hcju/proofwikips huggingface dataset:
Integral of Characteristic Function/Corollary
:$\displaystyle \int \chi_E \, \mathrm d\mu = \mu \left({E}\right)$ where the integral sign denotes the $\mu$-integral of $\chi_E$.
Integral of Positive Measurable Function is Positive Homogeneous
Let $\struct {X, \Sigma, \mu}$ be a measure space. Let $f: X \to \R, f \in \mathcal M_{\overline \R}^+$ be a positive measurable function. Let $\lambda \in \R_{\ge 0}$ be a positive real number. Then: :$\displaystyle \int \lambda f \rd \mu = \lambda \int f \rd \mu$ where: :$\lambda f$ is the pointwise $\lambda$-multiple of $f$ :The integral sign denotes $\mu$-integration This can be summarized by saying that $\displaystyle \int \cdot \rd \mu$ is positive homogeneous.
Linear Transformation as Matrix Product
Let $T: \R^n \to \R^m, \mathbf x \mapsto \map T {\mathbf x}$ be a linear transformation. Then: :$\map T {\mathbf x} = \mathbf A_T \mathbf x$ where $\mathbf A_T$ is the $m \times n$ matrix defined as: :$\mathbf A_T = \begin {bmatrix} \map T {\mathbf e_1} & \map T {\mathbf e_2} & \cdots & \map T {\mathbf e_n} \end {bmatrix}$ where $\tuple {\mathbf e_1, \mathbf e_2, \cdots, \mathbf e_n}$ is the standard ordered basis of $\R^n$.
Integral of Positive Measurable Function is Additive
Let $\left({X, \Sigma, \mu}\right)$ be a measure space. Let $f,g: X \to \overline{\R}$, $f,g \in \mathcal{M}_{\overline{\R}}^+$ be positive measurable functions. Then: :$\displaystyle \int f + g \, \mathrm d\mu = \displaystyle \int f \, \mathrm d\mu + \displaystyle \int g \, \mathrm d\mu$ where: :$f + g$ is the pointwise sum of $f$ and $g$ :The integral sign denotes $\mu$-integration This can be summarized by saying that $\displaystyle \int \cdot \, \mathrm d\mu$ is additive.
Integral of Positive Simple Function is Increasing
Let $\left({X, \Sigma, \mu}\right)$ be a measure space. Let $f, g: X \to \R$, $f, g \in \mathcal{E}^+$ be positive simple functions. Suppose that: : $f \le g$ where $\le$ denotes pointwise inequality. Then: :$I_\mu \left({f}\right) \le I_\mu \left({g}\right)$ where $I_\mu$ denotes $\mu$-integration This can be summarized by saying that $I_\mu$ is an increasing mapping.
Integral of Positive Measurable Function is Monotone
Let $\left({X, \Sigma, \mu}\right)$ be a measure space. Let $f,g: X \to \overline{\R}$, $f,g \in \mathcal{M}_{\overline{\R}}^+$ be positive measurable functions. Suppose that $f \le g$, where $\le$ denotes pointwise inequality. Then: :$\displaystyle \int f \, \mathrm d\mu \le \int g \, \mathrm d\mu$ where the integral sign denotes $\mu$-integration. This can be summarized by saying that $\displaystyle \int \cdot \, \mathrm d\mu$ is monotone.
Matrix Multiplication is Homogeneous of Degree 1
Let $\mathbf A$ be an $m \times n$ matrix and $\mathbf B$ be an $n \times p$ matrix such that the columns of $\mathbf A$ and $\mathbf B$ are members of $\R^m$ and $\R^n$, respectively. Let $\lambda \in \mathbb F \in \set {\R, \C}$ be a scalar. Then: :$\mathbf A \paren {\lambda \mathbf B} = \lambda \paren {\mathbf A \mathbf B}$
Series of Positive Measurable Functions is Positive Measurable Function
Let $\left({X, \Sigma}\right)$ be a measurable space. Let $\left({f_n}\right)_{n \in \N} \in \mathcal{M}_{\overline{\R}}^+$, $f_n: X \to \overline{\R}$ be a sequence of positive measurable functions. Let $\displaystyle \sum_{n \mathop \in \N} f_n: X \to \overline{\R}$ be the pointwise series of the $f_n$. Then $\displaystyle \sum_{n \mathop \in \N} f_n$ is also a positive measurable function.
Integral of Series of Positive Measurable Functions
Let $\left({X, \Sigma}\right)$ be a measurable space. Let $\left({f_n}\right)_{n \in \N} \in \mathcal{M}_{\overline{\R}}^+$, $f_n: X \to \overline{\R}$ be a sequence of positive measurable functions. Let $\displaystyle \sum_{n \mathop \in \N} f_n: X \to \overline{\R}$ be the pointwise series of the $f_n$. Then: :$\displaystyle \int \sum_{n \mathop \in \N} f_n \, \mathrm d \mu = \sum_{n \mathop \in \N} \int f_n \, \mathrm d \mu$ where the integral sign denotes $\mu$-integration.
Integral with respect to Discrete Measure
Let $\left({X, \Sigma}\right)$ be a measurable space. Let $\displaystyle \mu = \sum_{n \mathop \in \N} \lambda_n \delta_{x_n}$ be a discrete measure on $\left({X, \Sigma}\right)$. Let $f \in \mathcal{M}_{\overline{\R}}^+, f: X \to \overline{\R}$ be a positive measurable function. Then: :$\displaystyle \int f \, \mathrm d\mu = \sum_{n \mathop \in \N} \lambda_n f \left({x_n}\right)$ where the integral sign denotes $\mu$-integration.
Matrix Product as Linear Transformation
Let: :$ \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}$ :$\mathbf x_{n \times 1} = \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix}$ :$\mathbf y_{n \times 1} = \begin{bmatrix} y_1 \\ y_2 \\ \vdots \\ y_n \end{bmatrix}$ be matrices where each column is an element of a real vector space. Let $T$ be the mapping: :$T: \R^m \to \R^n, \mathbf x \mapsto \mathbf A \mathbf x$ Then $T$ is a linear transformation.
Fatou's Lemma for Integrals
Let $\struct {X, \Sigma, \mu}$ be a measure space.
Linear Transformation Maps Zero Vector to Zero Vector
Let $\mathbf V$ be a vector space, with zero $\mathbf 0$. Likewise let $\mathbf V\,'$ be another vector space, with zero $\mathbf 0'$. Let $T: \mathbf V \to \mathbf V\,'$ be a linear transformation. Then: :$T: \mathbf 0 \mapsto \mathbf 0'$
Infimum of Product
Let $\left({G, \circ, \preceq}\right)$ be an ordered group. Suppose that subsets $A$ and $B$ of $G$ admit infima in $G$. Then: :$\inf \left({A \circ_{\mathcal P} B}\right) = \inf A \circ \inf B$ where $\circ_{\mathcal P}$ denotes subset product.
Supremum of Product
Let $\struct {G, \circ, \preceq}$ be an ordered group. Suppose that subsets $A$ and $B$ of $G$ admit suprema in $G$. Then: :$\sup \paren {A \circ_{\PP} B} = \sup A \circ \sup B$ where $\circ_{\PP}$ denotes subset product.
Integral with respect to Series of Measures
Let $\left({X, \Sigma}\right)$ be a measurable space. Let $\displaystyle \mu := \sum_{n \mathop \in \N} \lambda_n \mu_n$ be a series of measures on $\left({X, \Sigma}\right)$. Then for all positive measurable functions $f: X \to \overline \R, f \in \mathcal M_{\overline{\R}}^+$: :$\displaystyle \int f \, \mathrm d \mu = \sum_{n \mathop \in \N} \int f \, \mathrm d \mu_n$ where the integral signs denote integration with respect to a measure.
Reverse Fatou's Lemma
Let $\struct {X, \Sigma, \mu}$ be a measure space.
Characteristic Function of Limit Inferior of Sequence of Sets
Let $\left({E_n}\right)_{n \in \N}$ be a sequence of sets. Let $E := \displaystyle \liminf_{n \mathop \to \infty} \, E_n$ be the limit inferior of the $E_n$. Then: :$\displaystyle \chi_E = \liminf_{n \mathop \to \infty} \, \chi_{E_n}$ where: :$\chi$ denotes characteristic function :$\displaystyle \liminf_{n \to \infty} \, \chi_{E_n}$ is the pointwise limit inferior of the $\chi_{E_n}$
Characteristic Function of Limit Superior of Sequence of Sets
Let $\left({E_n}\right)_{n \in \N}$ be a sequence of sets. Let $E := \displaystyle \limsup_{n \mathop \to \infty} \, E_n$ be the limit superior of the $E_n$. Then: :$\displaystyle \chi_E = \limsup_{n \to \infty} \, \chi_{E_n}$ where: :$\chi$ denotes characteristic function :$\displaystyle \liminf_{n \to \infty} \, \chi_{E_n}$ is the pointwise limit superior of the $\chi_{E_n}$
Fatou's Lemma for Measures
Let $\struct {X, \Sigma, \mu}$ be a measure space. Let $\sequence {E_n}_{n \mathop \in \N} \in \Sigma$ be a sequence of $\Sigma$-measurable sets. Then: :$\displaystyle \map \mu {\liminf_{n \mathop \to \infty} E_n} \le \liminf_{n \mathop \to \infty} \map \mu {E_n}$ where: :$\displaystyle \liminf_{n \mathop \to \infty} E_n$ is the limit inferior of the $E_n$ :the  limit inferior is taken in the extended real numbers $\overline \R$.
Kernel Transformation of Measure is Measure
Let $\left({X, \Sigma, \mu}\right)$ be a measure space. Let $N: X \times \Sigma \to \overline{\R}_{\ge0}$ be a kernel. Then $\mu N: X \to \overline{\R}$, the kernel transformation of $\mu$, is a measure.
Kernel Transformation of Positive Measurable Function is Positive Measurable Function
Let $\left({X, \Sigma, \mu}\right)$ be a measure space. Let $N: X \times \Sigma \to \overline{\R}_{\ge0}$ be a kernel. Let $f: X \to \overline{\R}$ be a positive measurable function. Then $N f: X \to \overline{\R}$, the transformation of $f$ by $N$, is also a positive measurable function.
Integral with respect to Kernel Transformation of Measure
Let $\struct {X, \Sigma, \mu}$ be a measure space. Let $N: X \times \Sigma \to \overline \R_{\ge 0}$ be a kernel. Let $f: X \to \overline \R$ be a positive measurable function. Then: :$\displaystyle \int f \map \rd {\mu N} = \int N f \rd \mu$ where: :The integral sign denotes integration with respect to a measure :$\mu N$ is the transformation of $\mu$ by $N$ :$N f$ is the transformation of $f$ by $N$ Writing $\map \mu f$ in place of $\displaystyle \int f \rd \mu$, the theorem statement can be conveniently expressed as: :$\map {\mu N} f = \map \mu {N f}$
Canonical Injection is Injection
Let $\struct {S_1, \circ_1}$ and $\struct {S_2, \circ_2}$ be algebraic structures with identities $e_1, e_2$ respectively. The canonical injections: :$\inj_1: \struct {S_1, \circ_1} \to \struct {S_1, \circ_1} \times \struct {S_2, \circ_2}: \forall x \in S_1: \map {\inj_1} x = \tuple {x, e_2}$ :$\inj_2: \struct {S_2, \circ_2} \to \struct {S_1, \circ_1} \times \struct {S_2, \circ_2}: \forall x \in S_2: \map {\inj_2} x = \tuple {e_1, x}$ are injections.
Canonical Injection is Injection/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 $e_1, e_2, \ldots, e_j, \ldots, e_n$ respectively. 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 an injection.
Finite Union of Countable Sets is Countable
The union of a finite number of countable sets is countable.
Particular Values of Binomial Coefficients
=== Binomial Coefficient $\dbinom 0 0$ === {{:Zero Choose Zero}} === Binomial Coefficient $\dbinom 0 n$ === {{:Zero Choose n}} === Binomial Coefficient $\dbinom 1 n$ === {{:One Choose n}} === N Choose Negative Number is Zero === {{:N Choose Negative Number is Zero}} === Binomial Coefficient with Zero === {{:Binomial Coefficient with Zero}} === Binomial Coefficient with One === {{:Binomial Coefficient with One}} === Binomial Coefficient with Self === {{:Binomial Coefficient with Self}} === Binomial Coefficient with Self minus One === {{:Binomial Coefficient with Self minus One}} === Binomial Coefficient with Two === {{:Binomial Coefficient with Two}}
Binomial Coefficient with Zero
:$\forall r \in \R: \dbinom r 0 = 1$
Binomial Coefficient with One
:$\forall r \in \R: \dbinom r 1 = r$
Binomial Coefficient with Self
:$\forall n \in \Z: \dbinom n n = \sqbrk {n \ge 0}$ where $\sqbrk {n \ge 0}$ denotes Iverson's convention. That is: :$\forall n \in \Z_{\ge 0}: \dbinom n n = 1$ :$\forall n \in \Z_{< 0}: \dbinom n n = 0$
Binomial Coefficient with Two
:$\forall r \in \R: \dbinom r 2 = \dfrac {r \left({r - 1}\right)} 2$
Binomial Theorem/Integral Index
Let $X$ be one of the set of numbers $\N, \Z, \Q, \R, \C$. Let $x, y \in X$. Then: {{begin-eqn}} {{eqn | ll= \forall n \in \Z_{\ge 0}:       | l = \paren {x + y}^n       | r = \sum_{k \mathop = 0}^n \binom n k x^{n - k} y^k       | c =  }} {{eqn | r = x^n + \binom n 1 x^{n - 1} y + \binom n 2 x^{n - 2} y^2 + \binom n 3 x^{n - 3} y^3 + \cdots       | c =  }} {{eqn | r = x^n + n x^{n - 1} y + \frac {n \paren {n - 1} } {2!} x^{n - 2} y^2 + \frac {n \paren {n - 1} \paren {n - 3} } {3!} x^{n - 3} y^3 + \cdots       | c =  }} {{end-eqn}} where $\dbinom n k$ is $n$ choose $k$.
Binomial Theorem/Ring Theory
Let $\left({R, +, \odot}\right)$ be a ringoid such that $\left({R, \odot}\right)$ is a commutative semigroup. Let $n \in \Z: n \ge 2$. Then: :$\displaystyle \forall x, y \in R: \odot^n \left({x + y}\right) = \odot^n x + \sum_{k \mathop = 1}^{n-1} \binom n k \left({\odot^{n-k} x}\right) \odot \left({\odot^k y}\right) + \odot^n y$ where $\dbinom n k = \dfrac {n!} {k! \ \left({n - k}\right)!}$ (see Binomial Coefficient). If $\left({R, \odot}\right)$ has an identity element $e$, then: :$\displaystyle \forall x, y \in R: \odot^n \left({x + y}\right) = \sum_{k \mathop = 0}^n \binom n k \left({\odot^{n - k} x}\right) \odot \left({\odot^k y}\right)$
Derivative of Absolute Value Function
Let $\size x$ be the absolute value of $x$ for real $x$. Then: :$\dfrac \d {\d x} \size x = \dfrac x {\size x}$ for $x \ne 0$. At $x = 0$, $\size x$ is not differentiable.
Inverse Element in Inverse Completion of Commutative Monoid
Let $\left({S, \circ}\right)$ be a commutative monoid. Let $\left ({C, \circ}\right) \subseteq \left({S, \circ}\right)$ be the subsemigroup of cancellable elements of $\left({S, \circ}\right)$. Let $\left({T, \circ'}\right)$ be an inverse completion of $\left({S, \circ}\right)$. Then the inverse of an element of $S$ which is invertible for $\circ$ is also its inverse for $\circ'$.
Construction of Inverse Completion/Congruence Relation
The cross-relation $\boxtimes$ is a congruence relation on $\left({S \times C, \oplus}\right)$. === Members of Equivalence Classes === {{:Construction of Inverse Completion/Equivalence Relation/Members of Equivalence Classes}} === Equivalence Class of Equal Elements === {{:Construction of Inverse Completion/Equivalence Relation/Equivalence Class of Equal Elements}}
Construction of Inverse Completion/Equivalence Relation/Members of Equivalence Classes
$\forall x, y \in S, a, b \in C:$ :$(1): \quad \tuple {x \circ a, a} \boxtimes \tuple {y \circ b, b} \iff x = y$ :$(2): \quad \eqclass {\tuple {x \circ a, y \circ a} } \boxtimes = \eqclass {\tuple {x, y} } \boxtimes$ where $\eqclass {\tuple {x, y} } \boxtimes$ is the equivalence class of $\tuple {x, y}$ under $\boxtimes$.
Construction of Inverse Completion/Equivalence Relation/Equivalence Class of Equal Elements
:$\forall c, d \in C: \left({c, c}\right) \boxtimes \left({d, d}\right)$
Construction of Inverse Completion/Quotient Structure is Commutative Semigroup
:$\left({T', \oplus'}\right)$ is a commutative semigroup.
Construction of Inverse Completion/Quotient Mapping is Injective
Let the mapping $\psi: S \to T'$ be defined as: :$\forall x \in S: \psi \left({x}\right) = \left[\!\left[{\left({x \circ a, a}\right)}\right]\!\right]_\boxtimes$ Then $\psi: S \to T'$ is an injection, and does not depend on the particular element $a$ chosen.
Construction of Inverse Completion/Quotient Mapping is Monomorphism
The mapping $\psi: S \to T'$ is a monomorphism.
Construction of Inverse Completion/Quotient Mapping to Image is Isomorphism
Let $S'$ be the image $\psi \left({S}\right)$ of $S$. Then $\psi$ is an isomorphism from $S$ onto $S'$.
Construction of Inverse Completion/Quotient Mapping/Image of Cancellable Elements
The set $C'$ of cancellable elements of the semigroup $S'$ is $\psi \left[{C}\right]$.
Construction of Inverse Completion/Image of Quotient Mapping is Subsemigroup
Let $S'$ be the image $\psi \left({S}\right)$ of $S$. Then $\left({S', \oplus'}\right)$ is a subsemigroup of $\left({T', \oplus'}\right)$.
Construction of Inverse Completion/Identity of Quotient Structure
Let $c \in C$ be arbitrary. Then: :$\eqclass {\tuple {c, c} } \boxtimes$ is the identity of $T'$.
Construction of Inverse Completion/Invertible Elements in Quotient Structure
Every cancellable element of $S'$ is invertible in $T'$.
Construction of Inverse Completion/Generator for Quotient Structure
$T' = S' \cup \left({C'}\right)^{-1}$ is a generator for the semigroup $T'$.
Construction of Inverse Completion/Quotient Structure is Inverse Completion
$T'$ is an inverse completion of its subsemigroup $S'$.
Ring of Integers Modulo Composite is not Integral Domain
Let $m \in \Z: m \ge 2$. Let $\struct {\Z_m, +, \times}$ be the ring of integers modulo $m$. Let $m$ be a composite number. Then $\struct {\Z_m, +, \times}$ is not an integral domain.
Ring Zero is not Cancellable
Let $\struct {R, +, \circ}$ be a ring which is not null. Let $0$ be the ring zero of $R$. Then $0$ is not a cancellable element for the ring product $\circ$.
Congruence Relation on Ring induces Ideal
Let $\left({R, +, \circ}\right)$ be a ring. Let $\mathcal E$ be a congruence relation on $R$. Let $J = \left[\!\left[{0_R}\right]\!\right]_\mathcal E$ be the equivalence class of $0_R$ under $\mathcal E$. Then $J$ is an ideal of $R$.
Ideal induces Congruence Relation on Ring
Let $\struct {R, +, \circ}$ be a ring. Let $J$ be an ideal of $R$ Then $J$ induces a congruence relation $\EE_J$ on $R$ such that $\struct {R / J, +, \circ}$ is a quotient ring.
Vector Space has Basis between Linearly Independent Set and Spanning Set
Let $V$ be a vector space over a field $F$. Let $L$ be a linearly independent subset of $V$. Let $S$ be a set that spans $V$. Suppose that $L \subseteq S$. Then $V$ has a basis $B$ such that $L \subseteq B \subseteq S$.
Null Ring is Trivial Ring
Let $R$ be the null ring. Then $R$ is a trivial ring and therefore a commutative ring.
Null Ring is Ring
Let $R$ be the null ring. That is, let: : $R := \left({\left\{{0_R}\right\}, +, \circ}\right)$ where ring addition and the ring product are defined as: * $0_R + 0_R = 0_R$ * $0_R \circ 0_R = 0_R$ Then $R$ is a ring.
Trivial Group is Cyclic Group
The trivial group is a cyclic group and therefore abelian.
Trivial Ring is Commutative Ring
Let $\struct {R, +, \circ}$ be a trivial ring. Then $\struct {R, +, \circ}$ is a commutative ring.
Congruence Relation and Ideal are Equivalent
Let $\left({R, +, \circ}\right)$ be a ring. Let $\mathcal E$ be an equivalence relation on $R$ compatible with both $\circ$ and $+$, i.e. a congruence relation on $R$. Let $J = \left[\!\left[{0_R}\right]\!\right]_\mathcal E$ be the equivalence class of $0_R$ under $\mathcal E$. Then: : $(1a): \quad J = \left[\!\left[{0_R}\right]\!\right]_\mathcal E$ is an ideal of $R$ : $(2a): \quad$ The equivalence defined by the quotient ring $R / J$ is $\mathcal E$ itself. Similarly, let $J$ be an ideal of $R$. Then: : $(1b): \quad J$ induces a congruence relation $\mathcal E_J$ on $R$ : $(2b): \quad$ The ideal of $R$ defined by $\mathcal E_J$ is $J$ itself.
Quotient Ring is Ring
Let $\struct {R, +, \circ}$ be a ring. Let $J$ be an ideal of $R$. Let $\struct {R / J, +, \circ}$ be the quotient ring of $R$ by $J$. Then $R / J$ is also a ring.
Ideal is Additive Normal Subgroup
Let $\left({R, +, \circ}\right)$ be a ring. Let $J$ be an ideal of $R$. Then $\left({J, +}\right)$ is a normal subgroup of $\left({R, +}\right)$.
Characterization of Integrable Functions
Let $\struct {X, \Sigma, \mu}$ be a measure space. Let $f: X \to \overline \R, f \in \MM_{\overline \R}$ be a $\Sigma$-measurable function. Then the following are equivalent: :$(1): \quad f \in \map {\LL_{\overline \R} } \mu$, that is, $f$ is $\mu$-integrable. :$(2): \quad$ The positive and negative parts $f^+$ and $f^-$ are $\mu$-integrable. :$(3): \quad$ The absolute value $\size f$ of $f$ is $\mu$-integrable. :$(4): \quad$ There exists an $\mu$-integrable function $g: X \to \overline \R$ such that $\size f \le g$ pointwise.
Quotient Ring is Ring/Quotient Ring Addition is Well-Defined
Let $\struct {R, +, \circ}$ be a ring whose zero is $0_R$ and whose unity is $1_R$. Let $J$ be an ideal of $R$. Let $\struct {R / J, +, \circ}$ be the quotient ring of $R$ by $J$. Then $+$ is well-defined on $R / J$, that is: :$x_1 + J = x_2 + J, y_1 + J = y_2 + J \implies \paren {x_1 + y_1} + J = \paren {x_2 + y_2} + J$
Implicitly Defined Real-Valued Function
Let $F: \struct {\mathbf X' \subseteq \R^{n + 1} } \to \struct {\mathbb I' \subseteq \R}$ have continuous partial derivatives.  {{explain|Can the language of this be brought into line with existing definitions of implicit functions?}} Let $\tuple {\mathbf x, z}$ denote an element of $\R^{n + 1}$, where $\mathbf x \in \R^n$ and $z \in \R$. Suppose $\exists \tuple {\mathbf x_0, z_0} \in \mathbf X'$ such that: :$\map F {\mathbf x_0, z_0} = 0$ :$\dfrac \partial {\partial z} \map F {\mathbf x_0, z_0} \ne 0$ Then there exists a unique mapping of the form: :$g: \mathbf X \to \mathbb I$ where $\mathbf X \subseteq \R^n$ contains $\mathbf x_0$ and $\mathbb I$ is an open real interval containing $z_0$, such that: :$\forall \mathbf x \in \mathbf X, z \in \mathbb I: \map F {\mathbf x, z} = 0 \iff z = \map g {\mathbf x}$ and $g$ itself has continuous partial derivatives.
Characteristic Function of Subset
Let $A \subseteq B \subseteq S$. Then: :$\forall s \in S: \chi_A \left({s}\right) \le \chi_B \left({s}\right)$ where $\chi$ denotes characteristic function.
Characteristic Function Determined by 0-Fiber
Let $A \subseteq S$. Let $f: S \to \set {0, 1}$ be a mapping. Denote by $\chi_A$ the characteristic function on $A$. Then the following are equivalent: :$(1): \quad f = \chi_A$ :$(2): \quad \forall s \in S: \map f s = 0 \iff s \notin A$ Using the notion of a fiber, $(2)$ may also be expressed as: :$(2'):\quad \map {f^{-1} } 0 = S \setminus A$
Ideal induced by Congruence Relation defines that Congruence
Let $\left({R, +, \circ}\right)$ be a ring. Let $\mathcal E$ be a congruence relation on $R$. Let $J = \left[\!\left[{0_R}\right]\!\right]_\mathcal E$ be the ideal induced by $\mathcal E$. Then the equivalence defined by the coset space $\left({R, +}\right) / \left({J, +}\right)$ is $\mathcal E$ itself.
Rational Numbers Null Set under Lebesgue Measure
Let $\lambda$ be $1$-dimensional Lebesgue measure on $\R$. Let $\Q$ be the set of rational numbers. Then $\lambda \left({\Q}\right) = 0$, i.e. $\Q$ is a $\lambda$-null set.
Cartesian Product of Intersections/General Case
:$\displaystyle \paren{ \prod_{i \mathop \in I} S_i } \cap \paren{ \prod_{i \mathop \in I} T_i } = \prod_{i \mathop \in I} \paren{S_i \cap T_i}$
Cartesian Product of Semirings of Sets
Let $\SS$ and $\TT$ be semirings of sets. Then $\SS \times \TT$ is also a semiring of sets. Here, $\times$ denotes Cartesian product.
Congruence Relation on Group induces Normal Subgroup
Let $\left({G, \circ}\right)$ be a group whose identity is $e$. Let $\mathcal R$ be a congruence relation for $\circ$. Let $H = \left[\!\left[{e}\right]\!\right]_\mathcal R$, where $\left[\!\left[{e}\right]\!\right]_\mathcal R$ is the equivalence class of $e$ under $\mathcal R$. Then: : $\left({H, \circ \restriction_H}\right)$ is a normal subgroup of $G$ where $\circ \restriction_H$ denotes the restriction of $\circ$ to $H$.
Normal Subgroup induced by Congruence Relation defines that Congruence
Let $\struct {G, \circ}$ be a group whose identity is $e$. Let $\mathcal R$ be a congruence relation for $\circ$. Let $\eqclass e {\mathcal R}$ be the equivalence class of $e$ under $\mathcal R$. Let $N = \eqclass e {\mathcal R}$ be the normal subgroup induced by $\mathcal R$. {{explain|Check the above because it does not look right}} Then $\mathcal R$ is the equivalence relation $\mathcal R_N$ defined by $N$.
Quotient Structure on Group defined by Congruence equals Quotient Group
Let $\struct {G, \circ}$ be a group whose identity is $e$. Let $\mathcal R$ be a congruence relation for $\circ$. Let $\struct {G / \mathcal R, \circ_\mathcal R}$ be the quotient structure defined by $\mathcal R$. Let $N = \eqclass e {\mathcal R}$ be the normal subgroup induced by $\mathcal R$. Let $\struct {G / N, \circ_N}$ be the quotient group of $G$ by $N$. Then $\struct {G / \mathcal R, \circ_\mathcal R}$ is the subgroup $\struct {G / N, \circ_N}$ of the semigroup $\struct {\powerset G, \circ_\mathcal P}$.
Vector Scaled by Zero is Zero Vector
Let $F$ be a field whose zero is $0_F$. Let $\struct {\mathbf V, +, \circ}_F$ be a vector space over $F$, as defined by the vector space axioms. Then: :$\forall \mathbf v \in \mathbf V: 0_F \circ \mathbf v = \bszero$
Zero Vector Scaled is Zero Vector
Let $\struct {\mathbf V, +, \circ}_F$ be a vector space over a field $F$, as defined by the vector space axioms. Then: :$\forall \lambda \in \mathbb F: \lambda \circ \bszero = \bszero$ where $\bszero \in \mathbf V$ is the zero vector.