hcju/proofwikips huggingface dataset:
Non-Null Cauchy Sequence in Non-Archimedean Norm is Eventually Stationary
Let $\struct {R, \norm{\,\cdot\,} }$ be a non-Archimedean normed division ring with zero $0_R$ Let $\sequence {x_n}$ be a Cauchy sequence such that $\sequence {x_n}$ does not converge to $0_R$. Then: :$\exists N \in \N: \forall n, m \ge N: \norm {x_n} = \norm {x_m}$
Equivalence of Definitions of Unit of Ring
Let $\struct {R, +, \circ}$ be a ring with unity whose unity is $1_R$. {{TFAE|def = Unit of Ring}}
Bézout's Lemma/Euclidean Domain
Let $\struct {D, +, \times}$ be a Euclidean domain whose zero is $0$ and whose unity is $1$. Let $\nu: D \setminus \set 0 \to \N$ be the Euclidean valuation on $D$. Let $a, b \in D$ such that $a$ and $b$ are not both equal to $0$. Let $\gcd \set {a, b}$ be the greatest common divisor of $a$ and $b$. Then: :$\exists x, y \in D: a \times x + b \times y = \gcd \set {a, b}$ such that $\gcd \set {a, b}$ is the element of $D$ such that: :$\forall c = a \times x + b \times y \in D: \map \nu {\gcd \set {a, b} } \le \map \nu c$
Euclidean Valuation of Non-Unit is less than that of Product
Let $\struct {D, +, \times}$ be a Euclidean domain whose zero is $0$, and unity is $1$. Let the valuation function of $D$ be $\nu$. Let $b, c \in D_{\ne 0}$. Then: :If $c$ is not a unit of $D$ then $\map \nu b < \map \nu {b c}$
Element is Unit iff its Euclidean Valuation equals that of 1
Let $\struct {D, +, \times}$ be a Euclidean domain whose zero is $0$, and unity is $1$. Let the valuation function of $D$ be $\nu$. Let $a \in D$. Then: :$a$ is a unit of $D$  $\map \nu a = \map \nu 1$
Gauss's Lemma on Primitive Polynomials over Ring
Let $R$ be a commutative ring with unity. Let $f, g \in R \sqbrk X$ be primitive polynomials. {{explain|Definition:Primitive Polynomial over Ring}} Then $f g$ is primitive.
Rational Polynomial is Content Times Primitive Polynomial/Existence
Let $\Q \sqbrk X$ be the ring of polynomial forms over the field of rational numbers in the indeterminate $X$. Let $\map f X \in \Q \sqbrk X$. Then: :$\map f X = \cont f \, \map {f^*} X$ where: :$\cont f$ is the content of $\map f X$ :$\map {f^*} X$ is a primitive polynomial.
Rational Polynomial is Content Times Primitive Polynomial/Uniqueness
Let $\Q \sqbrk X$ be the ring of polynomial forms over the field of rational numbers in the indeterminate $X$. Let $\map f X \in \Q \sqbrk X$ be given. Then there exist unique content $\cont f$ of $\map f X$ and primitive polynomial $\map {f^*} X$ such that: :$\map f X = \cont f \, \map {f^*} X$
Content of Polynomial in Dedekind Domain is Multiplicative
Let $R$ be a Dedekind domain. Let $f, g \in R \sqbrk X$ be polynomials. Let $\cont f$ denote the content of $f$. Then $\cont {f g} = \cont f \cont g$ is the product of $\cont f$ and $\cont g$.
Factors of Polynomial with Integer Coefficients have Integer Coefficients
Let $\Q \sqbrk X$ be the ring of polynomial forms over the field of rational numbers in the indeterminate $X$. Let $\map h X \in \Q \sqbrk X$ have coefficients all of which are integers. Let it be possible to express $\map h X$ as: :$\map h X = \map f X \, \map g X$ where $\map f X, \map g X \in \Q \sqbrk X$. Then it is also possible to express $\map h X$ as: :$\map h X = \map {f'} X \, \map {g'} X$ where: :$\map {f'} X, \map {g'} X \in \Q \sqbrk X$ :the coefficients of $\map {f'} X$ and $\map {g'} X$ are all integers :$\map {f'} X = a \map f X$ and $\map {g'} X = b \map f X$, for $a, b \in \Q$.
Polynomial which is Irreducible over Integers is Irreducible over Rationals
Let $\Z \sqbrk X$ be the ring of polynomial forms over the integers in the indeterminate $X$. Let $\Q \sqbrk X$ be the ring of polynomial forms over the field of rational numbers in the indeterminate $X$. Let $\map f X \in \Z \sqbrk X$ be irreducible in $\Z \sqbrk X$. Then $\map f X$ is also irreducible in $\Q \sqbrk X$.
Dirichlet Function is Periodic
Let $D: \R \to \R$ be a Dirichlet function: :$\forall x \in \R: \map D x = \begin{cases} c & : x \in \Q \\ d & : x \notin \Q \end{cases}$ Then $D$ is periodic. Namely, every non-zero rational number is a periodic element of $D$.
Dirichlet Function has no Period
The Dirichlet functions are periodic by Dirichlet Function is Periodic. However, they do not admit a period. That is, there does not exist a smallest value $L \in \R_{> 0}$ such that: :$\forall x \in \R: \map D x = \map D {x + L}$
Existence of Nonconstant Periodic Function with no Period
There exists a real, non-constant function $f$ such that: :$(1): \quad f$ is periodic. :$(2): \quad f$ does '''not''' have a period.
Vector Augend plus Addend equals Augend implies Addend is Zero
Let $V$ be a vector space over a field $F$. Let $\mathbf a, \mathbf b \in V$. Let $\mathbf a + \mathbf b = \mathbf a$. Then: :$\mathbf b = \bszero$ where $\bszero$ is the zero vector of $V$.
Vector Space of All Mappings is Vector Space
Let $\struct {K, +, \circ}$ be a division ring. Let $\struct {G, +_G, \circ}_K$ be a $K$-vector space. Let $S$ be a set. Let $\struct {G^S, +_G', \circ}_R$ be the vector space of all mappings from $S$ to $G$. Then $\struct {G^S, +_G', \circ}_K$ is a $K$-vector space.
Unitary Module of All Mappings is Unitary Module
Let $\struct {R, +_R, \times_R}$ be a ring. Let $\struct {G, +_G, \circ}_R$ be a unitary $R$-module. Let $S$ be a set. Let $\struct {G^S, +_G', \circ}_R$ be the module of all mappings from $S$ to $G$. Then $\struct {G^S, +_G', \circ}_R$ is a unitary module.
Finite Direct Product of Unitary Modules is Unitary Module
Let $\struct {R, +_R, \times_R}$ be a ring. Let $\struct {G_1, +_1, \circ_1}_R, \struct {G_2, +_2, \circ_2}_R, \ldots, \struct {G_n, +_n, \circ_n}_R$ be unitary $R$-modules. Let: :$\displaystyle G = \prod_{k \mathop = 1}^n G_k$ be their direct product. Then $G$ is a unitary module.
Nonconstant Periodic Function with no Period is Discontinuous Everywhere
Let $f$ be a real periodic function that does not have a period. Then $f$ is either constant or discontinuous everywhere.
Module on Cartesian Product is Module
Let $\struct {R, +_R, \times_R}$ be a ring. Let $n \in \N_{>0}$. Let $\struct {R^n, +, \times}_R$ be the '''$R$-module $R^n$'''. Then $\struct {R^n, +, \times}_R$ is an $R$-module.
Module on Cartesian Product of Ring with Unity is Unitary Module
Let $\struct {R, +_R, \times_R}$ be a ring with unity. Let $n \in \N_{>0}$. Let $\struct {R^n, +, \times}_R$ be the '''$R$-module $R^n$'''. Then $\struct {R^n, +, \times}_R$ is a unitary $R$-module.
Ring of Polynomial Forms over Field is Vector Space
Let $\struct {F, +, \times}$ be a field whose unity is $1_F$. Let $F \sqbrk X$ be the ring of polynomials over $F$. Then $F \sqbrk X$ is an vector space over $F$.
Norm is Complete Iff Equivalent Norm is Complete
Let $R$ be a division ring. Let $\norm {\,\cdot\,}_1$ and $\norm {\,\cdot\,}_2$ be equivalent norms on $R$. Then: :$\struct {R,\norm {\,\cdot\,}_1}$ is complete  $\struct {R,\norm {\,\cdot\,}_2}$ is complete.
Ring of Polynomial Forms over Field is Vector Space/Corollary
Let $S \subseteq F \sqbrk X$ denote the subset of $F \sqbrk X$ defined as: :$S = \set {\mathbf x \in F \sqbrk X: \map \deg {\mathbf x} < d}$ for some $d \in \Z_{>0}$. Then $S$ is an vector space over $F$.
P-adic Norm of p-adic Number is Power of p
Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $x \in \Q_p: x \ne 0$. Then: :$\exists v \in \Z: \norm x_p = p^{-v}$
Borel-TIS inequality
Let $T$ be a topological space. Let $\sequence {f_t}_{t \mathop \in T}$ be a centred (i.e. mean zero) Gaussian process on $T$, such that: :$\norm f_T := \sup_{t \mathop \in T} \size {f_t}$ is almost surely finite. Let: :$\sigma_T^2 := \sup_{t \mathop \in T} \operatorname E \size {f_t}^2$ Then $\map {\operatorname E} {\norm f_T}$ and $\sigma_T$ are both finite, and, for each $u > 0$: :$\map {\operatorname P} {\norm f_T > \map {\operatorname E} {\norm f_T} + u} \le \map \exp {\dfrac {-u^2} {2 \sigma_T^2} }$
Negation of Propositional Function in Two Variables
Let $\map P {x, y}$ be a propositional function of two Variables. Then: :$\neg \forall x: \exists y: \map P {x, y} \iff \exists x: \forall y: \neg \map P {x, y}$ That is: :''It is not the case that for all $x$ a value of $y$ can be found to satisfy $\map P {x, y}$'' means the same thing as: :''There exists at least one value of $x$ such that for all $y$ it is not possible to satisfy $\map P {x, y}$''
Valuation Ideal of P-adic Numbers
Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. The valuation ideal induced by norm $\norm {\,\cdot\,}_p$ is the principal ideal: :$p \Z_p = \set {x \in \Q_p: \norm x_p < 1}$
Integers are Arbitrarily Close to P-adic Integers
Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $\Z_p$ be the $p$-adic integers. Let $x \in \Z_p$. Then for $n \in \N$ there exists unique $\alpha \in \Z$: :$(1): \quad 0 \le \alpha \le p^n - 1$ :$(2): \quad \norm { x -\alpha}_p \le p^{-n}$
Integers are Dense in P-adic Integers
The integers $\Z$ are dense in the metric space $\struct{\Z_p, d_p}$.
P-adic Integer is Limit of Unique Coherent Sequence of Integers
Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $\Z_p$ be the $p$-adic integers. Let $x \in \Z_p$. Then there exists a unique coherent sequence $\sequence {\alpha_n}$: :$\displaystyle \lim_{n \mathop \to \infty} \alpha_n = x$
Linearly Independent Set is Contained in some Basis/Proof 1
Let $E$ be a vector space of $n$ dimensions. Let $H$ be a linearly independent subset of $E$. {{:Linearly Independent Set is Contained in some Basis}}
Linearly Independent Set is Contained in some Basis/Proof 2
Let $E$ be a vector space of $n$ dimensions. Let $H$ be a linearly independent subset of $E$. {{:Linearly Independent Set is Contained in some Basis}}
Vector Space over Division Subring is Vector Space
Let $\struct {R, +, \circ}$ be a ring with unity whose unity is $1_R$. Let $S$ be a division subring of $R$, such that $1_R \in S$. The vector space $\struct {R, +, \circ_S}_S$ over $\circ_S$ is a $S$-vector space.
Vector Space on Field Extension is Vector Space
Let $\struct {K, +, \times}$ be a field. Let $L / K$ be a field extension over $K$. Let $\struct {L, +, \times}_K$ be the a vector space of $L$ over $K$. Then $\struct {L, +, \times}_K$ is a vector space.
Sheldon Conjecture
There is only $1$ Sheldon prime, and that is $73$.
Condition for Linear Divisor of Polynomial
Let $\map P x$ be a polynomial in $x$. Let $a$ be a constant. Then $x - a$ is a divisor of $\map P x$  $a$ is a root of $P$.
Polynomial with Algebraic Number as Root is Multiple of Minimal Polynomial
Let $F$ be a field. Let $\map P x$ be a polynomial in $F$. Let $z$ be a root of $\map P x$. Then $\map P x$ is a multiple of the minimal polynomial $\map m x$ in $z$ over $F$.
Simple Algebraic Field Extension consists of Polynomials in Algebraic Number
Let $F$ be a field. Let $\theta \in \C$ be algebraic over $F$. Let $\map F \theta$ be the simple field extension of $F$ by $\theta$. Then $\map F \theta$ consists of polynomials that can be written in the form $\map f \theta$, where $\map f x$ is a polynomial over $F$.
Element of Simple Algebraic Field Extension of Degree n is Polynomial in Algebraic Number of Degree Less than n
Let $F$ be a field. Let $\theta \in \C$ be algebraic over $F$ of degree $n$. Let $\map F \theta$ be the simple field extension of $F$ by $\theta$. Then any element of $\map F \theta$ can be written as $\map f \theta$, where $\map f x$ is a polynomial over $F$ of degree at most $n - 1$.
Degree of Simple Algebraic Field Extension equals Degree of Algebraic Number
Let $F$ be a field. Let $\theta \in \C$ be algebraic over $F$ of degree $n$. Let $\map F \theta$ be the simple field extension of $F$ by $\theta$. Then $\map F \theta$ is a finite extension of $F$ whose degree is: :$\index {\map F \theta} F = n$
Degree of Element of Finite Field Extension divides Degree of Extension
Let $F$ be a field whose zero is $0$ and whose unity is $1$. Let $K / F$ be a finite field extension of degree $n$. Let $\alpha \in K$ be algebraic over $F$. Then the degree of $\alpha$ is a divisor of $n$.
Algebraic Element of Degree 3 is not Element of Field Extension of Degree Power of 2
Let $K / F$ be a finite field extension of degree $2^m$. Let $\alpha \in K$ be algebraic over $F$ with degree $3$. Then $\alpha \notin K$.
Construction of Point in Cartesian Plane with Rational Coordinates
Let $\CC$ be a Cartesian plane. Let $P = \tuple {x, y}$ be a rational point in $\CC$. Then $P$ is constructible using a compass and straightedge construction.
Construction of Integer Multiple of Line Segment
Let $AB$ be a line segment in the plane. Let $AC$ be a line segment in the plane through a point $C$ Let $D$ be a point on $AC$ such that $AD = n AB$ for some $n \in \Z$. Then $AD$ is constructible using a compass and straightedge construction.
Point in Plane is Constructible iff Coordinates in Extension of Degree Power of 2
Let $\CC$ be a Cartesian plane. Let $S$ be a set of points in $\CC$. Let $F$ be the smallest field containing all the coordinates of the points in $S$. Let $P = \tuple {a, b}$ be a point in $\CC$. Then: :$P$ is constructible from $S$ using a compass and straightedge construction : :the coordinates of $P$ are contained in a finite extension $K$ of $F$ whose degree is given by: ::$\index K F = 2^m$ :for some $m \in \Z_{\ge 0}$.
Construction of Lattice Point in Cartesian Plane
Let $\CC$ be a Cartesian plane. Let $P = \tuple {a, b}$ be a lattice point in $\CC$. Then $P$ is constructible using a compass and straightedge construction.
Construction of Regular Heptagon by Compass and Straightedge Construction is Impossible
There is no compass and straightedge construction for a regular heptagon.
Rational Numbers with Denominator Power of Two form Integral Domain
Let $\Q$ denote the set of rational numbers. Let $S \subseteq \Q$ denote the set of set of rational numbers of the form $\dfrac p q$ where $q$ is a power of $2$: :$S = \set {\dfrac p q: p \in \Z, q \in \set {2^m: m \in \Z_{\ge 0} } }$ Then $\struct {S, +, \times}$ is an integral domain.
Normal Subgroup of Order 25 in Group of Order 100
Let $G$ be a group of order $100$. Then $G$ has a normal subgroup of order $25$.
Normal p-Subgroup contained in All Sylow p-Subgroups
Let $G$ be a finite group. Let $p$ be a prime number. Let $H$ be a normal subgroup of $G$ which is a $p$-group. Then $H$ is a subset of every Sylow $p$-subgroup of $G$.
Group of Order p^2 q is not Simple
Let $p$ and $q$ be prime numbers such that $p \ne q$. Let $G$ be a group of order $p^2 q$. Then $G$ is not simple.
Simple Group of Order Less than 60 is Prime
Let $G$ be a simple group. Let $\order G < 60$, where $\order G$ denotes the order of $G$. Then $G$ is a prime group.
Group of Order 42 has Normal Subgroup of Order 7
Let $G$ be of order $42$. Then $G$ has a normal subgroup of order $7$.
Group of Order 54 has Normal Subgroup of Order 27
Let $G$ be of order $54$. Then $G$ has a normal subgroup of order $27$.
Group of Order 40 has Normal Subgroup of Order 5
Let $G$ be of order $40$. Then $G$ has a normal subgroup of order $5$.
Finite Group with One Sylow p-Subgroup per Prime Divisor is Isomorphic to Direct Product
Let $G$ be a finite group whose order is $n$ and whose identity element is $e$. Let $G$ be such that it has exactly $1$ Sylow $p$-subgroup for each prime divisor of $n$. Then $G$ is isomorphic to the internal direct product of all its Sylow $p$-subgroups.
Number of Subgroups of Prime Power Order is Congruent to 1 modulo Prime
Let $G$ be a finite group whose order is $n$. Let $p$ be a prime number such that $p^k$ is a divisor of $n$. Then the number of subgroups of order $p^k$ is congruent to $1$ modulo $p$.
Count of Distinct Homomorphisms between Additive Groups of Integers Modulo m
Let $m, n \in \Z_{>0}$ be (strictly) positive integers. Let $\struct {\Z_m, +}$ denote the additive group of integers modulo $m$. The number of distinct homomorphisms $\phi: \struct {\Z_m, +} \to \struct {\Z_n, +}$ is $\gcd \set {m, n}$.
P-adic Integer is Limit of Unique Coherent Sequence of Integers/Lemma 1
:$\forall n \in \N: \alpha_{n + 1} \equiv \alpha_n \pmod {p^{n + 1}}$
P-adic Integer is Limit of Unique Coherent Sequence of Integers/Lemma 3
$\sequence {\alpha_n}$ is a unique sequence satisfying properties $(1)$, $(2)$ and $(3)$ above.
P-adic Integer is Limit of Unique Coherent Sequence of Integers/Lemma 2
:$\displaystyle \lim_{n \mathop \to \infty} \alpha_n = x$
Group Types of Order Prime Squared
Let $p$ be a prime number. Let $G$ be a group of order $p^2$. Then $G$ is isomorphic either to $\Z_{p^2}$ or to $\Z_p \times \Z_p$, where $\Z_p$ denotes the additive group of integers modulo $p$.
Group/Examples/x+y over 1+xy/Isomorphic to Real Numbers
$\struct {G, \circ}$ is isomorphic to the additive group of real numbers $\struct {\R, +}$.
Open Set Characterization of Denseness/Analytic Basis
Let $\mathcal B \subseteq \tau$ be an analytic basis for $\tau$. Then $S$ is (everywhere) dense in $X$  every non-empty open set of $\mathcal B$ contains an element of $S$.
Open Set Characterization of Denseness/Open Ball
Let $\struct{X, d}$ be a metric space. Let $\tau_d$ be the topology induced by the metric $d$. Let $S \subseteq X$. Then $S$ is (everywhere) dense in $\struct{X, \tau_d}$  every open ball contains an element of $S$.
P-adic Integers is Metric Completion of Integers
Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $d$ be the subspace metric of the $p$-adic metric on the $p$-adic integers $\Z_p$. Then $\struct {\Z_p, d}$ is the metric completion of the integers $\Z$.
P-adic Metric on P-adic Numbers is Non-Archimedean Metric
Let $p \in \N$ be a prime. Let $\norm{\,\cdot\,}_p: \Q_p \to \R_{\ge 0}$ be the $p$-adic norm on the $p$-adic numbers $\Q_p$. Let $d_p$ be the $p$-adic metric on $\Q_p$: :$\forall x, y \in \Q_p: \map {d_p} {x, y} = \norm{x - y}_p$ Then $d_p$ is a non-Archimedean metric that extends the $p$-adic metric on the rationals $\Q$ to $\Q_p$.
Order of Automorphism Group of Cyclic Group
Let $C_n$ denote the cyclic group of order $n$. Let $\Aut {C_n}$ denote the automorphism group of $C_n$. Then: :$\order {\Aut {C_n} } = \map \phi n$ where: :$\order {\, \cdot \,}$ denotes the order of a group :$\map \phi n$ denotes the Euler $\phi$ function.
Order of Automorphism Group of Dihedral Group
Let $D_n$ denote the dihedral group of order $n$. Let $\Aut {D_n}$ denote the automorphism group of $D_n$. Then: :$\order {\Aut {D_n} } = 2 \map \phi n$ where: :$\order {\, \cdot \,}$ denotes the order of a group :$\map \phi n$ is the Euler $\phi$ function.
Valuation Ring of P-adic Norm on Rationals/Corollary 1
The set of integers $\Z$ is a subring of $\OO$.
Valuation Ring of P-adic Norm is Subring of P-adic Integers
Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Let $\Z_p$ be the $p$-adic integers. Let $\Z_{\ideal p}$ be the induced valuation ring on $\struct {\Q,\norm {\,\cdot\,}_p}$. Then: :$(1): \quad \Z_{\ideal p} = \Q \cap \Z_p$. :$(2): \quad \Z_{\ideal p}$ is a subring of $\Z_p$.
Valuation Ring of P-adic Norm is Subring of P-adic Integers/Corollary 1
The set of integers $\Z$ is a subring of $\Z_p$.
Laplace Transform of 1
Let $f: \R \to \R$ be the function defined as: :$\forall t \in \R: \map f t = 1$ Then the Laplace transform of $\map f t$ is given by: :$\laptrans {\map f t} = \dfrac 1 s$ for $\map \Re s > 0$.
Primitive of Exponential of a x by Hyperbolic Sine of b x
:$\displaystyle \int e^{a x} \sinh b x \rd x = \frac {e^{a x} \paren {a \sinh b x - b \cosh b x} } {a^2 - b^2} + C$
Primitive of Exponential of a x by Hyperbolic Cosine of b x
:$\displaystyle \int e^{a x} \cosh b x \rd x = \frac {e^{a x} \paren {a \cosh b x + b \sinh b x} } {a^2 - b^2} + C$
Laplace Transform of Derivative/Discontinuity at t = 0
Let $f$ fail to be continuous at $t = 0$, but let: :$\displaystyle \lim_{t \mathop \to 0} \map f t = \map f {0^+}$ exist. Then $\laptrans f$ exists for $\map \Re s > a$, and: :$\laptrans {\map {f'} t} = s \laptrans {\map f t} - \map f {0^+}$
Laplace Transform of Derivative/Discontinuity at t = a
Let $f$ have a jump discontinuity at $t = a$. Then: :$\laptrans {\map {f'} t} = s \laptrans {\map f t} - \map f 0 - e^{a s} \paren {\map f {a^+} - \map f {a^-} }$
Laplace Transform of Integral
:$\displaystyle \laptrans {\int_0^t \map f u \rd u} = \dfrac {\map F s} s$ wherever $\laptrans f$ exists.
Integral of Laplace Transform
:$\displaystyle \laptrans {\dfrac {\map f t} t} = \int_s^{\to \infty} \map F u \rd u$ wherever $\displaystyle \lim_{t \mathop \to 0} \dfrac {\map f t} t$ and $\laptrans f$ exist.
Inclusion Mapping on Subring is Homomorphism
Let $\struct {R, +, \circ}$ be a ring. Let $\struct {S, +{\restriction_S}, \circ {\restriction_S}}$ be a subring of $R$. Let $i_S: S \to R$ be the inclusion mapping from $S$ to $R$. Then ${i_S}$ is a ring homomorphism.
Inclusion Mapping on Subring is Monomorphism
Let $\struct {R, +, \circ}$ be a ring. Let $\struct {S, +{\restriction_S}, \circ {\restriction_S} }$ be a subring of $R$. Let $i_S: S \to R$ be the inclusion mapping from $S$ to $R$. Then $i_S$ is a ring monomorphism.
Negative of Subring is Negative of Ring
Let $\struct {R, +, \circ}$ be a ring. For each $x \in R$ let $-x$ denote the ring negative of $x$ in $R$. Let $\struct {S, + {\restriction_S}, \circ {\restriction_S}}$ be a subring of $R$. For each $x \in S$ let $\mathbin \sim x$ denote the ring negative of $x$ in $S$. Then: :$\forall x \in S: \mathbin \sim x = -x$
Limit to Infinity of Laplace Transform
Let $\laptrans {\map f t} = \map F s$ denote the Laplace transform of the real function $f$. Then: :$\displaystyle \lim_{s \mathop \to \infty} \map F s = 0$
Initial Value Theorem of Laplace Transform
Let $\laptrans {\map f t} = \map F s$ denote the Laplace transform of the real function $f$. Then: :$\displaystyle \lim_{t \mathop \to 0} \map f t = \lim_{s \mathop \to \infty} s \, \map F s$ if those limits exist.
Final Value Theorem of Laplace Transform
Let $\laptrans {\map f t} = \map F s$ denote the Laplace transform of the real function $f$. Then: :$\displaystyle \lim_{t \mathop \to \infty} \map f t = \lim_{s \mathop \to 0} s \, \map F s$ if those limits exist.
Evaluation of Integral using Laplace Transform
Let $\laptrans {\map f t} = \map F s$ denote the Laplace transform of the real function $f$. Then: :$\displaystyle \int_0^{\to \infty} \map f t \rd t = \map F 0$ assuming the integral is convergent.
Valuation Ring of Non-Archimedean Division Ring is Clopen
Let $\struct {R, \norm {\,\cdot\,} }$ be a non-Archimedean normed division ring with zero $0_R$. Let $\mathcal O$ be valuation ring induced by $\norm{\,\cdot\,}$. Then $\mathcal O$ is a both open and closed in the metric induced by $\norm{\,\cdot\,}$.
Valuation Ring of Non-Archimedean Division Ring is Clopen/Corollary 1
Let $p$ be a prime number. Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers. Then the $p$-adic integers $\Z_p$ is both open and closed in the $p$-adic metric.
Series Expansion of Bessel Function of the First Kind
{{begin-eqn}} {{eqn | l = \map {J_n} x       | r = \dfrac {x^n} {2^n \, \map \Gamma {n + 1} } \paren {1 - \dfrac {x^2} {2 \paren {2 n + 2} } + \dfrac {x^4} {2 \times 4 \paren {2 n + 2} \paren {2 n + 4} } - \cdots}       | c =  }} {{eqn | r = \sum_{k \mathop = 0}^\infty \dfrac {\paren {-1}^k} {k! \, \map \Gamma {n + k + 1} } \paren {\dfrac x 2}^{n + 2 k}       | c =  }} {{end-eqn}}
Recurrence Formula for Bessel Function of the First Kind
Let $\map {J_n} x$ denote the Bessel function of the first kind of order $n$. Then: :$\map {J_{n + 1} } x = \dfrac {2 n} x \map {J_n} x - \map {J_{n - 1} } x$ And:  :$\map {J_{n + 1} } x = -2 \map {J_n'} x + \map {J_{n - 1} } x$ {{refactor|page per result}}
Derivative of x^n by Bessel Function of the First Kind of Order n of x
Let $\map {J_n} x$ denote the Bessel function of the first kind of order $n$. Then: :$\map {\dfrac \d {\d x} } {x^n \map {J_n} x} = x^n \map {J_{n - 1} } x$
Generating Function for Bessel Function of the First Kind of Order n of x
Let $\map {J_n} x$ denote the Bessel function of the first kind of order $n$. Then: :$\map \exp {\dfrac {x \paren {t - \frac 1 t} } 2} = \displaystyle \sum_{n \mathop = -\infty}^\infty \map {J_n} x t^n$