hcju/proofwikips huggingface dataset:
Comparison Test for Convergence of Power Series
Let $A = \displaystyle \sum_{n \mathop \ge 0} a_n z^n$ and $B = \displaystyle \sum_{n \mathop \ge 0} b_n z^n$ be power series in $\C$. Let $R_A$ and $R_B$ be the radii of convergence of $A$ and $B$ respectively. Let $\cmod {b_n} \le \cmod {a_n}$ for all $n \in \N$. Then $R_A \le R_B$.
Hyperbolic Tangent of Complex Number
Let $a$ and $b$ be real numbers. Let $i$ be the imaginary unit. Then:
Hyperbolic Cosine of Complex Number
Let $a$ and $b$ be real numbers. Let $i$ be the imaginary unit. Then: :$\cosh \paren {a + b i} = \cosh a \cos b + i \sinh a \sin b$ where: :$\cos$ denotes the real cosine function :$\sin$ denotes the real sine function :$\sinh$ denotes the hyperbolic sine function :$\cosh$ denotes the hyperbolic cosine function
Hyperbolic Tangent of Complex Number/Formulation 1
:$\tanh \paren {a + b i} = \dfrac {\sinh a \cos b + i \cosh a \sin b} {\cosh a \cos b + i \sinh a \sin b}$
Hyperbolic Tangent of Complex Number/Formulation 2
:$\tanh \paren {a + b i} = \dfrac {\tanh a + i \tan b} {1 + i \tanh a \tan b}$
Hyperbolic Tangent of Complex Number/Formulation 3
:$\tanh \paren {a + b i} = \dfrac {\tanh a - \tanh a \tan^2 b} {1 + \tan^2 a \tanh^2 b} + \dfrac {\tan b + \tanh^2 a \tan b} {1 + \tanh^2 a \tan^2 b} i$
Hyperbolic Cosecant of Complex Number
Let $a$ and $b$ be real numbers. Let $i$ be the imaginary unit. Then: :$\map \csch {a + b i} = \dfrac {\sinh a \cos b - i \cosh a \sin b} {\sinh^2 a \cos^2 b + \cosh^2 a \sin^2 b}$ where: :$\csch$ denotes the hyperbolic cosecant function. :$\sin$ denotes the real sine function :$\cos$ denotes the real cosine function :$\sinh$ denotes the hyperbolic sine function :$\cosh$ denotes the hyperbolic cosine function
Hyperbolic Secant of Complex Number
Let $a$ and $b$ be real numbers. Let $i$ be the imaginary unit. Then: :$\sech \paren {a + b i} = \dfrac {\cosh a \cos b - i \sinh a \sin b} {\cosh^2 a \cos^2 b + \sinh^2 a \sin^2 b}$ where: :$\sech$ denotes the hyperbolic secant function. :$\sin$ denotes the real sine function :$\cos$ denotes the real cosine function :$\sinh$ denotes the hyperbolic sine function :$\cosh$ denotes the hyperbolic cosine function
Hyperbolic Cotangent of Complex Number
Let $a$ and $b$ be real numbers. Let $i$ be the imaginary unit. Then:
Hyperbolic Cotangent of Complex Number/Formulation 2
:$\map \coth {a + b i} = \dfrac {1 + i \coth a \cot b} {\coth a - i \cot b}$
Hyperbolic Cotangent of Complex Number/Formulation 3
:$\map \coth {a + b i} = \dfrac {\coth a - \coth a \cot^2 b} {\coth^2 a + \cot^2 b} + \dfrac {\coth b + \coth^2 a \cot b} {\coth^2 a + \cot^2 b} i$
Hyperbolic Tangent of Complex Number/Formulation 4
:$\tanh \paren {a + b i} = \dfrac {\sinh 2 a + i \sin 2 b} {\cosh 2 a + \cos 2 b}$
Tangent of Complex Number/Formulation 4
:$\tan \paren {a + b i} = \dfrac {\sin 2 a + i \sinh 2 b} {\cos 2 a + \cosh 2 b}$
Characterisation of Cauchy Sequence in Non-Archimedean Norm
Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring with non-Archimedean norm $\norm {\,\cdot\,}$. Let $\sequence {x_n}$ be a sequence in $R$. Then: :$\sequence {x_n}$ is a Cauchy sequence  $\displaystyle \lim_{n \mathop \to \infty} \norm {x_{n + 1} - x_n} = 0$.
Characterisation of Cauchy Sequence in Non-Archimedean Norm/Necessary Condition
Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring with non-Archimedean norm $\norm {\,\cdot\,}$. Let $\sequence {x_n}$ be a Cauchy sequence in $R$. Then: :$\lim_{n \mathop \to \infty} \norm {x_{n + 1} - x_n} = 0$
Characterisation of Cauchy Sequence in Non-Archimedean Norm/Sufficient Condition
Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring with non-Archimedean norm $\norm {\,\cdot\,}$. Let $\sequence {x_n}$ be a sequence in $R$. Let $\displaystyle \lim_{n \mathop \to \infty} \norm {x_{n + 1} - x_n} = 0$. Then: :$\sequence {x_n}$ is a Cauchy sequence.
Power Series of Sine of Odd Theta
Let $r \in \R$ such that $\size r < 1$. Let $\theta \in \R$ such that $\theta \ne m \pi$ for any $m \in \Z$. Then: {{begin-eqn}} {{eqn | l = \sum_{k \mathop \ge 0} \map \sin {2 k + 1} \theta r^k       | r = \sin \theta + r \sin 3 \theta + r^2 \sin 5 \theta + \cdots       | c =  }} {{eqn | r = \dfrac {\paren {1 + r} \sin \theta} {1 - 2 r \cos 2 \theta + r^2}       | c =  }} {{end-eqn}}
Modulus of Limit/Normed Division Ring
Let $\struct {R, \norm { \, \cdot \, } }$ be a normed division ring. Let $\sequence {x_n}$ be a convergent sequence in $R$ to the limit $l$. That is, let $\displaystyle \lim_{n \mathop \to \infty} x_n = l$. Then :$\displaystyle \lim_{n \mathop \to \infty} \norm {x_n} = \norm l$
Three Points in Ultrametric Space have Two Equal Distances
Let $\struct {X, d}$ be an ultrametric space. Let $x, y, z \in X$ with $\map d {x, z} \ne \map d {y, z}$. Then: :$\map d {x, y} = \max \set {\map d {x, z}, \map d {y, z} }$
Three Points in Ultrametric Space have Two Equal Distances/Corollary
Let $\struct {X, d}$ be an ultrametric space. Let $x, y, z \in X$. Then: :at least two of the distances $\map d {x, y}$, $\map d {x, z}$ and $\map d {y, z}$ are equal.
Non-Archimedean Norm iff Non-Archimedean Metric/Necessary Condition
Let $\struct {R, \norm {\,\cdot\,}}$ be a non-Archimedean normed division ring. Let $d$ be the metric induced by $\norm {\,\cdot\,}$. Then $d$ is a non-Archimedean metric.
Non-Archimedean Norm iff Non-Archimedean Metric/Sufficient Condition
Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring with zero $0$. Let $d$ be the metric induced by $\norm {\, \cdot \,}$. Let $d$ be non-Archimedean. Then: :$\norm {\, \cdot \,}$ is a non-Archimedean norm.
Non-Archimedean Norm iff Non-Archimedean Metric
Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring with zero $0$. Let $d$ be the metric induced by $\norm {\,\cdot\,}$. Then: :$\norm {\, \cdot \,}$ is a non-Archimedean norm  $d$ is a non-Archimedean metric.
Quotients of 3 Unequal Numbers are Unequal
Let $x, y, z \in \R_{\ne 0}$ be non-zero real numbers which are not all equal. Then $\dfrac x y, \dfrac y z, \dfrac z x$ are also not all equal.
Union of Power Sets not always Equal to Powerset of Union
The union of the power sets of two sets $S$ and $T$ is not necessarily equal to the power set of their union.
Limit of Intersection of Closed Intervals from Zero to Positive Integer Reciprocal
For all (strictly) positive integers $n \in \Z_{>0}$, let $A_n$ be the closed real interval: :$A_n = \closedint 0 {\dfrac 1 n}$ Let $A \subseteq \R$ be the subset of the real numbers defined as: :$A = \displaystyle \lim_{n \mathop \to \infty} \bigcap A_n$ Then: :$A = \set 0$
Distributive Laws/Examples/A cap B cap (C cup D) subset of (A cap D) cup (B cap C)
Let: :$P = A \cap B \cap \paren {C \cup D}$ :$Q = \paren {A \cap D} \cup \paren {B \cap C}$ Then: :$P \subseteq Q$
Total Number of Set Partitions
Let $S$ be a finite set of cardinality $n$. Then the number of different partitions of $S$ is $B_n$, where $B_n$ is the $n$th Bell number.
Combination Theorem for Sequences/Normed Division Ring/Inverse Rule
Suppose $l \ne 0$. Then: :$\exists k \in \N : \forall n \in \N: x_{k + n} \ne 0$ and the subsequence $\sequence { x_{k+n}^{-1} }$ is well-defined and convergent with: :$\displaystyle \lim_{n \mathop \to \infty} {x_{k + n} }^{-1} = l^{-1}$.
Bell Number as Summation over Lower Index of Stirling Numbers of the Second Kind
Let $B_n$ be the Bell number for $n \in \Z_{\ge 0}$. Then: :$B_n = \displaystyle \sum_{k \mathop = 0}^n {n \brace k}$ where $\displaystyle {n \brace k}$ denotes a Stirling number of the second kind.
Quotient Group of Solvable Group is Solvable
Let $G$ be a solvable group. Let $N$ be a normal subgroup of $G$. Then, $G / N$, the quotient group of $G$ by $N$ is solvable.
Equivalence of Well-Ordering Principle and Induction/Proof/PFI implies PCI
The Principle of Finite Induction implies the Principle of Complete Finite Induction. That is: :Principle of Finite Induction: Given a subset $S \subseteq \N$ of the natural numbers which has these properties: ::$0 \in S$ ::$n \in S \implies n + 1 \in S$ :then $S = \N$. implies: :Principle of Complete Finite Induction: Given a subset $S \subseteq \N$ of the natural numbers which has these properties: ::$0 \in S$ ::$\set {0, 1, \ldots, n} \subseteq S \implies n + 1 \in S$ :then $S = \N$.
Equivalence of Well-Ordering Principle and Induction/Proof/PCI implies WOP
The Principle of Complete Induction implies the Well-Ordering Principle. That is: :Principle of Complete Induction: Given a subset $S \subseteq \N$ of the natural numbers which has these properties: ::$0 \in S$ ::$\set {0, 1, \ldots, n} \subseteq S \implies n + 1 \in S$ :then $S = \N$. implies: :Well-Ordering Principle: Every nonempty subset of $\N$ has a minimal element.
Equivalence of Well-Ordering Principle and Induction/Proof/WOP implies PFI
The Well-Ordering Principle implies the Principle of Finite Induction. That is: :Well-Ordering Principle: Every non-empty subset of $\N$ has a minimal element implies: :Principle of Finite Induction: Given a subset $S \subseteq \N$ of the natural numbers which has these properties: ::$0 \in S$ ::$n \in S \implies n + 1 \in S$ :then $S = \N$.
Floor Function/Examples/Floor of 5 over 2
:$\floor {\dfrac 5 2} = 2$
Floor Function/Examples/Floor of Minus 5 over 2
:$\floor {-\dfrac 5 2} = -3$
Floor Function/Examples/Floor of 14
:$\floor {14} = 14$
Floor Function/Examples/Floor of Root 10
:$\floor {\sqrt {10} } = 3$
Combination Theorem for Cauchy Sequences/Inverse Rule
Suppose $\sequence {x_n}$ does not converge to $0$. Then: :$\exists K \in \N: \forall n > K : x_n \ne 0$ and the sequence: :$\sequence {\paren {x_{K + n} }^{-1} }_{n \mathop \in \N}$ is well-defined and a Cauchy sequence.
Subsequence of Cauchy Sequence in Normed Division Ring is Cauchy Sequence
Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring with zero: $0$. Let $\sequence {x_n}$ be a Cauchy sequence in $R$. Let $\sequence {x_{n_r} }$ be a subsequence of $\sequence {x_n}$. Then: :$\sequence {x_{n_r} }$ is a Cauchy sequence in $R$.
Coprime Divisors of Square Number are Square
Let $r$ be a square number. Let $r = s t$ where $s$ and $t$ are coprime. Then both $s$ and $t$ are square.
Combination Theorem for Cauchy Sequences/Constant Rule
:the constant sequence $\tuple {a, a, a, \dots}$ is a Cauchy sequence.
Combination Theorem for Cauchy Sequences/Multiple Rule
:$\sequence {a x_n}$ is a Cauchy sequence.
Combination Theorem for Cauchy Sequences/Difference Rule
:$\sequence {x_n - y_n}$ is a Cauchy sequence.
Congruent Integers in Same Residue Class
Let $m \in \Z_{>0}$ be a (strictly) positive integer. Let $\Z_m$ be the set of residue classes modulo $m$: :$Z_m = \set {\eqclass 0 m, \eqclass 1 m, \dotsc, \eqclass {m - 1} m}$ Let $a, b \in \set {0, 1, \ldots, m -1 }$. Then: :$\eqclass a m = \eqclass b m \iff a \equiv b \pmod m$
Residue Classes form Partition of Integers
Let $m \in \Z_{>0}$ be a (strictly) positive integer. Let $\Z_m$ be the set of residue classes modulo $m$: :$\Z_m = \set {\eqclass 0 m, \eqclass 1 m, \dotsc, \eqclass {m - 1} m}$ Then $\Z_m$ forms a partition of $\Z$.
Cardinality of Set of Residue Classes
Let $m \in \Z_{>0}$ be a (strictly) positive integer. Let $\Z_m$ be the set of residue classes modulo $m$. Then: :$\card {Z_m} = m$ where $\card { \, \cdot \,}$ denotes cardinality.
Structure Induced by Ring with Unity Operations is Ring with Unity
Let $\struct {R, +, \circ}$ be a ring with unity whose unity is $1_R$. Let $S$ be a set. Let $\struct {R^S, +', \circ'}$ be the structure on $R^S$ induced by $+'$ and $\circ'$. Then $\struct {R^S, +', \circ'}$ is a ring with unity whose unity is $f_{1_R}: S \to R$, defined by: :$\forall s \in S: \map {f_{1_R} } s = 1_R$
Cauchy Sequences form Ring with Unity
Let $\struct {R, +, \circ, \norm {\, \cdot \,} }$ be a normed division ring. Let $\struct {R^\N, +, \circ}$ be the ring of sequences over $R$ with unity $\tuple {1, 1, 1, \dotsc}$. Let $\CC \subset R^\N$ be the set of Cauchy sequences on $R$. Then: :$\struct {\CC, +, \circ}$ is a subring of $R^\N$ with unity $\tuple {1, 1, 1, \dotsc}$.
Equivalence Relation on Power Set induced by Intersection with Subset
Let $A, T$ be sets such that $A \subseteq T$. Let $S = \powerset T$ denote the power set of $T$. Let $\alpha$ denote the relation defined on $S$ by: :$\forall X, Y \in S: X \mathrel \alpha Y \iff X \cap A = Y \cap A$ Then $\alpha$ is an equivalence relation.
Equivalence Relation on Power Set induced by Intersection with Subset/Equivalence Class of Empty Set
The equivalence class of $\O$ in $S$ with respect to $\alpha$ is given by: :$\eqclass \O \alpha = \powerset {T \setminus A}$
Reflexive and Symmetric Relation is not necessarily Transitive
Let $S$ be a set. Let $\alpha \subseteq S \times S$ be a relation on $S$. Let $\alpha$ be both reflexive and symmetric. Then it is not necessarily the case that $\alpha$ is also transitive.
Symmetric and Transitive Relation is not necessarily Reflexive
Let $S$ be a set. Let $\alpha \subseteq S \times S$ be a relation on $S$. Let $\alpha$ be both symmetric and transitive. Then it is not necessarily the case that $\alpha$ is also reflexive.
Hilbert Proof System Instance 2 Independence Results
Let $\mathscr H_2$ be Instance 2 of the Hilbert proof systems. Then the following independence results hold:
Equivalence Relation on Natural Numbers such that Quotient is Power of Two
Let $\alpha$ denote the relation defined on the natural numbers $\N$ by: :$\forall x, y \in \N: x \mathrel \alpha y \iff \exists n \in \Z: x = 2^n y$ Then $\alpha$ is an equivalence relation.
Equivalence Relation on Natural Numbers such that Quotient is Power of Two/Equivalence Class of Prime
Let $\eqclass p \alpha$ be the $\alpha$-equivalence class of a prime number $p$. Then $\eqclass p \alpha$ contains no other prime number other than $p$.
Equivalence Relation on Natural Numbers such that Quotient is Power of Two/Smallest Equivalence Class with no Prime
Let $\eqclass x \alpha$ denote the $\alpha$-equivalence class of a natural number $x$. Let $r$ be the smallest natural number such that $\eqclass r \alpha$ contains no prime number. Then $r = 9$.
Equivalence Relation on Square Matrices induced by Positive Integer Powers
Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $S$ be the set of all square matrices of order $n$. Let $\alpha$ denote the relation defined on $S$ by: :$\forall \mathbf A, \mathbf B \in S: \mathbf A \mathrel \alpha \mathbf B \iff \exists r, s \in \N: \mathbf A^r = \mathbf B^s$ Then $\alpha$ is an equivalence relation.
Equivalence Relation on Integers Modulo 5 induced by Squaring
Let $\beta$ denote the relation defined on the integers $\Z$ by: :$\forall x, y \in \Z: x \mathrel \beta y \iff x^2 \equiv y^2 \pmod 5$ Then $\beta$ is an equivalence relation.
Equivalence Relation on Integers Modulo 5 induced by Squaring/Number of Equivalence Classes
The number of distinct $\beta$-equivalence classes is $3$: {{begin-eqn}} {{eqn | l = \eqclass 0 \beta       | o =  }} {{eqn | l = \eqclass 1 \beta       | r = \eqclass 4 \beta       | c =  }} {{eqn | l = \eqclass 2 \beta       | r = \eqclass 3 \beta       | c =  }} {{end-eqn}}
Equivalence Relation on Integers Modulo 5 induced by Squaring/Addition Modulo Beta is not Well-Defined
Let the $+_\beta$ operator ("addition") on the $\beta$-equivalence classes be defined as: :$\eqclass a \beta +_\beta \eqclass b \beta := \eqclass {a + b} \beta$ Then such an operation is not well-defined.
Constant Sequence Converges to Constant in Normed Division Ring
:the constant sequence $\tuple {\lambda, \lambda, \lambda, \dots}$ is convergent and $\displaystyle \lim_{n \mathop \to \infty} \lambda = \lambda$
Product of Sequence Converges to Zero with Cauchy Sequence Converges to Zero
Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring with zero $0$. Let $\sequence {x_n}$, $\sequence {y_n} $ be sequences in $R$. Let $\sequence {x_n}$ converge to $0$. Let $\sequence {y_n}$ be a Cauchy sequence. Then: :$\sequence {x_n y_n}$ and $\sequence {y_n x_n}$ converge to $0$.
Cauchy Sequence is Bounded/Normed Division Ring
Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring. Every Cauchy sequence in $R$ is bounded.
Cauchy Sequence with Finite Elements Prepended is Cauchy Sequence
Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring. Let $\sequence {x_n}$ be a sequence in $R$. Let $N \in \N$ Let $\sequence {y_n}$ be the sequence defined by: :$\forall n, y_n = x_{N + n}$ Let $\sequence {y_n}$ be a Cauchy sequence in $R$. Then: :$\sequence {x_n}$ is a Cauchy sequence in $R$.
Convergent Sequence with Finite Elements Prepended is Convergent Sequence
Let $\struct {R, \norm { \, \cdot \, } }$ be a normed division ring. Let $\sequence {x_n}$ be a sequence in $R$. Let $N \in \N$ Let $\sequence {y_n}$ be the sequence defined by: :$\forall n, y_n = x_{N+n}$ Let $\sequence {y_n}$ be a convergent sequence in $R$ with limit $l$. Then: :$\sequence {x_n}$ is a convergent sequence in $R$ with limit $l$.
Maximal Left and Right Ideal iff Quotient Ring is Division Ring
Let $R$ be a ring with unity. Let $J$ be an ideal of $R$. Then the following are equivalent: :$(1): \quad J$ is a maximal left ideal :$(2): \quad J$ is a maximal right ideal :$(3): \quad$ the quotient ring $R / J$ is a division ring.
Definition:Constructed Semantics/Instance 2/Rule of Addition
The Rule of Addition: :$q \implies (q \lor p)$ is a tautology in Instance 2 of constructed semantics.
Definition:Constructed Semantics/Instance 2/Rule of Commutation
The Rule of Commutation: :$\left({p \lor q}\right) \implies \left({q \lor p}\right)$ is a tautology in Instance 2 of constructed semantics.
Definition:Constructed Semantics/Instance 2/Factor Principle
The Factor Principle: :$\left({p \implies q}\right) \implies \left({\left({r \lor p}\right) \implies \left ({r \lor q}\right)}\right)$ is a tautology in Instance 2 of constructed semantics.
Test for Left Ideal
Let $J$ be a subset of a ring $\struct {R, +, \circ}$. Then $J$ is an left ideal of $\struct{R, +, \circ}$  these all hold: :$(1): \quad J \ne \O$ :$(2): \quad \forall x, y \in J: x + \paren {-y} \in J$ :$(3): \quad \forall j \in J, r \in R: r \circ j \in J$
Test for Right Ideal
Let $J$ be a subset of a ring $\struct {R, +, \circ}$. Then $J$ is a right ideal of $\struct {R, +, \circ}$  these all hold: :$(1): \quad J \ne \O$ :$(2): \quad \forall x, y \in J: x + \paren {-y} \in J$ :$(3): \quad \forall j \in J, r \in R: j \circ r \in J$
Maximal Left and Right Ideal iff Quotient Ring is Division Ring/Maximal Left Ideal implies Quotient Ring is Division Ring
Let $R$ be a ring with unity. Let $J$ be an ideal of $R$. If $J$ is a maximal left ideal then the quotient ring $R / J$ is a division ring.
Maximal Left and Right Ideal iff Quotient Ring is Division Ring/Maximal Right Ideal implies Quotient Ring is Division Ring
Let $R$ be a ring with unity. Let $J$ be an ideal of $R$. If $J$ is a maximal right ideal then the quotient ring $R / J$ is a division ring.
Inverse of Injective and Surjective Mapping is Mapping
Let $S$ and $T$ be sets. Let $f: S \to T$ be a mapping such that: :$(1): \quad f$ is an injection :$(2): \quad f$ is a surjection. Then the inverse $f^{-1}$ of $f$ is itself a mapping.
Inverse is Mapping implies Mapping is Injection and Surjection
Let $S$ and $T$ be sets. Let $f: S \to T$ be a mapping. Let the inverse $f^{-1} \subseteq T \times S$ itself be a mapping. Then: :$(1): \quad f$ is an injection :$(2): \quad f$ is a surjection.
Maximal Left and Right Ideal iff Quotient Ring is Division Ring/Quotient Ring is Division Ring implies Maximal Right Ideal
Let $R$ be a ring with unity. Let $J$ be an ideal of $R$. If the quotient ring $R / J$ is a division ring then $J$ is a maximal right ideal.
Maximal Left and Right Ideal iff Quotient Ring is Division Ring/Quotient Ring is Division Ring implies Maximal Left Ideal
Let $R$ be a ring with unity. Let $J$ be an ideal of $R$. If the quotient ring $R / J$ is a division ring then $J$ is a maximal left ideal.
Quotient Ring of Cauchy Sequences is Division Ring
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. Then the quotient ring $\CC / \NN$ is a division ring.
Hilbert Proof System Instance 2 Independence Results/Independence of A1
Axiom $(A1)$ is independent from $(A2)$, $(A3)$, $(A4)$.
Embedding Ring into Ring Structure Induced by Ring Operations
Let $\struct {R, +, \circ}$ be a ring. Let $S$ be a non-empty set. Let $\struct {R^S, +', \circ'}$ be the ring of mappings, where $+'$ and $\circ'$ are the pointwise operations induced on $R^S$ by $+$ and $\circ$. For each $r \in R$, let $f_r: S \to R$ be the mapping defined by: :$\forall s \in S, \map {f_r} s = r$ That is, $f_r$ is the constant mapping from $S$ to $r$. Let $\phi: R \to R^S$ be the mapping from the ring $R$ to the ring $R^S$ defined by: :$\forall r \in R: \map \phi r = f_r$ Then: :$\phi$ is a ring monomorphism.
Embedding Normed Division Ring into Ring of Cauchy Sequences
Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring. Let $\CC$ be the ring of Cauchy sequences over $R$ Let $\phi: R \to \CC$ be the mapping from $R$ to $\CC$ defined as: :$\forall a \in R: \map \phi a = \tuple {a, a, a, \dots}$ where $\tuple {a, a, a, \dots}$ is the constant sequence. Then $\phi$ is a ring monomorphism.
Hilbert Proof System Instance 2 Independence Results/Independence of A2
Axiom $(A2)$ is independent from $(A1)$, $(A3)$, $(A4)$.
Definition:Constructed Semantics/Instance 3/Rule of Idempotence
The Rule of Idempotence: :$(p \lor p) \implies p$ is a tautology in Instance 3 of constructed semantics.
Definition:Constructed Semantics/Instance 3/Rule of Commutation
The Rule of Commutation: :$\left({p \lor q}\right) \implies \left({q \lor p}\right)$ is a tautology in Instance 3 of constructed semantics.
Definition:Constructed Semantics/Instance 3/Factor Principle
The Factor Principle: :$\left({p \implies q}\right) \implies \left({\left({r \lor p}\right) \implies \left ({r \lor q}\right)}\right)$ is a tautology in Instance 3 of constructed semantics.
Hilbert Proof System Instance 2 Independence Results/Independence of A3
Axiom $(A3)$ is independent from $(A1)$, $(A2)$, $(A4)$.
Definition:Constructed Semantics/Instance 4/Rule of Idempotence
The Rule of Idempotence: :$(p \lor p) \implies p$ is a tautology in Instance 4 of constructed semantics.
Definition:Constructed Semantics/Instance 4/Rule of Addition
The Rule of Addition: :$q \implies (q \lor p)$ is a tautology in Instance 4 of constructed semantics.
Definition:Constructed Semantics/Instance 4/Factor Principle
The Factor Principle: :$\left({p \implies q}\right) \implies \left({\left({r \lor p}\right) \implies \left ({r \lor q}\right)}\right)$ is a tautology in Instance 4 of constructed semantics.
Cardinality of Set of Surjections
Let $S$ and $T$ be finite sets. Let $\card S = m, \card T = n$. Let $C$ be the number of surjections from $S$ to $T$. Then: :$C = n! \displaystyle {m \brace n}$ where $\displaystyle {m \brace n}$ denotes a Stirling number of the second kind.
Hilbert Proof System Instance 2 Independence Results/Independence of A4
Axiom $(\text A 4)$ is independent from $(\text A 1)$, $(\text A 2)$, $(\text A 3)$.
Image of Set Difference under Mapping/Corollary 3
Let $f: S \to T$ be a surjection. Let $A \subseteq S$ be a subset of $S$. Then: :$T \setminus f \sqbrk A \subseteq f \sqbrk {S \setminus A}$ where $\setminus$ denotes set difference.
Definition:Constructed Semantics/Instance 5/Rule of Idempotence
The Rule of Idempotence: :$(p \lor p) \implies p$ is a tautology in Instance 5 of constructed semantics.
Additive Function is Linear for Rational Factors
Let $f: \R \to \R$ be an additive function. Then: :$\forall r \in \Q, x \in \R: \map f {x r} = r \map f x$
Definition:Constructed Semantics/Instance 5/Rule of Addition
The Rule of Addition: :$q \implies (q \lor p)$ is a tautology in Instance 5 of constructed semantics.
Definition:Constructed Semantics/Instance 5/Rule of Commutation
The Rule of Commutation: :$\left({p \lor q}\right) \implies \left({q \lor p}\right)$ is a tautology in Instance 5 of constructed semantics.
Additive Function of Zero is Zero
Let $f: \R \to \R$ be an additive function. Then: :$f \paren 0 = 0$
Additive Function is Odd Function
Let $f: \R \to \R$ be an additive function. Then $f$ is an odd function.