TOC of Theorem List - Metamath Proof Explorer

Table of Contents Summary

PART 1  CLASSICAL FIRST-ORDER LOGIC WITH EQUALITY
      1.1  Pre-logic
      1.2  Propositional calculus
      1.3  Other axiomatizations related to classical propositional calculus
      1.4  Predicate calculus with equality: Tarski's system S2 (1 rule, 6 schemes)
      1.5  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes)
      1.6  Uniqueness and unique existence
      1.7  Other axiomatizations related to classical predicate calculus
PART 2  ZF (ZERMELO-FRAENKEL) SET THEORY
      2.1  ZF Set Theory - start with the Axiom of Extensionality
      2.2  ZF Set Theory - add the Axiom of Replacement
      2.3  ZF Set Theory - add the Axiom of Power Sets
      2.4  ZF Set Theory - add the Axiom of Union
      2.5  ZF Set Theory - add the Axiom of Regularity
      2.6  ZF Set Theory - add the Axiom of Infinity
PART 3  ZFC (ZERMELO-FRAENKEL WITH CHOICE) SET THEORY
      3.1  ZFC Set Theory - add Countable Choice and Dependent Choice
      3.2  ZFC Set Theory - add the Axiom of Choice
      3.3  ZFC Axioms with no distinct variable requirements
      3.4  The Generalized Continuum Hypothesis
PART 4  TG (TARSKI-GROTHENDIECK) SET THEORY
      4.1  Inaccessibles
      4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
PART 5  REAL AND COMPLEX NUMBERS
      5.1  Construction and axiomatization of real and complex numbers
      5.2  Derive the basic properties from the field axioms
      5.3  Real and complex numbers - basic operations
      5.4  Integer sets
      5.5  Order sets
      5.6  Elementary integer functions
      5.7  Words over a set
      5.8  Reflexive and transitive closures of relations
      5.9  Elementary real and complex functions
      5.10  Elementary limits and convergence
      5.11  Elementary trigonometry
      5.12  Cardinality of real and complex number subsets
PART 6  ELEMENTARY NUMBER THEORY
      6.1  Elementary properties of divisibility
      6.2  Elementary prime number theory
PART 7  BASIC STRUCTURES
      7.1  Extensible structures
      7.2  Moore spaces
PART 8  BASIC CATEGORY THEORY
      8.1  Categories
      8.2  Arrows (disjointified hom-sets)
      8.3  Examples of categories
      8.4  Categorical constructions
PART 9  BASIC ORDER THEORY
      9.1  Dual of an order structure
      9.2  Preordered sets and directed sets
      9.3  Partially ordered sets (posets)
      9.4  Totally ordered sets (tosets)
      9.5  Lattices
      9.6  Posets, directed sets, and lattices as relations
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  The complex numbers as an algebraic extensible structure
      10.9  Generalized pre-Hilbert and Hilbert spaces
PART 11  BASIC LINEAR ALGEBRA
      11.1  Vectors and free modules
      11.2  Associative algebras
      11.3  Abstract multivariate polynomials
      11.4  Matrices
      11.5  The determinant
      11.6  Polynomial matrices
      11.7  The characteristic polynomial
PART 12  BASIC TOPOLOGY
      12.1  Topology
      12.2  Filters and filter bases
      12.3  Uniform Structures and Spaces
      12.4  Metric spaces
      12.5  Metric subcomplex vector spaces
PART 13  BASIC REAL AND COMPLEX ANALYSIS
      13.1  Continuity
      13.2  Integrals
      13.3  Derivatives
PART 14  BASIC REAL AND COMPLEX FUNCTIONS
      14.1  Polynomials
      14.2  Sequences and series
      14.3  Basic trigonometry
      14.4  Basic number theory
PART 15  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 Zhi Wang
      20.43  Mathbox for Emmett Weisz
      20.44  Mathbox for David A. Wheeler
      20.45  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 209
            ​*1.2.6  Logical conjunction   wa 399
            ​*1.2.7  Logical disjunction   wo 847
            ​*1.2.8  Mixed connectives   jaao 955
            ​*1.2.9  The conditional operator for propositions   wif 1063
            ​*1.2.10  The weak deduction theorem for propositional calculus   elimh 1085
            1.2.11  Abbreviated conjunction and disjunction of three wff's   w3o 1088
            1.2.12  Logical "nand" (Sheffer stroke)   wnan 1487
            1.2.13  Logical "xor"   wxo 1507
            1.2.14  Logical "nor"   wnor 1526
            1.2.15  True and false constants   wal 1541
                  ​*1.2.15.1  Universal quantifier for use by df-tru   wal 1541
                  ​*1.2.15.2  Equality predicate for use by df-tru   cv 1542
                  1.2.15.3  The true constant   wtru 1544
                  1.2.15.4  The false constant   wfal 1555
            ​*1.2.16  Truth tables   truimtru 1566
                  1.2.16.1  Implication   truimtru 1566
                  1.2.16.2  Negation   nottru 1570
                  1.2.16.3  Equivalence   trubitru 1572
                  1.2.16.4  Conjunction   truantru 1576
                  1.2.16.5  Disjunction   truortru 1580
                  1.2.16.6  Alternative denial   trunantru 1584
                  1.2.16.7  Exclusive disjunction   truxortru 1588
                  1.2.16.8  Joint denial   trunortru 1592
            ​*1.2.17  Half adder and full adder in propositional calculus   whad 1599
                  1.2.17.1  Full adder: sum   whad 1599
                  1.2.17.2  Full adder: carry   wcad 1613
      ​1.3  Other axiomatizations related to classical propositional calculus
            ​*1.3.1  Minimal implicational calculus   minimp 1629
            ​*1.3.2  Implicational Calculus   impsingle 1635
            1.3.3  Derive the Lukasiewicz axioms from Meredith's sole axiom   meredith 1649
            1.3.4  Derive the standard axioms from the Lukasiewicz axioms   luklem1 1666
            ​*1.3.5  Derive Nicod's axiom from the standard axioms   nic-dfim 1677
            1.3.6  Derive the Lukasiewicz axioms from Nicod's axiom   nic-imp 1683
            1.3.7  Derive Nicod's Axiom from Lukasiewicz's First Sheffer Stroke Axiom   lukshef-ax1 1702
            1.3.8  Derive the Lukasiewicz Axioms from the Tarski-Bernays-Wajsberg Axioms   tbw-bijust 1706
            1.3.9  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's First CO Axiom   merco1 1721
            1.3.10  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's Second CO Axiom   merco2 1744
            1.3.11  Derive the Lukasiewicz axioms from the Russell-Bernays Axioms   rb-bijust 1757
            ​*1.3.12  Stoic logic non-modal portion (Chrysippus of Soli)   mptnan 1776
      ​​*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 1787
                  1.4.1.1  Existential quantifier   wex 1787
                  1.4.1.2  Nonfreeness predicate   wnf 1791
            1.4.2  Rule scheme ax-gen (Generalization)   ax-gen 1803
            1.4.3  Axiom scheme ax-4 (Quantified Implication)   ax-4 1817
                  ​*1.4.3.1  The empty domain of discourse   empty 1914
            1.4.4  Axiom scheme ax-5 (Distinctness) - first use of $d   ax-5 1918
            ​*1.4.5  Equality predicate (continued)   weq 1971
            1.4.6  Axiom scheme ax-6 (Existence)   ax-6 1976
            1.4.7  Axiom scheme ax-7 (Equality)   ax-7 2016
            1.4.8  Define proper substitution   sbjust 2071
            1.4.9  Membership predicate   wcel 2112
            1.4.10  Axiom scheme ax-8 (Left Equality for Binary Predicate)   ax-8 2114
            1.4.11  Axiom scheme ax-9 (Right Equality for Binary Predicate)   ax-9 2122
            ​*1.4.12  Logical redundancy of ax-10 , ax-11 , ax-12 , ax-13   ax6dgen 2130
      ​​*1.5  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes)
            1.5.1  Axiom scheme ax-10 (Quantified Negation)   ax-10 2143
            1.5.2  Axiom scheme ax-11 (Quantifier Commutation)   ax-11 2160
            1.5.3  Axiom scheme ax-12 (Substitution)   ax-12 2177
            1.5.4  Axiom scheme ax-13 (Quantified Equality)   ax-13 2373
      ​1.6  Uniqueness and unique existence
            1.6.1  Uniqueness: the at-most-one quantifier   wmo 2539
            1.6.2  Unique existence: the unique existential quantifier   weu 2569
      ​1.7  Other axiomatizations related to classical predicate calculus
            ​*1.7.1  Aristotelian logic: Assertic syllogisms   barbara 2665
            ​*1.7.2  Intuitionistic logic   axia1 2695
​​*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 2710
            2.1.2  Classes   cab 2716
                  2.1.2.1  Class abstractions   cab 2716
                  ​*2.1.2.2  Class equality   df-cleq 2731
                  2.1.2.3  Class membership   df-clel 2818
                  2.1.2.4  Elementary properties of class abstractions   abeq2 2872
            2.1.3  Class form not-free predicate   wnfc 2887
            2.1.4  Negated equality and membership   wne 2943
                  2.1.4.1  Negated equality   wne 2943
                  2.1.4.2  Negated membership   wnel 3049
            2.1.5  Restricted quantification   wral 3064
            2.1.6  The universal class   cvv 3423
            ​*2.1.7  Conditional equality (experimental)   wcdeq 3694
            2.1.8  Russell's Paradox   rru 3710
            2.1.9  Proper substitution of classes for sets   wsbc 3712
            2.1.10  Proper substitution of classes for sets into classes   csb 3829
            2.1.11  Define basic set operations and relations   cdif 3881
            2.1.12  Subclasses and subsets   df-ss 3901
            2.1.13  The difference, union, and intersection of two classes   dfdif3 4046
                  2.1.13.1  The difference of two classes   dfdif3 4046
                  2.1.13.2  The union of two classes   elun 4080
                  2.1.13.3  The intersection of two classes   elini 4124
                  2.1.13.4  The symmetric difference of two classes   csymdif 4173
                  2.1.13.5  Combinations of difference, union, and intersection of two classes   unabs 4186
                  2.1.13.6  Class abstractions with difference, union, and intersection of two classes   unabw 4229
                  2.1.13.7  Restricted uniqueness with difference, union, and intersection   reuun2 4246
            2.1.14  The empty set   c0 4254
            ​*2.1.15  The conditional operator for classes   cif 4457
            ​*2.1.16  The weak deduction theorem for set theory   dedth 4515
            2.1.17  Power classes   cpw 4531
            2.1.18  Unordered and ordered pairs   snjust 4558
            2.1.19  The union of a class   cuni 4837
            2.1.20  The intersection of a class   cint 4877
            2.1.21  Indexed union and intersection   ciun 4922
            2.1.22  Disjointness   wdisj 5036
            2.1.23  Binary relations   wbr 5071
            2.1.24  Ordered-pair class abstractions (class builders)   copab 5133
            2.1.25  Functions in maps-to notation   cmpt 5153
            2.1.26  Transitive classes   wtr 5186
      ​2.2  ZF Set Theory - add the Axiom of Replacement
            2.2.1  Introduce the Axiom of Replacement   ax-rep 5204
            2.2.2  Derive the Axiom of Separation   axsepgfromrep 5215
            2.2.3  Derive the Null Set Axiom   axnulALT 5222
            2.2.4  Theorems requiring subset and intersection existence   nalset 5231
            2.2.5  Theorems requiring empty set existence   class2set 5270
      ​2.3  ZF Set Theory - add the Axiom of Power Sets
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 5283
            2.3.2  Derive the Axiom of Pairing   axprlem1 5341
            2.3.3  Ordered pair theorem   opnz 5382
            2.3.4  Ordered-pair class abstractions (cont.)   opabidw 5431
            2.3.5  Power class of union and intersection   pwin 5474
            2.3.6  The identity relation   cid 5479
            2.3.7  The membership relation (or epsilon relation)   cep 5485
            ​*2.3.8  Partial and total orderings   wpo 5492
            2.3.9  Founded and well-ordering relations   wfr 5532
            2.3.10  Relations   cxp 5578
            2.3.11  The Predecessor Class   cpred 6190
            2.3.12  Well-founded induction (variant)   frpomin 6228
            2.3.13  Well-ordered induction   tz6.26 6235
            2.3.14  Ordinals   word 6250
            2.3.15  Definite description binder (inverted iota)   cio 6374
            2.3.16  Functions   wfun 6412
            2.3.17  Cantor's Theorem   canth 7209
            2.3.18  Restricted iota (description binder)   crio 7211
            2.3.19  Operations   co 7255
                  2.3.19.1  Variable-to-class conversion for operations   caovclg 7442
            2.3.20  Maps-to notation   mpondm0 7488
            2.3.21  Function operation   cof 7509
            2.3.22  Proper subset relation   crpss 7553
      ​2.4  ZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 7566
            2.4.2  Ordinals (continued)   epweon 7603
            2.4.3  Transfinite induction   tfi 7675
            2.4.4  The natural numbers (i.e., finite ordinals)   com 7687
            2.4.5  Peano's postulates   peano1 7710
            2.4.6  Finite induction (for finite ordinals)   find 7717
            2.4.7  Relations and functions (cont.)   dmexg 7724
            2.4.8  First and second members of an ordered pair   c1st 7802
            ​*2.4.9  The support of functions   csupp 7948
            ​*2.4.10  Special maps-to operations   opeliunxp2f 7997
            2.4.11  Function transposition   ctpos 8012
            2.4.12  Curry and uncurry   ccur 8052
            2.4.13  Undefined values   cund 8059
            2.4.14  Well-founded recursion   cfrecs 8067
            2.4.15  Well-ordered recursion   cwrecs 8098
            2.4.16  Functions on ordinals; strictly monotone ordinal functions   iunon 8141
            2.4.17  "Strong" transfinite recursion   crecs 8172
            2.4.18  Recursive definition generator   crdg 8211
            2.4.19  Finite recursion   frfnom 8236
            2.4.20  Ordinal arithmetic   c1o 8261
            2.4.21  Natural number arithmetic   nna0 8398
            2.4.22  Equivalence relations and classes   wer 8454
            2.4.23  The mapping operation   cmap 8574
            2.4.24  Infinite Cartesian products   cixp 8644
            2.4.25  Equinumerosity   cen 8689
            2.4.26  Schroeder-Bernstein Theorem   sbthlem1 8824
            2.4.27  Equinumerosity (cont.)   xpf1o 8876
            2.4.28  Pigeonhole Principle   phplem1 8893
            2.4.29  Finite sets   dif1enlem 8906
            2.4.30  Finitely supported functions   cfsupp 9057
            2.4.31  Finite intersections   cfi 9098
            2.4.32  Hall's marriage theorem   marypha1lem 9121
            2.4.33  Supremum and infimum   csup 9128
            2.4.34  Ordinal isomorphism, Hartogs's theorem   coi 9197
            2.4.35  Hartogs function   char 9244
            2.4.36  Weak dominance   cwdom 9252
      ​2.5  ZF Set Theory - add the Axiom of Regularity
            2.5.1  Introduce the Axiom of Regularity   ax-reg 9280
            2.5.2  Axiom of Infinity equivalents   inf0 9308
      ​2.6  ZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-inf 9325
            2.6.2  Existence of omega (the set of natural numbers)   omex 9330
            2.6.3  Cantor normal form   ccnf 9348
            2.6.4  Transitive closure under a relationship   ctrpred 9394
            2.6.5  Transitive closure   trcl 9416
            2.6.6  Well-Founded Induction   frmin 9437
            2.6.7  Well-Founded Recursion   frr3g 9444
            2.6.8  Rank   cr1 9450
            2.6.9  Scott's trick; collection principle; Hilbert's epsilon   scottex 9573
            2.6.10  Disjoint union   cdju 9586
            2.6.11  Cardinal numbers   ccrd 9623
            2.6.12  Axiom of Choice equivalents   wac 9801
            ​*2.6.13  Cardinal number arithmetic   undjudom 9853
            2.6.14  The Ackermann bijection   ackbij2lem1 9905
            2.6.15  Cofinality (without Axiom of Choice)   cflem 9932
            2.6.16  Eight inequivalent definitions of finite set   sornom 9963
            2.6.17  Hereditarily size-limited sets without Choice   itunifval 10102
​​*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 10121
            3.1.2  Introduce the Axiom of Dependent Choice   ax-dc 10132
      ​3.2  ZFC Set Theory - add the Axiom of Choice
            3.2.1  Introduce the Axiom of Choice   ax-ac 10145
            3.2.2  AC equivalents: well-ordering, Zorn's lemma   numthcor 10180
            3.2.3  Cardinal number theorems using Axiom of Choice   cardval 10232
            3.2.4  Cardinal number arithmetic using Axiom of Choice   iunctb 10260
            3.2.5  Cofinality using the Axiom of Choice   alephreg 10268
      ​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 10306
            3.4.2  Derivation of the Axiom of Choice   gchaclem 10364
​​*PART 4  TG (TARSKI-GROTHENDIECK) SET THEORY
      ​4.1  Inaccessibles
            4.1.1  Weakly and strongly inaccessible cardinals   cwina 10368
            4.1.2  Weak universes   cwun 10386
            4.1.3  Tarski classes   ctsk 10434
            4.1.4  Grothendieck universes   cgru 10476
      ​4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
            4.2.1  Introduce the Tarski-Grothendieck Axiom   ax-groth 10509
            4.2.2  Derive the Power Set, Infinity and Choice Axioms   grothpw 10512
            4.2.3  Tarski map function   ctskm 10523
​​*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 10530
            5.1.2  Final derivation of real and complex number postulates   axaddf 10831
            5.1.3  Real and complex number postulates restated as axioms   ax-cnex 10857
      ​5.2  Derive the basic properties from the field axioms
            5.2.1  Some deductions from the field axioms for complex numbers   cnex 10882
            5.2.2  Infinity and the extended real number system   cpnf 10936
            5.2.3  Restate the ordering postulates with extended real "less than"   axlttri 10976
            5.2.4  Ordering on reals   lttr 10981
            5.2.5  Initial properties of the complex numbers   mul12 11069
      ​5.3  Real and complex numbers - basic operations
            5.3.1  Addition   add12 11121
            5.3.2  Subtraction   cmin 11134
            5.3.3  Multiplication   kcnktkm1cn 11335
            5.3.4  Ordering on reals (cont.)   gt0ne0 11369
            5.3.5  Reciprocals   ixi 11533
            5.3.6  Division   cdiv 11561
            5.3.7  Ordering on reals (cont.)   elimgt0 11742
            5.3.8  Completeness Axiom and Suprema   fimaxre 11848
            5.3.9  Imaginary and complex number properties   inelr 11892
            5.3.10  Function operation analogue theorems   ofsubeq0 11899
      ​5.4  Integer sets
            5.4.1  Positive integers (as a subset of complex numbers)   cn 11902
            5.4.2  Principle of mathematical induction   nnind 11920
            ​*5.4.3  Decimal representation of numbers   c2 11957
            ​*5.4.4  Some properties of specific numbers   neg1cn 12016
            5.4.5  Simple number properties   halfcl 12127
            5.4.6  The Archimedean property   nnunb 12158
            5.4.7  Nonnegative integers (as a subset of complex numbers)   cn0 12162
            ​*5.4.8  Extended nonnegative integers   cxnn0 12234
            5.4.9  Integers (as a subset of complex numbers)   cz 12248
            5.4.10  Decimal arithmetic   cdc 12365
            5.4.11  Upper sets of integers   cuz 12510
            5.4.12  Well-ordering principle for bounded-below sets of integers   uzwo3 12611
            5.4.13  Rational numbers (as a subset of complex numbers)   cq 12616
            5.4.14  Existence of the set of complex numbers   rpnnen1lem2 12645
      ​5.5  Order sets
            5.5.1  Positive reals (as a subset of complex numbers)   crp 12658
            5.5.2  Infinity and the extended real number system (cont.)   cxne 12773
            5.5.3  Supremum and infimum on the extended reals   xrsupexmnf 12967
            5.5.4  Real number intervals   cioo 13007
            5.5.5  Finite intervals of integers   cfz 13167
            ​*5.5.6  Finite intervals of nonnegative integers   elfz2nn0 13275
            5.5.7  Half-open integer ranges   cfzo 13310
      ​5.6  Elementary integer functions
            5.6.1  The floor and ceiling functions   cfl 13437
            5.6.2  The modulo (remainder) operation   cmo 13516
            5.6.3  Miscellaneous theorems about integers   om2uz0i 13594
            5.6.4  Strong induction over upper sets of integers   uzsinds 13634
            5.6.5  Finitely supported functions over the nonnegative integers   fsuppmapnn0fiublem 13637
            5.6.6  The infinite sequence builder "seq" - extension   cseq 13648
            5.6.7  Integer powers   cexp 13709
            5.6.8  Ordered pair theorem for nonnegative integers   nn0le2msqi 13908
            5.6.9  Factorial function   cfa 13914
            5.6.10  The binomial coefficient operation   cbc 13943
            5.6.11  The ` # ` (set size) function   chash 13971
                  5.6.11.1  Proper unordered pairs and triples (sets of size 2 and 3)   hashprlei 14109
                  5.6.11.2  Functions with a domain containing at least two different elements   fundmge2nop0 14133
                  5.6.11.3  Finite induction on the size of the first component of a binary relation   hashdifsnp1 14137
      ​​*5.7  Words over a set
            5.7.1  Definitions and basic theorems   cword 14144
            5.7.2  Last symbol of a word   clsw 14192
            5.7.3  Concatenations of words   cconcat 14200
            5.7.4  Singleton words   cs1 14227
            5.7.5  Concatenations with singleton words   ccatws1cl 14248
            5.7.6  Subwords/substrings   csubstr 14280
            5.7.7  Prefixes of a word   cpfx 14310
            5.7.8  Subwords of subwords   swrdswrdlem 14344
            5.7.9  Subwords and concatenations   pfxcctswrd 14350
            5.7.10  Subwords of concatenations   swrdccatfn 14364
            5.7.11  Splicing words (substring replacement)   csplice 14389
            5.7.12  Reversing words   creverse 14398
            5.7.13  Repeated symbol words   creps 14408
            ​*5.7.14  Cyclical shifts of words   ccsh 14428
            5.7.15  Mapping words by a function   wrdco 14471
            5.7.16  Longer string literals   cs2 14481
      ​​*5.8  Reflexive and transitive closures of relations
            5.8.1  The reflexive and transitive properties of relations   coss12d 14610
            5.8.2  Basic properties of closures   cleq1lem 14620
            5.8.3  Definitions and basic properties of transitive closures   ctcl 14623
            5.8.4  Exponentiation of relations   crelexp 14657
            5.8.5  Reflexive-transitive closure as an indexed union   crtrcl 14693
            ​*5.8.6  Principle of transitive induction.   relexpindlem 14701
      ​5.9  Elementary real and complex functions
            5.9.1  The "shift" operation   cshi 14704
            5.9.2  Signum (sgn or sign) function   csgn 14724
            5.9.3  Real and imaginary parts; conjugate   ccj 14734
            5.9.4  Square root; absolute value   csqrt 14871
      ​5.10  Elementary limits and convergence
            5.10.1  Superior limit (lim sup)   clsp 15106
            5.10.2  Limits   cli 15120
            5.10.3  Finite and infinite sums   csu 15324
            5.10.4  The binomial theorem   binomlem 15468
            5.10.5  The inclusion/exclusion principle   incexclem 15475
            5.10.6  Infinite sums (cont.)   isumshft 15478
            5.10.7  Miscellaneous converging and diverging sequences   divrcnv 15491
            5.10.8  Arithmetic series   arisum 15499
            5.10.9  Geometric series   expcnv 15503
            5.10.10  Ratio test for infinite series convergence   cvgrat 15522
            5.10.11  Mertens' theorem   mertenslem1 15523
            5.10.12  Finite and infinite products   prodf 15526
                  5.10.12.1  Product sequences   prodf 15526
                  5.10.12.2  Non-trivial convergence   ntrivcvg 15536
                  5.10.12.3  Complex products   cprod 15542
                  5.10.12.4  Finite products   fprod 15578
                  5.10.12.5  Infinite products   iprodclim 15635
            5.10.13  Falling and Rising Factorial   cfallfac 15641
            5.10.14  Bernoulli polynomials and sums of k-th powers   cbp 15683
      ​5.11  Elementary trigonometry
            5.11.1  The exponential, sine, and cosine functions   ce 15698
                  5.11.1.1  The circle constant (tau = 2 pi)   ctau 15838
            5.11.2  _e is irrational   eirrlem 15840
      ​5.12  Cardinality of real and complex number subsets
            5.12.1  Countability of integers and rationals   xpnnen 15847
            5.12.2  The reals are uncountable   rpnnen2lem1 15850
​​*PART 6  ELEMENTARY NUMBER THEORY
      ​6.1  Elementary properties of divisibility
            6.1.1  Irrationality of square root of 2   sqrt2irrlem 15884
            6.1.2  Some Number sets are chains of proper subsets   nthruc 15888
            6.1.3  The divides relation   cdvds 15890
            ​*6.1.4  Even and odd numbers   evenelz 15972
            6.1.5  The division algorithm   divalglem0 16029
            6.1.6  Bit sequences   cbits 16053
            6.1.7  The greatest common divisor operator   cgcd 16128
            6.1.8  Bézout's identity   bezoutlem1 16174
            6.1.9  Algorithms   nn0seqcvgd 16202
            6.1.10  Euclid's Algorithm   eucalgval2 16213
            ​*6.1.11  The least common multiple   clcm 16220
            ​*6.1.12  Coprimality and Euclid's lemma   coprmgcdb 16281
            6.1.13  Cancellability of congruences   congr 16296
      ​6.2  Elementary prime number theory
            ​*6.2.1  Elementary properties   cprime 16303
            ​*6.2.2  Coprimality and Euclid's lemma (cont.)   coprm 16343
            6.2.3  Properties of the canonical representation of a rational   cnumer 16364
            6.2.4  Euler's theorem   codz 16391
            6.2.5  Arithmetic modulo a prime number   modprm1div 16425
            6.2.6  Pythagorean Triples   coprimeprodsq 16436
            6.2.7  The prime count function   cpc 16464
            6.2.8  Pocklington's theorem   prmpwdvds 16532
            6.2.9  Infinite primes theorem   unbenlem 16536
            6.2.10  Sum of prime reciprocals   prmreclem1 16544
            6.2.11  Fundamental theorem of arithmetic   1arithlem1 16551
            6.2.12  Lagrange's four-square theorem   cgz 16557
            6.2.13  Van der Waerden's theorem   cvdwa 16593
            6.2.14  Ramsey's theorem   cram 16627
            ​*6.2.15  Primorial function   cprmo 16659
            ​*6.2.16  Prime gaps   prmgaplem1 16677
            6.2.17  Decimal arithmetic (cont.)   dec2dvds 16691
            6.2.18  Cyclical shifts of words (cont.)   cshwsidrepsw 16722
            6.2.19  Specific prime numbers   prmlem0 16734
            6.2.20  Very large primes   1259lem1 16759
​PART 7  BASIC STRUCTURES
      ​7.1  Extensible structures
            ​*7.1.1  Basic definitions   cstr 16774
                  7.1.1.1  Extensible structures as structures with components   cstr 16774
                  7.1.1.2  Substitution of components   csts 16791
                  7.1.1.3  Slots   cslot 16809
                  7.1.1.4  Structure component indices   cnx 16821
                  7.1.1.5  Base sets   cbs 16839
                  7.1.1.6  Base set restrictions   cress 16866
            7.1.2  Slot definitions   cplusg 16887
            7.1.3  Definition of the structure product   crest 17047
            7.1.4  Definition of the structure quotient   cordt 17126
      ​7.2  Moore spaces
            7.2.1  Moore closures   mrcflem 17231
            7.2.2  Independent sets in a Moore system   mrisval 17255
            7.2.3  Algebraic closure systems   isacs 17276
​PART 8  BASIC CATEGORY THEORY
      ​8.1  Categories
            8.1.1  Categories   ccat 17289
            8.1.2  Opposite category   coppc 17336
            8.1.3  Monomorphisms and epimorphisms   cmon 17356
            8.1.4  Sections, inverses, isomorphisms   csect 17372
            ​*8.1.5  Isomorphic objects   ccic 17423
            8.1.6  Subcategories   cssc 17435
            8.1.7  Functors   cfunc 17484
            8.1.8  Full & faithful functors   cful 17533
            8.1.9  Natural transformations and the functor category   cnat 17572
            8.1.10  Initial, terminal and zero objects of a category   cinito 17611
      ​8.2  Arrows (disjointified hom-sets)
            8.2.1  Identity and composition for arrows   cida 17683
      ​8.3  Examples of categories
            8.3.1  The category of sets   csetc 17705
            8.3.2  The category of categories   ccatc 17728
            ​*8.3.3  The category of extensible structures   fncnvimaeqv 17751
      ​8.4  Categorical constructions
            8.4.1  Product of categories   cxpc 17800
            8.4.2  Functor evaluation   cevlf 17842
            8.4.3  Hom functor   chof 17881
​PART 9  BASIC ORDER THEORY
      ​9.1  Dual of an order structure
      ​9.2  Preordered sets and directed sets
      ​9.3  Partially ordered sets (posets)
      ​9.4  Totally ordered sets (tosets)
      ​9.5  Lattices
            9.5.1  Lattices   clat 18063
            9.5.2  Complete lattices   ccla 18130
            9.5.3  Distributive lattices   cdlat 18152
            9.5.4  Subset order structures   cipo 18159
      ​9.6  Posets, directed sets, and lattices as relations
            ​*9.6.1  Posets and lattices as relations   cps 18196
            9.6.2  Directed sets, nets   cdir 18226
​PART 10  BASIC ALGEBRAIC STRUCTURES
      ​10.1  Monoids
            ​*10.1.1  Magmas   cplusf 18237
            ​*10.1.2  Identity elements   mgmidmo 18258
            ​*10.1.3  Iterated sums in a magma   gsumvalx 18274
            ​*10.1.4  Semigroups   csgrp 18288
            ​*10.1.5  Definition and basic properties of monoids   cmnd 18299
            10.1.6  Monoid homomorphisms and submonoids   cmhm 18342
            ​*10.1.7  Iterated sums in a monoid   gsumvallem2 18386
            10.1.8  Free monoids   cfrmd 18400
                  ​*10.1.8.1  Monoid of endofunctions   cefmnd 18421
            10.1.9  Examples and counterexamples for magmas, semigroups and monoids   mgm2nsgrplem1 18471
      ​10.2  Groups
            10.2.1  Definition and basic properties   cgrp 18491
            ​*10.2.2  Group multiple operation   cmg 18614
            10.2.3  Subgroups and Quotient groups   csubg 18663
            ​*10.2.4  Cyclic monoids and groups   cycsubmel 18733
            10.2.5  Elementary theory of group homomorphisms   cghm 18745
            10.2.6  Isomorphisms of groups   cgim 18787
            10.2.7  Group actions   cga 18809
            10.2.8  Centralizers and centers   ccntz 18835
            10.2.9  The opposite group   coppg 18863
            10.2.10  Symmetric groups   csymg 18888
                  ​*10.2.10.1  Definition and basic properties   csymg 18888
                  10.2.10.2  Cayley's theorem   cayleylem1 18934
                  10.2.10.3  Permutations fixing one element   symgfix2 18938
                  ​*10.2.10.4  Transpositions in the symmetric group   cpmtr 18963
                  10.2.10.5  The sign of a permutation   cpsgn 19011
            10.2.11  p-Groups and Sylow groups; Sylow's theorems   cod 19046
            10.2.12  Direct products   clsm 19153
                  10.2.12.1  Direct products (extension)   smndlsmidm 19175
            10.2.13  Free groups   cefg 19226
            10.2.14  Abelian groups   ccmn 19300
                  10.2.14.1  Definition and basic properties   ccmn 19300
                  10.2.14.2  Cyclic groups   ccyg 19391
                  10.2.14.3  Group sum operation   gsumval3a 19418
                  10.2.14.4  Group sums over (ranges of) integers   fsfnn0gsumfsffz 19498
                  10.2.14.5  Internal direct products   cdprd 19510
                  10.2.14.6  The Fundamental Theorem of Abelian Groups   ablfacrplem 19582
            10.2.15  Simple groups   csimpg 19607
                  10.2.15.1  Definition and basic properties   csimpg 19607
                  10.2.15.2  Classification of abelian simple groups   ablsimpnosubgd 19621
      ​10.3  Rings
            10.3.1  Multiplicative Group   cmgp 19634
            10.3.2  Ring unit   cur 19651
                  10.3.2.1  Semirings   csrg 19655
                  ​*10.3.2.2  The binomial theorem for semirings   srgbinomlem1 19690
            10.3.3  Definition and basic properties of unital rings   crg 19697
            10.3.4  Opposite ring   coppr 19775
            10.3.5  Divisibility   cdsr 19794
            10.3.6  Ring primes   crpm 19868
            10.3.7  Ring homomorphisms   crh 19870
      ​10.4  Division rings and fields
            10.4.1  Definition and basic properties   cdr 19905
            10.4.2  Subrings of a ring   csubrg 19934
                  10.4.2.1  Sub-division rings   csdrg 19975
            10.4.3  Absolute value (abstract algebra)   cabv 19990
            10.4.4  Star rings   cstf 20017
      ​10.5  Left modules
            10.5.1  Definition and basic properties   clmod 20037
            10.5.2  Subspaces and spans in a left module   clss 20107
            10.5.3  Homomorphisms and isomorphisms of left modules   clmhm 20195
            10.5.4  Subspace sum; bases for a left module   clbs 20250
      ​10.6  Vector spaces
            10.6.1  Definition and basic properties   clvec 20278
      ​10.7  Ideals
            10.7.1  The subring algebra; ideals   csra 20344
            10.7.2  Two-sided ideals and quotient rings   c2idl 20414
            10.7.3  Principal ideal rings. Divisibility in the integers   clpidl 20424
            10.7.4  Nonzero rings and zero rings   cnzr 20440
            10.7.5  Left regular elements. More kinds of rings   crlreg 20462
      ​10.8  The complex numbers as an algebraic extensible structure
            10.8.1  Definition and basic properties   cpsmet 20493
            ​*10.8.2  Ring of integers   zring 20581
            10.8.3  Algebraic constructions based on the complex numbers   czrh 20612
            10.8.4  Signs as subgroup of the complex numbers   cnmsgnsubg 20693
            10.8.5  Embedding of permutation signs into a ring   zrhpsgnmhm 20700
            10.8.6  The ordered field of real numbers   crefld 20720
      ​10.9  Generalized pre-Hilbert and Hilbert spaces
            10.9.1  Definition and basic properties   cphl 20740
            10.9.2  Orthocomplements and closed subspaces   cocv 20776
            10.9.3  Orthogonal projection and orthonormal bases   cpj 20816
​​*PART 11  BASIC LINEAR ALGEBRA
      ​11.1  Vectors and free modules
            *11.1.1  Direct sum of left modules   cdsmm 20847
            ​*11.1.2  Free modules   cfrlm 20862
            ​*11.1.3  Standard basis (unit vectors)   cuvc 20898
            ​*11.1.4  Independent sets and families   clindf 20920
            11.1.5  Characterization of free modules   lmimlbs 20952
      ​11.2  Associative algebras
            11.2.1  Definition and basic properties   casa 20966
      ​11.3  Abstract multivariate polynomials
            11.3.1  Definition and basic properties   cmps 21016
            11.3.2  Polynomial evaluation   ces 21189
            11.3.3  Additional definitions for (multivariate) polynomials   cslv 21227
            ​*11.3.4  Univariate polynomials   cps1 21255
            11.3.5  Univariate polynomial evaluation   ces1 21388
      ​​*11.4  Matrices
            *11.4.1  The matrix multiplication   cmmul 21441
            ​*11.4.2  Square matrices   cmat 21463
            ​*11.4.3  The matrix algebra   matmulr 21494
            ​*11.4.4  Matrices of dimension 0 and 1   mat0dimbas0 21522
            ​*11.4.5  The subalgebras of diagonal and scalar matrices   cdmat 21544
            ​*11.4.6  Multiplication of a matrix with a "column vector"   cmvmul 21596
            11.4.7  Replacement functions for a square matrix   cmarrep 21612
            11.4.8  Submatrices   csubma 21632
      ​11.5  The determinant
            11.5.1  Definition and basic properties   cmdat 21640
            11.5.2  Determinants of 2 x 2 -matrices   m2detleiblem1 21680
            11.5.3  The matrix adjugate/adjunct   cmadu 21688
            ​*11.5.4  Laplace expansion of determinants (special case)   symgmatr01lem 21709
            11.5.5  Inverse matrix   invrvald 21732
            ​*11.5.6  Cramer's rule   slesolvec 21735
      ​​*11.6  Polynomial matrices
            11.6.1  Basic properties   pmatring 21748
            ​*11.6.2  Constant polynomial matrices   ccpmat 21759
            ​*11.6.3  Collecting coefficients of polynomial matrices   cdecpmat 21818
            ​*11.6.4  Ring isomorphism between polynomial matrices and polynomials over matrices   cpm2mp 21848
      ​​*11.7  The characteristic polynomial
            *11.7.1  Definition and basic properties   cchpmat 21882
            ​*11.7.2  The characteristic factor function G   fvmptnn04if 21905
            ​*11.7.3  The Cayley-Hamilton theorem   cpmadurid 21923
​PART 12  BASIC TOPOLOGY
      ​12.1  Topology
            ​*12.1.1  Topological spaces   ctop 21949
                  12.1.1.1  Topologies   ctop 21949
                  12.1.1.2  Topologies on sets   ctopon 21966
                  12.1.1.3  Topological spaces   ctps 21988
            12.1.2  Topological bases   ctb 22002
            12.1.3  Examples of topologies   distop 22052
            12.1.4  Closure and interior   ccld 22074
            12.1.5  Neighborhoods   cnei 22155
            12.1.6  Limit points and perfect sets   clp 22192
            12.1.7  Subspace topologies   restrcl 22215
            12.1.8  Order topology   ordtbaslem 22246
            12.1.9  Limits and continuity in topological spaces   ccn 22282
            12.1.10  Separated spaces: T0, T1, T2 (Hausdorff) ...   ct0 22364
            12.1.11  Compactness   ccmp 22444
            12.1.12  Bolzano-Weierstrass theorem   bwth 22468
            12.1.13  Connectedness   cconn 22469
            12.1.14  First- and second-countability   c1stc 22495
            12.1.15  Local topological properties   clly 22522
            12.1.16  Refinements   cref 22560
            12.1.17  Compactly generated spaces   ckgen 22591
            12.1.18  Product topologies   ctx 22618
            12.1.19  Continuous function-builders   cnmptid 22719
            12.1.20  Quotient maps and quotient topology   ckq 22751
            12.1.21  Homeomorphisms   chmeo 22811
      ​12.2  Filters and filter bases
            12.2.1  Filter bases   elmptrab 22885
            12.2.2  Filters   cfil 22903
            12.2.3  Ultrafilters   cufil 22957
            12.2.4  Filter limits   cfm 22991
            12.2.5  Extension by continuity   ccnext 23117
            12.2.6  Topological groups   ctmd 23128
            12.2.7  Infinite group sum on topological groups   ctsu 23184
            12.2.8  Topological rings, fields, vector spaces   ctrg 23214
      ​12.3  Uniform Structures and Spaces
            12.3.1  Uniform structures   cust 23258
            12.3.2  The topology induced by an uniform structure   cutop 23289
            12.3.3  Uniform Spaces   cuss 23312
            12.3.4  Uniform continuity   cucn 23334
            12.3.5  Cauchy filters in uniform spaces   ccfilu 23345
            12.3.6  Complete uniform spaces   ccusp 23356
      ​12.4  Metric spaces
            12.4.1  Pseudometric spaces   ispsmet 23364
            12.4.2  Basic metric space properties   cxms 23377
            12.4.3  Metric space balls   blfvalps 23443
            12.4.4  Open sets of a metric space   mopnval 23498
            12.4.5  Continuity in metric spaces   metcnp3 23601
            12.4.6  The uniform structure generated by a metric   metuval 23610
            12.4.7  Examples of metric spaces   dscmet 23633
            ​*12.4.8  Normed algebraic structures   cnm 23637
            12.4.9  Normed space homomorphisms (bounded linear operators)   cnmo 23774
            12.4.10  Topology on the reals   qtopbaslem 23827
            12.4.11  Topological definitions using the reals   cii 23943
            12.4.12  Path homotopy   chtpy 24035
            12.4.13  The fundamental group   cpco 24068
      ​12.5  Metric subcomplex vector spaces
            12.5.1  Subcomplex modules   cclm 24130
            ​*12.5.2  Subcomplex vector spaces   ccvs 24191
            ​*12.5.3  Normed subcomplex vector spaces   isncvsngp 24217
            12.5.4  Subcomplex pre-Hilbert spaces   ccph 24234
            12.5.5  Convergence and completeness   ccfil 24320
            12.5.6  Baire's Category Theorem   bcthlem1 24392
            12.5.7  Banach spaces and subcomplex Hilbert spaces   ccms 24400
                  12.5.7.1  The complete ordered field of the real numbers   retopn 24447
            12.5.8  Euclidean spaces   crrx 24451
            12.5.9  Minimizing Vector Theorem   minveclem1 24492
            12.5.10  Projection Theorem   pjthlem1 24505
​PART 13  BASIC REAL AND COMPLEX ANALYSIS
      ​13.1  Continuity
            13.1.1  Intermediate value theorem   pmltpclem1 24516
      ​13.2  Integrals
            13.2.1  Lebesgue measure   covol 24530
            13.2.2  Lebesgue integration   cmbf 24682
                  13.2.2.1  Lesbesgue integral   cmbf 24682
                  13.2.2.2  Lesbesgue directed integral   cdit 24914
      ​13.3  Derivatives
            13.3.1  Real and complex differentiation   climc 24930
                  13.3.1.1  Derivatives of functions of one complex or real variable   climc 24930
                  13.3.1.2  Results on real differentiation   dvferm1lem 25052
​PART 14  BASIC REAL AND COMPLEX FUNCTIONS
      ​14.1  Polynomials
            14.1.1  Polynomial degrees   cmdg 25119
            14.1.2  The division algorithm for univariate polynomials   cmn1 25194
            14.1.3  Elementary properties of complex polynomials   cply 25249
            14.1.4  The division algorithm for polynomials   cquot 25354
            14.1.5  Algebraic numbers   caa 25378
            14.1.6  Liouville's approximation theorem   aalioulem1 25396
      ​14.2  Sequences and series
            14.2.1  Taylor polynomials and Taylor's theorem   ctayl 25416
            14.2.2  Uniform convergence   culm 25439
            14.2.3  Power series   pserval 25473
      ​14.3  Basic trigonometry
            14.3.1  The exponential, sine, and cosine functions (cont.)   efcn 25506
            14.3.2  Properties of pi = 3.14159...   pilem1 25514
            14.3.3  Mapping of the exponential function   efgh 25601
            14.3.4  The natural logarithm on complex numbers   clog 25614
            ​*14.3.5  Logarithms to an arbitrary base   clogb 25818
            14.3.6  Theorems of Pythagoras, isosceles triangles, and intersecting chords   angval 25855
            14.3.7  Solutions of quadratic, cubic, and quartic equations   quad2 25893
            14.3.8  Inverse trigonometric functions   casin 25916
            14.3.9  The Birthday Problem   log2ublem1 26000
            14.3.10  Areas in R^2   carea 26009
            14.3.11  More miscellaneous converging sequences   rlimcnp 26019
            14.3.12  Inequality of arithmetic and geometric means   cvxcl 26038
            14.3.13  Euler-Mascheroni constant   cem 26045
            14.3.14  Zeta function   czeta 26066
            14.3.15  Gamma function   clgam 26069
      ​14.4  Basic number theory
            14.4.1  Wilson's theorem   wilthlem1 26121
            14.4.2  The Fundamental Theorem of Algebra   ftalem1 26126
            14.4.3  The Basel problem (ζ(2) = π²/6)   basellem1 26134
            14.4.4  Number-theoretical functions   ccht 26144
            14.4.5  Perfect Number Theorem   mersenne 26279
            14.4.6  Characters of Z/nZ   cdchr 26284
            14.4.7  Bertrand's postulate   bcctr 26327
            ​*14.4.8  Quadratic residues and the Legendre symbol   clgs 26346
            ​*14.4.9  Gauss' Lemma   gausslemma2dlem0a 26408
            14.4.10  Quadratic reciprocity   lgseisenlem1 26427
            14.4.11  All primes 4n+1 are the sum of two squares   2sqlem1 26469
            14.4.12  Chebyshev's Weak Prime Number Theorem, Dirichlet's Theorem   chebbnd1lem1 26521
            14.4.13  The Prime Number Theorem   mudivsum 26582
            14.4.14  Ostrowski's theorem   abvcxp 26667
​​*PART 15  ELEMENTARY GEOMETRY
      ​15.1  Definition and Tarski's Axioms of Geometry
            15.1.1  Justification for the congruence notation   tgjustf 26737
      ​15.2  Tarskian Geometry
            15.2.1  Congruence   tgcgrcomimp 26741
            15.2.2  Betweenness   tgbtwntriv2 26751
            15.2.3  Dimension   tglowdim1 26764
            15.2.4  Betweenness and Congruence   tgifscgr 26772
            15.2.5  Congruence of a series of points   ccgrg 26774
            15.2.6  Motions   cismt 26796
            15.2.7  Colinearity   tglng 26810
            15.2.8  Connectivity of betweenness   tgbtwnconn1lem1 26836
            15.2.9  Less-than relation in geometric congruences   cleg 26846
            15.2.10  Rays   chlg 26864
            15.2.11  Lines   btwnlng1 26883
            15.2.12  Point inversions   cmir 26916
            15.2.13  Right angles   crag 26957
            15.2.14  Half-planes   islnopp 27003
            15.2.15  Midpoints and Line Mirroring   cmid 27036
            15.2.16  Congruence of angles   ccgra 27071
            15.2.17  Angle Comparisons   cinag 27099
            15.2.18  Congruence Theorems   tgsas1 27118
            15.2.19  Equilateral triangles   ceqlg 27129
      ​15.3  Properties of geometries
            15.3.1  Isomorphisms between geometries   f1otrgds 27133
      ​15.4  Geometry in Hilbert spaces
            15.4.1  Geometry in the complex plane   cchhllem 27156
            15.4.2  Geometry in Euclidean spaces   cee 27158
                  15.4.2.1  Definition of the Euclidean space   cee 27158
                  15.4.2.2  Tarski's axioms for geometry for the Euclidean space   axdimuniq 27183
                  15.4.2.3  EE^n fulfills Tarski's Axioms   ceeng 27247
​​*PART 16  GRAPH THEORY
      ​*16.1  Vertices and edges
            16.1.1  The edge function extractor for extensible structures   cedgf 27258
            ​*16.1.2  Vertices and indexed edges   cvtx 27268
                  16.1.2.1  Definitions and basic properties   cvtx 27268
                  16.1.2.2  The vertices and edges of a graph represented as ordered pair   opvtxval 27275
                  16.1.2.3  The vertices and edges of a graph represented as extensible structure   funvtxdmge2val 27283
                  16.1.2.4  Representations of graphs without edges   snstrvtxval 27309
                  16.1.2.5  Degenerated cases of representations of graphs   vtxval0 27311
            16.1.3  Edges as range of the edge function   cedg 27319
      ​​*16.2  Undirected graphs
            16.2.1  Undirected hypergraphs   cuhgr 27328
            16.2.2  Undirected pseudographs and multigraphs   cupgr 27352
            ​*16.2.3  Loop-free graphs   umgrislfupgrlem 27394
            16.2.4  Edges as subsets of vertices of graphs   uhgredgiedgb 27398
            ​*16.2.5  Undirected simple graphs   cuspgr 27420
            16.2.6  Examples for graphs   usgr0e 27505
            16.2.7  Subgraphs   csubgr 27536
            16.2.8  Finite undirected simple graphs   cfusgr 27585
            16.2.9  Neighbors, complete graphs and universal vertices   cnbgr 27601
                  16.2.9.1  Neighbors   cnbgr 27601
                  16.2.9.2  Universal vertices   cuvtx 27654
                  16.2.9.3  Complete graphs   ccplgr 27678
            16.2.10  Vertex degree   cvtxdg 27734
            ​*16.2.11  Regular graphs   crgr 27824
      ​​*16.3  Walks, paths and cycles
            *16.3.1  Walks   cewlks 27864
            16.3.2  Walks for loop-free graphs   lfgrwlkprop 27956
            16.3.3  Trails   ctrls 27959
            16.3.4  Paths and simple paths   cpths 27980
            16.3.5  Closed walks   cclwlks 28038
            16.3.6  Circuits and cycles   ccrcts 28052
            ​*16.3.7  Walks as words   cwwlks 28090
            16.3.8  Walks/paths of length 2 (as length 3 strings)   2wlkdlem1 28190
            16.3.9  Walks in regular graphs   rusgrnumwwlkl1 28233
            ​*16.3.10  Closed walks as words   cclwwlk 28245
                  16.3.10.1  Closed walks as words   cclwwlk 28245
                  16.3.10.2  Closed walks of a fixed length as words   cclwwlkn 28288
                  16.3.10.3  Closed walks on a vertex of a fixed length as words   cclwwlknon 28351
            16.3.11  Examples for walks, trails and paths   0ewlk 28378
            16.3.12  Connected graphs   cconngr 28450
      ​16.4  Eulerian paths and the Konigsberg Bridge problem
            ​*16.4.1  Eulerian paths   ceupth 28461
            ​*16.4.2  The Königsberg Bridge problem   konigsbergvtx 28510
      ​16.5  The Friendship Theorem
            16.5.1  Friendship graphs - basics   cfrgr 28522
            16.5.2  The friendship theorem for small graphs   frgr1v 28535
            16.5.3  Theorems according to Mertzios and Unger   2pthfrgrrn 28546
            ​*16.5.4  Huneke's Proof of the Friendship Theorem   frgrncvvdeqlem1 28563
​PART 17  GUIDES AND MISCELLANEA
      ​17.1  Guides (conventions, explanations, and examples)
            ​*17.1.1  Conventions   conventions 28664
            17.1.2  Natural deduction   natded 28667
            ​*17.1.3  Natural deduction examples   ex-natded5.2 28668
            17.1.4  Definitional examples   ex-or 28685
            17.1.5  Other examples   aevdemo 28724
      ​17.2  Humor
            17.2.1  April Fool's theorem   avril1 28727
      ​17.3  (Future - to be reviewed and classified)
            17.3.1  Planar incidence geometry   cplig 28736
            ​*17.3.2  Aliases kept to prevent broken links   dummylink 28749
​​*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 28751
            18.1.2  Abelian groups   cablo 28806
      ​18.2  Complex vector spaces
            18.2.1  Definition and basic properties   cvc 28820
            18.2.2  Examples of complex vector spaces   cnaddabloOLD 28843
      ​18.3  Normed complex vector spaces
            18.3.1  Definition and basic properties   cnv 28846
            18.3.2  Examples of normed complex vector spaces   cnnv 28939
            18.3.3  Induced metric of a normed complex vector space   imsval 28947
            18.3.4  Inner product   cdip 28962
            18.3.5  Subspaces   css 28983
      ​18.4  Operators on complex vector spaces
            18.4.1  Definitions and basic properties   clno 29002
      ​18.5  Inner product (pre-Hilbert) spaces
            18.5.1  Definition and basic properties   ccphlo 29074
            18.5.2  Examples of pre-Hilbert spaces   cncph 29081
            18.5.3  Properties of pre-Hilbert spaces   isph 29084
      ​18.6  Complex Banach spaces
            18.6.1  Definition and basic properties   ccbn 29124
            18.6.2  Examples of complex Banach spaces   cnbn 29131
            18.6.3  Uniform Boundedness Theorem   ubthlem1 29132
            18.6.4  Minimizing Vector Theorem   minvecolem1 29136
      ​18.7  Complex Hilbert spaces
            18.7.1  Definition and basic properties   chlo 29147
            18.7.2  Standard axioms for a complex Hilbert space   hlex 29160
            18.7.3  Examples of complex Hilbert spaces   cnchl 29178
            18.7.4  Hellinger-Toeplitz Theorem   htthlem 29179
​​*PART 19  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      ​19.1  Axiomatization of complex pre-Hilbert spaces
            19.1.1  Basic Hilbert space definitions   chba 29181
            19.1.2  Preliminary ZFC lemmas   df-hnorm 29230
            ​*19.1.3  Derive the Hilbert space axioms from ZFC set theory   axhilex-zf 29243
            ​*19.1.4  Introduce the vector space axioms for a Hilbert space   ax-hilex 29261
            19.1.5  Vector operations   hvmulex 29273
            19.1.6  Inner product postulates for a Hilbert space   ax-hfi 29341
      ​19.2  Inner product and norms
            19.2.1  Inner product   his5 29348
            19.2.2  Norms   dfhnorm2 29384
            19.2.3  Relate Hilbert space to normed complex vector spaces   hilablo 29422
            19.2.4  Bunjakovaskij-Cauchy-Schwarz inequality   bcsiALT 29441
      ​19.3  Cauchy sequences and completeness axiom
            19.3.1  Cauchy sequences and limits   hcau 29446
            19.3.2  Derivation of the completeness axiom from ZF set theory   hilmet 29456
            19.3.3  Completeness postulate for a Hilbert space   ax-hcompl 29464
            19.3.4  Relate Hilbert space to ZFC pre-Hilbert and Hilbert spaces   hhcms 29465
      ​19.4  Subspaces and projections
            19.4.1  Subspaces   df-sh 29469
            19.4.2  Closed subspaces   df-ch 29483
            19.4.3  Orthocomplements   df-oc 29514
            19.4.4  Subspace sum, span, lattice join, lattice supremum   df-shs 29570
            19.4.5  Projection theorem   pjhthlem1 29653
            19.4.6  Projectors   df-pjh 29657
      ​19.5  Properties of Hilbert subspaces
            19.5.1  Orthomodular law   omlsilem 29664
            19.5.2  Projectors (cont.)   pjhtheu2 29678
            19.5.3  Hilbert lattice operations   sh0le 29702
            19.5.4  Span (cont.) and one-dimensional subspaces   spansn0 29803
            19.5.5  Commutes relation for Hilbert lattice elements   df-cm 29845
            19.5.6  Foulis-Holland theorem   fh1 29880
            19.5.7  Quantum Logic Explorer axioms   qlax1i 29889
            19.5.8  Orthogonal subspaces   chscllem1 29899
            19.5.9  Orthoarguesian laws 5OA and 3OA   5oalem1 29916
            19.5.10  Projectors (cont.)   pjorthi 29931
            19.5.11  Mayet's equation E_3   mayete3i 29990
      ​19.6  Operators on Hilbert spaces
            ​*19.6.1  Operator sum, difference, and scalar multiplication   df-hosum 29992
            19.6.2  Zero and identity operators   df-h0op 30010
            19.6.3  Operations on Hilbert space operators   hoaddcl 30020
            19.6.4  Linear, continuous, bounded, Hermitian, unitary operators and norms   df-nmop 30101
            19.6.5  Linear and continuous functionals and norms   df-nmfn 30107
            19.6.6  Adjoint   df-adjh 30111
            19.6.7  Dirac bra-ket notation   df-bra 30112
            19.6.8  Positive operators   df-leop 30114
            19.6.9  Eigenvectors, eigenvalues, spectrum   df-eigvec 30115
            19.6.10  Theorems about operators and functionals   nmopval 30118
            19.6.11  Riesz lemma   riesz3i 30324
            19.6.12  Adjoints (cont.)   cnlnadjlem1 30329
            19.6.13  Quantum computation error bound theorem   unierri 30366
            19.6.14  Dirac bra-ket notation (cont.)   branmfn 30367
            19.6.15  Positive operators (cont.)   leopg 30384
            19.6.16  Projectors as operators   pjhmopi 30408
      ​19.7  States on a Hilbert lattice and Godowski's equation
            19.7.1  States on a Hilbert lattice   df-st 30473
            19.7.2  Godowski's equation   golem1 30533
      ​19.8  Cover relation, atoms, exchange axiom, and modular symmetry
            19.8.1  Covers relation; modular pairs   df-cv 30541
            19.8.2  Atoms   df-at 30600
            19.8.3  Superposition principle   superpos 30616
            19.8.4  Atoms, exchange and covering properties, atomicity   chcv1 30617
            19.8.5  Irreducibility   chirredlem1 30652
            19.8.6  Atoms (cont.)   atcvat3i 30658
            19.8.7  Modular symmetry   mdsymlem1 30665
​PART 20  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      ​20.1  Mathboxes for user contributions
            20.1.1  Mathbox guidelines   mathbox 30704
      ​20.2  Mathbox for Stefan Allan
      ​20.3  Mathbox for Thierry Arnoux
            20.3.1  Propositional Calculus - misc additions   bian1d 30709
            20.3.2  Predicate Calculus   sbc2iedf 30715
                  20.3.2.1  Predicate Calculus - misc additions   sbc2iedf 30715
                  20.3.2.2  Restricted quantification - misc additions   ralcom4f 30718
                  20.3.2.3  Equality   eqtrb 30723
                  20.3.2.4  Double restricted existential uniqueness quantification   opsbc2ie 30724
                  20.3.2.5  Double restricted existential uniqueness quantification syntax   w2reu 30726
                  20.3.2.6  Substitution (without distinct variables) - misc additions   sbceqbidf 30735
                  20.3.2.7  Existential "at most one" - misc additions   mo5f 30737
                  20.3.2.8  Existential uniqueness - misc additions   reuxfrdf 30739
                  20.3.2.9  Restricted "at most one" - misc additions   rmoxfrd 30741
            20.3.3  General Set Theory   dmrab 30744
                  20.3.3.1  Class abstractions (a.k.a. class builders)   dmrab 30744
                  20.3.3.2  Image Sets   abrexdomjm 30752
                  20.3.3.3  Set relations and operations - misc additions   elunsn 30758
                  20.3.3.4  Unordered pairs   eqsnd 30777
                  20.3.3.5  Conditional operator - misc additions   ifeqeqx 30785
                  20.3.3.6  Set union   uniinn0 30790
                  20.3.3.7  Indexed union - misc additions   cbviunf 30795
                  20.3.3.8  Indexed intersection - misc additions   iinabrex 30808
                  20.3.3.9  Disjointness - misc additions   disjnf 30809
            20.3.4  Relations and Functions   xpdisjres 30837
                  20.3.4.1  Relations - misc additions   xpdisjres 30837
                  20.3.4.2  Functions - misc additions   ac6sf2 30860
                  20.3.4.3  Operations - misc additions   mpomptxf 30917
                  20.3.4.4  Explicit Functions with one or two points as a domain   cosnopne 30928
                  20.3.4.5  Isomorphisms - misc. add.   gtiso 30934
                  20.3.4.6  Disjointness (additional proof requiring functions)   disjdsct 30936
                  20.3.4.7  First and second members of an ordered pair - misc additions   df1stres 30937
                  20.3.4.8  Supremum - misc additions   supssd 30945
                  20.3.4.9  Finite Sets   imafi2 30947
                  20.3.4.10  Countable Sets   snct 30949
            20.3.5  Real and Complex Numbers   creq0 30971
                  20.3.5.1  Complex operations - misc. additions   creq0 30971
                  20.3.5.2  Ordering on reals - misc additions   lt2addrd 30975
                  20.3.5.3  Extended reals - misc additions   xrlelttric 30976
                  20.3.5.4  Extended nonnegative integers - misc additions   xnn0gt0 30993
                  20.3.5.5  Real number intervals - misc additions   joiniooico 30996
                  20.3.5.6  Finite intervals of integers - misc additions   uzssico 31006
                  20.3.5.7  Half-open integer ranges - misc additions   iundisjfi 31018
                  20.3.5.8  The ` # ` (set size) function - misc additions   hashunif 31027
                  20.3.5.9  The greatest common divisor operator - misc. add   dvdszzq 31030
                  20.3.5.10  Integers   nnindf 31034
                  20.3.5.11  Decimal numbers   dfdec100 31045
            ​*20.3.6  Decimal expansion   cdp2 31046
                  ​*20.3.6.1  Decimal point   cdp 31063
                  20.3.6.2  Division in the extended real number system   cxdiv 31092
            20.3.7  Words over a set - misc additions   wrdfd 31111
                  20.3.7.1  Splicing words (substring replacement)   splfv3 31131
                  20.3.7.2  Cyclic shift of words   1cshid 31132
            20.3.8  Extensible Structures   ressplusf 31136
                  20.3.8.1  Structure restriction operator   ressplusf 31136
                  20.3.8.2  The opposite group   oppgle 31139
                  20.3.8.3  Posets   ressprs 31142
                  20.3.8.4  Complete lattices   clatp0cl 31155
                  20.3.8.5  Order Theory   cmnt 31157
                  20.3.8.6  Extended reals Structure - misc additions   ax-xrssca 31183
                  20.3.8.7  The extended nonnegative real numbers commutative monoid   xrge0base 31195
            20.3.9  Algebra   abliso 31206
                  20.3.9.1  Monoids Homomorphisms   abliso 31206
                  20.3.9.2  Finitely supported group sums - misc additions   gsumsubg 31207
                  20.3.9.3  Centralizers and centers - misc additions   cntzun 31221
                  20.3.9.4  Totally ordered monoids and groups   comnd 31224
                  20.3.9.5  The symmetric group   symgfcoeu 31252
                  20.3.9.6  Transpositions   pmtridf1o 31262
                  20.3.9.7  Permutation Signs   psgnid 31265
                  20.3.9.8  Permutation cycles   ctocyc 31274
                  20.3.9.9  The Alternating Group   evpmval 31313
                  20.3.9.10  Signum in an ordered monoid   csgns 31326
                  20.3.9.11  The Archimedean property for generic ordered algebraic structures   cinftm 31331
                  20.3.9.12  Semiring left modules   cslmd 31354
                  20.3.9.13  Simple groups   prmsimpcyc 31382
                  20.3.9.14  Rings - misc additions   rngurd 31383
                  20.3.9.15  Subfields   primefldchr 31394
                  20.3.9.16  Totally ordered rings and fields   corng 31395
                  20.3.9.17  Ring homomorphisms - misc additions   rhmdvdsr 31418
                  20.3.9.18  Scalar restriction operation   cresv 31424
                  20.3.9.19  The commutative ring of gaussian integers   gzcrng 31444
                  20.3.9.20  The archimedean ordered field of real numbers   reofld 31445
                  20.3.9.21  The quotient map and quotient modules   qusker 31450
                  20.3.9.22  The ring of integers modulo ` N `   znfermltl 31463
                  20.3.9.23  Independent sets and families   islinds5 31464
                  ​*20.3.9.24  Subgroup sum / Sumset / Minkowski sum   elgrplsmsn 31479
                  20.3.9.25  The quotient map   quslsm 31494
                  20.3.9.26  Ideals   intlidl 31503
                  20.3.9.27  Prime Ideals   cprmidl 31511
                  20.3.9.28  Maximal Ideals   cmxidl 31532
                  20.3.9.29  The semiring of ideals of a ring   cidlsrg 31546
                  20.3.9.30  Unique factorization domains   cufd 31562
                  20.3.9.31  Associative algebras   asclmulg 31567
                  20.3.9.32  Univariate Polynomials   fply1 31568
                  20.3.9.33  The subring algebra   sra1r 31572
                  20.3.9.34  Division Ring Extensions   drgext0g 31578
                  20.3.9.35  Vector Spaces   lvecdimfi 31584
                  20.3.9.36  Vector Space Dimension   cldim 31585
            20.3.10  Field Extensions   cfldext 31614
            20.3.11  Matrices   csmat 31644
                  20.3.11.1  Submatrices   csmat 31644
                  20.3.11.2  Matrix literals   clmat 31662
                  20.3.11.3  Laplace expansion of determinants   mdetpmtr1 31674
            20.3.12  Topology   ist0cld 31684
                  20.3.12.1  Open maps   txomap 31685
                  20.3.12.2  Topology of the unit circle   qtopt1 31686
                  20.3.12.3  Refinements   reff 31690
                  20.3.12.4  Open cover refinement property   ccref 31693
                  20.3.12.5  Lindelöf spaces   cldlf 31703
                  20.3.12.6  Paracompact spaces   cpcmp 31706
                  ​*20.3.12.7  Spectrum of a ring   crspec 31713
                  20.3.12.8  Pseudometrics   cmetid 31737
                  20.3.12.9  Continuity - misc additions   hauseqcn 31749
                  20.3.12.10  Topology of the closed unit interval   elunitge0 31750
                  20.3.12.11  Topology of ` ( RR X. RR ) `   unicls 31754
                  20.3.12.12  Order topology - misc. additions   cnvordtrestixx 31764
                  20.3.12.13  Continuity in topological spaces - misc. additions   mndpluscn 31777
                  20.3.12.14  Topology of the extended nonnegative real numbers ordered monoid   xrge0hmph 31783
                  20.3.12.15  Limits - misc additions   lmlim 31798
                  20.3.12.16  Univariate polynomials   pl1cn 31806
            20.3.13  Uniform Stuctures and Spaces   chcmp 31807
                  20.3.13.1  Hausdorff uniform completion   chcmp 31807
            20.3.14  Topology and algebraic structures   zringnm 31809
                  20.3.14.1  The norm on the ring of the integer numbers   zringnm 31809
                  20.3.14.2  Topological ` ZZ ` -modules   zlm0 31811
                  20.3.14.3  Canonical embedding of the field of the rational numbers into a division ring   cqqh 31821
                  20.3.14.4  Canonical embedding of the real numbers into a complete ordered field   crrh 31842
                  20.3.14.5  Embedding from the extended real numbers into a complete lattice   cxrh 31865
                  20.3.14.6  Canonical embeddings into the ordered field of the real numbers   zrhre 31868
                  ​*20.3.14.7  Topological Manifolds   cmntop 31871
            20.3.15  Real and complex functions   nexple 31876
                  20.3.15.1  Integer powers - misc. additions   nexple 31876
                  20.3.15.2  Indicator Functions   cind 31877
                  20.3.15.3  Extended sum   cesum 31894
            20.3.16  Mixed Function/Constant operation   cofc 31962
            20.3.17  Abstract measure   csiga 31975
                  20.3.17.1  Sigma-Algebra   csiga 31975
                  20.3.17.2  Generated sigma-Algebra   csigagen 32005
                  ​*20.3.17.3  lambda and pi-Systems, Rings of Sets   ispisys 32019
                  20.3.17.4  The Borel algebra on the real numbers   cbrsiga 32048
                  20.3.17.5  Product Sigma-Algebra   csx 32055
                  20.3.17.6  Measures   cmeas 32062
                  20.3.17.7  The counting measure   cntmeas 32093
                  20.3.17.8  The Lebesgue measure - misc additions   voliune 32096
                  20.3.17.9  The Dirac delta measure   cdde 32099
                  20.3.17.10  The 'almost everywhere' relation   cae 32104
                  20.3.17.11  Measurable functions   cmbfm 32116
                  20.3.17.12  Borel Algebra on ` ( RR X. RR ) `   br2base 32135
                  ​*20.3.17.13  Caratheodory's extension theorem   coms 32157
            20.3.18  Integration   itgeq12dv 32192
                  20.3.18.1  Lebesgue integral - misc additions   itgeq12dv 32192
                  20.3.18.2  Bochner integral   citgm 32193
            20.3.19  Euler's partition theorem   oddpwdc 32220
            20.3.20  Sequences defined by strong recursion   csseq 32249
            20.3.21  Fibonacci Numbers   cfib 32262
            20.3.22  Probability   cprb 32273
                  20.3.22.1  Probability Theory   cprb 32273
                  20.3.22.2  Conditional Probabilities   ccprob 32297
                  20.3.22.3  Real-valued Random Variables   crrv 32306
                  20.3.22.4  Preimage set mapping operator   corvc 32321
                  20.3.22.5  Distribution Functions   orvcelval 32334
                  20.3.22.6  Cumulative Distribution Functions   orvclteel 32338
                  20.3.22.7  Probabilities - example   coinfliplem 32344
                  20.3.22.8  Bertrand's Ballot Problem   ballotlemoex 32351
            20.3.23  Signum (sgn or sign) function - misc. additions   sgncl 32404
                  20.3.23.1  Operations on words   ccatmulgnn0dir 32420
            20.3.24  Polynomials with real coefficients - misc additions   plymul02 32424
            20.3.25  Descartes's rule of signs   signspval 32430
                  20.3.25.1  Sign changes in a word over real numbers   signspval 32430
                  20.3.25.2  Counting sign changes in a word over real numbers   signslema 32440
            20.3.26  Number Theory   efcld 32470
                  20.3.26.1  Representations of a number as sums of integers   crepr 32487
                  20.3.26.2  Vinogradov Trigonometric Sums and the Circle Method   cvts 32514
                  20.3.26.3  The Ternary Goldbach Conjecture: Final Statement   ax-hgt749 32523
            20.3.27  Elementary Geometry   cstrkg2d 32543
                  ​*20.3.27.1  Two-dimensional geometry   cstrkg2d 32543
                  20.3.27.2  Outer Five Segment (not used, no need to move to main)   cafs 32548
            ​*20.3.28  LeftPad Project   clpad 32553
      ​​*20.4  Mathbox for Jonathan Ben-Naim
            20.4.1  First-order logic and set theory   bnj170 32576
            20.4.2  Well founded induction and recursion   bnj110 32737
            20.4.3  The existence of a minimal element in certain classes   bnj69 32889
            20.4.4  Well-founded induction   bnj1204 32891
            20.4.5  Well-founded recursion, part 1 of 3   bnj60 32941
            20.4.6  Well-founded recursion, part 2 of 3   bnj1500 32947
            20.4.7  Well-founded recursion, part 3 of 3   bnj1522 32951
      ​20.5  Mathbox for BTernaryTau
            20.5.1  ZF set theory   exdifsn 32952
                  20.5.1.1  Finitism   fineqvrep 32963
            20.5.2  Real and complex numbers   zltp1ne 32967
            20.5.3  Graph theory   lfuhgr 32978
                  20.5.3.1  Acyclic graphs   cacycgr 33003
      ​20.6  Mathbox for Mario Carneiro
            20.6.1  Predicate calculus with all distinct variables   ax-7d 33020
            20.6.2  Miscellaneous stuff   quartfull 33026
            20.6.3  Derangements and the Subfactorial   deranglem 33027
            20.6.4  The Erdős-Szekeres theorem   erdszelem1 33052
            20.6.5  The Kuratowski closure-complement theorem   kur14lem1 33067
            20.6.6  Retracts and sections   cretr 33078
            20.6.7  Path-connected and simply connected spaces   cpconn 33080
            20.6.8  Covering maps   ccvm 33116
            20.6.9  Normal numbers   snmlff 33190
            20.6.10  Godel-sets of formulas - part 1   cgoe 33194
            20.6.11  Godel-sets of formulas - part 2   cgon 33293
            20.6.12  Models of ZF   cgze 33307
            ​*20.6.13  Metamath formal systems   cmcn 33321
            20.6.14  Grammatical formal systems   cm0s 33446
            20.6.15  Models of formal systems   cmuv 33466
            20.6.16  Splitting fields   citr 33488
            20.6.17  p-adic number fields   czr 33504
      ​​*20.7  Mathbox for Filip Cernatescu
      ​20.8  Mathbox for Paul Chapman
            20.8.1  Real and complex numbers (cont.)   climuzcnv 33528
            20.8.2  Miscellaneous theorems   elfzm12 33532
      ​20.9  Mathbox for Scott Fenton
            20.9.1  ZFC Axioms in primitive form   axextprim 33541
            20.9.2  Untangled classes   untelirr 33548
            20.9.3  Extra propositional calculus theorems   3orel2 33555
            20.9.4  Misc. Useful Theorems   nepss 33563
            20.9.5  Properties of real and complex numbers   sqdivzi 33598
            20.9.6  Infinite products   iprodefisumlem 33611
            20.9.7  Factorial limits   faclimlem1 33614
            20.9.8  Greatest common divisor and divisibility   gcd32 33620
            20.9.9  Properties of relationships   brtp 33622
            20.9.10  Properties of functions and mappings   funpsstri 33644
            20.9.11  Set induction (or epsilon induction)   setinds 33659
            20.9.12  Ordinal numbers   elpotr 33662
            20.9.13  Defined equality axioms   axextdfeq 33678
            20.9.14  Hypothesis builders   hbntg 33686
            20.9.15  (Trans)finite Recursion Theorems   tfisg 33691
            20.9.16  Transitive closure of a relation   cttrcl 33692
            20.9.17  Well-Founded Induction   frpoins3xpg 33713
            20.9.18  Ordering Cross Products, Part 2   xpord2lem 33715
            20.9.19  Ordering Ordinal Sequences   orderseqlem 33727
            20.9.20  Well-founded zero, successor, and limits   cwsuc 33730
            20.9.21  Natural operations on ordinals   cnadd 33750
            20.9.22  Surreal Numbers   csur 33769
            20.9.23  Surreal Numbers: Ordering   sltsolem1 33804
            20.9.24  Surreal Numbers: Birthday Function   bdayfo 33806
            20.9.25  Surreal Numbers: Density   fvnobday 33807
            ​*20.9.26  Surreal Numbers: Full-Eta Property   bdayimaon 33822
            20.9.27  Surreal numbers - ordering theorems   csle 33873
            20.9.28  Surreal numbers - birthday theorems   bdayfun 33893
            20.9.29  Surreal numbers: Conway cuts   csslt 33901
            20.9.30  Surreal numbers - zero and one   c0s 33942
            20.9.31  Surreal numbers - cuts and options   cmade 33952
            20.9.32  Surreal numbers: Cofinality and coinitiality   cofsslt 34014
            20.9.33  Surreal numbers: Induction and recursion on one variable   cnorec 34020
            20.9.34  Surreal numbers: Induction and recursion on two variables   cnorec2 34031
            20.9.35  Surreal numbers - addition, negation, and subtraction   cadds 34042
            20.9.36  Quantifier-free definitions   ctxp 34058
            20.9.37  Alternate ordered pairs   caltop 34184
            20.9.38  Geometry in the Euclidean space   cofs 34210
                  20.9.38.1  Congruence properties   cofs 34210
                  20.9.38.2  Betweenness properties   btwntriv2 34240
                  20.9.38.3  Segment Transportation   ctransport 34257
                  20.9.38.4  Properties relating betweenness and congruence   cifs 34263
                  20.9.38.5  Connectivity of betweenness   btwnconn1lem1 34315
                  20.9.38.6  Segment less than or equal to   csegle 34334
                  20.9.38.7  Outside-of relationship   coutsideof 34347
                  20.9.38.8  Lines and Rays   cline2 34362
            20.9.39  Forward difference   cfwddif 34386
            20.9.40  Rank theorems   rankung 34394
            20.9.41  Hereditarily Finite Sets   chf 34400
      ​20.10  Mathbox for Jeff Hankins
            20.10.1  Miscellany   a1i14 34415
            20.10.2  Basic topological facts   topbnd 34439
            20.10.3  Topology of the real numbers   ivthALT 34450
            20.10.4  Refinements   cfne 34451
            20.10.5  Neighborhood bases determine topologies   neibastop1 34474
            20.10.6  Lattice structure of topologies   topmtcl 34478
            20.10.7  Filter bases   fgmin 34485
            20.10.8  Directed sets, nets   tailfval 34487
      ​20.11  Mathbox for Anthony Hart
            20.11.1  Propositional Calculus   tb-ax1 34498
            20.11.2  Predicate Calculus   nalfal 34518
            20.11.3  Miscellaneous single axioms   meran1 34526
            20.11.4  Connective Symmetry   negsym1 34532
      ​20.12  Mathbox for Chen-Pang He
            20.12.1  Ordinal topology   ontopbas 34543
      ​20.13  Mathbox for Jeff Hoffman
            20.13.1  Inferences for finite induction on generic function values   fveleq 34566
            20.13.2  gdc.mm   nnssi2 34570
      ​20.14  Mathbox for Asger C. Ipsen
            20.14.1  Continuous nowhere differentiable functions   dnival 34577
      ​​*20.15  Mathbox for BJ
            *20.15.1  Propositional calculus   bj-mp2c 34646
                  *20.15.1.1  Derived rules of inference   bj-mp2c 34646
                  ​*20.15.1.2  A syntactic theorem   bj-0 34648
                  20.15.1.3  Minimal implicational calculus   bj-a1k 34650
                  ​*20.15.1.4  Positive calculus   bj-syl66ib 34661
                  20.15.1.5  Implication and negation   bj-con2com 34667
                  ​*20.15.1.6  Disjunction   bj-jaoi1 34678
                  ​*20.15.1.7  Logical equivalence   bj-dfbi4 34680
                  20.15.1.8  The conditional operator for propositions   bj-consensus 34685
                  ​*20.15.1.9  Propositional calculus: miscellaneous   bj-imbi12 34690
            ​*20.15.2  Modal logic   bj-axdd2 34700
            ​*20.15.3  Provability logic   cprvb 34705
            ​*20.15.4  First-order logic   bj-genr 34714
                  20.15.4.1  Adding ax-gen   bj-genr 34714
                  20.15.4.2  Adding ax-4   bj-2alim 34718
                  20.15.4.3  Adding ax-5   bj-ax12wlem 34751
                  20.15.4.4  Equality and substitution   bj-ssbeq 34760
                  20.15.4.5  Adding ax-6   bj-spimvwt 34776
                  20.15.4.6  Adding ax-7   bj-cbvexw 34783
                  20.15.4.7  Membership predicate, ax-8 and ax-9   bj-ax89 34785
                  20.15.4.8  Adding ax-11   bj-alcomexcom 34788
                  20.15.4.9  Adding ax-12   axc11n11 34790
                  20.15.4.10  Nonfreeness   wnnf 34831
                  20.15.4.11  Adding ax-13   bj-axc10 34891
                  ​*20.15.4.12  Removing dependencies on ax-13 (and ax-11)   bj-axc10v 34901
                  ​*20.15.4.13  Distinct var metavariables   bj-hbaeb2 34927
                  ​*20.15.4.14  Around ~ equsal   bj-equsal1t 34931
                  ​*20.15.4.15  Some Principia Mathematica proofs   stdpc5t 34936
                  20.15.4.16  Alternate definition of substitution   bj-sbsb 34946
                  20.15.4.17  Lemmas for substitution   bj-sbf3 34948
                  20.15.4.18  Existential uniqueness   bj-eu3f 34951
                  ​*20.15.4.19  First-order logic: miscellaneous   bj-sblem1 34952
            20.15.5  Set theory   eliminable1 34969
                  ​*20.15.5.1  Eliminability of class terms   eliminable1 34969
                  ​*20.15.5.2  Classes without the axiom of extensionality   bj-denoteslem 34981
                  20.15.5.3  Characterization among sets versus among classes   elelb 35008
                  ​*20.15.5.4  The nonfreeness quantifier for classes   bj-nfcsym 35010
                  ​*20.15.5.5  Lemmas for class substitution   bj-sbeqALT 35011
                  20.15.5.6  Removing some axiom requirements and disjoint variable conditions   bj-exlimvmpi 35022
                  ​*20.15.5.7  Class abstractions   bj-elabd2ALT 35039
                  20.15.5.8  Generalized class abstractions   bj-cgab 35047
                  ​*20.15.5.9  Restricted nonfreeness   wrnf 35055
                  ​*20.15.5.10  Russell's paradox   bj-ru0 35057
                  20.15.5.11  Curry's paradox in set theory   currysetlem 35060
                  ​*20.15.5.12  Some disjointness results   bj-n0i 35066
                  ​*20.15.5.13  Complements on direct products   bj-xpimasn 35071
                  ​*20.15.5.14  "Singletonization" and tagging   bj-snsetex 35079
                  ​*20.15.5.15  Tuples of classes   bj-cproj 35106
                  ​*20.15.5.16  Set theory: elementary operations relative to a universe   bj-rcleqf 35141
                  ​*20.15.5.17  Set theory: miscellaneous   eleq2w2ALT 35146
                  ​*20.15.5.18  Evaluation at a class   bj-evaleq 35169
                  20.15.5.19  Elementwise operations   celwise 35176
                  ​*20.15.5.20  Elementwise intersection (families of sets induced on a subset)   bj-rest00 35178
                  20.15.5.21  Moore collections (complements)   bj-raldifsn 35197
                  20.15.5.22  Maps-to notation for functions with three arguments   bj-0nelmpt 35213
                  ​*20.15.5.23  Currying   csethom 35219
                  ​*20.15.5.24  Setting components of extensible structures   cstrset 35231
            ​*20.15.6  Extended real and complex numbers, real and complex projective lines   bj-nfald 35234
                  20.15.6.1  Complements on class abstractions of ordered pairs and binary relations   bj-nfald 35234
                  ​*20.15.6.2  Identity relation (complements)   bj-opabssvv 35247
                  ​*20.15.6.3  Functionalized identity (diagonal in a Cartesian square)   cdiag2 35269
                  ​*20.15.6.4  Direct image and inverse image   cimdir 35275
                  ​*20.15.6.5  Extended numbers and projective lines as sets   cfractemp 35293
                  ​*20.15.6.6  Addition and opposite   caddcc 35334
                  ​*20.15.6.7  Order relation on the extended reals   cltxr 35338
                  ​*20.15.6.8  Argument, multiplication and inverse   carg 35340
                  20.15.6.9  The canonical bijection from the finite ordinals   ciomnn 35346
                  20.15.6.10  Divisibility   cnnbar 35357
            ​*20.15.7  Monoids   bj-smgrpssmgm 35365
                  ​*20.15.7.1  Finite sums in monoids   cfinsum 35380
            ​*20.15.8  Affine, Euclidean, and Cartesian geometry   bj-fvimacnv0 35383
                  *20.15.8.1  Real vector spaces   bj-fvimacnv0 35383
                  ​*20.15.8.2  Complex numbers (supplements)   bj-subcom 35405
                  ​*20.15.8.3  Barycentric coordinates   bj-bary1lem 35407
            20.15.9  Monoid of endomorphisms   cend 35410
      ​20.16  Mathbox for Jim Kingdon
                  20.16.0.1  Circle constant   taupilem3 35416
                  20.16.0.2  Number theory   dfgcd3 35421
                  20.16.0.3  Real numbers   irrdifflemf 35422
      ​20.17  Mathbox for ML
            20.17.1  Miscellaneous   csbrecsg 35425
            20.17.2  Cartesian exponentiation   cfinxp 35480
            20.17.3  Topology   iunctb2 35500
                  ​*20.17.3.1  Pi-base theorems   pibp16 35510
      ​20.18  Mathbox for Wolf Lammen
            20.18.1  1. Bootstrapping   wl-section-boot 35519
            20.18.2  Implication chains   wl-section-impchain 35543
            20.18.3  Theorems around the conditional operator   wl-ifp-ncond1 35561
            20.18.4  Alternative development of hadd, cadd   wl-df-3xor 35565
            20.18.5  An alternative axiom ~ ax-13   ax-wl-13v 35590
            20.18.6  Other stuff   wl-mps 35592
      ​20.19  Mathbox for Brendan Leahy
      ​20.20  Mathbox for Jeff Madsen
            20.20.1  Logic and set theory   unirep 35797
            20.20.2  Real and complex numbers; integers   filbcmb 35824
            20.20.3  Sequences and sums   sdclem2 35826
            20.20.4  Topology   subspopn 35836
            20.20.5  Metric spaces   metf1o 35839
            20.20.6  Continuous maps and homeomorphisms   constcncf 35846
            20.20.7  Boundedness   ctotbnd 35850
            20.20.8  Isometries   cismty 35882
            20.20.9  Heine-Borel Theorem   heibor1lem 35893
            20.20.10  Banach Fixed Point Theorem   bfplem1 35906
            20.20.11  Euclidean space   crrn 35909
            20.20.12  Intervals (continued)   ismrer1 35922
            20.20.13  Operation properties   cass 35926
            20.20.14  Groups and related structures   cmagm 35932
            20.20.15  Group homomorphism and isomorphism   cghomOLD 35967
            20.20.16  Rings   crngo 35978
            20.20.17  Division Rings   cdrng 36032
            20.20.18  Ring homomorphisms   crnghom 36044
            20.20.19  Commutative rings   ccm2 36073
            20.20.20  Ideals   cidl 36091
            20.20.21  Prime rings and integral domains   cprrng 36130
            20.20.22  Ideal generators   cigen 36143
      ​20.21  Mathbox for Giovanni Mascellani
            ​*20.21.1  Tools for automatic proof building   efald2 36162
            ​*20.21.2  Tseitin axioms   fald 36213
            ​*20.21.3  Equality deductions   iuneq2f 36240
            ​*20.21.4  Miscellanea   orcomdd 36251
      ​20.22  Mathbox for Peter Mazsa
            20.22.1  Notations   cxrn 36258
            20.22.2  Preparatory theorems   el2v1 36296
            20.22.3  Range Cartesian product   df-xrn 36427
            20.22.4  Cosets by ` R `   df-coss 36463
            20.22.5  Relations   df-rels 36529
            20.22.6  Subset relations   df-ssr 36542
            20.22.7  Reflexivity   df-refs 36554
            20.22.8  Converse reflexivity   df-cnvrefs 36567
            20.22.9  Symmetry   df-syms 36582
            20.22.10  Reflexivity and symmetry   symrefref2 36603
            20.22.11  Transitivity   df-trs 36612
            20.22.12  Equivalence relations   df-eqvrels 36623
            20.22.13  Redundancy   df-redunds 36662
            20.22.14  Domain quotients   df-dmqss 36677
            20.22.15  Equivalence relations on domain quotients   df-ers 36701
            20.22.16  Functions   df-funss 36717
            20.22.17  Disjoints vs. converse functions   df-disjss 36740
      ​20.23  Mathbox for Rodolfo Medina
            20.23.1  Partitions   prtlem60 36793
      ​​*20.24  Mathbox for Norm Megill
            *20.24.1  Obsolete schemes ax-c4,c5,c7,c10,c11,c11n,c15,c9,c14,c16   ax-c5 36823
            ​*20.24.2  Rederive new axioms ax-4, ax-10, ax-6, ax-12, ax-13 from old   axc5 36833
            ​*20.24.3  Legacy theorems using obsolete axioms   ax5ALT 36847
            20.24.4  Experiments with weak deduction theorem   elimhyps 36901
            20.24.5  Miscellanea   cnaddcom 36912
            20.24.6  Atoms, hyperplanes, and covering in a left vector space (or module)   clsa 36914
            20.24.7  Functionals and kernels of a left vector space (or module)   clfn 36997
            20.24.8  Opposite rings and dual vector spaces   cld 37063
            20.24.9  Ortholattices and orthomodular lattices   cops 37112
            20.24.10  Atomic lattices with covering property   ccvr 37202
            20.24.11  Hilbert lattices   chlt 37290
            20.24.12  Projective geometries based on Hilbert lattices   clln 37431
            20.24.13  Construction of a vector space from a Hilbert lattice   cdlema1N 37731
            20.24.14  Construction of involution and inner product from a Hilbert lattice   clpoN 39420
      ​20.25  Mathbox for metakunt
            20.25.1  General helpful statements   leexp1ad 39906
            20.25.2  Some gcd and lcm results   12gcd5e1 39938
            20.25.3  Least common multiple inequality theorem   3factsumint1 39956
            20.25.4  Logarithm inequalities   3exp7 39988
            20.25.5  Miscellaneous results for AKS formalisation   intlewftc 39996
            20.25.6  Sticks and stones   sticksstones1 40022
            20.25.7  Permutation results   metakunt1 40045
            20.25.8  Unused lemmas scheduled for deletion   andiff 40079
      ​20.26  Mathbox for Steven Nguyen
            ​*20.26.1  Miscellaneous theorems   bicomdALT 40084
            20.26.2  Utility theorems   ioin9i8 40093
            20.26.3  Structures   nelsubginvcld 40138
            ​*20.26.4  Arithmetic theorems   c0exALT 40202
            20.26.5  Exponents and divisibility   oexpreposd 40234
            20.26.6  Real subtraction   cresub 40261
            ​*20.26.7  Projective spaces   cprjsp 40353
            20.26.8  Basic reductions for Fermat's Last Theorem   dffltz 40379
      ​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 40422
            20.29.2  Additional theory of functions   imaiinfv 40423
            20.29.3  Additional topology   elrfi 40424
            20.29.4  Characterization of closure operators. Kuratowski closure axioms   ismrcd1 40428
            20.29.5  Algebraic closure systems   cnacs 40432
            20.29.6  Miscellanea 1. Map utilities   constmap 40443
            20.29.7  Miscellanea for polynomials   mptfcl 40450
            20.29.8  Multivariate polynomials over the integers   cmzpcl 40451
            20.29.9  Miscellanea for Diophantine sets 1   coeq0i 40483
            20.29.10  Diophantine sets 1: definitions   cdioph 40485
            20.29.11  Diophantine sets 2 miscellanea   ellz1 40497
            20.29.12  Diophantine sets 2: union and intersection. Monotone Boolean algebra   diophin 40502
            20.29.13  Diophantine sets 3: construction   diophrex 40505
            20.29.14  Diophantine sets 4 miscellanea   2sbcrex 40514
            20.29.15  Diophantine sets 4: Quantification   rexrabdioph 40524
            20.29.16  Diophantine sets 5: Arithmetic sets   rabdiophlem1 40531
            20.29.17  Diophantine sets 6: reusability. renumbering of variables   eldioph4b 40541
            20.29.18  Pigeonhole Principle and cardinality helpers   fphpd 40546
            20.29.19  A non-closed set of reals is infinite   rencldnfilem 40550
            20.29.20  Lagrange's rational approximation theorem   irrapxlem1 40552
            20.29.21  Pell equations 1: A nontrivial solution always exists   pellexlem1 40559
            20.29.22  Pell equations 2: Algebraic number theory of the solution set   csquarenn 40566
            20.29.23  Pell equations 3: characterizing fundamental solution   infmrgelbi 40608
            ​*20.29.24  Logarithm laws generalized to an arbitrary base   reglogcl 40620
            20.29.25  Pell equations 4: the positive solution group is infinite cyclic   pellfund14 40628
            20.29.26  X and Y sequences 1: Definition and recurrence laws   crmx 40630
            20.29.27  Ordering and induction lemmas for the integers   monotuz 40671
            20.29.28  X and Y sequences 2: Order properties   rmxypos 40677
            20.29.29  Congruential equations   congtr 40695
            20.29.30  Alternating congruential equations   acongid 40705
            20.29.31  Additional theorems on integer divisibility   coprmdvdsb 40715
            20.29.32  X and Y sequences 3: Divisibility properties   jm2.18 40718
            20.29.33  X and Y sequences 4: Diophantine representability of Y   jm2.27a 40735
            20.29.34  X and Y sequences 5: Diophantine representability of X, ^, _C   rmxdiophlem 40745
            20.29.35  Uncategorized stuff not associated with a major project   setindtr 40754
            20.29.36  More equivalents of the Axiom of Choice   axac10 40763
            20.29.37  Finitely generated left modules   clfig 40800
            20.29.38  Noetherian left modules I   clnm 40808
            20.29.39  Addenda for structure powers   pwssplit4 40822
            20.29.40  Every set admits a group structure iff choice   unxpwdom3 40828
            20.29.41  Noetherian rings and left modules II   clnr 40842
            20.29.42  Hilbert's Basis Theorem   cldgis 40854
            20.29.43  Additional material on polynomials [DEPRECATED]   cmnc 40864
            20.29.44  Degree and minimal polynomial of algebraic numbers   cdgraa 40873
            20.29.45  Algebraic integers I   citgo 40890
            20.29.46  Endomorphism algebra   cmend 40908
            20.29.47  Cyclic groups and order   idomrootle 40928
            20.29.48  Cyclotomic polynomials   ccytp 40935
            20.29.49  Miscellaneous topology   fgraphopab 40943
      ​20.30  Mathbox for Jon Pennant
      ​20.31  Mathbox for Richard Penner
            20.31.1  Short Studies   ifpan123g 40956
                  20.31.1.1  Additional work on conditional logical operator   ifpan123g 40956
                  20.31.1.2  Sophisms   rp-fakeimass 41009
                  ​*20.31.1.3  Finite Sets   rp-isfinite5 41014
                  20.31.1.4  General Observations   intabssd 41016
                  20.31.1.5  Infinite Sets   pwelg 41048
                  ​*20.31.1.6  Finite intersection property   fipjust 41053
                  20.31.1.7  RP ADDTO: Subclasses and subsets   rababg 41062
                  20.31.1.8  RP ADDTO: The intersection of a class   elintabg 41063
                  20.31.1.9  RP ADDTO: Theorems requiring subset and intersection existence   elinintrab 41066
                  20.31.1.10  RP ADDTO: Relations   xpinintabd 41069
                  ​*20.31.1.11  RP ADDTO: Functions   elmapintab 41085
                  ​*20.31.1.12  RP ADDTO: Finite induction (for finite ordinals)   cnvcnvintabd 41089
                  20.31.1.13  RP ADDTO: First and second members of an ordered pair   elcnvlem 41090
                  20.31.1.14  RP ADDTO: The reflexive and transitive properties of relations   undmrnresiss 41093
                  20.31.1.15  RP ADDTO: Basic properties of closures   cleq2lem 41097
                  20.31.1.16  RP REPLACE: Definitions and basic properties of transitive closures   trcleq2lemRP 41119
                  ​*20.31.1.17  Additions for square root; absolute value   sqrtcvallem1 41120
            20.31.2  Additional statements on relations and subclasses   al3im 41136
                  20.31.2.1  Transitive relations (not to be confused with transitive classes).   trrelind 41154
                  20.31.2.2  Reflexive closures   crcl 41161
                  ​*20.31.2.3  Finite relationship composition.   relexp2 41166
                  20.31.2.4  Transitive closure of a relation   dftrcl3 41209
                  ​*20.31.2.5  Adapted from Frege   frege77d 41235
            ​*20.31.3  Propositions from _Begriffsschrift_   dfxor4 41255
                  *20.31.3.1  _Begriffsschrift_ Chapter I   dfxor4 41255
                  ​*20.31.3.2  _Begriffsschrift_ Notation hints   whe 41261
                  20.31.3.3  _Begriffsschrift_ Chapter II Implication   ax-frege1 41279
                  20.31.3.4  _Begriffsschrift_ Chapter II Implication and Negation   axfrege28 41318
                  ​*20.31.3.5  _Begriffsschrift_ Chapter II with logical equivalence   axfrege52a 41345
                  20.31.3.6  _Begriffsschrift_ Chapter II with equivalence of sets   axfrege52c 41376
                  ​*20.31.3.7  _Begriffsschrift_ Chapter II with equivalence of classes   frege53c 41403
                  ​*20.31.3.8  _Begriffsschrift_ Chapter III Properties hereditary in a sequence   dffrege69 41421
                  ​*20.31.3.9  _Begriffsschrift_ Chapter III Following in a sequence   dffrege76 41428
                  ​*20.31.3.10  _Begriffsschrift_ Chapter III Member of sequence   dffrege99 41451
                  ​*20.31.3.11  _Begriffsschrift_ Chapter III Single-valued procedures   dffrege115 41467
            ​*20.31.4  Exploring Topology via Seifert and Threlfall   enrelmap 41486
                  *20.31.4.1  Equinumerosity of sets of relations and maps   enrelmap 41486
                  ​*20.31.4.2  Generic Pseudoclosure Spaces, Pseudointerior Spaces, and Pseudoneighborhoods   or3or 41512
                  ​*20.31.4.3  Generic Neighborhood Spaces   gneispa 41621
            ​*20.31.5  Exploring Higher Homotopy via Kerodon   k0004lem1 41638
                  *20.31.5.1  Simplicial Sets   k0004lem1 41638
      ​20.32  Mathbox for Stanislas Polu
            20.32.1  IMO Problems   wwlemuld 41647
                  20.32.1.1  IMO 1972 B2   wwlemuld 41647
            ​*20.32.2  INT Inequalities Proof Generator   int-addcomd 41665
            ​*20.32.3  N-Digit Addition Proof Generator   unitadd 41687
            20.32.4  AM-GM (for k = 2,3,4)   gsumws3 41688
      ​20.33  Mathbox for Rohan Ridenour
            20.33.1  Misc   spALT 41693
            20.33.2  Monoid rings   cmnring 41705
            20.33.3  Shorter primitive equivalent of ax-groth   gru0eld 41728
                  20.33.3.1  Grothendieck universes are closed under collection   gru0eld 41728
                  20.33.3.2  Minimal universes   ismnu 41760
                  20.33.3.3  Primitive equivalent of ax-groth   expandan 41787
      ​20.34  Mathbox for Steve Rodriguez
            20.34.1  Miscellanea   nanorxor 41804
            20.34.2  Ratio test for infinite series convergence and divergence   dvgrat 41811
            20.34.3  Multiples   reldvds 41814
            20.34.4  Function operations   caofcan 41822
            20.34.5  Calculus   lhe4.4ex1a 41828
            20.34.6  The generalized binomial coefficient operation   cbcc 41835
            20.34.7  Binomial series   uzmptshftfval 41845
      ​20.35  Mathbox for Andrew Salmon
            20.35.1  Principia Mathematica * 10   pm10.12 41857
            20.35.2  Principia Mathematica * 11   2alanimi 41871
            20.35.3  Predicate Calculus   sbeqal1 41897
            20.35.4  Principia Mathematica * 13 and * 14   pm13.13a 41906
            20.35.5  Set Theory   elnev 41937
            20.35.6  Arithmetic   addcomgi 41955
            20.35.7  Geometry   cplusr 41956
      ​​*20.36  Mathbox for Alan Sare
            20.36.1  Auxiliary theorems for the Virtual Deduction tool   idiALT 41978
            20.36.2  Supplementary unification deductions   bi1imp 41982
            20.36.3  Conventional Metamath proofs, some derived from VD proofs   iidn3 42002
            20.36.4  What is Virtual Deduction?   wvd1 42070
            20.36.5  Virtual Deduction Theorems   df-vd1 42071
            20.36.6  Theorems proved using Virtual Deduction   trsspwALT 42319
            20.36.7  Theorems proved using Virtual Deduction with mmj2 assistance   simplbi2VD 42347
            20.36.8  Virtual Deduction transcriptions of textbook proofs   sb5ALTVD 42414
            20.36.9  Theorems proved using conjunction-form Virtual Deduction   elpwgdedVD 42418
            20.36.10  Theorems with a VD proof in conventional notation derived from a VD proof   suctrALT3 42425
            ​*20.36.11  Theorems with a proof in conventional notation derived from a VD proof   notnotrALT2 42428
      ​20.37  Mathbox for Glauco Siliprandi
            20.37.1  Miscellanea   evth2f 42439
            20.37.2  Functions   feq1dd 42584
            20.37.3  Ordering on real numbers - Real and complex numbers basic operations   sub2times 42694
            20.37.4  Real intervals   gtnelioc 42911
            20.37.5  Finite sums   fsummulc1f 42994
            20.37.6  Finite multiplication of numbers and finite multiplication of functions   fmul01 43003
            20.37.7  Limits   clim1fr1 43024
                  20.37.7.1  Inferior limit (lim inf)   clsi 43174
                  ​*20.37.7.2  Limits for sequences of extended real numbers   clsxlim 43241
            20.37.8  Trigonometry   coseq0 43287
            20.37.9  Continuous Functions   mulcncff 43293
            20.37.10  Derivatives   dvsinexp 43334
            20.37.11  Integrals   itgsin0pilem1 43373
            20.37.12  Stone Weierstrass theorem - real version   stoweidlem1 43424
            20.37.13  Wallis' product for π   wallispilem1 43488
            20.37.14  Stirling's approximation formula for ` n ` factorial   stirlinglem1 43497
            20.37.15  Dirichlet kernel   dirkerval 43514
            20.37.16  Fourier Series   fourierdlem1 43531
            20.37.17  e is transcendental   elaa2lem 43656
            20.37.18  n-dimensional Euclidean space   rrxtopn 43707
            20.37.19  Basic measure theory   csalg 43731
                  ​*20.37.19.1  σ-Algebras   csalg 43731
                  20.37.19.2  Sum of nonnegative extended reals   csumge0 43782
                  ​*20.37.19.3  Measures   cmea 43869
                  ​*20.37.19.4  Outer measures and Caratheodory's construction   come 43909
                  ​*20.37.19.5  Lebesgue measure on n-dimensional Real numbers   covoln 43956
                  ​*20.37.19.6  Measurable functions   csmblfn 44115
      ​20.38  Mathbox for Saveliy Skresanov
            20.38.1  Ceva's theorem   sigarval 44245
            20.38.2  Simple groups   simpcntrab 44265
      ​20.39  Mathbox for Jarvin Udandy
      ​20.40  Mathbox for Adhemar
            ​*20.40.1  Minimal implicational calculus   adh-minim 44375
      ​20.41  Mathbox for Alexander van der Vekens
            20.41.1  General auxiliary theorems (1)   eusnsn 44399
                  20.41.1.1  Unordered and ordered pairs - extension for singletons   eusnsn 44399
                  20.41.1.2  Unordered and ordered pairs - extension for unordered pairs   elprneb 44402
                  20.41.1.3  Unordered and ordered pairs - extension for ordered pairs   oppr 44403
                  20.41.1.4  Relations - extension   eubrv 44408
                  20.41.1.5  Definite description binder (inverted iota) - extension   iota0def 44411
                  20.41.1.6  Functions - extension   fveqvfvv 44413
            20.41.2  Alternative for Russell's definition of a description binder   caiota 44454
            20.41.3  Double restricted existential uniqueness   r19.32 44469
                  20.41.3.1  Restricted quantification (extension)   r19.32 44469
                  20.41.3.2  Restricted uniqueness and "at most one" quantification   reuf1odnf 44478
                  20.41.3.3  Analogs to Existential uniqueness (double quantification)   2reu3 44481
                  20.41.3.4  Additional theorems for double restricted existential uniqueness   2reu8i 44484
            ​*20.41.4  Alternative definitions of function and operation values   wdfat 44487
                  20.41.4.1  Restricted quantification (extension)   ralbinrald 44493
                  20.41.4.2  The universal class (extension)   nvelim 44494
                  20.41.4.3  Introduce the Axiom of Power Sets (extension)   alneu 44495
                  20.41.4.4  Predicate "defined at"   dfateq12d 44497
                  20.41.4.5  Alternative definition of the value of a function   dfafv2 44503
                  20.41.4.6  Alternative definition of the value of an operation   aoveq123d 44549
            ​*20.41.5  Alternative definitions of function values (2)   cafv2 44579
            20.41.6  General auxiliary theorems (2)   an4com24 44639
                  20.41.6.1  Logical conjunction - extension   an4com24 44639
                  20.41.6.2  Abbreviated conjunction and disjunction of three wff's - extension   3an4ancom24 44640
                  20.41.6.3  Negated membership (alternative)   cnelbr 44642
                  20.41.6.4  The empty set - extension   ralralimp 44649
                  20.41.6.5  Indexed union and intersection - extension   otiunsndisjX 44650
                  20.41.6.6  Functions - extension   fvifeq 44651
                  20.41.6.7  Maps-to notation - extension   fvmptrab 44663
                  20.41.6.8  Ordering on reals - extension   leltletr 44665
                  20.41.6.9  Subtraction - extension   cnambpcma 44666
                  20.41.6.10  Ordering on reals (cont.) - extension   leaddsuble 44669
                  20.41.6.11  Imaginary and complex number properties - extension   readdcnnred 44675
                  20.41.6.12  Nonnegative integers (as a subset of complex numbers) - extension   nn0resubcl 44680
                  20.41.6.13  Integers (as a subset of complex numbers) - extension   zgeltp1eq 44681
                  20.41.6.14  Decimal arithmetic - extension   1t10e1p1e11 44682
                  20.41.6.15  Upper sets of integers - extension   eluzge0nn0 44684
                  20.41.6.16  Infinity and the extended real number system (cont.) - extension   nltle2tri 44685
                  20.41.6.17  Finite intervals of integers - extension   ssfz12 44686
                  20.41.6.18  Half-open integer ranges - extension   fzopred 44694
                  20.41.6.19  The modulo (remainder) operation - extension   m1mod0mod1 44701
                  20.41.6.20  The infinite sequence builder "seq"   smonoord 44703
                  20.41.6.21  Finite and infinite sums - extension   fsummsndifre 44704
                  20.41.6.22  Extensible structures - extension   setsidel 44708
            ​*20.41.7  Preimages of function values   preimafvsnel 44711
            ​*20.41.8  Partitions of real intervals   ciccp 44745
            20.41.9  Shifting functions with an integer range domain   fargshiftfv 44771
            20.41.10  Words over a set (extension)   lswn0 44776
                  20.41.10.1  Last symbol of a word - extension   lswn0 44776
            20.41.11  Unordered pairs   wich 44777
                  20.41.11.1  Interchangeable setvar variables   wich 44777
                  20.41.11.2  Set of unordered pairs   sprid 44806
                  ​*20.41.11.3  Proper (unordered) pairs   prpair 44833
                  20.41.11.4  Set of proper unordered pairs   cprpr 44844
            20.41.12  Number theory (extension)   cfmtno 44859
                  ​*20.41.12.1  Fermat numbers   cfmtno 44859
                  ​*20.41.12.2  Mersenne primes   m2prm 44923
                  20.41.12.3  Proth's theorem   modexp2m1d 44944
                  20.41.12.4  Solutions of quadratic equations   quad1 44952
            ​*20.41.13  Even and odd numbers   ceven 44956
                  20.41.13.1  Definitions and basic properties   ceven 44956
                  20.41.13.2  Alternate definitions using the "divides" relation   dfeven2 44981
                  20.41.13.3  Alternate definitions using the "modulo" operation   dfeven3 44990
                  20.41.13.4  Alternate definitions using the "gcd" operation   iseven5 44996
                  20.41.13.5  Theorems of part 5 revised   zneoALTV 45001
                  20.41.13.6  Theorems of part 6 revised   odd2np1ALTV 45006
                  20.41.13.7  Theorems of AV's mathbox revised   0evenALTV 45020
                  20.41.13.8  Additional theorems   epoo 45035
                  20.41.13.9  Perfect Number Theorem (revised)   perfectALTVlem1 45053
            20.41.14  Number theory (extension 2)   cfppr 45056
                  ​*20.41.14.1  Fermat pseudoprimes   cfppr 45056
                  ​*20.41.14.2  Goldbach's conjectures   cgbe 45077
            20.41.15  Graph theory (extension)   cgrisom 45150
                  ​*20.41.15.1  Isomorphic graphs   cgrisom 45150
                  20.41.15.2  Loop-free graphs - extension   1hegrlfgr 45174
                  20.41.15.3  Walks - extension   cupwlks 45175
                  20.41.15.4  Edges of graphs expressed as sets of unordered pairs   upgredgssspr 45185
            20.41.16  Monoids (extension)   ovn0dmfun 45198
                  20.41.16.1  Auxiliary theorems   ovn0dmfun 45198
                  20.41.16.2  Magmas, Semigroups and Monoids (extension)   plusfreseq 45206
                  20.41.16.3  Magma homomorphisms and submagmas   cmgmhm 45211
                  20.41.16.4  Examples and counterexamples for magmas, semigroups and monoids (extension)   opmpoismgm 45241
                  20.41.16.5  Group sum operation (extension 1)   gsumsplit2f 45254
            ​*20.41.17  Magmas and internal binary operations (alternate approach)   ccllaw 45257
                  *20.41.17.1  Laws for internal binary operations   ccllaw 45257
                  ​*20.41.17.2  Internal binary operations   cintop 45270
                  20.41.17.3  Alternative definitions for magmas and semigroups   cmgm2 45289
            20.41.18  Categories (extension)   idfusubc0 45303
                  20.41.18.1  Subcategories (extension)   idfusubc0 45303
            20.41.19  Rings (extension)   lmod0rng 45306
                  20.41.19.1  Nonzero rings (extension)   lmod0rng 45306
                  ​*20.41.19.2  Non-unital rings ("rngs")   crng 45312
                  20.41.19.3  Rng homomorphisms   crngh 45323
                  20.41.19.4  Ring homomorphisms (extension)   rhmfn 45356
                  20.41.19.5  Ideals as non-unital rings   lidldomn1 45359
                  20.41.19.6  The non-unital ring of even integers   0even 45369
                  20.41.19.7  A constructed not unital ring   cznrnglem 45391
                  ​*20.41.19.8  The category of non-unital rings   crngc 45395
                  ​*20.41.19.9  The category of (unital) rings   cringc 45441
                  20.41.19.10  Subcategories of the category of rings   srhmsubclem1 45511
            20.41.20  Basic algebraic structures (extension)   opeliun2xp 45548
                  20.41.20.1  Auxiliary theorems   opeliun2xp 45548
                  20.41.20.2  The binomial coefficient operation (extension)   bcpascm1 45567
                  20.41.20.3  The ` ZZ `-module ` ZZ X. ZZ `   zlmodzxzlmod 45570
                  20.41.20.4  Group sum operation (extension 2)   mgpsumunsn 45577
                  20.41.20.5  Symmetric groups (extension)   exple2lt6 45580
                  20.41.20.6  Divisibility (extension)   invginvrid 45583
                  20.41.20.7  The support of functions (extension)   rmsupp0 45584
                  20.41.20.8  Finitely supported functions (extension)   rmsuppfi 45589
                  20.41.20.9  Left modules (extension)   lmodvsmdi 45598
                  20.41.20.10  Associative algebras (extension)   assaascl0 45600
                  20.41.20.11  Univariate polynomials (extension)   ply1vr1smo 45602
                  20.41.20.12  Univariate polynomials (examples)   linply1 45614
            20.41.21  Linear algebra (extension)   cdmatalt 45617
                  ​*20.41.21.1  The subalgebras of diagonal and scalar matrices (extension)   cdmatalt 45617
                  ​*20.41.21.2  Linear combinations   clinc 45625
                  ​*20.41.21.3  Linear independence   clininds 45661
                  20.41.21.4  Simple left modules and the ` ZZ `-module   lmod1lem1 45708
                  20.41.21.5  Differences between (left) modules and (left) vector spaces   lvecpsslmod 45728
            20.41.22  Complexity theory   suppdm 45731
                  20.41.22.1  Auxiliary theorems   suppdm 45731
                  20.41.22.2  The modulo (remainder) operation (extension)   fldivmod 45744
                  20.41.22.3  Even and odd integers   nn0onn0ex 45749
                  20.41.22.4  The natural logarithm on complex numbers (extension)   logcxp0 45761
                  20.41.22.5  Division of functions   cfdiv 45763
                  20.41.22.6  Upper bounds   cbigo 45773
                  20.41.22.7  Logarithm to an arbitrary base (extension)   rege1logbrege0 45784
                  ​*20.41.22.8  The binary logarithm   fldivexpfllog2 45791
                  20.41.22.9  Binary length   cblen 45795
                  ​*20.41.22.10  Digits   cdig 45821
                  20.41.22.11  Nonnegative integer as sum of its shifted digits   dignn0flhalflem1 45841
                  20.41.22.12  Algorithms for the multiplication of nonnegative integers   nn0mulfsum 45850
                  ​*20.41.22.13  N-ary functions   cnaryf 45852
                  ​*20.41.22.14  The Ackermann function   citco 45883
            20.41.23  Elementary geometry (extension)   fv1prop 45925
                  20.41.23.1  Auxiliary theorems   fv1prop 45925
                  20.41.23.2  Real euclidean space of dimension 2   rrx2pxel 45937
                  20.41.23.3  Spheres and lines in real Euclidean spaces   cline 45953
      ​20.42  Mathbox for Zhi Wang
            20.42.1  Propositional calculus   pm4.71da 46015
            20.42.2  Predicate calculus with equality   dtrucor3 46024
                  20.42.2.1  Axiom scheme ax-5 (Distinctness)   dtrucor3 46024
            20.42.3  ZF Set Theory - start with the Axiom of Extensionality   ralbidb 46025
                  20.42.3.1  Restricted quantification   ralbidb 46025
                  20.42.3.2  The empty set   ssdisjd 46033
                  20.42.3.3  Unordered and ordered pairs   vsn 46037
                  20.42.3.4  The union of a class   unilbss 46043
            20.42.4  ZF Set Theory - add the Axiom of Replacement   inpw 46044
                  20.42.4.1  Theorems requiring subset and intersection existence   inpw 46044
            20.42.5  ZF Set Theory - add the Axiom of Power Sets   mof0 46045
                  20.42.5.1  Functions   mof0 46045
                  20.42.5.2  Operations   fvconstr 46063
            20.42.6  ZF Set Theory - add the Axiom of Union   fvconst0ci 46066
                  20.42.6.1  Equinumerosity   fvconst0ci 46066
            20.42.7  Order sets   iccin 46070
                  20.42.7.1  Real number intervals   iccin 46070
            20.42.8  Moore spaces   mreuniss 46073
            ​*20.42.9  Topology   clduni 46074
                  20.42.9.1  Closure and interior   clduni 46074
                  20.42.9.2  Neighborhoods   neircl 46078
                  20.42.9.3  Subspace topologies   restcls2lem 46086
                  20.42.9.4  Limits and continuity in topological spaces   cnneiima 46090
                  20.42.9.5  Topological definitions using the reals   iooii 46091
                  20.42.9.6  Separated sets   sepnsepolem1 46095
                  20.42.9.7  Separated spaces: T0, T1, T2 (Hausdorff) ...   isnrm4 46104
            20.42.10  Preordered sets and directed sets using extensible structures   isprsd 46129
            20.42.11  Posets and lattices using extensible structures   lubeldm2 46130
                  20.42.11.1  Posets   lubeldm2 46130
                  20.42.11.2  Lattices   toslat 46148
                  20.42.11.3  Subset order structures   intubeu 46150
            20.42.12  Categories   catprslem 46171
                  20.42.12.1  Categories   catprslem 46171
                  20.42.12.2  Monomorphisms and epimorphisms   idmon 46177
                  20.42.12.3  Functors   funcf2lem 46179
            20.42.13  Examples of categories   cthinc 46180
                  20.42.13.1  Thin categories   cthinc 46180
                  20.42.13.2  Preordered sets as thin categories   cprstc 46223
                  20.42.13.3  Monoids as categories   cmndtc 46242
      ​20.43  Mathbox for Emmett Weisz
            ​*20.43.1  Miscellaneous Theorems   nfintd 46257
            20.43.2  Set Recursion   csetrecs 46267
                  ​*20.43.2.1  Basic Properties of Set Recursion   csetrecs 46267
                  20.43.2.2  Examples and properties of set recursion   elsetrecslem 46282
            ​*20.43.3  Construction of Games and Surreal Numbers   cpg 46292
      ​​*20.44  Mathbox for David A. Wheeler
            20.44.1  Natural deduction   sbidd 46298
            ​*20.44.2  Greater than, greater than or equal to.   cge-real 46300
            ​*20.44.3  Hyperbolic trigonometric functions   csinh 46310
            ​*20.44.4  Reciprocal trigonometric functions (sec, csc, cot)   csec 46321
            ​*20.44.5  Identities for "if"   ifnmfalse 46343
            ​*20.44.6  Logarithms generalized to arbitrary base using ` logb `   logb2aval 46344
            ​*20.44.7  Logarithm laws generalized to an arbitrary base - log_   clog- 46345
            ​*20.44.8  Formally define notions such as reflexivity   wreflexive 46347
            ​*20.44.9  Algebra helpers   comraddi 46351
            ​*20.44.10  Algebra helper examples   i2linesi 46360
            ​*20.44.11  Formal methods "surprises"   alimp-surprise 46362
            ​*20.44.12  Allsome quantifier   walsi 46368
            ​*20.44.13  Miscellaneous   5m4e1 46379
            20.44.14  Theorems about algebraic numbers   aacllem 46383
      ​20.45  Mathbox for Kunhao Zheng
            20.45.1  Weighted AM-GM inequality   amgmwlem 46384