hcju/proofwikips huggingface dataset:
Product Rule for Curl
Let $\map {\R^3} {x, y, z}$ denote the real Cartesian space of $3$ dimensions.. Let $\tuple {\mathbf i, \mathbf j, \mathbf k}$ be the standard ordered basis on $\R^3$. Let $\mathbf f := \tuple {\map {f_x} {\mathbf x}, \map {f_y} {\mathbf x}, \map {f_z} {\mathbf x} }: \R^3 \to \R^3$ be a vector-valued function on $\R^3$. Let $\map g {x, y, z}: \R^3 \to \R$ be a real-valued function on $\R^3$. Let $\nabla \times \mathbf f$ denote the curl of $f$. Then: :$\nabla \times \paren {g \, \mathbf f} = \map g {\nabla \times \mathbf f} + \paren {\nabla g} \times \mathbf f$
Moment Generating Function of Beta Distribution
Let $X \sim \operatorname{Beta} \left({\alpha, \beta}\right)$ for some $\alpha, \beta > 0$, where $\operatorname{Beta}$ is the Beta distribution.  Then the moment generating function $M_X$ of $X$ is given by:  :$\displaystyle M_X \left({t}\right) = 1 + \sum_{k \mathop = 1}^\infty \left({ \prod_{r \mathop = 0}^{k - 1} \frac {\alpha + r} {\alpha + \beta + r} }\right) \frac{t^k} {k!}$
Raw Moment of Beta Distribution
Let $X \sim \BetaDist \alpha \beta$ for some $\alpha, \beta > 0$, where $\operatorname{Beta}$ is the Beta distribution.  Then: :$\displaystyle \expect {X^n} = \prod_{r \mathop = 0}^{n - 1} \frac {\alpha + r} {\alpha + \beta + r}$ for positive integer $n$.
Linearity of Expectation Function/Discrete
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space. Let $X$ and $Y$ be random variables on $\struct {\Omega, \Sigma, \Pr}$. Let $\expect X$ denote the expectation of $X$. Then: :$\forall \alpha, \beta \in \R: \expect {\alpha X + \beta Y} = \alpha \, \expect X + \beta \, \expect Y$
Linearity of Expectation Function/Continuous
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space. Let $X$ and $Y$ be random variables on $\struct {\Omega, \Sigma, \Pr}$. Let $E$ denote the expectation function. Then: :$\forall \alpha, \beta \in \R: \expect {\alpha X + \beta Y} = \alpha \expect X + \beta \expect Y$
Derivative of Constant Multiple/Real
Let $f$ be a real function which is differentiable on $\R$. Let $c \in \R$ be a constant. Then: :$\map {\dfrac \d {\d x} } {c \map f x} = c \map {\dfrac \d {\d x} } {\map f x}$
Bound on Complex Values of Gamma Function
Let $\map \Gamma z$ denote the Gamma function. Then for any complex number $z = s + i t$, we have for $\size b \le \size t$: :$\size {\map \Gamma {s + i t} } \le \dfrac {\size {s + i b} } {\size {s + i t} } \size {\map \Gamma {s + i b} }$
Set of Vectors defined by Directed Line Segments in Space forms Vector Space
Let $\R^3$ be a real cartesian space of $3$ dimensions. Consider the set $S$ of directed line segments in $\R^3$. Let the equivalence relation $\sim$ be applied to $\R^3$ such that: :$\forall L_1, L_2 \in \R^3: L_1 \sim L_2$  there exists a translation $T$ such that $T \left({L_1}\right) = L_2$ Let $\mathbb V$ denote the set of equivalence classes of $\sim$ on $S$. The elements of $\mathbb V$ form a vector space where: :$\forall \mathbf v_1, \mathbf v_2 \in \mathbb V: \mathbf v_1 + \mathbf v_2$ denotes the element of $\mathbb V$ exemplified by the result of the operation of joining of a representative element of element of $\mathbf v_2$ to the end of a similarly representative element of element of $\mathbf v_1$. :$\forall \mathbf v \in \mathbb V, \lambda \in \R: \lambda \mathbf v$ denotes the element of $\mathbb V$ exemplified by the result of the operation of mulitplying a representative element of element of $\mathbf v$ by the scale factor $\lambda$.
Matrix is Invertible iff Determinant has Multiplicative Inverse/Sufficient Condition
Let $R$ be a commutative ring with unity. Let $\mathbf A \in R^{n \times n}$ be a square matrix of order $n$. Let  the determinant of $\mathbf A$ be invertible in $R$. Then $\mathbf A$ is an invertible matrix.
Chebyshev's Inequality
Let $X$ be a random variable.  Let $\expect X = \mu$ for some $\mu \in \R$.  Let $\var X = \sigma^2$ for some $\sigma^2 \in \R_{> 0}$. Then, for all $k > 0$:  :$\map \Pr {\size {X - \mu} \ge k \sigma} \le \dfrac 1 {k^2}$
Sign of Permutation on n Letters is Well-Defined
Let $n \in \N$ be a natural number. Let $S_n$ denote the symmetric group on $n$ letters. Let $\rho \in S_n$ be a permutation in $S_n$. Let $\sgn \paren \rho$ denote the sign of $\rho$. Then $\sgn \paren \rho$ is well-defined, in that it is either $1$ or $-1$.
Odd and Even Permutations of Set are Equivalent
Let $n \in \N_{> 0}$ be a natural number greater than $0$. Let $S$ be a set of cardinality $n$. Let $S_n$ denote the symmetric group on $S$ of order $n$. Let $R_e$ and $R_o$ denote the subsets of $S_n$ consisting of even permutations and odd permutations respectively. Then $R_e$ and $R_o$ are equivalent.
Cardinality of Even and Odd Permutations on Finite Set
Let $n \in \N_{> 0}$ be a natural number greater than $0$. Let $S$ be a set of cardinality $n$. Let $S_n$ denote the symmetric group on $S$ of order $n$. Let $R_e$ and $R_o$ denote the subsets of $S_n$ consisting of even permutations and odd permutations respectively. Then the cardinality of both $R_e$ and $R_o$ is $\dfrac {n!} 2$.
Congruence Modulo Integer is Equivalence Relation
For all $z \in \Z$, congruence modulo $z$ is an equivalence relation.
Integer Combination of Coprime Integers/General Result
Let $a_1, a_2, \ldots, a_n$ be integers. Then $\gcd \set {a_1, a_2, \ldots, a_n} = 1$  there exists an integer combination of them equal to $1$: :$\exists m_1, m_2, \ldots, m_n \in \Z: \ds \sum_{k \mathop = 1}^n m_k a_k = 1$
Properties of Beta Function
Let $\Beta \left({x, y}\right)$ denote the Beta function: {{:Definition:Beta Function/Definition 1}} $\Beta \left({x, y}\right)$ has the following properties:
Beta Function of x with y+m+1
Let $\map \Beta {x, y}$ denote the Beta function. Then: :$\map \Beta {x, y} = \dfrac {\map {\Gamma_m} y m^x} {\map {\Gamma_m} {x + y} } \map \Beta {x, y + m + 1}$ where $\Gamma_m$ is the partial Gamma function: :$\map {\Gamma_m} y := \dfrac {m^y m!} {y \paren {y + 1} \paren {y + 2} \cdots \paren {y + m} }$
Binomial Coefficient expressed using Beta Function
Let $\dbinom r k$ denote a binomial coefficient. Then: :$\dbinom r k = \dfrac 1 {\left({r + 1}\right) B \left({k + 1, r - k + 1}\right)}$
Symmetry Rule for Binomial Coefficients/Complex Numbers
For all $z, w \in \C$ such that it is not the case that $z$ is a negative integer and $w$ an integer: :$\dbinom z w = \dbinom z {z - w}$
Factors of Binomial Coefficient/Complex Numbers
For all $z, w \in \C$ such that it is not the case that $z$ is a negative integer and $w$ an integer: :$\dbinom z w = \dfrac z w \dbinom {z - 1} {w - 1}$ where $\dbinom z w$ is a binomial coefficient.
Pascal's Rule/Complex Numbers
For all $z, w \in \C$ such that it is not the case that $z$ is a negative integer and $w$ an integer: :$\dbinom z {w - 1} + \dbinom z w = \dbinom {z + 1} w$ where $\dbinom z w$ is a binomial coefficient.
Binomial Theorem/Extended
Let $r, \alpha \in \C$ be complex numbers. Let $z \in \C$ be a complex number such that $\left|{z}\right| < 1$. Then: :$\displaystyle \paren {1 + z}^r = \sum_{k \mathop \in \Z} \dbinom r {\alpha + k} z^{\alpha + k}$ where $\dbinom r {\alpha + k}$ denotes a binomial coefficient.
Chu-Vandermonde Identity/Extended
Let $r, s, \alpha, \beta \in \C$ be complex numbers. Then: :$\displaystyle \sum_{k \mathop \in \Z} \dbinom r {\alpha + k} \dbinom s {\beta - k}$ where $\dbinom r {\alpha + k}$ denotes a binomial coefficient.
Beta Function of Half with Half
:$\Beta \left({\dfrac 1 2, \dfrac 1 2}\right) = \pi$ where $\Beta$ denotes the Beta function.
Binomial Coefficient of Real Number with Half
:$\dbinom r {1 / 2} = \dfrac {2^{2 r + 1} } {\dbinom {2 r} r \pi}$ where $\dbinom r {1 / 2}$ denotes a binomial coefficient.
Limit to Infinity of Binomial Coefficient over Power
:$\displaystyle \lim_{r \mathop \to \infty} \dfrac {\dbinom r k} {r^k} = \frac 1 {\map \Gamma {k + 1}}$
Approximation to x+y Choose y
:$\displaystyle \lim_{x, y \mathop \to \infty} \dbinom {x + y} y = \sqrt {\dfrac 1 {2 \pi} \paren {\frac 1 x + \frac 1 y} } \paren {1 + \dfrac y x}^x \paren {1 + \dfrac x y}^y$
Product of r Choose k with r Minus Half Choose k/Formulation 1
Let $k \in \Z$, $r \in \R$. :$\dbinom r k \dbinom {r - \frac 1 2} k = \dfrac {\dbinom {2 r} k \dbinom {2 r - k} k} {4^k}$ where $\dbinom r k$ denotes a binomial coefficient.
Provable Consequence of Theorems is Theorem
Let $\mathcal P$ be a proof system for a formal language $\mathcal L$. Let $\mathcal F$ be a collection of theorems of $\mathcal P$. Denote with $\mathscr P \left({\mathcal F}\right)$ the proof system obtained from $\mathscr P$ by adding all the WFFs from $\mathcal F$ as axioms. Let $\phi$ be a provable consequence of $\mathcal F$: :$\vdash_{\mathscr P} \mathcal F$ :$\mathcal F \vdash_{\mathscr P} \phi$ Then $\phi$ is also a theorem of $\mathscr P$: :$\vdash_{\mathscr P} \phi$
Binomial Coefficient of Minus Half
Let $k \in \Z$. :$\dbinom {-\frac 1 2} k = \dfrac {\paren {-1}^k} {4^k} \dbinom {2 k} k$ where $\dbinom {-\frac 1 2} k$ denotes a binomial coefficient.
Falling Factorial of Sum of Integers
Let $r \in \R$ be a real number. Let $a, b \in \Z$ be (positive) integers. Then: :$r^{\underline {a + b} } = r^{\underline a} \left({r - a}\right)^{\underline b}$ where $r^{\underline a}$ denotes the $a$th falling factorial of $r$.
Approximate Size of Sum of Harmonic Series
Let $H_n$ denote the sum of the harmonic series: :$H_n = \displaystyle \sum_{k \mathop = 1}^n \frac 1 k$ Then $H_n$ can be approximated as follows: :$H_n \approx \ln n + \gamma + \dfrac 1 {2 n} - \dfrac 1 {12 n^2} + \dfrac 1 {120 n^4} - \epsilon$ where: :$\gamma$ denotes the Euler-Mascheroni constant: $\gamma \approx 0 \cdotp 57721 \, 56649 \, \ldots$ :$0 < \epsilon < \dfrac 1 {252 n^6}$
Riemann Zeta Function of 8
The Riemann zeta function of $8$ is given by: {{begin-eqn}} {{eqn | l = \map \zeta 8       | r = \dfrac 1 {1^8} +  \dfrac 1 {2^8} +  \dfrac 1 {3^8} +  \dfrac 1 {4^8} + \cdots       | c =  }} {{eqn | r = \dfrac {\pi^8} {9450}       | c =  }} {{eqn | o = \approx       | r = 1 \cdotp 00408 \, 3 \ldots       | c =  }} {{end-eqn}}
Sum of Sequence of Harmonic Numbers
:$\displaystyle \sum_{k \mathop = 1}^n H_k = \left({n + 1}\right) H_n - n$ where $H_k$ denotes the $k$th harmonic number.
Sum over k to n of k Choose m by kth Harmonic Number
:$\displaystyle \sum_{k \mathop = 1}^n \binom k m H_k = \binom {n + 1} {m + 1} \left({H_{n + 1} - \frac 1 {m + 1} }\right)$ where: : $\dbinom k m$ denotes a binomial coefficient : $H_k$ denotes the $k$th harmonic number.
Sum over k of n Choose k by x to the k by kth Harmonic Number
Let $x \in \R_{> 0}$ be a real number. Then: :$\displaystyle \sum_{k \mathop \in \Z} \binom n k x^k H_k = \left({x + 1}\right)^n \left({H_n - \ln \left({1 + \frac 1 x}\right)}\right) + \epsilon$ where: : $\dbinom n k$ denotes a binomial coefficient : $H_k$ denotes the $k$th harmonic number : $0 < \epsilon < \dfrac 1 {x \left({n + 1}\right)}$
Power to Real Number by Decimal Expansion is Uniquely Defined
Let $r \in \R_{> 1}$ be a real number greater than $1$, expressed by its decimal expansion: :$r = n \cdotp d_1 d_2 d_3 \ldots$ The power $x^r$ of a (strictly) positive real number $x$ defined as: :$(1): \quad \displaystyle \lim_{k \mathop \to \infty} x^{\psi_1} \le \xi \le x^{\psi_2}$ where: {{begin-eqn}} {{eqn | l = \psi_1       | r = n + \sum_{j \mathop = 1}^k \frac {d_1} {10^k} = n + \frac {d_1} {10} + \cdots + \frac {d_k} {10^k} }} {{eqn | l = \psi_2       | r = \psi_1 + \dfrac 1 {10^k} }} {{end-eqn}} is unique.
Change of Base of Logarithm/Base 2 to Base 10
:$\log_{10} x = \left({\lg x}\right) \left({\log_{10} 2}\right) = 0 \cdotp 30102 \, 99956 \, 63981 \, 19521 \, 37389 \ldots \lg x$
Ordering on Real Numbers from Decimal Expansion
Let $x, y \in \R$ be real numbers. Let $x$ and $y$ be expressed by their decimal expansions: {{begin-eqn}} {{eqn | l = x       | r = m \cdotp d_1 d_2 d_3 \ldots }} {{eqn | l = y       | r = n \cdotp e_1 e_2 e_3 \ldots       | c =  }} {{end-eqn}} Let $\preccurlyeq$ be the lexicographic ordering on $\R$ defined as: :$x \preccurlyeq y$ : ::$m \prec n$ :or: ::$m = n$ and $\exists k \in \Z_{>0}: \left({\forall j: 1 \le j < k: d_j = e_j}\right) \land d_k < e_k$ :or: ::$m = n$ and $\forall j \in \Z_{>0}: d_j = e_j$. Then: : $x \le y$ where $\le$ denotes the usual ordering on $\R$.
Necessary Precision for x equal to log base 10 of 2 to determine Decimal expansion of 10 to the x
Let $b = 10$. Let $x \approx \log_{10} 2$. Let it be necessary to calculate the decimal expansion of $x$ to determine the first $3$ decimal places of $b^x$. An infinite number of decimal places of $x$ would in fact be necessary.
Change of Base of Logarithm/Base 10 to Base e/Form 1
:$\ln x = \paren {\ln 10} \paren {\log_{10} x} = 2 \cdotp 30258 \, 50929 \, 94 \ldots \log_{10} x$
Change of Base of Logarithm/Base 10 to Base e/Form 2
:$\ln x = \dfrac {\log_{10} x} {\log_{10} e} = \dfrac {\log_{10} x} {0 \cdotp 43429 \, 44819 \, 03 \ldots}$
Change of Base of Logarithm/Base e to Base 10/Form 1
:$\log_{10} x = \paren {\log_{10} e} \paren {\ln x} = 0 \cdotp 43429 \, 44819 \, 03 \ldots \ln x$
Change of Base of Logarithm/Base e to Base 10/Form 2
:$\log_{10} x = \dfrac {\ln x} {\ln 10} = \dfrac {\ln x} {2 \cdotp 30258 \, 50929 \, 94 \ldots}$
Change of Base of Logarithm/Base 10 to Base 2
:$\lg x = \dfrac {\log_{10} x} {\log_{10} 2} = \dfrac {\log_{10} x} {0 \cdotp 30102 \, 99956 \, 63981 \, 19521 \, 37389 \ldots}$
Root of Quotient equals Quotient of Roots
:$\sqrt [n] {\dfrac a b} = \dfrac {\sqrt [n] a} {\sqrt [n] b}$
Logarithm to Own Base equals 1
Let $b \in \R_{>0}$ be a strictly positive real number such that $b \ne 1$. Let $\log_b$ denote the logarithm to base $b$. Then: :$\log_b b = 1$
General Logarithm/Examples/Base b of 1
:$\log_b 1 = 0$
General Logarithm/Examples/Base b of -1
:$\log_b \left({-1}\right)$ is undefined in the real number line.
Change of Base of Logarithm/Base 2 to Base 8
:$\log_8 x = \dfrac {\lg x} 3$
Number of Digits to Represent Integer in Given Number Base
Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $b \in \Z$ be an integer such that $b > 1$. Let $d$ denote the number of digits of $n$ when represented in base $b$. Then: :$d = \ceiling {\map {\log_b} {n + 1} }$ where $\ceiling {\, \cdot \,}$ denotes the ceiling function.
Number of Bits for Decimal Integer
Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $n$ have $m$ digits when expressed in decimal notation. Then $n$ may require as many as $\ceiling {\dfrac m {\log_{10} 2} }$ bits to represent it.
Sum of Summations over Overlapping Domains/Infinite Series
Let the fiber of truth of both $R$ and $S$ be infinite. Then provided that any $3$ of the $4$ summations converge: :$\ds \sum_{\map R j} a_j + \sum_{\map S j} a_j = \sum_{\map R j \mathop \lor \map S J} a_j + \sum_{\map R j \mathop \land \map S j} a_j$ where $\lor$ and $\land$ signify logical disjunction and logical conjunction respectively.
Translation of Index Variable of Summation/Infinite Series
Let $R: \Z \to \left\{ {\mathrm T, \mathrm F}\right\}$ be a propositional function on the set of integers. Let $\displaystyle \sum_{R \left({j}\right)} a_j$ denote a summation over $R$. Let the fiber of truth of $R$ be infinite. Then: :$\displaystyle \sum_{R \left({j}\right)} a_j = \sum_{R \left({c \mathop + j}\right)} a_{c \mathop + j} = \sum_{R \left({c \mathop - j}\right)} a_{c \mathop - j}$ where $c$ is an integer constant which is not dependent upon $j$.
Exchange of Order of Summation/Example
Let the fiber of truth of both $R$ and $S$ be infinite. Then it is not necessarily the case that: :$\displaystyle \sum_{\map R i} \sum_{\map S j} a_{i j} = \sum_{\map S j} \sum_{\map R i} a_{i j}$
Sum of Geometric Sequence/Examples/Index to Minus 1
Let $x$ be an element of one of the standard number fields: $\Q, \R, \C$ such that $x \ne 1$. Then the formula for Sum of Geometric Sequence: :$\displaystyle \sum_{j \mathop = 0}^n x^j = \frac {x^{n + 1} - 1} {x - 1}$ still holds when $n = -1$: :$\displaystyle \sum_{j \mathop = 0}^{-1} x^j = \frac {x^0 - 1} {x - 1}$
Sum of Geometric Sequence/Examples/Common Ratio 1
Consider the Sum of Geometric Sequence defined on the standard number fields for all $x \ne 1$. :$\displaystyle \sum_{j \mathop = 0}^n a x^j = a \paren {\frac {1 - x^{n + 1} } {1 - x} }$ When $x = 1$, the formula reduces to: :$\displaystyle \sum_{j \mathop = 0}^n a 1^j = a \paren {n + 1}$
Sum of Geometric Sequence/Examples/One Seventh from 1 to n
:$\displaystyle \sum_{j \mathop = 0}^n \dfrac 1 {7^j} = \frac 7 6 \left({1 - \frac 1 {7^{n + 1} } }\right)$
Summation of Unity over Elements
Let $S \subseteq \Z$ be a set of integers. Let: :$n := \displaystyle \sum_{j \mathop \in  S} 1$ Then $n$ is equal to the cardinality of $S$.
Hausdorff Maximal Principle Implies Zorn's Lemma
The Hausdorff Maximal Principle implies Zorn's Lemma.
Zorn's Lemma Implies Well Ordering Theorem
Zorn's Lemma implies the Well-Ordering Theorem.
Uncountable Sum as Series
Let $X$ be an uncountable set. Let $f: X \to \closedint 0 {+\infty}$ be an extended real-valued function. The uncountable sum: :$\displaystyle \sum_{x \mathop \in X} \map f x = \sup \set {\sum_{x \mathop \in F} \map f x : F \subseteq X, F \text{ finite} }$ is $+\infty$, or can be expressed as a (possibly divergent) series.
Sum with Maximum is Maximum of Sum
Let $a, b, c \in \R$ be real numbers. Then: :$a + \max \set {b, c} = \max \set {a + b, a + c}$
Sum of Indexed Suprema
Let $\left \langle {a_i} \right \rangle_{i \mathop \in I}$ be a family of elements of the real numbers $\R$ indexed by $I$. Let $\left \langle {b_j} \right \rangle_{j \mathop \in J}$ be a family of elements of the real numbers $\R$ indexed by $J$. Let $R \left({i}\right)$ and $S \left({j}\right)$ be propositional functions of $i \in I$, $j \in J$. Let $\displaystyle \sup_{R \left({i}\right)} a_i$ and $\displaystyle \sup_{S \left({j}\right)} b_j$ be the indexed suprema on $\left \langle {a_i} \right \rangle$ and $\left \langle {b_j} \right \rangle$ respectively. Then: :$\displaystyle \left({\sup_{R \left({i}\right)} a_i}\right) + \left({\sup_{S \left({j}\right)} b_j}\right) = \sup_{R \left({i}\right)} \left({\sup_{S \left({j}\right)} \left({a_i + b_j}\right)}\right)$
Product of Indexed Suprema of Non-Negative Numbers
Let $\left \langle {a_i} \right \rangle_{i \mathop \in I}$ be a family of elements of the non-negative real numbers $\R_{\ge 0}$ indexed by $I$. Let $\left \langle {b_j} \right \rangle_{j \mathop \in J}$ be a family of elements of the non-negative real numbers $\R_{\ge 0}$ indexed by $J$. Let $R \left({i}\right)$ and $S \left({j}\right)$ be propositional functions of $i \in I$, $j \in J$. Let $\displaystyle \sup_{R \left({i}\right)} a_i$ and $\displaystyle \sup_{S \left({j}\right)} b_j$ be the indexed suprema on $\left \langle {a_i} \right \rangle$ and $\left \langle {b_j} \right \rangle$ respectively. Then: :$\displaystyle \left({\sup_{R \left({i}\right)} a_i}\right) \left({\sup_{S \left({j}\right)} b_j}\right) = \sup_{R \left({i}\right)} \left({\sup_{S \left({j}\right)} \left({a_i b_j}\right)}\right)$
Change of Index Variable of Supremum
Let $\family {a_i}{i \mathop \in I}$ be a family of elements of the non-negative real numbers $\R_{\ge 0}$ indexed by $I$. Let $\map R i$ be a propositional functions of $i \in I$. Let $\displaystyle \sup_{\map R i} a_i$ be the indexed supremum on $\family {a_i}$. Then: :$\displaystyle \sup_{\map R i} a_i = \sup_{\map R j} a_j$
Permutation of Indices of Supremum
Let $\left \langle {a_i} \right \rangle_{i \mathop \in I}$ be a family of elements of the non-negative real numbers $\R_{\ge 0}$ indexed by $I$. Let $R \left({i}\right)$ be a propositional functions of $i \in I$. Let $\displaystyle \sup_{R \left({i}\right)} a_i$ be the indexed supremum on $\left \langle {a_i} \right \rangle$. Then: :$\displaystyle \sum_{R \left({i}\right)} a_i = \sum_{R \left({\pi \left({i}\right)}\right)} a_{\pi \left({i}\right)}$ where $\pi$ is a permutation on the fiber of truth of $R$.
Exchange of Order of Supremum Operators
Let $\left \langle {a_i} \right \rangle_{i \mathop \in I}$ be a family of elements of the non-negative real numbers $\R_{\ge 0}$ indexed by $I$. Let $\left \langle {b_j} \right \rangle_{j \mathop \in J}$ be a family of elements of the non-negative real numbers $\R_{\ge 0}$ indexed by $J$. Let $R \left({i}\right)$ and $S \left({j}\right)$ be propositional functions of $i \in I$, $j \in J$. Let $\displaystyle \sup_{R \left({i}\right)} a_i$ and $\displaystyle \sup_{S \left({j}\right)} b_j$ be the indexed suprema on $\left \langle {a_i} \right \rangle$ and $\left \langle {b_j} \right \rangle$ respectively. Then: :$\displaystyle \sup_{R \left({i}\right)} \left({\sup_{S \left({j}\right)} a_{i j} }\right) = \sup_{S \left({j}\right)} \left({\sup_{R \left({i}\right)} a_{i j} }\right)$
Supremum of Suprema over Overlapping Domains
Let $\family {a_i}_{i \mathop \in I}$ be a family of elements of the non-negative real numbers $\R_{\ge 0}$ indexed by $I$. Let $\map R i$ and $\map S i$ be propositional functions of $i \in I$. Let $\displaystyle \sup_{\map R i} a_i$ and $\displaystyle \sup_{\map S i} a_i$ be the indexed suprema on $\family {a_i}$ over $\map R i$ and $\map S i$ respectively. Then: :$\displaystyle \map \sup {\sup_{\map R i} a_i, \sup_{\map S i} a_i} = \sup_{\map R i \mathop \lor \map S i} a_i$
Contour Integration by Substitution
Let $f$ be a holomorphic function on a simply connected domain $V \subseteq \mathbb C$. Let $\gamma$ be a contour in $V$ starting at $z_1$ and ending at $z_2$. Let $U$ be a connected domain. Let $\phi: U \to V$ be a holomorphic function with $\phi^{-1} \sqbrk {\set {z_1, z_2} } \ne \O$. Let $\omega$ be a contour in $U$ starting at $u_1$ and ending at $u_2$, such that $\map \phi {u_1} = z_1$ and $\map \phi {u_2} = z_2$. Then the contour integral of $f$ over $\gamma$ satisfies the following substitution: :$\displaystyle \int_\gamma \map f z \rd z = \int_\omega \map f {\map \phi u} \, \map {\phi'} u \rd u$
Infimum of Set of Integers is Integer
Let $S \subset \Z$ be a non-empty subset of the set of integers. Let $S$ be bounded below in the set of real numbers. Then its infimum $\inf S$ is an integer.
Infimum of Set of Integers equals Smallest Element
Let $S \subset \Z$ be a non-empty subset of the set of integers. Let $S$ be bounded below in the set of real numbers $\R$. Then $S$ has a smallest element, and it is equal to the infimum $\sup S$.
Greatest Element is Supremum
Let $\struct {S, \preceq}$ be an ordered set. Let $T \subseteq S$. Let $T$ have a greatest element $M$. Then $M$ is the supremum of $T$ in $S$.
Smallest Element is Infimum
Let $\left({S, \preceq}\right)$ be an ordered set. Let $T \subseteq S$. Let $T$ have a smallest element $m$. Then $m$ is the infimum of $T$ in $S$.
Floor Function/Examples/Floor of 1.1
:$\floor {1 \cdotp 1} = 1$
Floor Function/Examples/Floor of -1.1
:$\floor {-1 \cdotp 1} = -2$
Ceiling Function/Examples/Ceiling of -1.1
:$\ceiling {-1 \cdotp 1} = -1$
Floor Function/Examples/Floor of 0.99999
:$\floor {0 \cdotp 99999} = 0$
Floor Function/Examples/Floor of Binary Logarithm of 35
:$\floor {\lg 35} = 5$ where: : $\lg x$ denotes the binary logarithm ($\log_2$) of $x$
Modulo Operation/Examples/100 mod 3
:$100 \bmod 3 = 1$
Modulo Operation/Examples/-100 mod 7
:$-100 \bmod 7 = 5$
Modulo Operation/Examples/-100 mod 0
:$-100 \bmod 0 = -100$
Modulo Operation/Examples/5 mod -3
:$5 \bmod -3 = 5$
Modulo Operation/Examples/18 mod -3
:$18 \bmod -3 = 0$
Modulo Operation/Examples/-2 mod -3
:$-2 \bmod -3 = -2$
Modulo Operation/Examples/0.11 mod -0.1
:$0 \cdotp 11 \bmod -0 \cdotp 1 = -0 \cdotp 09$
Common Factor Cancelling in Congruence/Corollary 1/Warning
Let $a$ ''not'' be coprime to $m$. Then it is not necessarily the case that: :$x \equiv y \pmod m$
Chinese Remainder Theorem/Warning
Let $r$ ''not'' be coprime to $s$. Then it is not necessarily the case that: :$a \equiv b \pmod {r s}$  $a \equiv b \pmod r$ and $a \equiv b \pmod s$ where $a \equiv b \pmod r$ denotes that $a$ is congruent modulo $r$ to $b$.
Euler Phi Function of 2
:$\map \phi 2 = 1$