Addition and Multiplication on [0,∞]

In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, let $\mathbb{R}$, its order $\le$, its sum, its product and its elements $0$ and $1$ be as in The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness (ii), and let $\infty$ and $[0,\infty]$ be as in The Extended Real Line, Its Order, and [0,∞] (i) and The Extended Real Line, Its Order, and [0,∞] (ii). For $p\in[0,\infty]\times[0,\infty]$, $p=(p_{1},p_{2})$ with the components $p_{1},p_{2}\in[0,\infty]$ (Membership in a Cartesian Product, and the Cartesian Product of Two Sets Is a Set (i)).

(i) The sum on $[0,\infty]$ is the binary operation $+$ on $[0,\infty]$ with $x+y$ the sum in $\mathbb{R}$ if $x,y\in\mathbb{R}$, and $x+y=\infty$ otherwise, for $x,y\in[0,\infty]$. It exists and is unique by Maps Defined by Cases (i), applied on $[0,\infty]\times[0,\infty]$ to the condition $p_{1},p_{2}\in\mathbb{R}$: for real $x,y$ with $0\le x$ and $0\le y$, the real sum $x+y$ is real with $0\le x+y$, so it lies in $[0,\infty]$, and $\infty\in[0,\infty]$.

(ii) The product on $[0,\infty]$ is the binary operation $\cdot$ on $[0,\infty]$, with $x\cdot y$ also written $xy$, such that for $x,y\in[0,\infty]$: $xy$ is the product in $\mathbb{R}$ if $x,y\in\mathbb{R}$; $xy=0$ if $x=0$ or $y=0$; and $xy=\infty$ otherwise. The elements $0$ and $1$ of $\mathbb{R}$ lie in $[0,\infty]$, as $0\le0$ and $0\le1$ in $\mathbb{R}$. The first two cases agree where they overlap, as a real product with a factor $0$ is $0$. It exists and is unique by two applications of Maps Defined by Cases (i): first to the map $h:[0,\infty]\times[0,\infty]\to[0,\infty]$ with $h(p)=\infty$ if $p_{1}=\infty$ and $p_{2}\neq0$, or $p_{2}=\infty$ and $p_{1}\neq0$, and $h(p)=0$ otherwise; then to the map with value the real product $p_{1}p_{2}$ if $p_{1},p_{2}\in\mathbb{R}$, which is real with $0\le p_{1}p_{2}$, and $h(p)$ otherwise. If not both of $x,y$ are real, then one of them is $\infty$, and $h(x,y)=0$ exactly when the other is $0$; so the values are those stated.