Addition and Multiplication of Real Numbers Are Binary Operations: Values and Membership

This lemma adopts the setting of The Integers and the Rational Numbers, with the Natural Numbers and the Integers Identified with Subsets of the Rationals. Let $\mathbb{R}$ be the set of real numbers, let $\oplus$, $\ominus$, $\odot$ and $u^{*}$ for $u\in\mathbb{Q}$ be as in Construction of the Dedekind Cuts: They Form a Set Totally Ordered by Inclusion, Closed under Sums, Negatives and Products of Nonnegative Cuts §constructions, and let $|x|_{C}$ for $x\in\mathbb{R}$, the sum $x+y=x\oplus y$ and the product $x\cdot y$ for $x,y\in\mathbb{R}$ be as in The Real Numbers §operations. Let $+$ and $\cdot$ be the maps $\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ whose values at $(x,y)$, for all $x,y\in\mathbb{R}$, are $x+y$ and $x\cdot y$ respectively, which Maps and Relations Given by Formulas §binary gives in The Real Numbers §operations.

¶ For all $x,y\in\mathbb{R}$: $x+y\in\mathbb{R}$, $x\cdot y\in\mathbb{R}$, $|x|_{C}\in\mathbb{R}$, and $0^{*}\subseteq|x|_{C}$ in the sense of Subclasses and Subsets §subclass.

¶ $+$ and $\cdot$ are binary operations on $\mathbb{R}$, with values $x+y$ and $x\cdot y$ at $(x,y)$ for all $x,y\in\mathbb{R}$; and they are the only maps $\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ with these values.

¶ For all sets $x$, $y$ and $z$: $((x,y),z)\in{+}$ if and only if $x\in\mathbb{R}$, $y\in\mathbb{R}$ and $z=x+y$; and likewise $((x,y),z)\in{\cdot}$ if and only if $x\in\mathbb{R}$, $y\in\mathbb{R}$ and $z=x\cdot y$, with the sum $x+y$ and the product $x\cdot y$ as defined in The Real Numbers §operations.