hcju/proofwikips huggingface dataset:
Reflexive Euclidean Relation is Equivalence
A relation is an equivalence  it is either left-Euclidean or right-Euclidean, and also reflexive.
Reflexive Relation is Serial
Every reflexive relation is also a serial relation.
Complement of Complete Bipartite Graph
Let $K_{p, q}$ be a complete bipartite graph. The complement of $K_{p, q}$ consists of a disconnected graph with two components: :The complete graph $K_p$ :The complete graph $K_q$.
Complement of Strict Total Ordering
Let $\left({S, \prec}\right)$ be a relational structure such that $\prec$ is a strict total ordering. Then the complement of $\prec$ is a weak total ordering.
Mahler's Inequality
The geometric mean of the termwise sum of two finite sequences of positive real numbers is never less than the sum of their two separate geometric means: :$\displaystyle \prod_{k \mathop = 1}^n \paren {x_k + y_k}^{1/n} \ge \prod_{k \mathop = 1}^n x_k^{1/n} + \prod_{k \mathop = 1}^n y_k^{1/n}$ where $x_k, y_k > 0$ for all $k$.
Equivalence of Definitions of Transitive Closure (Relation Theory)/Intersection is Smallest
Let $\RR$ be a relation on a set $S$. Then the intersection of all transitive relations on $S$ that contain $\RR$ is the smallest transitive relation on $S$ that contains $\RR$.
Intersection of Transitive Relations is Transitive
The intersection of two transitive relations is also a transitive relation.
Intersection of Reflexive Relations is Reflexive
The intersection of two reflexive relations is also a reflexive relation.
Intersection of Symmetric Relations is Symmetric
The intersection of two symmetric relations is also a symmetric relation.
Union of Reflexive Relations is Reflexive
The union of two reflexive relations is also a reflexive relation.
Union of Symmetric Relations is Symmetric
The union of two symmetric relations is also a symmetric relation.
Union of Transitive Relations Not Always Transitive
The union of transitive relations is not necessarily itself transitive.
Set System Closed under Intersection is Commutative Semigroup
Let $\mathcal S$ be a system of sets. Let $\mathcal S$ be such that: :$\forall A, B \in \mathcal S: A \cap B \in \mathcal S$ Then $\left({\mathcal S, \cap}\right)$ is a commutative semigroup.
Set System Closed under Union is Commutative Semigroup
Let $\mathcal S$ be a system of sets. Let $\mathcal S$ be such that: :$\forall A, B \in \mathcal S: A \cup B \in \mathcal S$ Then $\struct {\mathcal S, \cup}$ is a commutative semigroup.
Set System Closed under Symmetric Difference is Abelian Group
Let $\SS$ be a system of sets. Let $\SS$ be such that: :$\forall A, B \in \SS: A * B \in \SS$ where $A * B$ denotes the symmetric difference between $A$ and $B$. Then $\struct {\SS, *}$ is an abelian group.
Ring of Sets is Commutative Ring
A ring of sets $\left({\mathcal R, *, \cap}\right)$ is a commutative ring whose zero is $\varnothing$.
Union as Symmetric Difference with Intersection
Let $A$ and $B$ be sets. Then: :$A \cup B = \left({A * B}\right) * \left({A \cap B}\right)$ where: : $A \cup B$ denotes set union : $A \cap B$ denotes set intersection : $A * B$ denotes set symmetric difference
Set Difference as Symmetric Difference with Intersection
:$S \setminus T = S * \paren {S \cap T}$ where: :$S \setminus T$ denotes set difference :$S * T$ denotes set symmetric difference :$S \cap T$ denotes set intersection.
Unit of System of Sets is Unique
The unit of a system of sets, if it exists, is unique. If $U$ is the unit of a system of sets $\mathcal S$, then $\forall A \in \mathcal S: A \subseteq U$.
Power Set is Algebra of Sets
Let $S$ be a set. Let $\mathcal P \left({S}\right)$ be the power set of $S$. Then $\mathcal P \left({S}\right)$ is an algebra of sets where $S$ is the unit.
Power Set is Closed under Intersection
Let $S$ be a set. Let $\powerset S$ be the power set of $S$. Then: :$\forall A, B \in \powerset S: A \cap B \in \powerset S$
Power Set is Closed under Symmetric Difference
Let $S$ be a set. Let $\powerset S$ be the power set of $S$. Then: :$\forall A, B \in \powerset S: A * B \in \powerset S$ where $A * B$ is the symmetric difference between $A$ and $B$.
Intersection of Rings of Sets
Let $\mathcal R_k$ be a ring of sets, where $k$ is an element of an arbitrary set of indices. Then their intersection $\displaystyle \mathcal R = \bigcap_k \mathcal R_k$ is itself a ring of sets.
Minimal Ring Generated by System of Sets
Let $\SS$ be a non-empty system of sets. Then there is a unique ring of sets $\map \RR \SS$ which: :$(1): \quad$ contains $\SS$ :$(2): \quad$ is contained by every ring of sets which also contains $\SS$. This ring of sets $\map \RR \SS$ is called the '''minimal ring generated by $\SS$'''.
Menelaus's Theorem
Let $ABC$ be a triangle. Let points $D, E, F$ lie on lines $BC, AC, AB$ respectively (produced if necessary). Then $D, E$ and $F$ are collinear : : $\dfrac {AF} {FB} \cdot \dfrac {BD} {DC} \cdot \dfrac {CE} {EA} = -1$ In the above, the line segments $AF, BD, EA$ are determined to have negative length if they lie outside the line segments $AB, BC, CA$.
Lines Joining Equal and Parallel Straight Lines are Parallel
The straight lines joining equal and parallel straight lines at their endpoints, in the same direction, are themselves equal and parallel. {{:Euclid:Proposition/I/33}}
Opposite Sides and Angles of Parallelogram are Equal
The opposite sides and angles of a parallelogram are equal to one another, and either of its diameters bisects its area. {{:Euclid:Proposition/I/34}}
Equal Sized Triangles on Same Base have Same Height
Triangles of equal area which are on the same base, and on the same side of it, are also in the same parallels. {{:Euclid:Proposition/I/39}}
Parallelogram on Same Base as Triangle has Twice its Area
A parallelogram on the same base as a triangle, and in the same parallels, has twice the area of the triangle. {{:Euclid:Proposition/I/41}}
Construction of Parallelogram equal to Triangle in Given Angle
A parallelogram can be constructed in a given angle the same size as any given triangle. {{:Euclid:Proposition/I/42}}
Construction of Parallelogram in Given Angle equal to Given Polygon
A parallelogram can be constructed in a given angle the same size as any given polygon. {{:Euclid:Proposition/I/45}}
Square of Sum
:$\forall x, y \in \R: \paren {x + y}^2 = x^2 + 2 x y + y^2$
Square of Sum less Square
:$\forall x, y \in \R: \paren {2x + y} y = \paren {x + y}^2 - x^2$
Square of Difference
:$\forall x, y \in \R: \paren {x - y}^2 = x^2 - 2 x y + y^2$
Square of Sum with Double
:$\forall a, b \in \R: \paren {a + 2 b}^2 = a^2 + 4 a b + 4 b^2$
Construction of Square equal to Given Polygon
A square can be constructed the same size as any given polygon. {{:Euclid:Proposition/II/14}}
Finding Center of Circle
For any given circle, it is possible to find its center. {{:Euclid:Proposition/III/1}}
Perpendicular Bisector of Chord Passes Through Center
The perpendicular bisector of any chord of any given circle must pass through the center of that circle. {{:Euclid:Proposition/III/1/Porism}}
Conditions for Diameter to be Perpendicular Bisector
If in a circle a diameter bisects a chord (which is itself not a diameter), then it cuts it at right angles, and if it cuts it at right angles then it bisects it. {{:Euclid:Proposition/III/3}}
Chords do not Bisect Each Other
If in a circle two chords (which are not diameters) cut one another, then they do not bisect one another. {{:Euclid:Proposition/III/4}}
Concentric Circles do not Intersect
If two circles are concentric, they share no points on their circumferences. Alternatively, this can be worded: :If two concentric circles share one point on their circumferences, then they share them all (that is, they are the same circle).
Baire Category Theorem
Let $M = \struct {A, d}$ be a complete metric space. Then $M = \struct {A, d}$ is also a Baire space.
Baire Characterisation Theorem
A real-valued function $f$ defined on a Banach space $X$ is a Baire-$1$ function : : for every non-empty closed subset $K$ of $X$, the restriction of $f$ to $K$ has a point of continuity relative to the topology of $K$.
Ramsey's Theorem
In any coloring of the edges of a sufficiently large complete graph, one will find monochromatic complete subgraphs. For 2 colors, Ramsey's theorem states that for any pair of positive integers $\tuple {r, s}$, there exists a least positive integer $\map R {r, s}$ such that for any complete graph on $\map R {r, s}$ vertices, whose edges are colored red or blue, there exists either a complete subgraph on $r$ vertices which is entirely red, or a complete subgraph on $s$ vertices which is entirely blue. More generally, for any given number of colors $c$, and any given integers $n_1, \ldots, n_c$, there is a number $\map R {n_1, \ldots, n_c}$ such that: :if the edges of a complete graph of order $\map R {n_1, \ldots, n_c}$ are colored with $c$ different colours, then for some $i$ between $1$ and $c$, it must contain a complete subgraph of order $n_i$ whose edges are all color $i$. This number $\map R {n_1, \ldots, n_c}$ is called the Ramsey number for $n_1, \ldots, n_c$. The special case above has $c = 2$ (and $n_1 = r$ and $n_2 = s$). Here $\map R {r, s}$ signifies an integer that depends on both $r$ and $s$. It is understood to represent the smallest integer for which the theorem holds.
Schur's Theorem (Ramsey Theory)
Let $r$ be a positive integer. Then there exists a positive integer $S$ such that: :for every partition of the integers $\set {1, \ldots, S}$ into $r$ parts, one of the parts contains integers $x$, $y$ and $z$ such that: ::$x + y = z$
Brahmagupta's Formula
The area of a cyclic quadrilateral with sides of lengths $a, b, c, d$ is: :$\sqrt {\paren {s - a} \paren {s - b} \paren {s - c} \paren {s - d} }$ where $s$ is the semiperimeter: :$s = \dfrac {a + b + c + d} 2$
Euler-Binet Formula
The Fibonacci numbers have a closed-form solution: :$F_n = \dfrac {\phi^n - \paren {1 - \phi}^n} {\sqrt 5} = \dfrac {\phi^n - \paren {-1 / \phi}^n} {\sqrt 5} = \dfrac {\phi^n - \paren {-1}^n \phi^{-n} } {\sqrt 5}$ where $\phi$ is the golden mean. Putting $\hat \phi = 1 - \phi = -\dfrac 1 \phi$ this can be written: :$F_n = \dfrac {\phi^n - \hat \phi^n} {\sqrt 5}$
Hölder's Inequality for Sums
Let $p, q \in \R_{>0}$ be strictly positive real numbers such that: :$\dfrac 1 p + \dfrac 1 q = 1$ Let: :$\mathbf x = \sequence {x_n} \in \ell^p$ :$\mathbf y = \sequence {y_n} \in \ell^q$ where $\ell^p$ denotes the $p$-sequence space. Let $\norm {\mathbf x}_p$ denote the $p$-norm of $\mathbf x$. Then $\mathbf x \mathbf y = \sequence {x_n y_n} \in \ell^1$, and: :$\norm {\mathbf x \mathbf y}_1 \le \norm {\mathbf x}_p \norm {\mathbf y}_q$
Binet Form
Let $m \in \R$. Define: {{begin-eqn}} {{eqn | l = \Delta       | r = \sqrt {m^2 + 4} }} {{eqn | l = \alpha       | r = \frac {m + \Delta} 2 }} {{eqn | l = \beta       | r = \frac {m - \Delta} 2 }} {{end-eqn}}
Binet-Cauchy Identity
:$\displaystyle \left({\sum_{i \mathop = 1}^n a_i c_i}\right) \left({\sum_{j \mathop = 1}^n b_j d_j}\right) = \left({\sum_{i \mathop = 1}^n a_i d_i}\right) \left({\sum_{j \mathop = 1}^n b_j c_j}\right) + \sum_{1 \mathop \le i \mathop < j \mathop \le n} \left({a_i b_j - a_j b_i}\right) \left({c_i d_j - c_j d_i}\right)$ where all of the $a, b, c, d$ are elements of a commutative ring. Thus the identity holds for $\Z, \Q, \R, \C$.
Nicomachus's Theorem
{{begin-eqn}} {{eqn | l = 1^3       | r = 1 }} {{eqn | l = 2^3       | r = 3 + 5 }} {{eqn | l = 3^3       | r = 7 + 9 + 11 }} {{eqn | l = 4^3       | r = 13 + 15 + 17 + 19 }} {{eqn | l = \vdots       | o =  }} {{end-eqn}} In general: :$\forall n \in \N_{>0}: n^3 = \left({n^2 - n + 1}\right) + \left({n^2 - n + 3}\right) + \ldots + \left({n^2 + n - 1}\right)$ In particular, the first term for $\left({n + 1}\right)^3$ is $2$ greater than the last term for $n^3$.
Conservation of Angular Momentum
Newton's Laws of Motion imply the conservation of angular momentum in systems of masses in which no external force is acting.
Determinant of Combinatorial Matrix
Let $C_n$ be the combinatorial matrix of order $n$ given by: :$C_n = \begin{bmatrix} x + y & y & \cdots & y \\ y & x + y & \cdots & y \\ \vdots & \vdots & \ddots & \vdots \\ y & y & \cdots & x + y \end{bmatrix}$ Then the determinant of $C_n$ is given by: :$\det \left({C_n}\right) = x^{n-1} \left ({x + n y}\right)$
Power Set is Closed under Union
Let $S$ be a set. Let $\powerset S$ be the power set of $S$. Then: :$\forall A, B \in \powerset S: A \cup B \in \powerset S$
Matrix Product with Adjugate Matrix
Let $R$ be a commutative ring with unity. Let $\mathbf A \in R^{n \times n}$ be a square matrix of order $n$. Let $\adj {\mathbf A}$ be its adjugate matrix. Then: {{begin-eqn}} {{eqn | l = \mathbf A \cdot \adj {\mathbf A}       | r = \map \det {\mathbf A} \cdot \mathbf I_n }} {{eqn | l = \adj {\mathbf A} \cdot \mathbf A       | r = \map \det {\mathbf A} \cdot \mathbf I_n }} {{end-eqn}} where $\map \det {\mathbf A}$ is the determinant of $\mathbf A$.
Square of Ones Matrix
Let $\mathbf J = \sqbrk 1_n$ be a square ones matrix of order $n$. Then $\mathbf J^2 = n \mathbf J$. That is: :$\begin{bmatrix} 1 & 1 & \cdots & 1 \\ 1 & 1 & \cdots & 1 \\ \vdots & \vdots & \ddots & \vdots \\ 1 & 1 & \cdots & 1 \end{bmatrix}^2 = \begin{bmatrix} n & n & \cdots & n \\ n & n & \cdots & n \\ \vdots & \vdots & \ddots & \vdots \\ n & n & \cdots & n \end{bmatrix}$
Inverse of Combinatorial Matrix
Let $C_n$ be the combinatorial matrix of order $n$ given by: :$C_n = \begin{bmatrix} x + y & y & \cdots & y \\ y & x + y & \cdots & y \\ \vdots & \vdots & \ddots & \vdots \\ y & y & \cdots & x + y \end{bmatrix}$ Then its inverse $C_n^{-1} = \sqbrk b_n$ can be specified as: :$b_{i j} = \dfrac {-y + \delta_{i j} \paren {x + n y} } {x \paren {x + n y} }$ where $\delta_{i j}$ is the Kronecker delta.
Inverse of Vandermonde Matrix
Let $V_n$ be the Vandermonde matrix of order $n$ given by: :$V_n = \begin {bmatrix}   x_1 & x_2 & \cdots & x_n \\   x_1^2 & x_2^2 & \cdots & x_n^2 \\ \vdots & \vdots & \ddots & \vdots \\   x_1^n & x_2^n & \cdots & x_n^n \end{bmatrix}$ Then its inverse $V_n^{-1} = \sqbrk b_n$ can be specified as: :$b_{i j} = \begin{cases} \paren {-1}^{j - 1} \paren {\dfrac {\displaystyle \sum_{\substack {1 \mathop \le m_1 \mathop < \ldots \mathop < m_{n - j} \mathop \le n \\ m_1, \ldots, m_{n - j} \mathop \ne i} } x_{m_1} \cdots x_{m_{n - j} } } {x_i \displaystyle \prod_{\substack {1 \mathop \le m \mathop \le n \\ m \mathop \ne i} } \paren {x_m - x_i} } } & : 1 \le  j < n \\ \qquad \qquad \qquad \dfrac 1 {x_i \displaystyle \prod_{\substack {1 \mathop \le m \mathop \le n \\ m \mathop \ne i} } \paren {x_i - x_m} } & : j = n \end{cases}$
Floor Function is Idempotent
:$\floor {\floor x} = \floor x$
Relation between Floor and Ceiling
Let $x \in \R$ be a real number. Let $\left \lfloor {x}\right \rfloor$ be the floor of $x$, and $\left \lceil {x}\right \rceil$ be the ceiling of $x$. Then the following results apply:
Range of Values of Floor Function
=== Number less than Integer iff Floor less than Integer === {{:Number less than Integer iff Floor less than Integer}} === Number not less than Integer iff Floor not less than Integer === {{:Number not less than Integer iff Floor not less than Integer}} === Integer equals Floor iff between Number and One Less === {{:Integer equals Floor iff between Number and One Less}} === Integer equals Floor iff Number between Integer and One More === {{:Integer equals Floor iff Number between Integer and One More}}
Range of Values of Ceiling Function
=== Number greater than Integer iff Ceiling greater than Integer === {{:Number greater than Integer iff Ceiling greater than Integer}} === Number not greater than Integer iff Ceiling not greater than Integer === {{:Number not greater than Integer iff Ceiling not greater than Integer}} === Integer equals Ceiling iff between Number and One More === {{:Integer equals Ceiling iff between Number and One More}} === Integer equals Ceiling iff Number between Integer and One Less === {{:Integer equals Ceiling iff Number between Integer and One Less}}
Steiner Inellipse is Unique
For every triangle, there is one and only one Steiner inellipse.
Kronecker's Lemma
Let $\sequence {x_n}$ be an infinite sequence of real numbers such that: :$\displaystyle \sum_{n \mathop = 1}^\infty x_n = s$  exists and is finite. Then for $0 < b_1 \le b_2 \le b_3 \le \ldots$ and $b_n \to \infty$:  :$\displaystyle \lim_{n \mathop \to \infty} \frac 1 {b_n} \sum_{k \mathop = 1}^n b_k x_k = 0$
Number of Permutations
Let $S$ be a set of $n$ elements. Let $r \in \N: r \le n$. Then the number of $r$-permutations of $S$ is: :${}^r P_n = \dfrac {n!} {\paren {n - r}!}$ When $r = n$, this becomes: :${}^n P_n = \dfrac {n!} {\paren {n - n}!} = n!$ Using the falling factorial symbol, this can also be expressed: :${}^r P_n = n^{\underline r}$
Construction of Permutations
The ${}^n P_n$ permutations of $n$ objects can be generated algorithmically. By Number of Permutations, that number is given by: :${}^n P_n = n!$ where $n!$ denotes the factorial of $n$. This will be demonstrated to hold.
De Polignac's Formula
Let $n!$ be the factorial of $n$. Let $p$ be a prime number. Then $p^\mu$ is a divisor of $n!$, and $p^{\mu + 1}$ is not, where: :$\displaystyle \mu = \sum_{k \mathop > 0} \floor {\frac n {p^k} }$ where $\floor {\, \cdot \,}$ denotes the floor function.
Sum of Binomial Coefficients over Lower Index
:$\displaystyle \sum_{i \mathop = 0}^n \binom n i = 2^n$
Alternating Sum and Difference of Binomial Coefficients for Given n
:$\displaystyle \forall n \in \Z: \sum_{i \mathop = 0}^n \left({-1}\right)^i \binom n i = \delta_{n 0}$
Symmetry Rule for Binomial Coefficients
Let $n \in \Z_{>0}, k \in \Z$. Then: :$\dbinom n k = \dbinom n {n - k}$
Factors of Binomial Coefficient
For all $r \in \R, k \in \Z$: :$k \dbinom r k = r \dbinom {r - 1} {k - 1}$ where $\dbinom r k$ is a binomial coefficient. Hence: :$\dbinom r k = \dfrac r k \dbinom {r - 1} {k - 1}$ (if $k \ne 0$) and: :$\dfrac 1 r \dbinom r k = \dfrac 1 k \dbinom {r - 1} {k - 1}$ (if $k \ne 0$ and $r \ne 0$)
Sum of r+k Choose k up to n
:$\displaystyle \forall n \in \Z: n \ge 0: \sum_{k \mathop = 0}^n \binom {r + k} k = \binom {r + n + 1} n$
Negated Upper Index of Binomial Coefficient
:$\dbinom r k = \paren {-1}^k \dbinom {k - r - 1} k$
Alternating Sum and Difference of r Choose k up to n
:$\displaystyle \sum_{k \mathop \le n} \paren {-1}^k \binom r k = \paren {-1}^n \binom {r - 1} n$
Product of r Choose m with m Choose k
:$\dbinom r m \dbinom m k = \dbinom r k \dbinom {r - k} {m - k}$
Chu-Vandermonde Identity
:$\displaystyle \sum_k \binom r k \binom s {n - k} = \binom {r + s} n$
Lucas' Theorem
Let $p$ be a prime number. Let $n, k \in \Z_{\ge 0}$. Then: :$\dbinom n k \equiv \dbinom {\floor {n / p} } {\floor {k / p} } \dbinom {n \bmod p} {k \bmod p} \pmod p$ where: :$\dbinom n k$ denotes a binomial coefficient :$n \bmod p$ denotes the modulo operation :$\floor \cdot$ denotes the floor function.
Congruence Modulo Real Number is Equivalence Relation
For all $z \in \R$, congruence modulo $z$ is an equivalence relation.
Filter Basis Generates Filter
Let $S$ be a set. Let $\powerset S$ be the power set of $S$. Let $\BB \subset \powerset S$. Then: :$\FF = \set {V \subseteq S: \exists U \in \BB: U \subseteq V}$ is a filter on $S$ : :$(1): \quad \forall V_1, V_2 \in \BB: \exists U \in \BB: U \subseteq V_1 \cap V_2$ :$(2): \quad \O \notin \BB, \BB \ne \O$ That is, $\BB$ is a filter basis of the filter $\FF$, which is generated by $\BB$.
Limit Point iff Superfilter Converges
Let $T = \left({S, \tau}\right)$ be a topological space. Let $\mathcal F$ be a filter on $S$. Let $x \in S$. Then $x$ is a limit point of $\mathcal F$  there exists a superfilter $\mathcal F'$ of $\mathcal F$ on $S$ which converges to $x$.
Zorn's Lemma
Let $\struct {X, \preceq}, X \ne \O$ be a non-empty ordered set such that every non-empty chain in $X$ has an upper bound in $X$. Then $X$ has at least one maximal element.
Ultrafilter Lemma
Let $S$ be a set. Every filter on $S$ is contained in an ultrafilter on $S$.
Generated Topology is a Topology
Let $X$ be a set. Let $\mathcal S \subseteq \mathcal P \left({X}\right)$, where $\mathcal P \left({X}\right)$ is the power set of $X$. Let $\mathcal T_\mathcal S$ be the generated topology for $\mathcal S$. Then $\mathcal T_\mathcal S$ is a topology on $X$.
Equivalent Definitions of Synthetic Basis
Let $X$ be a set. Then for a subset $\BB \subseteq \powerset X$ of the power set of $X$, the following are equivalent: :$(1): \quad$ $\displaystyle \forall A, B \in \BB: \exists \AA \subseteq \BB: A \cap B = \bigcup \AA$. :$(2): \quad$ $\displaystyle \forall A, B \in \BB: \forall x \in A \cap B: \exists W \in \BB: x \in W \subseteq A \cap B$.
Product Distributes over Modulo Operation
Let $x, y, z \in \R$ be real numbers. Let $x \bmod y$ denote the modulo operation. Then: :$z \left({x \bmod y}\right) = \left({z x}\right) \bmod \left({z y}\right)$
Congruence by Product of Moduli
Let $a, b, m \in \Z$. Let $a \equiv b \pmod m$ denote that $a$ is congruent to $b$ modulo $m$. Then $\forall n \in \Z, n \ne 0$: : $a \equiv b \pmod m \iff a n \equiv b n \pmod {m n}$
Definite Integral of Partial Derivative
Let $\map f {x, y}$ and $\map {\dfrac {\partial f} {\partial x} } {x, y}$ be continuous functions of $x$ and $y$ on $D = \closedint {x_1} {x_2} \times \closedint a b$. Then: :$\displaystyle \frac \d {\d x} \int_a^b \map f {x, y} \rd y = \int_a^b \map {\frac {\partial f} {\partial x} } {x, y} \rd y$ for $x \in \closedint {x_1} {x_2}$.
Law of Inverses (Modulo Arithmetic)
Let $m, n \in \Z$. Then: :$\exists n' \in \Z: n n' \equiv d \pmod m$ where $d = \gcd \set {m, n}$.
Common Factor Cancelling in Congruence
Let $a, b, x, y, m \in \Z$. Let: :$a x \equiv b y \pmod m$ and $a \equiv b \pmod m$ where $a \equiv b \pmod m$ denotes that $a$ is congruent modulo $m$ to $b$. Then: :$x \equiv y \pmod {m / d}$ where $d = \gcd \set {a, m}$.
Chinese Remainder Theorem
Let $a, b, r, s \in \Z$. Let $r \perp s$ (that is, let $r$ and $s$ be coprime). Then: :$a \equiv b \pmod {r s}$  $a \equiv b \pmod r$ and $a \equiv b \pmod s$ where $a \equiv b \pmod r$ denotes that $a$ is congruent modulo $r$ to $b$.
Sum of Floors not greater than Floor of Sum
Let $\left \lfloor {x} \right \rfloor$ be the floor function. Then: :$\left \lfloor {x} \right \rfloor + \left \lfloor {y} \right \rfloor \le \left \lfloor {x + y} \right \rfloor$ The equality holds: :$\left \lfloor {x} \right \rfloor + \left \lfloor {y} \right \rfloor = \left \lfloor {x + y} \right \rfloor$ : :$x \bmod 1 + y \bmod 1 < 1$ where $x \bmod 1$ denotes the modulo operation.
Sum of Ceilings not less than Ceiling of Sum
Let $\left \lceil {x} \right \rceil$ be the ceiling function. Then: :$\left \lceil {x} \right \rceil + \left \lceil {y} \right \rceil \ge \left \lceil {x + y} \right \rceil$ The equality holds: :$\left \lceil {x} \right \rceil + \left \lceil {y} \right \rceil = \left \lceil {x + y} \right \rceil$  either: :$x \in \Z$ or $y \in \Z$ or: :$x \bmod 1 + y \bmod 1 > 1$ where $x \bmod 1$ denotes the modulo operation.
Integer to Power of p-1 over 2 Modulo p
Let $a \in \Z$. Let $p$ be an odd prime. Let $b = a^{\frac {\paren {p - 1} } 2}$. Then one of the following cases holds: :$b \bmod p = 0$ which happens exactly when $a \equiv 0 \pmod p$, or: :$b \bmod p = 1$ or: :$b \bmod p = p - 1$ where: :$b \bmod p$ denotes the modulo operation :$x \equiv y \pmod p$ denotes that $x$ is congruent modulo $p$ to $y$.
Image Filter is Filter
Let $X, Y$ be sets. Let $\mathcal P \left({X}\right)$ and $\mathcal P \left({Y}\right)$ be the power sets of $X$ and $Y$ respectively. Let $f: X \to Y$ be a mapping. Let $\mathcal F \subset \mathcal P \left({X}\right)$ be a filter on $X$. Then the image filter of $\mathcal F$ with respect to $f$: :$f \left[{\mathcal F}\right] := \left\{ {U \subseteq Y: f^{-1} \left[{U}\right] \in \mathcal F}\right\}$ is a filter on $Y$.