hcju/proofwikips huggingface dataset:
Max Operation Representation on Real Numbers
Let $x, y \in \R$. Then: :$\max \left\{{x, y}\right\} = \frac 1 2 \left({x + y + \left\vert{x - y}\right\vert}\right)$ where $\operatorname{max}$ denotes the max operation.
Reparameterization of Directed Smooth Curve Preserves Image
Let $\left[{a \,.\,.\, b}\right]$ and $\left[{c\,.\,.\, d}\right]$ be closed real intervals. Let $\gamma : \left[{a \,.\,.\, b}\right] \to \C$ be a smooth path. Let $C$ be a directed smooth curve with parameterization $\gamma$. Suppose that $\sigma : \left[{c \,.\,.\, d}\right] \to \C$ is a reparameterization of $C$. Then $\operatorname{Im} \left({\gamma}\right) = \operatorname{Im} \left({\sigma}\right)$, where $\operatorname{Im} \left({\gamma}\right)$ denotes the image of $\gamma$.
Reparameterization of Directed Smooth Curve Maps Endpoints To Endpoints
Let $\R^n$ be a real cartesian space of $n$ dimensions. Let $\left[{a \,.\,.\, b}\right]$ and $\left[{c \,.\,.\, d}\right]$ be closed real intervals. Let $\gamma: \left[{a \,.\,.\, b}\right] \to \R^n$ be a smooth path in $\R^n$. Let $C$ be a directed smooth curve with parameterization $\gamma$. Suppose that $\sigma: \left[{c \,.\,.\, d}\right] \to \C$ is a reparameterization of $C$. Then the start points and end points of $\gamma$ and $\sigma$ are identical: :$\gamma \left({a}\right) = \sigma \left({c}\right)$ :$\gamma \left({b}\right) = \sigma \left({d}\right)$
Product of Ring Negatives
Let $\struct {R, +, \circ}$ be a ring. Then: :$\forall x, y \in \struct {R, +, \circ}: \paren {-x} \circ \paren {-y} = x \circ y$ where $\paren {-x}$ denotes the negative of $x$.
Distance Function for Distinct Elements in Metric Space is Strictly Positive
Let $A$ be a set. Let $d: A \times A \to \R$ be a real-valued function on $A$ with the following properties: {{begin-axiom}} {{axiom|n = M1'        |q = \forall x, y \in A        |m = d \left({x, y}\right) = 0 \iff x = y }} {{axiom|n = M2        |q = \forall x, y, z \in A        |m = d \left({x, y}\right) + d \left({y, z}\right) \ge d \left({x, z}\right) }} {{axiom|n = M3        |q = \forall x, y \in A        |m = d \left({x, y}\right) = d \left({y, x}\right) }} {{end-axiom}} which can be considered as being an alternative formulation of the metric space axioms. Then: :$\forall x, y \in A: x \ne y \implies d \left({x, y}\right) > 0$ which is metric space axiom $(M4)$. Thus $d$ is a distance function, so making $M := \left({A, d}\right)$ a metric space.
Transpose of Upper Triangular Matrix is Lower Triangular
The transpose of an upper triangular matrix is a lower triangular matrix.
Derivative iff Right and Left Derivative
Let $f$ be a real function. Then $f$ is differentiable  it has both right- and left-hand derivatives which agree.
Composition of Ring Homomorphisms is Ring Homomorphism
Let: :$\struct {R_1, +_1, \circ_1}$ :$\struct {R_2, +_2, \circ_2}$ :$\struct {R_3, +_3, \circ_3}$ be rings. Let: :$\phi: \struct {R_1, +_1, \circ_1} \to \struct {R_2, +_2, \circ_2}$ :$\psi: \struct {R_2, +_2, \circ_2} \to \struct {R_3, +_3, \circ_3}$ be homomorphisms. Then the composite of $\phi$ and $\psi$ is also a homomorphism.
Composition of Ring Epimorphisms is Ring Epimorphism
Let: :$\struct {R_1, +_1, \circ_1}$ :$\struct {R_2, +_2, \circ_2}$ :$\struct {R_3, +_3, \circ_3}$ be rings. Let: :$\phi: \struct {R_1, +_1, \circ_1} \to \struct {R_2, +_2, \circ_2}$ :$\psi: \struct {R_2, +_2, \circ_2} \to \struct {R_3, +_3, \circ_3}$ be (ring) epimorphisms. Then the composite of $\phi$ and $\psi$ is also a (ring) epimorphism.
Composition of Ring Isomorphisms is Ring Isomorphism
Let: * $\left({R_1, +_1, \circ_1}\right)$ * $\left({R_2, +_2, \circ_2}\right)$ * $\left({R_3, +_3, \circ_3}\right)$ be rings. Let: * $\phi: \left({R_1, +_1, \circ_1}\right) \to \left({R_2, +_2, \circ_2}\right)$ * $\psi: \left({R_2, +_2, \circ_2}\right) \to \left({R_3, +_3, \circ_3}\right)$ be (ring) isomorphisms. Then the composite of $\phi$ and $\psi$ is also a (ring) isomorphism.
Composition of Ring Endomorphisms is Ring Endomorphism
Let $R$ be a set. Let: : $\left({R, +_1, \circ_1}\right)$ : $\left({R, +_2, \circ_2}\right)$ : $\left({R, +_3, \circ_3}\right)$ be rings. Let: : $\phi: \left({R, +_1, \circ_1}\right) \to \left({R, +_2, \circ_2}\right)$ : $\psi: \left({R, +_2, \circ_2}\right) \to \left({R, +_3, \circ_3}\right)$ be (ring) endomorphisms. Then the composite of $\phi$ and $\psi$ is also a (ring) endomorphism.
Composition of Ring Automorphisms is Ring Automorphism
Let $R$ be a set. Let: * $\left({R, +_1, \circ_1}\right)$ * $\left({R, +_2, \circ_2}\right)$ * $\left({R, +_3, \circ_3}\right)$ be rings. Let: * $\phi: \left({R, +_1, \circ_1}\right) \to \left({R, +_2, \circ_2}\right)$ * $\psi: \left({R, +_2, \circ_2}\right) \to \left({R, +_3, \circ_3}\right)$ be (ring) automorphisms. Then the composite of $\phi$ and $\psi$ is also a (ring) automorphism.
Set has Rank
Let $S$ be a set. Then $S$ has a rank.
Quotient Epimorphism is Epimorphism/Group
Let $G$ be a group. Let $N$ be a normal subgroup of $G$. Let $G / N$ be the quotient group of $G$ by $N$. Let $q_N: G \to G / N$ be the quotient epimorphism from $G$ to $G / N$: :$\forall x \in G: \map {q_N} x = x N$ Then $q_N$ is a group epimorphism whose kernel is $N$.
Strictly Increasing Mapping on Well-Ordered Class
Let $ \left({S, \prec}\right)$ be a strictly well-ordered class. Let $\left({T, <}\right)$ be a strictly ordered class. Let $f$ be a mapping from $S$ to $T$. For each $i \in S$ such that $i$ is not maximal in $S$, let: : $f \left({i}\right) < f \left({\operatorname{succ} \left({i}\right)}\right)$ where $\operatorname{succ} \left({i}\right)$ is the immediate successor element of $i$. Let: :$\forall i, j \in S: i \preceq j \implies f \left({i}\right) \le f \left({j}\right)$ Then for each $i, j \in S$ such that $i \prec j$: :$f \left({i}\right) < f \left({j}\right)$ {{MissingLinks|In the above, link to definitions of increasing mapping and strictly increasing mapping, in order to correlate with page title.}}
Non-Maximal Element of Well-Ordered Class has Immediate Successor
Let $\left({C, \le}\right)$ be a well-ordered class. Let $x \in C$. Suppose that $x$ is maximal in $C$. {{mistake|maximal?}} Then $x$ has an immediate successor in $C$.
Set Contained in Smallest Transitive Set
Let $S$ be a set. Then there exists a transitive set $G$ such that: :$S \subseteq G$ and: :if $Q$ is any transitive set such that $S \subseteq Q$, then $G \subseteq Q$.
Transitive Set Contained in Von Neumann Hierarchy Level
Let $G$ be a transitive set. Then for some ordinal $i$, $G \subseteq V_i$.
External Direct Product Closure/General Result
Let $\displaystyle \struct {S, \circ} = \prod_{k \mathop = 1}^n S_k$ be the external direct product of the algebraic structures $\struct {S_1, \circ_1}, \struct {S_2, \circ_2}, \ldots, \struct {S_n, \circ_n}$. Let $\struct {S_1, \circ_1}, \struct {S_2, \circ_2}, \ldots, \struct {S_n, \circ_n}$ all be closed algebraic structures. Then $\struct {S, \circ}$ is also a closed algebraic structure.
External Direct Product of Abelian Groups is Abelian Group/General Result
The external direct product of a finite sequence of abelian groups is itself an abelian group.
Homomorphism of External Direct Products/General Result
Let $n \in \N_{>0}$. Let: : $\displaystyle \left({\mathcal S_n, \circledcirc_n}\right) := \prod_{k \mathop = 1}^n S_k = \left({S_1, \circ_1}\right) \times \left({S_2, \circ_2}\right) \times \cdots \times \left({S_n, \circ_n}\right)$ : $\displaystyle \left({\mathcal T_n, \circledast_n}\right) := \prod_{k \mathop = 1}^n T_k = \left({T_1, \ast_1}\right) \times \left({T_2, \ast_2}\right) \times \cdots \times \left({T_n, \ast_n}\right)$ be external direct products of algebraic structures. Let $\Phi_n: \left({\mathcal S_n, \circledcirc_n}\right) \to \left({\mathcal T_n, \circledast_n}\right)$ be the mapping defined as: :$\Phi_n: \left({s_1, \ldots, s_n}\right) := \begin{cases} \phi_1 \left({s_1}\right) & : n = 1 \\ \left({\phi_1 \left({s_1}\right), \phi_2 \left({s_2}\right)}\right) & : n = 2 \\ \left({\Phi_n \left({s_1, \ldots, s_{n - 1} }\right), \phi_n \left({s_n}\right)}\right) & : n > 2 \\ \end{cases}$ That is: :$\Phi_n: \left({s_1, \ldots, s_n}\right) := \left({\phi_1 \left({s_1}\right), \phi_2 \left({s_2}\right), \ldots, \phi_n \left({s_n}\right)}\right)$ Let $\phi_k: \left({S_k, \circ_k}\right) \to \left({T_k, \ast_k}\right)$ be a homomorphism for each $k \in \left\{{1, 2, \ldots, n}\right\}$. Then $\Phi_n$ is a homomorphism from $\left({\mathcal S_n, \circledcirc_n}\right)$ to $\left({\mathcal T_n, \circledast_n}\right)$.
Cesàro Mean
Let $\sequence {a_n}$ be a sequence of complex numbers. Suppose that $\sequence {a_n}$ converges to $\ell$ in $\C$: :$\displaystyle \lim_{n \mathop \to \infty} a_n = \ell$ Then also: :$\displaystyle \lim_{n \mathop \to \infty} \frac {a_1 + \dotsb + a_n} n = \ell$
Characterization of Minimal Element
Let $C$ be a class. Let $\prec$ be a relation on $C$. Let $B$ be a subclass of $C$. Let $x \in B$. Let $S_x = \left\{ {y \in C: y \prec x \text{ and } y \ne x}\right\}$ be the initial segment of $x$ in $C$. Then $x$ is a minimal element of $B$  $B \cap S_x = \varnothing$.
Non-Empty Class has Element of Least Rank
Let $C$ be a class. Let $C \ne \varnothing$. Then $C$ has an element of least rank. That is: :$\exists x \in C: \forall y \in C: \operatorname{rank} \left({x}\right) \le \operatorname {rank}\left({y}\right)$ where $\operatorname{rank}\left({x}\right)$ is the rank of $x$.
Reversed Directed Smooth Curve is Directed Smooth Curve
Let $C$ be a directed smooth curve in $\C$. Let $C$ be parameterized by the smooth path $\gamma: \left[{a \,.\,.\, b}\right] \to \C$. Define $\psi: \left[{a \,.\,.\, b}\right] \to \left[{a \,.\,.\, b}\right]$ by $\psi \left({t}\right) = a + b - t$. Define $\rho: \left[{a \,.\,.\, b}\right] \to \C$ by $\rho = \gamma \circ \psi$. Then $\rho$ is a smooth path which parameterizes a directed smooth curve $-C$. The directed smooth curve $-C$ is independent of the parameterization $\gamma$.
Reversed Contour is Contour
Let $\R^n$ be a real cartesian space of $n$ dimensions. Let $C$ be a contour in $\R^n$ that is defined as a concatenation of a finite sequence $C_1, \ldots, C_n$ of directed smooth curves in $\R^n$. Then the finite sequence of reversed directed smooth curves: :$-C_n, -C_{n - 1}, \ldots, -C_1$ defines a contour that is independent of the parameterization of $C_1, \ldots, C_n$.
Contour Integral along Reversed Contour
Let $C$ be a contour. Let $f: \Img C \to \C$ be a continuous complex functions, where $\Img C$ denotes the image of $C$. Then the contour integral of $f$ along the reversed contour $-C$ is: :$\displaystyle \int_{-C} \map f z \rd z = -\int_C \map f z \rd z$
Well-Founded Relation is Strongly Well-Founded/Lemma
Let $A$ be a non-empty class. Let $\prec$ be a foundational relation on $A$. Then $A$ has a $\prec$-minimal element.
Alternative Differentiability Condition
Let $\mathbb K$ be either $\R$ or $\C$. Let $f: D \to \mathbb K$ be a continuous mapping, where $D \subseteq \mathbb K$ is an open set. Let $z \in \mathbb K$. Then $f$ is differentiable at $z$  there exist $\alpha \in \mathbb K$ and $r \in \R_{>0}$ such that for all $h \in \map {B_r} 0 \setminus \set 0$: :$\map f {z + h} = \map f z + h \paren {\alpha + \map \epsilon h}$ where $\map {B_r} 0$ denotes an open ball of $0$, and $\epsilon: \map {B_r} 0 \setminus \set 0 \to \mathbb K$ is a continuous mapping with $\displaystyle \lim_{h \mathop \to 0} \map \epsilon h = 0$. If the conditions are true, then $\alpha = \map {f'} z$.
Von Neumann Hierarchy is Cumulative
For any two ordinals $x$ and $y$, If $x < y$ then $V(x) \subsetneqq V(y)$.
Odd Number Theorem/Corollary
A recurrence relation for the square numbers is: :$S_n = S_{n - 1} + 2 n - 1$
Effect of Sequence of Elementary Row Operations on Determinant
Let $\hat o_1, \ldots, \hat o_m$ be a finite sequence of elementary row operations.  Here, $\hat o_i$ denotes an elementary row operation on a square matrix of order $n$ over a commutative ring with unity $\struct {R, +, \circ}$. Here, $i \in \set {1, \ldots, m}$. Then there exists $c \in R$ such that for all square matrices of order $n$ $\mathbf A$ over $R$: :$\map \det {\mathbf A} = c \map \det {\mathbf A'}$ where $\mathbf A'$ is the square matrix of order $n$ that results from applying the elementary row operations $\hat o_1, \ldots, \hat o_m$ on $\mathbf A$.
Alternative Differentiability Condition/Proof 1
Let $\mathbb K$ be either $\R$ or $\C$. Let $f: D \to \mathbb K$ be a continuous mapping, where $D \subseteq \mathbb K$ is an open set. Let $z \in \mathbb K$. Then: : $f$ is differentiable at $z$ : : there exist $\alpha \in \mathbb K$ and $r \in \R_{>0}$ such that for all $h \in B_r \left({0}\right) \setminus \left\{ {0}\right\}$: ::$f \left({z + h}\right) = f \left({z}\right) + h \left({\alpha + \epsilon \left({h}\right) }\right)$ where: : $B_r \left({0}\right)$ denotes an open ball of $0$ : $\epsilon: B_r \left({0}\right) \setminus \left\{ {0}\right\} \to \mathbb K$ is a continuous mapping with $\displaystyle \lim_{h \to 0} \ \epsilon \left({h}\right) = 0$. If the conditions are true, then $\alpha = f' \left({z}\right)$.
Alternative Differentiability Condition/Proof 2
Let $\mathbb K$ be either $\R$ or $\C$. Let $f: D \to \mathbb K$ be a continuous mapping, where $D \subseteq \mathbb K$ is an open set. Let $z \in \mathbb K$. Then $f$ is differentiable at $z$  there exist $\alpha \in \mathbb K$ and $r \in \R_{>0}$ such that for all $h \in B_r \left({0}\right) \setminus \left\{ {0}\right\}$: :$f \left({z + h}\right) = f \left({z}\right) + h \left({\alpha + \epsilon \left({h}\right) }\right)$ where $B_r \left({0}\right)$ denotes an open ball of $0$, and $\epsilon: B_r \left({0}\right) \setminus \left\{ {0}\right\} \to \mathbb K$ is a continuous mapping with $\displaystyle \lim_{h \to 0} \ \epsilon \left({h}\right) = 0$. If the conditions are true, then $\alpha = f' \left({z}\right)$.
Every Filter has Limit Point implies Every Ultrafilter Converges
Let $T = \left({S, \tau}\right)$ be a topological space. Let each filter on $S$ have a limit point in $S$. Then each ultrafilter on $S$ converges to a point in $S$.
Every Ultrafilter Converges implies Every Filter has Limit Point
Let $T = \left({S, \tau}\right)$ be a topological space. Let every ultrafilter on $S$ be convergent. Then every filter on $S$ has a limit point.
Left Distributive and Commutative implies Distributive
Let $\struct {S, \circ, *}$ be an algebraic structure. Let the operation $\circ$ be left distributive over the operation $*$. Let $\circ$ be commutative. Then $\circ$ is distributive over $*$.
Filter on Product Space Converges iff Projections Converge
Let $\family{X_i}_{i \mathop \in I}$ be an indexed family of non-empty topological spaces where $I$ is an arbitrary index set. Let $\displaystyle X := \prod_{i \mathop \in I} X_i$ be the corresponding product space. Let $\pr_i: X \to X_i$ denote the projection from $X$ onto $X_i$. Let $\FF \subset \powerset X$ be a filter on $X$. Then $\FF$ converges  for all $i \in I$ the image filter $\map {\pr_i} \FF$ converges.
Tychonoff's Theorem for Hausdorff Spaces
Let $\family {X_i}_{i \mathop \in I}$ be an indexed family of non-empty Hausdorff spaces, where $I$ is an arbitrary index set. Let $\displaystyle X = \prod_{i \mathop \in I} X_i$ be the corresponding product space. Then $X$ is compact  each $X_i$ is compact. == Proof == First assume that $X$ is compact.  From Projection from Product Topology is Continuous, the projections $\pr_i : X \to X_i$ are continuous. From Continuous Image of Compact Space is Compact, it follows that the $X_i$ are compact. Assume now that each $X_i$ is compact. By Equivalent Definitions of Compactness it is enough to show that every ultrafilter on $X$ converges. Thus let $\FF$ be an ultrafilter on $X$. From Image of Ultrafilter is Ultrafilter, for each $i \in I$, the image filter $\map {\pr_i} \FF$ is an ultrafilter on $X_i$. Each $X_i$ is compact by assumption. So by Equivalent Definitions of Compactness, each $\map {\pr_i} \FF$ converges. From Filter on Product of Hausdorff Spaces Converges iff Projections Converge, $\FF$ converges. {{explain|Find the appropriate link for "converges" and explain why this proves the result.}}
Filter on Product of Hausdorff Spaces Converges iff Projections Converge
Let $\family{X_i}_{i \mathop \in I}$ be an indexed family of non-empty Hausdorff spaces where $I$ is an arbitrary index set. Let $\displaystyle X := \prod_{i \mathop \in I} X_i$ be the corresponding product space. Let $\pr_i: X \to X_i$ denote the projection from $X$ onto $X_i$. Let $\FF \subset \powerset X$ be a filter on $X$. Then $\FF$ converges  for each $i \in I$ the image filter $\map {\pr_i} \FF$ converges.
Tychonoff's Theorem/General Case
Let $\family {X_i}_{i \mathop \in I}$ be an indexed family of non-empty topological spaces, where $I$ is an arbitrary index set. Let $\displaystyle X = \prod_{i \mathop \in I} X_i$ be the corresponding product space. Then $X$ is compact  each $X_i$ is compact.
Positive Real has Real Square Root
Let $x \in \R_{>0}$ be a (strictly) positive real number. Then: :$\exists y \in \R: x = y^2$
Join is Idempotent
Let $\struct {S, \vee, \preceq}$ be a join semilattice. Then $\vee$ is idempotent.
Supremum of Singleton
Let $\struct {S, \preceq}$ be an ordered set. Then for all $a \in S$: :$\sup \set a = a$ where $\sup$ denotes supremum.
Infimum of Singleton
Let $\struct {S, \preceq}$ be an ordered set. Then for all $a \in S$: :$\inf \set a = a$ where $\inf$ denotes infimum.
Meet is Idempotent
Let $\struct {S, \wedge, \preceq}$ be a meet semilattice. Then $\wedge$ is idempotent.
GCD from Prime Decomposition
Let $a, b \in \Z$. From Expression for Integers as Powers of Same Primes, let: {{begin-eqn}} {{eqn | l = a       | r = p_1^{k_1} p_2^{k_2} \ldots p_r^{k_r} }} {{eqn | l = b       | r = p_1^{l_1} p_2^{l_2} \ldots p_r^{l_r} }} {{eqn | ll= \forall i \in \set {1, 2, \dotsc, r}:       | l = p_i       | o = \divides       | r = a       | c =  }} {{eqn | lo= \lor       | l = p_i       | o = \divides       | r = b }} {{end-eqn}} That is, the primes given in these prime decompositions may be divisors of ''either'' of the numbers $a$ or $b$. Then: :$\gcd \set {a, b} = p_1^{\min \set {k_1, l_1} } p_2^{\min \set {k_2, l_2} } \ldots p_r^{\min \set {k_r, l_r} }$ where $\gcd \set {a, b}$ denotes the greatest common divisor of $a$ and $b$.
LCM from Prime Decomposition
Let $a, b \in \Z$. From Expression for Integers as Powers of Same Primes, let: {{begin-eqn}} {{eqn | l = a       | r = p_1^{k_1} p_2^{k_2} \ldots p_r^{k_r} }} {{eqn | l = b       | r = p_1^{l_1} p_2^{l_2} \ldots p_r^{l_r} }} {{eqn | ll= \forall i \in \set {1, 2, \dotsc, r}:       | l = p_i       | o = \divides       | r = a       | c =  }} {{eqn | lo= \lor       | l = p_i       | o = \divides       | r = b }} {{end-eqn}} That is, the primes given in these prime decompositions may be divisors of ''either'' of the numbers $a$ or $b$. Then: :$\lcm \set {a, b} = p_1^{\max \set {k_1, l_1} } p_2^{\max \set {k_2, l_2} } \ldots p_r^{\max \set {k_r, l_r} }$ where $\lcm \set {a, b}$ denotes the lowest common multiple of $a$ and $b$.
Product of Positive Element and Element Greater than One
Let $\left({R, +, \circ, \le}\right)$ be an ordered ring with unity $1_R$ and zero $0_R$. Let $x,y \in R$. Suppose that $x > 0_R$ and $y > 1_R$. Then $x \circ y > x$ and $y \circ x > x$.
Strictly Positive Power of Strictly Positive Element Greater than One Succeeds Element
Let $\left({R, +, \circ, \le}\right)$ be an ordered ring with unity. Let $x \in R$ with $x > 1$ and $x > 0$. Let $n \in \N_{>0}$. Then: : $\circ^n x \ge x$
Directed Set has Strict Successors iff Unbounded Above
Let $\left({S, \le}\right)$ be a directed set. Then every element of $S$ has a strict successor in $S$ iff $S$ has no upper bound in $S$.
Complex Numbers are Uncountable
The set of complex numbers $\C$ is uncountably infinite.
Infimum of Infima
Let $\left({S, \preceq}\right)$ be an ordered set. Let $\mathbb T$ be a collection of subsets of $S$. Suppose all $T \in \mathbb T$ admit an infimum $\inf T$ in $S$. Then: :$\inf \bigcup \mathbb T = \inf \left\{{\inf T: T \in \mathbb T}\right\}$ as soon as one of these two quantities exists.
Join Semilattice is Ordered Structure
Let $\left({S, \vee, \preceq}\right)$ be a join semilattice. Then $\left({S, \vee, \preceq}\right)$ is an ordered structure. That is, $\preceq$ is compatible with $\vee$.
Meet Semilattice is Ordered Structure
Let $\left({S, \wedge, \preceq}\right)$ be a meet semilattice. Then $\left({S, \wedge, \preceq}\right)$ is an ordered structure.
Join Semilattice is Semilattice
Let $\struct {S, \vee, \preceq}$ be a join semilattice. Then $\struct {S, \vee}$ is a semilattice.
Meet Semilattice is Semilattice
Let $\struct {S, \wedge, \preceq}$ be a meet semilattice. Then $\struct {S, \wedge}$ is a semilattice.
Semilattice Induces Ordering
Let $\struct {S, \circ}$ be a semilattice. Let $\preceq$ be the relation on $S$ defined by, for all $a, b \in S$: :$a \preceq b$  $a \circ b = b$ Then $\preceq$ is an ordering.
Elementary Row Operations Commute with Matrix Multiplication
Let $\struct {R, +, \circ}$ be a commutative ring. Let $\mathbf A = \sqbrk a_{m n}$ be an $m \times n$ matrix over $R$. Let $\mathbf B = \sqbrk b_{n p}$ be an $n \times p$ matrix over $R$. Let $\hat o_1, \ldots, \hat o_{\hat n}$ be a finite sequence of elementary row operations that can be performed on a matrix over $R$ with $m$ rows. Let $\mathbf A'$ denote the $m \times n$-matrix that results from using $\hat o_1, \ldots, \hat o_{\hat n}$ on $\mathbf A$. Let $\mathbf C = \mathbf A \mathbf B$ be the matrix product of $\mathbf A$ and $\mathbf B$. Let $\mathbf C'$ denote the $m \times p$-matrix that results from using $\hat o_1, \ldots, \hat o_{\hat n}$ on $\mathbf C$. Then: :$\mathbf C' = \mathbf A' \mathbf B$
Hypothetical Syllogism/Formulation 1
:$p \implies q, q \implies r \vdash p \implies r$
Hypothetical Syllogism/Formulation 2/Proof 1
: $p \implies q, q \implies r, p \vdash r$
Hypothetical Syllogism/Formulation 2/Proof 3
: $p \implies q, q \implies r, p \vdash r$
Union of Relations Compatible with Operation is Compatible
Let $\left({S, \circ}\right)$ be a closed algebraic structure. Let $\mathcal F$ be a family of relations on $S$. Let each element of $\mathcal F$ be compatible with $\circ$. Let $\mathcal Q = \bigcup \mathcal F$. Then $Q$ is a relation compatible with $\circ$.
Complement of Relation Compatible with Group is Compatible
Let $\left({G, \circ}\right)$ be a group. Let $\mathcal R$ be a relation on $G$. Let $\mathcal R$ be compatible with $\circ$. Let $\mathcal Q = \complement_{G \times G} \mathcal R$, so that: :$\forall a, b \in G: a \mathop {\mathcal Q} b \iff \neg \left({a \mathop {\mathcal R} b}\right)$ Then $\mathcal Q$ is a relation compatible with $\circ$.
Relation Compatible with Group Operation is Strongly Compatible
Let $\left({S,\circ}\right)$ be a group. Let $\mathcal R$ be an endorelation on $S$ compatible with $\circ$. Let $x, y, z \in S$. Then $\mathcal R$ is strongly compatible with $\circ$. That is, the following equivalences hold: :$x \mathrel{\mathcal R} y \iff x \circ z \mathrel{\mathcal R} y \circ z$ :$x \mathrel{\mathcal R} y \iff z \circ x \mathrel{\mathcal R} z \circ y$
Intersection of Relations Compatible with Operation is Compatible
Let $\struct {S, \circ}$ be a closed algebraic structure. Let $\mathscr F$ be a indexed family of relations on $S$. Suppose that each element of $\mathscr F$ is compatible with $\circ$. Let $\QQ = \bigcap \mathscr F$ be the intersection of $\mathscr F$. Then $\QQ$ is a relation compatible with $\circ$.
Inverse of Relation Compatible with Operation is Compatible
Let $\left({S, \circ}\right)$ be a closed algebraic structure. Let $\mathcal R$ be a relation on $S$ which is compatible with $\circ$. Let $\mathcal Q$ be the inverse relation of $\mathcal R$. Then $\mathcal Q$ is compatible with $\circ$.
Properties of Ordered Group/OG1
{{begin-axiom}} {{axiom|n = \operatorname{OG}1.1        |m = x \preceq y \iff x \circ z \preceq y \circ z }} {{axiom|n = \operatorname{OG}1.2        |m = x \preceq y \iff z \circ x \preceq z \circ y }} {{axiom|n = \operatorname{OG}1.1'        |m = x \prec y \iff x \circ z \prec y \circ z }} {{axiom|n = \operatorname{OG}1.2'        |m = x \prec y \iff z \circ x \prec z \circ y }} {{end-axiom}}