hcju/proofwikips huggingface dataset:
Null Space is Subspace
Let: :$\operatorname{N} \left({\mathbf A}\right) = \left\{{\mathbf x \in \R^n: \mathbf {Ax} = \mathbf 0}\right\}$ be the null space of $\mathbf A$, where: :$ \mathbf A_{m \times n} = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \\ \end{bmatrix}$,  $\mathbf x_{n \times 1} = \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix}$, $\mathbf 0_{m \times 1} = \begin{bmatrix} 0 \\ 0 \\ \vdots \\ 0 \end{bmatrix}$ are matrices. Then $\operatorname{N} \left({\mathbf A}\right)$ is a linear subspace of $\R^n$.
Norm is Continuous
Let $\struct {V, \norm {\,\cdot\,} }$ be a normed vector space. Then the mapping $x \mapsto \norm x$ is continuous. Here, the metric used is the metric $d$ induced by $\norm {\,\cdot\,}$.
P-adic Valuation is Valuation
The $p$-adic valuation $\nu_p: \Q \to \Z \cup \left\{{+\infty}\right\}$ is a valuation on $\Q$.
P-adic Norm not Complete on Rational Numbers
Let $\norm {\,\cdot\,}_p$ be the $p$-adic norm on the rationals $\Q$ for some prime $p$. Then: :the valued field $\struct {\Q, \norm {\,\cdot\,}_p}$ is not complete. That is, there exists a Cauchy sequence in $\struct {\Q, \norm{\,\cdot\,}_p}$ which does not converge to a limit in $\Q$.
P-adic Valuation on Integers is Valuation
Let $\nu_p^\Z: \Z \to \Z \cup \set {+\infty}$ be the $p$-adic valuation restricted to the integers. Then $\nu_p^\Z$ is a valuation.
Infinite Measure is Measure
Let $\struct {X, \Sigma}$ be a measurable space. Then the infinite measure $\mu$ on $\struct {X, \Sigma}$ is a measure.
Linear Combination of Measures
Let $\left({X, \Sigma}\right)$ be a measurable space. Let $\mu, \nu$ be measures on $\left({X, \Sigma}\right)$. Then for all positive real numbers $a, b \in \R_{\ge 0}$, the pointwise sum: :$a \mu + b \nu: \Sigma \to \overline \R, \ \left({a \mu + b \nu}\right) \left({E}\right) := a \mu \left({E}\right) + b \nu \left({E}\right)$ is also a measure on $\left({X, \Sigma}\right)$.
Series of Measures is Measure
Let $\left({X, \Sigma}\right)$ be a measurable space. Let $\left({\mu_n}\right)_{n \mathop \in \N}$ be a sequence of measures on $\left({X, \Sigma}\right)$. Let $\left({a_n}\right)_{n \mathop \in \N} \subseteq \R_{\ge 0}$ be a sequence of positive real numbers. Then the series of measures $\mu: \Sigma \to \overline{\R}$, defined by: :$\displaystyle \mu \left({E}\right) := \sum_{n \mathop \in \N} a_n \mu_n \left({E}\right)$ is also a measure on $\left({X, \Sigma}\right)$.
Quotient Set Determined by Relation Induced by Partition is That Partition
Let $S$ be a set. Let $\mathcal P$ be a partition of $S$. Let $\mathcal R$ be the relation induced by $\mathcal P$. Then the quotient set $S / \mathcal R$ of $S$ is $\mathcal P$ itself.
Null Sets Closed under Subset
Let $\left({X, \Sigma, \mu}\right)$ be a measure space. Let $N \in \Sigma$ be a $\mu$-null set, and let $M \in \Sigma$ be a subset of $N$. Then $M$ is also a $\mu$-null set.
Null Sets Closed under Countable Union
Let $\left({X, \Sigma, \mu}\right)$ be a measure space. Let $\left({N_n}\right)_{n \in \N}$ be a sequence of $\mu$-null sets. Then $N := \displaystyle \bigcup_{n \mathop \in \N} N_n$ is also a $\mu$-null set.
Absolute Value induces Equivalence Compatible with Integer Multiplication
Let $\Z$ be the set of integers. Let $\RR$ be the relation on $\Z$ defined as: :$\forall x, y \in \Z: \struct {x, y} \in \RR \iff \size x = \size y$ where $\size x$ denotes the absolute value of $x$. Then $\RR$ is a congruence relation for integer multiplication.
Completion Theorem (Measure Spaces)
Let $\left({X, \Sigma, \mu}\right)$ be a measure space. Then there exists a completion $\left({X, \Sigma^*, \bar \mu}\right)$ of $\left({X, \Sigma, \mu}\right)$.
Restricted Measure is Measure
Let $\left({X, \Sigma, \mu}\right)$ be a measure space. Let $\Sigma'$ be a sub-$\sigma$-algebra of $\Sigma$. Then the restricted measure $\mu \restriction_{\Sigma'}$ is a measure on the measurable space $\left({X, \Sigma'}\right)$.
Restricting Measure Preserves Finiteness
Let $\left({X, \Sigma, \mu}\right)$ be a measure space. Let $\mu$ be a finite measure. Let $\Sigma'$ be a sub-$\sigma$-algebra of $\Sigma$. Then the restricted measure $\mu \restriction_{\Sigma'}$ is also a finite measure.
Central Subgroup is Normal
Let $G$ be a group. Let $H$ be a central subgroup of $G$. Then $H$ is a normal subgroup of $G$.
Angle Between Non-Zero Vectors Always Defined
The angle between two non-zero vectors in $\R^n$ is always defined.
Dynkin System Contains Empty Set
Let $X$ be a set, and let $\mathcal D$ be a Dynkin system on $X$. Then the empty set $\varnothing$ is an element of $\mathcal D$.
Dynkin System Closed under Disjoint Union
Let $X$ be a set, and let $\mathcal D$ be a Dynkin system on $X$. Let $D, E \in \mathcal D$ be disjoint. Then the union $D \cup E$ is also an element of $\mathcal D$.
Sigma-Algebra is Dynkin System
Let $X$ be a set, and let $\Sigma$ be a $\sigma$-algebra on $X$. Then $\Sigma$ is a Dynkin system on $X$.
Existence and Uniqueness of Dynkin System Generated by Collection of Subsets
Let $X$ be a set. Let $\mathcal G \subseteq \mathcal P \left({X}\right)$ be a collection of subsets of $X$. Then $\delta \left({\mathcal G}\right)$, the Dynkin system generated by $\mathcal G$, exists and is unique.
Generated Sigma-Algebra Contains Generated Dynkin System
Let $X$ be a set. Let $\mathcal G \subseteq \mathcal P \left({X}\right)$ be a collection of subsets of $X$. Then $\delta \left({\mathcal G}\right) \subseteq \sigma \left({\mathcal G}\right)$. Here $\delta$ denotes generated Dynkin system, and $\sigma$ denotes generated $\sigma$-algebra.
Dynkin System Closed under Intersections is Sigma-Algebra
Let $X$ be a set, and let $\DD$ be a Dynkin system on $X$. Suppose that $\DD$ satisfies the following condition: :$(1):\quad \forall D, E \in \DD: D \cap E \in \DD$ That is, $\DD$ is closed under intersection. Then $\DD$ is a $\sigma$-algebra.
Dynkin System with Generator Closed under Intersection is Sigma-Algebra
Let $X$ be a set. Let $\GG \subseteq \powerset X$ be a collection of subsets of $X$. Suppose that $\GG$ satisfies the following condition: :$(1):\quad \forall G, H \in \GG: G \cap H \in \GG$ That is, $\GG$ is closed under intersection. Then: :$\map \delta \GG = \map \sigma \GG$ where $\delta$ denotes generated Dynkin system, and $\sigma$ denotes generated $\sigma$-algebra.
Carathéodory's Theorem (Measure Theory)
Let $X$ be a set. Let $\SS \subseteq \powerset X$ be a semi-ring of subsets of $X$. Let $\mu: \SS \to \overline \R$ be a pre-measure on $\SS$. Let $\map \sigma \SS$ be the $\sigma$-algebra generated by $\SS$. Then $\mu$ extends to a measure $\mu^*$ on $\map \sigma \SS$. {{Refactor|extract corollary}}
Congruence Modulo Normal Subgroup is Congruence Relation
Let $\struct {G, \circ}$ be a group. Let $N$ be a normal subgroup of $G$. Then congruence modulo $N$ is a congruence relation for the group operation $\circ$.
Intersection Measure is Measure
Let $\left({X, \Sigma, \mu}\right)$ be a measure space. Let $F \in \Sigma$ be a measurable set. Then the intersection measure $\mu_F$ is a measure on the measurable space $\left({X, \Sigma}\right)$.
Uniqueness of Measures/Proof 1
Let $\left({X, \Sigma}\right)$ be a measurable space. Let $\mathcal G \subseteq \mathcal P \left({X}\right)$ be a generator for $\Sigma$; i.e., $\Sigma = \sigma \left({\mathcal G}\right)$. Suppose that $\mathcal G$ satisfies the following conditions: :$(1):\quad \forall G, H \in \mathcal G: G \cap H \in \mathcal G$ :$(2):\quad$ There exists an exhausting sequence $\left({G_n}\right)_{n \in \N} \uparrow X$ in $\mathcal G$ Let $\mu, \nu$ be measures on $\left({X, \Sigma}\right)$, and suppose that: :$(3):\quad \forall G \in \mathcal G: \mu \left({G}\right) = \nu \left({G}\right)$ :$(4):\quad \forall n \in \N: \mu \left({G_n}\right)$ is finite Then $\mu = \nu$. Alternatively, by Countable Cover induces Exhausting Sequence, the exhausting sequence in $(2)$ may be replaced by a countable $\mathcal G$-cover $\left({G_n}\right)_{n \in \N}$, still subject to $(4)$.
Lebesgue Measure Invariant under Translations
Let $\lambda^n$ be the $n$-dimensional Lebesgue measure on $\R^n$ equipped with the Borel $\sigma$-algebra $\mathcal B \left({\R^n}\right)$. Let $\mathbf x \in \R^n$. Then $\lambda^n$ is translation-invariant; i.e., for all $B \in \mathcal B \left({\R^n}\right)$, have: :$\lambda^n \left({\mathbf x + B}\right) = \lambda^n \left({B}\right)$ where $\mathbf x + B$ is the set $\left\{{\mathbf x + \mathbf b: \mathbf b \in B}\right\}$.
Translation-Invariant Measure on Euclidean Space is Multiple of Lebesgue Measure
Let $\mu$ be a measure on $\R^n$ equipped with the Borel $\sigma$-algebra $\mathcal B \left({\R^n}\right)$. Suppose that $\mu$ is translation-invariant. Also, suppose that $\kappa := \mu \left({\left[{0 \,.\,.\, 1}\right)^n }\right) < +\infty$. Then $\mu = \kappa \lambda^n$, where $\lambda^n$ is the $n$-dimensional Lebesgue measure.
Dynkin System Closed under Set Difference with Subset
Let $X$ be a set, and let $\mathcal D$ be a Dynkin system on $X$. Let $D, E \in \mathcal D$ and suppose that $E \subseteq D$. Then the set difference $D \setminus E$ is also an element of $\mathcal D$.
Coset Product of Normal Subgroup is Consistent with Subset Product Definition
Let $\struct {G, \circ}$ be a group. Let $N$ be a normal subgroup of $G$. Let $a, b \in G$. Let $a \circ N$ and $b \circ N$ be the left cosets of $a$ and $b$ by $N$. Then the coset product: :$\paren {a \circ N} \circ \paren {b \circ N} = \paren {a \circ b} \circ N$ is consistent with the definition of the coset product as the subset product of $a \circ N$ and $b \circ  N$: :$\paren {a \circ N} \paren {b \circ N} = \set {x \circ y: x \in a \circ N, y \in b \circ N}$
Congruence Relation iff Compatible with Operation
Let $\struct {S, \circ}$ be an algebraic structure. Let $\RR$ be an equivalence relation on $S$. Then $\RR$ is a congruence relation for $\circ$ : {{begin-eqn}} {{eqn | ll= \forall x, y, z \in S:       | l = x \mathrel \RR y       | o = \implies       | r = \paren {x \circ z} \mathrel \RR \paren {y \circ z} }} {{eqn | l = x \mathrel \RR y       | o = \implies       | r = \paren {z \circ x} \mathrel \RR \paren {z \circ y} }} {{end-eqn}} That is,  $\RR$ is compatible with $\circ$.
Outer Measure of Limit of Increasing Sequence of Sets
Let $\mu^*$ be an outer measure on a set $X$. Let $\left\langle{S_n}\right\rangle$ be an increasing sequence of $\mu^*$-measurable sets, and let $S_n \uparrow S$ (as $n \to \infty$). Then for any subset $A \subseteq X$: : $\displaystyle \mu^* \left({A \cap S}\right) = \lim_{n \to \infty} \mu^* \left({A \cap S_n}\right)$
Half-Open Rectangles form Semiring of Sets
The half-open $n$-rectangles $\JJ_{ho}^n$ form a semiring of sets.
Lebesgue Pre-Measure is Pre-Measure
The Lebesgue pre-measure $\lambda^n$ on the half-open $n$-rectangles $\mathcal{J}_{ho}^n$ is a pre-measure.
Existence and Uniqueness of Lebesgue Measure
Let $\lambda^n$ be the Lebesgue pre-measure on the half-open $n$-rectangles $\mathcal{J}_{ho}^n$. Then Lebesgue measure, the extension of $\lambda^n$ to the Borel $\sigma$-algebra $\mathcal B \left({\R^n}\right)$, exists and is unique.
Lebesgue Measure is Diffuse
Let $\lambda^n$ be Lebesgue measure on $\R^n$. Then $\lambda^n$ is a diffuse measure.
Half-Open Rectangles Closed under Intersection
Let $\lefthalf-open $n$-rectangles. Then $\lefthalf-open $n$-rectangle.
Equivalence of Definitions of Null Set in Euclidean Space
Let $\lambda^n$ be $n$-dimensional Lebesgue measure on $\R^n$. Let $E \subseteq \R^n$. Then the following are equivalent: :$(1):\quad \exists B \in \mathcal B \left({\R^n}\right): E \subseteq B, \lambda^n \left({B}\right) = 0$ :$(2):\quad$ For every $\epsilon > 0$, there exists a countable cover $\left({J_i}\right)_{i \mathop \in \N}$ of $E$ by open $n$-rectangles, such that: ::$\displaystyle \sum_{i \mathop = 1}^\infty \operatorname{vol} \left({J_i}\right) \le \epsilon$ {{explain|link to some definition (might not exist) for $\operatorname{vol}$}}
Homomorphism to Group Preserves Inverses
Let $\struct {S, \circ}$ be an algebraic structure. Let $\struct {T, *}$ be a group. Let $\phi: \struct {S, \circ} \to \struct {T, *}$ be a homomorphism. Let $\struct {S, \circ}$ have an identity $e_S$. Let $x^{-1}$ be an inverse of $x$ for $\circ$. Then $\map \phi {x^{-1} }$ is an inverse of $\map \phi x$ for $*$.
Equivalence Induced by Epimorphism is Congruence Relation
Let $\left({S, \circ}\right)$ and $\left({T, *}\right)$ be algebraic structures. Let $\phi: \left({S, \circ}\right) \to \left({T, *}\right)$ be an epimorphism. Let $\mathcal R_\phi$ be the equivalence induced by $\phi$. Then the induced equivalence $\mathcal R_\phi$ is a congruence relation for $\circ$.
Unique Isomorphism from Quotient Mapping to Epimorphism Domain
Let $\struct {S, \circ}$ and $\struct {T, *}$ be algebraic structures. Let $\phi: \struct {S, \circ} \to \struct {T, *}$ be an epimorphism. Let $\mathcal R_\phi$ be the equivalence induced by $\phi$. Let $S / \mathcal R_\phi$ be the quotient of $S$ by $\mathcal R_\phi$. Let $q_{\mathcal R_\phi}: S \to S / \mathcal R_\phi$ be the quotient mapping induced by $\mathcal R_\phi$. Let $\struct {S / \mathcal R_\phi}, {\circ_{\mathcal R_\phi} }$ be the quotient structure defined by $\mathcal R_\phi$. Then there is one and only one isomorphism: :$\psi: \struct {S / \mathcal R_\phi}, {\circ_{\mathcal R_\phi} } \to \struct {T, *}$ which satisfies: :$\psi \bullet q_{\mathcal R_\phi} = \phi$ where, in order not to cause notational confusion, $\bullet$ is used as the symbol to denote composition of mappings.
Identity is in Kernel of Group Homomorphism
Let $G$ and $H$ be groups. Let $e_G$ and $e_H$ be the identity elements of $G$ and $H$ respectively. Let $\phi: G \to H$ be a (group) homomorphism from $G$ to $H$. Then: :$e_G \in \map \ker \phi$ where $\map \ker \phi$ is the kernel of $\phi$.
Union is Dominated by Disjoint Union
Let $I$ be an indexing set. For all $i \in I$, let $S_i$ be a set. Then: :$\displaystyle \bigcup_{i \mathop \in I} S_i \preccurlyeq \bigsqcup_{i \mathop \in I} S_i$ where $\preccurlyeq$ denotes domination, $\bigcup$ denotes union, and $\bigsqcup$ denotes disjoint union.
Disjoint Union Preserves Domination
Let $I$ be an indexing set. For all $i \in I$, let $A_i$ and $B_i$ be sets such that $A_i \preccurlyeq B_i$. Here, $\preccurlyeq$ denotes domination. Then: :$\displaystyle \bigsqcup_{i \mathop \in I} A_i \preccurlyeq \bigsqcup_{i \mathop \in I} B_i$ where $\bigsqcup$ denotes disjoint union.
Epimorphism from Real Numbers to Circle Group
Let $\struct {K, \times}$ be the circle group, that is: :$K = \set {z \in \C: \cmod z = 1}$ under complex multiplication. Let $f: \R \to K$ be the mapping from the real numbers to $K$ defined as: :$\forall x \in \R: \map f x = \cos x + i \sin x$ Then $f: \struct {\R, +} \to \struct {K, \times}$ is a group epimorphism. Its kernel is: :$\map \ker f = \set {2 \pi n: n \in \Z}$
Increasing Sequence of Sets induces Partition on Limit
Let $\sequence {S_n}_{n \mathop \in \N} \uparrow S$ be an increasing sequence of sets with limit $S$. Define $T_1 = S_1$, and, for $n \in \N$, $T_{n + 1} = S_{n + 1} \setminus S_n$, where $\setminus$ denotes set difference. Then $\sequence {T_n}_{n \mathop \in \N}$ is a countable partition of $S$.
Power Function on Complex Numbers is Epimorphism
Let $n \in \Z_{>0}$ be a strictly positive integer. Let $\struct {\C_{\ne 0}, \times}$ be the multiplicative group of complex numbers. Let $f_n: \C_{\ne 0} \to \C_{\ne 0}$ be the mapping from the set of complex numbers less zero to itself defined as: :$\forall z \in \C_{\ne 0}: \map {f_n} z = z^n$ Then $f_n: \struct {\C_{\ne 0}, \times} \to \struct {\C_{\ne 0}, \times}$ is a group epimorphism. The kernel of $f_n$ is the set of complex $n$th roots of unity.
Real Part as Mapping is Endomorphism for Complex Addition
Let $\struct {\C, +}$ be the additive group of complex numbers. Let $\struct {\R, +}$ be the additive group of real numbers. Let $f: \C \to \R$ be the mapping from the complex numbers to the real numbers defined as: :$\forall z \in \C: \map f z = \map \Re z$ where $\map \Re z$ denotes the real part of $z$. Then $f: \struct {\C, +} \to \struct {\R, +}$ is a group epimorphism. Its kernel is the set: :$\map \ker f = \set {i x: x \in \R}$ of wholly imaginary numbers.
Equivalence of Definitions of Semiring of Sets
{{TFAE|def = Semiring of Sets}} A collection $\SS$ of subsets of a set $X$ is a semiring (of sets) : :$(1):\quad \O \in \SS$ :$(2):\quad A, B \in \SS \implies A \cap B \in \SS$; that is, $\SS$ is $\cap$-stable :$(3):\quad$ If $A, A_1 \in \SS$ such that $A_1 \subseteq A$, then there exists a finite sequence $A_2, A_3, \ldots, A_n \in \SS$ such that: ::$(3a):\quad \displaystyle A = \bigcup_{k \mathop = 1}^n A_k$ ::$(3b):\quad$ The $A_k$ are pairwise disjoint We prove that criterion $(3)$ can be replaced by: :$(3'):\quad$ If $A, B \in \SS$, then there exist finite sequence of pairwise disjoint sets $A_1, A_2, \ldots, A_n \in \SS$ such that $\displaystyle A \setminus B = \bigcup_{k \mathop = 1}^n A_k$.
Imaginary Part as Mapping is Endomorphism for Complex Addition
Let $\struct {\C, +}$ be the additive group of complex numbers. Let $\struct {\R, +}$ be the additive group of real numbers. Let $f: \C \to \R$ be the mapping from the complex numbers to the real numbers defined as: :$\forall z \in \C: \map f z = \map \Im z$ where $\map \Im z$ denotes the imaginary part of $z$. Then $f: \struct {\C, +} \to \struct {\R, +}$ is a group epimorphism. Its kernel is the set: :$\map \ker f = \R$ of (wholly) real numbers.
Reduced Echelon Matrix is Unique
Every $m \times n$ matrix is row equivalent to exactly one $m \times n$ reduced echelon matrix. That is, the reduced echelon form of a matrix is unique.
Measure of Set Difference with Subset
Let $\struct {X, \Sigma, \mu}$ be a measure space. Let $S, T \in \Sigma$ be such that $S \subseteq T$, and suppose that $\mu \paren S < +\infty$. Then: :$\mu \paren {T \setminus S} = \mu \paren T - \mu \paren S$ where $T \setminus S$ denotes set difference.
Subset of Codomain is Superset of Image of Preimage
Let $f: S \to T$ be a mapping. Then: :$B \subseteq T \implies \paren {f \circ f^{-1} } \sqbrk B \subseteq B$ where: :$f \sqbrk B$ denotes the image of $B$ under $f$ :$f^{-1}$ denotes the inverse of $f$ :$f \circ f^{-1}$ denotes composition of $f$ and $f^{-1}$. This can be expressed in the language and notation of direct image mappings and inverse image mappings as: :$\forall B \in \powerset T: \map {\paren {f^\to \circ f^\gets} } B \subseteq B$
Induced Outer Measure Restricted to Semiring is Pre-Measure
Let $\mathcal S$ be a semiring over a set $X$. Let $\mu: \mathcal S \to \overline \R_{\ge 0}$ be a pre-measure on $\mathcal S$, where $\overline \R_{\ge 0}$ denotes the set of positive extended real numbers. Let $\mu^*: \mathcal P \left({X}\right) \to \overline \R_{\ge 0}$ be the outer measure induced by $\mu$. Then: :$\displaystyle \mu^*\restriction_{\mathcal S} \, = \mu$ where $\restriction$ denotes restriction.
Ordering on Extended Real Numbers is Ordering
Denote with $\le$ the usual ordering on the extended real numbers $\overline \R$. Then $\le$ is an ordering, and so $\overline \R$ is an ordered set.
Ordering on Extended Real Numbers is Total Ordering
Let $\le$ denote the ordering on the extended real numbers $\overline \R$. Then $\le$ is a total ordering, and so $\overline \R$ is a toset.
Extended Real Number Space is Compact
The extended real number space is compact.
Euclidean Space is Subspace of Extended Real Number Space
Let $\struct {\overline \R, \tau}$ be the extended real number space. Then $\tau {\restriction_\R}$, the subspace topology on $\R$, is the Euclidean topology. That is, Euclidean $1$-space is a subspace of the extended real number space.
Lagrange's Formula
Let: :$\mathbf a = \begin{bmatrix} a_x \\ a_y \\ a_z \end{bmatrix}$, $\mathbf b = \begin{bmatrix} b_x \\ b_y \\ b_z \end{bmatrix}$, $\mathbf c = \begin{bmatrix} c_x \\ c_y \\ c_z \end{bmatrix}$ be vectors in a vector space of $3$ dimensions. Then: :$\mathbf a \times \paren {\mathbf b \times \mathbf c} = \paren {\mathbf a \cdot \mathbf c} \mathbf b - \paren {\mathbf a \cdot \mathbf b} \mathbf c$
External Direct Product of Groups is Group/Finite Product
The external direct product of a finite sequence of groups is itself a group.
External Direct Product Inverses/General Result
Let $\displaystyle \left({S, \circ}\right) = \prod_{k \mathop = 1}^n S_k$ be the external direct product of the algebraic structures $\left({S_1, \circ_1}\right), \left({S_2, \circ_2}\right), \ldots, \left({S_n, \circ_n}\right)$. Let $\left({x_1, x_2, \ldots, x_n}\right) \in S$. Let $y_k$ be an inverse of $x_k$ in $\left({S_k, \circ_k}\right)$ for each of $k \in \N^*_n$. Then $\left({y_1, y_2, \ldots, y_n}\right)$ is the inverse of $\left({x_1, x_2, \ldots, x_n}\right) \in S$ in $\left({S, \circ}\right)$.
External Direct Product Identity/General Result
Let $\displaystyle \left({S, \circ}\right) = \prod_{k \mathop = 1}^n S_k$ be the external direct product of the algebraic structures $\left({S_1, \circ_1}\right), \left({S_2, \circ_2}\right), \ldots, \left({S_n, \circ_n}\right)$. Let $e_1, e_2, \ldots, e_n$ be the identity elements of $\left({S_1, \circ_1}\right), \left({S_2, \circ_2}\right), \ldots, \left({S_n, \circ_n}\right)$ respectively. Then $\left({e_1, e_2, \ldots, e_n}\right)$ is the identity element of $\left({S, \circ}\right)$.
External Direct Product Commutativity/General Result
Let $\displaystyle \left({S, \circ}\right) = \prod_{k \mathop = 1}^n S_k$ be the external direct product of the algebraic structures $\left({S_1, \circ_1}\right), \left({S_2, \circ_2}\right), \ldots, \left({S_n, \circ_n}\right)$. If $\circ_1, \ldots, \circ_n$ are all commutative, then so is $\circ$.
External Direct Product Associativity/General Result
Let $\displaystyle \left({S, \circ}\right) = \prod_{k \mathop = 1}^n S_k$ be the external direct product of the algebraic structures $\left({S_1, \circ_1}\right), \left({S_2, \circ_2}\right), \ldots, \left({S_n, \circ_n}\right)$. If $\circ_1, \ldots, \circ_n$ are all associative, then so is $\circ$.
Vector Cross Product is not Associative
The vector cross product is ''not'' associative. That is, in general: :$\mathbf a \times \left({\mathbf b \times \mathbf c}\right) \ne \left({\mathbf a \times \mathbf b}\right) \times \mathbf c$ for $\mathbf {a}, \mathbf {b}, \mathbf {c} \in \R^3$.
Vector Cross Product is Anticommutative
The vector cross product is anticommutative: :$\forall \mathbf a, \mathbf b \in \R^3: \mathbf a \times \mathbf b = -\left({\mathbf b \times \mathbf a}\right)$
Canonical Injection is Monomorphism/General Result
Let $\struct {S_1, \circ_1}, \struct {S_2, \circ_2}, \dotsc, \struct {S_j, \circ_j}, \dotsc, \struct {S_n, \circ_n}$ be algebraic structures with identities $e_1, e_2, \dotsc, e_j, \dotsc, e_n$ respectively. Then the canonical injection: :$\displaystyle \inj_j: \struct {S_j, \circ_j} \to \prod_{i \mathop = 1}^n \struct {S_i, \circ_i}$ defined as: :$\map {\inj_j} x = \tuple {e_1, e_2, \dotsc, e_{j - 1}, x, e_{j + 1}, \dotsc, e_n}$ is a monomorphism.