees with the order of $\mathbb{R}$ on $\mathbb{R}$, every $x\in[0,\infty]$ satisfies $0\le x\le\infty$, and $\infty\le x$ only for $x=\infty$. In the real cases below, real arithmetic means the laws of the real numbers for their sum, product and order (The Real Numbers as a Type (iv), The Real Numbers as a Type (v), The Real Numbers as a Type (vi)). We use these facts without repeating the references, together with the following, for $x,y\in[0,\infty]$.

(P0) If $x\le y$ and $y\in\mathbb{R}$, then $x\in\mathbb{R}$: otherwise $x=\infty$ and $\infty\le y$ gives $y=\infty\notin\mathbb{R}$.

(P1) If $x,y\in\mathbb{R}$, then $x+y$ is the real sum; otherwise $x+y=\infty$. So $x+y\in\mathbb{R}$ if and only if $x,y\in\mathbb{R}$. This is Addition and Multiplication on [0,∞] (i).

(P2) $xy=0$ if and only if $x=0$ or $y=0$. If $x,y\in\mathbb{R}$, this holds as $\mathbb{R}$ is a field, which has no zero divisors by Rules of Arithmetic in a Commutative Ring: Zero, Signs and Squares, and No Zero Divisors in a Field (iv), and a real product with a factor $0$ is $0$; otherwise, by Addition and Multiplication on [0,∞] (ii), $xy=0$ if $x=0$ or $y=0$ and $xy=\infty\neq0$ otherwise, as $0\in\mathbb{R}$.

(P3) If $x\neq0$ and $y\neq0$, then $xy$ is the real product if $x,y\in\mathbb{R}$, and $xy=\infty$ otherwise, by Addition and Multiplication on [0,∞] (ii).

Proof of (i). Commutativity holds since both the real sum and the conditions in Addition and Multiplication on [0,∞] (i) are symmetric in $x$ and $y$. For $x\in[0,\infty]$, $0+x=x$: for real $x$ this is real arithmetic, and $0+\infty=\infty$ by (P1); and $x+0=0+x$ by commutativity. Associativity: if $a,b,c\in\mathbb{R}$, $(a+b)+c=a+(b+c)$ is real arithmetic, all sums being real by (P1). Otherwise one of $a,b,c$ is $\infty$; if $a$ or $b$ is $\infty$, then $a+b=\infty$ and $(a+b)+c=\infty$, and if $c=\infty$, then $(a+b)+c=\infty$, by (P1); likewise $a+(b+c)=\infty$.

Proof of (ii). Commutativity holds since the real product and the cases of Addition and Multiplication on [0,∞] (ii) are symmetric. For $x\in[0,\infty]$, $1x=x$: for real $x$ this is real arithmetic, and $1\cdot\infty=\infty$ by (P3), as $1\neq0$ and $\infty\neq0$; and $x1=1x$. Associativity: if one of $a,b,c$ is $0$, then $(ab)c$ and $a(bc)$ each have a factor $0$ somewhere, so both are $0$ by (P2), applied once or twice. Let $a,b,c\neq0$. If $a,b,c\in\mathbb{R}$, associativity is real arithmetic, all products being real by (P3). Otherwise one of them is $\infty$. By (P2), $ab\neq0$ and $bc\neq0$; by (P3), $ab=\infty$ if $a$ or $b$ is $\infty$, and then $(ab)c=\infty$, while if $c=\infty$ then $(ab)c=\infty$ as well; likewise $a(bc)=\infty$.

Proof of (iii). If $a,b,c\in\mathbb{R}$, this is real arithmetic, all terms being real by (P1) and Addition and Multiplication on [0,∞] (ii). If $a=0$, both sides are $0$ by (P2), as $0+0=0$. Let $a\neq0$. If $b=\infty$ or $c=\infty$, then $b+c=\infty$ by (P1), so $a(b+c)=\infty$ by (P3); and $ab=\infty$ or $ac=\infty$ by (P3), so $ab+ac=\infty$ by (P1). There remains $a=\infty$ with $b,c\in\mathbb{R}$. If $b=c=0$, both sides are $0$ by (P2). Otherwise $b+c$ is real and nonzero, as $0\le b$, $0\le c$ and $b+c=0$ force $b=c=0$ in $\mathbb{R}$; so $a(b+c)=\infty$ by (P3); and one of $b,c$ is nonzero, so one of $ab,ac$ is $\infty$ by (P3), and $ab+ac=\infty$ by (P1).

Proof of (iv). Let $a\le b$. If $b+c=\infty$, then $a+c\le b+c$. Otherwise $b,c\in\mathbb{R}$ by (P1), $a\in\mathbb{R}$ by (P0), and $a+c\le b+c$ holds in $\mathbb{R}$, all sums being real sums by (P1). For the product: if $a=0$ or $c=0$, then $ac=0\le bc$ by (P2). If $bc=\infty$, then $ac\le bc$. Otherwise $a\neq0$, $c\neq0$ and $bc\in\mathbb{R}$; also $b\neq0$, since $a\le b=0$ would give $a=0$; so $b,c\in\mathbb{R}$ by (P3), $a\in\mathbb{R}$ by (P0), and $ac\le bc$ holds in $\mathbb{R}$ as $0\le c$, all products being real products.

Proof of (v). As $0\le c$, (iv) gives $0+a\le c+a$, that is $a\le a+c$ by (i).

Proof of (vi). Let $c\in\mathbb{R}$ and $a+c\le b+c$. If $b=\infty$, then $a\le b$. Otherwise $b\in\mathbb{R}$, so $b+c\in\mathbb{R}$ by (P1), $a+c\in\mathbb{R}$ by (P0), $a\in\mathbb{R}$ by (P1), and $a\le b$ follows from $a+c\le b+c$ in $\mathbb{R}$.