Proof of Arithmetic and Order Rules in the Real Numbers, Stated for ℝ

Each result cited is universally quantified over the data in its own statement.

The structure. By The Real Numbers Form an Ordered Field in Which Every Nonempty Set Bounded Above Has a Supremum §ordered-field, $\mathbb{R}$, with $+$, $\cdot$, $0$, $1$ and $\le$, is an ordered field, these being the data of The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness §reals. Ordered fields are defined in Ordered Fields for a field with a total order, so $\mathbb{R}$, with $+$, $\cdot$, $0$ and $1$, is a field; and by Fields §field a field is a commutative ring. So $\mathbb{R}$, with $+$, $\cdot$, $0$ and $1$, is a commutative ring, and by Commutative Rings §ring the following ring laws hold for all $u,v,w\in\mathbb{R}$: associativity of addition, $(u+v)+w=u+(v+w)$; commutativity of addition, $u+v=v+u$; the neutral law for $0$, $u+0=u$; associativity of multiplication, $(u\cdot v)\cdot w=u\cdot(v\cdot w)$; commutativity of multiplication, $u\cdot v=v\cdot u$; the neutral law for $1$, $u\cdot1=u$; and distributivity, $u\cdot(v+w)=u\cdot v+u\cdot w$. Below they are used by these names, with $u$, $v$ and $w$ taken among $x$, $y$, $z$ and the elements of $\mathbb{R}$ formed from them in the clause operations.

Clause operations. By Addition and Multiplication of Real Numbers Are Binary Operations: Values and Membership §operation, $+$ and $\cdot$ are binary operations on $\mathbb{R}$; they are the operations of The Real Numbers §operations, which The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness §reals adopts. By Addition and Multiplication of Real Numbers Are Binary Operations: Values and Membership §values, $x+y\in\mathbb{R}$ and $x\cdot y\in\mathbb{R}$. By Closure Facts in an Ordered Field: Negatives, Differences, Quotients, Absolute Values, Powers, Natural Numbers, Halves and Minima §negative, applied to the ordered field $\mathbb{R}$, $-x\in\mathbb{R}$. By The Real Numbers §constants, which The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness §reals adopts for the elements $0$ and $1$, $0\in\mathbb{R}$ and $1\in\mathbb{R}$.

Clauses commutative and associative. These are commutativity and associativity of addition and of multiplication with $u=x$, $v=y$ and $w=z$.

Clause distributive. The first equation is distributivity with $u=x$, $v=y$, $w=z$. For the second, $x+y\in\mathbb{R}$ by the clause operations, so commutativity of multiplication gives $(x+y)\cdot z=z\cdot(x+y)$; distributivity with $u=z$, $v=x$, $w=y$ gives $z\cdot(x+y)=z\cdot x+z\cdot y$; and commutativity of multiplication gives $z\cdot x=x\cdot z$ and $z\cdot y=y\cdot z$, so $z\cdot x+z\cdot y=x\cdot z+y\cdot z$.

Clause neutral. $x+0=x$ and $x\cdot1=x$ are the neutral laws for $0$ and $1$. Since $0,1\in\mathbb{R}$ by the clause operations, commutativity of addition and of multiplication gives $0+x=x+0=x$ and $1\cdot x=x\cdot1=x$.

Clause negative. By Negatives, Differences, Reciprocals and Quotients §negative, applied to the commutative ring $\mathbb{R}$, $-x$ is the element $w\in\mathbb{R}$ with $x+w=0$; so $x+(-x)=0$. Since $-x\in\mathbb{R}$ by the clause operations, commutativity of addition gives $(-x)+x=x+(-x)=0$.

Clause zero-product. By Rules of Arithmetic in a Commutative Ring: Zero, Signs and Squares, and No Zero Divisors in a Field §zero, applied to the commutative ring $\mathbb{R}$, $0\cdot x=x\cdot0=0$.

Clause order. By The Real Numbers §reals, $\le$ is a total order on $\mathbb{R}$. By Partial and Total Orders on a Set and the Associated Strict Relation §total it is a partial order on $\mathbb{R}$ with $u\le v$ or $v\le u$ for all $u,v\in\mathbb{R}$; this gives $x\le y$ or $y\le x$. By Partial and Total Orders on a Set and the Associated Strict Relation §partial it is reflexive, antisymmetric and transitive, which by Reflexive, Symmetric, Antisymmetric and Transitive Relations on a Set §reflexive, Reflexive, Symmetric, Antisymmetric and Transitive Relations on a Set §antisymmetric and Reflexive, Symmetric, Antisymmetric and Transitive Relations on a Set §transitive mean, with $u$, $v$ and $w$ taken as $x$, $y$ and $z$: $x\le x$; $x\le y$ and $y\le x$ imply $x=y$; and $x\le y$ and $y\le z$ imply $x\le z$.

Clause absolute-value. By Closure Facts in an Ordered Field: Negatives, Differences, Quotients, Absolute Values, Powers, Natural Numbers, Halves and Minima §absolute-value, applied to the ordered field $\mathbb{R}$, $|x|\in\mathbb{R}$; by Rules of Arithmetic and Order in an Ordered Field §absolute-value, applied to the same ordered field, $0\le|x|$.