TOC of Theorem List - Metamath Proof Explorer

Table of Contents Summary

PART 1  CLASSICAL FIRST-ORDER LOGIC WITH EQUALITY
      1.1  Pre-logic
      1.2  Propositional calculus
      1.3  Other axiomatizations related to classical propositional calculus
      1.4  Predicate calculus with equality: Tarski's system S2 (1 rule, 6 schemes)
      1.5  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes)
      1.6  Uniqueness and unique existence
      1.7  Other axiomatizations related to classical predicate calculus
PART 2  ZF (ZERMELO-FRAENKEL) SET THEORY
      2.1  ZF Set Theory - start with the Axiom of Extensionality
      2.2  ZF Set Theory - add the Axiom of Replacement
      2.3  ZF Set Theory - add the Axiom of Power Sets
      2.4  ZF Set Theory - add the Axiom of Union
      2.5  ZF Set Theory - add the Axiom of Regularity
      2.6  ZF Set Theory - add the Axiom of Infinity
PART 3  ZFC (ZERMELO-FRAENKEL WITH CHOICE) SET THEORY
      3.1  ZFC Set Theory - add Countable Choice and Dependent Choice
      3.2  ZFC Set Theory - add the Axiom of Choice
      3.3  ZFC Axioms with no distinct variable requirements
      3.4  The Generalized Continuum Hypothesis
PART 4  TG (TARSKI-GROTHENDIECK) SET THEORY
      4.1  Inaccessibles
      4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
PART 5  REAL AND COMPLEX NUMBERS
      5.1  Construction and axiomatization of real and complex numbers
      5.2  Derive the basic properties from the field axioms
      5.3  Real and complex numbers - basic operations
      5.4  Integer sets
      5.5  Order sets
      5.6  Elementary integer functions
      5.7  Words over a set
      5.8  Reflexive and transitive closures of relations
      5.9  Elementary real and complex functions
      5.10  Elementary limits and convergence
      5.11  Elementary trigonometry
      5.12  Cardinality of real and complex number subsets
PART 6  ELEMENTARY NUMBER THEORY
      6.1  Elementary properties of divisibility
      6.2  Elementary prime number theory
PART 7  BASIC STRUCTURES
      7.1  Extensible structures
      7.2  Moore spaces
PART 8  BASIC CATEGORY THEORY
      8.1  Categories
      8.2  Arrows (disjointified hom-sets)
      8.3  Examples of categories
      8.4  Categorical constructions
PART 9  BASIC ORDER THEORY
      9.1  Dual of an order structure
      9.2  Preordered sets and directed sets
      9.3  Partially ordered sets (posets)
      9.4  Totally ordered sets (tosets)
      9.5  Lattices
      9.6  Posets, directed sets, and lattices as relations
      9.7  Chains
PART 10  BASIC ALGEBRAIC STRUCTURES
      10.1  Monoids
      10.2  Groups
      10.3  Rings
      10.4  Division rings and fields
      10.5  Left modules
      10.6  Vector spaces
      10.7  Subring algebras and ideals
      10.8  The complex numbers as an algebraic extensible structure
      10.9  Generalized pre-Hilbert and Hilbert spaces
PART 11  BASIC LINEAR ALGEBRA
      11.1  Vectors and free modules
      11.2  Associative algebras
      11.3  Abstract multivariate polynomials
      11.4  Matrices
      11.5  The determinant
      11.6  Polynomial matrices
      11.7  The characteristic polynomial
PART 12  BASIC TOPOLOGY
      12.1  Topology
      12.2  Filters and filter bases
      12.3  Uniform Structures and Spaces
      12.4  Metric spaces
      12.5  Metric subcomplex vector spaces
PART 13  BASIC REAL AND COMPLEX ANALYSIS
      13.1  Continuity
      13.2  Integrals
      13.3  Derivatives
PART 14  BASIC REAL AND COMPLEX FUNCTIONS
      14.1  Polynomials
      14.2  Sequences and series
      14.3  Basic trigonometry
      14.4  Basic number theory
PART 15  SURREAL NUMBERS
      15.1  Sign sequence representation and Alling's axioms
      15.2  Initial consequences of Alling's axioms
      15.3  Conway cut representation
      15.4  Induction and recursion
      15.5  Surreal arithmetic
      15.6  Subsystems of surreals
PART 16  ELEMENTARY GEOMETRY
      16.1  Definition and Tarski's Axioms of Geometry
      16.2  Tarskian Geometry
      16.3  Properties of geometries
      16.4  Geometry in Hilbert spaces
PART 17  GRAPH THEORY
      17.1  Vertices and edges
      17.2  Undirected graphs
      17.3  Walks, paths and cycles
      17.4  Eulerian paths and the Konigsberg Bridge problem
      17.5  The Friendship Theorem
PART 18  GUIDES AND MISCELLANEA
      18.1  Guides (conventions, explanations, and examples)
      18.2  Humor
      18.3  (Future - to be reviewed and classified)
PART 19  COMPLEX TOPOLOGICAL VECTOR SPACES (DEPRECATED)
      19.1  Additional material on group theory (deprecated)
      19.2  Complex vector spaces
      19.3  Normed complex vector spaces
      19.4  Operators on complex vector spaces
      19.5  Inner product (pre-Hilbert) spaces
      19.6  Complex Banach spaces
      19.7  Complex Hilbert spaces
PART 20  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      20.1  Axiomatization of complex pre-Hilbert spaces
      20.2  Inner product and norms
      20.3  Cauchy sequences and completeness axiom
      20.4  Subspaces and projections
      20.5  Properties of Hilbert subspaces
      20.6  Operators on Hilbert spaces
      20.7  States on a Hilbert lattice and Godowski's equation
      20.8  Cover relation, atoms, exchange axiom, and modular symmetry
PART 21  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      ​21.1  Mathboxes for user contributions
      21.2  Mathbox for Jorge Agra
      21.3  Mathbox for Stefan Allan
      21.4  Mathbox for Thierry Arnoux
      21.5  Mathbox for Jonathan Ben-Naim
      21.6  Mathbox for BTernaryTau
      21.7  Mathbox for Mario Carneiro
      21.8  Mathbox for Filip Cernatescu
      21.9  Mathbox for Paul Chapman
      21.10  Mathbox for Hongxiu Chen
      21.11  Mathbox for Adrian Ducourtial
      21.12  Mathbox for Scott Fenton
      21.13  Mathbox for Gino Giotto
      21.14  Mathbox for Jeff Hankins
      21.15  Mathbox for Anthony Hart
      21.16  Mathbox for Chen-Pang He
      21.17  Mathbox for Jeff Hoffman
      21.18  Mathbox for Matthew House
      21.19  Mathbox for Asger C. Ipsen
      21.20  Mathbox for BJ
      21.21  Mathbox for Jim Kingdon
      21.22  Mathbox for ML
      21.23  Mathbox for Wolf Lammen
      21.24  Mathbox for Brendan Leahy
      21.25  Mathbox for Thomas van Maaren
      21.26  Mathbox for Jeff Madsen
      21.27  Mathbox for Giovanni Mascellani
      21.28  Mathbox for Peter Mazsa
      21.29  Mathbox for Rodolfo Medina
      21.30  Mathbox for Norm Megill
      21.31  Mathbox for metakunt
      21.32  Mathbox for Luke Murphy
      21.33  Mathbox for Steven Nguyen
      21.34  Mathbox for Igor Ieskov
      21.35  Mathbox for Timothy J. O'Malley
      21.36  Mathbox for OpenAI
      21.37  Mathbox for Stefan O'Rear
      21.38  Mathbox for Noam Pasman
      21.39  Mathbox for Jon Pennant
      21.40  Mathbox for Richard Penner
      21.41  Mathbox for Stanislas Polu
      21.42  Mathbox for Rohan Ridenour
      21.43  Mathbox for Steve Rodriguez
      21.44  Mathbox for Andrew Salmon
      21.45  Mathbox for Alan Sare
      21.46  Mathbox for Eric Schmidt
      21.47  Mathbox for Glauco Siliprandi
      21.48  Mathbox for Saveliy Skresanov
      21.49  Mathbox for Ender Ting
      21.50  Mathbox for Jarvin Udandy
      21.51  Mathbox for Adhemar
      21.52  Mathbox for Alexander van der Vekens
      21.53  Mathbox for Zhi Wang
      21.54  Mathbox for Emmett Weisz
      21.55  Mathbox for David A. Wheeler
      21.56  Mathbox for Mingli Yuan
      21.57  Mathbox for Jiamin Zhao
      21.58  Mathbox for Kunhao Zheng

Detailed Table of Contents
(* means the section header has a description)

​​*PART 1  CLASSICAL FIRST-ORDER LOGIC WITH EQUALITY
      ​*1.1  Pre-logic
            *1.1.1  Inferences for assisting proof development   idi 1
      ​​*1.2  Propositional calculus
            1.2.1  Recursively define primitive wffs for propositional calculus   wn 3
            ​*1.2.2  The axioms of propositional calculus   ax-mp 5
            ​*1.2.3  Logical implication   mp2 9
            ​*1.2.4  Logical negation   con4 114
            ​*1.2.5  Logical equivalence   wb 209
            ​*1.2.6  Logical conjunction   wa 401
            ​*1.2.7  Logical disjunction   wo 861
            ​*1.2.8  Mixed connectives   jaao 969
            ​*1.2.9  The conditional operator for propositions   wif 1078
            ​*1.2.10  The weak deduction theorem for propositional calculus   elimh 1099
            1.2.11  Abbreviated conjunction and disjunction of three wff's   w3o 1102
            1.2.12  Logical "nand" (Sheffer stroke)   wnan 1521
            1.2.13  Logical "xor"   wxo 1541
            1.2.14  Logical "nor"   wnor 1558
            1.2.15  True and false constants   wal 1568
                  ​*1.2.15.1  Universal quantifier for use by df-tru   wal 1568
                  ​*1.2.15.2  Equality predicate for use by df-tru   cv 1569
                  1.2.15.3  The true constant   wtru 1571
                  1.2.15.4  The false constant   wfal 1582
            ​*1.2.16  Truth tables   truimtru 1593
                  1.2.16.1  Implication   truimtru 1593
                  1.2.16.2  Negation   nottru 1597
                  1.2.16.3  Equivalence   trubitru 1599
                  1.2.16.4  Conjunction   truantru 1603
                  1.2.16.5  Disjunction   truortru 1607
                  1.2.16.6  Alternative denial   trunantru 1611
                  1.2.16.7  Exclusive disjunction   truxortru 1615
                  1.2.16.8  Joint denial   trunortru 1619
            ​*1.2.17  Half adder and full adder in propositional calculus   whad 1623
                  1.2.17.1  Full adder: sum   whad 1623
                  1.2.17.2  Full adder: carry   wcad 1639
      ​1.3  Other axiomatizations related to classical propositional calculus
            ​*1.3.1  Minimal implicational calculus   minimp 1654
            ​*1.3.2  Implicational Calculus   impsingle 1660
            1.3.3  Derive the Lukasiewicz axioms from Meredith's sole axiom   meredith 1674
            1.3.4  Derive the standard axioms from the Lukasiewicz axioms   luklem1 1691
            ​*1.3.5  Derive Nicod's axiom from the standard axioms   nic-dfim 1702
            1.3.6  Derive the Lukasiewicz axioms from Nicod's axiom   nic-imp 1708
            1.3.7  Derive Nicod's Axiom from Lukasiewicz's First Sheffer Stroke Axiom   lukshef-ax1 1727
            1.3.8  Derive the Lukasiewicz Axioms from the Tarski-Bernays-Wajsberg Axioms   tbw-bijust 1731
            1.3.9  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's First CO Axiom   merco1 1746
            1.3.10  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's Second CO Axiom   merco2 1769
            1.3.11  Derive the Lukasiewicz axioms from the Russell-Bernays Axioms   rb-bijust 1782
            ​*1.3.12  Stoic logic non-modal portion (Chrysippus of Soli)   mptnan 1801
      ​​*1.4  Predicate calculus with equality: Tarski's system S2 (1 rule, 6 schemes)
            *1.4.1  Universal quantifier (continued); define "exists" and "not free"   wex 1812
                  1.4.1.1  Existential quantifier   wex 1812
                  1.4.1.2  Nonfreeness predicate   wnf 1816
            1.4.2  Rule scheme ax-gen (Generalization)   ax-gen 1828
            1.4.3  Axiom scheme ax-4 (Quantified Implication)   ax-4 1842
                  ​*1.4.3.1  The empty domain of discourse   empty 1941
            1.4.4  Axiom scheme ax-5 (Distinctness) - first use of $d   ax-5 1945
            ​*1.4.5  Equality predicate (continued)   weq 1997
            1.4.6  Axiom scheme ax-6 (Existence)   ax-6 2002
            1.4.7  Axiom scheme ax-7 (Equality)   ax-7 2043
            1.4.8  Define proper substitution   sbjust 2100
            1.4.9  Membership predicate   wcel 2147
            1.4.10  Axiom scheme ax-8 (Left Equality for Binary Predicate)   ax-8 2149
            1.4.11  Axiom scheme ax-9 (Right Equality for Binary Predicate)   ax-9 2157
            ​*1.4.12  Logical redundancy of ax-10 , ax-11 , ax-12 , ax-13   ax6dgen 2167
      ​​*1.5  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes)
            1.5.1  Axiom scheme ax-10 (Quantified Negation)   ax-10 2180
            1.5.2  Axiom scheme ax-11 (Quantifier Commutation)   ax-11 2197
            1.5.3  Axiom scheme ax-12 (Substitution)   ax-12 2216
            1.5.4  Axiom scheme ax-13 (Quantified Equality)   ax-13 2404
      ​1.6  Uniqueness and unique existence
            1.6.1  Uniqueness: the at-most-one quantifier   wmo 2565
            1.6.2  Unique existence: the unique existential quantifier   weu 2596
      ​1.7  Other axiomatizations related to classical predicate calculus
            ​*1.7.1  Aristotelian logic: Assertic syllogisms   barbara 2690
            ​*1.7.2  Intuitionistic logic   axia1 2720
​​*PART 2  ZF (ZERMELO-FRAENKEL) SET THEORY
      ​2.1  ZF Set Theory - start with the Axiom of Extensionality
            2.1.1  Introduce the Axiom of Extensionality   ax-ext 2735
            2.1.2  Classes   cab 2741
                  2.1.2.1  Class abstractions   cab 2741
                  ​*2.1.2.2  Class equality   df-cleq 2755
                  2.1.2.3  Class membership   df-clel 2839
                  2.1.2.4  Elementary properties of class abstractions   eqabdv 2897
            2.1.3  Class form not-free predicate   wnfc 2911
            2.1.4  Negated equality and membership   wne 2959
                  2.1.4.1  Negated equality   wne 2959
                  2.1.4.2  Negated membership   wnel 3065
            2.1.5  Restricted quantification   wral 3080
                  2.1.5.1  Restricted universal and existential quantification   wral 3080
                  2.1.5.2  Restricted existential uniqueness and at-most-one quantifier   wreu 3367
                  2.1.5.3  Restricted class abstraction   crab 3416
            2.1.6  The universal class   cvv 3454
            ​*2.1.7  Conditional equality (experimental)   wcdeq 3724
            2.1.8  Russell's Paradox   rru 3740
            2.1.9  Proper substitution of classes for sets   wsbc 3742
            2.1.10  Proper substitution of classes for sets into classes   csb 3851
            2.1.11  Subclasses and subsets, proper subclasses and subsets   wss 3900
                  2.1.11.1  Subclasses and subsets   wss 3900
                  2.1.11.2  Proper subclasses and subsets   wpss 4016
            2.1.12  Basic operations on classes   cdif 4048
                  2.1.12.1  The difference of two classes   cdif 4048
                  2.1.12.2  The intersection of two classes   cin 4092
                  2.1.12.3  The union of two classes   cun 4152
                  2.1.12.4  The symmetric difference of two classes   csymdif 4200
                  2.1.12.5  Combinations of difference, union, and intersection of two classes   unabs 4213
                  2.1.12.6  Class abstractions with difference, union, and intersection of two classes   unabw 4255
                  2.1.12.7  Restricted uniqueness with difference, union, and intersection   reuun2 4273
            2.1.13  The empty set   c0 4281
            ​*2.1.14  The conditional operator for classes   cif 4485
            ​*2.1.15  The weak deduction theorem for set theory   dedth 4544
            2.1.16  Power classes   cpw 4560
            2.1.17  Singletons, unordered pairs, unordered triples   snjust 4587
            2.1.18  Ordered pairs and triples   cop 4838
                  2.1.18.1  Ordered pairs   cop 4838
                  2.1.18.2  Ordered triples   cotp 4861
            2.1.19  The union of a class   cuni 4870
            2.1.20  The intersection of a class   cint 4910
            2.1.21  Indexed union and intersection   ciun 4954
            2.1.22  Disjointness   wdisj 5073
            2.1.23  Binary relations   wbr 5106
            2.1.24  Ordered-pair class abstractions (class builders)   copab 5170
            2.1.25  Functions in maps-to notation   cmpt 5189
            2.1.26  Transitive classes   wtr 5215
      ​2.2  ZF Set Theory - add the Axiom of Replacement
            2.2.1  Introduce the Axiom of Replacement   ax-rep 5235
            2.2.2  Derive the Axiom of Separation   axsepgfromrep 5250
            2.2.3  Derive the Null Set Axiom   axnulALT 5261
            2.2.4  Theorems requiring subset and intersection existence   exnelv 5270
            2.2.5  Theorems requiring empty set existence   class2set 5320
      ​2.3  ZF Set Theory - add the Axiom of Power Sets
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 5331
            2.3.2  Derive the Axiom of Pairing   axprlem1 5389
            2.3.3  Ordered pair theorem   opnz 5446
            2.3.4  Ordered-pair class abstractions (cont.)   opabidw 5502
            2.3.5  Power class of union and intersection   pwin 5546
            2.3.6  The identity relation   cid 5549
            2.3.7  The membership relation (or epsilon relation)   cep 5554
            ​*2.3.8  Partial and total orderings   wpo 5561
            2.3.9  Founded and well-ordering relations   wfr 5605
            2.3.10  Relations   cxp 5653
            2.3.11  The Predecessor Class   cpred 6321
            2.3.12  Well-founded induction (variant)   frpomin 6361
            2.3.13  Well-ordered induction   tz6.26 6368
            2.3.14  Ordinals   word 6379
            2.3.15  Definite description binder (inverted iota)   cio 6510
            2.3.16  Functions   wfun 6552
            2.3.17  Cantor's Theorem   canth 7397
            2.3.18  Restricted iota (description binder)   crio 7399
            2.3.19  Operations   co 7443
                  2.3.19.1  Variable-to-class conversion for operations   caovclg 7637
            2.3.20  Operations in maps-to notation   mpondm0 7685
            2.3.21  Functions with three arguments in maps-to notation   cmpt3 7707
            2.3.22  Function operation   cof 7715
            2.3.23  Proper subset relation   crpss 7762
      ​2.4  ZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 7775
            2.4.2  Ordinals (continued)   epweon 7813
            2.4.3  Transfinite induction   tfi 7888
            2.4.4  The natural numbers (i.e., finite ordinals)   com 7901
            2.4.5  Peano's postulates   peano1 7924
            2.4.6  Finite induction (for finite ordinals)   find 7931
            2.4.7  Relations and functions (cont.)   dmexg 7937
            2.4.8  First and second members of an ordered pair   c1st 8023
            2.4.9  Induction on Cartesian products   frpoins3xpg 8184
            2.4.10  Ordering on Cartesian products   xpord2lem 8186
            2.4.11  Ordering Ordinal Sequences   orderseqlem 8201
            ​*2.4.12  The support of functions   csupp 8204
            ​*2.4.13  Special maps-to operations   opeliunxp2f 8255
            2.4.14  Function transposition   ctpos 8270
            2.4.15  Curry and uncurry   ccur 8310
            2.4.16  Undefined values   cund 8317
            2.4.17  Well-founded recursion   cfrecs 8326
            2.4.18  Well-ordered recursion   cwrecs 8357
            2.4.19  Functions on ordinals; strictly monotone ordinal functions   iunon 8375
            2.4.20  "Strong" transfinite recursion   crecs 8406
            2.4.21  Recursive definition generator   crdg 8445
            2.4.22  Finite recursion   frfnom 8471
            2.4.23  Ordinal arithmetic   c1o 8497
            2.4.24  Natural number arithmetic   nna0 8641
            2.4.25  Natural addition   cnadd 8702
            2.4.26  Equivalence relations and classes   wer 8742
            2.4.27  The mapping operation   cmap 8875
            2.4.28  Infinite Cartesian products   cixp 8953
            2.4.29  Equinumerosity   cen 8998
            2.4.30  Schroeder-Bernstein Theorem   sbthlem1 9134
            2.4.31  Equinumerosity (cont.)   xpf1o 9186
            2.4.32  Finite sets   dif1enlem 9203
            2.4.33  Pigeonhole Principle   phplem1 9247
            2.4.34  Finite sets (cont.)   onomeneq 9257
            2.4.35  Finitely supported functions   cfsupp 9381
            2.4.36  Finite intersections   cfi 9429
            2.4.37  Hall's marriage theorem   marypha1lem 9452
            2.4.38  Supremum and infimum   csup 9459
            2.4.39  Ordinal isomorphism, Hartogs's theorem   coi 9530
            2.4.40  Hartogs function   char 9577
            2.4.41  Weak dominance   cwdom 9585
      ​2.5  ZF Set Theory - add the Axiom of Regularity
            2.5.1  Introduce the Axiom of Regularity   ax-reg 9613
            2.5.2  Axiom of Infinity equivalents   inf0 9649
      ​2.6  ZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-inf 9666
            2.6.2  Existence of omega (the set of natural numbers)   omex 9671
            2.6.3  Cantor normal form   ccnf 9689
            2.6.4  Transitive closure of a relation   cttrcl 9735
            2.6.5  Transitive closure   trcl 9756
            2.6.6  Set induction (or epsilon induction)   setind 9775
            2.6.7  Well-Founded Induction   frmin 9780
            2.6.8  Well-Founded Recursion   frr3g 9787
            2.6.9  Rank   cr1 9793
            2.6.10  Hereditarily finite sets   chf 9931
            2.6.11  Scott's trick; collection principle; Hilbert's epsilon   cscott 9955
            2.6.12  Set Recursion   csetrecs 9991
                  ​*2.6.12.1  Basic Properties of Set Recursion   csetrecs 9991
            2.6.13  Disjoint union   cdju 10006
            2.6.14  Cardinal numbers   ccrd 10043
            2.6.15  Axiom of Choice equivalents   wac 10221
            ​*2.6.16  Cardinal number arithmetic   undjudom 10273
            2.6.17  The Ackermann bijection   ackbij2lem1 10323
            2.6.18  Cofinality (without Axiom of Choice)   cflem 10350
            2.6.19  Eight inequivalent definitions of finite set   sornom 10382
            2.6.20  Hereditarily size-limited sets without Choice   itunifval 10521
​​*PART 3  ZFC (ZERMELO-FRAENKEL WITH CHOICE) SET THEORY
      ​3.1  ZFC Set Theory - add Countable Choice and Dependent Choice
            3.1.1  Introduce the Axiom of Countable Choice   ax-cc 10540
            3.1.2  Introduce the Axiom of Dependent Choice   ax-dc 10551
      ​3.2  ZFC Set Theory - add the Axiom of Choice
            3.2.1  Introduce the Axiom of Choice   ax-ac 10564
            3.2.2  AC equivalents: well-ordering, Zorn's lemma   numthcor 10599
            3.2.3  Cardinal number theorems using Axiom of Choice   cardval 10657
            3.2.4  Cardinal number arithmetic using Axiom of Choice   iunctb 10686
            3.2.5  Cofinality using the Axiom of Choice   alephreg 10694
      ​3.3  ZFC Axioms with no distinct variable requirements
      ​3.4  The Generalized Continuum Hypothesis
            3.4.1  Sets satisfying the Generalized Continuum Hypothesis   cgch 10732
            3.4.2  Derivation of the Axiom of Choice   gchaclem 10790
​​*PART 4  TG (TARSKI-GROTHENDIECK) SET THEORY
      ​4.1  Inaccessibles
            4.1.1  Weakly and strongly inaccessible cardinals   cwina 10794
            4.1.2  Weak universes   cwun 10812
            4.1.3  Tarski classes   ctsk 10860
            4.1.4  Grothendieck universes   cgru 10902
      ​4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
            4.2.1  Introduce the Tarski-Grothendieck Axiom   ax-groth 10935
            4.2.2  Derive the Power Set, Infinity and Choice Axioms   grothpw 10938
            4.2.3  Tarski map function   ctskm 10949
​​*PART 5  REAL AND COMPLEX NUMBERS
      ​5.1  Construction and axiomatization of real and complex numbers
            5.1.1  Dedekind-cut construction of real and complex numbers   cnpi 10956
            5.1.2  Final derivation of real and complex number postulates   axaddf 11257
            5.1.3  Real and complex number postulates restated as axioms   ax-cnex 11283
      ​5.2  Derive the basic properties from the field axioms
            5.2.1  Some deductions from the field axioms for complex numbers   cnex 11308
            5.2.2  Infinity and the extended real number system   cpnf 11367
            5.2.3  Restate the ordering postulates with extended real "less than"   axlttri 11408
            5.2.4  Ordering on reals   lttr 11413
            5.2.5  Initial properties of the complex numbers   mul12 11502
      ​5.3  Real and complex numbers - basic operations
            5.3.1  Addition   add12 11555
            5.3.2  Subtraction   cmin 11568
            5.3.3  Multiplication   kcnktkm1cn 11774
            5.3.4  Ordering on reals (cont.)   gt0ne0 11808
            5.3.5  Reciprocals   ixi 11972
            5.3.6  Division   cdiv 12000
            5.3.7  Ordering on reals (cont.)   elimgt0 12182
            5.3.8  Completeness Axiom and Suprema   fimaxre 12288
            5.3.9  Imaginary and complex number properties   neg1cn 12332
            5.3.10  Function operation analogue theorems   ofsubeq0 12344
            ​*5.3.11  Indicator Functions   cind 12347
      ​5.4  Integer sets
            5.4.1  Positive integers (as a subset of complex numbers)   cn 12362
            5.4.2  Principle of mathematical induction   nnind 12380
            ​*5.4.3  Decimal representation of numbers   c2 12424
            ​*5.4.4  Some properties of specific numbers   1pneg1e0 12487
            5.4.5  Simple number properties   halfcl 12599
            5.4.6  The Archimedean property   nnunb 12629
            5.4.7  Nonnegative integers (as a subset of complex numbers)   cn0 12633
            ​*5.4.8  Extended nonnegative integers   cxnn0 12706
            5.4.9  Integers (as a subset of complex numbers)   cz 12720
            5.4.10  Decimal arithmetic   cdc 12841
            5.4.11  Upper sets of integers   cuz 12992
            5.4.12  Well-ordering principle for bounded-below sets of integers   uzwo3 13097
            5.4.13  Rational numbers (as a subset of complex numbers)   cq 13102
            5.4.14  Existence of the set of complex numbers   rpnnen1lem2 13132
      ​5.5  Order sets
            5.5.1  Positive reals (as a subset of complex numbers)   crp 13147
            5.5.2  Infinity and the extended real number system (cont.)   cxne 13265
            5.5.3  Supremum and infimum on the extended reals   xrsupexmnf 13462
            5.5.4  Real number intervals   cioo 13503
            5.5.5  Finite intervals of integers   cfz 13666
            ​*5.5.6  Finite intervals of nonnegative integers   elfz2nn0 13779
            5.5.7  Half-open integer ranges   cfzo 13815
      ​5.6  Elementary integer functions
            5.6.1  The floor and ceiling functions   cfl 13957
            5.6.2  The modulo (remainder) operation   cmo 14036
            5.6.3  Miscellaneous theorems about integers   om2uz0i 14117
            5.6.4  Strong induction over upper sets of integers   uzsinds 14157
            5.6.5  Finitely supported functions over the nonnegative integers   fsuppmapnn0fiublem 14160
            5.6.6  The infinite sequence builder "seq" - extension   cseq 14171
            5.6.7  Integer powers   cexp 14231
            5.6.8  Ordered pair theorem for nonnegative integers   nn0le2msqi 14438
            5.6.9  Factorial function   cfa 14444
            5.6.10  The binomial coefficient operation   cbc 14473
            5.6.11  The ` # ` (set size) function   chash 14501
                  5.6.11.1  Proper unordered pairs and triples (sets of size 2 and 3)   hashprlei 14640
                  5.6.11.2  Functions with a domain containing at least two different elements   fundmge2nop0 14674
                  5.6.11.3  Finite induction on the size of the first component of a binary relation   hashdifsnp1 14678
      ​​*5.7  Words over a set
            5.7.1  Definitions and basic theorems   cword 14685
            5.7.2  Last symbol of a word   clsw 14734
            5.7.3  Concatenations of words   cconcat 14742
            5.7.4  Singleton words   cs1 14769
            5.7.5  Concatenations with singleton words   ccatws1cl 14792
            5.7.6  Subwords/substrings   csubstr 14816
            5.7.7  Prefixes of a word   cpfx 14848
            5.7.8  Subwords of subwords   swrdswrdlem 14881
            5.7.9  Subwords and concatenations   pfxcctswrd 14887
            5.7.10  Subwords of concatenations   swrdccatfn 14901
            5.7.11  Splicing words (substring replacement)   csplice 14926
            5.7.12  Reversing words   creverse 14935
            5.7.13  Repeated symbol words   creps 14947
            ​*5.7.14  Cyclical shifts of words   ccsh 14967
            5.7.15  Mapping words by a function   wrdco 15010
            5.7.16  Longer string literals   cs2 15020
      ​​*5.8  Reflexive and transitive closures of relations
            5.8.1  The reflexive and transitive properties of relations   coss12d 15154
            5.8.2  Basic properties of closures   cleq1lem 15164
            5.8.3  Definitions and basic properties of transitive closures   ctcl 15167
            5.8.4  Exponentiation of relations   crelexp 15201
            5.8.5  Reflexive-transitive closure as an indexed union   crtrcl 15237
            ​*5.8.6  Principle of transitive induction   relexpindlem 15245
      ​5.9  Elementary real and complex functions
            5.9.1  The "shift" operation   cshi 15248
            5.9.2  Signum (sgn or sign) function   csgn 15268
            5.9.3  Real and imaginary parts; conjugate   ccj 15292
            5.9.4  Square root; absolute value   csqrt 15429
      ​5.10  Elementary limits and convergence
            5.10.1  Superior limit (lim sup)   clsp 15666
            5.10.2  Limits   cli 15680
            5.10.3  Finite and infinite sums   csu 15882
            5.10.4  The binomial theorem   binomlem 16027
            5.10.5  The inclusion/exclusion principle   incexclem 16034
            5.10.6  Infinite sums (cont.)   isumshft 16037
            5.10.7  Miscellaneous converging and diverging sequences   divrcnv 16050
            5.10.8  Arithmetic series   arisum 16058
            5.10.9  Geometric series   expcnv 16062
            5.10.10  Ratio test for infinite series convergence   cvgrat 16081
            5.10.11  Mertens' theorem   mertenslem1 16082
            5.10.12  Finite and infinite products   prodf 16085
                  5.10.12.1  Product sequences   prodf 16085
                  5.10.12.2  Non-trivial convergence   ntrivcvg 16095
                  5.10.12.3  Complex products   cprod 16101
                  5.10.12.4  Finite products   fprod 16137
                  5.10.12.5  Infinite products   iprodclim 16194
            5.10.13  Falling and Rising Factorial   cfallfac 16200
            5.10.14  Bernoulli polynomials and sums of k-th powers   cbp 16241
      ​5.11  Elementary trigonometry
            5.11.1  The exponential, sine, and cosine functions   ce 16256
                  5.11.1.1  The circle constant (tau = 2 pi)   ctau 16399
            5.11.2  _e is irrational   eirrlem 16401
      ​5.12  Cardinality of real and complex number subsets
            5.12.1  Countability of integers and rationals   xpnnen 16408
            5.12.2  The reals are uncountable   rpnnen2lem1 16411
​​*PART 6  ELEMENTARY NUMBER THEORY
      ​6.1  Elementary properties of divisibility
            6.1.1  Irrationality of square root of 2   sqrt2irrlem 16445
            6.1.2  Some Number sets are chains of proper subsets   nthruc 16449
            6.1.3  The divides relation   cdvds 16451
            ​*6.1.4  Even and odd numbers   evenelz 16535
            6.1.5  The division algorithm   divalglem0 16592
            6.1.6  Bit sequences   cbits 16618
            6.1.7  The greatest common divisor operator   cgcd 16693
            6.1.8  Bézout's identity   bezoutlem1 16741
            6.1.9  Algorithms   nn0seqcvgd 16774
            6.1.10  Euclid's Algorithm   eucalgval2 16785
            ​*6.1.11  The least common multiple   clcm 16792
            ​*6.1.12  Coprimality and Euclid's lemma   coprmgcdb 16853
            6.1.13  Cancellability of congruences   congr 16868
      ​6.2  Elementary prime number theory
            ​*6.2.1  Elementary properties   cprime 16875
            ​*6.2.2  Coprimality and Euclid's lemma (cont.)   coprm 16916
            6.2.3  Properties of the canonical representation of a rational   cnumer 16938
            6.2.4  Euler's theorem   codz 16969
            6.2.5  Arithmetic modulo a prime number   modprm1div 17004
            6.2.6  Pythagorean Triples   coprimeprodsq 17015
            6.2.7  The prime count function   cpc 17043
            6.2.8  Pocklington's theorem   prmpwdvds 17111
            6.2.9  Infinite primes theorem   unbenlem 17115
            6.2.10  Sum of prime reciprocals   prmreclem1 17123
            6.2.11  Fundamental theorem of arithmetic   1arithlem1 17130
            6.2.12  Lagrange's four-square theorem   cgz 17136
            6.2.13  Van der Waerden's theorem   cvdwa 17172
            6.2.14  Ramsey's theorem   cram 17206
            ​*6.2.15  Primorial function   cprmo 17238
            ​*6.2.16  Prime gaps   prmgaplem1 17256
            6.2.17  Decimal arithmetic (cont.)   dec2dvds 17270
            6.2.18  Cyclical shifts of words (cont.)   cshwsidrepsw 17300
            6.2.19  Specific prime numbers   prmlem0 17312
            6.2.20  Very large primes   1259lem1 17338
​PART 7  BASIC STRUCTURES
      ​7.1  Extensible structures
            ​*7.1.1  Basic definitions   cstr 17353
                  7.1.1.1  Extensible structures as structures with components   cstr 17353
                  7.1.1.2  Substitution of components   csts 17370
                  7.1.1.3  Slots   cslot 17388
                  ​*7.1.1.4  Structure component indices   cnx 17400
                  7.1.1.5  Base sets   cbs 17416
                  7.1.1.6  Base set restrictions   cress 17437
            7.1.2  Slot definitions   cplusg 17457
            7.1.3  Definition of the structure product   crest 17620
            7.1.4  Definition of the structure quotient   cordt 17700
      ​7.2  Moore spaces
            7.2.1  Moore closures   mrcflem 17809
            7.2.2  Independent sets in a Moore system   mrisval 17833
            7.2.3  Algebraic closure systems   isacs 17854
​PART 8  BASIC CATEGORY THEORY
      ​8.1  Categories
            8.1.1  Categories   ccat 17867
            8.1.2  Opposite category   coppc 17914
            8.1.3  Monomorphisms and epimorphisms   cmon 17932
            8.1.4  Sections, inverses, isomorphisms   csect 17948
            ​*8.1.5  Isomorphic objects   ccic 17999
            8.1.6  Subcategories   cssc 18011
            8.1.7  Functors   cfunc 18058
            8.1.8  Full & faithful functors   cful 18108
            8.1.9  Natural transformations and the functor category   cnat 18148
            8.1.10  Initial, terminal and zero objects of a category   cinito 18185
      ​8.2  Arrows (disjointified hom-sets)
            8.2.1  Identity and composition for arrows   cida 18257
      ​8.3  Examples of categories
            8.3.1  The category of sets   csetc 18279
            8.3.2  The category of categories   ccatc 18302
            ​*8.3.3  The category of extensible structures   fncnvimaeqv 18323
      ​8.4  Categorical constructions
            8.4.1  Product of categories   cxpc 18371
            8.4.2  Functor evaluation   cevlf 18412
            8.4.3  Hom functor   chof 18451
​PART 9  BASIC ORDER THEORY
      ​9.1  Dual of an order structure
      ​9.2  Preordered sets and directed sets
      ​9.3  Partially ordered sets (posets)
      ​9.4  Totally ordered sets (tosets)
      ​9.5  Lattices
            9.5.1  Lattices   clat 18634
            9.5.2  Complete lattices   ccla 18701
            9.5.3  Distributive lattices   cdlat 18723
            9.5.4  Subset order structures   cipo 18730
      ​9.6  Posets, directed sets, and lattices as relations
            ​*9.6.1  Posets and lattices as relations   cps 18767
            9.6.2  Directed sets, nets   cdir 18797
      ​9.7  Chains
​PART 10  BASIC ALGEBRAIC STRUCTURES
      ​10.1  Monoids
            ​*10.1.1  Magmas   cplusf 18842
            ​*10.1.2  Identity elements   mgmidmo 18867
            ​*10.1.3  Iterated sums in a magma   gsumvalx 18894
            10.1.4  Magma homomorphisms and submagmas   cmgmhm 18908
            ​*10.1.5  Semigroups   csgrp 18936
            ​*10.1.6  Definition and basic properties of monoids   cmnd 18952
            10.1.7  Monoid homomorphisms and submonoids   cmhm 19005
            ​*10.1.8  Iterated sums in a monoid   gsumvallem2 19059
            10.1.9  Free monoids   cfrmd 19072
                  ​*10.1.9.1  Monoid of endofunctions   cefmnd 19093
            10.1.10  Examples and counterexamples for magmas, semigroups and monoids   mgm2nsgrplem1 19146
      ​10.2  Groups
            10.2.1  Definition and basic properties   cgrp 19173
            ​*10.2.2  Group multiple operation   cmg 19306
            10.2.3  Subgroups and Quotient groups   csubg 19359
            ​*10.2.4  Cyclic monoids and groups   cycsubmel 19444
            10.2.5  Elementary theory of group homomorphisms   cghm 19456
            10.2.6  Isomorphisms of groups   cgim 19500
                  10.2.6.1  The first isomorphism theorem of groups   ghmqusnsglem1 19524
            10.2.7  Group actions   cga 19533
            10.2.8  Centralizers and centers   ccntz 19559
            10.2.9  The opposite group   coppg 19589
            10.2.10  Symmetric groups   csymg 19613
                  ​*10.2.10.1  Definition and basic properties   csymg 19613
                  10.2.10.2  Cayley's theorem   cayleylem1 19656
                  10.2.10.3  Permutations fixing one element   symgfix2 19660
                  ​*10.2.10.4  Transpositions in the symmetric group   cpmtr 19685
                  10.2.10.5  The sign of a permutation   cpsgn 19733
            10.2.11  p-Groups and Sylow groups; Sylow's theorems   cod 19768
            10.2.12  Direct products   clsm 19878
                  10.2.12.1  Direct products (extension)   smndlsmidm 19900
            10.2.13  Free groups   cefg 19950
            10.2.14  Abelian groups   ccmn 20024
                  10.2.14.1  Definition and basic properties   ccmn 20024
                  10.2.14.2  Cyclic groups   ccyg 20121
                  10.2.14.3  Group sum operation   gsumval3a 20147
                  10.2.14.4  Group sums over (ranges of) integers   fsfnn0gsumfsffz 20227
                  10.2.14.5  Internal direct products   cdprd 20239
                  10.2.14.6  The Fundamental Theorem of Abelian Groups   ablfacrplem 20311
            10.2.15  Simple groups   csimpg 20336
                  10.2.15.1  Definition and basic properties   csimpg 20336
                  10.2.15.2  Classification of abelian simple groups   ablsimpnosubgd 20350
            10.2.16  Totally ordered monoids and groups   comnd 20363
      ​10.3  Rings
            10.3.1  Multiplicative Group   cmgp 20390
            ​*10.3.2  Non-unital rings ("rngs")   crng 20404
            ​*10.3.3  Ring unity (multiplicative identity)   cur 20437
            10.3.4  Semirings   csrg 20442
                  ​*10.3.4.1  The binomial theorem for semirings   srgbinomlem1 20482
            10.3.5  Unital rings   crg 20489
            10.3.6  Opposite ring   coppr 20596
            10.3.7  Divisibility   cdsr 20614
            10.3.8  Ring primes   crpm 20692
            10.3.9  Homomorphisms of non-unital rings   crnghm 20694
            10.3.10  Ring homomorphisms   crh 20729
            10.3.11  Nonzero rings and zero rings   cnzr 20792
            10.3.12  Local rings   clring 20820
            10.3.13  Subrings   csubrng 20827
                  10.3.13.1  Subrings of non-unital rings   csubrng 20827
                  10.3.13.2  Subrings of unital rings   csubrg 20851
                  10.3.13.3  Subrings generated by a subset   crgspn 20892
            10.3.14  Categories of rings   crngc 20898
                  ​*10.3.14.1  The category of non-unital rings   crngc 20898
                  ​*10.3.14.2  The category of (unital) rings   cringc 20927
                  10.3.14.3  Subcategories of the category of rings   srhmsubclem1 20959
            10.3.15  Left regular elements and domains   crlreg 20973
      ​10.4  Division rings and fields
            10.4.1  Definition and basic properties   cdr 21010
            10.4.2  Sub-division rings   csdrg 21073
            10.4.3  Absolute value (abstract algebra)   cabv 21095
            10.4.4  Star rings   cstf 21124
            10.4.5  Totally ordered rings and fields   corng 21144
      ​10.5  Left modules
            10.5.1  Definition and basic properties   clmod 21165
            10.5.2  Subspaces and spans in a left module   clss 21236
            10.5.3  Homomorphisms and isomorphisms of left modules   clmhm 21324
            10.5.4  Subspace sum; bases for a left module   clbs 21379
      ​10.6  Vector spaces
            10.6.1  Definition and basic properties   clvec 21407
      ​10.7  Subring algebras and ideals
            10.7.1  Subring algebras   csra 21476
            ​*10.7.2  Left ideals and spans   clidl 21514
            10.7.3  Two-sided ideals and quotient rings   c2idl 21572
                  ​*10.7.3.1  Condition for a non-unital ring to be unital   rngqiprng1elbas 21612
                  10.7.3.2  Prime Ideals   cprmidl 21646
            10.7.4  Principal ideal rings. Divisibility in the integers   clpidl 21674
            10.7.5  Principal ideal domains   cpid 21690
      ​10.8  The complex numbers as an algebraic extensible structure
            10.8.1  Definition and basic properties   cpsmet 21692
            ​*10.8.2  Ring of integers   czring 21782
                  ​*10.8.2.1  Example for a condition for a non-unital ring to be unital   pzriprnglem1 21817
            10.8.3  Algebraic constructions based on the complex numbers   czrh 21835
            10.8.4  Signs as subgroup of the complex numbers   cnmsgnsubg 21913
            10.8.5  Embedding of permutation signs into a ring   zrhpsgnmhm 21920
            10.8.6  The ordered field of real numbers   crefld 21940
      ​10.9  Generalized pre-Hilbert and Hilbert spaces
            10.9.1  Definition and basic properties   cphl 21960
            10.9.2  Orthocomplements and closed subspaces   cocv 21996
            10.9.3  Orthogonal projection and orthonormal bases   cpj 22036
​​*PART 11  BASIC LINEAR ALGEBRA
      ​11.1  Vectors and free modules
            *11.1.1  Direct sum of left modules   cdsmm 22067
            ​*11.1.2  Free modules   cfrlm 22082
            ​*11.1.3  Standard basis (unit vectors)   cuvc 22118
            ​*11.1.4  Independent sets and families   clindf 22140
            11.1.5  Characterization of free modules   lmimlbs 22172
      ​11.2  Associative algebras
            11.2.1  Definition and basic properties   casa 22188
      ​11.3  Abstract multivariate polynomials
            11.3.1  Definition and basic properties   cmps 22242
            11.3.2  Polynomial evaluation   ces 22411
            11.3.3  The "variable selection" function   cslv 22455
            11.3.4  Additional definitions for (multivariate) polynomials   cmhp 22484
            ​*11.3.5  Univariate polynomials   cps1 22523
            11.3.6  Univariate polynomial evaluation   ces1 22661
                  11.3.6.1  Specialization of polynomial evaluation as a ring homomorphism   evls1scafv 22714
      ​​*11.4  Matrices
            *11.4.1  The matrix multiplication   cmmul 22735
            ​*11.4.2  Square matrices   cmat 22752
            ​*11.4.3  The matrix algebra   matmulr 22783
            ​*11.4.4  Matrices of dimension 0 and 1   mat0dimbas0 22811
            ​*11.4.5  The subalgebras of diagonal and scalar matrices   cdmat 22833
            ​*11.4.6  Multiplication of a matrix with a "column vector"   cmvmul 22885
            11.4.7  Replacement functions for a square matrix   cmarrep 22901
            11.4.8  Submatrices   csubma 22921
      ​11.5  The determinant
            11.5.1  Definition and basic properties   cmdat 22929
            11.5.2  Determinants of 2 x 2 -matrices   m2detleiblem1 22969
            11.5.3  The matrix adjugate/adjunct   cmadu 22977
            ​*11.5.4  Laplace expansion of determinants (special case)   symgmatr01lem 22998
            11.5.5  Inverse matrix   invrvald 23021
            ​*11.5.6  Cramer's rule   slesolvec 23027
      ​​*11.6  Polynomial matrices
            11.6.1  Basic properties   pmatring 23040
            ​*11.6.2  Constant polynomial matrices   ccpmat 23051
            ​*11.6.3  Collecting coefficients of polynomial matrices   cdecpmat 23110
            ​*11.6.4  Ring isomorphism between polynomial matrices and polynomials over matrices   cpm2mp 23140
      ​​*11.7  The characteristic polynomial
            *11.7.1  Definition and basic properties   cchpmat 23174
            ​*11.7.2  The characteristic factor function G   fvmptnn04if 23197
            ​*11.7.3  The Cayley-Hamilton theorem   cpmadurid 23215
​PART 12  BASIC TOPOLOGY
      ​12.1  Topology
            ​*12.1.1  Topological spaces   ctop 23241
                  12.1.1.1  Topologies   ctop 23241
                  12.1.1.2  Topologies on sets   ctopon 23258
                  12.1.1.3  Topological spaces   ctps 23280
            12.1.2  Topological bases   ctb 23293
            12.1.3  Examples of topologies   distop 23343
            12.1.4  Closure and interior   ccld 23364
            12.1.5  Neighborhoods   cnei 23445
            12.1.6  Limit points and perfect sets   clp 23482
            12.1.7  Subspace topologies   restrcl 23505
            12.1.8  Order topology   ordtbaslem 23536
            12.1.9  Limits and continuity in topological spaces   ccn 23572
            12.1.10  Separated spaces: T0, T1, T2 (Hausdorff) ...   ct0 23654
            12.1.11  Compactness   ccmp 23734
            12.1.12  Bolzano-Weierstrass theorem   bwth 23758
            12.1.13  Connectedness   cconn 23759
            12.1.14  First- and second-countability   c1stc 23785
            12.1.15  Local topological properties   clly 23813
            12.1.16  Refinements   cref 23851
            12.1.17  Compactly generated spaces   ckgen 23882
            12.1.18  Product topologies   ctx 23909
            12.1.19  Continuous function-builders   cnmptid 24010
            12.1.20  Quotient maps and quotient topology   ckq 24042
            12.1.21  Homeomorphisms   chmeo 24102
      ​12.2  Filters and filter bases
            12.2.1  Filter bases   elmptrab 24176
            12.2.2  Filters   cfil 24194
            12.2.3  Ultrafilters   cufil 24248
            12.2.4  Filter limits   cfm 24282
            12.2.5  Extension by continuity   ccnext 24408
            12.2.6  Topological groups   ctmd 24419
            12.2.7  Infinite group sum on topological groups   ctsu 24475
            12.2.8  Topological rings, fields, vector spaces   ctrg 24505
      ​12.3  Uniform Structures and Spaces
            12.3.1  Uniform structures   cust 24549
            12.3.2  The topology induced by an uniform structure   cutop 24579
            12.3.3  Uniform Spaces   cuss 24602
            12.3.4  Uniform continuity   cucn 24623
            12.3.5  Cauchy filters in uniform spaces   ccfilu 24634
            12.3.6  Complete uniform spaces   ccusp 24645
      ​12.4  Metric spaces
            12.4.1  Pseudometric spaces   ispsmet 24653
            12.4.2  Basic metric space properties   cxms 24666
            12.4.3  Metric space balls   blfvalps 24732
            12.4.4  Open sets of a metric space   mopnval 24787
            12.4.5  Continuity in metric spaces   metcnp3 24889
            12.4.6  The uniform structure generated by a metric   metuval 24898
            12.4.7  Examples of metric spaces   dscmet 24921
            ​*12.4.8  Normed algebraic structures   cnm 24925
            12.4.9  Normed space homomorphisms (bounded linear operators)   cnmo 25054
            12.4.10  Topology on the reals   qtopbaslem 25107
            12.4.11  Topological definitions using the reals   cii 25226
            12.4.12  Path homotopy   chtpy 25318
            12.4.13  The fundamental group   cpco 25351
      ​12.5  Metric subcomplex vector spaces
            12.5.1  Subcomplex modules   cclm 25413
            ​*12.5.2  Subcomplex vector spaces   ccvs 25474
            ​*12.5.3  Normed subcomplex vector spaces   isncvsngp 25500
            12.5.4  Subcomplex pre-Hilbert spaces   ccph 25517
            12.5.5  Convergence and completeness   ccfil 25603
            12.5.6  Baire's Category Theorem   bcthlem1 25675
            12.5.7  Banach spaces and subcomplex Hilbert spaces   ccms 25683
                  12.5.7.1  The complete ordered field of the real numbers   retopn 25730
            12.5.8  Euclidean spaces   crrx 25734
            12.5.9  Minimizing Vector Theorem   minveclem1 25775
            12.5.10  Projection Theorem   pjthlem1 25788
​PART 13  BASIC REAL AND COMPLEX ANALYSIS
      ​13.1  Continuity
            13.1.1  Intermediate value theorem   pmltpclem1 25799
      ​13.2  Integrals
            13.2.1  Lebesgue measure   covol 25813
            13.2.2  Lebesgue integration   cmbf 25965
                  13.2.2.1  Lesbesgue integral   cmbf 25965
                  13.2.2.2  Lesbesgue directed integral   cdit 26196
      ​13.3  Derivatives
            13.3.1  Real and complex differentiation   climc 26212
                  13.3.1.1  Derivatives of functions of one complex or real variable   climc 26212
                  13.3.1.2  Results on real differentiation   dvferm1lem 26334
​PART 14  BASIC REAL AND COMPLEX FUNCTIONS
      ​14.1  Polynomials
            14.1.1  Polynomial degrees   cmdg 26401
            14.1.2  The division algorithm for univariate polynomials   cmn1 26474
            14.1.3  Elementary properties of complex polynomials   cply 26532
            14.1.4  The division algorithm for polynomials   cquot 26641
            14.1.5  Algebraic numbers   caa 26667
            14.1.6  Liouville's approximation theorem   aalioulem1 26689
      ​14.2  Sequences and series
            14.2.1  Taylor polynomials and Taylor's theorem   ctayl 26710
            14.2.2  Uniform convergence   culm 26733
            14.2.3  Power series   pserval 26767
      ​14.3  Basic trigonometry
            14.3.1  The exponential, sine, and cosine functions (cont.)   efcn 26800
            14.3.2  Properties of pi = 3.14159...   pilem1 26808
            14.3.3  Mapping of the exponential function   efgh 26899
            14.3.4  The natural logarithm on complex numbers   clog 26912
            ​*14.3.5  Logarithms to an arbitrary base   clogb 27122
            14.3.6  Theorems of Pythagoras, isosceles triangles, and intersecting chords   angval 27159
            14.3.7  Solutions of quadratic, cubic, and quartic equations   quad2 27197
            14.3.8  Inverse trigonometric functions   casin 27220
            14.3.9  The Birthday Problem   log2ublem1 27304
            14.3.10  Areas in R^2   carea 27313
            14.3.11  More miscellaneous converging sequences   rlimcnp 27323
            14.3.12  Inequality of arithmetic and geometric means   cvxcl 27342
            14.3.13  Euler-Mascheroni constant   cem 27349
            14.3.14  Zeta function   czeta 27370
            14.3.15  Gamma function   clgam 27373
      ​14.4  Basic number theory
            14.4.1  Wilson's theorem   wilthlem1 27425
            14.4.2  The Fundamental Theorem of Algebra   ftalem1 27430
            14.4.3  The Basel problem (ζ(2) = π²/6)   basellem1 27438
            14.4.4  Number-theoretical functions   ccht 27448
            14.4.5  Perfect Number Theorem   mersenne 27584
            14.4.6  Characters of Z/nZ   cdchr 27589
            14.4.7  Bertrand's postulate   bcctr 27632
            ​*14.4.8  Quadratic residues and the Legendre symbol   clgs 27651
            ​*14.4.9  Gauss' Lemma   gausslemma2dlem0a 27713
            14.4.10  Quadratic reciprocity   lgseisenlem1 27732
            14.4.11  All primes 4n+1 are the sum of two squares   2sqlem1 27774
            14.4.12  Chebyshev's Weak Prime Number Theorem, Dirichlet's Theorem   chebbnd1lem1 27826
            14.4.13  The Prime Number Theorem   mudivsum 27887
            14.4.14  Ostrowski's theorem   abvcxp 27972
            14.4.15  Fermat's Last Theorem   infdesc 27997
                  14.4.15.1  Basic reductions for Fermat's Last Theorem   infdesc 27997
                  ​*14.4.15.2  Proof of Fermat's Last Theorem (work in progress)   fltoprmlem1 28023
​PART 15  SURREAL NUMBERS
      ​​*15.1  Sign sequence representation and Alling's axioms
            15.1.1  Definitions and initial properties   csur 28027
            15.1.2  Ordering   ltssolem1 28062
            15.1.3  Birthday Function   bdayfo 28064
            15.1.4  Density   fvnobday 28065
            ​*15.1.5  Full-Eta Property   bdayimaon 28080
      ​15.2  Initial consequences of Alling's axioms
            15.2.1  Ordering Theorems   cles 28131
            15.2.2  Birthday Theorems   bdayfun 28163
      ​​*15.3  Conway cut representation
            15.3.1  Conway cuts   cslts 28173
            15.3.2  Zero and One   c0s 28221
            15.3.3  Cuts and Options   cmade 28238
            15.3.4  Cofinality and coinitiality   cofslts 28334
      ​15.4  Induction and recursion
            15.4.1  Induction and recursion on one variable   cnorec 28353
            15.4.2  Induction and recursion on two variables   cnorec2 28364
      ​15.5  Surreal arithmetic
            15.5.1  Addition   cadds 28375
            15.5.2  Negation and Subtraction   cnegs 28435
            15.5.3  Multiplication   cmuls 28522
            15.5.4  Division   cdivs 28603
            15.5.5  Absolute value   cabss 28653
      ​15.6  Subsystems of surreals
            15.6.1  Ordinal numbers   cons 28667
            15.6.2  Surreal recursive sequences   cseqs 28699
            15.6.3  Natural numbers   cn0s 28728
            15.6.4  Integers   czs 28794
            15.6.5  Dyadic fractions   c2s 28826
            15.6.6  Real numbers   creno 28905
​​*PART 16  ELEMENTARY GEOMETRY
      ​16.1  Definition and Tarski's Axioms of Geometry
            16.1.1  Justification for the congruence notation   tgjustf 28965
      ​16.2  Tarskian Geometry
            16.2.1  Congruence   tgcgrcomimp 28969
            16.2.2  Betweenness   tgbtwntriv2 28980
            16.2.3  Dimension   tglowdim1 28993
            16.2.4  Betweenness and Congruence   tgifscgr 29001
            16.2.5  Congruence of a series of points   ccgrg 29003
            16.2.6  Motions   cismt 29025
            16.2.7  Colinearity   tglng 29039
            16.2.8  Connectivity of betweenness   tgbtwnconn1lem1 29065
            16.2.9  Less-than relation in geometric congruences   cleg 29075
            16.2.10  Rays   chlg 29093
            16.2.11  Lines   btwnlng1 29117
            16.2.12  Point inversions   cmir 29154
            16.2.13  Right angles   crag 29198
            16.2.14  Half-planes   islnopp 29245
            16.2.15  Planes   cplng 29281
            16.2.16  Midpoints and Line Mirroring   cmid 29307
            16.2.17  Congruence of angles   ccgra 29344
            16.2.18  Angle Comparisons   cinag 29384
            16.2.19  Angle Addition   cangmgm 29403
            16.2.20  Congruence Theorems   tgsas1 29429
            16.2.21  Equilateral triangles   ceqlg 29440
            16.2.22  Parallel lines   cprlng 29444
      ​16.3  Properties of geometries
            16.3.1  Isomorphisms between geometries   f1otrgds 29476
      ​16.4  Geometry in Hilbert spaces
            16.4.1  Geometry in the complex plane   cchhllem 29494
            16.4.2  Geometry in Euclidean spaces   cee 29495
                  16.4.2.1  Definition of the Euclidean space   cee 29495
                  16.4.2.2  Tarski's axioms for geometry for the Euclidean space   axdimuniq 29521
                  16.4.2.3  EE^n fulfills Tarski's Axioms   ceeng 29585
​​*PART 17  GRAPH THEORY
      ​*17.1  Vertices and edges
            17.1.1  The edge function extractor for extensible structures   cedgf 29596
            ​*17.1.2  Vertices and indexed edges   cvtx 29604
                  17.1.2.1  Definitions and basic properties   cvtx 29604
                  17.1.2.2  The vertices and edges of a graph represented as ordered pair   opvtxval 29611
                  17.1.2.3  The vertices and edges of a graph represented as extensible structure   funvtxdmge2val 29619
                  17.1.2.4  Representations of graphs without edges   snstrvtxval 29645
                  17.1.2.5  Degenerated cases of representations of graphs   vtxval0 29647
            17.1.3  Edges as range of the edge function   cedg 29655
      ​​*17.2  Undirected graphs
            17.2.1  Undirected hypergraphs   cuhgr 29664
            17.2.2  Undirected pseudographs and multigraphs   cupgr 29688
            ​*17.2.3  Loop-free graphs   umgrislfupgrlem 29730
            17.2.4  Edges as subsets of vertices of graphs   uhgredgiedgb 29734
            ​*17.2.5  Undirected simple graphs   cuspgr 29759
            17.2.6  Examples for graphs   usgr0e 29847
            17.2.7  Subgraphs   csubgr 29878
            17.2.8  Finite undirected simple graphs   cfusgr 29927
            17.2.9  Neighbors, complete graphs and universal vertices   cnbgr 29943
                  17.2.9.1  Neighbors   cnbgr 29943
                  17.2.9.2  Universal vertices   cuvtx 29996
                  17.2.9.3  Complete graphs   ccplgr 30020
            17.2.10  Vertex degree   cvtxdg 30076
            ​*17.2.11  Regular graphs   crgr 30166
      ​​*17.3  Walks, paths and cycles
            *17.3.1  Walks   cewlks 30206
            17.3.2  Walks for loop-free graphs   lfgrwlkprop 30300
            17.3.3  Trails   ctrls 30303
            17.3.4  Paths and simple paths   cpths 30325
            17.3.5  Closed walks   cclwlks 30387
            17.3.6  Circuits and cycles   ccrcts 30401
            ​*17.3.7  Walks as words   cwwlks 30444
            17.3.8  Walks/paths of length 2 (as length 3 strings)   2wlkdlem1 30544
            17.3.9  Walks in regular graphs   rusgrnumwwlkl1 30590
            ​*17.3.10  Closed walks as words   cclwwlk 30602
                  17.3.10.1  Closed walks as words   cclwwlk 30602
                  17.3.10.2  Closed walks of a fixed length as words   cclwwlkn 30645
                  17.3.10.3  Closed walks on a vertex of a fixed length as words   cclwwlknon 30708
            17.3.11  Examples for walks, trails and paths   0ewlk 30735
            17.3.12  Acyclic graphs   cacycgr 30778
            17.3.13  Connected graphs   cconngr 30817
      ​17.4  Eulerian paths and the Konigsberg Bridge problem
            ​*17.4.1  Eulerian paths   ceupth 30828
            ​*17.4.2  The Königsberg Bridge problem   konigsbergvtx 30877
      ​17.5  The Friendship Theorem
            17.5.1  Friendship graphs - basics   cfrgr 30889
            17.5.2  The friendship theorem for small graphs   frgr1v 30902
            17.5.3  Theorems according to Mertzios and Unger   2pthfrgrrn 30913
            ​*17.5.4  Huneke's Proof of the Friendship Theorem   frgrncvvdeqlem1 30930
​PART 18  GUIDES AND MISCELLANEA
      ​18.1  Guides (conventions, explanations, and examples)
            ​*18.1.1  Conventions   conventions 31031
            18.1.2  Natural deduction   natded 31034
            ​*18.1.3  Natural deduction examples   ex-natded5.2 31035
            18.1.4  Definitional examples   ex-or 31052
            18.1.5  Other examples   aevdemo 31091
      ​18.2  Humor
            18.2.1  April Fool's theorem   avril1 31094
      ​18.3  (Future - to be reviewed and classified)
            18.3.1  Planar incidence geometry   cplig 31106
            ​*18.3.2  Aliases kept to prevent broken links   dummylink 31119
​​*PART 19  COMPLEX TOPOLOGICAL VECTOR SPACES (DEPRECATED)
      ​*19.1  Additional material on group theory (deprecated)
            19.1.1  Definitions and basic properties for groups   cgr 31121
            19.1.2  Abelian groups   cablo 31176
      ​19.2  Complex vector spaces
            19.2.1  Definition and basic properties   cvc 31190
            19.2.2  Examples of complex vector spaces   cnaddabloOLD 31213
      ​19.3  Normed complex vector spaces
            19.3.1  Definition and basic properties   cnv 31216
            19.3.2  Examples of normed complex vector spaces   cnnv 31309
            19.3.3  Induced metric of a normed complex vector space   imsval 31317
            19.3.4  Inner product   cdip 31332
            19.3.5  Subspaces   css 31353
      ​19.4  Operators on complex vector spaces
            19.4.1  Definitions and basic properties   clno 31372
      ​19.5  Inner product (pre-Hilbert) spaces
            19.5.1  Definition and basic properties   ccphlo 31444
            19.5.2  Examples of pre-Hilbert spaces   cncph 31451
            19.5.3  Properties of pre-Hilbert spaces   isph 31454
      ​19.6  Complex Banach spaces
            19.6.1  Definition and basic properties   ccbn 31494
            19.6.2  Examples of complex Banach spaces   cnbn 31501
            19.6.3  Uniform Boundedness Theorem   ubthlem1 31502
            19.6.4  Minimizing Vector Theorem   minvecolem1 31506
      ​19.7  Complex Hilbert spaces
            19.7.1  Definition and basic properties   chlo 31517
            19.7.2  Standard axioms for a complex Hilbert space   hlex 31530
            19.7.3  Examples of complex Hilbert spaces   cnchl 31548
            19.7.4  Hellinger-Toeplitz Theorem   htthlem 31549
​​*PART 20  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      ​20.1  Axiomatization of complex pre-Hilbert spaces
            20.1.1  Basic Hilbert space definitions   chba 31551
            20.1.2  Preliminary ZFC lemmas   df-hnorm 31600
            ​*20.1.3  Derive the Hilbert space axioms from ZFC set theory   axhilex-zf 31613
            ​*20.1.4  Introduce the vector space axioms for a Hilbert space   ax-hilex 31631
            20.1.5  Vector operations   hvmulex 31643
            20.1.6  Inner product postulates for a Hilbert space   ax-hfi 31711
      ​20.2  Inner product and norms
            20.2.1  Inner product   his5 31718
            20.2.2  Norms   dfhnorm2 31754
            20.2.3  Relate Hilbert space to normed complex vector spaces   hilablo 31792
            20.2.4  Bunjakovaskij-Cauchy-Schwarz inequality   bcsiALT 31811
      ​20.3  Cauchy sequences and completeness axiom
            20.3.1  Cauchy sequences and limits   hcau 31816
            20.3.2  Derivation of the completeness axiom from ZF set theory   hilmet 31826
            20.3.3  Completeness postulate for a Hilbert space   ax-hcompl 31834
            20.3.4  Relate Hilbert space to ZFC pre-Hilbert and Hilbert spaces   hhcms 31835
      ​20.4  Subspaces and projections
            20.4.1  Subspaces   df-sh 31839
            20.4.2  Closed subspaces   df-ch 31853
            20.4.3  Orthocomplements   df-oc 31884
            20.4.4  Subspace sum, span, lattice join, lattice supremum   df-shs 31940
            20.4.5  Projection theorem   pjhthlem1 32023
            20.4.6  Projectors   df-pjh 32027
      ​20.5  Properties of Hilbert subspaces
            20.5.1  Orthomodular law   omlsilem 32034
            20.5.2  Projectors (cont.)   pjhtheu2 32048
            20.5.3  Hilbert lattice operations   sh0le 32072
            20.5.4  Span (cont.) and one-dimensional subspaces   spansn0 32173
            20.5.5  Commutes relation for Hilbert lattice elements   df-cm 32215
            20.5.6  Foulis-Holland theorem   fh1 32250
            20.5.7  Quantum Logic Explorer axioms   qlax1i 32259
            20.5.8  Orthogonal subspaces   chscllem1 32269
            20.5.9  Orthoarguesian laws 5OA and 3OA   5oalem1 32286
            20.5.10  Projectors (cont.)   pjorthi 32301
            20.5.11  Mayet's equation E_3   mayete3i 32360
      ​20.6  Operators on Hilbert spaces
            ​*20.6.1  Operator sum, difference, and scalar multiplication   df-hosum 32362
            20.6.2  Zero and identity operators   df-h0op 32380
            20.6.3  Operations on Hilbert space operators   hoaddcl 32390
            20.6.4  Linear, continuous, bounded, Hermitian, unitary operators and norms   df-nmop 32471
            20.6.5  Linear and continuous functionals and norms   df-nmfn 32477
            20.6.6  Adjoint   df-adjh 32481
            20.6.7  Dirac bra-ket notation   df-bra 32482
            20.6.8  Positive operators   df-leop 32484
            20.6.9  Eigenvectors, eigenvalues, spectrum   df-eigvec 32485
            20.6.10  Theorems about operators and functionals   nmopval 32488
            20.6.11  Riesz lemma   riesz3i 32694
            20.6.12  Adjoints (cont.)   cnlnadjlem1 32699
            20.6.13  Quantum computation error bound theorem   unierri 32736
            20.6.14  Dirac bra-ket notation (cont.)   branmfn 32737
            20.6.15  Positive operators (cont.)   leopg 32754
            20.6.16  Projectors as operators   pjhmopi 32778
      ​20.7  States on a Hilbert lattice and Godowski's equation
            20.7.1  States on a Hilbert lattice   df-st 32843
            20.7.2  Godowski's equation   golem1 32903
      ​20.8  Cover relation, atoms, exchange axiom, and modular symmetry
            20.8.1  Covers relation; modular pairs   df-cv 32911
            20.8.2  Atoms   df-at 32970
            20.8.3  Superposition principle   superpos 32986
            20.8.4  Atoms, exchange and covering properties, atomicity   chcv1 32987
            20.8.5  Irreducibility   chirredlem1 33022
            20.8.6  Atoms (cont.)   atcvat3i 33028
            20.8.7  Modular symmetry   mdsymlem1 33035
​PART 21  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      ​21.1  Mathboxes for user contributions
            21.1.1  Mathbox guidelines   mathbox 33074
      ​21.2  Mathbox for Jorge Agra
      ​21.3  Mathbox for Stefan Allan
      ​21.4  Mathbox for Thierry Arnoux
            21.4.1  Propositional Calculus - misc additions   ad11antr 33121
            21.4.2  Predicate Calculus   sbc2iedf 33134
                  21.4.2.1  Predicate Calculus - misc additions   sbc2iedf 33134
                  21.4.2.2  Restricted quantification - misc additions   ralcom4f 33136
                  21.4.2.3  Equality   eqtrb 33142
                  21.4.2.4  Double restricted existential uniqueness quantification   opsbc2ie 33144
                  21.4.2.5  Double restricted existential uniqueness quantification syntax   w2reu 33146
                  21.4.2.6  Substitution (without distinct variables) - misc additions   sbceqbidf 33155
                  21.4.2.7  Existential "at most one" - misc additions   mo5f 33157
                  21.4.2.8  Existential uniqueness - misc additions   reuxfrdf 33159
                  21.4.2.9  Restricted "at most one" - misc additions   rmoxfrd 33161
                  21.4.2.10  Restricted iota (description binder)   riotaeqbidva 33164
            21.4.3  General Set Theory   dmrab 33165
                  21.4.3.1  Class abstractions (a.k.a. class builders)   dmrab 33165
                  21.4.3.2  Image Sets   abrexdomjm 33175
                  21.4.3.3  Set relations and operations - misc additions   nelun 33181
                  21.4.3.4  Unordered pairs   elpreq 33196
                  21.4.3.5  Unordered triples   tpssg 33205
                  21.4.3.6  Conditional operator - misc additions   ifeqeqx 33210
                  21.4.3.7  Set union   uniinn0 33219
                  21.4.3.8  Indexed union - misc additions   cbviunf 33222
                  21.4.3.9  Indexed intersection - misc additions   iinabrex 33235
                  21.4.3.10  Disjointness - misc additions   disjnf 33236
            21.4.4  Relations and Functions   xpdisjres 33264
                  21.4.4.1  Relations - misc additions   xpdisjres 33264
                  21.4.4.2  Functions - misc additions   fconst7v 33286
                  21.4.4.3  Operations - misc additions   mpomptxf 33343
                  21.4.4.4  Support of a function   suppovss 33345
                  21.4.4.5  Explicit Functions with one or two points as a domain   cosnopne 33358
                  21.4.4.6  Isomorphisms - misc. additions   gtiso 33365
                  21.4.4.7  Disjointness (additional proof requiring functions)   disjdsct 33367
                  21.4.4.8  First and second members of an ordered pair - misc additions   df1stres 33368
                  21.4.4.9  Countable Sets   snct 33376
            21.4.5  Real and Complex Numbers   sgnval2 33397
                  21.4.5.1  Complex operations - misc. additions   creq0 33398
                  21.4.5.2  Ordering on reals - misc additions   lt2addrd 33412
                  21.4.5.3  Extended reals - misc additions   nn0mnfxrd 33413
                  21.4.5.4  Extended nonnegative integers - misc additions   xnn0gt0 33431
                  21.4.5.5  Real number intervals - misc additions   joiniooico 33436
                  21.4.5.6  Finite intervals of integers - misc additions   uzssico 33446
                  21.4.5.7  Half-open integer ranges - misc additions   iundisjfi 33458
                  21.4.5.8  The ` # ` (set size) function - misc additions   hashunif 33468
                  21.4.5.9  The greatest common divisor operator - misc. additions   elq2 33473
                  21.4.5.10  Integers   nn0split01 33479
                  21.4.5.11  Decimal numbers   dfdec100 33491
            21.4.6  Real and complex functions   sgnsgn 33492
                  21.4.6.1  Signum (sgn or sign) function - misc. additions   sgnsgn 33492
                  21.4.6.2  Integer powers - misc. additions   nexple 33494
                  21.4.6.3  Indicator Functions (continued)   indsumin 33498
            ​*21.4.7  Decimal expansion   cdp2 33507
                  ​*21.4.7.1  Decimal point   cdp 33524
                  21.4.7.2  Division in the extended real number system   cxdiv 33553
            21.4.8  Words over a set - misc additions   wrdres 33572
                  21.4.8.1  Splicing words (substring replacement)   splfv3 33589
                  21.4.8.2  Cyclic shift of words   1cshid 33590
            21.4.9  Extensible Structures   ressplusf 33594
                  21.4.9.1  Structure restriction operator   ressplusf 33594
                  21.4.9.2  Posets   ressprs 33597
                  21.4.9.3  Complete lattices   clatp0cl 33607
                  21.4.9.4  Order Theory   cmnt 33609
                  21.4.9.5  Extended reals Structure - misc additions   ax-xrssca 33635
                  21.4.9.6  The extended nonnegative real numbers commutative monoid   xrge00 33645
            21.4.10  Algebra   mndcld 33653
                  21.4.10.1  Monoids   mndcld 33653
                  21.4.10.2  Monoids Homomorphisms   abliso 33666
                  21.4.10.3  Groups - misc additions   grpidcld 33670
                  21.4.10.4  Abelian Groups - misc additions   ablcomd 33676
                  21.4.10.5  Finitely supported group sums - misc additions   gsumsubg 33677
                  21.4.10.6  Group or monoid sums over words   gsumwun 33707
                  21.4.10.7  Centralizers and centers - misc additions   cntzun 33710
                  21.4.10.8  The symmetric group   symgfcoeu 33713
                  21.4.10.9  Transpositions   pmtridf1o 33725
                  21.4.10.10  Permutation Signs   psgnid 33728
                  21.4.10.11  Permutation cycles   ctocyc 33737
                  21.4.10.12  The Alternating Group   evpmval 33776
                  21.4.10.13  Signum in an ordered monoid   csgns 33789
                  21.4.10.14  Fixed points   cfxp 33794
                  21.4.10.15  The Archimedean property for generic ordered algebraic structures   cinftm 33807
                  21.4.10.16  Semiring left modules   cslmd 33831
                  21.4.10.17  Simple groups   prmsimpcyc 33859
                  21.4.10.18  Rings - misc additions   ringrngd 33860
                  21.4.10.19  Subrings generated by a set   elrgspnlem1 33873
                  21.4.10.20  The zero ring   irrednzr 33881
                  21.4.10.21  Localization of rings   cerl 33884
                  21.4.10.22  Integral Domains   domnmuln0rd 33908
                  21.4.10.23  Euclidean Domains   ceuf 33922
                  21.4.10.24  Division Rings   rndrhmcl 33928
                  21.4.10.25  The field of rational numbers   qfld 33929
                  21.4.10.26  Subfields   subsdrg 33930
                  21.4.10.27  Field of fractions   cfrac 33934
                  21.4.10.28  Field extensions generated by a set   cfldgen 33942
                  21.4.10.29  Ring homomorphisms - misc additions   rhmdvd 33955
                  21.4.10.30  Scalar restriction operation   cresv 33957
                  21.4.10.31  The commutative ring of gaussian integers   gzcrng 33972
                  21.4.10.32  The archimedean ordered field of real numbers   cnfldfld 33973
                  21.4.10.33  The quotient map and quotient modules   qusker 33980
                  21.4.10.34  The ring of integers modulo ` N `   znfermltl 33992
                  21.4.10.35  Independent sets and families   islinds5 33993
                  21.4.10.36  Ring associates, ring units   dvdsruassoi 34009
                  ​*21.4.10.37  Subgroup sum / Sumset / Minkowski sum   elgrplsmsn 34015
                  21.4.10.38  The quotient map   quslsm 34026
                  21.4.10.39  Ideals   intlidl 34040
                  21.4.10.40  Maximal Ideals   cmxidl 34054
                  21.4.10.41  Local rings   drnglring 34094
                  21.4.10.42  The semiring of ideals of a ring   cidlsrg 34102
                  21.4.10.43  Prime Elements   rprmval 34118
                  21.4.10.44  Unique factorization domains   cufd 34140
                  21.4.10.45  The ring of integers   zringidom 34153
                  21.4.10.46  Associative Algebra   assaassd 34157
                  21.4.10.47  Univariate Polynomials   0ringmon1p 34159
                  21.4.10.48  Polynomial quotient and polynomial remainder   q1pdir 34205
                  21.4.10.49  Multivariate Polynomials   psrbasfsupp 34213
                  21.4.10.50  The ring of symmetric polynomials   csply 34257
                  21.4.10.51  The subring algebra   sra1r 34283
                  21.4.10.52  Division Ring Extensions   drgext0g 34292
                  21.4.10.53  Vector Spaces   lvecdimfi 34298
                  21.4.10.54  Vector Space Dimension   cldim 34301
            21.4.11  Field Extensions   cfldext 34340
                  21.4.11.1  Algebraic numbers   cirng 34385
                  21.4.11.2  Algebraic extensions   calgext 34397
                  21.4.11.3  Minimal polynomials   cminply 34401
                  21.4.11.4  Quadratic Field Extensions   rtelextdg2lem 34428
                  21.4.11.5  Towers of quadratic extentions   fldext2chn 34430
            ​*21.4.12  Constructible Numbers   cconstr 34431
                  21.4.12.1  Impossible constructions   2sqr3minply 34482
            21.4.13  Matrices   csmat 34495
                  21.4.13.1  Submatrices   csmat 34495
                  21.4.13.2  Matrix literals   clmat 34513
                  21.4.13.3  Laplace expansion of determinants   mdetpmtr1 34525
            21.4.14  Topology   ist0cld 34535
                  21.4.14.1  Open maps   txomap 34536
                  21.4.14.2  Topology of the unit circle   qtopt1 34537
                  21.4.14.3  Refinements   reff 34541
                  21.4.14.4  Open cover refinement property   ccref 34544
                  21.4.14.5  Lindelöf spaces   cldlf 34554
                  21.4.14.6  Paracompact spaces   cpcmp 34557
                  ​*21.4.14.7  Spectrum of a ring   crspec 34564
                  21.4.14.8  Pseudometrics   cmetid 34588
                  21.4.14.9  Continuity - misc additions   hauseqcn 34600
                  21.4.14.10  Topology of the closed unit interval   elunitge0 34601
                  21.4.14.11  Topology of ` ( RR X. RR ) `   unicls 34605
                  21.4.14.12  Order topology - misc. additions   cnvordtrestixx 34615
                  21.4.14.13  Continuity in topological spaces - misc. additions   mndpluscn 34628
                  21.4.14.14  Topology of the extended nonnegative real numbers ordered monoid   xrge0hmph 34634
                  21.4.14.15  Limits - misc additions   lmlim 34649
                  21.4.14.16  Univariate polynomials   pl1cn 34657
            21.4.15  Uniform Stuctures and Spaces   chcmp 34658
                  21.4.15.1  Hausdorff uniform completion   chcmp 34658
            21.4.16  Topology and algebraic structures   zringnm 34660
                  21.4.16.1  The norm on the ring of the integer numbers   zringnm 34660
                  21.4.16.2  Topological ` ZZ ` -modules   zlm0 34662
                  21.4.16.3  Canonical embedding of the field of the rational numbers into a division ring   cqqh 34672
                  21.4.16.4  Canonical embedding of the real numbers into a complete ordered field   crrh 34695
                  21.4.16.5  Embedding from the extended real numbers into a complete lattice   cxrh 34718
                  21.4.16.6  Canonical embeddings into the ordered field of the real numbers   zrhre 34721
                  ​*21.4.16.7  Topological Manifolds   cmntop 34724
                  21.4.16.8  Extended sum   cesum 34729
            21.4.17  Mixed Function/Constant operation   cofc 34797
            21.4.18  Abstract measure   csiga 34810
                  21.4.18.1  Sigma-Algebra   csiga 34810
                  21.4.18.2  Generated sigma-Algebra   csigagen 34841
                  ​*21.4.18.3  lambda and pi-Systems, Rings of Sets   ispisys 34855
                  21.4.18.4  The Borel algebra on the real numbers   cbrsiga 34884
                  21.4.18.5  Product Sigma-Algebra   csx 34891
                  21.4.18.6  Measures   cmeas 34898
                  21.4.18.7  The counting measure   cntmeas 34929
                  21.4.18.8  The Lebesgue measure - misc additions   voliune 34932
                  21.4.18.9  The Dirac delta measure   cdde 34935
                  21.4.18.10  The 'almost everywhere' relation   cae 34940
                  21.4.18.11  Measurable functions   cmbfm 34952
                  21.4.18.12  Borel Algebra on ` ( RR X. RR ) `   br2base 34971
                  ​*21.4.18.13  Caratheodory's extension theorem   coms 34993
            21.4.19  Integration   itgeq12dv 35028
                  21.4.19.1  Lebesgue integral - misc additions   itgeq12dv 35028
                  21.4.19.2  Bochner integral   citgm 35029
            21.4.20  Euler's partition theorem   oddpwdc 35056
            21.4.21  Sequences defined by strong recursion   csseq 35085
            21.4.22  Fibonacci Numbers   cfib 35098
            21.4.23  Probability   cprb 35109
                  21.4.23.1  Probability Theory   cprb 35109
                  21.4.23.2  Conditional Probabilities   ccprob 35133
                  21.4.23.3  Real-valued Random Variables   crrv 35142
                  21.4.23.4  Preimage set mapping operator   corvc 35158
                  21.4.23.5  Distribution Functions   orvcelval 35171
                  21.4.23.6  Cumulative Distribution Functions   orvclteel 35175
                  21.4.23.7  Probabilities - example   coinfliplem 35181
                  21.4.23.8  Bertrand's Ballot Problem   ballotlemoex 35188
            21.4.24  Signum (sgn or sign) function - misc. additions   fzssfzo 35241
                  21.4.24.1  Operations on words   ccatmulgnn0dir 35244
            21.4.25  Polynomials with real coefficients - misc additions   plyrecld 35248
            21.4.26  Descartes's rule of signs   signspval 35251
                  21.4.26.1  Sign changes in a word over real numbers   signspval 35251
                  21.4.26.2  Counting sign changes in a word over real numbers   signslema 35261
            21.4.27  Number Theory   iblidicc 35291
                  21.4.27.1  Representations of a number as sums of integers   crepr 35307
                  21.4.27.2  Vinogradov Trigonometric Sums and the Circle Method   cvts 35334
                  21.4.27.3  The Ternary Goldbach Conjecture: Final Statement   ax-hgt749 35343
            21.4.28  Elementary Geometry   cstrkg2d 35363
                  ​*21.4.28.1  Two-dimensional geometry   cstrkg2d 35363
                  21.4.28.2  Morley's Miracle   cgranbtwn 35368
                  21.4.28.3  Outer Five Segment (not used, no need to move to main)   cafs 35371
            ​*21.4.29  LeftPad Project   clpad 35376
      ​​*21.5  Mathbox for Jonathan Ben-Naim
            21.5.1  First-order logic and set theory   bnj170 35399
            21.5.2  Well founded induction and recursion   bnj110 35558
            21.5.3  The existence of a minimal element in certain classes   bnj69 35710
            21.5.4  Well-founded induction   bnj1204 35712
            21.5.5  Well-founded recursion, part 1 of 3   bnj60 35762
            21.5.6  Well-founded recursion, part 2 of 3   bnj1500 35768
            21.5.7  Well-founded recursion, part 3 of 3   bnj1522 35772
      ​21.6  Mathbox for BTernaryTau
            21.6.1  First-order logic   nfan1c 35773
                  21.6.1.1  Auxiliary axiom schemes   nfan1c 35773
            21.6.2  ZF set theory   reu6f 35779
                  21.6.2.1  Ordinals 5 through 9   c5o 35853
                  21.6.2.2  Finitism   prcinf 35881
                  21.6.2.3  Introduce ax-regs   ax-regs 35894
                  21.6.2.4  Derive ax-regs   axregs 35907
                  21.6.2.5  ZFC axioms with reduced distinct variable conditions   axsepg2 35908
                  21.6.2.6  Cardinality without the Axiom of Choice   ckard 35917
                  21.6.2.7  Gödel operations   ccnv2 35941
                  21.6.2.8  The constructible universe   clexo 35980
                  21.6.2.9  Global choice   gblacfnacd 36029
            21.6.3  Real and complex numbers   zltp1ne 36057
            21.6.4  Graph theory   cplgredgex 36062
                  21.6.4.1  Acyclic graphs   acycgr0v 36070
            21.6.5  Model theory   wsmext 36081
      ​21.7  Mathbox for Mario Carneiro
            21.7.1  Predicate calculus with all distinct variables   ax-7d 36091
            21.7.2  Miscellaneous stuff   quartfull 36097
            21.7.3  Derangements and the Subfactorial   deranglem 36098
            21.7.4  The Erdős-Szekeres theorem   erdszelem1 36123
            21.7.5  The Kuratowski closure-complement theorem   kur14lem1 36138
            21.7.6  Retracts and sections   cretr 36149
            21.7.7  Path-connected and simply connected spaces   cpconn 36151
            21.7.8  Covering maps   ccvm 36187
            21.7.9  Normal numbers   snmlff 36261
            21.7.10  Godel-sets of formulas - part 1   cgoe 36265
            21.7.11  Godel-sets of formulas - part 2   cgon 36364
            21.7.12  Models of ZF   cgze 36378
            ​*21.7.13  Metamath formal systems   cmcn 36392
            21.7.14  Grammatical formal systems   cm0s 36517
            21.7.15  Models of formal systems   cmuv 36537
            21.7.16  Splitting fields   ccpms 36559
            21.7.17  p-adic number fields   czr 36579
      ​​*21.8  Mathbox for Filip Cernatescu
      ​21.9  Mathbox for Paul Chapman
            21.9.1  Real and complex numbers (cont.)   climuzcnv 36603
            21.9.2  Miscellaneous theorems   elfzm12 36607
      ​21.10  Mathbox for Hongxiu Chen
      ​21.11  Mathbox for Adrian Ducourtial
            21.11.1  Propositional calculus   currybi 36620
            21.11.2  Set theory   mirimanoffr 36627
            21.11.3  Clone theory   ccloneop 36638
      ​21.12  Mathbox for Scott Fenton
            21.12.1  ZFC Axioms in primitive form   axextprim 36644
            21.12.2  Untangled classes   untelirr 36651
            21.12.3  Extra propositional calculus theorems   3jaodd 36658
            21.12.4  Misc. Useful Theorems   nepss 36661
            21.12.5  Properties of real and complex numbers   sqdivzi 36671
            21.12.6  Infinite products   iprodefisumlem 36683
            21.12.7  Factorial limits   faclimlem1 36686
            21.12.8  Greatest common divisor and divisibility   gcd32 36692
            21.12.9  Properties of relationships   dftr6 36694
            21.12.10  Properties of functions and mappings   funpsstri 36709
            21.12.11  Ordinal numbers   elpotr 36721
            21.12.12  Defined equality axioms   axextdfeq 36737
            21.12.13  Hypothesis builders   hbntg 36745
            21.12.14  Well-founded zero, successor, and limits   cwsuc 36750
            21.12.15  Quantifier-free definitions   ctxp 36770
            21.12.16  Alternate ordered pairs   caltop 36899
            21.12.17  Geometry in the Euclidean space   cofs 36925
                  21.12.17.1  Congruence properties   cofs 36925
                  21.12.17.2  Betweenness properties   btwntriv2 36955
                  21.12.17.3  Segment Transportation   ctransport 36972
                  21.12.17.4  Properties relating betweenness and congruence   cifs 36978
                  21.12.17.5  Connectivity of betweenness   btwnconn1lem1 37030
                  21.12.17.6  Segment less than or equal to   csegle 37049
                  21.12.17.7  Outside-of relationship   coutsideof 37062
                  21.12.17.8  Lines and Rays   cline2 37077
            21.12.18  Forward difference   cfwddif 37101
            21.12.19  Rank theorems   rank0 37109
            21.12.20  Hereditarily Finite Sets   hftr 37111
            21.12.21  Natural ordinal operations   cnmul 37114
      ​21.13  Mathbox for Gino Giotto
            21.13.1  Equality theorems   rmoeqi 37154
                  21.13.1.1  Inference versions   rmoeqi 37154
                  21.13.1.2  Deduction versions   rmoeqdv 37179
            21.13.2  Change bound variables   in-ax8 37191
                  21.13.2.1  Change bound variables and domains   cbvralvw2 37193
                  21.13.2.2  Change bound variables, deduction versions   cbvmodavw 37217
                  21.13.2.3  Change bound variables and domains, deduction versions   cbvrmodavw2 37250
            21.13.3  Study of ax-mulf usage   mpomulnzcnf 37266
      ​21.14  Mathbox for Jeff Hankins
            21.14.1  Miscellany   a1i14 37267
            21.14.2  Basic topological facts   topbnd 37290
            21.14.3  Topology of the real numbers   ivthALT 37301
            21.14.4  Refinements   cfne 37302
            21.14.5  Neighborhood bases determine topologies   neibastop1 37325
            21.14.6  Lattice structure of topologies   topmtcl 37329
            21.14.7  Filter bases   fgmin 37336
            21.14.8  Directed sets, nets   tailfval 37338
      ​21.15  Mathbox for Anthony Hart
            21.15.1  Propositional Calculus   tb-ax1 37349
            21.15.2  Predicate Calculus   nalfal 37369
            21.15.3  Miscellaneous single axioms   meran1 37377
            21.15.4  Connective Symmetry   negsym1 37383
      ​21.16  Mathbox for Chen-Pang He
            21.16.1  Ordinal topology   ontopbas 37394
      ​21.17  Mathbox for Jeff Hoffman
            21.17.1  Inferences for finite induction on generic function values   fveleq 37417
            21.17.2  gdc.mm   nnssi2 37421
      ​21.18  Mathbox for Matthew House
            21.18.1  Relations on well-ordered indexed unions   weiunval 37428
            21.18.2  Axiom of Transitive Containment   axtco 37437
            21.18.3  Transitive closure of a class   tr0elw 37450
            ​*21.18.4  Stronger axioms of regularity   mh-setind 37502
            21.18.5  Short axioms written in primitive symbols   mh-inf3f1 37507
      ​21.19  Mathbox for Asger C. Ipsen
            21.19.1  Continuous nowhere differentiable functions   dnival 37515
      ​​*21.20  Mathbox for BJ
            *21.20.1  Propositional calculus   bj-mp2c 37584
                  *21.20.1.1  Derived rules of inference   bj-mp2c 37584
                  ​*21.20.1.2  A syntactic theorem   bj-0 37586
                  ​*21.20.1.3  Minimal implicational calculus   bj-poni 37588
                  ​*21.20.1.4  Positive calculus   bj-bisimpl 37600
                  ​*21.20.1.5  Implication and negation   bj-con2com 37608
                  ​*21.20.1.6  Disjunction   bj-jaoi1 37619
                  ​*21.20.1.7  Logical equivalence   bj-dfbi4 37621
                  21.20.1.8  The conditional operator for propositions   bj-consensus 37626
                  ​*21.20.1.9  Propositional calculus: miscellaneous   bj-imbi12 37631
            ​*21.20.2  Modal logic   bj-axdd2 37640
            ​*21.20.3  Provability logic   cprvb 37645
            ​*21.20.4  First-order logic   bj-exexalal 37654
                  21.20.4.1  Universal and existential quantifiers, nonfreeness predicate   bj-exexalal 37654
                  21.20.4.2  Adding ax-gen   bj-genr 37655
                  21.20.4.3  Adding ax-4   bj-almp 37659
                  21.20.4.4  Adding ax-5   bj-spvw 37712
                  21.20.4.5  Equality and substitution   bj-df-sb 37727
                  21.20.4.6  Adding ax-6   bj-spim0 37746
                  21.20.4.7  Adding ax-7   bj-cbvexw 37754
                  21.20.4.8  Membership predicate, ax-8 and ax-9   bj-ax89 37756
                  21.20.4.9  Adding ax-11   bj-alcomexcom 37758
                  21.20.4.10  Adding ax-12   axc11n11 37762
                  ​*21.20.4.11  Really adding ax-12   bj-substax12 37804
                  21.20.4.12  Nonfreeness   wnnf 37806
                  21.20.4.13  Adding ax-13   bj-axc10 37873
                  ​*21.20.4.14  Removing dependencies on ax-13 (and ax-11)   bj-axc10v 37883
                  ​*21.20.4.15  Distinct var metavariables   bj-hbaeb2 37908
                  ​*21.20.4.16  Around ~ equsal   bj-equsal1t 37912
                  ​*21.20.4.17  Some Principia Mathematica proofs   stdpc5t 37917
                  21.20.4.18  Alternate definition of substitution   bj-sbsb 37927
                  21.20.4.19  Lemmas for substitution   bj-sbf3 37929
                  21.20.4.20  Existential uniqueness   bj-eu3f 37931
                  ​*21.20.4.21  First-order logic: miscellaneous   bj-sblem1 37932
            21.20.5  Set theory   eliminable1 37949
                  ​*21.20.5.1  Eliminability of class terms   eliminable1 37949
                  ​*21.20.5.2  Classes without the axiom of extensionality   bj-denoteslem 37961
                  21.20.5.3  Characterization among sets versus among classes   elelb 37987
                  ​*21.20.5.4  The nonfreeness quantifier for classes   bj-nfcsym 37989
                  ​*21.20.5.5  Lemmas for class substitution   bj-sbeqALT 37990
                  21.20.5.6  Removing some axiom requirements and disjoint variable conditions   bj-exlimvmpi 38001
                  ​*21.20.5.7  Class abstractions   bj-elabd2ALT 38016
                  21.20.5.8  Generalized class abstractions   bj-cgab 38024
                  ​*21.20.5.9  Restricted nonfreeness   wrnf 38032
                  ​*21.20.5.10  Russell's paradox   bj-ru1 38034
                  21.20.5.11  Curry's paradox in set theory   currysetlem 38036
                  ​*21.20.5.12  Some disjointness results   bj-n0i 38042
                  ​*21.20.5.13  Complements on direct products   bj-xpimasn 38046
                  ​*21.20.5.14  "Singletonization" and tagging   bj-snsetex 38054
                  ​*21.20.5.15  Tuples of classes   bj-cproj 38081
                  ​*21.20.5.16  Set theory: elementary operations relative to a universe   bj-rcleqf 38116
                  ​*21.20.5.17  Axioms for finite unions   bj-abex 38121
                  ​*21.20.5.18  Relations, functions, composition: supplements   bj-elinxp 38138
                  ​*21.20.5.19  Set theory: miscellaneous   eleq2w2ALT 38153
                  ​*21.20.5.20  Axioms of separation and replacement   bj-axnul 38179
                  ​*21.20.5.21  Evaluation at a class   bj-evaleq 38183
                  21.20.5.22  Elementwise operations   celwise 38191
                  ​*21.20.5.23  Elementwise intersection (families of sets induced on a subset)   bj-rest00 38193
                  21.20.5.24  Moore collections (complements)   bj-raldifsn 38212
                  21.20.5.25  Maps-to notation for functions with three arguments   bj-0nelmpt 38229
                  ​*21.20.5.26  Currying   csethom 38233
                  ​*21.20.5.27  Setting components of extensible structures   cstrset 38245
            ​*21.20.6  Extended real and complex numbers, real and complex projective lines   bj-nfald 38248
                  21.20.6.1  Complements on class abstractions of ordered pairs and binary relations   bj-nfald 38248
                  ​*21.20.6.2  Identity relation (complements)   bj-opabssvv 38263
                  ​*21.20.6.3  Functionalized identity (diagonal in a Cartesian square)   cdiag2 38285
                  ​*21.20.6.4  Direct image and inverse image   cimdir 38291
                  ​*21.20.6.5  Extended numbers and projective lines as sets   cfractemp 38309
                  ​*21.20.6.6  Addition and opposite   caddcc 38350
                  ​*21.20.6.7  Order relation on the extended reals   cltxr 38354
                  ​*21.20.6.8  Argument, multiplication and inverse   carg 38356
                  21.20.6.9  The canonical bijection from the finite ordinals   ciomnn 38362
                  21.20.6.10  Divisibility   cnnbar 38373
            ​*21.20.7  Monoids   bj-smgrpssmgm 38381
                  ​*21.20.7.1  Finite sums in monoids   cfinsum 38396
            ​*21.20.8  Affine, Euclidean, and Cartesian geometry   bj-fvimacnv0 38399
                  *21.20.8.1  Real vector spaces   bj-fvimacnv0 38399
                  ​*21.20.8.2  Complex numbers (supplements)   bj-subcom 38421
                  ​*21.20.8.3  Barycentric coordinates   bj-bary1lem 38423
            21.20.9  Monoid of endomorphisms   cend 38426
      ​21.21  Mathbox for Jim Kingdon
            21.21.1  Circle constant   taupilem3 38432
            21.21.2  Number theory   dfgcd3 38437
            21.21.3  Real numbers   irrdifflemf 38438
      ​21.22  Mathbox for ML
            21.22.1  Miscellaneous   csbrecsg 38443
            21.22.2  Cartesian exponentiation   cfinxp 38498
            21.22.3  Topology   iunctb2 38518
                  ​*21.22.3.1  Pi-base theorems   pibp16 38528
      ​21.23  Mathbox for Wolf Lammen
            21.23.1  1. Bootstrapping   wl-section-boot 38537
            21.23.2  Implication chains   wl-section-impchain 38561
            21.23.3  Theorems around the conditional operator   wl-ifp-ncond1 38579
            21.23.4  Alternative development of hadd, cadd   wl-df-3xor 38583
            21.23.5  An alternative axiom ~ ax-13   ax-wl-13v 38608
            21.23.6  Bootstrapping set theory with classes   wl-cleq-0 38610
            21.23.7  Other stuff   wl-mps 38631
      ​21.24  Mathbox for Brendan Leahy
      ​21.25  Mathbox for Thomas van Maaren
            ​*21.25.1  The language of Propositional Calculus.   cpropvar 38825
      ​21.26  Mathbox for Jeff Madsen
            21.26.1  Logic and set theory   unirep 38840
            21.26.2  Real and complex numbers; integers   filbcmb 38866
            21.26.3  Sequences and sums   sdclem2 38868
            21.26.4  Topology   subspopn 38878
            21.26.5  Metric spaces   metf1o 38881
            21.26.6  Continuous maps and homeomorphisms   constcncf 38888
            21.26.7  Boundedness   ctotbnd 38892
            21.26.8  Isometries   cismty 38924
            21.26.9  Heine-Borel Theorem   heibor1lem 38935
            21.26.10  Banach Fixed Point Theorem   bfplem1 38948
            21.26.11  Euclidean space   crrn 38951
            21.26.12  Intervals (continued)   ismrer1 38964
            ​*21.26.13  Operation properties   cass 38968
            21.26.14  Groups and related structures   cmagm 38974
            21.26.15  Group homomorphism and isomorphism   cghomOLD 39009
            21.26.16  Rings   crngo 39020
            21.26.17  Division Rings   cdrng 39074
            21.26.18  Ring homomorphisms   crngohom 39086
            21.26.19  Commutative rings   ccm2 39115
            21.26.20  Ideals   cidl 39133
            21.26.21  Prime rings and integral domains   cprrng 39172
            21.26.22  Ideal generators   cigen 39185
      ​21.27  Mathbox for Giovanni Mascellani
            ​*21.27.1  Tools for automatic proof building   efald2 39204
            ​*21.27.2  Tseitin axioms   fald 39253
            ​*21.27.3  Equality deductions   iuneq2f 39280
            ​*21.27.4  Miscellanea   orcomdd 39291
      ​21.28  Mathbox for Peter Mazsa
            21.28.1  Notations   cxrn 39298
            21.28.2  Preparatory theorems   el2v1 39353
            21.28.3  Range Cartesian product   df-xrn 39504
            21.28.4  Relations   df-rels 39564
            21.28.5  Quotient map (coset map)   df-qmap 39570
            21.28.6  Lifts, shifts, successor, and predecessor   df-adjliftmap 39579
            21.28.7  Cosets by ` R `   df-coss 39625
            21.28.8  Subset relations   df-ssr 39702
            21.28.9  Reflexivity   df-refs 39714
            21.28.10  Converse reflexivity   df-cnvrefs 39729
            21.28.11  Symmetry   df-syms 39746
            21.28.12  Reflexivity and symmetry   symrefref2 39771
            21.28.13  Transitivity   df-trs 39780
            21.28.14  Equivalence relations   df-eqvrels 39792
            21.28.15  Redundancy   df-redunds 39831
            21.28.16  Domain quotients   df-dmqss 39846
            21.28.17  Equivalence relations on domain quotients   df-ers 39872
            21.28.18  Functions   df-funss 39889
            21.28.19  Disjoints vs. converse functions   df-disjss 39912
            21.28.20  Antisymmetry   df-antisymrel 39987
            21.28.21  Partitions: disjoints on domain quotients   df-parts 39992
            21.28.22  Partition-Equivalence Theorems   disjim 40008
            21.28.23  Type-safe Partition-Equivalence: PetParts, PetErs, Pet2Parts, Pet2Ers   df-petparts 40092
      ​21.29  Mathbox for Rodolfo Medina
            21.29.1  Partitions   prtlem60 40102
      ​​*21.30  Mathbox for Norm Megill
            *21.30.1  Obsolete schemes ax-c4,c5,c7,c10,c11,c11n,c15,c9,c14,c16   ax-c5 40132
            ​*21.30.2  Rederive new axioms ax-4, ax-10, ax-6, ax-12, ax-13 from old   axc5 40142
            ​*21.30.3  Legacy theorems using obsolete axioms   ax5ALT 40156
            21.30.4  Experiments with weak deduction theorem   elimhyps 40210
            21.30.5  Miscellanea   cnaddcom 40221
            21.30.6  Atoms, hyperplanes, and covering in a left vector space (or module)   clsa 40223
            21.30.7  Functionals and kernels of a left vector space (or module)   clfn 40306
            21.30.8  Opposite rings and dual vector spaces   cld 40372
            21.30.9  Ortholattices and orthomodular lattices   cops 40421
            21.30.10  Atomic lattices with covering property   ccvr 40511
            21.30.11  Hilbert lattices   chlt 40599
            21.30.12  Projective geometries based on Hilbert lattices   clln 40740
            21.30.13  Construction of a vector space from a Hilbert lattice   cdlema1N 41040
            21.30.14  Construction of involution and inner product from a Hilbert lattice   clpoN 42729
      ​21.31  Mathbox for metakunt
            21.31.1  Commutative Semiring   ccsrg 43211
            21.31.2  General helpful statements   rhmzrhval 43214
            21.31.3  Some gcd and lcm results   12gcd5e1 43245
            21.31.4  Least common multiple inequality theorem   3factsumint1 43263
            21.31.5  Logarithm inequalities   3exp7 43295
            21.31.6  Miscellaneous results for AKS formalisation   intlewftc 43303
            21.31.7  Sticks and stones   sticksstones1 43388
            21.31.8  Continuation AKS   aks6d1c6lem1 43412
      ​21.32  Mathbox for Luke Murphy
            21.32.1  Solutions of quadratic equations   quadfac 43447
            21.32.2  April Fool's theorem   25or6to4 43448
      ​21.33  Mathbox for Steven Nguyen
            21.33.1  Utility theorems   jarrii 43449
            ​*21.33.2  Arithmetic theorems   c0exALT 43496
            21.33.3  Exponents and divisibility   oexpreposd 43572
            21.33.4  Trigonometry and Calculus   tanhalfpim 43593
            ​*21.33.5  Independence of ax-mulcom   cresub 43609
            21.33.6  Structures   sn-base0 43752
            ​*21.33.7  Projective spaces   cprjsp 43822
            21.33.8  Basic reductions for Fermat's Last Theorem   dffltz 43863
            ​*21.33.9  Exemplar theorems   iddii 43868
                  ​*21.33.9.1  Standard replacements of ax-10 , ax-11 , ax-12   nfa1w 43879
      ​21.34  Mathbox for Igor Ieskov
      ​21.35  Mathbox for Timothy J. O'Malley
      ​21.36  Mathbox for OpenAI
      ​21.37  Mathbox for Stefan O'Rear
            21.37.1  Additional elementary logic and set theory   moxfr 43896
            21.37.2  Additional theory of functions   imaiinfv 43897
            21.37.3  Additional topology   elrfi 43898
            21.37.4  Characterization of closure operators. Kuratowski closure axioms   ismrcd1 43902
            21.37.5  Algebraic closure systems   cnacs 43906
            21.37.6  Miscellanea 1. Map utilities   constmap 43917
            21.37.7  Miscellanea for polynomials   mptfcl 43924
            21.37.8  Multivariate polynomials over the integers   cmzpcl 43925
            21.37.9  Miscellanea for Diophantine sets 1   coeq0i 43957
            21.37.10  Diophantine sets 1: definitions   cdioph 43959
            21.37.11  Diophantine sets 2 miscellanea   ellz1 43971
            21.37.12  Diophantine sets 2: union and intersection. Monotone Boolean algebra   diophin 43976
            21.37.13  Diophantine sets 3: construction   diophrex 43979
            21.37.14  Diophantine sets 4 miscellanea   2sbcrex 43988
            21.37.15  Diophantine sets 4: Quantification   rexrabdioph 43994
            21.37.16  Diophantine sets 5: Arithmetic sets   rabdiophlem1 44001
            21.37.17  Diophantine sets 6: reusability. renumbering of variables   eldioph4b 44011
            21.37.18  Pigeonhole Principle and cardinality helpers   fphpd 44016
            21.37.19  A non-closed set of reals is infinite   rencldnfilem 44020
            21.37.20  Lagrange's rational approximation theorem   irrapxlem1 44022
            21.37.21  Pell equations 1: A nontrivial solution always exists   pellexlem1 44029
            21.37.22  Pell equations 2: Algebraic number theory of the solution set   csquarenn 44036
            21.37.23  Pell equations 3: characterizing fundamental solution   infmrgelbi 44078
            ​*21.37.24  Logarithm laws generalized to an arbitrary base   reglogcl 44090
            21.37.25  Pell equations 4: the positive solution group is infinite cyclic   pellfund14 44098
            21.37.26  X and Y sequences 1: Definition and recurrence laws   crmx 44100
            21.37.27  Ordering and induction lemmas for the integers   monotuz 44141
            21.37.28  X and Y sequences 2: Order properties   rmxypos 44147
            21.37.29  Congruential equations   congtr 44165
            21.37.30  Alternating congruential equations   acongid 44175
            21.37.31  Additional theorems on integer divisibility   coprmdvdsb 44185
            21.37.32  X and Y sequences 3: Divisibility properties   jm2.18 44188
            21.37.33  X and Y sequences 4: Diophantine representability of Y   jm2.27a 44205
            21.37.34  X and Y sequences 5: Diophantine representability of X, ^, _C   rmxdiophlem 44215
            21.37.35  Uncategorized stuff not associated with a major project   setindtr 44224
            21.37.36  More equivalents of the Axiom of Choice   axac10 44233
            21.37.37  Finitely generated left modules   clfig 44262
            21.37.38  Noetherian left modules I   clnm 44270
            21.37.39  Addenda for structure powers   pwssplit4 44284
            21.37.40  Every set admits a group structure iff choice   unxpwdom3 44290
            21.37.41  Noetherian rings and left modules II   clnr 44303
            21.37.42  Hilbert's Basis Theorem   cldgis 44315
            21.37.43  Additional material on polynomials [DEPRECATED]   cmnc 44325
            21.37.44  Degree and minimal polynomial of algebraic numbers   cdgraa 44334
            21.37.45  Algebraic integers I   citgo 44351
            21.37.46  Endomorphism algebra   cmend 44365
            21.37.47  Cyclic groups and order   idomodle 44385
            21.37.48  Cyclotomic polynomials   ccytp 44391
            21.37.49  Miscellaneous topology   fgraphopab 44397
      ​21.38  Mathbox for Noam Pasman
      ​21.39  Mathbox for Jon Pennant
      ​21.40  Mathbox for Richard Penner
            21.40.1  Set Theory and Ordinal Numbers   uniel 44411
            21.40.2  Natural addition of Cantor normal forms   oawordex2 44520
            21.40.3  Surreal Contributions   abeqabi 44601
            21.40.4  Short Studies   nlimsuc 44634
                  21.40.4.1  Additional work on conditional logical operator   ifpan123g 44652
                  21.40.4.2  Sophisms   rp-fakeimass 44705
                  ​*21.40.4.3  Finite Sets   rp-isfinite5 44710
                  21.40.4.4  General Observations   intabssd 44712
                  21.40.4.5  Infinite Sets   pwelg 44753
                  ​*21.40.4.6  Finite intersection property   fipjust 44758
                  21.40.4.7  RP ADDTO: Subclasses and subsets   rababg 44767
                  21.40.4.8  RP ADDTO: The intersection of a class   elinintab 44768
                  21.40.4.9  RP ADDTO: Theorems requiring subset and intersection existence   elinintrab 44770
                  21.40.4.10  RP ADDTO: Relations   xpinintabd 44773
                  ​*21.40.4.11  RP ADDTO: Functions   elmapintab 44788
                  ​*21.40.4.12  RP ADDTO: Finite induction (for finite ordinals)   cnvcnvintabd 44792
                  21.40.4.13  RP ADDTO: First and second members of an ordered pair   elcnvlem 44793
                  21.40.4.14  RP ADDTO: The reflexive and transitive properties of relations   undmrnresiss 44796
                  21.40.4.15  RP ADDTO: Basic properties of closures   cleq2lem 44800
                  21.40.4.16  RP REPLACE: Definitions and basic properties of transitive closures   trcleq2lemRP 44822
                  ​*21.40.4.17  Additions for square root; absolute value   sqrtcvallem1 44823
            21.40.5  Additional statements on relations and subclasses   al3im 44839
                  21.40.5.1  Transitive relations (not to be confused with transitive classes)   trrelind 44857
                  21.40.5.2  Reflexive closures   crcl 44864
                  ​*21.40.5.3  Finite relationship composition   relexp2 44869
                  21.40.5.4  Transitive closure of a relation   dftrcl3 44912
                  ​*21.40.5.5  Adapted from Frege   frege77d 44938
            ​*21.40.6  Propositions from _Begriffsschrift_   dfxor4 44958
                  *21.40.6.1  _Begriffsschrift_ Chapter I   dfxor4 44958
                  ​*21.40.6.2  _Begriffsschrift_ Notation hints   whe 44964
                  21.40.6.3  _Begriffsschrift_ Chapter II Implication   ax-frege1 44982
                  21.40.6.4  _Begriffsschrift_ Chapter II Implication and Negation   axfrege28 45021
                  ​*21.40.6.5  _Begriffsschrift_ Chapter II with logical equivalence   axfrege52a 45048
                  21.40.6.6  _Begriffsschrift_ Chapter II with equivalence of sets   axfrege52c 45079
                  ​*21.40.6.7  _Begriffsschrift_ Chapter II with equivalence of classes   frege53c 45106
                  ​*21.40.6.8  _Begriffsschrift_ Chapter III Properties hereditary in a sequence   dffrege69 45124
                  ​*21.40.6.9  _Begriffsschrift_ Chapter III Following in a sequence   dffrege76 45131
                  ​*21.40.6.10  _Begriffsschrift_ Chapter III Member of sequence   dffrege99 45154
                  ​*21.40.6.11  _Begriffsschrift_ Chapter III Single-valued procedures   dffrege115 45170
            ​*21.40.7  Exploring Topology via Seifert and Threlfall   enrelmap 45189
                  *21.40.7.1  Equinumerosity of sets of relations and maps   enrelmap 45189
                  ​*21.40.7.2  Generic Pseudoclosure Spaces, Pseudointerior Spaces, and Pseudoneighborhoods   or3or 45215
                  ​*21.40.7.3  Generic Neighborhood Spaces   gneispa 45322
            ​*21.40.8  Exploring Higher Homotopy via Kerodon   k0004lem1 45339
                  *21.40.8.1  Simplicial Sets   k0004lem1 45339
      ​21.41  Mathbox for Stanislas Polu
            21.41.1  IMO Problems   wwlemuld 45348
                  21.41.1.1  IMO 1972 B2   wwlemuld 45348
            ​*21.41.2  INT Inequalities Proof Generator   int-addcomd 45365
            ​*21.41.3  N-Digit Addition Proof Generator   unitadd 45387
            21.41.4  AM-GM (for k = 2,3,4)   gsumws3 45388
      ​21.42  Mathbox for Rohan Ridenour
            21.42.1  Misc   spALT 45393
            21.42.2  Monoid rings   cmnring 45401
            21.42.3  Shorter primitive equivalent of ax-groth   gru0eld 45419
                  21.42.3.1  Grothendieck universes are closed under collection   gru0eld 45419
                  21.42.3.2  Minimal universes   ismnu 45437
                  21.42.3.3  Primitive equivalent of ax-groth   expandan 45464
      ​21.43  Mathbox for Steve Rodriguez
            21.43.1  Miscellanea   nanorxor 45481
            21.43.2  Ratio test for infinite series convergence and divergence   dvgrat 45488
            21.43.3  Multiples   reldvds 45491
            21.43.4  Function operations   caofcan 45499
            21.43.5  Calculus   lhe4.4ex1a 45505
            21.43.6  The generalized binomial coefficient operation   cbcc 45512
            21.43.7  Binomial series   uzmptshftfval 45522
      ​21.44  Mathbox for Andrew Salmon
            21.44.1  Principia Mathematica * 10   pm10.12 45534
            21.44.2  Principia Mathematica * 11   2alanimi 45548
            21.44.3  Predicate Calculus   sbeqal1 45574
            21.44.4  Principia Mathematica * 13 and * 14   pm13.13a 45583
            21.44.5  Set Theory   elnev 45612
            21.44.6  Arithmetic   addcomgi 45629
            21.44.7  Geometry   cplusr 45630
      ​​*21.45  Mathbox for Alan Sare
            21.45.1  Auxiliary theorems for the Virtual Deduction tool   idiALT 45652
            21.45.2  Supplementary unification deductions   bi1imp 45656
            21.45.3  Conventional Metamath proofs, some derived from VD proofs   iidn3 45675
            21.45.4  What is Virtual Deduction?   wvd1 45743
            21.45.5  Virtual Deduction Theorems   df-vd1 45744
            21.45.6  Theorems proved using Virtual Deduction   trsspwALT 45991
            21.45.7  Theorems proved using Virtual Deduction with mmj2 assistance   simplbi2VD 46019
            21.45.8  Virtual Deduction transcriptions of textbook proofs   sb5ALTVD 46086
            21.45.9  Theorems proved using conjunction-form Virtual Deduction   elpwgdedVD 46090
            21.45.10  Theorems with a VD proof in conventional notation derived from a VD proof   suctrALT3 46097
            ​*21.45.11  Theorems with a proof in conventional notation derived from a VD proof   notnotrALT2 46100
      ​21.46  Mathbox for Eric Schmidt
            21.46.1  Miscellany   rspesbcd 46111
            21.46.2  Study of dfbi1ALT   dfbi1ALTa 46124
            21.46.3  Relation-preserving functions   wrelp 46127
            21.46.4  Orbits   orbitex 46140
            21.46.5  Well-founded sets   trwf 46144
            21.46.6  Absoluteness in transitive models   ralabso 46153
            21.46.7  Lemmas for showing axioms hold in models   traxext 46162
            21.46.8  The class of well-founded sets is a model for ZFC   wfaxext 46178
            21.46.9  Permutation models   brpermmodel 46188
            21.46.10  Isomorphism of finite ordinals and non-negative integers   hashnna 46204
            21.46.11  Hereditarily finite sets   hfpr 46211
            21.46.12  Ackermann bijection   cackbij 46218
            21.46.13  Hereditarily finite structures   chfstruct 46221
      ​21.47  Mathbox for Glauco Siliprandi
            21.47.1  Miscellanea   evth2f 46237
            21.47.2  Functions   fnresdmss 46388
            21.47.3  Ordering on real numbers - Real and complex numbers basic operations   sub2times 46494
            21.47.4  Real intervals   gtnelioc 46708
            21.47.5  Finite sums   fsummulc1f 46788
            21.47.6  Finite multiplication of numbers and finite multiplication of functions   fmul01 46797
            21.47.7  Limits   clim1fr1 46818
                  21.47.7.1  Inferior limit (lim inf)   clsi 46966
                  ​*21.47.7.2  Limits for sequences of extended real numbers   clsxlim 47033
            21.47.8  Trigonometry   coseq0 47079
            21.47.9  Continuous Functions   mulcncff 47085
            21.47.10  Derivatives   dvsinexp 47126
            21.47.11  Integrals   itgsin0pilem1 47165
            21.47.12  Stone Weierstrass theorem - real version   stoweidlem1 47216
            21.47.13  Wallis' product for π   wallispilem1 47280
            21.47.14  Stirling's approximation formula for ` n ` factorial   stirlinglem1 47289
            21.47.15  Dirichlet kernel   dirkerval 47306
            21.47.16  Fourier Series   fourierdlem1 47323
            21.47.17  e is transcendental   elaa2lem 47448
            21.47.18  n-dimensional Euclidean space   rrxtopn 47499
            21.47.19  Basic measure theory   csalg 47523
                  ​*21.47.19.1  σ-Algebras   csalg 47523
                  21.47.19.2  Sum of nonnegative extended reals   csumge0 47577
                  ​*21.47.19.3  Measures   cmea 47664
                  ​*21.47.19.4  Outer measures and Caratheodory's construction   come 47704
                  ​*21.47.19.5  Lebesgue measure on n-dimensional Real numbers   covoln 47751
                  ​*21.47.19.6  Measurable functions   csmblfn 47910
      ​21.48  Mathbox for Saveliy Skresanov
            21.48.1  Ceva's theorem   sigarval 48065
            21.48.2  Simple groups   simpcntrab 48085
      ​21.49  Mathbox for Ender Ting
            21.49.1  Interesting facts   et-ltneverrefl 48086
            21.49.2  Increasing sequences and subsequences   ormklocald 48091
            21.49.3  Scratchpad for number theory   evenwodadd 48116
            21.49.4  Scratchpad for math on real numbers   squeezedltsq 48117
            21.49.5  Turing Machine Finite Reach theorem   tmachlem-extapes 48149
      ​21.50  Mathbox for Jarvin Udandy
      ​21.51  Mathbox for Adhemar
            ​*21.51.1  Minimal implicational calculus   adh-minim 48276
      ​21.52  Mathbox for Alexander van der Vekens
            21.52.1  General auxiliary theorems (1)   n0nsn2el 48300
                  21.52.1.1  Unordered and ordered pairs - extension for singletons   n0nsn2el 48300
                  21.52.1.2  Unordered and ordered pairs - extension for unordered pairs   elprneb 48304
                  21.52.1.3  Unordered and ordered pairs - extension for ordered pairs   oppr 48305
                  21.52.1.4  Relations - extension   eubrv 48310
                  21.52.1.5  Definite description binder (inverted iota) - extension   iota0def 48313
                  21.52.1.6  Functions - extension   fveqvfvv 48315
            21.52.2  Alternative for Russell's definition of a description binder   caiota 48358
            21.52.3  Double restricted existential uniqueness   r19.32 48374
                  21.52.3.1  Restricted quantification (extension)   r19.32 48374
                  21.52.3.2  Restricted uniqueness and "at most one" quantification   reuf1odnf 48383
                  21.52.3.3  Analogs to Existential uniqueness (double quantification)   2reu3 48386
                  21.52.3.4  Additional theorems for double restricted existential uniqueness   2reu8i 48389
            ​*21.52.4  Alternative definitions of function and operation values   wdfat 48392
                  21.52.4.1  Restricted quantification (extension)   ralbinrald 48398
                  21.52.4.2  Introduce the Axiom of Power Sets (extension)   alneu 48399
                  21.52.4.3  Predicate "defined at"   dfateq12d 48401
                  21.52.4.4  Alternative definition of the value of a function   dfafv2 48407
                  21.52.4.5  Alternative definition of the value of an operation   aoveq123d 48453
            ​*21.52.5  Alternative definitions of function values (2)   cafv2 48483
            21.52.6  General auxiliary theorems (2)   an4com24 48543
                  21.52.6.1  Logical conjunction - extension   an4com24 48543
                  21.52.6.2  Abbreviated conjunction and disjunction of three wff's - extension   3an4ancom24 48544
                  21.52.6.3  Negated membership (alternative)   cnelbr 48546
                  21.52.6.4  The empty set - extension   ralralimp 48553
                  21.52.6.5  Indexed union and intersection - extension   otiunsndisjX 48554
                  21.52.6.6  Functions - extension   fvifeq 48555
                  21.52.6.7  Maps-to notation - extension   fvmptrab 48567
                  21.52.6.8  Subtraction - extension   cnambpcma 48569
                  21.52.6.9  Ordering on reals (cont.) - extension   leaddsuble 48572
                  21.52.6.10  Imaginary and complex number properties - extension   readdcnnred 48578
                  21.52.6.11  Nonnegative integers (as a subset of complex numbers) - extension   nn0resubcl 48583
                  21.52.6.12  Integers (as a subset of complex numbers) - extension   zgeltp1eq 48584
                  21.52.6.13  Decimal arithmetic - extension   1t10e1p1e11 48585
                  21.52.6.14  Upper sets of integers - extension   eluzge0nn0 48587
                  21.52.6.15  Infinity and the extended real number system (cont.) - extension   nltle2tri 48588
                  21.52.6.16  Finite intervals of integers - extension   ssfz12 48589
                  21.52.6.17  Half-open integer ranges - extension   fzopred 48598
                  21.52.6.18  The floor and ceiling functions - extension   2ltceilhalf 48607
                  21.52.6.19  The modulo (remainder) operation - extension   fldivmod 48619
                  21.52.6.20  The infinite sequence builder "seq"   smonoord 48652
                  21.52.6.21  Integer powers - extension   2timesltsq 48653
                  21.52.6.22  Finite and infinite sums - extension   fsummsndifre 48655
                  21.52.6.23  The divides relation - extension   nndivides2 48659
                  21.52.6.24  Extensible structures - extension   setsidel 48663
            ​*21.52.7  Preimages of function values   preimafvsnel 48666
            ​*21.52.8  Partitions of real intervals   ciccp 48700
            21.52.9  Shifting functions with an integer range domain   fargshiftfv 48726
            21.52.10  Words over a set (extension)   lswn0 48731
                  21.52.10.1  Last symbol of a word - extension   lswn0 48731
            21.52.11  Unordered pairs   wich 48732
                  21.52.11.1  Interchangeable setvar variables   wich 48732
                  21.52.11.2  Set of unordered pairs   sprid 48761
                  ​*21.52.11.3  Proper (unordered) pairs   prpair 48788
                  21.52.11.4  Set of proper unordered pairs   cprpr 48799
            21.52.12  Number theory (extension)   nprmmul1 48814
                  21.52.12.1  Properties of non-prime numbers   nprmmul1 48814
                  ​*21.52.12.2  Fermat numbers   cfmtno 48817
                  ​*21.52.12.3  Mersenne primes   m2prm 48881
                  21.52.12.4  Proth's theorem   modexp2m1d 48902
                  21.52.12.5  The prime-counting function according to Ján Mináč   nprmdvdsfacm1lem1 48910
                  21.52.12.6  Solutions of quadratic equations   quad1 48923
            ​*21.52.13  Even and odd numbers   ceven 48927
                  21.52.13.1  Definitions and basic properties   ceven 48927
                  21.52.13.2  Alternate definitions using the "divides" relation   dfeven2 48952
                  21.52.13.3  Alternate definitions using the "modulo" operation   dfeven3 48961
                  21.52.13.4  Alternate definitions using the "gcd" operation   iseven5 48967
                  21.52.13.5  Theorems of part 5 revised   zneoALTV 48972
                  21.52.13.6  Theorems of part 6 revised   odd2np1ALTV 48977
                  21.52.13.7  Theorems of AV's mathbox revised   0evenALTV 48991
                  21.52.13.8  Additional theorems   epoo 49006
                  21.52.13.9  Perfect Number Theorem (revised)   perfectALTVlem1 49024
            21.52.14  Number theory (extension 2)   cfppr 49027
                  ​*21.52.14.1  Fermat pseudoprimes   cfppr 49027
                  ​*21.52.14.2  Goldbach's conjectures   cgbe 49048
            21.52.15  Graph theory (extension)   cclnbgr 49121
                  21.52.15.1  Closed neighborhood of a vertex   cclnbgr 49121
                  ​*21.52.15.2  Semiclosed and semiopen neighborhoods (experimental)   dfsclnbgr2 49149
                  21.52.15.3  Induced subgraphs   cisubgr 49163
                  ​*21.52.15.4  Isomorphisms of graphs   cgrisom 49177
                  ​*21.52.15.5  Triangles in graphs   cgrtri 49240
                  ​*21.52.15.6  Star graphs   cstgr 49254
                  ​*21.52.15.7  Local isomorphisms of graphs   cgrlim 49279
                  ​*21.52.15.8  Generalized Petersen graphs   cgpg 49343
                  21.52.15.9  Loop-free graphs - extension   1hegrlfgr 49435
                  21.52.15.10  Walks - extension   cupwlks 49436
                  21.52.15.11  Edges of graphs expressed as sets of unordered pairs   upgredgssspr 49446
            21.52.16  Monoids (extension)   ovn0dmfun 49459
                  21.52.16.1  Auxiliary theorems   ovn0dmfun 49459
                  21.52.16.2  Magmas, Semigroups and Monoids (extension)   plusfreseq 49466
                  21.52.16.3  Examples and counterexamples for magmas, semigroups and monoids (extension)   opmpoismgm 49469
                  21.52.16.4  Group sum operation (extension 1)   gsumsplit2f 49482
            ​*21.52.17  Magmas and internal binary operations (alternate approach)   ccllaw 49485
                  *21.52.17.1  Laws for internal binary operations   ccllaw 49485
                  ​*21.52.17.2  Internal binary operations   cintop 49498
                  21.52.17.3  Alternative definitions for magmas and semigroups   cmgm2 49517
            21.52.18  Rings (extension)   lmod0rng 49531
                  21.52.18.1  Nonzero rings (extension)   lmod0rng 49531
                  21.52.18.2  Ideals as non-unital rings   lidldomn1 49533
                  21.52.18.3  The non-unital ring of even integers   0even 49539
                  21.52.18.4  A constructed not unital ring   cznrnglem 49561
                  ​*21.52.18.5  The category of non-unital rings (alternate definition)   crngcALTV 49565
                  ​*21.52.18.6  The category of (unital) rings (alternate definition)   cringcALTV 49589
            ​*21.52.19  Prime rings (and integral domains)   cprmrng 49636
            21.52.20  Basic algebraic structures (extension)   eliunxp2 49651
                  21.52.20.1  Auxiliary theorems   eliunxp2 49651
                  21.52.20.2  The binomial coefficient operation (extension)   bcpascm1 49668
                  21.52.20.3  The ` ZZ `-module ` ZZ X. ZZ `   zlmodzxzlmod 49671
                  21.52.20.4  Group sum operation (extension 2)   mgpsumunsn 49678
                  21.52.20.5  Symmetric groups (extension)   exple2lt6 49681
                  21.52.20.6  Divisibility (extension)   invginvrid 49684
                  21.52.20.7  The support of functions (extension)   rmsupp0 49685
                  21.52.20.8  Finitely supported functions (extension)   rmsuppfi 49689
                  21.52.20.9  Left modules (extension)   lmodvsmdi 49696
                  21.52.20.10  Associative algebras (extension)   assaascl0 49698
                  21.52.20.11  Univariate polynomials (extension)   ply1vr1smo 49700
                  21.52.20.12  Univariate polynomials (examples)   linply1 49710
            21.52.21  Linear algebra (extension)   cdmatalt 49713
                  ​*21.52.21.1  The subalgebras of diagonal and scalar matrices (extension)   cdmatalt 49713
                  ​*21.52.21.2  Linear combinations   clinc 49721
                  ​*21.52.21.3  Linear independence   clininds 49757
                  21.52.21.4  Simple left modules and the ` ZZ `-module   lmod1lem1 49804
                  21.52.21.5  Differences between (left) modules and (left) vector spaces   lvecpsslmod 49824
            21.52.22  Complexity theory   suppdm 49827
                  21.52.22.1  Auxiliary theorems   suppdm 49827
                  21.52.22.2  Even and odd integers   nn0onn0ex 49840
                  21.52.22.3  The natural logarithm on complex numbers (extension)   logcxp0 49852
                  21.52.22.4  Division of functions   cfdiv 49854
                  21.52.22.5  Upper bounds   cbigo 49864
                  21.52.22.6  Logarithm to an arbitrary base (extension)   rege1logbrege0 49875
                  ​*21.52.22.7  The binary logarithm   fldivexpfllog2 49882
                  21.52.22.8  Binary length   cblen 49886
                  ​*21.52.22.9  Digits   cdig 49912
                  21.52.22.10  Nonnegative integer as sum of its shifted digits   dignn0flhalflem1 49932
                  21.52.22.11  Algorithms for the multiplication of nonnegative integers   nn0mulfsum 49941
                  ​*21.52.22.12  N-ary functions   cnaryf 49943
                  ​*21.52.22.13  The Ackermann function   citco 49974
            21.52.23  Elementary geometry (extension)   fv1prop 50016
                  21.52.23.1  Auxiliary theorems   fv1prop 50016
                  21.52.23.2  Real euclidean space of dimension 2   rrx2pxel 50028
                  21.52.23.3  Spheres and lines in real Euclidean spaces   cline 50045
      ​21.53  Mathbox for Zhi Wang
            21.53.1  Propositional calculus   imbi12d2 50106
            21.53.2  Predicate calculus with equality   dtrucor3 50114
                  21.53.2.1  Axiom scheme ax-5 (Distinctness)   dtrucor3 50114
            21.53.3  ZF Set Theory - start with the Axiom of Extensionality   ralbidb 50115
                  21.53.3.1  Restricted quantification   ralbidb 50115
                  21.53.3.2  The universal class   reuxfr1dd 50122
                  21.53.3.3  The empty set   ssdisjd 50123
                  21.53.3.4  The conditional operator for classes   elif0 50127
                  21.53.3.5  Unordered and ordered pairs   vsn 50128
                  21.53.3.6  The union of a class   unilbss 50134
                  21.53.3.7  Indexed union and intersection   iuneq0 50135
            21.53.4  ZF Set Theory - add the Axiom of Replacement   inpw 50141
                  21.53.4.1  Theorems requiring subset and intersection existence   inpw 50141
            21.53.5  ZF Set Theory - add the Axiom of Power Sets   opth1neg 50142
                  21.53.5.1  Ordered pair theorem   opth1neg 50142
                  21.53.5.2  Ordered-pair class abstractions (cont.)   brab2dd 50144
                  21.53.5.3  Relations   iinxp 50147
                  21.53.5.4  Functions   mof0 50154
                  21.53.5.5  Operations   ovsng 50174
            21.53.6  ZF Set Theory - add the Axiom of Union   fonex 50183
                  21.53.6.1  Relations and functions (cont.)   fonex 50183
                  21.53.6.2  First and second members of an ordered pair   eloprab1st2nd 50184
                  21.53.6.3  Operations in maps-to notation (continued)   elovmpoeq 50185
                  21.53.6.4  Function transposition   resinsnlem 50189
                  21.53.6.5  Infinite Cartesian products   ixpv 50208
                  21.53.6.6  Equinumerosity   fvconst0ci 50210
            21.53.7  Order sets   iccin 50215
                  21.53.7.1  Real number intervals   iccin 50215
            21.53.8  Extensible structures   slotresfo 50218
                  21.53.8.1  Basic definitions   slotresfo 50218
            21.53.9  Moore spaces   mreuniss 50219
            ​*21.53.10  Topology   clduni 50220
                  21.53.10.1  Closure and interior   clduni 50220
                  21.53.10.2  Neighborhoods   neircl 50224
                  21.53.10.3  Subspace topologies   restcls2lem 50232
                  21.53.10.4  Limits and continuity in topological spaces   cnneiima 50236
                  21.53.10.5  Topological definitions using the reals   iooii 50237
                  21.53.10.6  Separated sets   sepnsepolem1 50241
                  21.53.10.7  Separated spaces: T0, T1, T2 (Hausdorff) ...   isnrm4 50250
            21.53.11  Preordered sets and directed sets using extensible structures   isprsd 50274
            21.53.12  Posets and lattices using extensible structures   lubeldm2 50275
                  21.53.12.1  Posets   lubeldm2 50275
                  21.53.12.2  Lattices   toslat 50301
                  21.53.12.3  Subset order structures   intubeu 50303
            21.53.13  Rings   elmgpcntrd 50324
                  21.53.13.1  Multiplicative Group   elmgpcntrd 50324
            21.53.14  Associative algebras   asclelbasALT 50325
                  21.53.14.1  Definition and basic properties   asclelbasALT 50325
            21.53.15  Categories   homf0 50328
                  21.53.15.1  Categories   homf0 50328
                  21.53.15.2  Opposite category   oppccatb 50335
                  21.53.15.3  Monomorphisms and epimorphisms   idmon 50339
                  21.53.15.4  Sections, inverses, isomorphisms   sectrcl 50341
                  21.53.15.5  Isomorphic objects   cicfn 50361
                  21.53.15.6  Subcategories   dmdm 50372
                  21.53.15.7  Functors   reldmfunc 50394
                  21.53.15.8  Opposite functors   coppf 50443
                  21.53.15.9  Full & faithful functors   imasubc 50472
                  21.53.15.10  Universal property   upciclem1 50487
                  21.53.15.11  Natural transformations and the functor category   isnatd 50544
                  21.53.15.12  Initial, terminal and zero objects of a category   initoo2 50559
                  21.53.15.13  Product of categories   reldmxpc 50573
                  21.53.15.14  Swap functors   cswapf 50586
                  21.53.15.15  Functor evaluation   oppc1stflem 50614
                  21.53.15.16  Transposed curry functors   cofuswapfcl 50620
                  21.53.15.17  Constant functors   diag1 50631
                  21.53.15.18  Functor composition bifunctors   fucofulem1 50637
                  21.53.15.19  Post-composition functors   postcofval 50701
                  21.53.15.20  Pre-composition functors   precofvallem 50706
            21.53.16  Examples of categories   catcrcl 50735
                  21.53.16.1  The category of categories   catcrcl 50735
                  21.53.16.2  Thin categories   cthinc 50757
                  21.53.16.3  Terminal categories   ctermc 50812
                  21.53.16.4  Preordered sets as thin categories   cprstc 50892
                  21.53.16.5  Monoids as categories   cmndtc 50920
                  21.53.16.6  Categories with at most one object and at most two morphisms   2arwcatlem1 50938
                  21.53.16.7  Parallel-arrow categories   parhomval 50948
                  21.53.16.8  Nonuniqueness of the morphism part of functor composition   fuconuniqlem 50986
            21.53.17  Kan extensions and related concepts   clan 50993
                  21.53.17.1  Kan extensions   clan 50993
                  21.53.17.2  Limits and colimits   clmd 51031
      ​21.54  Mathbox for Emmett Weisz
            ​*21.54.1  Miscellaneous Theorems   nfintd 51061
            21.54.2  Examples and properties of set recursion   setrecseq 51068
            ​*21.54.3  Construction of Games and Surreal Numbers   cpg 51082
      ​​*21.55  Mathbox for David A. Wheeler
            21.55.1  Natural deduction   sbidd 51091
            ​*21.55.2  Greater than, greater than or equal to   cge-real 51093
            ​*21.55.3  Hyperbolic trigonometric functions   csinh 51103
            ​*21.55.4  Reciprocal trigonometric functions (sec, csc, cot)   csec 51114
            ​*21.55.5  Identities for "if"   ifnmfalse 51139
            ​*21.55.6  Logarithms generalized to arbitrary base using ` logb `   logb2aval 51140
            ​*21.55.7  Logarithm laws generalized to an arbitrary base - log_   clog- 51141
            ​*21.55.8  Formally define notions such as reflexivity   wreflexive 51143
            ​*21.55.9  Algebra helpers   mvlraddi 51147
            ​*21.55.10  Algebra helper examples   i2linesi 51154
            ​*21.55.11  Formal methods "surprises"   alimp-surprise 51156
            ​*21.55.12  Allsome quantifier   wals 51162
            ​*21.55.13  Allsome one quantifier   walseu 51195
            ​*21.55.14  Miscellaneous   5m4e1 51215
            21.55.15  Theorems about algebraic numbers   aacllem 51219
      ​21.56  Mathbox for Mingli Yuan
      ​21.57  Mathbox for Jiamin Zhao
            21.57.1  Cross product and scalar triple product in RR^3   1ne3 51221
            21.57.2  Veronese map and linear dependence   cveronese 51248
      ​21.58  Mathbox for Kunhao Zheng
            21.58.1  Weighted AM-GM inequality   amgmwlem 51267