hcju/proofwikips huggingface dataset:
Sturm-Liouville Problem/Unit Weight Function
Let $P, Q: \R \to \R$ be real mappings such that $P$ is smooth and positive, while $Q$ is continuous: :$\map P x \in C^\infty$  :$\map P x > 0$ :$\map Q x \in C^0$ Let the Sturm-Liouville equation, with $\map w x = 1$, be of the form: :$-\paren {P y'}' + Q y = \lambda y$ where $\lambda \in \R$. Let it satisfy the following boundary conditions: :$\map y a = \map y b = 0$ Then all solutions of the Sturm-Liouville equation, together with their eigenvalues, form infinite sequences $\sequence {y^{\paren n} }$ and $\sequence {\lambda^{\paren n} }$. Furthermore, each $\lambda^{\paren n}$ corresponds to an eigenfunction $y^{\paren n}$, unique up to a constant factor.
Discrepancy between Julian Year and Tropical Year
The Julian year and the tropical year differ such that the Julian calendar becomes $1$ day further out approximately every $128$ years.
Completely Irreducible and Subset Admits Infimum Equals Element implies Element Belongs to Subset
Let $L = \left({S, \preceq}\right)$ be an ordered set. Let $x \in S$ such that :$x$ is completely irreducible. Let $X \subseteq S$ such that :$X$ admits an infimum and $x = \inf X$ Then $x \in X$
Order Generating Includes Completely Irreducible Elements
Let $\left({S, \preceq}\right)$ be an ordered set. Let $X \subseteq S$ be an order generating subset of $S$. Let $X \in S$ be a completely irreducible element of $S$. Then $x \in X$.
399 is not Expressible as Sum of Fewer than 19 Fourth Powers
:$399 = 14 \times 1^4 + 3 \times 2^4 + 3^4 + 4^4$ or: :$399 = 11 \times 1^4 + 4 \times 2^4 + 4 \times 3^4$
Maximal implies Difference equals Intersection
Let $\left({S, \preceq}\right)$ be an ordered set. Let $x, y \in S$ such that :$x$ is maximal in $S \setminus y^\succeq$ Then $x^\succeq \setminus \left\{ {x}\right\} = x^\succeq \cap y^\succeq$
Square of n such that 2n-1 is Composite is not Sum of Square and Prime
Let $n^2$ be a square such that $2 n - 1$ is composite. Then $n^2$ cannot be expressed as the sum of a square and a prime.
40 times Heptagonal Numbers plus 9 gives Squares of Numbers ending in 7
Consider the heptagonal numbers: :$\displaystyle H_n = \sum_{k \mathop = 1}^n \left({5 k - 4}\right)$ Let $S_n$ be the sequence defined as: :$\forall n \in \Z_{>1}: S_n = 40 \times H_n + 9$ Then $S_n$ consists of the squares of all the positive integers which end in a $7$: :$49, 289, 729, 1369, 2209, 3249, 4489, 5929, 7569, \ldots$ that is: :$7^2, 17^2, 27^2, 37^2, 47^2, 57^2, 67^2, 77^2, 87^2, 97^2, \ldots$ {{OEIS|A017354}}
Maximal implies Completely Irreducible
Let $L = \struct {S, \vee, \wedge, \preceq}$ be a complete lattice. Let $p \in S$ such that :$\exists k \in S: p$ is maximal in $S \setminus k^\succeq$ Then $p$ is completely irreducible.
Infimum of Intersection of Upper Closures equals Join Operands
Let $L = \left({S, \vee, \preceq}\right)$ be a join semilattice. Let $x, y \in S$. Then $\inf\left({x^\succeq \cap y^\succeq}\right) = x \vee y$
Sturm-Liouville Problem/Unit Weight Function/Lemma
Let $\map \alpha x: \R \to \R$ such that $\map \alpha x \in C^2 \closedint a b$. Suppose: :$\displaystyle \forall \map h x \in C^2 \closedint a b: \map h a = \map h b = \map {h'} a = \map {h'} b = 0: \int_a^b \map \alpha x \, \map {h''} x \rd x = 0$ Then: :$\forall x \in \closedint a b: \map \alpha x = c_0 + c_1 x$ where $ c_0, c_1 \in \R $ are constants.
Intersection of Upper Closures is Upper Closure of Join Operands
Let $L = \left({S, \vee, \preceq}\right)$ be a join semilattice. Let $x, y \in S$. Then $\left({x \vee y}\right)^\succeq = x^\succeq \cap y^\succeq$
Strictly Precede and Step Condition and not Precede implies Joins are equal
Let $\struct {S, \vee, \preceq}$ be a join semilattice. Let $p, q, u \in S$ be such that: :$p \prec q$ and $\paren {\forall s \in S: p \prec s \implies q \preceq s}$ and $u \npreceq p$ Then: :$p \vee u = q \vee u$
Not Preceding implies Exists Completely Irreducible Element in Algebraic Lattice
Let $L = \struct {S, \vee, \wedge, \preceq}$ be a bounded below algebraic lattice. Let $x, y \in S$ such that :$y \npreceq x$ Then :$\exists p \in S: p$ is completely irreducible $\land ~x \preceq p \land y \npreceq p$
Set of All Completely Irreducible Elements is Smallest Order Generating
Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a bounded below algebraic lattice. Then $\mathit{Irr}\left({L}\right)$ is order generating and :$\forall X \subseteq S: X$ is order generating $\implies \mathit{Irr}\left({L}\right) \subseteq X$ where $\mathit{Irr}\left({L}\right)$ denotes the set of all completely irreducible elements of $L$.
Preimage of Lower Set under Increasing Mapping is Lower
Let $\struct {S, \preceq}$, $\struct {T, \precsim}$ be preordered sets. Let $f: S \to T$ be an increasing mapping. Let $X \subseteq T$ be a lower subset of $T$. Then $f^{-1} \sqbrk X$ is lower where $f^{-1} \sqbrk X$ denotes the preimage of $X$ under $f$.
Preimage of Upper Set under Increasing Mapping is Upper
Let $\struct {S, \preceq}$, $\struct {T, \precsim}$ be preordered sets. Let $f: S \to T$ be an increasing mapping. Let $X \subseteq T$ be a upper subset of $T$. Then $f^{-1} \sqbrk X$ is upper where $f^{-1} \sqbrk X$ denotes the preimage of $X$ under $f$.
Directed Suprema Preserving Mapping at Element is Supremum
Let $\left({S, \vee, \wedge, \preceq}\right)$ and $\left({T, \vee_2, \wedge_2, \precsim}\right)$ be bounded below continuous lattices. Let $f: S \to T$ be a mapping such that :$f$ preserves directed suprema. Let $x \in S$. Then $f\left({x}\right) = \sup \left\{ {f\left({w}\right): w \in S \land w \ll x}\right\}$
Preceding implies Way Below Closure is Subset of Way Below Closure
Let $\struct {S, \preceq}$ be an ordered set. Let $x, y \in S$ such that :$x \preceq y$ Then $x^\ll \subseteq y^\ll$ where $x^\ll$ denotes the way below closure of $x$.
Mapping at Element is Supremum implies Mapping is Increasing
Let $\left({S, \vee, \wedge, \preceq}\right)$ be a lattice. Let $\left({T, \vee_2, \wedge_2, \precsim}\right)$ be a complete lattice. Let $f: S \to T$ be a mapping such that :$\forall x \in S: f\left({x}\right) = \sup \left\{ {f\left({w}\right): w \in S \land w \ll x}\right\}$ Then $f$ is an increasing mapping.
Semiperfect Number is not Deficient
Let $n \in \Z_{>0}$ be a semiperfect number. Then $n$ is not deficient.
Perfect Number is Primitive Semiperfect
Let $n \in \Z_{>0}$ be a perfect number. Then $n$ is also a primitive semiperfect number.
Mapping at Element is Supremum implies Way Below iff There Exists Element that Way Below and Way Below
Let $\left({S, \vee, \wedge, \preceq}\right)$ be a complete lattice. Let $\left({T, \vee_2, \wedge_2, \precsim}\right)$ be a continuous complete lattice. Let $f: S \to T$ be a mapping such that :$\forall x \in S: f\left({x}\right) = \sup \left\{ {f\left({w}\right): w \in S \land w \ll x}\right\}$ Let $x \in S, y \in T$. Then :$y \ll f\left({x}\right) \iff \exists w \in S: w \ll x \land y \ll f\left({w}\right)$
Subset and Image Admit Suprema and Mapping is Increasing implies Supremum of Image Precedes Mapping at Supremum
Let $\left({S, \preceq}\right)$, $\left({T, \precsim}\right)$ be ordered sets. Let $f: S \to T$ be a increasing mapping. Let $D \subseteq S$ such that :$D$ admits a supremum in $S$ and $f\left[{D}\right]$ admits a supremum in $T$. Then $\sup \left({f\left[{D}\right]}\right) \precsim f\left({\sup D}\right)$
Characteristic Subgroup of Normal Subgroup is Normal
Let $G$ be a group. Let $N\leq G$ be normal. Let $H\leq N$ be characteristic. Then $H$ is normal in $G$.
Center is Characteristic Subgroup
Let $G$ be a group. Then its center $\map Z G$ is characteristic in $G$.
Semidirect Product of Groups is Group
Let $H$ and $N$ be groups. Let $\operatorname{Aut}(N)$ denote the automorphism group of $N$. Let $\phi : H\to \operatorname{Aut}(N)$ be a group homomorphism, that is, let $H$ act on $N$. Let $N\rtimes_\phi H$ be the semidirect product of $N$ and $H$ with respect to $\phi$, that is: :$N\rtimes_\phi H = (N\times H, \circ)$ where :$(n_1, h_1) \circ (n_2, h_2) = (n_1\cdot \phi(h_1)(n_2), h_1\cdot h_2)$ Then $N\rtimes_\phi H$ is a group.
Inverse of Element in Semidirect Product
Let $N$ and $H$ be groups. Let $H$ act by automorphisms on $N$ via $\phi$. Let $N \rtimes_\phi H$ be the corresponding (outer) semidirect product. Let $\tuple {n, h} \in N \rtimes_\phi H$. Then: {{begin-eqn}} {{eqn | l = \tuple {n, h}^{-1}       | r = \tuple {\map {\phi_{h^{-1} } } {n^{-1} }, h^{-1} }       | c =  }} {{eqn | r = \tuple {\paren {\map {\phi_{h^{-1} } } n}^{-1}, h^{-1} }       | c =  }} {{eqn | r = \tuple {\map { {\phi_h}^{-1} } {n^{-1} }, h^{-1} }       | c =  }} {{eqn | r = \tuple {\paren {\map { {\phi_h}^{-1} } n}^{-1}, h^{-1} }       | c =  }} {{end-eqn}}
Semidirect Product with Trivial Action is Direct Product
Let $H$ and $N$ be groups. Let $\Aut N$ denote the automorphism group of $N$. Let $\phi: H \to \Aut N$ be defined as: :$\forall h \in H: \map \phi h = I_N$ for all $h \in H$ where $I_N$ denotes the identity mapping on $N$. Let $N \rtimes_\phi H$ be the corresponding semidirect product. Then $N \rtimes_\phi H$ is the direct product of $N$ and $H$.
Semidirect Product is Abelian iff Components are Abelian and Action is Trivial
Let $N$ and $H$ be groups. Let $H$ act by automorphisms on $N$ via $\phi$. Let $N\rtimes_\phi H$ be the corresponding semidirect product. Then the following are equivalent: :$(1): \quad$ $N\rtimes_\phi H$ is abelian :$(2): \quad$ $N$ and $H$ are abelian and $H$ acts trivially
Integers whose Squares end in 444
The sequence of positive integers whose square ends in $444$ begins: :$38, 462, 538, 962, 1038, 1462, 1538, 1962, 2038, 2462, 2538, 2962, 3038, 3462, \ldots$ {{OEIS|A039685}}
Subset and Image Admit Infima and Mapping is Increasing implies Infimum of Image Succeeds Mapping at Infimum
Let $\struct {S, \preceq}$ and $\struct {T, \precsim}$ be ordered sets. Let $f: S \to T$ be a increasing mapping. Let $D \subseteq S$ such that :$D$ admits a infimum in $S$ and $f \sqbrk D$ admits a infimum in $T$. Then $\map f {\inf D} \precsim \map \inf {f \sqbrk D}$
Equivalence of Definitions of Finite Galois Extension
Let $L/K$ be a finite field extension. {{TFAE|def = Finite Galois Extension}}
Finite Field Extension has Finite Galois Group
Let $E / F$ be a finite field extension. Then its automorphism group is finite.
Primitive Element Theorem
Let $E/F$ be a separable field extension of finite degree. Then $E/F$ is simple: there exists $\alpha\in E$ such that $E=F(\alpha)$.
Largest Number not Expressible as Sum of Less than 32 Positive Fifth Powers
The largest positive integer which cannot be expressed as the sum of less than $32$ positive fifth powers is $466$: :$466 = 18 \times 1^5 + 14 \times 2^5$
Smallest Square which is Sum of 3 Fourth Powers
The smallest positive integer whose square is the sum of $3$ fourth powers is $481$: :$481^2 = 12^4 + 15^4 + 20^4$
Solutions to p^2 Divides 10^p - 10
The known prime numbers $p$ which satisfy the equation: :$p^2 \divides \paren {10^p - 10}$ where $\divides$ denotes divisibility, are: :$3, 487, 56 \, 598 \, 313$ {{OEIS|A045616}}
Kaprekar's Process on 3 Digit Number ends in 495
Let $n$ be a $3$-digit integer whose digits are not all the same. Kaprekar's process, when applied to $n$, results in $495$ after no more than $6$ iterations.
Nilpotent Elements of Commutative Ring form Ideal
Let $\struct {R, +, \circ}$ be a commutative ring whose zero is $0_R$ and whose unity is $1_R$. The subset of nilpotent elements of $R$ form an ideal of $R$.
Algebraic Closure of Field is Unique
Let $F$ be a field. Let $K$ and $L$ be algebraic closures of $F$. Then $K$ and $L$ are $F$-isomorphic.
Multiplicative Group of Galois Field is Cyclic
Let $\GF$ be a Galois field of order $q$. Then its multiplicative group is cyclic of order $q-1$: :$\GF^\times \cong C_{q - 1}$
Automorphism Group of C Over R
The field extension $\C / \R$ of complex numbers $\C$ over real numbers $\R$ has automorphism group $\operatorname{Aut}$: :$\operatorname{Aut} \paren {\C / \R} = \set {\operatorname{id}, \sigma}$ where: :$\operatorname{id}$ denotes the identity mapping :$\sigma$ denotes complex conjugation
Image under Inclusion Mapping
Let $X$ be a set. Let $S \subseteq X$, $Z \subseteq S$. Then $i_S\left[{Z}\right] = Z$ where :$i_S$ denotes the inclusion mapping of $S$, :$i_S\left[{Z}\right]$ denotes the image of $Z$ under $i_S$.
Limit Inferior of Inclusion Moore-Smith Sequence is Supremum of Directed Subset
Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be an up-complete lattice. Let $D \subseteq S$ be a directed subset of $S$. Let $\left({D, \preceq'}\right)$ be a directed ordered subset of $L$. Let $i_D: D \to S$, the inclusion mapping, be a Moore-Smith sequence in $S$. Then $\liminf i_D = \sup D$
Correspondence Between Group Actions and Permutation Representations
Let $G$ be a group. Let $X$ be a set. There is a one-to-one correspondence between group actions of $G$ on $X$ and permutation representations of $G$ in $X$, as follows: Let $\phi : G \times X \to X$ be a group action. Let $\rho : G \to \struct {\map \Gamma X, \circ}$ be a permutation representation. The following are equivalent: :$(1): \quad \rho$ is the permutation representation associated to $\phi$ :$(2): \quad \phi$ is the group action associated to $\rho$
Trivial Field Extension is Galois
Let $F$ be a field. The trivial field extension $F/F$ is Galois.
Automorphism Group Acts Faithfully on Generating Set
Let $E/F$ be a field extension. Let $\operatorname{Aut}(E/F)$ be its automorphism group. Let $S\subset E$ be a generating set of the extension. Let $S$ be stable under the group action of $\operatorname{Aut}(E/F)$. Then the induced group action on $S$ is faithful.
Upper Closure in Ordered Subset is Intersection of Subset and Upper Closure
Let $L = \left({S, \preceq}\right)$ be an ordered set. Let $\left({T, \precsim}\right)$ be an ordered subset of $L$. Let $t \in T$. Then $t^\succsim = T \cap t^\succeq$
Approximation to Power of 7 by Power of 10
:$7^{510} \approx 1 \cdotp 00000 \, 09377 \, 76536 \ldots \times 10^{431}$ This is the closest known approximation of a power of $7$ by a power of $10$.
Limit Inferior of Repetition Moore-Smith Sequence
Let $L = \left({S, \wedge, \preceq}\right)$ be a meet semilattice. Let $N = \left({\N, \le}\right)$ be a directed ordered set. Let $a, b \in S$. Let $f = \left({c_i}\right)_{i \in \N} = \left({a, b, a, b, \dots}\right):\N \to S$ be a Moore-Smith sequence. Then $\liminf \left({c_i}\right)_{i \in \N} = a \wedge b$
Mapping at Limit Inferior Precedes Limit Inferior of Composition Mapping and Sequence implies Mapping is Increasing
Let $\left({S, \vee_1, \wedge_1, \preceq_1}\right)$ and $\left({T, \vee_2, \wedge_2, \preceq_2}\right)$ be lattices. Let $f: S \to T$ be a mapping such that :for all directed set $\left({D, \precsim}\right)$ and Moore-Smith sequence $N:D \to S$ in $S$: $f\left({\liminf N}\right) \preceq_2 \liminf\left({f \circ N}\right)$ Then $f$ is an increasing mapping.
Mapping at Limit Inferior Precedes Limit Inferior of Composition Mapping and Sequence implies Supremum of Image is Mapping at Supremum of Directed Subset
Let $\left({S, \vee_1, \wedge_1, \preceq_1}\right)$ and $\left({T, \vee_2, \wedge_2, \preceq_2}\right)$ be up-complete lattices. Let $f: S \to T$ be a mapping such that :for all directed set $\left({D, \precsim}\right)$ and Moore-Smith sequence $N:D \to S$ in $S$: $f\left({\liminf N}\right) \preceq_2 \liminf\left({f \circ N}\right)$ Let $D$ be a directed subset of $S$. Then $\sup \left({f\left[{D}\right]}\right) = f\left({\sup D}\right)$ where $f\left[{D}\right]$ denotes the image of $D$ under $f$.
Prime Values of Double Factorial plus 1
Let $n!!$ denote the double factorial function. The sequence of positive integers $n$ such that $n!! + 1$ is prime begins: :$0, 1, 2, 518, 33 \, 416, 37 \, 310, 52 \, 608, 123 \, 998, 220 \, 502, \ldots$ {{OEIS|A080778}}
Limit Inferior of Restriction Moore-Smith Sequence is Supremum of Image of Directed Subset
Let $L = \left({S, \vee_1, \wedge_1, \preceq_1}\right)$ and $\left({T, \vee_2, \wedge_2, \preceq_2}\right)$ be up-complete lattices. Let $f:S \to T$ be an increasing mapping. Let $D \subseteq S$ be a directed subset of $S$. Let $\left({D, \preceq'}\right)$ be a directed ordered subset of $L$. Let $f \restriction D: D \to T$, the restriction of mapping, be a Moore-Smith sequence in $T$. Then $\liminf \left({f \restriction D}\right) = \sup \left({f\left[{D}\right]}\right)$
Infimum of Image of Upper Closure of Element under Increasing Mapping
Let $\left({S, \preceq}\right)$ and $\left({T, \precsim}\right)$ be ordered set. Let $f:S \to T$ be an increasing mapping. Let $x \in S$. Then $\inf \left({f\left[{x^\succeq}\right]}\right) = f\left({x}\right)$
Composition of Mapping and Inclusion is Restriction of Mapping
Let $S, T$ be sets. Let $f:S \to T$ be a mapping. Let $A \subseteq S$. Then $f \circ i_A = f \restriction A$ where :$i_A$ denotes the inclusion mapping of $A$, :$f \restriction A$ denotes the restriction of $f$ to $A$.
Mapping at Limit Inferior Precedes Limit Inferior of Composition Mapping and Sequence implies Mapping Preserves Directed Suprema
Let $\struct {S, \vee_1, \wedge_1, \preceq_1}$ and $\struct {T, \vee_2, \wedge_2, \preceq_2}$ be complete lattices. Let $f: S \to T$ be a mapping such that :for all directed sets $\struct {D, \precsim}$ and Moore-Smith sequences $N:D \to S$ in $S$: $\map f {\liminf N} \preceq_2 \map \liminf {f \circ N}$ Then $f$ preserves directed suprema.
Thurston's Geometrization Conjecture
When a topological manifold of dimension $3$ has been split into its connected sum and the Jaco-Shalen-Johannson torus decomposition, the remaining components each admit exactly one of the following geometries: :$(1): \quad$ Euclidean geometry :$(2): \quad$ Hyperbolic geometry :$(3): \quad$ Spherical geometry :$(4): \quad$ The geometry of $\mathbb S^2 \times \R$ :$(5): \quad$ The geometry of $\mathbb H^2 \times \R$ :$(6): \quad$ The geometry of the universal cover $S L_2 \R^~$ of the Lie group $S L_2 \R$ :$(7): \quad$ Nil geometry :$(8): \quad$ Sol geometry where: :$\mathbb S^2$ is the 2-sphere :$\mathbb H^2$ is the hyperbolic plane.
Poincaré Conjecture/Historical Note
The Poincaré Conjecture was first posed in $1904$ by {{AuthorRef|Jules Henri Poincaré}}. It was finally resolved by the work of {{AuthorRef|Grigori Perelman}}, who solved Thurston's Geometrization Conjecture in $2003$.
Definition:Stabilizer/Linguistic Note
The British English spelling for '''stabilizer''' is '''stabiliser'''.
Mapping Preserves Directed Suprema implies Mapping is Continuous
Let $\left({S, \preceq_1, \tau_1}\right)$ and $\left({T, \preceq_2, \tau_2}\right)$ be up-complete ordered sets with Scott topologies. Let $f: S \to T$ be a directed suprema preserving mapping. Then $f$ is continuous.
Secant of Complex Number
Let $a$ and $b$ be real numbers. Let $i$ be the imaginary unit. Then: :$\sec \paren {a + b i} = \dfrac {\cos a \cosh b + i \sin a \sinh b} {\cos^2 a \cosh^2 b + \sin^2 a \sinh^2 b}$ where: :$\sec$ denotes the complex 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
Tangent of Complex Number/Formulation 1
:$\tan \paren {a + b i} = \dfrac {\sin a \cosh b + i \cos a \sinh b} {\cos a \cosh b - i \sin a \sinh b}$
Tangent of Complex Number/Formulation 2
:$\tan \paren {a + b i} = \dfrac {\tan a + i \tanh b} {1 - i \tan a \tanh b}$
Tangent of Complex Number/Formulation 3
:$\tan \paren {a + b i} = \dfrac {\tan a - \tan a \tanh ^2 b} {1 + \tan ^2 a \tanh ^2 b} + \dfrac {\tanh b + \tan ^2 a \tanh b} {1 + \tan ^2 a \tanh ^2 b} i$
Cosecant of Complex Number
Let $a$ and $b$ be real numbers. Let $i$ be the imaginary unit. Then: :$\csc \paren {a + b i} = \dfrac {\sin a \cosh b - i \cos a \sinh b} {\sin^2 a \cosh^2 b + \cos^2 a \sinh^2 b}$ where: :$\csc$ denotes the complex 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
Cotangent of Complex Number
Let $a$ and $b$ be real numbers. Let $i$ be the imaginary unit. Then:
Cotangent of Complex Number/Formulation 1
:$\cot \paren {a + b i} = \dfrac {\cos a \cosh b - i \sin a \sinh b} {\sin a \cosh b + i \cos a \sinh b}$
Cotangent of Complex Number/Formulation 2
:$\map \cot {a + b i} = \dfrac {i \cot a \coth b - 1} {\cot a - i \coth b}$
Continuous iff Mapping at Limit Inferior Precedes Limit Inferior of Composition of Mapping and Sequence
Let $\left({S, \preceq_1, \tau_1}\right)$ and $\left({T, \preceq_2, \tau_2}\right)$ be complete topological lattices with Scott topologies. Let $f: S \to T$ be a mapping. Then $f$ is continuous  :for all directed set $\left({D, \precsim}\right)$ and Moore-Smith sequence $N:D \to S$ in $S$: $f\left({\liminf N}\right) \preceq_2 \liminf\left({f \circ N}\right)$
Mapping is Increasing implies Mapping at Infimum for Sequence Precedes Infimum for Composition of Mapping and Sequence
Let $\left({S, \vee_1, \wedge_1, \preceq_1}\right)$ and $\left({T, \vee_2, \wedge_2, \preceq_2}\right)$ be complete lattices. Let $f: S \to T$ be an increasing mapping. Let $\left({D, \precsim}\right)$ be a directed set. Let $N: D \to S$ be a Moore-Smith sequence in $S$. Let $j \in D$. Then $f\left({\inf\left({N\left[{\precsim \left({j}\right)}\right]}\right)}\right) \preceq_2 \inf \left({\left({f \circ N}\right)\left[{\precsim \left({j}\right)}\right]}\right)$
Set of Infima for Sequence is Directed
Let $\left({S, \vee, \wedge, \preceq}\right)$ be a complete lattice. Let $\left({A, \precsim}\right)$ be a non-empty directed set. Let $Z: A \to S$ be a Moore-Smith sequence. Let $D = \left\{ {\inf \left({Z\left[{\precsim \left({j}\right)}\right]}\right): j \in A}\right\}$ be a subset of $S$. Then $D$ is directed.
Preceding implies Image is Subset of Image
Let $\left({S, \precsim}\right)$ be a preordered set. Let $x, y \in S$ such that :$x \precsim y$ Then $\precsim\left({y}\right) \subseteq \mathord\precsim\left({x}\right)$ where $\precsim\left({y}\right)$ denotes the image of $y$ under $\precsim$.
Universal Property of Group Ring
Let $R$ be a commutative ring with unity. Let $G$ be a group. Let $R \sqbrk G$ be the corresponding group ring. Let $S$ be a commutative ring with unity. Let $\phi: R \to S$ be a ring homomorphism. Let $\beta : G \to R^\times$ be a group homomorphism, where $R^\times$ is the multiplicative group of $R$. Then there exists a unique ring homomorphism from $R \sqbrk G$ to $S$ which extends $\phi$ and $\beta$.
Continuous iff Directed Suprema Preserving
Let $\left({S, \preceq_1, \tau_1}\right)$ and $\left({T, \preceq_2, \tau_2}\right)$ be complete topological lattices with Scott topologies. Let $f: S \to T$ be a mapping. Then $f$ is continuous  $f$ preserves directed suprema.
Universal Property of Quotient of Topological Group
Let $G$ and $H$ be topological groups. Let $N$ be a normal subgroup of $G$. Let $\pi : G \to G/N$ be the quotient mapping. Let $f : G \to H$ be a continuous group homomorphism whose kernel contains $N$. Then there exists a unique continuous group homomorphism $\overline f : G/N \to H$ such that $f = \overline f \circ \pi$.
Universal Property of Quotient Set
Let $X$ and $Y$ be sets. Let $\sim$ be an equivalence relation on $X$. Let $\pi : X \to X/\sim$ be the quotient mapping to the quotient set. Let $f : X \to Y$ be $\sim$-invariant. Then there exists a unique mapping $\overline f : X/\!\sim \to Y$ such that $f = \overline f \circ \pi$. :$\xymatrix{ X \ar[d]_\pi \ar[r]^f & Y\\ X/\sim \ar@{.>}[ru]_{\overline f} }$
Continuous iff Way Below iff There Exists Element that Way Below and Way Below
Let $\struct {S, \preceq_1, \tau_1}$ and $\struct {T, \preceq_2, \tau_2}$ be complete continuous topological lattices with Scott topologies. Let $f: S \to T$ be a mapping. Then $f$ is continuous  :$\forall x \in S, y \in T: y \ll \map f x \iff \exists w \in S: w \ll x \land y \ll \map f w$ {{explain|link to definition of $\ll$ and $\gg$}}
Finite Symmetric Group is Ambivalent
Let $n$ be a natural number. Let $S_n$ be a symmetric group of order $n$. Then $S_n$ is ambivalent.
Alternating Groups that are Ambivalent
Let $n$ be a natural number. Then the $n$th alternating group $A_n$ is ambivalent  $n\in \{1, 2, 5, 6, 10, 14\}$. {{OEIS|A115200}}