hcju/proofwikips huggingface dataset:
Singleton Classes are Equal iff Sets are Equal
Let $a$ and $b$ be sets. Let $\set a$ and $\set b$ denote the singleton classes of $a$ and $b$. Then: :$\set a = \set b \iff a = b$
Doubleton Class can be Formed from Two Sets
Let $V$ be a basic universe. Let $a, b \in V$ be sets. Then the doubleton class $\set {a, b}$ can be formed, which is a subclass of $V$.
Doubleton Class of Equal Sets is Singleton Class
Let $V$ be a basic universe. Let $a, b \in V$ be sets. Consider the doubleton class $\set {a, b}$. Let $a = b$. Then: :$\set {a, b} = \set a$ where $\set a$ denotes the singleton class of $a$.
Equivalence of Definitions of Axiom of Pairing
The following formulations of the '''axiom of pairing''' in the context of '''axiomatic set theory''' are equivalent:
Singleton Class of Set is Set
Let $x$ be a set. Then the singleton class $\set x$ is likewise a set.
Basic Universe has Infinite Number of Elements
Let $V$ be a basic universe. Then $V$ has an infinite number of elements.
Equivalence of Definitions of Axiom of Pairing for Classes
The following formulations of the '''axiom of pairing''' in the context of '''class theory''' are equivalent:
Equality of Ordered Pairs/Lemma
Let $\set {a, b}$ and $\set {a, d}$ be doubletons such that $\set {a, b} = \set {a, d}$. Then: :$b = d$
Equality of Ordered Pairs/Necessary Condition
Let $\tuple {a, b}$ and $\tuple {c, d}$ be ordered pairs such that $\tuple {a, b} = \tuple {c, d}$. Then $a = c$ and $b = d$.
Equality of Ordered Pairs/Sufficient Condition
Let $\tuple {a, b}$ and $\tuple {c, d}$ be ordered pairs. Let $a = c$ and $b = d$. Then: :$\tuple {a, b} = \tuple {c, d}$
Intersection of Class Exists and is Unique
Let $V$ be a basic universe. Let $A \subseteq V$ be a class. Let $\displaystyle \bigcap A$ denote the intersection of $A$. Then $\displaystyle \bigcap A$ is guaranteed to exist and is unique.
Intersection of Non-Empty Class is Set
Let $V$ be a basic universe. Let $A \subseteq V$ be a non-empty class. Let $\displaystyle \bigcap A$ denote the intersection of $A$. Then $\displaystyle \bigcap A$ is a set.
Intersection of Empty Set/Class Theory
Let $V$ be a basic universe. Let $\O$ denote the empty class. Then the intersection of $\O$ is $V$: :$\displaystyle \bigcap \O = V$
Union of Subclass is Subset of Union of Class
Let $A$ and $B$ be classes. Let $\displaystyle \bigcup A$ and $\displaystyle \bigcup B$ denote the union of $A$ and union of $B$ respectively. Let $A$ be a subclass of $B$: :$A \subseteq B$ Then $\displaystyle \bigcup A$ is a subset of $\displaystyle \bigcup B$: :$\displaystyle \bigcup A \subseteq \displaystyle \bigcup B$
Union of Transitive Class is Subset
Let $A$ be a transitive class. Let $\displaystyle \bigcup A$ denote the union of $A$. Then: :$\displaystyle \bigcup A \subseteq A$
Union of Class is Subset implies Class is Transitive
Let $A$ be a class. Let $\displaystyle \bigcup A$ denote the union of $A$. Let: :$\displaystyle \bigcup A \subseteq A$ Then $A$ is transitive.
Union of Transitive Class is Transitive
Let $A$ be a class. Let $\displaystyle \bigcup A$ denote the union of $A$. Let $A$ be transitive. Then $\displaystyle \bigcup A$ is also transitive.
Union of Class is Transitive if Every Element is Transitive
Let $A$ be a class. Let $\bigcup A$ denote the union of $A$. Let $A$ be such that every element of $A$ is transitive. Then $\bigcup A$ is also transitive.
Set Difference with Set Difference is Union of Set Difference with Intersection/Corollary
:$T \setminus \paren {S \setminus T} = T$
Power Set Exists and is Unique
Let $V$ be a basic universe. Let $x \in V$ be a set. Let $\powerset x$ denote the power set of $x$. Then $\powerset x$ is guaranteed to exist and is unique.
Element of Class is Subset of Union of Class
Let $A$ be a class. Let $\displaystyle \bigcup A$ denote the union of $A$. Let $x \in A$. Then: :$x \subseteq \displaystyle \bigcup A$
Set equals Union of Power Set
Let $x$ be a set of sets. Let $\powerset x$ denote the power set of $x$. Let $\displaystyle \map \bigcup {\powerset x}$ denote the union of $\powerset x$. Then: :$x = \displaystyle \map \bigcup {\powerset x}$
Cartesian Product Exists and is Unique
Let $A$ and $B$ be classes. Let $A \times B$ be the '''cartesian product''' of $A$ and $B$. Then $A \times B$ exists and is unique.
Cartesian Product of Sets is Set
Let $V$ be a basic universe. Let $A$ and $B$ be sets in $V$. Then $A \times B$ is also a set.
Real Function with Positive Derivative is Increasing
If $\forall x \in \openint a b: \map {f'} x \ge 0$, then $f$ is increasing on $\closedint a b$.
Real Function with Strictly Positive Derivative is Strictly Increasing
If $\forall x \in \openint a b: \map {f'} x > 0$, then $f$ is strictly increasing on $\closedint a b$.
Continuous Real Function Differentiable on Borel Set
Let $\map \BB {\R, \size {\, \cdot \,} }$ be the Borel Sigma-Algebra on $\R$ with the usual topology. Let $f: \R \to \R$ be a continuous real function. Let $\map D f$ be the set of all points at which $f$ is differentiable. Then $\map D f$ is a Borel Set with respect to $\map \BB {\R, \size {\, \cdot \,} }$.
Limit with Rational Epsilon and Delta
Let $\openint a b$ be an open real interval. Let $c \in \openint a b$. Let $f: \openint a b \setminus \set c \to \R$ be a real function. Let $L \in \R$. Suppose that: :$\forall \epsilon > 0 \in \Q_{>0}: \exists \delta \in \Q_{>0}: \forall x \in \R: 0 < \size {x - c} < \delta \implies \size {\map f x - L} < \epsilon$ Then the limit of $f$ exists as $x$ tends to $c$, and is equal to $L$.
Chain Rule for Probability
Let $\EE$ be an experiment with probability space $\struct {\Omega, \Sigma, \Pr}$. Let $A, B \in \Sigma$ be events of $\EE$. The '''conditional probability of $A$ given $B$''' is: :$\map \Pr {A \mid B} = \dfrac {\map \Pr {A \cap B} } {\map \Pr B}$
Limit with Epsilon Powers of 2
Let $\openint a b$ be an open real interval. Let $c \in \openint a b$. Let $f: \openint a b \setminus \set c \to \R$ be a real function. Let $L \in \R$. Suppose that: :$\forall n > 0 \in \N: \exists \delta \in \R_{>0}: \forall x \in \R: 0 < \size {x - c} < \delta \implies \size {\map f x - L} < 2^{-n} $ Then the limit of $f$ exists as $x$ tends to $c$, and is equal to $L$.
Characterization of Probability Density Function
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space. Let $X: \Omega \to \R$ be a continuous random variable on $\struct {\Omega, \Sigma, \Pr}$. Let $\Omega_X = \Img X$, the image of $X$. Let the '''probability density function''' of $X$ is the mapping $f_X: \R \to \closedint 0 1$ be defined as: :$\forall x \in \R: \map {f_X} x = \begin {cases} \displaystyle \lim_{\epsilon \mathop \to 0^+} \frac {\map \Pr {x - \frac \epsilon 2 \le X \le x + \frac \epsilon 2} } \epsilon & : x \in \Omega_X \\ 0 & : x \notin \Omega_X \end {cases}$ Suppose that the cumulative distribution function of $X$ defines a continuously differentiable real function $F_X: x \mapsto \map \Pr {X \le x}$. Then the '''probability density function''' of $X$ is the mapping $f_X: \R \to \closedint 0 1$ satisfies: :$\dfrac{\d}{\d x} \map{F_X}{x} = \map {f_X} x$.
Abel's Test
Let $\displaystyle \sum a_n$ be a convergent real series. Let $\sequence {b_n}$ be a decreasing sequence of positive real numbers. Then the series $\displaystyle \sum a_n b_n$ is also convergent.
Equivalence of Definitions of Topology Generated by Synthetic Basis/Definition 1 iff Definition 2
Let $S$ be a set. Let $\BB$ be a synthetic basis on $S$. Let $\tau$ be the topology on $S$ generated by the synthetic basis $\mathcal B$: :$\tau = \set{\bigcup \AA: \AA \subseteq \BB}$ Then: :$\forall U \subseteq S: U \in \tau \iff U = \bigcup \set {B \in \BB: B \subseteq U}$
Equivalence of Definitions of Topology Generated by Synthetic Basis/Definition 1 iff Definition 3
Let $S$ be a set. Let $\mathcal B$ be a synthetic basis on $S$. Let $\tau$ be the topology on $S$ generated by the synthetic basis $\mathcal B$: :$\tau = \left\{{\bigcup \mathcal A: \mathcal A \subseteq \mathcal B}\right\}$ Then: :$\forall U \subseteq S: U \in \tau \iff \forall x \in U: \exists B \in \mathcal B: x \in B \subseteq U$
Projection from Product Topology is Open and Continuous
Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be topological spaces. Let $T = \struct {T_1 \times T_2, \tau}$ be the product space of $T_1$ and $T_2$, where $\tau$ is the Tychonoff topology on $S$. Let $\pr_1: T \to T_1$ and $\pr_2: T \to T_2$ be the first and second projections from $T$ onto its factors. Then both $\pr_1$ and $\pr_2$ are open and continuous.
Projection from Product Topology is Open and Continuous/General Result
Let $\family {T_i}_{i \mathop \in I} = \family {\struct{S_i, \tau_i}}_{i \mathop \in I}$ be an indexed family of topological spaces where $I$ is an arbitrary index set. Let $\displaystyle S = \prod_{i \mathop \in I} S_i$ be the corresponding product space. Let $\tau$ denote the Tychonoff topology on $S$. Let $\pr_i: S \to S_i$ be the corresponding projection from $S$ onto $S_i$. Then $\pr_i$ is open and continuous for all $i \in I$.
Projection from Product Topology is Continuous/General Result
Let $\family {T_i}_{i \mathop \in I} = \family {\struct{S_i, \tau_i}}_{i \mathop \in I}$ be an indexed family of topological spaces where $I$ is an arbitrary index set. Let $\displaystyle S = \prod_{i \mathop \in I} S_i$ be the corresponding product space. Let $\tau$ denote the Tychonoff topology on $S$. Let $\pr_i: S \to S_i$ be the corresponding projection from $S$ onto $S_i$. Then $\pr_i$ is continuous for all $i \in I$.
Projection from Product Topology is Open/General Result
Let $\family {T_i}_{i \mathop \in I} = \family {\struct {S_i, \tau_i} }_{i \mathop \in I}$ be an indexed family of topological spaces where $I$ is an arbitrary index set. Let $\displaystyle S = \prod_{i \mathop \in I} S_i$ be the corresponding product space. Let $\tau$ denote the Tychonoff topology on $S$. Let $\pr_i: S \to S_i$ be the corresponding projection from $S$ onto $S_i$. Then $\pr_i$ is open for all $i \in I$.
Product Space Basis Induced from Factor Space Bases
Let $\family {\struct{S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be an indexed family of topological spaces for $\alpha$ in some indexing set $I$. Let $\BB_\alpha$ be a basis for the topology $\tau_\alpha$ for each $\alpha \in I$. Let $\struct {S, \tau} = \displaystyle \prod_{\alpha \mathop \in I} \struct {S_\alpha, \tau_\alpha}$ be the product space of $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$. Let $\BB$ be the set of all cartesian products of the form $\displaystyle \prod_{\alpha \mathop \in I} U_\alpha$ where: :for all but finitely many indices $\alpha : U_\alpha = S_\alpha$ :for all $\alpha \in I : U_\alpha \ne S_\alpha \implies U_\alpha \in \BB_\alpha$ Then $\BB$ is a basis for the topology on the product space $\struct{S, \tau}$.
Existence and Uniqueness of Domain of Relation
Let $V$ be a basic universe. Let $\RR \subseteq V \times V$ be a relation. Then the domain $\Dom \RR$ of $\RR$ exists and is unique.
Existence and Uniqueness of Image of Relation
Let $V$ be a basic universe. Let $\RR \subseteq V \times V$ be a relation. Then the image $\Img \RR$ of $\RR$ exists and is unique.
Union of Union of Relation is Union of Domain with Image
Let $V$ be a basic universe. Let $\RR \subseteq V \times V$ be a relation. Let $\Dom \RR$ denote the domain of $\RR$. Then: :$\map \bigcup {\bigcup \RR} = \Dom \RR \cup \Img \RR$ where: :$\bigcup \RR$ denotes the union of $\RR$ :$\Dom \RR$ denotes the domain of $\RR$ :$\Img \RR$ denotes the image of $\RR$.
Domain of Relation is Subset of Union of Union of Relation
Let $V$ be a basic universe. Let $\RR \subseteq V \times V$ be a relation. Let $\Dom \RR$ denote the domain of $\RR$. Then: :$\Dom \RR \subseteq \map \bigcup {\bigcup \RR}$ where $\bigcup \RR$ denotes the union of $\RR$.
Image of Relation is Subset of Union of Union of Relation
Let $V$ be a basic universe. Let $\RR \subseteq V \times V$ be a relation. Let $\Img \RR$ denote the image of $\RR$. Then: :$\Img \RR \subseteq \map \bigcup {\bigcup \RR}$ where $\bigcup \RR$ denotes the union of $\RR$.
Relation is Set implies Domain and Image are Sets
Let $V$ be a basic universe. Let $\RR \subseteq V \times V$ be a relation. Let $\RR$ be a set. Then $\Dom \RR$ and $\Img \RR$ are also sets.
Definition:Mapping/Class Theory
Let $V$ be a basic universe. A '''mapping''' $f$ in the context of Class Theory is a relation such that: :$f \subseteq V \times V$: :$\forall x \in \Dom f: \exists! y \in \Img f: \tuple {x, y} \in f$ That is, for every $x$ in the domain of $f$, there exists exactly one $y$ in the image of $f$ such that $\tuple {x, y} \in f$.
Set is Transitive iff Subset of Power Set
A set $S$ is transitive : :$S \subseteq \powerset S$ where $\powerset S$ denotes the power set of $S$.
Universal Class less Set is not Transitive
Let $V$ be a basic universe. Let $a \in V$ be a set. Then: :$V \setminus \set a$ is not a transitive class where $\setminus$ denotes class difference.
Equivalence of Formulations of Axiom of Infinity for Zermelo Universe
The following formulations of the '''Axiom of Infinity''' in the context of '''class theory''' are equivalent:
Inductive Construction of Natural Numbers fulfils Peano's Axioms
Let $P$ denote the set of natural numbers by definition as an inductive set. Then $P$ fulfils Peano's axioms.
Natural Number is Transitive Set
Let $n$ be a natural number. Then $n$ is a transitive set.
Natural Number is Ordinary Set
Let $n$ be a natural number. Then $n$ is an ordinary set.
Product Space Local Basis Induced from Factor Spaces Local Bases
Let $\family {\struct{S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be an indexed family of topological spaces for $\alpha$ in some indexing set $I$. Let $\struct {S, \tau} = \displaystyle \prod_{\alpha \mathop \in I} \struct {S_\alpha, \tau_\alpha}$ be the product space of $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$. Let $x = \family{x_\alpha} \in S$. Let $\BB_\alpha$ be a local basis for $x_\alpha$ in the topological space $\struct {S_\alpha, \tau_\alpha}$ for each $\alpha \in I$. Let $\BB_x$ be the set of all cartesian products of the form $\displaystyle \prod_{\alpha \mathop \in I} U_\alpha$ where: :for all but finitely many indices $\alpha : U_\alpha = S_\alpha$ :for all $\alpha \in I : U_\alpha \ne S_\alpha \implies U_\alpha \in \BB_\alpha$ Then $\BB_x$ is a local basis for $x$ in the product space $\struct{S, \tau}$.
Natural Numbers cannot be Elements of Each Other
Let $m$ and $n$ be natural numbers. Then it cannot be the case that both $m \in n$ and $n \in m$.
Inductive Construction of Natural Numbers fulfils Peano's Axiom of Injectivity
Let $P$ denote the set of natural numbers by definition as an inductive set. Then $P$ fulfils: :{{PeanoAxiom|3}} where $s$ denotes the successor mapping.
Element of Natural Number is Natural Number
Let $n$ be a natural number. Let $m \in n$. Then $m$ is also a natural number.
Natural Number is Superset of its Union
Let $n \in \N$ be a natural number as defined by the von Neumann construction. Then: :$\bigcup n \subseteq n$
Subspace of Product Space Homeomorphic to Factor Space/Product with Singleton
Let $T_1$ and $T_2$ be non-empty topological spaces. Let $b \in T_2$. Let $T_1 \times T_2$ be the product space of $T_1$ and $T_2$. Let $T_2 \times T_1$ be the product space of $T_2$ and $T_1$. Then: :$T_1$ is homeomorphic to the subspace $T_1 \times \set b$ of $T_1 \times T_2$ :$T_1$ is homeomorphic to the subspace $\set b \times T_1$ of $T_2 \times T_1$
Set of Natural Numbers Equals its Union
Let $\omega$ denote the set of natural numbers as defined by the von Neumann construction on a Zermelo universe $V$. Then: :$\bigcup \omega = \omega$
Set of Natural Numbers Equals Union of its Successor
Let $\omega$ denote the set of natural numbers as defined by the von Neumann construction on a Zermelo universe $V$. Then: :$\bigcup \omega^+ = \omega$
Equivalence of Definitions of Minimally Inductive Class
Let $A$ be a class. Let $g$ be a mapping on $A$. {{TFAE|def = Minimally Inductive Class under General Mapping|view = minimally inductive class under $g$}}
Principle of General Induction
Let $M$ be a class. Let $g: M \to M$ be a mapping on $M$. Let $M$ be minimally inductive under $g$. Let $P: M \to \set {\T, \F}$ be a propositional function on $M$. Suppose that: :$(1): \quad \map P \O = \T$ :$(2): \quad \forall x \in M: \map P x = \T \implies \map P {\map g x} = \T$ Then: :$\forall x \in M: \map P x = \T$
Von Neumann Construction of Natural Numbers is Minimally Inductive
Let $\omega$ denote the set of natural numbers as defined by the von Neumann construction. $\omega$ is a minimally inductive class under the successor mapping.
Cartesian Product of Subsets/Family of Subsets
Let $\family {S_i}_{i \mathop \in I}$ be a family of sets where $I$ is an arbitrary index set. Let $S = \displaystyle \prod_{i \mathop \in I} S_i$ be the Cartesian product of $\family {S_i}_{i \mathop \in I}$. Let $\family {T_i}_{i \mathop \in I}$ be a family of sets. Let $T = \displaystyle \prod_{i \mathop \in I} T_i$ be the Cartesian product of $\family {T_i}_{i \mathop \in I}$. Then: :$\paren{\forall i \in I: T_i \subseteq S_i} \implies T \subseteq S$.
Double Induction Principle
Let $M$ be a class. Let $g: M \to M$ be a mapping on $M$. Let $M$ be a minimally inductive class under $g$. Let $\RR$ be a relation on $M$ which satisfies: {{begin-axiom}} {{axiom | n = \text D_1         | q = \forall x \in M         | m = \map \RR {x, \O} }} {{axiom | n = \text D_2         | q = \forall x, y \in M         | m = \map \RR {x, y} \land \map \RR {y, x} \implies \map \RR {x, \map g y} }} {{end-axiom}} Then $\map \RR {x, y}$ holds for all $x, y \in M$.
Double Induction Principle/Lemma
Let $x$ be a right normal element of $M$ with respect to $\RR$. Then $x$ is also a left normal element of $M$ with respect to $\RR$.
Cartesian Product of Subsets/Family of Nonempty Subsets
Let $T_i \ne \O$ for all $i \in I$. Then: :$T \subseteq S \iff \forall i \in I: T_i \subseteq S_i$.
Projection is Injection iff Factor is Singleton/Family of Sets
Let $\family {S_i}_{i \mathop \in I}$ be a non-empty family of non-empty sets where $I$ is an arbitrary index set. Let $S = \displaystyle \prod_{i \mathop \in I} S_i$ be the Cartesian product of $\family {S_i}_{i \mathop \in I}$. Let $\pr_j: S \to S_j$ be the $j$th projection on $S$. Then $\pr_j$ is an injection  $S_i$ is a singleton for all $i \in I \setminus \set j$.
Progressing Function Lemma
Let $A$ be a class. Let $g$ be a progressing mapping on $A$. Let $\RR$ be the relation defined as: :$\map \RR {x, y} \iff \map g x \subseteq y \lor y \subseteq x$ where $\lor$ denotes disjunction (inclusive "or"). Then: {{begin-axiom}} {{axiom | n = 1         | q = \forall y \in \Dom g         | ml= \map \RR {y, \O} }} {{axiom | n = 2         | q = \forall x, y \in \Dom g         | ml= \map \RR {x, y} \land \map \RR {y, x}         | mo= \implies         | mr= \map \RR {x, \map g y} }} {{end-axiom}} where $\land$ denotes conjunction ("and").
Minimally Inductive Class under Progressing Mapping induces Nest
Let $M$ be a class which is minimally inductive under a progressing mapping $g$. Then $M$ is a nest in which: :$\forall x, y \in M: \map g x \subseteq y \lor y \subseteq x$
Sandwich Principle
Let $A$ be a class. Let $g: A \to A$ be a mapping on $A$ such that: :for all $x, y \in A$, either $\map g x \subseteq y$ or $y \subseteq x$. Then: :$\forall x, y \in A: x \subseteq y \subseteq \map g x \implies x = y \lor y = \map g x$ That is, there is no element $y$ of $A$ such that: :$x \subset y \subset \map g x$ where $\subset$ denotes a proper subset.
Sandwich Principle/Corollary 1
Let: :$x \subset y$ where $\subset$ denotes a proper subset. Then: :$\map g x \subseteq y$
Sandwich Principle/Corollary 2
Let $g$ be a progressing mapping. Let $x \subseteq y$. Then: :$\map g x \subseteq \map g y$
Class under Progressing Mapping such that Elements are Sandwiched is Nest
Let $A$ be a class. Let $g: A \to A$ be a progressing mapping on $A$ such that: :$\forall x, y \in A: \map g x \subseteq y \lor y \subseteq x$ Then $A$ is a nest: :$\forall x, y \in A: x \subseteq y \lor y \subseteq x$
Characteristics of Minimally Inductive Class under Progressing Mapping
Let $M$ be a class which is minimally inductive under a progressing mapping $g$. Then for all $x, y \in M$:
Characteristics of Minimally Inductive Class under Progressing Mapping/Sandwich Principle
:$x \subseteq y \subseteq \map g x \implies x = y \lor y = \map g x$
Characteristics of Minimally Inductive Class under Progressing Mapping/Image of Proper Subset is Subset
:$x \subset y \implies \map g x \subseteq y$
Characteristics of Minimally Inductive Class under Progressing Mapping/Mapping Preserves Subsets
:$x \subseteq y \implies \map g x \subseteq \map g y$
Non-Empty Bounded Subset of Minimally Inductive Class under Progressing Mapping has Greatest Element
Let $M$ be a class which is minimally inductive under a progressing mapping $g$. Then every non-empty bounded subset of $M$ has a greatest element.
Projection is Injection iff Factor is Singleton/Family of Sets/Necessary Condition
Let $\family {S_i}_{i \mathop \in I}$ be a non-empty family of non-empty sets where $I$ is an arbitrary index set. Let $S = \displaystyle \prod_{i \mathop \in I} S_i$ be the Cartesian product of $\family {S_i}_{i \mathop \in I}$. Let $\pr_j: S \to S_j$ be the $j$th projection on $S$. Let $\pr_j$ be an injection. Then $S_i$ is a singleton for all $i \in I \setminus \set j$.
Projection is Injection iff Factor is Singleton/Family of Sets/Sufficient Condition
Let $\family {S_i}_{i \mathop \in I}$ be a non-empty family of non-empty sets where $I$ is an arbitrary index set. Let $S = \displaystyle \prod_{i \mathop \in I} S_i$ be the Cartesian product of $\family {S_i}_{i \mathop \in I}$. Let $\pr_j: S \to S_j$ be the $j$th projection on $S$. Let $S_i$ be a singleton for all $i \in I \setminus \set {j}$. Then $\pr_j$ is an injection.
Product Space of Subspaces is Subspace of Product Space
Let $\family {\struct {X_i, \tau_i} }_{i \mathop \in I}$ be a family of topological spaces where $I$ is an arbitrary index set. Let $\displaystyle T = \struct {X, \tau} = \prod_{i \mathop \in I} \struct {X_i, \tau_i}$ be the product space of $\family {\struct {X_i, \tau_i} }_{i \mathop \in I}$. Let $\family {\struct {Y_i, \upsilon_i} }_{i \mathop \in I}$ be a family of topological spaces such that: :$\forall i \in I : \struct {Y_i, \upsilon_i}$ is a topological subspace of $\struct {X_i, \tau_i}$ Let $\displaystyle S = \struct {Y, \upsilon} = \prod_{i \mathop \in I} \struct {Y_i, \upsilon_i}$ be the product space of $\family {\struct {Y_i, \upsilon_i} }_{i \mathop \in I}$. Let $T_Y = \struct {Y, \tau_Y}$ be the topological subspace of $T$. Then $S = T_Y$.
Difference of Complex Conjugates
Let $z_1, z_2 \in \C$ be complex numbers. Let $\overline z$ denote the complex conjugate of the complex number $z$. Then: :$\overline {z_1 - z_2} = \overline {z_1} - \overline {z_2}$
Nonzero natural number is another natural number successor
Let $\N$ be the 0-based natural numbers: :$\N = \left\{{0, 1, 2, \ldots}\right\}$ Let $s: \N \to \N: \map s n = n + 1$ be the successor function. Then: :$\forall n \in \N \setminus \set 0 \paren {\exists m \in \N: \map s m = n }$
Double Angle Formulas/Hyperbolic Functions
=== Double Angle Formula for Hyperbolic Sine === {{:Double Angle Formulas/Hyperbolic Sine}} === Double Angle Formula for Hyperbolic Cosine === {{:Double Angle Formulas/Hyperbolic Cosine}} === Double Angle Formula for Hyperbolic Tangent === {{:Double Angle Formulas/Hyperbolic Tangent}} where $\sinh, \cosh, \tanh$ denote hyperbolic sine, hyperbolic cosine and hyperbolic tangent respectively.
Subspace of Product Space Homeomorphic to Factor Space/Proof 1/Lemma 2
:$\pr_i {\restriction_{Y_i} } = p_i$
Subspace of Product Space Homeomorphic to Factor Space/Proof 1/Lemma 1
:$Y_i = \prod_{j \mathop \in I} Z_j$
Fixed Point of Progressing Mapping on Minimally Inductive Class is Greatest Element
Let $M$ be a class which is minimally inductive under a progressing mapping $g$. Let $x$ be a fixed point of $g$. Then $x$ is the greatest element of $M$.