hcju/proofwikips huggingface dataset:
Greatest Common Divisor in Principal Ideal Domain is Expressible as Linear Combination
Let $\struct {D, +, \circ}$ be a principal ideal domain. Let $S = \set {a_1, a_2, \dotsc, a_n}$ be a set of non-zero elements of $D$. Let $y$ be a greatest common divisor of $S$. Then $y$ is expressible in the form: :$y = d_1 a_1 + d_2 a_2 + \dotsb + d_n a_n$ where $d_1, d_2, \dotsc, d_n \in D$.
Complete Factorizations of Proper Element in Principal Ideal Domain are Equivalent
Let $\struct {D, +, \circ}$ be a principal ideal domain. Let $x \in D$ be a proper element of $D$. Let there be two complete factorizations of $x$: :$x = u_y \circ y_1 \circ y_2 \circ \cdots \circ y_m = F_1$ :$x = u_z \circ z_1 \circ z_2 \circ \cdots \circ z_n = F_2$ Then $F_1$ and $F_2$ are equivalent.
Moment Generating Function of Poisson Distribution
Let $X$ be a discrete random variable with a Poisson distribution with parameter $\lambda$ for some $\lambda \in \R_{> 0}$. Then the moment generating function $M_X$ of $X$ is given by:  :$\map {M_X} t = e^{\lambda \paren {e^t - 1} }$
Moment Generating Function of Binomial Distribution
Let $X$ be a discrete random variable with a binomial distribution with parameters $n$ and $p$ for some $n \in \N$ and $0 \le p \le 1$: :$X \sim \Binomial n p$ Then the moment generating function $M_X$ of $X$ is given by:  :$\map {M_X} t = \paren {1 - p + p e^t}^n$
Moment Generating Function of Exponential Distribution
Let $X$ be a continuous random variable with an exponential distribution with parameter $\beta$ for some $\beta \in \R_{> 0}$. Then the moment generating function $M_X$ of $X$ is given by:  :$\displaystyle \map {M_X} t = \frac 1 {1 - \beta t}$ for $t < \dfrac 1 \beta$, and is undefined otherwise.
Moment Generating Function of Geometric Distribution
Let $X$ be a discrete random variable with a geometric distribution with parameter $p$ for some $0 < p \le 1$.  Then the moment generating function $M_X$ of $X$ is given by:  :$\displaystyle \map {M_X} t = \frac p {1 - \paren {1 - p} e^t}$ for $t < -\map \ln {1 - p}$, and is undefined otherwise.
Maximal Ideal iff Quotient Ring is Field/Proof 1/Maximal Ideal implies Quotient Ring is Field
Let $\struct {R, +, \circ}$ be a commutative ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $J$ be an ideal of $R$. Let $J$ be a maximal ideal. Then the quotient ring $R / J$ is a field.
Maximal Ideal iff Quotient Ring is Field/Proof 1/Quotient Ring is Field implies Ideal is Maximal
Let $\struct {R, +, \circ}$ be a commutative ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $J$ be an ideal of $R$. Let the quotient ring $R / J$ be a field. Then $J$ is a maximal ideal.
Ring of Polynomial Forms is not necessarily Isomorphic to Ring of Polynomial Functions
Let $D$ be an integral domain. Let $D \sqbrk X$ be the ring of polynomial forms in $X$ over $D$. Let $\map P D$ be the ring of polynomial functions over $D$. Then it is not necessarily the case that $D \sqbrk X$ is isomorphic with $\map P D$.
Moment Generating Function of Bernoulli Distribution
Let $X$ be a discrete random variable with a Bernoulli distribution with parameter $p$ for some $0 \le p \le 1$.  Then the moment generating function $M_X$ of $X$ is given by:  :$\map {M_X} t = q + p e^t$ where $q = 1 - p$.
Double of Antiperiodic Element is Periodic
Let $f: X \to X$ be a function, where $X$ is either $\R$ or $\C$. Let $L \in X_{\ne 0}$ be an anti-periodic element of $f$. Then $2L$ is a periodic element of $f$. In other words, every anti-periodic function is also periodic.
Skewness of Gaussian Distribution
Let $X$ be a continuous random variable with a Gaussian distribution with parameters $\mu$ and $\sigma^2$ for some $\mu \in \R$ and $\sigma \in \R_{> 0}$.  Then the skewness $\gamma_1$ of $X$ is equal to $0$.
Inverse of Unit in Centralizer of Ring is in Centralizer
Let $\struct {R, +, \circ}$ be a ring. Let $S$ be a subset of $R$. Let $\map {C_R} S$ denote the centralizer of $S$ in $R$ Let $u \in R$ be a unit of $R$. Then: :$u \in \map {C_R} S \implies u^{-1} \in \map {C_R} S$
Skewness in terms of Non-Central Moments
Let $X$ be a random variable with mean $\mu$ and standard deviation $\sigma$. Then the skewness $\gamma_1$ of $X$ is given by:  :$\gamma_1 = \dfrac {\expect {X^3} - 3 \mu \sigma^2 - \mu^3} {\sigma^3}$
Intersection of Ring Ideals is Ideal
Let $\struct {R, +, \circ}$ be a ring Let $\mathbb L$ be a non-empty set of ideals of $R$. Then the intersection $\bigcap \mathbb L$ of the members of $\mathbb L$ is itself an ideal of $R$.
Set of Ring Elements forming Zero Product with given Element is Ideal
Let $\struct {R, +, \circ}$ be a commutative ring whose zero is $0_R$. Let $a \in R$ be an arbitrary element of $R$. Let $A$ be the subset of $R$ defined as: :$A = \set {x \in R: x \circ a = 0_R}$ Then $A$ is an ideal of $A$.
Skewness of Bernoulli Distribution
Let $X$ be a discrete random variable with a Bernoulli distribution with parameter $p$. Then the skewness $\gamma_1$ of $X$ is given by:  :$\gamma_1 = \dfrac {1 - 2 p} {\sqrt {p q} }$ where $q = 1 - p$.
Intersection of Subrings is Subring
Let $\struct {R, +, \circ}$ be a ring. Let $\mathbb L$ be a non-empty set of subrings of $R$. Then the intersection $\displaystyle \bigcap \mathbb L$ of the members of $\mathbb L$ is itself a subring of $R$.
Intersection of Ring Ideals is Largest Ideal Contained in all Ideals
Let $\struct {R, +, \circ}$ be a ring Let $\mathbb L$ be a non-empty set of ideals of $R$. Then the intersection $\bigcap \mathbb L$ of the members of $\mathbb L$ is the largest ideal of $R$ contained in each member of $\mathbb L$.
Intersection of Subrings is Largest Subring Contained in all Subrings
Let $\struct {R, +, \circ}$ be a ring. Let $\mathbb L$ be a non-empty set of subrings of $R$. Then the intersection $\displaystyle \bigcap \mathbb L$ of the members of $\mathbb L$ is the largest subring of $R$ contained in each member of $\mathbb L$.
Median of Continuous Uniform Distribution
Let $X$ be a continuous random variable which is uniformly distributed on a closed real interval $\closedint a b$.  Then the median $M$ of $X$ is given by:  :$M = \dfrac {a + b} 2$
Intersection of Ring Ideals Containing Subset is Smallest
Let $\struct {R, +, \circ}$ be a ring Let $S \subseteq R$ be a subset of $R$. Let $L$ be the intersection of the set of all ideals of $R$ containing $S$. Then $L$ is the smallest ideal of $R$ containing $S$.
Intersection of Subrings Containing Subset is Smallest
Let $\struct {R, +, \circ}$ be a ring. Let $S \subseteq R$ be a subset of $R$. Let $L$ be the intersection of the set of all subrings of $R$ containing $S$. Then $L$ is the smallest subring of $R$ containing $S$.
Intersection of Division Subrings is Division Subring
Let $\struct {D, +, \circ}$ be a division ring. Let $\mathbb K$ be a non-empty set of division subrings of $D$. Then the intersection $\bigcap \mathbb K$ of the members of $\mathbb K$ is itself a division subring of $D$.
Intersection of Subfields is Subfield
Let $\struct {F, +, \circ}$ be a field. Let $\mathbb K$ be a non-empty set of subfields of $F$. Then the intersection $\bigcap \mathbb K$ of the members of $\mathbb K$ is itself a subfield of $F$.
Intersection of Division Subrings is Largest Division Subring Contained in all Division Subrings
Let $\struct {D, +, \circ}$ be a division ring. Let $\mathbb K$ be a non-empty set of division subrings of $D$. Let $\bigcap \mathbb K$ be the intersection of the elements of $\mathbb K$. Then $\bigcap \mathbb K$ is the largest division subring of $D$ contained in each element of $\mathbb K$.
Intersection of Division Subrings Containing Subset is Smallest
Let $\struct {D, +, \circ}$ be a division ring. Let $S \subseteq D$ be a subset of $D$. Let $L$ be the intersection of the set of all division subrings of $D$ containing $S$. Then $L$ is the smallest division subring of $D$ containing $S$.
Intersection of Subfields Containing Subset is Smallest
Let $\struct {F, +, \circ}$ be a field. Let $S \subseteq F$ be a subset of $F$. Let $L$ be the intersection of the set of all subfields of $F$ containing $S$. Then $L$ is the smallest subfield of $F$ containing $S$.
Intersection of Subfields is Largest Subfield Contained in all Subfields
Let $\struct {F, +, \circ}$ be a field. Let $\mathbb K$ be a non-empty set of subfields of $F$. Let $\bigcap \mathbb K$ be the intersection of the elements of $\mathbb K$. Then $\bigcap \mathbb K$ is the largest subfield of $F$ contained in each element of $\mathbb K$.
Equivalent Norms on Rational Numbers/Necessary Condition
Let $\norm {\, \cdot \,}_1$ and $\norm {\, \cdot \,}_2$ be norms on the rational numbers $\Q$. Let $\norm {\, \cdot \,}_1$ and $\norm {\, \cdot \,}_2$ be equivalent norms. Then: :$\exists \alpha \in \R_{\gt 0}: \forall n \in \N: \norm n_1 = \norm n_2^\alpha$
Equivalent Norms on Rational Numbers/Sufficient Condition
Let $\norm {\, \cdot \,}_1$ and $\norm {\, \cdot \,}_2$ be norms on the rational numbers $\Q$. Let $\norm {\, \cdot \,}_1$ and $\norm {\, \cdot \,}_2$ satisfy: :$\exists \alpha \in \R_{\gt 0}: \forall n \in \N: \norm n_1 = \norm n_2^\alpha$ Then $\norm {\, \cdot \,}_1$ and $\norm {\, \cdot \,}_2$ are equivalent
Commutative and Unitary Ring with 2 Ideals is Field
Let $\struct {R, +, \circ}$ be a commutative ring with unity whose zero is $0_R$. Let $\struct {R, +, \circ}$ be such that the only ideals of $\struct {R, +, \circ}$ are: :$\set {0_R}$ and: $\struct {R, +, \circ}$ itself. That is, such that $\struct {R, +, \circ}$ has no non-null proper ideals. Then $\struct {R, +, \circ}$ is a field.
Field has 2 Ideals
Let $\struct {F, +, \circ}$ be a field whose zero is $0_F$ and whose unity is $1_F$. Then the only ideals of $\struct {F, +, \circ}$ are:$\struct {F, +, \circ}$ and $\set {0_F}$. That is, $\struct {F, +, \circ}$ has no non-null proper ideals.
General Antiperiodicity Property
Let $f: X \to X$ be an antiperiodic function, where $X$ is either $\R$ or $\C$. Let $L$ be an antiperiodic element of $f$. Let $n \in \Z$ be an integer. :If $n$ is even, then $n L$ is a periodic element of $f$. :If $n$ is odd, then $n L$ is an antiperiodic element of $f$.
Antiperiodic Element is Multiple of Antiperiod
Let $f: \R \to \R$ be a real anti-periodic function with anti-period $A$. Let $L$ be an anti-periodic element of $f$. Then $A \divides L$.
Skewness of Poisson Distribution
Let $X$ be a discrete random variable with a Poisson distribution with parameter $\lambda$. Then the skewness $\gamma_1$ of $X$ is given by:  :$\gamma_1 = \dfrac 1 {\sqrt \lambda}$
Periodic Element is Multiple of Antiperiod
Let $f: \R \to \R$ be a real anti-periodic function with anti-period $A$. Let $L$ be a periodic element of $f$. Then $A \divides L$.
Skewness of Continuous Uniform Distribution
Let $X$ be a continuous random variable which is uniformly distributed on a closed real interval $\closedint a b$.  Then the skewness $\gamma_1$ of $X$ is equal to $0$.
Principal Ideal in Integral Domain generated by Power Plus One is Subset of Principal Ideal generated by Power
Let $\struct {D, +, \circ}$ be an integral domain whose unity is $1_D$. Let $a \in D$ be a proper element of $D$. Then: :$\forall n \in \Z_{\ge 0}: \ideal {a^{n + 1} } \subsetneq \ideal {a_n}$ where $\ideal x$ denotes the principal ideal of $D$ generated by $x$.
Non-Field Integral Domain has Infinite Number of Ideals
Let $\struct {D, +, \circ}$ be an integral domain which is not a field. Then $\struct {D, +, \circ}$ has an infinite number of distinct ideals.
Ring Homomorphism from Ring with Unity to Integral Domain Preserves Unity
Let $\struct {R, +_R, \circ_R}$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $\struct {D, +_D, \circ_D}$ be an integral domain whose zero is $0_D$ and whose unity is $1_D$. Let $\phi: R \to D$ be a ring homomorphism such that: :$\map \ker \phi \ne R$ where $\map \ker \phi$ denotes the kernel of $\phi$. Then $\map \phi {1_R} = 1_D$.
Unity plus Negative of Nilpotent Ring Element is Unit
Let $\struct {R, +, \circ}$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $x \in R$ be nilpotent. Then $1_R - x$ is a unit of $R$.
Quotient of Commutative Ring by Nilradical is Reduced
Let $\struct {R, +, \circ}$ be a commutative ring whose zero is $0_R$ and whose unity is $1_R$. Let $\struct {N, +, \circ}$ denote the nilradical of $R$. The quotient ring $R / N$ is a reduced ring.
Self-Inverse Element of Integral Domain is Unity or its Negative
Let $\struct {D, +, \circ}$ be an integral domain whose zero is $0_D$ and whose unity is $1_D$. Let $x \in D$ such that $x^2 = 1_D$. Then either $x = 1_D$ or $x = -1_D$.
Product of Units of Integral Domain with Finite Number of Units
Let $\struct {D, +, \circ}$ be an integral domain whose zero is $0_D$ and whose unity is $1_D$. Let $D$ have a finite number of units. Let $U_D$ be the set of units of $\struct {D, +, \circ}$. Then: :$\displaystyle \prod_{x \mathop \in U_D} x = -1_D$
Prime Ideals of Ring of Integers
Let $\struct {\Z, +, \times}$ denote the ring of integers. Let $J$ be a prime ideal of $\Z$. Then either: :$J = \set 0$ or: :$J = \ideal p$ where: :$p$ is a prime number :$\ideal p$ denotes the principal ideal of $\Z$ generated by $p$.
Additive Group and Multiplicative Group of Field are not Isomorphic
Let $\struct {F, +, \times}$ be a field whose zero is $0_F$ and whose unity is $1_F$. Let $\struct {F, +}$ denote the additive group of $F$. Let $\struct {F_{\ne 0_F}, \times}$ denote the multiplicative group of $F$. Then $\struct {F, +}$ and $\struct {F_{\ne 0_F}, \times}$ are not isomorphic to each other.
Field Norm of Complex Number is Positive Definite
Let $\C$ denote the set of complex numbers. Let $N: \C \to \R_{\ge 0}$ denote the field norm on complex numbers: :$\forall z \in \C: \map N z = \cmod z^2$ where $\cmod z$ denotes the complex modulus of $z$. Then $N$ is positive definite on $\C$.
Field Norm of Complex Number is Multiplicative Function
Let $\C$ denote the set of complex numbers. Let $N: \C \to \R_{\ge 0}$ denote the field norm on complex numbers: :$\forall z \in \C: \map N z = \cmod z^2$ where $\cmod z$ denotes the complex modulus of $z$. Then $N$ is a multiplicative function on $\C$.
Units of 5th Cyclotomic Ring
Let $\struct {\Z \sqbrk {i \sqrt 5}, +, \times}$ denote the $5$th cyclotomic ring. The units of $\struct {\Z \sqbrk {i \sqrt 5}, +, \times}$ are $1$ and $-1$.
Field Norm on 5th Cyclotomic Ring
Let $\struct {\Z \sqbrk {i \sqrt 5}, +, \times}$ denote the $5$th cyclotomic ring. Let $\alpha = a + i b \sqrt 5$ be an arbitrary element of $\Z \sqbrk {i \sqrt 5}$. The field norm of $\alpha$ is given by: :$\map N \alpha = a^2 + 5 b^2$
5th Cyclotomic Ring has no Elements with Field Norm of 2 or 3
Let $\struct {\Z \sqbrk {i \sqrt 5}, +, \times}$ denote the $5$th cyclotomic ring. There are no elements of $\struct {\Z \sqbrk {i \sqrt 5}, +, \times}$ whose field norm is either $2$ or $3$.
Irreducible Elements of 5th Cyclotomic Ring
Let $\struct {\Z \sqbrk {i \sqrt 5}, +, \times}$ denote the $5$th cyclotomic ring. The following elements of $\struct {\Z \sqbrk {i \sqrt 5}, +, \times}$ are irreducible: :$2$ :$3$ :$1 + i \sqrt 5$ :$1 - i \sqrt 5$
Value of Field Norm on 5th Cyclotomic Ring is Integer
Let $\struct {\Z \sqbrk {i \sqrt 5}, +, \times}$ denote the $5$th cyclotomic ring. Let $\alpha = a + i b \sqrt 5$ be an arbitrary element of $\Z \sqbrk {i \sqrt 5}$. Let $\map N \alpha$ denoted the field norm of $\alpha$. Then $\map N \alpha$ is an integer.
Elements of 5th Cyclotomic Ring with Field Norm 1
Let $\struct {\Z \sqbrk {i \sqrt 5}, +, \times}$ denote the $5$th cyclotomic ring. The only elements of $\struct {\Z \sqbrk {i \sqrt 5}, +, \times}$ whose field norm equals $1$ are the units of $\struct {\Z \sqbrk {i \sqrt 5}, +, \times}$: $1$ and $-1$.
Polynomials in Integers with Even Constant Term forms Ideal
Let $\Z \sqbrk X$ be the ring of polynomials in $X$ over $\Z$. Let $S \subseteq \Z \sqbrk X$ be the set of polynomials over $\Z$ in $X$ which have a constant term which is even. Then $S$ is an ideal of $\Z \sqbrk X$.
Polynomials in Integers is Unique Factorization Domain
Let $\Z \sqbrk X$ be the ring of polynomials in $X$ over $\Z$. Then $\Z \sqbrk X$ is a unique factorization domain.
Ring of Gaussian Integers is Principal Ideal Domain
The ring of Gaussian integers: :$\struct {\Z \sqbrk i, +, \times}$ forms a principal ideal domain.
Matrix Multiplication is not Commutative
Let $R$ be a ring with unity. Let $n \in \Z_{>0}$ be a (strictly) positive integer such that $n \ne 1$. Let $\map {\MM_R} n$ denote the $n \times n$ matrix space over $R$. Then (conventional) matrix multiplication over $\map {\MM_R} n$ is not commutative: :$\exists \mathbf A, \mathbf B \in \map {\MM_R} n: \mathbf {A B} \ne \mathbf {B A}$ If $R$ is specifically not commutative, then the result holds when $n = 1$ as well.
Matrix Multiplication on Square Matrices over Trivial Ring is Commutative
Let $\struct {R, +, \circ}$ be the trivial ring over an underlying set. Let $\map {\MM_R} n$ denote the $n \times n$ matrix space over $R$. Then (conventional) matrix multiplication is commutative over $\map {\MM_R} n$: :$\forall \mathbf A, \mathbf B \in \map {\MM_R} n: \mathbf {A B} = \mathbf {B A}$
Area between Smooth Curve and Line with Fixed Endpoints is Maximized by Arc of Circle
Let $y$ be a smooth curve, embedded in $2$-dimensional Euclidean space. Let $y$ have a total length of $l$. Let it be contained in the upper halfplane with an exception of endpoints, which are on the x-axis and are given. Suppose, $y$, together with a line segment connecting $y$'s endpoints, maximizes the enclosed area. Then $y$ is an arc of a circle.
Preordering of Products under Operation Compatible with Preordering
Let $\struct {S, \circ}$ be an algebraic structure. Let $\precsim$ be a preordering on $S$. Then $\precsim$ is compatible with $\circ$ : :$\forall x_1, x_2, y_1, y_2 \in S: x_1 \precsim x_2 \land y_1 \precsim y_2 \implies \paren {x_1 \circ y_1} \precsim \paren {x_2 \circ y_2}$
P-adic Norm not Complete on Rational Numbers/Proof 2/Lemma 1
:$\exists x \in \Z_{>0}:  p \nmid x, x \ge \dfrac {p + 1} 2$
P-adic Norm not Complete on Rational Numbers/Proof 2/Lemma 5
:$\displaystyle \lim_{n \mathop \to \infty} x_n^k = a$ in $\struct {\Q, \norm {\,\cdot\,}_p}$
Characterisation of Cauchy Sequence in Non-Archimedean Norm/Corollary 1
Let $\norm {\,\cdot\,}_p$ be the $p$-adic norm on the rationals $\Q$ for some prime $p$. Let $\sequence {x_n}$ be a sequence of integers such that: :$\forall n: x_{n + 1} \equiv x_n \pmod {p^n}$ Then: :$\sequence {x_n}$ is a Cauchy sequence in $\struct {\Q, \norm {\,\cdot\,}_p}$.
P-adic Norm not Complete on Rational Numbers/Proof 2/Lemma 2
:$a \in \Z_{> 0}: \nexists \,c \in \Z : c^k = a$
P-adic Norm not Complete on Rational Numbers/Proof 2/Lemma 3
:$\map f {x_1} \equiv 0 \pmod p$
P-adic Norm not Complete on Rational Numbers/Proof 2/Lemma 4
:$\map {f'} {x_1} \not \equiv 0 \pmod p$
Surjection from Natural Numbers iff Countable/Corollary 1
Let $T$ be a countably infinite set. Let $S$ be a non-empty set. Then $S$ is countable  there exists a surjection $f: T \to S$.
Surjection from Natural Numbers iff Countable/Corollary 2
Let $T$ be a countably infinite set. Let $S$ be an uncountable set. Let $f:T \to S$ be a mapping. Then $f$ is not a surjection.
P-adic Numbers are Uncountable
Let $p$ be any prime number. The set of $p$-adic numbers $\Q_p$ is an uncountable set.
Ordered Integral Domain is Totally Ordered Ring
Let $\struct {D, +, \times, \le}$ be an ordered integral domain. Then $\struct {D, +, \times, \le}$ is a totally ordered ring.
Strict Negativity is equivalent to Strictly Preceding Zero
:$\map N a \iff a < 0$
Sum of Strictly Negative Elements is Strictly Negative
:$\map N a, \map N b \implies \map N {a + b}$
Product of Two Strictly Negative Elements is Strictly Positive
:$\map N a, \map N b \implies \map P {a \times b}$
Product of Strictly Negative Element with Strictly Positive Element is Strictly Negative
:$\map N a, \map P b \implies \map N {a \times b}$
Principle of Mathematical Induction/Zero-Based
Let $\map P n$ be a propositional function depending on $n \in \N$. Suppose that: :$(1): \quad \map P 0$ is true :$(2): \quad \forall k \in \N: k \ge 0 : \map P k \implies \map P {k + 1}$ Then: :$\map P n$ is true for all $n \in \N$.
Principle of Mathematical Induction/One-Based
Let $\map P n$ be a propositional function depending on $n \in \N_{>0}$. Suppose that: :$(1): \quad \map P 1$ is true :$(2): \quad \forall k \in \N_{>0}: k \ge 1 : \map P k \implies \map P {k + 1}$ Then: :$\map P n$ is true for all $n \in \N_{>0}$.
Principle of Finite Induction/One-Based
Let $S \subseteq \N_{>0}$ be a subset of the $1$-based natural numbers. Suppose that: :$(1): \quad 1 \in S$ :$(2): \quad \forall n \in \N_{>0} : n \in S \implies n + 1 \in S$ Then: :$S = \N_{>0}$
Second Principle of Finite Induction/One-Based
Let $S \subseteq \N_{>0}$ be a subset of the $1$-based natural numbers. Suppose that: :$(1): \quad 1 \in S$ :$(2): \quad \forall n \in \N_{>0}: \paren {\forall k: 1 \le k \le n \implies k \in S} \implies n + 1 \in S$ Then: :$S = \N_{>0}$
Second Principle of Mathematical Induction/Zero-Based
Let $\map P n$ be a propositional function depending on $n \in \N$. Suppose that: :$(1): \quad \map P 0$ is true :$(2): \quad \forall k \in \N: \map P 0 \land \map P 1 \land \ldots \land \map P {k - 1} \land \map P k \implies \map P {k + 1}$ Then: :$\map P n$ is true for all $n \in \N$.
Second Principle of Mathematical Induction/One-Based
Let $\map P n$ be a propositional function depending on $n \in \N_{>0}$. Suppose that: :$(1): \quad \map P 1$ is true :$(2): \quad \forall k \in \N_{>0}: \map P 1 \land \map P 2 \land \ldots \land \map P {k - 1} \land \map P k \implies \map P {k + 1}$ Then: :$\map P n$ is true for all $n \in \N_{>0}$.
Null Sequences form Maximal Left and Right Ideal/Corollary 1
Then $\NN$ is a maximal ring ideal of $\CC$.
Non-Zero Integer has Finite Number of Divisors
Let $n \in \Z_{\ne 0}$ be a non-zero integer. Then $n$ has a finite number of divisors.
Coprimality Relation is Non-Reflexive
:$\perp$ is non-reflexive.
Ring of Polynomials over Reals is not Field
Let $\R \sqbrk X$ be the ring of polynomials in an indeterminate $X$ over $\R$. Then $\R \sqbrk X$ is not a field.
Ideals of Ring of Integers Modulo m
Let $m \in \Z_{>0}$ be a (strictly) positive integer. Let $\struct {\Z_m, +, \times}$ denote the ring of integers modulo $m$. The ideals of $\struct {\Z_m, +, \times}$ are of the form: :$d \Z / m \Z$ where $d$ is a divisor of $m$.
Quotient Ring of Cauchy Sequences is Division Ring/Corollary 1
Then the quotient ring $\CC \,\big / \NN$ is a field.
Quotient Ring of Cauchy Sequences is Normed Division Ring/Corollary 1
Then $\struct {\CC \,\big / \NN, \norm {\, \cdot \,}_1 }$ is a valued field.
Cauchy Sequence Converges Iff Equivalent to Constant Sequence
Let $\struct {R, \norm {\,\cdot\,}}$ be a normed division ring. Let $\CC$ be the ring of Cauchy sequences over $R$ Let $\NN$ be the set of null sequences. Let $\\CC \,\big / \NN$ be the quotient ring of Cauchy sequences of $\CC$ by the maximal ideal $\NN$. Let $\sequence {x_n} \in \CC$. Then $\sequence {x_n}$ converges in $\struct {R, \norm {\,\cdot\,} }$  :$\exists a \in R: \sequence {x_n} \in \sequence {a, a, a, \dotsc} + \NN$ where $\sequence {a, a, a, \dotsc} + \NN$ is the left coset in $\CC \, \big / \NN$ that contains the constant sequence $\sequence {a, a, a, \dotsc}$.
Homomorphism of Ring Subtraction
Let $\phi: \struct {R_1, +_1, \circ_1} \to \struct {R_2, +_2, \circ_2}$ be a ring homomorphism. Then: :$\forall a, b \in R_1: \map \phi {a -_1 b} = \map \phi a -_2 \map \phi b$ where $a -_1 b$ denotes subtraction of $b$ from $a$.
Limit of Modulo Operation/Limit 1
Let $x$ and $y$ be real numbers. Let $x \bmod y$ denote the modulo operation. Then $\displaystyle \lim_{y \mathop \to 0} x \bmod y = 0$.
Limit of Modulo Operation/Limit 2
Let $x$ and $y$ be real numbers. Let $x \bmod y$ denote the modulo operation. Then $\displaystyle \lim_{y \mathop \to \infty} x \bmod y = x$ if $x \ge 0$.
Limit of Modulo Operation
Let $x$ and $y$ be real numbers. Let $x \bmod y$ denote the modulo operation. Then holding $x$ fixed gives: :$\displaystyle \lim_{y \mathop \to 0} x \bmod y = 0$ :$\displaystyle \lim_{y \mathop \to \infty} x \bmod y = x$ if $x \ge 0$ {{expand|What about the limits with respect to $x$?}}