hcju/proofwikips huggingface dataset:
Well-Ordering Principle
Every non-empty subset of $\N$ has a smallest (or '''first''') element. This is called the '''well-ordering principle'''. The '''well-ordering principle''' also holds for $\N_{\ne 0}$.
Equality of Natural Numbers
Let $m, n \in \N$. Then: : $\N_m \sim \N_n \iff m = n$ where $\sim$ denotes set equivalence and $\N_n$ denotes the set of all natural numbers less than $n$.
Cardinality Less One
Let $S$ be a finite set. Let: :$\left\lvert{S}\right\rvert = n + 1$ where $\left\lvert{S}\right\rvert$ is the cardinality of $S$. Let $a \in S$. Then: :$\left\lvert{S \setminus \left\{ {a}\right\} }\right\rvert = n$ where $\setminus$ denotes set difference.
Cardinality of Empty Set
:$\card S = 0 \iff S = \O$ That is, the empty set is finite, and has a cardinality of zero.
Cardinality of Subset of Finite Set
Let $A$ and $B$ be finite sets such that $A \subseteq B$. Let :$\card B = n$ where $\card {\, \cdot \,}$ denotes cardinality. Then $\card A \le n$.
Cardinality of Surjection
Let $S$ be a set. Let: :$\card S = n$ where $\card S$ denotes the cardinality of $S$. Let $f: S \to T$ be a surjection. Then $\card T \le n$. The equality: :$\card T = n$ occurs  $f$ is a bijection.
Equivalence of Mappings between Sets of Same Cardinality
Let $S$ and $T$ be finite sets such that $\card S = \card T$. Let $f: S \to T$ be a mapping. Then the following statements are equivalent: : $(1): \quad f$ is bijective : $(2): \quad f$ is injective : $(3): \quad f$ is surjective.
Natural Numbers are Infinite
The set $\N$ of natural numbers is infinite.
Finite Non-Empty Subset of Totally Ordered Set has Smallest and Greatest Elements
Let $\left({S, \preceq}\right)$ be a totally ordered set. Then every finite $T$ such that $\varnothing \subset T \subseteq S$ has both a smallest and a greatest element.
Unique Isomorphism between Finite Totally Ordered Sets
Let $S$ and $T$ be finite sets such that: :$\card S = \card T$ Let $\struct {S, \preceq}$ and $\struct {T, \preccurlyeq}$ be totally ordered sets. Then there is exactly one order isomorphism from $\struct {S, \preceq}$ to $\struct {T, \preccurlyeq}$.
Isomorphism to Closed Interval
Let $m, n \in \N$ such that $m < n$. Then: : $\left|{\left[{m + 1 \,.\,.\, n}\right]}\right| = n - m$ Let $h: \N_{n - m} \to \left[{m + 1 \,.\,.\, n}\right]$ be the mapping defined as: :$\forall x \in \N_{n - m}: h \left({x}\right) = x + m + 1$ Let the orderings on $\left[{m + 1 \,.\,.\, n}\right]$ and $\N_{n - m}$ be those induced by the ordering of $\N$. Then $h$ a unique order isomorphism.
Regular Representation wrt Cancellable Element on Finite Semigroup is Bijection
Let $\left({S, \circ}\right)$ be a finite semigroup. Let $a \in S$ be cancellable. Then: :the left regular representation $\lambda_a$ and: :the right regular representation $\rho_a$ of $\left({S, \circ}\right)$ with respect to $a$ are both bijections.
Power Set of Natural Numbers is not Countable
The power set $\powerset \N$ of the natural numbers $\N$ is not countable.
Subset of Countably Infinite Set is Countable
Every subset of a countably infinite set is countable.
Infinite Set has Countably Infinite Subset
Every infinite set has a countably infinite subset.
No Bijection between Finite Set and Proper Subset
A finite set can not be in one-to-one correspondence with one of its proper subsets. That is, a finite set is not Dedekind-infinite.
Infinite Set is Equivalent to Proper Subset
A set is infinite  it is equivalent to one of its proper subsets.
Cartesian Product of Countable Sets is Countable
The cartesian product of two countable sets is countable.
Composition of Sequence with Mapping
Let $\left \langle {a_j} \right \rangle_{j \in B}$ be a sequence. Let $\sigma: A \to B$ be a mapping, where $A \subseteq \N$. Then $\left \langle {a_j} \right \rangle \circ \sigma$ is a sequence whose value at each $k \in A$ is $a_{\sigma \left({k}\right)}$. Thus $\left \langle {a_j} \right \rangle \circ \sigma$ is denoted $\left \langle {a_{\sigma \left({k}\right)}} \right \rangle_{k \in A}$.
General Operation from Binary Operation
Let $\left({S, \oplus}\right)$ be a magma. Then there a unique sequence $\left \langle {\oplus_k} \right \rangle_{k \mathop \ge 1}$ such that: :$(1): \quad \forall n \in \N_{>0}: \oplus_n$ is an $n$-ary operation on $S$ such that: :$(2): \quad \forall \left({a_1, \ldots, a_k}\right) \in S^k: \oplus_k \left({a_1, \ldots, a_k}\right) = \begin{cases} a: & k = 1 \\ \oplus_n \left({a_1, \ldots, a_n}\right) \oplus a_{n+1}: & k = n + 1 \end{cases}$ In particular, $\oplus_2$ is the same as the given binary operation $\oplus$. The $n$th term $\oplus_n$ of the sequence $\left \langle {\oplus} \right \rangle$ is called the '''$n$-ary operation defined by $\oplus$'''.
Strictly Increasing Sequence induces Partition
Let $\left \langle {r_k} \right \rangle_{0 \mathop \le k \mathop \le n}$ be a strictly increasing finite sequence of natural numbers. Let: :$\forall k \in \left[{1 \,.\,.\, n}\right]: A_k := \left[{r_{k-1} + 1 \,.\,.\, r_k}\right]$ Then: :$\left\{{A_k: k \in \left[{1 \,.\,.\, n}\right]}\right\}$ is a partition of $\left[{r_0 + 1 \,.\,.\, r_n}\right]$.
Fundamental Principle of Counting
Let $A$ be a finite set. Let $\sequence {B_n}$ be a sequence of distinct finite subsets of $A$ which form a partition of $A$. Let $p_k = \size {B_k}$ for each $k \in \closedint 1 n$. Then: :$\displaystyle \size A = \sum_{k \mathop = 1}^n p_k$ That is, the sum of the numbers of elements in the subsets of a partition of a set is equal to the total number of elements in the set.
Odd Number Theorem
:$\displaystyle \sum_{j \mathop = 1}^n \paren {2 j - 1} = n^2$ That is, the sum of the first $n$ odd numbers is the $n$th square number.
General Associativity Theorem
If an operation is associative on $3$ entities, then it is associative on any number of them.
General Commutativity Theorem
Let $\struct {S, \circ}$ be a semigroup. Let $\family {a_k}_{1 \mathop \le k \mathop \le n}$ be a sequence of elements of $S$. Suppose that: :$\forall i, j \in \closedint 1 n: a_i \circ a_j = a_j \circ a_i$ Then for every permutation $\sigma: \N_n \to \N_n$: :$a_{\map \sigma 1} \circ \cdots \circ a_{\map \sigma n} = a_1 \circ \cdots \circ a_n$ where $\N_n$ is used here to denote the initial segment of $\N_{>0}$: :$\N_n = \set {1, 2, \ldots, n}$
General Distributivity Theorem
:$\displaystyle \paren {\sum_{i \mathop = 1}^m a_i} * \paren {\sum_{j \mathop = 1}^n b_j} = \sum_{ {1 \mathop \le i \mathop \le m} \atop {1 \mathop \le j \mathop \le n} } \paren {a_i * b_j}$
Associativity on Indexing Set
Let $\left({S, \circ}\right)$ be a commutative semigroup. Let $\left \langle {x_\alpha} \right \rangle_{\alpha \mathop \in A}$ be a family of terms of $S$ indexed by a finite non-empty set $A$. Let $\left \langle {B_k} \right \rangle_{1 \mathop \le k \mathop \le n}$ be a family of distinct subsets of $A$ forming a partition of $A$. Then: : $\displaystyle \prod_{k \mathop = 1}^n \left({\prod_{a \mathop \in B_k} x_\alpha}\right) = \prod_{\alpha \mathop \in A} x_\alpha$
External Direct Product of Groups is Group
Let $\struct {G_1, \circ_1}$ and $\struct {G_2, \circ_2}$ be groups whose identity elements are $e_1$ and $e_2$ respectively. Let $\struct {G_1 \times G_2, \circ}$ be the external direct product of $G_1$ and $G_2$. Then $\struct {G_1 \times G_2, \circ}$ is a group whose identity element is $\tuple {e_1, e_2}$.
External Direct Product of Projection with Canonical Injection
Let $\struct {S_1, \circ_1}$ and $\struct {S_2, \circ_2}$ be algebraic structures with identity elements $e_1$ and $e_2$ respectively. Let $\struct {S_1 \times S_2, \circ}$ be the external direct product of $\struct {S_1, \circ_1}$ and $\struct {S_2, \circ_2}$ Let: :$\pr_1: \struct {S_1 \times S_2, \circ} \to \struct {S_1, \circ_1}$ be the first projection from $\struct {S_1 \times S_2, \circ}$ to $\struct {S_1, \circ_1}$ :$\pr_2: \struct {S_1 \times S_2, \circ} \to \struct {S_2, \circ_2}$ be the second projection from $\struct {S_1 \times S_2, \circ}$ to $\struct {S_2, \circ_2}$. Let: :$\inj_1: \struct {S_1, \circ_1} \to \struct {S_1 \times S_2, \circ}$ be the canonical injection from $\struct {S_1, \circ_1}$ to $\struct {S_1 \times S_2, \circ}$ :$\inj_2: \struct {S_2, \circ_2} \to \struct {S_1 \times S_2, \circ}$ be the canonical injection from $\struct {S_2, \circ_2}$ to $\struct {S_1 \times S_2, \circ}$. Then: :$(1): \quad \pr_1 \circ \inj_1 = I_{S_1}$ :$(2): \quad \pr_2 \circ \inj_2 = I_{S_2}$ where $I_{S_1}$ and $I_{S_2}$ are the identity mappings on $S_1$ and $S_2$ respectively.
Inverse Completion of Natural Numbers
There exists an inverse completion of the natural numbers under addition.
L'Hôpital's Rule
Let $f$ and $g$ be real functions which are continuous on the closed interval $\closedint a b$ and differentiable on the open interval $\openint a b$. Let: :$\forall x \in \openint a b: \map {g'} x \ne 0$ where $g'$ denotes the derivative of $g$ {{WRT|Differentiation}} $x$. Let: :$\map f a = \map g a = 0$ Then: :$\displaystyle \lim_{x \mathop \to a^+} \frac {\map f x} {\map g x} = \lim_{x \mathop \to a^+} \frac {\map {f'} x} {\map {g'} x}$ provided that the second limit exists.
Natural Numbers under Addition form Commutative Monoid
The algebraic structure $\struct {\N, +}$ consisting of the set of natural numbers $\N$ under addition $+$ is a commutative monoid whose identity is zero.
Natural Numbers form Commutative Semiring
The semiring of natural numbers $\struct {\N, +, \times}$ forms a commutative semiring.
Integer Multiplication is Closed
The set of integers is closed under multiplication: :$\forall a, b \in \Z: a \times b \in \Z$
Integer Multiplication is Commutative
The operation of multiplication on the set of integers $\Z$ is commutative: :$\forall x, y \in \Z: x \times y = y \times x$
Integer Multiplication is Associative
The operation of multiplication on the set of integers $\Z$ is associative: :$\forall x, y, z \in \Z: x \times \paren {y \times z} = \paren {x \times y} \times z$
Integer Multiplication Distributes over Addition
The operation of multiplication on the set of integers $\Z$ is distributive over addition: :$\forall x, y, z \in \Z: x \times \paren {y + z} = \paren {x \times y} + \paren {x \times z}$ :$\forall x, y, z \in \Z: \paren {y + z} \times x = \paren {y \times x} + \paren {z \times x}$
Construction of Inverse Completion
This page consists of a series of linked theorems, each of which builds towards one result. To access the proofs for the individual theorems, click on the links which form the titles of each major section.
Integer Multiplication Identity is One
The identity of integer multiplication is $1$: :$\exists 1 \in \Z: \forall a \in \Z: a \times 1 = a = 1 \times a$
Integer Multiplication has Zero
The set of integers under multiplication $\struct {\Z, \times}$ has a zero element, which is $0$.
Ring of Integers has no Zero Divisors
The integers have no zero divisors: :$\forall x, y, \in \Z: x \times y = 0 \implies x = 0 \lor y = 0$
Integers form Integral Domain
The integers $\Z$ form an integral domain under addition and multiplication.
Natural Numbers are Non-Negative Integers
Let $m \in \Z$. Then: :$(1): \quad m \in \N \iff 0 \le m$ :$(2): \quad m \in \N_{> 0} \iff 0 < m$ :$(3): \quad m \notin \N \iff -m \in \N_{> 0}$ That is, the natural numbers are precisely those integers which are greater than or equal to zero.
Subtraction on Integers is Extension of Natural Numbers
Integer subtraction is an extension of the definition of subtraction on the natural numbers.
Multiplicative Ordering on Integers
Let $x, y, z \in \Z$ such that $z > 0$. Then: :$x < y \iff z x < z y$ :$x \le y \iff z x \le z y$
Invertible Integers under Multiplication
The only invertible elements of $\Z$ for multiplication (that is, units of $\Z$) are $1$ and $-1$.
Index Laws for Monoids
These results are an extension of the results in Index Laws for Semigroup in which the domain of the indices is extended to include all integers. Let $\struct {S, \circ}$ be a monoid whose identity is $e$. Let $a \in S$ be invertible for $\circ$. Let $n \in \N$. Let $a^n$ be the $n$th power of $a$: :$a^n = \begin{cases} e : & n = 0 \\ a^{n - 1} \circ a : & n > 0 \\ \paren {a^{-n}}^{-1} : & n < 0 \end{cases}$ Then we have the following results:
Totally Ordered Abelian Group Isomorphism
Let $\left({\Z', +', \le'}\right)$ be a totally ordered abelian group. Let $0'$ be the identity of $\left({\Z', +', \le'}\right)$. Let $\N' = \left\{{x \in \Z': x \ge' 0'}\right\}$. Let $\Z'$ contain at least two elements. Let $\N'$ be well-ordered for the ordering induced on $\N'$ by $\le'$. Then the mapping $g: \Z \to \Z'$ defined by: :$\forall n \in \Z: g \left({n}\right) = \left({+'}\right)^n 1'$ is an isomorphism from $\left({\Z, +, \le}\right)$ onto $\left({\Z', +', \le'}\right)$, where $1'$ is the smallest element of $\N' \setminus \left\{{0'}\right\}$.
Integers under Addition form Totally Ordered Group
Then the ordered structure $\struct {\Z, +, \le}$ is a totally ordered group.
Integers form Totally Ordered Ring
The structure $\struct {\Z, +, \times, \le}$ is a totally ordered ring.
Congruences on Rational Numbers
There are only two congruence relations on the field of rational numbers $\left({\Q, +, \times}\right)$: :$(1): \quad$ The diagonal relation $\Delta_\Q$ :$(2): \quad$ The trivial relation $\Q \times \Q$.
Number of Elements in Partition
Let $S$ be a set. Let there be a partition on $S$ of $n$ subsets, each of which has $m$ elements. Then: :$\card S = n m$
Cardinality of Complement
Let $T \subseteq S$ such that $\card S = n, \card T = m$. Then: :$\card {\relcomp S T} = \card {S \setminus T} = n - m$ where: :$\relcomp S T$ denotes the complement of $T$ relative to $S$ :$S \setminus T$ denotes the difference between $S$ and $T$.
Cardinality of Cartesian Product
Let $S \times T$ be the cartesian product of two finite sets $S$ and $T$. Then: :$\card {S \times T} = \card S \times \card T$ where $\card S$ denotes cardinality.
Cardinality of Set of All Mappings
Let $S$ and $T$ be sets. The cardinality of the set of all mappings from $S$ to $T$ (that is, the total number of mappings from $S$ to $T$) is: :$\card {T^S} = \card T^{\card S}$
Cardinality of Power Set of Finite Set
Let $S$ be a set such that: :$\card S = n$ where $\card S$ denotes the cardinality of $S$, Then: :$\card {\powerset S} = 2^n$ where $\powerset S$ denotes the power set of $S$.
Cardinality of Set of Injections
Let $S$ and $T$ be finite sets. The number of injections from $S$ to $T$, where $\card S = m, \card T = n$ is often denoted ${}^m P_n$, and is: :${}^m P_n = \begin{cases} \dfrac {n!} {\paren {n - m}!} & : m \le n \\ 0 & : m > n \end{cases}$
Cardinality of Set of Bijections
Let $S$ and $T$ be sets such that $\size S = \size T = n$. Then there are $n!$ bijections from $S$ to $T$.
Cardinality of Set of Subsets
Let $S$ be a set such that $\card S = n$. Let $m \le n$. Then the number of subsets $T$ of $S$ such that $\card T = m$ is: : ${}^m C_n = \dfrac {n!} {m! \paren {n - m}!}$
Set of Integers Bounded Below by Integer has Smallest Element
Let $\Z$ be the set of integers. Let $\le$ be the ordering on the integers. Let $\O \subset S \subseteq \Z$ such that $S$ is bounded below in $\struct {\Z, \le}$. Then $S$ has a smallest element.
Set of Integers Bounded Above by Integer has Greatest Element
Let $\Z$ be the set of integers. Let $\le$ be the ordering on the integers. Let $\O \subset S \subseteq \Z$ such that $S$ is bounded above in $\struct {\Z, \le}$. Then $S$ has a greatest element.
Principle of Least Counterexample
Suppose $P \paren n$ is a condition on $n \in \set {x \in \Z: x \ge m \in \Z}$. Suppose next that: $\neg \paren {\forall n \ge m: P \paren n}$. (That is, not all $n \ge m$ satisfy $P \paren n$.) Then there is a '''least counterexample''', that is a smallest integral value of $n$ for which $\neg P \paren n$.
Absolute Value is Bounded Below by Zero
Let $x \in \R$ be a real number. Then the absolute value $\size x$ of $x$ is bounded below by $0$.
Division Theorem
For every pair of integers $a, b$ where $b \ne 0$, there exist unique integers $q, r$ such that $a = q b + r$ and $0 \le r < \size b$: :$\forall a, b \in \Z, b \ne 0: \exists! q, r \in \Z: a = q b + r, 0 \le r < \size b$
Odd Integer 2n + 1
Let $m$ be an odd integer. Then there exists exactly one integer $n$ such that $2 n + 1 = m$.
Integer Divisor Results
Let $m, n \in \Z$ be integers. Let $m \divides n$ denote that $m$ is a divisor of $n$. The following results all hold:
Zero Divides Zero
Let $n \in \Z$ be an integer. Then: :$0 \divides n \implies n = 0$ That is, zero is the only integer divisible by zero.
Absolute Value of Integer is not less than Divisors
A (non-zero) integer is greater than or equal to its divisors in magnitude: :$\forall c \in \Z_{\ne 0}: a \divides c \implies a \le \size a \le \size c$
Divisor Relation on Positive Integers is Partial Ordering
The divisor relation is a partial ordering of $\Z_{>0}$.
Common Divisor in Integral Domain Divides Linear Combination
Let $\struct {D, +, \times}$ be an integral domain. Let $c$ be a common divisor of two elements $a$ and $b$ of $D$. That is: :$a, b, c \in D: c \divides a \land c \divides b$ Then: :$\forall p, q \in D: c \divides \paren {p \times a + q \times b}$
Greatest Common Divisor is at least 1
Let $a, b \in \Z$ be integers. The greatest common divisor of $a$ and $b$ is at least $1$: :$\forall a, b \in \Z_{\ne 0}: \gcd \set {a, b} \ge 1$
GCD of Integer and Divisor
Let $a, b \in \Z_{>0}$, i.e. integers such that $a, b > 0$. Then: : $a \divides b \implies \gcd \set {a, b} = a$
GCD for Negative Integers
:$\gcd \set {a, b} = \gcd \set {\size a, b} = \gcd \set {a, \size b} = \gcd \set {\size a, \size b}$ Alternatively, this can be put: :$\gcd \set {a, b} = \gcd \set {-a, b} = \gcd \set {a, -b} = \gcd \set {-a, -b}$   which follows directly from the above.
GCD with Zero
Let $a \in \Z$ be an integer such that $a \ne 0$. Then: :$\gcd \left\{{a, 0}\right\} = \left\lvert{a}\right\rvert$ where $\gcd$ denotes greatest common divisor (GCD).
Set of Integer Combinations equals Set of Multiples of GCD
The set of all integer combinations of $a$ and $b$ is precisely the set of all integer multiples of the GCD of $a$ and $b$: :$\gcd \set {a, b} \divides c \iff \exists x, y \in \Z: c = x a + y b$
GCD with Remainder
Let $a, b \in \Z$. Let $q, r \in \Z$ such that $a = q b + r$. Then: :$\gcd \set {a, b} = \gcd \set {b, r}$ where $\gcd \set {a, b}$ is the greatest common divisor of $a$ and $b$.
Integer Combination of Coprime Integers
Two integers are coprime  there exists an integer combination of them equal to $1$: :$\forall a, b \in \Z: a \perp b \iff \exists m, n \in \Z: m a + n b = 1$
Integers Divided by GCD are Coprime
Any pair of integers, not both zero, can be reduced to a pair of coprime ones by dividing them by their GCD: :$\gcd \set {a, b} = d \iff \dfrac a d, \dfrac b d \in \Z \land \gcd \set {\dfrac a d, \dfrac b d} = 1$ That is: :$\dfrac a {\gcd \set {a, b} } \perp \dfrac b {\gcd \set {a, b} }$
Product of Coprime Factors
Let $a, b, c \in \Z$ such that $a$ and $b$ are coprime. Let both $a$ and $b$ be divisors of $c$. Then $a b$ is also a divisor of $c$. That is: :$a \perp b \land a \divides c \land b \divides c \implies a b \divides c$
Existence of Lowest Common Multiple
Let $a, b \in \Z: a b \ne 0$. The lowest common multiple of $a$ and $b$, denoted $\lcm \set {a, b}$, always exists.
Product of GCD and LCM
:$\lcm \set {a, b} \times \gcd \set {a, b} = \size {a b}$ where: :$\lcm \set {a, b}$ denotes the lowest common multiple of $a$ and $b$ :$\gcd \set {a, b}$ denotes the greatest common divisor of $a$ and $b$.
Congruent to Zero if Modulo is Divisor
Let $a, z \in \R$. Then $a$ is congruent to $0$ modulo $z$  $a$ is an integer multiple of $z$. :$\exists k \in \Z: k z = a \iff a \equiv 0 \pmod z$ If $z \in \Z$, then further: :$z \divides a \iff a \equiv 0 \pmod z$
Integer is Congruent Modulo Divisor to Remainder
Let $a \in \Z$. Let $a$ have a remainder $r$ on division by $m$. Then: : $a \equiv r \pmod m$ where the notation denotes that $a$ and $r$ are congruent modulo $m$.
Integer is Congruent to Integer less than Modulus
Let $m \in \Z$. Then each integer is congruent (modulo $m$) to precisely one of the integers $0, 1, \ldots, m - 1$.
Modulo Addition is Well-Defined
Let $m \in \Z$ be an integer. Let $\Z_m$ be the set of integers modulo $m$. The modulo addition operation on $\Z_m$, defined by the rule: :$\eqclass a m +_m \eqclass b m = \eqclass {a + b} m$ is a well-defined operation. That is: :If $a \equiv b \pmod m$ and $x \equiv y \pmod m$, then $a + x \equiv b + y \pmod m$.
Modulo Multiplication is Well-Defined
The multiplication modulo $m$ operation on $\Z_m$, the set of integers modulo $m$, defined by the rule: :$\eqclass x m \times_m \eqclass y m = \eqclass {x y} m$ is a well-defined operation. That is: :If $a \equiv b \pmod m$ and $x \equiv y \pmod m$, then $a x \equiv b y \pmod m$.
Congruence of Product
Let $a, b, z \in \R$. Let $a$ be congruent to $b$ modulo $z$, that is: :$a \equiv b \pmod z$ Then: :$\forall m \in \Z: m a \equiv m b \pmod z$
Congruence of Powers
Let $a, b \in \R$ and $m \in \Z$. Let $a$ be congruent to $b$ modulo $m$, that is: :$a \equiv b \pmod m$ Then: :$\forall n \in \Z_{\ge 0}: a^n \equiv b^n \pmod m$
Modulo Addition is Associative
Addition modulo $m$ is associative: :$\forall \eqclass x m, \eqclass y m, \eqclass z m \in \Z_m: \paren {\eqclass x m +_m \eqclass y m} +_m \eqclass z m = \eqclass x m +_m \paren {\eqclass y m +_m \eqclass z m}$ where $\Z_m$ is the set of integers modulo $m$. That is: :$\forall x, y, z \in \Z: \paren {x + y} + z \equiv x + \paren {y + z} \pmod m$
Modulo Addition is Commutative
Modulo addition is commutative: :$\forall x, y, z \in \Z: x + y \pmod m = y + x \pmod m$
Modulo Addition has Identity
Let $m \in \Z$ be an integer. Then addition modulo $m$ has an identity: :$\forall \eqclass x m \in \Z_m: \eqclass x m +_m \eqclass 0 m = \eqclass x m = \eqclass 0 m +_m \eqclass x m$ That is: :$\forall a \in \Z: a + 0 \equiv a \equiv 0 + a \pmod m$
Modulo Addition has Inverses
Let $m \in \Z$ be an integer. Then addition modulo $m$ has inverses: For each element $\eqclass x m \in \Z_m$, there exists the element $\eqclass {-x} m \in \Z_m$ with the property: :$\eqclass x m +_m \eqclass {-x} m = \eqclass 0 m = \eqclass {-x} m +_m \eqclass x m$ where $\Z_m$ is the set of integers modulo $m$. That is: :$\forall a \in \Z: a + \paren {-a} \equiv 0 \equiv \paren {-a} + a \pmod m$
Modulo Multiplication is Closed
Multiplication modulo $m$ is closed on the set of integers modulo $m$: :$\forall \eqclass x m, \eqclass y m \in \Z_m: \eqclass x m \times_m \eqclass y m \in \Z_m$.
Modulo Multiplication is Associative
Multiplication modulo $m$ is associative: :$\forall \eqclass x m, \eqclass y m, \eqclass z m \in \Z_m: \paren {\eqclass x m \times_m \eqclass y m} \times_m \eqclass z m = \eqclass x m \times_m \paren {\eqclass y m \times_m \eqclass z m}$ That is: :$\forall x, y, z \in \Z_m: \paren {x \cdot_m y} \cdot_m z = x \cdot_m \paren {y \cdot_m z}$
Modulo Multiplication is Commutative
Multiplication modulo $m$ is commutative: :$\forall \eqclass x m, \eqclass y m \in \Z_m: \eqclass x m \times_m \eqclass y m = \eqclass y m \times_m \eqclass x m$
Modulo Multiplication has Identity
Multiplication modulo $m$ has an identity: :$\forall \eqclass x m \in \Z_m: \eqclass x m \times_m \eqclass 1 m = \eqclass x m = \eqclass 1 m \times_m \eqclass x m$
Modulo Multiplication Distributes over Modulo Addition
Multiplication modulo $m$ is distributive over addition modulo $m$: :$\forall \eqclass x m, \eqclass y m, \eqclass z m \in \Z_m$: :: $\eqclass x m \times_m \paren {\eqclass y m +_m \eqclass z m} = \paren {\eqclass x m \times_m \eqclass y m} +_m \paren {\eqclass x m \times_m \eqclass z m}$ :: $\paren {\eqclass x m +_m \eqclass y m} \times_m \eqclass z m = \paren {\eqclass x m \times_m \eqclass z m} +_m \paren {\eqclass y m \times_m \eqclass z m}$ where $\Z_m$ is the set of integers modulo $m$. That is, $\forall x, y, z, m \in \Z$: : $x \paren {y + z} \equiv x y + x z \pmod m$ : $\paren {x + y} z \equiv x z + y z \pmod m$