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  Preordered sets and directed sets using extensible structures
      9.2  Posets and lattices using extensible structures
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  Ideals
      10.8  Associative algebras
      10.9  Abstract multivariate polynomials
      10.10  The complex numbers as an algebraic extensible structure
      10.11  Generalized pre-Hilbert and Hilbert spaces
PART 11  BASIC LINEAR ALGEBRA
      11.1  Vectors and free modules
      11.2  Matrices
      11.3  The determinant
      11.4  Polynomial matrices
      11.5  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  ELEMENTARY GEOMETRY
      15.1  Definition and Tarski's Axioms of Geometry
      15.2  Tarskian Geometry
      15.3  Properties of geometries
      15.4  Geometry in Hilbert spaces
PART 16  GRAPH THEORY
      16.1  Vertices and edges
      16.2  Undirected graphs
      16.3  Walks, paths and cycles
      16.4  Eulerian paths and the Konigsberg Bridge problem
      16.5  The Friendship Theorem
PART 17  GUIDES AND MISCELLANEA
      17.1  Guides (conventions, explanations, and examples)
      17.2  Humor
      17.3  (Future - to be reviewed and classified)
PART 18  COMPLEX TOPOLOGICAL VECTOR SPACES (DEPRECATED)
      18.1  Additional material on group theory (deprecated)
      18.2  Complex vector spaces
      18.3  Normed complex vector spaces
      18.4  Operators on complex vector spaces
      18.5  Inner product (pre-Hilbert) spaces
      18.6  Complex Banach spaces
      18.7  Complex Hilbert spaces
PART 19  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      19.1  Axiomatization of complex pre-Hilbert spaces
      19.2  Inner product and norms
      19.3  Cauchy sequences and completeness axiom
      19.4  Subspaces and projections
      19.5  Properties of Hilbert subspaces
      19.6  Operators on Hilbert spaces
      19.7  States on a Hilbert lattice and Godowski's equation
      19.8  Cover relation, atoms, exchange axiom, and modular symmetry
PART 20  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      ​20.1  Mathboxes for user contributions
      20.2  Mathbox for Stefan Allan
      20.3  Mathbox for Thierry Arnoux
      20.4  Mathbox for Jonathan Ben-Naim
      20.5  Mathbox for BTernaryTau
      20.6  Mathbox for Mario Carneiro
      20.7  Mathbox for Filip Cernatescu
      20.8  Mathbox for Paul Chapman
      20.9  Mathbox for Scott Fenton
      20.10  Mathbox for Jeff Hankins
      20.11  Mathbox for Anthony Hart
      20.12  Mathbox for Chen-Pang He
      20.13  Mathbox for Jeff Hoffman
      20.14  Mathbox for Asger C. Ipsen
      20.15  Mathbox for BJ
      20.16  Mathbox for Jim Kingdon
      20.17  Mathbox for ML
      20.18  Mathbox for Wolf Lammen
      20.19  Mathbox for Brendan Leahy
      20.20  Mathbox for Jeff Madsen
      20.21  Mathbox for Giovanni Mascellani
      20.22  Mathbox for Peter Mazsa
      20.23  Mathbox for Rodolfo Medina
      20.24  Mathbox for Norm Megill
      20.25  Mathbox for metakunt
      20.26  Mathbox for Steven Nguyen
      20.27  Mathbox for Igor Ieskov
      20.28  Mathbox for OpenAI
      20.29  Mathbox for Stefan O'Rear
      20.30  Mathbox for Jon Pennant
      20.31  Mathbox for Richard Penner
      20.32  Mathbox for Stanislas Polu
      20.33  Mathbox for Rohan Ridenour
      20.34  Mathbox for Steve Rodriguez
      20.35  Mathbox for Andrew Salmon
      20.36  Mathbox for Alan Sare
      20.37  Mathbox for Glauco Siliprandi
      20.38  Mathbox for Saveliy Skresanov
      20.39  Mathbox for Jarvin Udandy
      20.40  Mathbox for Adhemar
      20.41  Mathbox for Alexander van der Vekens
      20.42  Mathbox for Emmett Weisz
      20.43  Mathbox for David A. Wheeler
      20.44  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 113
            ​*1.2.5  Logical equivalence   wb 208
            ​*1.2.6  Logical conjunction   wa 398
            ​*1.2.7  Logical disjunction   wo 843
            ​*1.2.8  Mixed connectives   jaao 951
            ​*1.2.9  The conditional operator for propositions   wif 1057
            ​*1.2.10  The weak deduction theorem for propositional calculus   elimh 1076
            1.2.11  Abbreviated conjunction and disjunction of three wff's   w3o 1082
            1.2.12  Logical "nand" (Sheffer stroke)   wnan 1481
            1.2.13  Logical "xor"   wxo 1501
            1.2.14  Logical "nor"   wnor 1520
            1.2.15  True and false constants   wal 1535
                  ​*1.2.15.1  Universal quantifier for use by df-tru   wal 1535
                  ​*1.2.15.2  Equality predicate for use by df-tru   cv 1536
                  1.2.15.3  The true constant   wtru 1538
                  1.2.15.4  The false constant   wfal 1549
            ​*1.2.16  Truth tables   truimtru 1560
                  1.2.16.1  Implication   truimtru 1560
                  1.2.16.2  Negation   nottru 1564
                  1.2.16.3  Equivalence   trubitru 1566
                  1.2.16.4  Conjunction   truantru 1570
                  1.2.16.5  Disjunction   truortru 1574
                  1.2.16.6  Alternative denial   trunantru 1578
                  1.2.16.7  Exclusive disjunction   truxortru 1582
                  1.2.16.8  Joint denial   trunortru 1586
            ​*1.2.17  Half adder and full adder in propositional calculus   whad 1593
                  1.2.17.1  Full adder: sum   whad 1593
                  1.2.17.2  Full adder: carry   wcad 1607
      ​1.3  Other axiomatizations related to classical propositional calculus
            ​*1.3.1  Minimal implicational calculus   minimp 1622
            ​*1.3.2  Implicational Calculus   impsingle 1628
            1.3.3  Derive the Lukasiewicz axioms from Meredith's sole axiom   meredith 1642
            1.3.4  Derive the standard axioms from the Lukasiewicz axioms   luklem1 1659
            ​*1.3.5  Derive Nicod's axiom from the standard axioms   nic-dfim 1670
            1.3.6  Derive the Lukasiewicz axioms from Nicod's axiom   nic-imp 1676
            1.3.7  Derive Nicod's Axiom from Lukasiewicz's First Sheffer Stroke Axiom   lukshef-ax1 1695
            1.3.8  Derive the Lukasiewicz Axioms from the Tarski-Bernays-Wajsberg Axioms   tbw-bijust 1699
            1.3.9  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's First CO Axiom   merco1 1714
            1.3.10  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's Second CO Axiom   merco2 1737
            1.3.11  Derive the Lukasiewicz axioms from the Russell-Bernays Axioms   rb-bijust 1750
            ​*1.3.12  Stoic logic non-modal portion (Chrysippus of Soli)   mptnan 1769
      ​​*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 1780
                  1.4.1.1  Existential quantifier   wex 1780
                  1.4.1.2  Non-freeness predicate   wnf 1784
            1.4.2  Rule scheme ax-gen (Generalization)   ax-gen 1796
            1.4.3  Axiom scheme ax-4 (Quantified Implication)   ax-4 1810
                  ​*1.4.3.1  The empty domain of discourse   empty 1907
            1.4.4  Axiom scheme ax-5 (Distinctness) - first use of $d   ax-5 1911
            ​*1.4.5  Equality predicate (continued)   weq 1964
            1.4.6  Axiom scheme ax-6 (Existence)   ax-6 1970
            1.4.7  Axiom scheme ax-7 (Equality)   ax-7 2015
            1.4.8  Define proper substitution   sbjust 2068
            1.4.9  Membership predicate   wcel 2114
            1.4.10  Axiom scheme ax-8 (Left Equality for Binary Predicate)   ax-8 2116
            1.4.11  Axiom scheme ax-9 (Right Equality for Binary Predicate)   ax-9 2124
            ​*1.4.12  Logical redundancy of ax-10 , ax-11 , ax-12 , ax-13   ax6dgen 2132
      ​​*1.5  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes)
            1.5.1  Axiom scheme ax-10 (Quantified Negation)   ax-10 2145
            1.5.2  Axiom scheme ax-11 (Quantifier Commutation)   ax-11 2161
            1.5.3  Axiom scheme ax-12 (Substitution)   ax-12 2177
            1.5.4  Axiom scheme ax-13 (Quantified Equality)   ax-13 2390
            ​*1.5.5  Alternate definition of substitution   sbimiALT 2577
      ​1.6  Uniqueness and unique existence
            1.6.1  Uniqueness: the at-most-one quantifier   wmo 2620
            1.6.2  Unique existence: the unique existential quantifier   weu 2653
      ​1.7  Other axiomatizations related to classical predicate calculus
            ​*1.7.1  Aristotelian logic: Assertic syllogisms   barbara 2748
            ​*1.7.2  Intuitionistic logic   axia1 2778
​​*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 2793
            2.1.2  Classes   cab 2799
                  2.1.2.1  Class abstractions   cab 2799
                  ​*2.1.2.2  Class equality   df-cleq 2814
                  2.1.2.3  Class membership   df-clel 2893
                  2.1.2.4  Elementary properties of class abstractions   abeq2 2945
            2.1.3  Class form not-free predicate   wnfc 2961
            2.1.4  Negated equality and membership   wne 3016
                  2.1.4.1  Negated equality   wne 3016
                  2.1.4.2  Negated membership   wnel 3123
            2.1.5  Restricted quantification   wral 3138
            2.1.6  The universal class   cvv 3494
            ​*2.1.7  Conditional equality (experimental)   wcdeq 3754
            2.1.8  Russell's Paradox   rru 3770
            2.1.9  Proper substitution of classes for sets   wsbc 3772
            2.1.10  Proper substitution of classes for sets into classes   csb 3883
            2.1.11  Define basic set operations and relations   cdif 3933
            2.1.12  Subclasses and subsets   df-ss 3952
            2.1.13  The difference, union, and intersection of two classes   dfdif3 4091
                  2.1.13.1  The difference of two classes   dfdif3 4091
                  2.1.13.2  The union of two classes   elun 4125
                  2.1.13.3  The intersection of two classes   elin 4169
                  2.1.13.4  The symmetric difference of two classes   csymdif 4218
                  2.1.13.5  Combinations of difference, union, and intersection of two classes   unabs 4231
                  2.1.13.6  Class abstractions with difference, union, and intersection of two classes   unab 4270
                  2.1.13.7  Restricted uniqueness with difference, union, and intersection   reuss2 4283
            2.1.14  The empty set   c0 4291
            ​*2.1.15  The conditional operator for classes   cif 4467
            ​*2.1.16  The weak deduction theorem for set theory   dedth 4523
            2.1.17  Power classes   cpw 4539
            2.1.18  Unordered and ordered pairs   snjust 4566
            2.1.19  The union of a class   cuni 4838
            2.1.20  The intersection of a class   cint 4876
            2.1.21  Indexed union and intersection   ciun 4919
            2.1.22  Disjointness   wdisj 5031
            2.1.23  Binary relations   wbr 5066
            2.1.24  Ordered-pair class abstractions (class builders)   copab 5128
            2.1.25  Functions in maps-to notation   cmpt 5146
            2.1.26  Transitive classes   wtr 5172
      ​2.2  ZF Set Theory - add the Axiom of Replacement
            2.2.1  Introduce the Axiom of Replacement   ax-rep 5190
            2.2.2  Derive the Axiom of Separation   axsepgfromrep 5201
            2.2.3  Derive the Null Set Axiom   axnulALT 5208
            2.2.4  Theorems requiring subset and intersection existence   nalset 5217
            2.2.5  Theorems requiring empty set existence   class2set 5254
      ​2.3  ZF Set Theory - add the Axiom of Power Sets
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 5266
            2.3.2  Derive the Axiom of Pairing   axprlem1 5324
            2.3.3  Ordered pair theorem   opnz 5365
            2.3.4  Ordered-pair class abstractions (cont.)   opabidw 5412
            2.3.5  Power class of union and intersection   pwin 5454
            2.3.6  The identity relation   cid 5459
            2.3.7  The membership relation (or epsilon relation)   cep 5464
            ​*2.3.8  Partial and total orderings   wpo 5472
            2.3.9  Founded and well-ordering relations   wfr 5511
            2.3.10  Relations   cxp 5553
            2.3.11  The Predecessor Class   cpred 6147
            2.3.12  Well-founded induction   tz6.26 6179
            2.3.13  Ordinals   word 6190
            2.3.14  Definite description binder (inverted iota)   cio 6312
            2.3.15  Functions   wfun 6349
            2.3.16  Cantor's Theorem   canth 7111
            2.3.17  Restricted iota (description binder)   crio 7113
            2.3.18  Operations   co 7156
                  2.3.18.1  Variable-to-class conversion for operations   caovclg 7340
            2.3.19  Maps-to notation   mpondm0 7386
            2.3.20  Function operation   cof 7407
            2.3.21  Proper subset relation   crpss 7448
      ​2.4  ZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 7461
            2.4.2  Ordinals (continued)   epweon 7497
            2.4.3  Transfinite induction   tfi 7568
            2.4.4  The natural numbers (i.e., finite ordinals)   com 7580
            2.4.5  Peano's postulates   peano1 7601
            2.4.6  Finite induction (for finite ordinals)   find 7607
            2.4.7  Relations and functions (cont.)   dmexg 7613
            2.4.8  First and second members of an ordered pair   c1st 7687
            ​*2.4.9  The support of functions   csupp 7830
            ​*2.4.10  Special maps-to operations   opeliunxp2f 7876
            2.4.11  Function transposition   ctpos 7891
            2.4.12  Curry and uncurry   ccur 7931
            2.4.13  Undefined values   cund 7938
            2.4.14  Well-founded recursion   cwrecs 7946
            2.4.15  Functions on ordinals; strictly monotone ordinal functions   iunon 7976
            2.4.16  "Strong" transfinite recursion   crecs 8007
            2.4.17  Recursive definition generator   crdg 8045
            2.4.18  Finite recursion   frfnom 8070
            2.4.19  Ordinal arithmetic   c1o 8095
            2.4.20  Natural number arithmetic   nna0 8230
            2.4.21  Equivalence relations and classes   wer 8286
            2.4.22  The mapping operation   cmap 8406
            2.4.23  Infinite Cartesian products   cixp 8461
            2.4.24  Equinumerosity   cen 8506
            2.4.25  Schroeder-Bernstein Theorem   sbthlem1 8627
            2.4.26  Equinumerosity (cont.)   xpf1o 8679
            2.4.27  Pigeonhole Principle   phplem1 8696
            2.4.28  Finite sets   onomeneq 8708
            2.4.29  Finitely supported functions   cfsupp 8833
            2.4.30  Finite intersections   cfi 8874
            2.4.31  Hall's marriage theorem   marypha1lem 8897
            2.4.32  Supremum and infimum   csup 8904
            2.4.33  Ordinal isomorphism, Hartogs's theorem   coi 8973
            2.4.34  Hartogs function, order types, weak dominance   char 9020
      ​2.5  ZF Set Theory - add the Axiom of Regularity
            2.5.1  Introduce the Axiom of Regularity   ax-reg 9056
            2.5.2  Axiom of Infinity equivalents   inf0 9084
      ​2.6  ZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-inf 9101
            2.6.2  Existence of omega (the set of natural numbers)   omex 9106
            2.6.3  Cantor normal form   ccnf 9124
            2.6.4  Transitive closure   trcl 9170
            2.6.5  Rank   cr1 9191
            2.6.6  Scott's trick; collection principle; Hilbert's epsilon   scottex 9314
            2.6.7  Disjoint union   cdju 9327
            2.6.8  Cardinal numbers   ccrd 9364
            2.6.9  Axiom of Choice equivalents   wac 9541
            ​*2.6.10  Cardinal number arithmetic   undjudom 9593
            2.6.11  The Ackermann bijection   ackbij2lem1 9641
            2.6.12  Cofinality (without Axiom of Choice)   cflem 9668
            2.6.13  Eight inequivalent definitions of finite set   sornom 9699
            2.6.14  Hereditarily size-limited sets without Choice   itunifval 9838
​​*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 9857
            3.1.2  Introduce the Axiom of Dependent Choice   ax-dc 9868
      ​3.2  ZFC Set Theory - add the Axiom of Choice
            3.2.1  Introduce the Axiom of Choice   ax-ac 9881
            3.2.2  AC equivalents: well-ordering, Zorn's lemma   numthcor 9916
            3.2.3  Cardinal number theorems using Axiom of Choice   cardval 9968
            3.2.4  Cardinal number arithmetic using Axiom of Choice   iunctb 9996
            3.2.5  Cofinality using the Axiom of Choice   alephreg 10004
      ​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 10042
            3.4.2  Derivation of the Axiom of Choice   gchaclem 10100
​​*PART 4  TG (TARSKI-GROTHENDIECK) SET THEORY
      ​4.1  Inaccessibles
            4.1.1  Weakly and strongly inaccessible cardinals   cwina 10104
            4.1.2  Weak universes   cwun 10122
            4.1.3  Tarski classes   ctsk 10170
            4.1.4  Grothendieck universes   cgru 10212
      ​4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
            4.2.1  Introduce the Tarski-Grothendieck Axiom   ax-groth 10245
            4.2.2  Derive the Power Set, Infinity and Choice Axioms   grothpw 10248
            4.2.3  Tarski map function   ctskm 10259
​​*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 10266
            5.1.2  Final derivation of real and complex number postulates   axaddf 10567
            5.1.3  Real and complex number postulates restated as axioms   ax-cnex 10593
      ​5.2  Derive the basic properties from the field axioms
            5.2.1  Some deductions from the field axioms for complex numbers   cnex 10618
            5.2.2  Infinity and the extended real number system   cpnf 10672
            5.2.3  Restate the ordering postulates with extended real "less than"   axlttri 10712
            5.2.4  Ordering on reals   lttr 10717
            5.2.5  Initial properties of the complex numbers   mul12 10805
      ​5.3  Real and complex numbers - basic operations
            5.3.1  Addition   add12 10857
            5.3.2  Subtraction   cmin 10870
            5.3.3  Multiplication   kcnktkm1cn 11071
            5.3.4  Ordering on reals (cont.)   gt0ne0 11105
            5.3.5  Reciprocals   ixi 11269
            5.3.6  Division   cdiv 11297
            5.3.7  Ordering on reals (cont.)   elimgt0 11478
            5.3.8  Completeness Axiom and Suprema   fimaxre 11584
            5.3.9  Imaginary and complex number properties   inelr 11628
            5.3.10  Function operation analogue theorems   ofsubeq0 11635
      ​5.4  Integer sets
            5.4.1  Positive integers (as a subset of complex numbers)   cn 11638
            5.4.2  Principle of mathematical induction   nnind 11656
            ​*5.4.3  Decimal representation of numbers   c2 11693
            ​*5.4.4  Some properties of specific numbers   neg1cn 11752
            5.4.5  Simple number properties   halfcl 11863
            5.4.6  The Archimedean property   nnunb 11894
            5.4.7  Nonnegative integers (as a subset of complex numbers)   cn0 11898
            ​*5.4.8  Extended nonnegative integers   cxnn0 11968
            5.4.9  Integers (as a subset of complex numbers)   cz 11982
            5.4.10  Decimal arithmetic   cdc 12099
            5.4.11  Upper sets of integers   cuz 12244
            5.4.12  Well-ordering principle for bounded-below sets of integers   uzwo3 12344
            5.4.13  Rational numbers (as a subset of complex numbers)   cq 12349
            5.4.14  Existence of the set of complex numbers   rpnnen1lem2 12377
      ​5.5  Order sets
            5.5.1  Positive reals (as a subset of complex numbers)   crp 12390
            5.5.2  Infinity and the extended real number system (cont.)   cxne 12505
            5.5.3  Supremum and infimum on the extended reals   xrsupexmnf 12699
            5.5.4  Real number intervals   cioo 12739
            5.5.5  Finite intervals of integers   cfz 12893
            ​*5.5.6  Finite intervals of nonnegative integers   elfz2nn0 12999
            5.5.7  Half-open integer ranges   cfzo 13034
      ​5.6  Elementary integer functions
            5.6.1  The floor and ceiling functions   cfl 13161
            5.6.2  The modulo (remainder) operation   cmo 13238
            5.6.3  Miscellaneous theorems about integers   om2uz0i 13316
            5.6.4  Strong induction over upper sets of integers   uzsinds 13356
            5.6.5  Finitely supported functions over the nonnegative integers   fsuppmapnn0fiublem 13359
            5.6.6  The infinite sequence builder "seq" - extension   cseq 13370
            5.6.7  Integer powers   cexp 13430
            5.6.8  Ordered pair theorem for nonnegative integers   nn0le2msqi 13628
            5.6.9  Factorial function   cfa 13634
            5.6.10  The binomial coefficient operation   cbc 13663
            5.6.11  The ` # ` (set size) function   chash 13691
                  5.6.11.1  Proper unordered pairs and triples (sets of size 2 and 3)   hashprlei 13827
                  5.6.11.2  Functions with a domain containing at least two different elements   fundmge2nop0 13851
                  5.6.11.3  Finite induction on the size of the first component of a binary relation   hashdifsnp1 13855
      ​​*5.7  Words over a set
            5.7.1  Definitions and basic theorems   cword 13862
            5.7.2  Last symbol of a word   clsw 13914
            5.7.3  Concatenations of words   cconcat 13922
            5.7.4  Singleton words   cs1 13949
            5.7.5  Concatenations with singleton words   ccatws1cl 13970
            5.7.6  Subwords/substrings   csubstr 14002
            5.7.7  Prefixes of a word   cpfx 14032
            5.7.8  Subwords of subwords   swrdswrdlem 14066
            5.7.9  Subwords and concatenations   pfxcctswrd 14072
            5.7.10  Subwords of concatenations   swrdccatfn 14086
            5.7.11  Splicing words (substring replacement)   csplice 14111
            5.7.12  Reversing words   creverse 14120
            5.7.13  Repeated symbol words   creps 14130
            ​*5.7.14  Cyclical shifts of words   ccsh 14150
            5.7.15  Mapping words by a function   wrdco 14193
            5.7.16  Longer string literals   cs2 14203
      ​​*5.8  Reflexive and transitive closures of relations
            5.8.1  The reflexive and transitive properties of relations   coss12d 14332
            5.8.2  Basic properties of closures   cleq1lem 14342
            5.8.3  Definitions and basic properties of transitive closures   ctcl 14345
            5.8.4  Exponentiation of relations   crelexp 14379
            5.8.5  Reflexive-transitive closure as an indexed union   crtrcl 14414
            ​*5.8.6  Principle of transitive induction.   relexpindlem 14422
      ​5.9  Elementary real and complex functions
            5.9.1  The "shift" operation   cshi 14425
            5.9.2  Signum (sgn or sign) function   csgn 14445
            5.9.3  Real and imaginary parts; conjugate   ccj 14455
            5.9.4  Square root; absolute value   csqrt 14592
      ​5.10  Elementary limits and convergence
            5.10.1  Superior limit (lim sup)   clsp 14827
            5.10.2  Limits   cli 14841
            5.10.3  Finite and infinite sums   csu 15042
            5.10.4  The binomial theorem   binomlem 15184
            5.10.5  The inclusion/exclusion principle   incexclem 15191
            5.10.6  Infinite sums (cont.)   isumshft 15194
            5.10.7  Miscellaneous converging and diverging sequences   divrcnv 15207
            5.10.8  Arithmetic series   arisum 15215
            5.10.9  Geometric series   expcnv 15219
            5.10.10  Ratio test for infinite series convergence   cvgrat 15239
            5.10.11  Mertens' theorem   mertenslem1 15240
            5.10.12  Finite and infinite products   prodf 15243
                  5.10.12.1  Product sequences   prodf 15243
                  5.10.12.2  Non-trivial convergence   ntrivcvg 15253
                  5.10.12.3  Complex products   cprod 15259
                  5.10.12.4  Finite products   fprod 15295
                  5.10.12.5  Infinite products   iprodclim 15352
            5.10.13  Falling and Rising Factorial   cfallfac 15358
            5.10.14  Bernoulli polynomials and sums of k-th powers   cbp 15400
      ​5.11  Elementary trigonometry
            5.11.1  The exponential, sine, and cosine functions   ce 15415
                  5.11.1.1  The circle constant (tau = 2 pi)   ctau 15555
            5.11.2  _e is irrational   eirrlem 15557
      ​5.12  Cardinality of real and complex number subsets
            5.12.1  Countability of integers and rationals   xpnnen 15564
            5.12.2  The reals are uncountable   rpnnen2lem1 15567
​​*PART 6  ELEMENTARY NUMBER THEORY
      ​6.1  Elementary properties of divisibility
            6.1.1  Irrationality of square root of 2   sqrt2irrlem 15601
            6.1.2  Some Number sets are chains of proper subsets   nthruc 15605
            6.1.3  The divides relation   cdvds 15607
            ​*6.1.4  Even and odd numbers   evenelz 15685
            6.1.5  The division algorithm   divalglem0 15744
            6.1.6  Bit sequences   cbits 15768
            6.1.7  The greatest common divisor operator   cgcd 15843
            6.1.8  Bézout's identity   bezoutlem1 15887
            6.1.9  Algorithms   nn0seqcvgd 15914
            6.1.10  Euclid's Algorithm   eucalgval2 15925
            ​*6.1.11  The least common multiple   clcm 15932
            ​*6.1.12  Coprimality and Euclid's lemma   coprmgcdb 15993
            6.1.13  Cancellability of congruences   congr 16008
      ​6.2  Elementary prime number theory
            ​*6.2.1  Elementary properties   cprime 16015
            ​*6.2.2  Coprimality and Euclid's lemma (cont.)   coprm 16055
            6.2.3  Properties of the canonical representation of a rational   cnumer 16073
            6.2.4  Euler's theorem   codz 16100
            6.2.5  Arithmetic modulo a prime number   modprm1div 16134
            6.2.6  Pythagorean Triples   coprimeprodsq 16145
            6.2.7  The prime count function   cpc 16173
            6.2.8  Pocklington's theorem   prmpwdvds 16240
            6.2.9  Infinite primes theorem   unbenlem 16244
            6.2.10  Sum of prime reciprocals   prmreclem1 16252
            6.2.11  Fundamental theorem of arithmetic   1arithlem1 16259
            6.2.12  Lagrange's four-square theorem   cgz 16265
            6.2.13  Van der Waerden's theorem   cvdwa 16301
            6.2.14  Ramsey's theorem   cram 16335
            ​*6.2.15  Primorial function   cprmo 16367
            ​*6.2.16  Prime gaps   prmgaplem1 16385
            6.2.17  Decimal arithmetic (cont.)   dec2dvds 16399
            6.2.18  Cyclical shifts of words (cont.)   cshwsidrepsw 16427
            6.2.19  Specific prime numbers   prmlem0 16439
            6.2.20  Very large primes   1259lem1 16464
​PART 7  BASIC STRUCTURES
      ​7.1  Extensible structures
            ​*7.1.1  Basic definitions   cstr 16479
            7.1.2  Slot definitions   cplusg 16565
            7.1.3  Definition of the structure product   crest 16694
            7.1.4  Definition of the structure quotient   cordt 16772
      ​7.2  Moore spaces
            7.2.1  Moore closures   mrcflem 16877
            7.2.2  Independent sets in a Moore system   mrisval 16901
            7.2.3  Algebraic closure systems   isacs 16922
​PART 8  BASIC CATEGORY THEORY
      ​8.1  Categories
            8.1.1  Categories   ccat 16935
            8.1.2  Opposite category   coppc 16981
            8.1.3  Monomorphisms and epimorphisms   cmon 16998
            8.1.4  Sections, inverses, isomorphisms   csect 17014
            ​*8.1.5  Isomorphic objects   ccic 17065
            8.1.6  Subcategories   cssc 17077
            8.1.7  Functors   cfunc 17124
            8.1.8  Full & faithful functors   cful 17172
            8.1.9  Natural transformations and the functor category   cnat 17211
            8.1.10  Initial, terminal and zero objects of a category   cinito 17248
      ​8.2  Arrows (disjointified hom-sets)
            8.2.1  Identity and composition for arrows   cida 17313
      ​8.3  Examples of categories
            8.3.1  The category of sets   csetc 17335
            8.3.2  The category of categories   ccatc 17354
            ​*8.3.3  The category of extensible structures   fncnvimaeqv 17370
      ​8.4  Categorical constructions
            8.4.1  Product of categories   cxpc 17418
            8.4.2  Functor evaluation   cevlf 17459
            8.4.3  Hom functor   chof 17498
​PART 9  BASIC ORDER THEORY
      ​9.1  Preordered sets and directed sets using extensible structures
      ​9.2  Posets and lattices using extensible structures
            9.2.1  Posets   cpo 17550
            9.2.2  Lattices   clat 17655
            9.2.3  The dual of an ordered set   codu 17738
            9.2.4  Subset order structures   cipo 17761
            9.2.5  Distributive lattices   latmass 17798
            9.2.6  Posets and lattices as relations   cps 17808
            9.2.7  Directed sets, nets   cdir 17838
​PART 10  BASIC ALGEBRAIC STRUCTURES
      ​10.1  Monoids
            ​*10.1.1  Magmas   cplusf 17849
            ​*10.1.2  Identity elements   mgmidmo 17870
            ​*10.1.3  Iterated sums in a magma   gsumvalx 17886
            ​*10.1.4  Semigroups   csgrp 17900
            ​*10.1.5  Definition and basic properties of monoids   cmnd 17911
            10.1.6  Monoid homomorphisms and submonoids   cmhm 17954
            ​*10.1.7  Iterated sums in a monoid   gsumvallem2 17998
            10.1.8  Free monoids   cfrmd 18012
                  ​*10.1.8.1  Monoid of endofunctions   cefmnd 18033
            10.1.9  Examples and counterexamples for magmas, semigroups and monoids   mgm2nsgrplem1 18083
      ​10.2  Groups
            10.2.1  Definition and basic properties   cgrp 18103
            ​*10.2.2  Group multiple operation   cmg 18224
            10.2.3  Subgroups and Quotient groups   csubg 18273
            ​*10.2.4  Cyclic monoids and groups   cycsubmel 18343
            10.2.5  Elementary theory of group homomorphisms   cghm 18355
            10.2.6  Isomorphisms of groups   cgim 18397
            10.2.7  Group actions   cga 18419
            10.2.8  Centralizers and centers   ccntz 18445
            10.2.9  The opposite group   coppg 18473
            10.2.10  Symmetric groups   csymg 18495
                  ​*10.2.10.1  Definition and basic properties   csymg 18495
                  10.2.10.2  Cayley's theorem   cayleylem1 18540
                  10.2.10.3  Permutations fixing one element   symgfix2 18544
                  ​*10.2.10.4  Transpositions in the symmetric group   cpmtr 18569
                  10.2.10.5  The sign of a permutation   cpsgn 18617
            10.2.11  p-Groups and Sylow groups; Sylow's theorems   cod 18652
            10.2.12  Direct products   clsm 18759
                  10.2.12.1  Direct products (extension)   smndlsmidm 18781
            10.2.13  Free groups   cefg 18832
            10.2.14  Abelian groups   ccmn 18906
                  10.2.14.1  Definition and basic properties   ccmn 18906
                  10.2.14.2  Cyclic groups   ccyg 18996
                  10.2.14.3  Group sum operation   gsumval3a 19023
                  10.2.14.4  Group sums over (ranges of) integers   fsfnn0gsumfsffz 19103
                  10.2.14.5  Internal direct products   cdprd 19115
                  10.2.14.6  The Fundamental Theorem of Abelian Groups   ablfacrplem 19187
            10.2.15  Simple groups   csimpg 19212
                  10.2.15.1  Definition and basic properties   csimpg 19212
                  10.2.15.2  Classification of abelian simple groups   ablsimpnosubgd 19226
      ​10.3  Rings
            10.3.1  Multiplicative Group   cmgp 19239
            10.3.2  Ring unit   cur 19251
                  10.3.2.1  Semirings   csrg 19255
                  ​*10.3.2.2  The binomial theorem for semirings   srgbinomlem1 19290
            10.3.3  Definition and basic properties of unital rings   crg 19297
            10.3.4  Opposite ring   coppr 19372
            10.3.5  Divisibility   cdsr 19388
            10.3.6  Ring primes   crpm 19462
            10.3.7  Ring homomorphisms   crh 19464
      ​10.4  Division rings and fields
            10.4.1  Definition and basic properties   cdr 19502
            10.4.2  Subrings of a ring   csubrg 19531
                  10.4.2.1  Sub-division rings   csdrg 19572
            10.4.3  Absolute value (abstract algebra)   cabv 19587
            10.4.4  Star rings   cstf 19614
      ​10.5  Left modules
            10.5.1  Definition and basic properties   clmod 19634
            10.5.2  Subspaces and spans in a left module   clss 19703
            10.5.3  Homomorphisms and isomorphisms of left modules   clmhm 19791
            10.5.4  Subspace sum; bases for a left module   clbs 19846
      ​10.6  Vector spaces
            10.6.1  Definition and basic properties   clvec 19874
      ​10.7  Ideals
            10.7.1  The subring algebra; ideals   csra 19940
            10.7.2  Two-sided ideals and quotient rings   c2idl 20004
            10.7.3  Principal ideal rings. Divisibility in the integers   clpidl 20014
            10.7.4  Nonzero rings and zero rings   cnzr 20030
            10.7.5  Left regular elements. More kinds of rings   crlreg 20052
      ​10.8  Associative algebras
            10.8.1  Definition and basic properties   casa 20082
      ​10.9  Abstract multivariate polynomials
            10.9.1  Definition and basic properties   cmps 20131
            10.9.2  Polynomial evaluation   ces 20284
            10.9.3  Additional definitions for (multivariate) polynomials   cslv 20321
            ​*10.9.4  Univariate polynomials   cps1 20343
            10.9.5  Univariate polynomial evaluation   ces1 20476
      ​10.10  The complex numbers as an algebraic extensible structure
            10.10.1  Definition and basic properties   cpsmet 20529
            ​*10.10.2  Ring of integers   zring 20617
            10.10.3  Algebraic constructions based on the complex numbers   czrh 20647
            10.10.4  Signs as subgroup of the complex numbers   cnmsgnsubg 20721
            10.10.5  Embedding of permutation signs into a ring   zrhpsgnmhm 20728
            10.10.6  The ordered field of real numbers   crefld 20748
      ​10.11  Generalized pre-Hilbert and Hilbert spaces
            10.11.1  Definition and basic properties   cphl 20768
            10.11.2  Orthocomplements and closed subspaces   cocv 20804
            10.11.3  Orthogonal projection and orthonormal bases   cpj 20844
​​*PART 11  BASIC LINEAR ALGEBRA
      ​11.1  Vectors and free modules
            *11.1.1  Direct sum of left modules   cdsmm 20875
            ​*11.1.2  Free modules   cfrlm 20890
            ​*11.1.3  Standard basis (unit vectors)   cuvc 20926
            ​*11.1.4  Independent sets and families   clindf 20948
            11.1.5  Characterization of free modules   lmimlbs 20980
      ​​*11.2  Matrices
            *11.2.1  The matrix multiplication   cmmul 20994
            ​*11.2.2  Square matrices   cmat 21016
            ​*11.2.3  The matrix algebra   matmulr 21047
            ​*11.2.4  Matrices of dimension 0 and 1   mat0dimbas0 21075
            ​*11.2.5  The subalgebras of diagonal and scalar matrices   cdmat 21097
            ​*11.2.6  Multiplication of a matrix with a "column vector"   cmvmul 21149
            11.2.7  Replacement functions for a square matrix   cmarrep 21165
            11.2.8  Submatrices   csubma 21185
      ​11.3  The determinant
            11.3.1  Definition and basic properties   cmdat 21193
            11.3.2  Determinants of 2 x 2 -matrices   m2detleiblem1 21233
            11.3.3  The matrix adjugate/adjunct   cmadu 21241
            ​*11.3.4  Laplace expansion of determinants (special case)   symgmatr01lem 21262
            11.3.5  Inverse matrix   invrvald 21285
            ​*11.3.6  Cramer's rule   slesolvec 21288
      ​​*11.4  Polynomial matrices
            11.4.1  Basic properties   pmatring 21301
            ​*11.4.2  Constant polynomial matrices   ccpmat 21311
            ​*11.4.3  Collecting coefficients of polynomial matrices   cdecpmat 21370
            ​*11.4.4  Ring isomorphism between polynomial matrices and polynomials over matrices   cpm2mp 21400
      ​​*11.5  The characteristic polynomial
            *11.5.1  Definition and basic properties   cchpmat 21434
            ​*11.5.2  The characteristic factor function G   fvmptnn04if 21457
            ​*11.5.3  The Cayley-Hamilton theorem   cpmadurid 21475
​PART 12  BASIC TOPOLOGY
      ​12.1  Topology
            ​*12.1.1  Topological spaces   ctop 21501
                  12.1.1.1  Topologies   ctop 21501
                  12.1.1.2  Topologies on sets   ctopon 21518
                  12.1.1.3  Topological spaces   ctps 21540
            12.1.2  Topological bases   ctb 21553
            12.1.3  Examples of topologies   distop 21603
            12.1.4  Closure and interior   ccld 21624
            12.1.5  Neighborhoods   cnei 21705
            12.1.6  Limit points and perfect sets   clp 21742
            12.1.7  Subspace topologies   restrcl 21765
            12.1.8  Order topology   ordtbaslem 21796
            12.1.9  Limits and continuity in topological spaces   ccn 21832
            12.1.10  Separated spaces: T0, T1, T2 (Hausdorff) ...   ct0 21914
            12.1.11  Compactness   ccmp 21994
            12.1.12  Bolzano-Weierstrass theorem   bwth 22018
            12.1.13  Connectedness   cconn 22019
            12.1.14  First- and second-countability   c1stc 22045
            12.1.15  Local topological properties   clly 22072
            12.1.16  Refinements   cref 22110
            12.1.17  Compactly generated spaces   ckgen 22141
            12.1.18  Product topologies   ctx 22168
            12.1.19  Continuous function-builders   cnmptid 22269
            12.1.20  Quotient maps and quotient topology   ckq 22301
            12.1.21  Homeomorphisms   chmeo 22361
      ​12.2  Filters and filter bases
            12.2.1  Filter bases   elmptrab 22435
            12.2.2  Filters   cfil 22453
            12.2.3  Ultrafilters   cufil 22507
            12.2.4  Filter limits   cfm 22541
            12.2.5  Extension by continuity   ccnext 22667
            12.2.6  Topological groups   ctmd 22678
            12.2.7  Infinite group sum on topological groups   ctsu 22734
            12.2.8  Topological rings, fields, vector spaces   ctrg 22764
      ​12.3  Uniform Structures and Spaces
            12.3.1  Uniform structures   cust 22808
            12.3.2  The topology induced by an uniform structure   cutop 22839
            12.3.3  Uniform Spaces   cuss 22862
            12.3.4  Uniform continuity   cucn 22884
            12.3.5  Cauchy filters in uniform spaces   ccfilu 22895
            12.3.6  Complete uniform spaces   ccusp 22906
      ​12.4  Metric spaces
            12.4.1  Pseudometric spaces   ispsmet 22914
            12.4.2  Basic metric space properties   cxms 22927
            12.4.3  Metric space balls   blfvalps 22993
            12.4.4  Open sets of a metric space   mopnval 23048
            12.4.5  Continuity in metric spaces   metcnp3 23150
            12.4.6  The uniform structure generated by a metric   metuval 23159
            12.4.7  Examples of metric spaces   dscmet 23182
            ​*12.4.8  Normed algebraic structures   cnm 23186
            12.4.9  Normed space homomorphisms (bounded linear operators)   cnmo 23314
            12.4.10  Topology on the reals   qtopbaslem 23367
            12.4.11  Topological definitions using the reals   cii 23483
            12.4.12  Path homotopy   chtpy 23571
            12.4.13  The fundamental group   cpco 23604
      ​12.5  Metric subcomplex vector spaces
            12.5.1  Subcomplex modules   cclm 23666
            ​*12.5.2  Subcomplex vector spaces   ccvs 23727
            ​*12.5.3  Normed subcomplex vector spaces   isncvsngp 23753
            12.5.4  Subcomplex pre-Hilbert space   ccph 23770
            12.5.5  Convergence and completeness   ccfil 23855
            12.5.6  Baire's Category Theorem   bcthlem1 23927
            12.5.7  Banach spaces and subcomplex Hilbert spaces   ccms 23935
                  12.5.7.1  The complete ordered field of the real numbers   retopn 23982
            12.5.8  Euclidean spaces   crrx 23986
            12.5.9  Minimizing Vector Theorem   minveclem1 24027
            12.5.10  Projection Theorem   pjthlem1 24040
​PART 13  BASIC REAL AND COMPLEX ANALYSIS
      ​13.1  Continuity
            13.1.1  Intermediate value theorem   pmltpclem1 24049
      ​13.2  Integrals
            13.2.1  Lebesgue measure   covol 24063
            13.2.2  Lebesgue integration   cmbf 24215
                  13.2.2.1  Lesbesgue integral   cmbf 24215
                  13.2.2.2  Lesbesgue directed integral   cdit 24444
      ​13.3  Derivatives
            13.3.1  Real and complex differentiation   climc 24460
                  13.3.1.1  Derivatives of functions of one complex or real variable   climc 24460
                  13.3.1.2  Results on real differentiation   dvferm1lem 24581
​PART 14  BASIC REAL AND COMPLEX FUNCTIONS
      ​14.1  Polynomials
            14.1.1  Polynomial degrees   cmdg 24647
            14.1.2  The division algorithm for univariate polynomials   cmn1 24719
            14.1.3  Elementary properties of complex polynomials   cply 24774
            14.1.4  The division algorithm for polynomials   cquot 24879
            14.1.5  Algebraic numbers   caa 24903
            14.1.6  Liouville's approximation theorem   aalioulem1 24921
      ​14.2  Sequences and series
            14.2.1  Taylor polynomials and Taylor's theorem   ctayl 24941
            14.2.2  Uniform convergence   culm 24964
            14.2.3  Power series   pserval 24998
      ​14.3  Basic trigonometry
            14.3.1  The exponential, sine, and cosine functions (cont.)   efcn 25031
            14.3.2  Properties of pi = 3.14159...   pilem1 25039
            14.3.3  Mapping of the exponential function   efgh 25125
            14.3.4  The natural logarithm on complex numbers   clog 25138
            ​*14.3.5  Logarithms to an arbitrary base   clogb 25342
            14.3.6  Theorems of Pythagoras, isosceles triangles, and intersecting chords   angval 25379
            14.3.7  Solutions of quadratic, cubic, and quartic equations   quad2 25417
            14.3.8  Inverse trigonometric functions   casin 25440
            14.3.9  The Birthday Problem   log2ublem1 25524
            14.3.10  Areas in R^2   carea 25533
            14.3.11  More miscellaneous converging sequences   rlimcnp 25543
            14.3.12  Inequality of arithmetic and geometric means   cvxcl 25562
            14.3.13  Euler-Mascheroni constant   cem 25569
            14.3.14  Zeta function   czeta 25590
            14.3.15  Gamma function   clgam 25593
      ​14.4  Basic number theory
            14.4.1  Wilson's theorem   wilthlem1 25645
            14.4.2  The Fundamental Theorem of Algebra   ftalem1 25650
            14.4.3  The Basel problem (ζ(2) = π²/6)   basellem1 25658
            14.4.4  Number-theoretical functions   ccht 25668
            14.4.5  Perfect Number Theorem   mersenne 25803
            14.4.6  Characters of Z/nZ   cdchr 25808
            14.4.7  Bertrand's postulate   bcctr 25851
            ​*14.4.8  Quadratic residues and the Legendre symbol   clgs 25870
            ​*14.4.9  Gauss' Lemma   gausslemma2dlem0a 25932
            14.4.10  Quadratic reciprocity   lgseisenlem1 25951
            14.4.11  All primes 4n+1 are the sum of two squares   2sqlem1 25993
            14.4.12  Chebyshev's Weak Prime Number Theorem, Dirichlet's Theorem   chebbnd1lem1 26045
            14.4.13  The Prime Number Theorem   mudivsum 26106
            14.4.14  Ostrowski's theorem   abvcxp 26191
​​*PART 15  ELEMENTARY GEOMETRY
      ​15.1  Definition and Tarski's Axioms of Geometry
            15.1.1  Justification for the congruence notation   tgjustf 26259
      ​15.2  Tarskian Geometry
            15.2.1  Congruence   tgcgrcomimp 26263
            15.2.2  Betweenness   tgbtwntriv2 26273
            15.2.3  Dimension   tglowdim1 26286
            15.2.4  Betweenness and Congruence   tgifscgr 26294
            15.2.5  Congruence of a series of points   ccgrg 26296
            15.2.6  Motions   cismt 26318
            15.2.7  Colinearity   tglng 26332
            15.2.8  Connectivity of betweenness   tgbtwnconn1lem1 26358
            15.2.9  Less-than relation in geometric congruences   cleg 26368
            15.2.10  Rays   chlg 26386
            15.2.11  Lines   btwnlng1 26405
            15.2.12  Point inversions   cmir 26438
            15.2.13  Right angles   crag 26479
            15.2.14  Half-planes   islnopp 26525
            15.2.15  Midpoints and Line Mirroring   cmid 26558
            15.2.16  Congruence of angles   ccgra 26593
            15.2.17  Angle Comparisons   cinag 26621
            15.2.18  Congruence Theorems   tgsas1 26640
            15.2.19  Equilateral triangles   ceqlg 26651
      ​15.3  Properties of geometries
            15.3.1  Isomorphisms between geometries   f1otrgds 26655
      ​15.4  Geometry in Hilbert spaces
            15.4.1  Geometry in the complex plane   cchhllem 26673
            15.4.2  Geometry in Euclidean spaces   cee 26674
                  15.4.2.1  Definition of the Euclidean space   cee 26674
                  15.4.2.2  Tarski's axioms for geometry for the Euclidean space   axdimuniq 26699
                  15.4.2.3  EE^n fulfills Tarski's Axioms   ceeng 26763
​​*PART 16  GRAPH THEORY
      ​*16.1  Vertices and edges
            16.1.1  The edge function extractor for extensible structures   cedgf 26774
            ​*16.1.2  Vertices and indexed edges   cvtx 26781
                  16.1.2.1  Definitions and basic properties   cvtx 26781
                  16.1.2.2  The vertices and edges of a graph represented as ordered pair   opvtxval 26788
                  16.1.2.3  The vertices and edges of a graph represented as extensible structure   funvtxdmge2val 26796
                  16.1.2.4  Representations of graphs without edges   snstrvtxval 26822
                  16.1.2.5  Degenerated cases of representations of graphs   vtxval0 26824
            16.1.3  Edges as range of the edge function   cedg 26832
      ​​*16.2  Undirected graphs
            16.2.1  Undirected hypergraphs   cuhgr 26841
            16.2.2  Undirected pseudographs and multigraphs   cupgr 26865
            ​*16.2.3  Loop-free graphs   umgrislfupgrlem 26907
            16.2.4  Edges as subsets of vertices of graphs   uhgredgiedgb 26911
            ​*16.2.5  Undirected simple graphs   cuspgr 26933
            16.2.6  Examples for graphs   usgr0e 27018
            16.2.7  Subgraphs   csubgr 27049
            16.2.8  Finite undirected simple graphs   cfusgr 27098
            16.2.9  Neighbors, complete graphs and universal vertices   cnbgr 27114
                  16.2.9.1  Neighbors   cnbgr 27114
                  16.2.9.2  Universal vertices   cuvtx 27167
                  16.2.9.3  Complete graphs   ccplgr 27191
            16.2.10  Vertex degree   cvtxdg 27247
            ​*16.2.11  Regular graphs   crgr 27337
      ​​*16.3  Walks, paths and cycles
            *16.3.1  Walks   cewlks 27377
            16.3.2  Walks for loop-free graphs   lfgrwlkprop 27469
            16.3.3  Trails   ctrls 27472
            16.3.4  Paths and simple paths   cpths 27493
            16.3.5  Closed walks   cclwlks 27551
            16.3.6  Circuits and cycles   ccrcts 27565
            ​*16.3.7  Walks as words   cwwlks 27603
            16.3.8  Walks/paths of length 2 (as length 3 strings)   2wlkdlem1 27704
            16.3.9  Walks in regular graphs   rusgrnumwwlkl1 27747
            ​*16.3.10  Closed walks as words   cclwwlk 27759
                  16.3.10.1  Closed walks as words   cclwwlk 27759
                  16.3.10.2  Closed walks of a fixed length as words   cclwwlkn 27802
                  16.3.10.3  Closed walks on a vertex of a fixed length as words   cclwwlknon 27866
            16.3.11  Examples for walks, trails and paths   0ewlk 27893
            16.3.12  Connected graphs   cconngr 27965
      ​16.4  Eulerian paths and the Konigsberg Bridge problem
            ​*16.4.1  Eulerian paths   ceupth 27976
            ​*16.4.2  The Königsberg Bridge problem   konigsbergvtx 28025
      ​16.5  The Friendship Theorem
            16.5.1  Friendship graphs - basics   cfrgr 28037
            16.5.2  The friendship theorem for small graphs   frgr1v 28050
            16.5.3  Theorems according to Mertzios and Unger   2pthfrgrrn 28061
            ​*16.5.4  Huneke's Proof of the Friendship Theorem   frgrncvvdeqlem1 28078
​PART 17  GUIDES AND MISCELLANEA
      ​17.1  Guides (conventions, explanations, and examples)
            ​*17.1.1  Conventions   conventions 28179
            17.1.2  Natural deduction   natded 28182
            ​*17.1.3  Natural deduction examples   ex-natded5.2 28183
            17.1.4  Definitional examples   ex-or 28200
            17.1.5  Other examples   aevdemo 28239
      ​17.2  Humor
            17.2.1  April Fool's theorem   avril1 28242
      ​17.3  (Future - to be reviewed and classified)
            17.3.1  Planar incidence geometry   cplig 28251
            ​*17.3.2  Aliases kept to prevent broken links   dummylink 28264
​​*PART 18  COMPLEX TOPOLOGICAL VECTOR SPACES (DEPRECATED)
      ​*18.1  Additional material on group theory (deprecated)
            18.1.1  Definitions and basic properties for groups   cgr 28266
            18.1.2  Abelian groups   cablo 28321
      ​18.2  Complex vector spaces
            18.2.1  Definition and basic properties   cvc 28335
            18.2.2  Examples of complex vector spaces   cnaddabloOLD 28358
      ​18.3  Normed complex vector spaces
            18.3.1  Definition and basic properties   cnv 28361
            18.3.2  Examples of normed complex vector spaces   cnnv 28454
            18.3.3  Induced metric of a normed complex vector space   imsval 28462
            18.3.4  Inner product   cdip 28477
            18.3.5  Subspaces   css 28498
      ​18.4  Operators on complex vector spaces
            18.4.1  Definitions and basic properties   clno 28517
      ​18.5  Inner product (pre-Hilbert) spaces
            18.5.1  Definition and basic properties   ccphlo 28589
            18.5.2  Examples of pre-Hilbert spaces   cncph 28596
            18.5.3  Properties of pre-Hilbert spaces   isph 28599
      ​18.6  Complex Banach spaces
            18.6.1  Definition and basic properties   ccbn 28639
            18.6.2  Examples of complex Banach spaces   cnbn 28646
            18.6.3  Uniform Boundedness Theorem   ubthlem1 28647
            18.6.4  Minimizing Vector Theorem   minvecolem1 28651
      ​18.7  Complex Hilbert spaces
            18.7.1  Definition and basic properties   chlo 28662
            18.7.2  Standard axioms for a complex Hilbert space   hlex 28675
            18.7.3  Examples of complex Hilbert spaces   cnchl 28693
            18.7.4  Hellinger-Toeplitz Theorem   htthlem 28694
​​*PART 19  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      ​19.1  Axiomatization of complex pre-Hilbert spaces
            19.1.1  Basic Hilbert space definitions   chba 28696
            19.1.2  Preliminary ZFC lemmas   df-hnorm 28745
            ​*19.1.3  Derive the Hilbert space axioms from ZFC set theory   axhilex-zf 28758
            ​*19.1.4  Introduce the vector space axioms for a Hilbert space   ax-hilex 28776
            19.1.5  Vector operations   hvmulex 28788
            19.1.6  Inner product postulates for a Hilbert space   ax-hfi 28856
      ​19.2  Inner product and norms
            19.2.1  Inner product   his5 28863
            19.2.2  Norms   dfhnorm2 28899
            19.2.3  Relate Hilbert space to normed complex vector spaces   hilablo 28937
            19.2.4  Bunjakovaskij-Cauchy-Schwarz inequality   bcsiALT 28956
      ​19.3  Cauchy sequences and completeness axiom
            19.3.1  Cauchy sequences and limits   hcau 28961
            19.3.2  Derivation of the completeness axiom from ZF set theory   hilmet 28971
            19.3.3  Completeness postulate for a Hilbert space   ax-hcompl 28979
            19.3.4  Relate Hilbert space to ZFC pre-Hilbert and Hilbert spaces   hhcms 28980
      ​19.4  Subspaces and projections
            19.4.1  Subspaces   df-sh 28984
            19.4.2  Closed subspaces   df-ch 28998
            19.4.3  Orthocomplements   df-oc 29029
            19.4.4  Subspace sum, span, lattice join, lattice supremum   df-shs 29085
            19.4.5  Projection theorem   pjhthlem1 29168
            19.4.6  Projectors   df-pjh 29172
      ​19.5  Properties of Hilbert subspaces
            19.5.1  Orthomodular law   omlsilem 29179
            19.5.2  Projectors (cont.)   pjhtheu2 29193
            19.5.3  Hilbert lattice operations   sh0le 29217
            19.5.4  Span (cont.) and one-dimensional subspaces   spansn0 29318
            19.5.5  Commutes relation for Hilbert lattice elements   df-cm 29360
            19.5.6  Foulis-Holland theorem   fh1 29395
            19.5.7  Quantum Logic Explorer axioms   qlax1i 29404
            19.5.8  Orthogonal subspaces   chscllem1 29414
            19.5.9  Orthoarguesian laws 5OA and 3OA   5oalem1 29431
            19.5.10  Projectors (cont.)   pjorthi 29446
            19.5.11  Mayet's equation E_3   mayete3i 29505
      ​19.6  Operators on Hilbert spaces
            ​*19.6.1  Operator sum, difference, and scalar multiplication   df-hosum 29507
            19.6.2  Zero and identity operators   df-h0op 29525
            19.6.3  Operations on Hilbert space operators   hoaddcl 29535
            19.6.4  Linear, continuous, bounded, Hermitian, unitary operators and norms   df-nmop 29616
            19.6.5  Linear and continuous functionals and norms   df-nmfn 29622
            19.6.6  Adjoint   df-adjh 29626
            19.6.7  Dirac bra-ket notation   df-bra 29627
            19.6.8  Positive operators   df-leop 29629
            19.6.9  Eigenvectors, eigenvalues, spectrum   df-eigvec 29630
            19.6.10  Theorems about operators and functionals   nmopval 29633
            19.6.11  Riesz lemma   riesz3i 29839
            19.6.12  Adjoints (cont.)   cnlnadjlem1 29844
            19.6.13  Quantum computation error bound theorem   unierri 29881
            19.6.14  Dirac bra-ket notation (cont.)   branmfn 29882
            19.6.15  Positive operators (cont.)   leopg 29899
            19.6.16  Projectors as operators   pjhmopi 29923
      ​19.7  States on a Hilbert lattice and Godowski's equation
            19.7.1  States on a Hilbert lattice   df-st 29988
            19.7.2  Godowski's equation   golem1 30048
      ​19.8  Cover relation, atoms, exchange axiom, and modular symmetry
            19.8.1  Covers relation; modular pairs   df-cv 30056
            19.8.2  Atoms   df-at 30115
            19.8.3  Superposition principle   superpos 30131
            19.8.4  Atoms, exchange and covering properties, atomicity   chcv1 30132
            19.8.5  Irreducibility   chirredlem1 30167
            19.8.6  Atoms (cont.)   atcvat3i 30173
            19.8.7  Modular symmetry   mdsymlem1 30180
​PART 20  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      ​20.1  Mathboxes for user contributions
            20.1.1  Mathbox guidelines   mathbox 30219
      ​20.2  Mathbox for Stefan Allan
      ​20.3  Mathbox for Thierry Arnoux
            20.3.1  Propositional Calculus - misc additions   bian1d 30224
            20.3.2  Predicate Calculus   sbc2iedf 30230
                  20.3.2.1  Predicate Calculus - misc additions   sbc2iedf 30230
                  20.3.2.2  Restricted quantification - misc additions   ralcom4f 30233
                  20.3.2.3  Equality   eqtrb 30238
                  20.3.2.4  Double restricted existential uniqueness quantification   opsbc2ie 30239
                  20.3.2.5  Double restricted existential uniqueness quantification syntax   w2reu 30241
                  20.3.2.6  Substitution (without distinct variables) - misc additions   sbceqbidf 30250
                  20.3.2.7  Existential "at most one" - misc additions   moel 30252
                  20.3.2.8  Existential uniqueness - misc additions   reuxfrdf 30255
                  20.3.2.9  Restricted "at most one" - misc additions   rmoxfrd 30257
            20.3.3  General Set Theory   dmrab 30260
                  20.3.3.1  Class abstractions (a.k.a. class builders)   dmrab 30260
                  20.3.3.2  Image Sets   abrexdomjm 30267
                  20.3.3.3  Set relations and operations - misc additions   elunsn 30273
                  20.3.3.4  Unordered pairs   eqsnd 30289
                  20.3.3.5  Conditional operator - misc additions   ifeqeqx 30297
                  20.3.3.6  Set union   uniinn0 30302
                  20.3.3.7  Indexed union - misc additions   cbviunf 30307
                  20.3.3.8  Disjointness - misc additions   disjnf 30320
            20.3.4  Relations and Functions   xpdisjres 30348
                  20.3.4.1  Relations - misc additions   xpdisjres 30348
                  20.3.4.2  Functions - misc additions   ac6sf2 30370
                  20.3.4.3  Operations - misc additions   mpomptxf 30425
                  20.3.4.4  Explicit Functions with one or two points as a domain   brsnop 30429
                  20.3.4.5  Isomorphisms - misc. add.   gtiso 30436
                  20.3.4.6  Disjointness (additional proof requiring functions)   disjdsct 30438
                  20.3.4.7  First and second members of an ordered pair - misc additions   df1stres 30439
                  20.3.4.8  Supremum - misc additions   supssd 30445
                  20.3.4.9  Finite Sets   imafi2 30447
                  20.3.4.10  Countable Sets   snct 30449
            20.3.5  Real and Complex Numbers   creq0 30471
                  20.3.5.1  Complex operations - misc. additions   creq0 30471
                  20.3.5.2  Ordering on reals - misc additions   lt2addrd 30475
                  20.3.5.3  Extended reals - misc additions   xrlelttric 30476
                  20.3.5.4  Extended nonnegative integers - misc additions   xnn0gt0 30494
                  20.3.5.5  Real number intervals - misc additions   joiniooico 30497
                  20.3.5.6  Finite intervals of integers - misc additions   uzssico 30507
                  20.3.5.7  Half-open integer ranges - misc additions   iundisjfi 30519
                  20.3.5.8  The ` # ` (set size) function - misc additions   hashunif 30528
                  20.3.5.9  The greatest common divisor operator - misc. add   dvdszzq 30531
                  20.3.5.10  Integers   nnindf 30535
                  20.3.5.11  Decimal numbers   dfdec100 30546
            ​*20.3.6  Decimal expansion   cdp2 30547
                  ​*20.3.6.1  Decimal point   cdp 30564
                  20.3.6.2  Division in the extended real number system   cxdiv 30593
            20.3.7  Words over a set - misc additions   wrdfd 30612
                  20.3.7.1  Splicing words (substring replacement)   splfv3 30632
                  20.3.7.2  Cyclic shift of words   1cshid 30633
            20.3.8  Extensible Structures   ressplusf 30637
                  20.3.8.1  Structure restriction operator   ressplusf 30637
                  20.3.8.2  The opposite group   oppgle 30640
                  20.3.8.3  Posets   ressprs 30642
                  20.3.8.4  Complete lattices   clatp0cl 30658
                  20.3.8.5  Extended reals Structure - misc additions   ax-xrssca 30660
                  20.3.8.6  The extended nonnegative real numbers commutative monoid   xrge0base 30672
            20.3.9  Algebra   abliso 30683
                  20.3.9.1  Monoids Homomorphisms   abliso 30683
                  20.3.9.2  Finitely supported group sums - misc additions   gsumsubg 30684
                  20.3.9.3  Centralizers and centers - misc additions   cntzun 30695
                  20.3.9.4  Totally ordered monoids and groups   comnd 30698
                  20.3.9.5  The symmetric group   symgfcoeu 30726
                  20.3.9.6  Transpositions   pmtridf1o 30736
                  20.3.9.7  Permutation Signs   psgnid 30739
                  20.3.9.8  Permutation cycles   ctocyc 30748
                  20.3.9.9  The Alternating Group   evpmval 30787
                  20.3.9.10  Signum in an ordered monoid   csgns 30800
                  20.3.9.11  The Archimedean property for generic ordered algebraic structures   cinftm 30805
                  20.3.9.12  Semiring left modules   cslmd 30828
                  20.3.9.13  Simple groups   prmsimpcyc 30856
                  20.3.9.14  Rings - misc additions   rngurd 30857
                  20.3.9.15  Subfields   primefldchr 30867
                  20.3.9.16  Totally ordered rings and fields   corng 30868
                  20.3.9.17  Ring homomorphisms - misc additions   rhmdvdsr 30891
                  20.3.9.18  Scalar restriction operation   cresv 30897
                  20.3.9.19  The commutative ring of gaussian integers   gzcrng 30912
                  20.3.9.20  The archimedean ordered field of real numbers   reofld 30913
                  20.3.9.21  The quotient map and quotient modules   qusker 30918
                  20.3.9.22  Univariate Polynomials   fply1 30931
                  20.3.9.23  Independent sets and families   islinds5 30932
                  ​*20.3.9.24  Subgroup sum / Sumset / Minkowski sum   elgrplsmsn 30944
                  20.3.9.25  Prime Ideals   cprmidl 30952
                  20.3.9.26  Maximal Ideals   cmxidl 30968
                  20.3.9.27  The semiring of ideals of a ring   cidlsrg 30982
                  20.3.9.28  The subring algebra   sra1r 30986
                  20.3.9.29  Division Ring Extensions   drgext0g 30992
                  20.3.9.30  Vector Spaces   lvecdimfi 30998
                  20.3.9.31  Vector Space Dimension   cldim 30999
            20.3.10  Field Extensions   cfldext 31028
            20.3.11  Matrices   csmat 31058
                  20.3.11.1  Submatrices   csmat 31058
                  20.3.11.2  Matrix literals   clmat 31076
                  20.3.11.3  Laplace expansion of determinants   mdetpmtr1 31088
            20.3.12  Topology   txomap 31098
                  20.3.12.1  Open maps   txomap 31098
                  20.3.12.2  Topology of the unit circle   qtopt1 31099
                  20.3.12.3  Refinements   reff 31103
                  20.3.12.4  Open cover refinement property   ccref 31106
                  20.3.12.5  Lindelöf spaces   cldlf 31116
                  20.3.12.6  Paracompact spaces   cpcmp 31119
                  20.3.12.7  Pseudometrics   cmetid 31126
                  20.3.12.8  Continuity - misc additions   hauseqcn 31138
                  20.3.12.9  Topology of the closed unit interval   unitsscn 31139
                  20.3.12.10  Topology of ` ( RR X. RR ) `   unicls 31146
                  20.3.12.11  Order topology - misc. additions   cnvordtrestixx 31156
                  20.3.12.12  Continuity in topological spaces - misc. additions   mndpluscn 31169
                  20.3.12.13  Topology of the extended nonnegative real numbers ordered monoid   xrge0hmph 31175
                  20.3.12.14  Limits - misc additions   lmlim 31190
                  20.3.12.15  Univariate polynomials   pl1cn 31198
            20.3.13  Uniform Stuctures and Spaces   chcmp 31199
                  20.3.13.1  Hausdorff uniform completion   chcmp 31199
            20.3.14  Topology and algebraic structures   zringnm 31201
                  20.3.14.1  The norm on the ring of the integer numbers   zringnm 31201
                  20.3.14.2  Topological ` ZZ ` -modules   zlm0 31203
                  20.3.14.3  Canonical embedding of the field of the rational numbers into a division ring   cqqh 31213
                  20.3.14.4  Canonical embedding of the real numbers into a complete ordered field   crrh 31234
                  20.3.14.5  Embedding from the extended real numbers into a complete lattice   cxrh 31257
                  20.3.14.6  Canonical embeddings into the ordered field of the real numbers   zrhre 31260
                  ​*20.3.14.7  Topological Manifolds   cmntop 31263
            20.3.15  Real and complex functions   nexple 31268
                  20.3.15.1  Integer powers - misc. additions   nexple 31268
                  20.3.15.2  Indicator Functions   cind 31269
                  20.3.15.3  Extended sum   cesum 31286
            20.3.16  Mixed Function/Constant operation   cofc 31354
            20.3.17  Abstract measure   csiga 31367
                  20.3.17.1  Sigma-Algebra   csiga 31367
                  20.3.17.2  Generated sigma-Algebra   csigagen 31397
                  ​*20.3.17.3  lambda and pi-Systems, Rings of Sets   ispisys 31411
                  20.3.17.4  The Borel algebra on the real numbers   cbrsiga 31440
                  20.3.17.5  Product Sigma-Algebra   csx 31447
                  20.3.17.6  Measures   cmeas 31454
                  20.3.17.7  The counting measure   cntmeas 31485
                  20.3.17.8  The Lebesgue measure - misc additions   voliune 31488
                  20.3.17.9  The Dirac delta measure   cdde 31491
                  20.3.17.10  The 'almost everywhere' relation   cae 31496
                  20.3.17.11  Measurable functions   cmbfm 31508
                  20.3.17.12  Borel Algebra on ` ( RR X. RR ) `   br2base 31527
                  ​*20.3.17.13  Caratheodory's extension theorem   coms 31549
            20.3.18  Integration   itgeq12dv 31584
                  20.3.18.1  Lebesgue integral - misc additions   itgeq12dv 31584
                  20.3.18.2  Bochner integral   citgm 31585
            20.3.19  Euler's partition theorem   oddpwdc 31612
            20.3.20  Sequences defined by strong recursion   csseq 31641
            20.3.21  Fibonacci Numbers   cfib 31654
            20.3.22  Probability   cprb 31665
                  20.3.22.1  Probability Theory   cprb 31665
                  20.3.22.2  Conditional Probabilities   ccprob 31689
                  20.3.22.3  Real-valued Random Variables   crrv 31698
                  20.3.22.4  Preimage set mapping operator   corvc 31713
                  20.3.22.5  Distribution Functions   orvcelval 31726
                  20.3.22.6  Cumulative Distribution Functions   orvclteel 31730
                  20.3.22.7  Probabilities - example   coinfliplem 31736
                  20.3.22.8  Bertrand's Ballot Problem   ballotlemoex 31743
            20.3.23  Signum (sgn or sign) function - misc. additions   sgncl 31796
                  20.3.23.1  Operations on words   ccatmulgnn0dir 31812
            20.3.24  Polynomials with real coefficients - misc additions   plymul02 31816
            20.3.25  Descartes's rule of signs   signspval 31822
                  20.3.25.1  Sign changes in a word over real numbers   signspval 31822
                  20.3.25.2  Counting sign changes in a word over real numbers   signslema 31832
            20.3.26  Number Theory   efcld 31862
                  20.3.26.1  Representations of a number as sums of integers   crepr 31879
                  20.3.26.2  Vinogradov Trigonometric Sums and the Circle Method   cvts 31906
                  20.3.26.3  The Ternary Goldbach Conjecture: Final Statement   ax-hgt749 31915
            20.3.27  Elementary Geometry   cstrkg2d 31935
                  ​*20.3.27.1  Two-dimensional geometry   cstrkg2d 31935
                  20.3.27.2  Outer Five Segment (not used, no need to move to main)   cafs 31940
            ​*20.3.28  LeftPad Project   clpad 31945
      ​​*20.4  Mathbox for Jonathan Ben-Naim
            20.4.1  First-order logic and set theory   bnj170 31968
            20.4.2  Well founded induction and recursion   bnj110 32130
            20.4.3  The existence of a minimal element in certain classes   bnj69 32282
            20.4.4  Well-founded induction   bnj1204 32284
            20.4.5  Well-founded recursion, part 1 of 3   bnj60 32334
            20.4.6  Well-founded recursion, part 2 of 3   bnj1500 32340
            20.4.7  Well-founded recursion, part 3 of 3   bnj1522 32344
      ​20.5  Mathbox for BTernaryTau
            20.5.1  Acyclic graphs   cacycgr 32389
      ​20.6  Mathbox for Mario Carneiro
            20.6.1  Predicate calculus with all distinct variables   ax-7d 32406
            20.6.2  Miscellaneous stuff   quartfull 32412
            20.6.3  Derangements and the Subfactorial   deranglem 32413
            20.6.4  The Erdős-Szekeres theorem   erdszelem1 32438
            20.6.5  The Kuratowski closure-complement theorem   kur14lem1 32453
            20.6.6  Retracts and sections   cretr 32464
            20.6.7  Path-connected and simply connected spaces   cpconn 32466
            20.6.8  Covering maps   ccvm 32502
            20.6.9  Normal numbers   snmlff 32576
            20.6.10  Godel-sets of formulas - part 1   cgoe 32580
            20.6.11  Godel-sets of formulas - part 2   cgon 32679
            20.6.12  Models of ZF   cgze 32693
            ​*20.6.13  Metamath formal systems   cmcn 32707
            20.6.14  Grammatical formal systems   cm0s 32832
            20.6.15  Models of formal systems   cmuv 32852
            20.6.16  Splitting fields   citr 32874
            20.6.17  p-adic number fields   czr 32890
      ​​*20.7  Mathbox for Filip Cernatescu
      ​20.8  Mathbox for Paul Chapman
            20.8.1  Real and complex numbers (cont.)   climuzcnv 32914
            20.8.2  Miscellaneous theorems   elfzm12 32918
      ​20.9  Mathbox for Scott Fenton
            20.9.1  ZFC Axioms in primitive form   axextprim 32927
            20.9.2  Untangled classes   untelirr 32934
            20.9.3  Extra propositional calculus theorems   3orel2 32941
            20.9.4  Misc. Useful Theorems   nepss 32948
            20.9.5  Properties of real and complex numbers   sqdivzi 32959
            20.9.6  Infinite products   iprodefisumlem 32972
            20.9.7  Factorial limits   faclimlem1 32975
            20.9.8  Greatest common divisor and divisibility   pdivsq 32981
            20.9.9  Properties of relationships   brtp 32985
            20.9.10  Properties of functions and mappings   funpsstri 33008
            20.9.11  Set induction (or epsilon induction)   setinds 33023
            20.9.12  Ordinal numbers   elpotr 33026
            20.9.13  Defined equality axioms   axextdfeq 33042
            20.9.14  Hypothesis builders   hbntg 33050
            20.9.15  (Trans)finite Recursion Theorems   tfisg 33055
            20.9.16  Transitive closure under a relationship   ctrpred 33056
            20.9.17  Founded Induction   frpomin 33078
            20.9.18  Ordering Ordinal Sequences   orderseqlem 33094
            20.9.19  Well-founded zero, successor, and limits   cwsuc 33097
            20.9.20  Founded Partial Recursion   cfrecs 33117
            20.9.21  Surreal Numbers   csur 33147
            20.9.22  Surreal Numbers: Ordering   sltsolem1 33180
            20.9.23  Surreal Numbers: Birthday Function   bdayfo 33182
            20.9.24  Surreal Numbers: Density   fvnobday 33183
            20.9.25  Surreal Numbers: Full-Eta Property   bdayimaon 33197
            20.9.26  Surreal numbers - ordering theorems   csle 33223
            20.9.27  Surreal numbers - birthday theorems   bdayfun 33242
            20.9.28  Surreal numbers: Conway cuts   csslt 33250
            20.9.29  Surreal numbers - cuts and options   cmade 33279
            20.9.30  Quantifier-free definitions   ctxp 33291
            20.9.31  Alternate ordered pairs   caltop 33417
            20.9.32  Geometry in the Euclidean space   cofs 33443
                  20.9.32.1  Congruence properties   cofs 33443
                  20.9.32.2  Betweenness properties   btwntriv2 33473
                  20.9.32.3  Segment Transportation   ctransport 33490
                  20.9.32.4  Properties relating betweenness and congruence   cifs 33496
                  20.9.32.5  Connectivity of betweenness   btwnconn1lem1 33548
                  20.9.32.6  Segment less than or equal to   csegle 33567
                  20.9.32.7  Outside-of relationship   coutsideof 33580
                  20.9.32.8  Lines and Rays   cline2 33595
            20.9.33  Forward difference   cfwddif 33619
            20.9.34  Rank theorems   rankung 33627
            20.9.35  Hereditarily Finite Sets   chf 33633
      ​20.10  Mathbox for Jeff Hankins
            20.10.1  Miscellany   a1i14 33648
            20.10.2  Basic topological facts   topbnd 33672
            20.10.3  Topology of the real numbers   ivthALT 33683
            20.10.4  Refinements   cfne 33684
            20.10.5  Neighborhood bases determine topologies   neibastop1 33707
            20.10.6  Lattice structure of topologies   topmtcl 33711
            20.10.7  Filter bases   fgmin 33718
            20.10.8  Directed sets, nets   tailfval 33720
      ​20.11  Mathbox for Anthony Hart
            20.11.1  Propositional Calculus   tb-ax1 33731
            20.11.2  Predicate Calculus   nalfal 33751
            20.11.3  Miscellaneous single axioms   meran1 33759
            20.11.4  Connective Symmetry   negsym1 33765
      ​20.12  Mathbox for Chen-Pang He
            20.12.1  Ordinal topology   ontopbas 33776
      ​20.13  Mathbox for Jeff Hoffman
            20.13.1  Inferences for finite induction on generic function values   fveleq 33799
            20.13.2  gdc.mm   nnssi2 33803
      ​20.14  Mathbox for Asger C. Ipsen
            20.14.1  Continuous nowhere differentiable functions   dnival 33810
      ​​*20.15  Mathbox for BJ
            *20.15.1  Propositional calculus   bj-mp2c 33879
                  *20.15.1.1  Derived rules of inference   bj-mp2c 33879
                  ​*20.15.1.2  A syntactic theorem   bj-0 33881
                  20.15.1.3  Minimal implicational calculus   bj-a1k 33883
                  ​*20.15.1.4  Positive calculus   bj-syl66ib 33890
                  20.15.1.5  Implication and negation   bj-con2com 33896
                  ​*20.15.1.6  Disjunction   bj-jaoi1 33904
                  ​*20.15.1.7  Logical equivalence   bj-dfbi4 33906
                  20.15.1.8  The conditional operator for propositions   bj-consensus 33911
                  ​*20.15.1.9  Propositional calculus: miscellaneous   bj-imbi12 33916
            ​*20.15.2  Modal logic   bj-axdd2 33926
            ​*20.15.3  Provability logic   cprvb 33931
            ​*20.15.4  First-order logic   bj-genr 33940
                  20.15.4.1  Adding ax-gen   bj-genr 33940
                  20.15.4.2  Adding ax-4   bj-2alim 33944
                  20.15.4.3  Adding ax-5   bj-ax12wlem 33977
                  20.15.4.4  Equality and substitution   bj-ssbeq 33986
                  20.15.4.5  Adding ax-6   bj-spimvwt 34002
                  20.15.4.6  Adding ax-7   bj-cbvexw 34009
                  20.15.4.7  Membership predicate, ax-8 and ax-9   bj-ax89 34011
                  20.15.4.8  Adding ax-11   bj-alcomexcom 34014
                  20.15.4.9  Adding ax-12   axc11n11 34016
                  20.15.4.10  Nonfreeness   wnnf 34055
                  20.15.4.11  Adding ax-13   bj-axc10 34105
                  ​*20.15.4.12  Removing dependencies on ax-13 (and ax-11)   bj-axc10v 34115
                  ​*20.15.4.13  Distinct var metavariables   bj-hbaeb2 34141
                  ​*20.15.4.14  Around ~ equsal   bj-equsal1t 34145
                  ​*20.15.4.15  Some Principia Mathematica proofs   stdpc5t 34150
                  20.15.4.16  Alternate definition of substitution   bj-sbsb 34160
                  20.15.4.17  Lemmas for substitution   bj-sbf3 34162
                  20.15.4.18  Existential uniqueness   bj-eu3f 34165
                  ​*20.15.4.19  First-order logic: miscellaneous   bj-sblem1 34166
            20.15.5  Set theory   eliminable1 34182
                  ​*20.15.5.1  Eliminability of class terms   eliminable1 34182
                  ​*20.15.5.2  Classes without the axiom of extensionality   bj-denoteslem 34188
                  20.15.5.3  Characterization among sets versus among classes   elelb 34216
                  ​*20.15.5.4  The nonfreeness quantifier for classes   bj-nfcsym 34218
                  ​*20.15.5.5  Proposal for the definitions of class membership and class equality   bj-ax9 34219
                  ​*20.15.5.6  Lemmas for class substitution   bj-sbeqALT 34220
                  20.15.5.7  Removing some axiom requirements and disjoint variable conditions   bj-exlimvmpi 34230
                  ​*20.15.5.8  Class abstractions   bj-unrab 34247
                  ​*20.15.5.9  Restricted nonfreeness   wrnf 34254
                  ​*20.15.5.10  Russell's paradox   bj-ru0 34256
                  20.15.5.11  Curry's paradox in set theory   currysetlem 34259
                  ​*20.15.5.12  Some disjointness results   bj-n0i 34265
                  ​*20.15.5.13  Complements on direct products   bj-xpimasn 34270
                  ​*20.15.5.14  "Singletonization" and tagging   bj-snsetex 34278
                  ​*20.15.5.15  Tuples of classes   bj-cproj 34305
                  ​*20.15.5.16  Set theory: elementary operations relative to a universe   bj-rcleqf 34340
                  ​*20.15.5.17  Set theory: miscellaneous   bj-pw0ALT 34345
                  ​*20.15.5.18  Evaluation   bj-evaleq 34366
                  20.15.5.19  Elementwise operations   celwise 34373
                  ​*20.15.5.20  Elementwise intersection (families of sets induced on a subset)   bj-rest00 34375
                  20.15.5.21  Moore collections (complements)   bj-intss 34394
                  20.15.5.22  Maps-to notation for functions with three arguments   bj-0nelmpt 34411
                  ​*20.15.5.23  Currying   csethom 34417
                  ​*20.15.5.24  Setting components of extensible structures   cstrset 34429
            ​*20.15.6  Extended real and complex numbers, real and complex projective lines   bj-nfald 34432
                  20.15.6.1  Complements on class abstractions of ordered pairs and binary relations   bj-nfald 34432
                  ​*20.15.6.2  Identity relation (complements)   bj-opabssvv 34445
                  ​*20.15.6.3  Functionalized identity (diagonal in a Cartesian square)   cdiag2 34467
                  ​*20.15.6.4  Direct image and inverse image   cimdir 34473
                  ​*20.15.6.5  Extended numbers and projective lines as sets   cfractemp 34481
                  ​*20.15.6.6  Addition and opposite   caddcc 34522
                  ​*20.15.6.7  Order relation on the extended reals   cltxr 34526
                  ​*20.15.6.8  Argument, multiplication and inverse   carg 34528
                  20.15.6.9  The canonical bijection from the finite ordinals   ciomnn 34534
                  20.15.6.10  Divisibility   cnnbar 34545
            ​*20.15.7  Monoids   bj-smgrpssmgm 34553
                  ​*20.15.7.1  Finite sums in monoids   cfinsum 34568
            ​*20.15.8  Affine, Euclidean, and Cartesian geometry   bj-fvimacnv0 34571
                  *20.15.8.1  Real vector spaces   bj-fvimacnv0 34571
                  ​*20.15.8.2  Complex numbers (supplements)   bj-subcom 34592
                  ​*20.15.8.3  Barycentric coordinates   bj-bary1lem 34594
            20.15.9  Monoid of endomorphisms   cend 34597
      ​20.16  Mathbox for Jim Kingdon
                  20.16.0.1  Circle constant   taupilem3 34603
                  20.16.0.2  Number theory   dfgcd3 34608
      ​20.17  Mathbox for ML
            20.17.1  Miscellaneous   csbdif 34609
            20.17.2  Cartesian exponentiation   cfinxp 34667
            20.17.3  Topology   iunctb2 34687
                  ​*20.17.3.1  Pi-base theorems   pibp16 34697
      ​20.18  Mathbox for Wolf Lammen
            20.18.1  1. Bootstrapping   wl-section-boot 34706
            20.18.2  Implication chains   wl-section-impchain 34730
            20.18.3  Alternative development of hadd   wl-df-had 34748
            20.18.4  An alternative axiom ~ ax-13   ax-wl-13v 34760
            20.18.5  Other stuff   wl-mps 34762
            20.18.6  1. Restricted Quantifiers   wl-ral 34846
      ​20.19  Mathbox for Brendan Leahy
      ​20.20  Mathbox for Jeff Madsen
            20.20.1  Logic and set theory   unirep 35003
            20.20.2  Real and complex numbers; integers   filbcmb 35030
            20.20.3  Sequences and sums   sdclem2 35032
            20.20.4  Topology   subspopn 35042
            20.20.5  Metric spaces   metf1o 35045
            20.20.6  Continuous maps and homeomorphisms   constcncf 35052
            20.20.7  Boundedness   ctotbnd 35059
            20.20.8  Isometries   cismty 35091
            20.20.9  Heine-Borel Theorem   heibor1lem 35102
            20.20.10  Banach Fixed Point Theorem   bfplem1 35115
            20.20.11  Euclidean space   crrn 35118
            20.20.12  Intervals (continued)   ismrer1 35131
            20.20.13  Operation properties   cass 35135
            20.20.14  Groups and related structures   cmagm 35141
            20.20.15  Group homomorphism and isomorphism   cghomOLD 35176
            20.20.16  Rings   crngo 35187
            20.20.17  Division Rings   cdrng 35241
            20.20.18  Ring homomorphisms   crnghom 35253
            20.20.19  Commutative rings   ccm2 35282
            20.20.20  Ideals   cidl 35300
            20.20.21  Prime rings and integral domains   cprrng 35339
            20.20.22  Ideal generators   cigen 35352
      ​20.21  Mathbox for Giovanni Mascellani
            ​*20.21.1  Tools for automatic proof building   efald2 35371
            ​*20.21.2  Tseitin axioms   fald 35422
            ​*20.21.3  Equality deductions   iuneq2f 35449
            ​*20.21.4  Miscellanea   orcomdd 35460
      ​20.22  Mathbox for Peter Mazsa
            20.22.1  Notations   cxrn 35467
            20.22.2  Preparatory theorems   el2v1 35505
            20.22.3  Range Cartesian product   df-xrn 35638
            20.22.4  Cosets by ` R `   df-coss 35674
            20.22.5  Relations   df-rels 35740
            20.22.6  Subset relations   df-ssr 35753
            20.22.7  Reflexivity   df-refs 35765
            20.22.8  Converse reflexivity   df-cnvrefs 35778
            20.22.9  Symmetry   df-syms 35793
            20.22.10  Reflexivity and symmetry   symrefref2 35814
            20.22.11  Transitivity   df-trs 35823
            20.22.12  Equivalence relations   df-eqvrels 35834
            20.22.13  Redundancy   df-redunds 35873
            20.22.14  Domain quotients   df-dmqss 35888
            20.22.15  Equivalence relations on domain quotients   df-ers 35912
            20.22.16  Functions   df-funss 35928
            20.22.17  Disjoints vs. converse functions   df-disjss 35951
      ​20.23  Mathbox for Rodolfo Medina
            20.23.1  Partitions   prtlem60 36004
      ​​*20.24  Mathbox for Norm Megill
            *20.24.1  Obsolete schemes ax-c4,c5,c7,c10,c11,c11n,c15,c9,c14,c16   ax-c5 36034
            ​*20.24.2  Rederive new axioms ax-4, ax-10, ax-6, ax-12, ax-13 from old   axc5 36044
            ​*20.24.3  Legacy theorems using obsolete axioms   ax5ALT 36058
            20.24.4  Experiments with weak deduction theorem   elimhyps 36112
            20.24.5  Miscellanea   cnaddcom 36123
            20.24.6  Atoms, hyperplanes, and covering in a left vector space (or module)   clsa 36125
            20.24.7  Functionals and kernels of a left vector space (or module)   clfn 36208
            20.24.8  Opposite rings and dual vector spaces   cld 36274
            20.24.9  Ortholattices and orthomodular lattices   cops 36323
            20.24.10  Atomic lattices with covering property   ccvr 36413
            20.24.11  Hilbert lattices   chlt 36501
            20.24.12  Projective geometries based on Hilbert lattices   clln 36642
            20.24.13  Construction of a vector space from a Hilbert lattice   cdlema1N 36942
            20.24.14  Construction of involution and inner product from a Hilbert lattice   clpoN 38631
      ​20.25  Mathbox for metakunt
      ​20.26  Mathbox for Steven Nguyen
            20.26.1  Utility theorems   ioin9i8 39148
            ​*20.26.2  Arithmetic theorems   c0exALT 39201
            20.26.3  Exponents   oexpreposd 39228
            20.26.4  Real subtraction   cresub 39244
            ​*20.26.5  Projective spaces   cprjsp 39300
            20.26.6  Equivalent formulations of Fermat's Last Theorem   dffltz 39320
      ​20.27  Mathbox for Igor Ieskov
      ​20.28  Mathbox for OpenAI
      ​20.29  Mathbox for Stefan O'Rear
            20.29.1  Additional elementary logic and set theory   moxfr 39338
            20.29.2  Additional theory of functions   imaiinfv 39339
            20.29.3  Additional topology   elrfi 39340
            20.29.4  Characterization of closure operators. Kuratowski closure axioms   ismrcd1 39344
            20.29.5  Algebraic closure systems   cnacs 39348
            20.29.6  Miscellanea 1. Map utilities   constmap 39359
            20.29.7  Miscellanea for polynomials   mptfcl 39366
            20.29.8  Multivariate polynomials over the integers   cmzpcl 39367
            20.29.9  Miscellanea for Diophantine sets 1   coeq0i 39399
            20.29.10  Diophantine sets 1: definitions   cdioph 39401
            20.29.11  Diophantine sets 2 miscellanea   ellz1 39413
            20.29.12  Diophantine sets 2: union and intersection. Monotone Boolean algebra   diophin 39418
            20.29.13  Diophantine sets 3: construction   diophrex 39421
            20.29.14  Diophantine sets 4 miscellanea   2sbcrex 39430
            20.29.15  Diophantine sets 4: Quantification   rexrabdioph 39440
            20.29.16  Diophantine sets 5: Arithmetic sets   rabdiophlem1 39447
            20.29.17  Diophantine sets 6: reusability. renumbering of variables   eldioph4b 39457
            20.29.18  Pigeonhole Principle and cardinality helpers   fphpd 39462
            20.29.19  A non-closed set of reals is infinite   rencldnfilem 39466
            20.29.20  Lagrange's rational approximation theorem   irrapxlem1 39468
            20.29.21  Pell equations 1: A nontrivial solution always exists   pellexlem1 39475
            20.29.22  Pell equations 2: Algebraic number theory of the solution set   csquarenn 39482
            20.29.23  Pell equations 3: characterizing fundamental solution   infmrgelbi 39524
            ​*20.29.24  Logarithm laws generalized to an arbitrary base   reglogcl 39536
            20.29.25  Pell equations 4: the positive solution group is infinite cyclic   pellfund14 39544
            20.29.26  X and Y sequences 1: Definition and recurrence laws   crmx 39546
            20.29.27  Ordering and induction lemmas for the integers   monotuz 39587
            20.29.28  X and Y sequences 2: Order properties   rmxypos 39593
            20.29.29  Congruential equations   congtr 39611
            20.29.30  Alternating congruential equations   acongid 39621
            20.29.31  Additional theorems on integer divisibility   coprmdvdsb 39631
            20.29.32  X and Y sequences 3: Divisibility properties   jm2.18 39634
            20.29.33  X and Y sequences 4: Diophantine representability of Y   jm2.27a 39651
            20.29.34  X and Y sequences 5: Diophantine representability of X, ^, _C   rmxdiophlem 39661
            20.29.35  Uncategorized stuff not associated with a major project   setindtr 39670
            20.29.36  More equivalents of the Axiom of Choice   axac10 39679
            20.29.37  Finitely generated left modules   clfig 39716
            20.29.38  Noetherian left modules I   clnm 39724
            20.29.39  Addenda for structure powers   pwssplit4 39738
            20.29.40  Every set admits a group structure iff choice   unxpwdom3 39744
            20.29.41  Noetherian rings and left modules II   clnr 39758
            20.29.42  Hilbert's Basis Theorem   cldgis 39770
            20.29.43  Additional material on polynomials [DEPRECATED]   cmnc 39780
            20.29.44  Degree and minimal polynomial of algebraic numbers   cdgraa 39789
            20.29.45  Algebraic integers I   citgo 39806
            20.29.46  Endomorphism algebra   cmend 39824
            20.29.47  Cyclic groups and order   idomrootle 39844
            20.29.48  Cyclotomic polynomials   ccytp 39851
            20.29.49  Miscellaneous topology   fgraphopab 39859
      ​20.30  Mathbox for Jon Pennant
      ​20.31  Mathbox for Richard Penner
            20.31.1  Short Studies   ifpan123g 39873
                  20.31.1.1  Additional work on conditional logical operator   ifpan123g 39873
                  20.31.1.2  Sophisms   rp-fakeimass 39927
                  ​*20.31.1.3  Finite Sets   rp-isfinite5 39932
                  20.31.1.4  General Observations   intabssd 39934
                  20.31.1.5  Infinite Sets   pwelg 39968
                  ​*20.31.1.6  Finite intersection property   fipjust 39973
                  20.31.1.7  RP ADDTO: Subclasses and subsets   rababg 39982
                  20.31.1.8  RP ADDTO: The intersection of a class   elintabg 39983
                  20.31.1.9  RP ADDTO: Theorems requiring subset and intersection existence   elinintrab 39986
                  20.31.1.10  RP ADDTO: Relations   xpinintabd 39989
                  ​*20.31.1.11  RP ADDTO: Functions   elmapintab 40005
                  ​*20.31.1.12  RP ADDTO: Finite induction (for finite ordinals)   cnvcnvintabd 40009
                  20.31.1.13  RP ADDTO: First and second members of an ordered pair   elcnvlem 40010
                  20.31.1.14  RP ADDTO: The reflexive and transitive properties of relations   undmrnresiss 40013
                  20.31.1.15  RP ADDTO: Basic properties of closures   cleq2lem 40017
                  20.31.1.16  RP REPLACE: Definitions and basic properties of transitive closures   trcleq2lemRP 40039
            20.31.2  Additional statements on relations and subclasses   al3im 40040
                  20.31.2.1  Transitive relations (not to be confused with transitive classes).   trrelind 40059
                  20.31.2.2  Reflexive closures   crcl 40066
                  ​*20.31.2.3  Finite relationship composition.   relexp2 40071
                  20.31.2.4  Transitive closure of a relation   dftrcl3 40114
                  ​*20.31.2.5  Adapted from Frege   frege77d 40140
            ​*20.31.3  Propositions from _Begriffsschrift_   dfxor4 40160
                  *20.31.3.1  _Begriffsschrift_ Chapter I   dfxor4 40160
                  ​*20.31.3.2  _Begriffsschrift_ Notation hints   rp-imass 40166
                  20.31.3.3  _Begriffsschrift_ Chapter II Implication   ax-frege1 40185
                  20.31.3.4  _Begriffsschrift_ Chapter II Implication and Negation   axfrege28 40224
                  ​*20.31.3.5  _Begriffsschrift_ Chapter II with logical equivalence   axfrege52a 40251
                  20.31.3.6  _Begriffsschrift_ Chapter II with equivalence of sets   axfrege52c 40282
                  ​*20.31.3.7  _Begriffsschrift_ Chapter II with equivalence of classes   frege53c 40309
                  ​*20.31.3.8  _Begriffsschrift_ Chapter III Properties hereditary in a sequence   dffrege69 40327
                  ​*20.31.3.9  _Begriffsschrift_ Chapter III Following in a sequence   dffrege76 40334
                  ​*20.31.3.10  _Begriffsschrift_ Chapter III Member of sequence   dffrege99 40357
                  ​*20.31.3.11  _Begriffsschrift_ Chapter III Single-valued procedures   dffrege115 40373
            ​*20.31.4  Exploring Topology via Seifert and Threlfall   enrelmap 40392
                  *20.31.4.1  Equinumerosity of sets of relations and maps   enrelmap 40392
                  ​*20.31.4.2  Generic Pseudoclosure Spaces, Pseudointerior Spaces, and Pseudoneighborhoods   sscon34b 40418
                  ​*20.31.4.3  Generic Neighborhood Spaces   gneispa 40529
            ​*20.31.5  Exploring Higher Homotopy via Kerodon   k0004lem1 40546
                  *20.31.5.1  Simplicial Sets   k0004lem1 40546
      ​20.32  Mathbox for Stanislas Polu
            20.32.1  IMO Problems   wwlemuld 40555
                  20.32.1.1  IMO 1972 B2   wwlemuld 40555
            ​*20.32.2  INT Inequalities Proof Generator   int-addcomd 40575
            ​*20.32.3  N-Digit Addition Proof Generator   unitadd 40597
            20.32.4  AM-GM (for k = 2,3,4)   gsumws3 40598
      ​20.33  Mathbox for Rohan Ridenour
            20.33.1  Misc   spALT 40603
            20.33.2  Shorter primitive equivalent of ax-groth   gru0eld 40614
                  20.33.2.1  Grothendieck universes are closed under collection   gru0eld 40614
                  20.33.2.2  Minimal universes   ismnu 40646
                  20.33.2.3  Primitive equivalent of ax-groth   expandan 40673
      ​20.34  Mathbox for Steve Rodriguez
            20.34.1  Miscellanea   nanorxor 40686
            20.34.2  Ratio test for infinite series convergence and divergence   dvgrat 40693
            20.34.3  Multiples   reldvds 40696
            20.34.4  Function operations   caofcan 40704
            20.34.5  Calculus   lhe4.4ex1a 40710
            20.34.6  The generalized binomial coefficient operation   cbcc 40717
            20.34.7  Binomial series   uzmptshftfval 40727
      ​20.35  Mathbox for Andrew Salmon
            20.35.1  Principia Mathematica * 10   pm10.12 40739
            20.35.2  Principia Mathematica * 11   2alanimi 40753
            20.35.3  Predicate Calculus   sbeqal1 40779
            20.35.4  Principia Mathematica * 13 and * 14   pm13.13a 40788
            20.35.5  Set Theory   elnev 40819
            20.35.6  Arithmetic   addcomgi 40837
            20.35.7  Geometry   cplusr 40838
      ​​*20.36  Mathbox for Alan Sare
            20.36.1  Auxiliary theorems for the Virtual Deduction tool   idiALT 40860
            20.36.2  Supplementary unification deductions   bi1imp 40864
            20.36.3  Conventional Metamath proofs, some derived from VD proofs   iidn3 40884
            20.36.4  What is Virtual Deduction?   wvd1 40952
            20.36.5  Virtual Deduction Theorems   df-vd1 40953
            20.36.6  Theorems proved using Virtual Deduction   trsspwALT 41201
            20.36.7  Theorems proved using Virtual Deduction with mmj2 assistance   simplbi2VD 41229
            20.36.8  Virtual Deduction transcriptions of textbook proofs   sb5ALTVD 41296
            20.36.9  Theorems proved using conjunction-form Virtual Deduction   elpwgdedVD 41300
            20.36.10  Theorems with a VD proof in conventional notation derived from a VD proof   suctrALT3 41307
            ​*20.36.11  Theorems with a proof in conventional notation derived from a VD proof   notnotrALT2 41310
      ​20.37  Mathbox for Glauco Siliprandi
            20.37.1  Miscellanea   evth2f 41321
            20.37.2  Functions   feq1dd 41472
            20.37.3  Ordering on real numbers - Real and complex numbers basic operations   sub2times 41589
            20.37.4  Real intervals   gtnelioc 41814
            20.37.5  Finite sums   fsumclf 41899
            20.37.6  Finite multiplication of numbers and finite multiplication of functions   fmul01 41910
            20.37.7  Limits   clim1fr1 41931
                  20.37.7.1  Inferior limit (lim inf)   clsi 42081
                  ​*20.37.7.2  Limits for sequences of extended real numbers   clsxlim 42148
            20.37.8  Trigonometry   coseq0 42194
            20.37.9  Continuous Functions   mulcncff 42200
            20.37.10  Derivatives   dvsinexp 42244
            20.37.11  Integrals   itgsin0pilem1 42284
            20.37.12  Stone Weierstrass theorem - real version   stoweidlem1 42335
            20.37.13  Wallis' product for π   wallispilem1 42399
            20.37.14  Stirling's approximation formula for ` n ` factorial   stirlinglem1 42408
            20.37.15  Dirichlet kernel   dirkerval 42425
            20.37.16  Fourier Series   fourierdlem1 42442
            20.37.17  e is transcendental   elaa2lem 42567
            20.37.18  n-dimensional Euclidean space   rrxtopn 42618
            20.37.19  Basic measure theory   csalg 42642
                  ​*20.37.19.1  σ-Algebras   csalg 42642
                  20.37.19.2  Sum of nonnegative extended reals   csumge0 42693
                  ​*20.37.19.3  Measures   cmea 42780
                  ​*20.37.19.4  Outer measures and Caratheodory's construction   come 42820
                  ​*20.37.19.5  Lebesgue measure on n-dimensional Real numbers   covoln 42867
                  ​*20.37.19.6  Measurable functions   csmblfn 43026
      ​20.38  Mathbox for Saveliy Skresanov
            20.38.1  Ceva's theorem   sigarval 43156
            20.38.2  Simple groups   simpcntrab 43176
      ​20.39  Mathbox for Jarvin Udandy
      ​20.40  Mathbox for Adhemar
            ​*20.40.1  Minimal implicational calculus   adh-minim 43286
      ​20.41  Mathbox for Alexander van der Vekens
            20.41.1  General auxiliary theorems (1)   eusnsn 43310
                  20.41.1.1  Unordered and ordered pairs - extension for singletons   eusnsn 43310
                  20.41.1.2  Unordered and ordered pairs - extension for unordered pairs   elprneb 43313
                  20.41.1.3  Unordered and ordered pairs - extension for ordered pairs   oppr 43314
                  20.41.1.4  Relations - extension   eubrv 43319
                  20.41.1.5  Definite description binder (inverted iota) - extension   iota0def 43322
                  20.41.1.6  Functions - extension   fveqvfvv 43324
            20.41.2  Alternative for Russell's definition of a description binder   caiota 43332
            20.41.3  Double restricted existential uniqueness   r19.32 43345
                  20.41.3.1  Restricted quantification (extension)   r19.32 43345
                  20.41.3.2  Restricted uniqueness and "at most one" quantification   reuf1odnf 43355
                  20.41.3.3  Analogs to Existential uniqueness (double quantification)   2reu3 43358
                  20.41.3.4  Additional theorems for double restricted existential uniqueness   2reu8i 43361
            ​*20.41.4  Alternative definitions of function and operation values   wdfat 43364
                  20.41.4.1  Restricted quantification (extension)   ralbinrald 43370
                  20.41.4.2  The universal class (extension)   nvelim 43371
                  20.41.4.3  Introduce the Axiom of Power Sets (extension)   alneu 43372
                  20.41.4.4  Predicate "defined at"   dfateq12d 43374
                  20.41.4.5  Alternative definition of the value of a function   dfafv2 43380
                  20.41.4.6  Alternative definition of the value of an operation   aoveq123d 43426
            ​*20.41.5  Alternative definitions of function values (2)   cafv2 43456
            20.41.6  General auxiliary theorems (2)   an4com24 43516
                  20.41.6.1  Logical conjunction - extension   an4com24 43516
                  20.41.6.2  Abbreviated conjunction and disjunction of three wff's - extension   3an4ancom24 43517
                  20.41.6.3  Negated membership (alternative)   cnelbr 43519
                  20.41.6.4  The empty set - extension   ralralimp 43526
                  20.41.6.5  Indexed union and intersection - extension   otiunsndisjX 43527
                  20.41.6.6  Functions - extension   fvifeq 43528
                  20.41.6.7  Maps-to notation - extension   fvmptrab 43540
                  20.41.6.8  Ordering on reals - extension   leltletr 43542
                  20.41.6.9  Subtraction - extension   cnambpcma 43543
                  20.41.6.10  Ordering on reals (cont.) - extension   leaddsuble 43546
                  20.41.6.11  Imaginary and complex number properties - extension   readdcnnred 43552
                  20.41.6.12  Nonnegative integers (as a subset of complex numbers) - extension   nn0resubcl 43557
                  20.41.6.13  Integers (as a subset of complex numbers) - extension   zgeltp1eq 43558
                  20.41.6.14  Decimal arithmetic - extension   1t10e1p1e11 43559
                  20.41.6.15  Upper sets of integers - extension   eluzge0nn0 43561
                  20.41.6.16  Infinity and the extended real number system (cont.) - extension   nltle2tri 43562
                  20.41.6.17  Finite intervals of integers - extension   ssfz12 43563
                  20.41.6.18  Half-open integer ranges - extension   fzopred 43571
                  20.41.6.19  The modulo (remainder) operation - extension   m1mod0mod1 43578
                  20.41.6.20  The infinite sequence builder "seq"   smonoord 43580
                  20.41.6.21  Finite and infinite sums - extension   fsummsndifre 43581
                  20.41.6.22  Extensible structures - extension   setsidel 43585
            ​*20.41.7  Preimages of function values   preimafvsnel 43588
            ​*20.41.8  Partitions of real intervals   ciccp 43622
            20.41.9  Shifting functions with an integer range domain   fargshiftfv 43648
            20.41.10  Words over a set (extension)   lswn0 43653
                  20.41.10.1  Last symbol of a word - extension   lswn0 43653
            20.41.11  Unordered pairs   wich 43654
                  20.41.11.1  Interchangeable setvar variables   wich 43654
                  20.41.11.2  Set of unordered pairs   sprid 43685
                  ​*20.41.11.3  Proper (unordered) pairs   prpair 43712
                  20.41.11.4  Set of proper unordered pairs   cprpr 43723
            20.41.12  Number theory (extension)   cfmtno 43738
                  ​*20.41.12.1  Fermat numbers   cfmtno 43738
                  ​*20.41.12.2  Mersenne primes   m2prm 43802
                  20.41.12.3  Proth's theorem   modexp2m1d 43826
                  20.41.12.4  Solutions of quadratic equations   quad1 43834
            ​*20.41.13  Even and odd numbers   ceven 43838
                  20.41.13.1  Definitions and basic properties   ceven 43838
                  20.41.13.2  Alternate definitions using the "divides" relation   dfeven2 43863
                  20.41.13.3  Alternate definitions using the "modulo" operation   dfeven3 43872
                  20.41.13.4  Alternate definitions using the "gcd" operation   iseven5 43878
                  20.41.13.5  Theorems of part 5 revised   zneoALTV 43883
                  20.41.13.6  Theorems of part 6 revised   odd2np1ALTV 43888
                  20.41.13.7  Theorems of AV's mathbox revised   0evenALTV 43902
                  20.41.13.8  Additional theorems   epoo 43917
                  20.41.13.9  Perfect Number Theorem (revised)   perfectALTVlem1 43935
            20.41.14  Number theory (extension 2)   cfppr 43938
                  ​*20.41.14.1  Fermat pseudoprimes   cfppr 43938
                  ​*20.41.14.2  Goldbach's conjectures   cgbe 43959
            20.41.15  Graph theory (extension)   cgrisom 44032
                  ​*20.41.15.1  Isomorphic graphs   cgrisom 44032
                  20.41.15.2  Loop-free graphs - extension   1hegrlfgr 44056
                  20.41.15.3  Walks - extension   cupwlks 44057
                  20.41.15.4  Edges of graphs expressed as sets of unordered pairs   upgredgssspr 44067
            20.41.16  Monoids (extension)   ovn0dmfun 44080
                  20.41.16.1  Auxiliary theorems   ovn0dmfun 44080
                  20.41.16.2  Magmas, Semigroups and Monoids (extension)   plusfreseq 44088
                  20.41.16.3  Magma homomorphisms and submagmas   cmgmhm 44093
                  20.41.16.4  Examples and counterexamples for magmas, semigroups and monoids (extension)   opmpoismgm 44123
                  20.41.16.5  Group sum operation (extension 1)   gsumsplit2f 44136
            ​*20.41.17  Magmas and internal binary operations (alternate approach)   ccllaw 44139
                  *20.41.17.1  Laws for internal binary operations   ccllaw 44139
                  ​*20.41.17.2  Internal binary operations   cintop 44152
                  20.41.17.3  Alternative definitions for magmas and semigroups   cmgm2 44171
            20.41.18  Categories (extension)   idfusubc0 44185
                  20.41.18.1  Subcategories (extension)   idfusubc0 44185
            20.41.19  Rings (extension)   lmod0rng 44188
                  20.41.19.1  Nonzero rings (extension)   lmod0rng 44188
                  ​*20.41.19.2  Non-unital rings ("rngs")   crng 44194
                  20.41.19.3  Rng homomorphisms   crngh 44205
                  20.41.19.4  Ring homomorphisms (extension)   rhmfn 44238
                  20.41.19.5  Ideals as non-unital rings   lidldomn1 44241
                  20.41.19.6  The non-unital ring of even integers   0even 44251
                  20.41.19.7  A constructed not unital ring   cznrnglem 44273
                  ​*20.41.19.8  The category of non-unital rings   crngc 44277
                  ​*20.41.19.9  The category of (unital) rings   cringc 44323
                  20.41.19.10  Subcategories of the category of rings   srhmsubclem1 44393
            20.41.20  Basic algebraic structures (extension)   opeliun2xp 44430
                  20.41.20.1  Auxiliary theorems   opeliun2xp 44430
                  20.41.20.2  The binomial coefficient operation (extension)   bcpascm1 44448
                  20.41.20.3  The ` ZZ `-module ` ZZ X. ZZ `   zlmodzxzlmod 44451
                  20.41.20.4  Group sum operation (extension 2)   mgpsumunsn 44458
                  20.41.20.5  Symmetric groups (extension)   exple2lt6 44461
                  20.41.20.6  Divisibility (extension)   invginvrid 44464
                  20.41.20.7  The support of functions (extension)   rmsupp0 44465
                  20.41.20.8  Finitely supported functions (extension)   rmsuppfi 44470
                  20.41.20.9  Left modules (extension)   lmodvsmdi 44479
                  20.41.20.10  Associative algebras (extension)   ascl1 44481
                  20.41.20.11  Univariate polynomials (extension)   ply1vr1smo 44484
                  20.41.20.12  Univariate polynomials (examples)   linply1 44496
            20.41.21  Linear algebra (extension)   cdmatalt 44500
                  ​*20.41.21.1  The subalgebras of diagonal and scalar matrices (extension)   cdmatalt 44500
                  ​*20.41.21.2  Linear combinations   clinc 44508
                  ​*20.41.21.3  Linear independence   clininds 44544
                  20.41.21.4  Simple left modules and the ` ZZ `-module   lmod1lem1 44591
                  20.41.21.5  Differences between (left) modules and (left) vector spaces   lvecpsslmod 44611
            20.41.22  Complexity theory   suppdm 44614
                  20.41.22.1  Auxiliary theorems   suppdm 44614
                  20.41.22.2  The modulo (remainder) operation (extension)   fldivmod 44627
                  20.41.22.3  Even and odd integers   nn0onn0ex 44632
                  20.41.22.4  The natural logarithm on complex numbers (extension)   logcxp0 44644
                  20.41.22.5  Division of functions   cfdiv 44646
                  20.41.22.6  Upper bounds   cbigo 44656
                  20.41.22.7  Logarithm to an arbitrary base (extension)   rege1logbrege0 44667
                  ​*20.41.22.8  The binary logarithm   fldivexpfllog2 44674
                  20.41.22.9  Binary length   cblen 44678
                  ​*20.41.22.10  Digits   cdig 44704
                  20.41.22.11  Nonnegative integer as sum of its shifted digits   dignn0flhalflem1 44724
                  20.41.22.12  Algorithms for the multiplication of nonnegative integers   nn0mulfsum 44733
            20.41.23  Elementary geometry (extension)   fv1prop 44735
                  20.41.23.1  Auxiliary theorems   fv1prop 44735
                  20.41.23.2  Real euclidean space of dimension 2   rrx2pxel 44747
                  20.41.23.3  Spheres and lines in real Euclidean spaces   cline 44763
      ​20.42  Mathbox for Emmett Weisz
            ​*20.42.1  Miscellaneous Theorems   nfintd 44825
            20.42.2  Set Recursion   csetrecs 44835
                  ​*20.42.2.1  Basic Properties of Set Recursion   csetrecs 44835
                  20.42.2.2  Examples and properties of set recursion   elsetrecslem 44850
            ​*20.42.3  Construction of Games and Surreal Numbers   cpg 44860
      ​​*20.43  Mathbox for David A. Wheeler
            20.43.1  Natural deduction   sbidd 44866
            ​*20.43.2  Greater than, greater than or equal to.   cge-real 44868
            ​*20.43.3  Hyperbolic trigonometric functions   csinh 44878
            ​*20.43.4  Reciprocal trigonometric functions (sec, csc, cot)   csec 44889
            ​*20.43.5  Identities for "if"   ifnmfalse 44911
            ​*20.43.6  Logarithms generalized to arbitrary base using ` logb `   logb2aval 44912
            ​*20.43.7  Logarithm laws generalized to an arbitrary base - log_   clog- 44913
            ​*20.43.8  Formally define terms such as Reflexivity   wreflexive 44915
            ​*20.43.9  Algebra helpers   comraddi 44919
            ​*20.43.10  Algebra helper examples   i2linesi 44928
            ​*20.43.11  Formal methods "surprises"   alimp-surprise 44930
            ​*20.43.12  Allsome quantifier   walsi 44936
            ​*20.43.13  Miscellaneous   5m4e1 44947
            20.43.14  Theorems about algebraic numbers   aacllem 44951
      ​20.44  Mathbox for Kunhao Zheng
            20.44.1  Weighted AM-GM inequality   amgmwlem 44952