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
      20.46  Mathbox for Ender Ting

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 205
            ​*1.2.6  Logical conjunction   wa 396
            ​*1.2.7  Logical disjunction   wo 844
            ​*1.2.8  Mixed connectives   jaao 952
            ​*1.2.9  The conditional operator for propositions   wif 1060
            ​*1.2.10  The weak deduction theorem for propositional calculus   elimh 1082
            1.2.11  Abbreviated conjunction and disjunction of three wff's   w3o 1085
            1.2.12  Logical "nand" (Sheffer stroke)   wnan 1486
            1.2.13  Logical "xor"   wxo 1506
            1.2.14  Logical "nor"   wnor 1525
            1.2.15  True and false constants   wal 1536
                  ​*1.2.15.1  Universal quantifier for use by df-tru   wal 1536
                  ​*1.2.15.2  Equality predicate for use by df-tru   cv 1537
                  1.2.15.3  The true constant   wtru 1539
                  1.2.15.4  The false constant   wfal 1550
            ​*1.2.16  Truth tables   truimtru 1561
                  1.2.16.1  Implication   truimtru 1561
                  1.2.16.2  Negation   nottru 1565
                  1.2.16.3  Equivalence   trubitru 1567
                  1.2.16.4  Conjunction   truantru 1571
                  1.2.16.5  Disjunction   truortru 1575
                  1.2.16.6  Alternative denial   trunantru 1579
                  1.2.16.7  Exclusive disjunction   truxortru 1583
                  1.2.16.8  Joint denial   trunortru 1587
            ​*1.2.17  Half adder and full adder in propositional calculus   whad 1591
                  1.2.17.1  Full adder: sum   whad 1591
                  1.2.17.2  Full adder: carry   wcad 1604
      ​1.3  Other axiomatizations related to classical propositional calculus
            ​*1.3.1  Minimal implicational calculus   minimp 1620
            ​*1.3.2  Implicational Calculus   impsingle 1626
            1.3.3  Derive the Lukasiewicz axioms from Meredith's sole axiom   meredith 1640
            1.3.4  Derive the standard axioms from the Lukasiewicz axioms   luklem1 1657
            ​*1.3.5  Derive Nicod's axiom from the standard axioms   nic-dfim 1668
            1.3.6  Derive the Lukasiewicz axioms from Nicod's axiom   nic-imp 1674
            1.3.7  Derive Nicod's Axiom from Lukasiewicz's First Sheffer Stroke Axiom   lukshef-ax1 1693
            1.3.8  Derive the Lukasiewicz Axioms from the Tarski-Bernays-Wajsberg Axioms   tbw-bijust 1697
            1.3.9  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's First CO Axiom   merco1 1712
            1.3.10  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's Second CO Axiom   merco2 1735
            1.3.11  Derive the Lukasiewicz axioms from the Russell-Bernays Axioms   rb-bijust 1748
            ​*1.3.12  Stoic logic non-modal portion (Chrysippus of Soli)   mptnan 1767
      ​​*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 1778
                  1.4.1.1  Existential quantifier   wex 1778
                  1.4.1.2  Nonfreeness predicate   wnf 1782
            1.4.2  Rule scheme ax-gen (Generalization)   ax-gen 1794
            1.4.3  Axiom scheme ax-4 (Quantified Implication)   ax-4 1808
                  ​*1.4.3.1  The empty domain of discourse   empty 1906
            1.4.4  Axiom scheme ax-5 (Distinctness) - first use of $d   ax-5 1910
            ​*1.4.5  Equality predicate (continued)   weq 1963
            1.4.6  Axiom scheme ax-6 (Existence)   ax-6 1968
            1.4.7  Axiom scheme ax-7 (Equality)   ax-7 2008
            1.4.8  Define proper substitution   sbjust 2063
            1.4.9  Membership predicate   wcel 2103
            1.4.10  Axiom scheme ax-8 (Left Equality for Binary Predicate)   ax-8 2105
            1.4.11  Axiom scheme ax-9 (Right Equality for Binary Predicate)   ax-9 2113
            ​*1.4.12  Logical redundancy of ax-10 , ax-11 , ax-12 , ax-13   ax6dgen 2121
      ​​*1.5  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes)
            1.5.1  Axiom scheme ax-10 (Quantified Negation)   ax-10 2134
            1.5.2  Axiom scheme ax-11 (Quantifier Commutation)   ax-11 2151
            1.5.3  Axiom scheme ax-12 (Substitution)   ax-12 2168
            1.5.4  Axiom scheme ax-13 (Quantified Equality)   ax-13 2369
      ​1.6  Uniqueness and unique existence
            1.6.1  Uniqueness: the at-most-one quantifier   wmo 2535
            1.6.2  Unique existence: the unique existential quantifier   weu 2565
      ​1.7  Other axiomatizations related to classical predicate calculus
            ​*1.7.1  Aristotelian logic: Assertic syllogisms   barbara 2661
            ​*1.7.2  Intuitionistic logic   axia1 2691
​​*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 2706
            2.1.2  Classes   cab 2712
                  2.1.2.1  Class abstractions   cab 2712
                  ​*2.1.2.2  Class equality   df-cleq 2727
                  2.1.2.3  Class membership   df-clel 2813
                  2.1.2.4  Elementary properties of class abstractions   abeq2 2869
            2.1.3  Class form not-free predicate   wnfc 2884
            2.1.4  Negated equality and membership   wne 2940
                  2.1.4.1  Negated equality   wne 2940
                  2.1.4.2  Negated membership   wnel 3046
            2.1.5  Restricted quantification   wral 3061
            2.1.6  The universal class   cvv 3431
            ​*2.1.7  Conditional equality (experimental)   wcdeq 3697
            2.1.8  Russell's Paradox   rru 3713
            2.1.9  Proper substitution of classes for sets   wsbc 3715
            2.1.10  Proper substitution of classes for sets into classes   csb 3831
            2.1.11  Define basic set operations and relations   cdif 3883
            2.1.12  Subclasses and subsets   df-ss 3903
            2.1.13  The difference, union, and intersection of two classes   dfdif3 4048
                  2.1.13.1  The difference of two classes   dfdif3 4048
                  2.1.13.2  The union of two classes   elun 4082
                  2.1.13.3  The intersection of two classes   elini 4126
                  2.1.13.4  The symmetric difference of two classes   csymdif 4174
                  2.1.13.5  Combinations of difference, union, and intersection of two classes   unabs 4187
                  2.1.13.6  Class abstractions with difference, union, and intersection of two classes   unabw 4230
                  2.1.13.7  Restricted uniqueness with difference, union, and intersection   reuun2 4247
            2.1.14  The empty set   c0 4255
            ​*2.1.15  The conditional operator for classes   cif 4458
            ​*2.1.16  The weak deduction theorem for set theory   dedth 4516
            2.1.17  Power classes   cpw 4532
            2.1.18  Unordered and ordered pairs   snjust 4559
            2.1.19  The union of a class   cuni 4838
            2.1.20  The intersection of a class   cint 4878
            2.1.21  Indexed union and intersection   ciun 4923
            2.1.22  Disjointness   wdisj 5038
            2.1.23  Binary relations   wbr 5073
            2.1.24  Ordered-pair class abstractions (class builders)   copab 5135
            2.1.25  Functions in maps-to notation   cmpt 5156
            2.1.26  Transitive classes   wtr 5190
      ​2.2  ZF Set Theory - add the Axiom of Replacement
            2.2.1  Introduce the Axiom of Replacement   ax-rep 5208
            2.2.2  Derive the Axiom of Separation   axsepgfromrep 5220
            2.2.3  Derive the Null Set Axiom   axnulALT 5227
            2.2.4  Theorems requiring subset and intersection existence   nalset 5236
            2.2.5  Theorems requiring empty set existence   class2set 5275
      ​2.3  ZF Set Theory - add the Axiom of Power Sets
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 5287
            2.3.2  Derive the Axiom of Pairing   axprlem1 5345
            2.3.3  Ordered pair theorem   opnz 5387
            2.3.4  Ordered-pair class abstractions (cont.)   opabidw 5436
            2.3.5  Power class of union and intersection   pwin 5482
            2.3.6  The identity relation   cid 5487
            2.3.7  The membership relation (or epsilon relation)   cep 5493
            ​*2.3.8  Partial and total orderings   wpo 5500
            2.3.9  Founded and well-ordering relations   wfr 5540
            2.3.10  Relations   cxp 5586
            2.3.11  The Predecessor Class   cpred 6203
            2.3.12  Well-founded induction (variant)   frpomin 6245
            2.3.13  Well-ordered induction   tz6.26 6252
            2.3.14  Ordinals   word 6267
            2.3.15  Definite description binder (inverted iota)   cio 6391
            2.3.16  Functions   wfun 6429
            2.3.17  Cantor's Theorem   canth 7236
            2.3.18  Restricted iota (description binder)   crio 7238
            2.3.19  Operations   co 7282
                  2.3.19.1  Variable-to-class conversion for operations   caovclg 7471
            2.3.20  Maps-to notation   mpondm0 7517
            2.3.21  Function operation   cof 7538
            2.3.22  Proper subset relation   crpss 7582
      ​2.4  ZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 7595
            2.4.2  Ordinals (continued)   epweon 7632
            2.4.3  Transfinite induction   tfi 7707
            2.4.4  The natural numbers (i.e., finite ordinals)   com 7719
            2.4.5  Peano's postulates   peano1 7742
            2.4.6  Finite induction (for finite ordinals)   find 7750
            2.4.7  Relations and functions (cont.)   dmexg 7757
            2.4.8  First and second members of an ordered pair   c1st 7836
            ​*2.4.9  The support of functions   csupp 7984
            ​*2.4.10  Special maps-to operations   opeliunxp2f 8033
            2.4.11  Function transposition   ctpos 8048
            2.4.12  Curry and uncurry   ccur 8088
            2.4.13  Undefined values   cund 8095
            2.4.14  Well-founded recursion   cfrecs 8103
            2.4.15  Well-ordered recursion   cwrecs 8134
            2.4.16  Functions on ordinals; strictly monotone ordinal functions   iunon 8177
            2.4.17  "Strong" transfinite recursion   crecs 8208
            2.4.18  Recursive definition generator   crdg 8247
            2.4.19  Finite recursion   frfnom 8273
            2.4.20  Ordinal arithmetic   c1o 8297
            2.4.21  Natural number arithmetic   nna0 8442
            2.4.22  Equivalence relations and classes   wer 8502
            2.4.23  The mapping operation   cmap 8622
            2.4.24  Infinite Cartesian products   cixp 8692
            2.4.25  Equinumerosity   cen 8737
            2.4.26  Schroeder-Bernstein Theorem   sbthlem1 8877
            2.4.27  Equinumerosity (cont.)   xpf1o 8933
            2.4.28  Finite sets   dif1enlem 8950
            2.4.29  Pigeonhole Principle   phplem1 8997
            2.4.30  Finite sets (cont.)   onomeneq 9018
            2.4.31  Finitely supported functions   cfsupp 9135
            2.4.32  Finite intersections   cfi 9176
            2.4.33  Hall's marriage theorem   marypha1lem 9199
            2.4.34  Supremum and infimum   csup 9206
            2.4.35  Ordinal isomorphism, Hartogs's theorem   coi 9275
            2.4.36  Hartogs function   char 9322
            2.4.37  Weak dominance   cwdom 9330
      ​2.5  ZF Set Theory - add the Axiom of Regularity
            2.5.1  Introduce the Axiom of Regularity   ax-reg 9358
            2.5.2  Axiom of Infinity equivalents   inf0 9386
      ​2.6  ZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-inf 9403
            2.6.2  Existence of omega (the set of natural numbers)   omex 9408
            2.6.3  Cantor normal form   ccnf 9426
            2.6.4  Transitive closure of a relation   cttrcl 9472
            2.6.5  Transitive closure   trcl 9493
            2.6.6  Well-Founded Induction   frmin 9514
            2.6.7  Well-Founded Recursion   frr3g 9521
            2.6.8  Rank   cr1 9527
            2.6.9  Scott's trick; collection principle; Hilbert's epsilon   scottex 9650
            2.6.10  Disjoint union   cdju 9663
            2.6.11  Cardinal numbers   ccrd 9700
            2.6.12  Axiom of Choice equivalents   wac 9878
            ​*2.6.13  Cardinal number arithmetic   undjudom 9930
            2.6.14  The Ackermann bijection   ackbij2lem1 9982
            2.6.15  Cofinality (without Axiom of Choice)   cflem 10009
            2.6.16  Eight inequivalent definitions of finite set   sornom 10040
            2.6.17  Hereditarily size-limited sets without Choice   itunifval 10179
​​*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 10198
            3.1.2  Introduce the Axiom of Dependent Choice   ax-dc 10209
      ​3.2  ZFC Set Theory - add the Axiom of Choice
            3.2.1  Introduce the Axiom of Choice   ax-ac 10222
            3.2.2  AC equivalents: well-ordering, Zorn's lemma   numthcor 10257
            3.2.3  Cardinal number theorems using Axiom of Choice   cardval 10309
            3.2.4  Cardinal number arithmetic using Axiom of Choice   iunctb 10337
            3.2.5  Cofinality using the Axiom of Choice   alephreg 10345
      ​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 10383
            3.4.2  Derivation of the Axiom of Choice   gchaclem 10441
​​*PART 4  TG (TARSKI-GROTHENDIECK) SET THEORY
      ​4.1  Inaccessibles
            4.1.1  Weakly and strongly inaccessible cardinals   cwina 10445
            4.1.2  Weak universes   cwun 10463
            4.1.3  Tarski classes   ctsk 10511
            4.1.4  Grothendieck universes   cgru 10553
      ​4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
            4.2.1  Introduce the Tarski-Grothendieck Axiom   ax-groth 10586
            4.2.2  Derive the Power Set, Infinity and Choice Axioms   grothpw 10589
            4.2.3  Tarski map function   ctskm 10600
​​*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 10607
            5.1.2  Final derivation of real and complex number postulates   axaddf 10908
            5.1.3  Real and complex number postulates restated as axioms   ax-cnex 10934
      ​5.2  Derive the basic properties from the field axioms
            5.2.1  Some deductions from the field axioms for complex numbers   cnex 10959
            5.2.2  Infinity and the extended real number system   cpnf 11013
            5.2.3  Restate the ordering postulates with extended real "less than"   axlttri 11053
            5.2.4  Ordering on reals   lttr 11058
            5.2.5  Initial properties of the complex numbers   mul12 11147
      ​5.3  Real and complex numbers - basic operations
            5.3.1  Addition   add12 11199
            5.3.2  Subtraction   cmin 11212
            5.3.3  Multiplication   kcnktkm1cn 11413
            5.3.4  Ordering on reals (cont.)   gt0ne0 11447
            5.3.5  Reciprocals   ixi 11611
            5.3.6  Division   cdiv 11639
            5.3.7  Ordering on reals (cont.)   elimgt0 11820
            5.3.8  Completeness Axiom and Suprema   fimaxre 11926
            5.3.9  Imaginary and complex number properties   inelr 11970
            5.3.10  Function operation analogue theorems   ofsubeq0 11977
      ​5.4  Integer sets
            5.4.1  Positive integers (as a subset of complex numbers)   cn 11980
            5.4.2  Principle of mathematical induction   nnind 11998
            ​*5.4.3  Decimal representation of numbers   c2 12035
            ​*5.4.4  Some properties of specific numbers   neg1cn 12094
            5.4.5  Simple number properties   halfcl 12205
            5.4.6  The Archimedean property   nnunb 12236
            5.4.7  Nonnegative integers (as a subset of complex numbers)   cn0 12240
            ​*5.4.8  Extended nonnegative integers   cxnn0 12312
            5.4.9  Integers (as a subset of complex numbers)   cz 12326
            5.4.10  Decimal arithmetic   cdc 12444
            5.4.11  Upper sets of integers   cuz 12589
            5.4.12  Well-ordering principle for bounded-below sets of integers   uzwo3 12690
            5.4.13  Rational numbers (as a subset of complex numbers)   cq 12695
            5.4.14  Existence of the set of complex numbers   rpnnen1lem2 12724
      ​5.5  Order sets
            5.5.1  Positive reals (as a subset of complex numbers)   crp 12737
            5.5.2  Infinity and the extended real number system (cont.)   cxne 12852
            5.5.3  Supremum and infimum on the extended reals   xrsupexmnf 13046
            5.5.4  Real number intervals   cioo 13086
            5.5.5  Finite intervals of integers   cfz 13246
            ​*5.5.6  Finite intervals of nonnegative integers   elfz2nn0 13354
            5.5.7  Half-open integer ranges   cfzo 13389
      ​5.6  Elementary integer functions
            5.6.1  The floor and ceiling functions   cfl 13517
            5.6.2  The modulo (remainder) operation   cmo 13596
            5.6.3  Miscellaneous theorems about integers   om2uz0i 13674
            5.6.4  Strong induction over upper sets of integers   uzsinds 13714
            5.6.5  Finitely supported functions over the nonnegative integers   fsuppmapnn0fiublem 13717
            5.6.6  The infinite sequence builder "seq" - extension   cseq 13728
            5.6.7  Integer powers   cexp 13789
            5.6.8  Ordered pair theorem for nonnegative integers   nn0le2msqi 13988
            5.6.9  Factorial function   cfa 13994
            5.6.10  The binomial coefficient operation   cbc 14023
            5.6.11  The ` # ` (set size) function   chash 14051
                  5.6.11.1  Proper unordered pairs and triples (sets of size 2 and 3)   hashprlei 14189
                  5.6.11.2  Functions with a domain containing at least two different elements   fundmge2nop0 14213
                  5.6.11.3  Finite induction on the size of the first component of a binary relation   hashdifsnp1 14217
      ​​*5.7  Words over a set
            5.7.1  Definitions and basic theorems   cword 14224
            5.7.2  Last symbol of a word   clsw 14272
            5.7.3  Concatenations of words   cconcat 14280
            5.7.4  Singleton words   cs1 14307
            5.7.5  Concatenations with singleton words   ccatws1cl 14328
            5.7.6  Subwords/substrings   csubstr 14360
            5.7.7  Prefixes of a word   cpfx 14390
            5.7.8  Subwords of subwords   swrdswrdlem 14424
            5.7.9  Subwords and concatenations   pfxcctswrd 14430
            5.7.10  Subwords of concatenations   swrdccatfn 14444
            5.7.11  Splicing words (substring replacement)   csplice 14469
            5.7.12  Reversing words   creverse 14478
            5.7.13  Repeated symbol words   creps 14488
            ​*5.7.14  Cyclical shifts of words   ccsh 14508
            5.7.15  Mapping words by a function   wrdco 14551
            5.7.16  Longer string literals   cs2 14561
      ​​*5.8  Reflexive and transitive closures of relations
            5.8.1  The reflexive and transitive properties of relations   coss12d 14690
            5.8.2  Basic properties of closures   cleq1lem 14700
            5.8.3  Definitions and basic properties of transitive closures   ctcl 14703
            5.8.4  Exponentiation of relations   crelexp 14737
            5.8.5  Reflexive-transitive closure as an indexed union   crtrcl 14773
            ​*5.8.6  Principle of transitive induction.   relexpindlem 14781
      ​5.9  Elementary real and complex functions
            5.9.1  The "shift" operation   cshi 14784
            5.9.2  Signum (sgn or sign) function   csgn 14804
            5.9.3  Real and imaginary parts; conjugate   ccj 14814
            5.9.4  Square root; absolute value   csqrt 14951
      ​5.10  Elementary limits and convergence
            5.10.1  Superior limit (lim sup)   clsp 15186
            5.10.2  Limits   cli 15200
            5.10.3  Finite and infinite sums   csu 15404
            5.10.4  The binomial theorem   binomlem 15548
            5.10.5  The inclusion/exclusion principle   incexclem 15555
            5.10.6  Infinite sums (cont.)   isumshft 15558
            5.10.7  Miscellaneous converging and diverging sequences   divrcnv 15571
            5.10.8  Arithmetic series   arisum 15579
            5.10.9  Geometric series   expcnv 15583
            5.10.10  Ratio test for infinite series convergence   cvgrat 15602
            5.10.11  Mertens' theorem   mertenslem1 15603
            5.10.12  Finite and infinite products   prodf 15606
                  5.10.12.1  Product sequences   prodf 15606
                  5.10.12.2  Non-trivial convergence   ntrivcvg 15616
                  5.10.12.3  Complex products   cprod 15622
                  5.10.12.4  Finite products   fprod 15658
                  5.10.12.5  Infinite products   iprodclim 15715
            5.10.13  Falling and Rising Factorial   cfallfac 15721
            5.10.14  Bernoulli polynomials and sums of k-th powers   cbp 15763
      ​5.11  Elementary trigonometry
            5.11.1  The exponential, sine, and cosine functions   ce 15778
                  5.11.1.1  The circle constant (tau = 2 pi)   ctau 15918
            5.11.2  _e is irrational   eirrlem 15920
      ​5.12  Cardinality of real and complex number subsets
            5.12.1  Countability of integers and rationals   xpnnen 15927
            5.12.2  The reals are uncountable   rpnnen2lem1 15930
​​*PART 6  ELEMENTARY NUMBER THEORY
      ​6.1  Elementary properties of divisibility
            6.1.1  Irrationality of square root of 2   sqrt2irrlem 15964
            6.1.2  Some Number sets are chains of proper subsets   nthruc 15968
            6.1.3  The divides relation   cdvds 15970
            ​*6.1.4  Even and odd numbers   evenelz 16052
            6.1.5  The division algorithm   divalglem0 16109
            6.1.6  Bit sequences   cbits 16133
            6.1.7  The greatest common divisor operator   cgcd 16208
            6.1.8  Bézout's identity   bezoutlem1 16254
            6.1.9  Algorithms   nn0seqcvgd 16282
            6.1.10  Euclid's Algorithm   eucalgval2 16293
            ​*6.1.11  The least common multiple   clcm 16300
            ​*6.1.12  Coprimality and Euclid's lemma   coprmgcdb 16361
            6.1.13  Cancellability of congruences   congr 16376
      ​6.2  Elementary prime number theory
            ​*6.2.1  Elementary properties   cprime 16383
            ​*6.2.2  Coprimality and Euclid's lemma (cont.)   coprm 16423
            6.2.3  Properties of the canonical representation of a rational   cnumer 16444
            6.2.4  Euler's theorem   codz 16471
            6.2.5  Arithmetic modulo a prime number   modprm1div 16505
            6.2.6  Pythagorean Triples   coprimeprodsq 16516
            6.2.7  The prime count function   cpc 16544
            6.2.8  Pocklington's theorem   prmpwdvds 16612
            6.2.9  Infinite primes theorem   unbenlem 16616
            6.2.10  Sum of prime reciprocals   prmreclem1 16624
            6.2.11  Fundamental theorem of arithmetic   1arithlem1 16631
            6.2.12  Lagrange's four-square theorem   cgz 16637
            6.2.13  Van der Waerden's theorem   cvdwa 16673
            6.2.14  Ramsey's theorem   cram 16707
            ​*6.2.15  Primorial function   cprmo 16739
            ​*6.2.16  Prime gaps   prmgaplem1 16757
            6.2.17  Decimal arithmetic (cont.)   dec2dvds 16771
            6.2.18  Cyclical shifts of words (cont.)   cshwsidrepsw 16802
            6.2.19  Specific prime numbers   prmlem0 16814
            6.2.20  Very large primes   1259lem1 16839
​PART 7  BASIC STRUCTURES
      ​7.1  Extensible structures
            ​*7.1.1  Basic definitions   cstr 16854
                  7.1.1.1  Extensible structures as structures with components   cstr 16854
                  7.1.1.2  Substitution of components   csts 16871
                  7.1.1.3  Slots   cslot 16889
                  ​*7.1.1.4  Structure component indices   cnx 16901
                  7.1.1.5  Base sets   cbs 16919
                  7.1.1.6  Base set restrictions   cress 16948
            7.1.2  Slot definitions   cplusg 16969
            7.1.3  Definition of the structure product   crest 17138
            7.1.4  Definition of the structure quotient   cordt 17217
      ​7.2  Moore spaces
            7.2.1  Moore closures   mrcflem 17322
            7.2.2  Independent sets in a Moore system   mrisval 17346
            7.2.3  Algebraic closure systems   isacs 17367
​PART 8  BASIC CATEGORY THEORY
      ​8.1  Categories
            8.1.1  Categories   ccat 17380
            8.1.2  Opposite category   coppc 17427
            8.1.3  Monomorphisms and epimorphisms   cmon 17447
            8.1.4  Sections, inverses, isomorphisms   csect 17463
            ​*8.1.5  Isomorphic objects   ccic 17514
            8.1.6  Subcategories   cssc 17526
            8.1.7  Functors   cfunc 17576
            8.1.8  Full & faithful functors   cful 17625
            8.1.9  Natural transformations and the functor category   cnat 17664
            8.1.10  Initial, terminal and zero objects of a category   cinito 17703
      ​8.2  Arrows (disjointified hom-sets)
            8.2.1  Identity and composition for arrows   cida 17775
      ​8.3  Examples of categories
            8.3.1  The category of sets   csetc 17797
            8.3.2  The category of categories   ccatc 17820
            ​*8.3.3  The category of extensible structures   fncnvimaeqv 17843
      ​8.4  Categorical constructions
            8.4.1  Product of categories   cxpc 17892
            8.4.2  Functor evaluation   cevlf 17934
            8.4.3  Hom functor   chof 17973
​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 18156
            9.5.2  Complete lattices   ccla 18223
            9.5.3  Distributive lattices   cdlat 18245
            9.5.4  Subset order structures   cipo 18252
      ​9.6  Posets, directed sets, and lattices as relations
            ​*9.6.1  Posets and lattices as relations   cps 18289
            9.6.2  Directed sets, nets   cdir 18319
​PART 10  BASIC ALGEBRAIC STRUCTURES
      ​10.1  Monoids
            ​*10.1.1  Magmas   cplusf 18330
            ​*10.1.2  Identity elements   mgmidmo 18351
            ​*10.1.3  Iterated sums in a magma   gsumvalx 18367
            ​*10.1.4  Semigroups   csgrp 18381
            ​*10.1.5  Definition and basic properties of monoids   cmnd 18392
            10.1.6  Monoid homomorphisms and submonoids   cmhm 18435
            ​*10.1.7  Iterated sums in a monoid   gsumvallem2 18479
            10.1.8  Free monoids   cfrmd 18493
                  ​*10.1.8.1  Monoid of endofunctions   cefmnd 18514
            10.1.9  Examples and counterexamples for magmas, semigroups and monoids   mgm2nsgrplem1 18564
      ​10.2  Groups
            10.2.1  Definition and basic properties   cgrp 18584
            ​*10.2.2  Group multiple operation   cmg 18707
            10.2.3  Subgroups and Quotient groups   csubg 18756
            ​*10.2.4  Cyclic monoids and groups   cycsubmel 18826
            10.2.5  Elementary theory of group homomorphisms   cghm 18838
            10.2.6  Isomorphisms of groups   cgim 18880
            10.2.7  Group actions   cga 18902
            10.2.8  Centralizers and centers   ccntz 18928
            10.2.9  The opposite group   coppg 18956
            10.2.10  Symmetric groups   csymg 18981
                  ​*10.2.10.1  Definition and basic properties   csymg 18981
                  10.2.10.2  Cayley's theorem   cayleylem1 19027
                  10.2.10.3  Permutations fixing one element   symgfix2 19031
                  ​*10.2.10.4  Transpositions in the symmetric group   cpmtr 19056
                  10.2.10.5  The sign of a permutation   cpsgn 19104
            10.2.11  p-Groups and Sylow groups; Sylow's theorems   cod 19139
            10.2.12  Direct products   clsm 19246
                  10.2.12.1  Direct products (extension)   smndlsmidm 19268
            10.2.13  Free groups   cefg 19319
            10.2.14  Abelian groups   ccmn 19393
                  10.2.14.1  Definition and basic properties   ccmn 19393
                  10.2.14.2  Cyclic groups   ccyg 19484
                  10.2.14.3  Group sum operation   gsumval3a 19511
                  10.2.14.4  Group sums over (ranges of) integers   fsfnn0gsumfsffz 19591
                  10.2.14.5  Internal direct products   cdprd 19603
                  10.2.14.6  The Fundamental Theorem of Abelian Groups   ablfacrplem 19675
            10.2.15  Simple groups   csimpg 19700
                  10.2.15.1  Definition and basic properties   csimpg 19700
                  10.2.15.2  Classification of abelian simple groups   ablsimpnosubgd 19714
      ​10.3  Rings
            10.3.1  Multiplicative Group   cmgp 19727
            10.3.2  Ring unit   cur 19744
                  10.3.2.1  Semirings   csrg 19748
                  ​*10.3.2.2  The binomial theorem for semirings   srgbinomlem1 19783
            10.3.3  Definition and basic properties of unital rings   crg 19790
            10.3.4  Opposite ring   coppr 19868
            10.3.5  Divisibility   cdsr 19887
            10.3.6  Ring primes   crpm 19961
            10.3.7  Ring homomorphisms   crh 19963
      ​10.4  Division rings and fields
            10.4.1  Definition and basic properties   cdr 19998
            10.4.2  Subrings of a ring   csubrg 20027
                  10.4.2.1  Sub-division rings   csdrg 20068
            10.4.3  Absolute value (abstract algebra)   cabv 20083
            10.4.4  Star rings   cstf 20110
      ​10.5  Left modules
            10.5.1  Definition and basic properties   clmod 20130
            10.5.2  Subspaces and spans in a left module   clss 20200
            10.5.3  Homomorphisms and isomorphisms of left modules   clmhm 20288
            10.5.4  Subspace sum; bases for a left module   clbs 20343
      ​10.6  Vector spaces
            10.6.1  Definition and basic properties   clvec 20371
      ​10.7  Ideals
            10.7.1  The subring algebra; ideals   csra 20437
            10.7.2  Two-sided ideals and quotient rings   c2idl 20509
            10.7.3  Principal ideal rings. Divisibility in the integers   clpidl 20519
            10.7.4  Nonzero rings and zero rings   cnzr 20535
            10.7.5  Left regular elements. More kinds of rings   crlreg 20557
      ​10.8  The complex numbers as an algebraic extensible structure
            10.8.1  Definition and basic properties   cpsmet 20588
            ​*10.8.2  Ring of integers   czring 20677
            10.8.3  Algebraic constructions based on the complex numbers   czrh 20708
            10.8.4  Signs as subgroup of the complex numbers   cnmsgnsubg 20789
            10.8.5  Embedding of permutation signs into a ring   zrhpsgnmhm 20796
            10.8.6  The ordered field of real numbers   crefld 20816
      ​10.9  Generalized pre-Hilbert and Hilbert spaces
            10.9.1  Definition and basic properties   cphl 20836
            10.9.2  Orthocomplements and closed subspaces   cocv 20872
            10.9.3  Orthogonal projection and orthonormal bases   cpj 20914
​​*PART 11  BASIC LINEAR ALGEBRA
      ​11.1  Vectors and free modules
            *11.1.1  Direct sum of left modules   cdsmm 20945
            ​*11.1.2  Free modules   cfrlm 20960
            ​*11.1.3  Standard basis (unit vectors)   cuvc 20996
            ​*11.1.4  Independent sets and families   clindf 21018
            11.1.5  Characterization of free modules   lmimlbs 21050
      ​11.2  Associative algebras
            11.2.1  Definition and basic properties   casa 21064
      ​11.3  Abstract multivariate polynomials
            11.3.1  Definition and basic properties   cmps 21114
            11.3.2  Polynomial evaluation   ces 21287
            11.3.3  Additional definitions for (multivariate) polynomials   cslv 21325
            ​*11.3.4  Univariate polynomials   cps1 21353
            11.3.5  Univariate polynomial evaluation   ces1 21486
      ​​*11.4  Matrices
            *11.4.1  The matrix multiplication   cmmul 21539
            ​*11.4.2  Square matrices   cmat 21561
            ​*11.4.3  The matrix algebra   matmulr 21594
            ​*11.4.4  Matrices of dimension 0 and 1   mat0dimbas0 21622
            ​*11.4.5  The subalgebras of diagonal and scalar matrices   cdmat 21644
            ​*11.4.6  Multiplication of a matrix with a "column vector"   cmvmul 21696
            11.4.7  Replacement functions for a square matrix   cmarrep 21712
            11.4.8  Submatrices   csubma 21732
      ​11.5  The determinant
            11.5.1  Definition and basic properties   cmdat 21740
            11.5.2  Determinants of 2 x 2 -matrices   m2detleiblem1 21780
            11.5.3  The matrix adjugate/adjunct   cmadu 21788
            ​*11.5.4  Laplace expansion of determinants (special case)   symgmatr01lem 21809
            11.5.5  Inverse matrix   invrvald 21832
            ​*11.5.6  Cramer's rule   slesolvec 21835
      ​​*11.6  Polynomial matrices
            11.6.1  Basic properties   pmatring 21848
            ​*11.6.2  Constant polynomial matrices   ccpmat 21859
            ​*11.6.3  Collecting coefficients of polynomial matrices   cdecpmat 21918
            ​*11.6.4  Ring isomorphism between polynomial matrices and polynomials over matrices   cpm2mp 21948
      ​​*11.7  The characteristic polynomial
            *11.7.1  Definition and basic properties   cchpmat 21982
            ​*11.7.2  The characteristic factor function G   fvmptnn04if 22005
            ​*11.7.3  The Cayley-Hamilton theorem   cpmadurid 22023
​PART 12  BASIC TOPOLOGY
      ​12.1  Topology
            ​*12.1.1  Topological spaces   ctop 22049
                  12.1.1.1  Topologies   ctop 22049
                  12.1.1.2  Topologies on sets   ctopon 22066
                  12.1.1.3  Topological spaces   ctps 22088
            12.1.2  Topological bases   ctb 22102
            12.1.3  Examples of topologies   distop 22152
            12.1.4  Closure and interior   ccld 22174
            12.1.5  Neighborhoods   cnei 22255
            12.1.6  Limit points and perfect sets   clp 22292
            12.1.7  Subspace topologies   restrcl 22315
            12.1.8  Order topology   ordtbaslem 22346
            12.1.9  Limits and continuity in topological spaces   ccn 22382
            12.1.10  Separated spaces: T0, T1, T2 (Hausdorff) ...   ct0 22464
            12.1.11  Compactness   ccmp 22544
            12.1.12  Bolzano-Weierstrass theorem   bwth 22568
            12.1.13  Connectedness   cconn 22569
            12.1.14  First- and second-countability   c1stc 22595
            12.1.15  Local topological properties   clly 22622
            12.1.16  Refinements   cref 22660
            12.1.17  Compactly generated spaces   ckgen 22691
            12.1.18  Product topologies   ctx 22718
            12.1.19  Continuous function-builders   cnmptid 22819
            12.1.20  Quotient maps and quotient topology   ckq 22851
            12.1.21  Homeomorphisms   chmeo 22911
      ​12.2  Filters and filter bases
            12.2.1  Filter bases   elmptrab 22985
            12.2.2  Filters   cfil 23003
            12.2.3  Ultrafilters   cufil 23057
            12.2.4  Filter limits   cfm 23091
            12.2.5  Extension by continuity   ccnext 23217
            12.2.6  Topological groups   ctmd 23228
            12.2.7  Infinite group sum on topological groups   ctsu 23284
            12.2.8  Topological rings, fields, vector spaces   ctrg 23314
      ​12.3  Uniform Structures and Spaces
            12.3.1  Uniform structures   cust 23358
            12.3.2  The topology induced by an uniform structure   cutop 23389
            12.3.3  Uniform Spaces   cuss 23412
            12.3.4  Uniform continuity   cucn 23434
            12.3.5  Cauchy filters in uniform spaces   ccfilu 23445
            12.3.6  Complete uniform spaces   ccusp 23456
      ​12.4  Metric spaces
            12.4.1  Pseudometric spaces   ispsmet 23464
            12.4.2  Basic metric space properties   cxms 23477
            12.4.3  Metric space balls   blfvalps 23543
            12.4.4  Open sets of a metric space   mopnval 23598
            12.4.5  Continuity in metric spaces   metcnp3 23703
            12.4.6  The uniform structure generated by a metric   metuval 23712
            12.4.7  Examples of metric spaces   dscmet 23735
            ​*12.4.8  Normed algebraic structures   cnm 23739
            12.4.9  Normed space homomorphisms (bounded linear operators)   cnmo 23876
            12.4.10  Topology on the reals   qtopbaslem 23929
            12.4.11  Topological definitions using the reals   cii 24045
            12.4.12  Path homotopy   chtpy 24137
            12.4.13  The fundamental group   cpco 24170
      ​12.5  Metric subcomplex vector spaces
            12.5.1  Subcomplex modules   cclm 24232
            ​*12.5.2  Subcomplex vector spaces   ccvs 24293
            ​*12.5.3  Normed subcomplex vector spaces   isncvsngp 24320
            12.5.4  Subcomplex pre-Hilbert spaces   ccph 24337
            12.5.5  Convergence and completeness   ccfil 24423
            12.5.6  Baire's Category Theorem   bcthlem1 24495
            12.5.7  Banach spaces and subcomplex Hilbert spaces   ccms 24503
                  12.5.7.1  The complete ordered field of the real numbers   retopn 24550
            12.5.8  Euclidean spaces   crrx 24554
            12.5.9  Minimizing Vector Theorem   minveclem1 24595
            12.5.10  Projection Theorem   pjthlem1 24608
​PART 13  BASIC REAL AND COMPLEX ANALYSIS
      ​13.1  Continuity
            13.1.1  Intermediate value theorem   pmltpclem1 24619
      ​13.2  Integrals
            13.2.1  Lebesgue measure   covol 24633
            13.2.2  Lebesgue integration   cmbf 24785
                  13.2.2.1  Lesbesgue integral   cmbf 24785
                  13.2.2.2  Lesbesgue directed integral   cdit 25017
      ​13.3  Derivatives
            13.3.1  Real and complex differentiation   climc 25033
                  13.3.1.1  Derivatives of functions of one complex or real variable   climc 25033
                  13.3.1.2  Results on real differentiation   dvferm1lem 25155
​PART 14  BASIC REAL AND COMPLEX FUNCTIONS
      ​14.1  Polynomials
            14.1.1  Polynomial degrees   cmdg 25222
            14.1.2  The division algorithm for univariate polynomials   cmn1 25297
            14.1.3  Elementary properties of complex polynomials   cply 25352
            14.1.4  The division algorithm for polynomials   cquot 25457
            14.1.5  Algebraic numbers   caa 25481
            14.1.6  Liouville's approximation theorem   aalioulem1 25499
      ​14.2  Sequences and series
            14.2.1  Taylor polynomials and Taylor's theorem   ctayl 25519
            14.2.2  Uniform convergence   culm 25542
            14.2.3  Power series   pserval 25576
      ​14.3  Basic trigonometry
            14.3.1  The exponential, sine, and cosine functions (cont.)   efcn 25609
            14.3.2  Properties of pi = 3.14159...   pilem1 25617
            14.3.3  Mapping of the exponential function   efgh 25704
            14.3.4  The natural logarithm on complex numbers   clog 25717
            ​*14.3.5  Logarithms to an arbitrary base   clogb 25921
            14.3.6  Theorems of Pythagoras, isosceles triangles, and intersecting chords   angval 25958
            14.3.7  Solutions of quadratic, cubic, and quartic equations   quad2 25996
            14.3.8  Inverse trigonometric functions   casin 26019
            14.3.9  The Birthday Problem   log2ublem1 26103
            14.3.10  Areas in R^2   carea 26112
            14.3.11  More miscellaneous converging sequences   rlimcnp 26122
            14.3.12  Inequality of arithmetic and geometric means   cvxcl 26141
            14.3.13  Euler-Mascheroni constant   cem 26148
            14.3.14  Zeta function   czeta 26169
            14.3.15  Gamma function   clgam 26172
      ​14.4  Basic number theory
            14.4.1  Wilson's theorem   wilthlem1 26224
            14.4.2  The Fundamental Theorem of Algebra   ftalem1 26229
            14.4.3  The Basel problem (ζ(2) = π²/6)   basellem1 26237
            14.4.4  Number-theoretical functions   ccht 26247
            14.4.5  Perfect Number Theorem   mersenne 26382
            14.4.6  Characters of Z/nZ   cdchr 26387
            14.4.7  Bertrand's postulate   bcctr 26430
            ​*14.4.8  Quadratic residues and the Legendre symbol   clgs 26449
            ​*14.4.9  Gauss' Lemma   gausslemma2dlem0a 26511
            14.4.10  Quadratic reciprocity   lgseisenlem1 26530
            14.4.11  All primes 4n+1 are the sum of two squares   2sqlem1 26572
            14.4.12  Chebyshev's Weak Prime Number Theorem, Dirichlet's Theorem   chebbnd1lem1 26624
            14.4.13  The Prime Number Theorem   mudivsum 26685
            14.4.14  Ostrowski's theorem   abvcxp 26770
​​*PART 15  ELEMENTARY GEOMETRY
      ​15.1  Definition and Tarski's Axioms of Geometry
            15.1.1  Justification for the congruence notation   tgjustf 26841
      ​15.2  Tarskian Geometry
            15.2.1  Congruence   tgcgrcomimp 26845
            15.2.2  Betweenness   tgbtwntriv2 26855
            15.2.3  Dimension   tglowdim1 26868
            15.2.4  Betweenness and Congruence   tgifscgr 26876
            15.2.5  Congruence of a series of points   ccgrg 26878
            15.2.6  Motions   cismt 26900
            15.2.7  Colinearity   tglng 26914
            15.2.8  Connectivity of betweenness   tgbtwnconn1lem1 26940
            15.2.9  Less-than relation in geometric congruences   cleg 26950
            15.2.10  Rays   chlg 26968
            15.2.11  Lines   btwnlng1 26987
            15.2.12  Point inversions   cmir 27020
            15.2.13  Right angles   crag 27061
            15.2.14  Half-planes   islnopp 27107
            15.2.15  Midpoints and Line Mirroring   cmid 27140
            15.2.16  Congruence of angles   ccgra 27175
            15.2.17  Angle Comparisons   cinag 27203
            15.2.18  Congruence Theorems   tgsas1 27222
            15.2.19  Equilateral triangles   ceqlg 27233
      ​15.3  Properties of geometries
            15.3.1  Isomorphisms between geometries   f1otrgds 27237
      ​15.4  Geometry in Hilbert spaces
            15.4.1  Geometry in the complex plane   cchhllem 27261
            15.4.2  Geometry in Euclidean spaces   cee 27263
                  15.4.2.1  Definition of the Euclidean space   cee 27263
                  15.4.2.2  Tarski's axioms for geometry for the Euclidean space   axdimuniq 27288
                  15.4.2.3  EE^n fulfills Tarski's Axioms   ceeng 27352
​​*PART 16  GRAPH THEORY
      ​*16.1  Vertices and edges
            16.1.1  The edge function extractor for extensible structures   cedgf 27363
            ​*16.1.2  Vertices and indexed edges   cvtx 27373
                  16.1.2.1  Definitions and basic properties   cvtx 27373
                  16.1.2.2  The vertices and edges of a graph represented as ordered pair   opvtxval 27380
                  16.1.2.3  The vertices and edges of a graph represented as extensible structure   funvtxdmge2val 27388
                  16.1.2.4  Representations of graphs without edges   snstrvtxval 27414
                  16.1.2.5  Degenerated cases of representations of graphs   vtxval0 27416
            16.1.3  Edges as range of the edge function   cedg 27424
      ​​*16.2  Undirected graphs
            16.2.1  Undirected hypergraphs   cuhgr 27433
            16.2.2  Undirected pseudographs and multigraphs   cupgr 27457
            ​*16.2.3  Loop-free graphs   umgrislfupgrlem 27499
            16.2.4  Edges as subsets of vertices of graphs   uhgredgiedgb 27503
            ​*16.2.5  Undirected simple graphs   cuspgr 27525
            16.2.6  Examples for graphs   usgr0e 27610
            16.2.7  Subgraphs   csubgr 27641
            16.2.8  Finite undirected simple graphs   cfusgr 27690
            16.2.9  Neighbors, complete graphs and universal vertices   cnbgr 27706
                  16.2.9.1  Neighbors   cnbgr 27706
                  16.2.9.2  Universal vertices   cuvtx 27759
                  16.2.9.3  Complete graphs   ccplgr 27783
            16.2.10  Vertex degree   cvtxdg 27839
            ​*16.2.11  Regular graphs   crgr 27929
      ​​*16.3  Walks, paths and cycles
            *16.3.1  Walks   cewlks 27969
            16.3.2  Walks for loop-free graphs   lfgrwlkprop 28062
            16.3.3  Trails   ctrls 28065
            16.3.4  Paths and simple paths   cpths 28087
            16.3.5  Closed walks   cclwlks 28145
            16.3.6  Circuits and cycles   ccrcts 28159
            ​*16.3.7  Walks as words   cwwlks 28197
            16.3.8  Walks/paths of length 2 (as length 3 strings)   2wlkdlem1 28297
            16.3.9  Walks in regular graphs   rusgrnumwwlkl1 28340
            ​*16.3.10  Closed walks as words   cclwwlk 28352
                  16.3.10.1  Closed walks as words   cclwwlk 28352
                  16.3.10.2  Closed walks of a fixed length as words   cclwwlkn 28395
                  16.3.10.3  Closed walks on a vertex of a fixed length as words   cclwwlknon 28458
            16.3.11  Examples for walks, trails and paths   0ewlk 28485
            16.3.12  Connected graphs   cconngr 28557
      ​16.4  Eulerian paths and the Konigsberg Bridge problem
            ​*16.4.1  Eulerian paths   ceupth 28568
            ​*16.4.2  The Königsberg Bridge problem   konigsbergvtx 28617
      ​16.5  The Friendship Theorem
            16.5.1  Friendship graphs - basics   cfrgr 28629
            16.5.2  The friendship theorem for small graphs   frgr1v 28642
            16.5.3  Theorems according to Mertzios and Unger   2pthfrgrrn 28653
            ​*16.5.4  Huneke's Proof of the Friendship Theorem   frgrncvvdeqlem1 28670
​PART 17  GUIDES AND MISCELLANEA
      ​17.1  Guides (conventions, explanations, and examples)
            ​*17.1.1  Conventions   conventions 28771
            17.1.2  Natural deduction   natded 28774
            ​*17.1.3  Natural deduction examples   ex-natded5.2 28775
            17.1.4  Definitional examples   ex-or 28792
            17.1.5  Other examples   aevdemo 28831
      ​17.2  Humor
            17.2.1  April Fool's theorem   avril1 28834
      ​17.3  (Future - to be reviewed and classified)
            17.3.1  Planar incidence geometry   cplig 28843
            ​*17.3.2  Aliases kept to prevent broken links   dummylink 28856
​​*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 28858
            18.1.2  Abelian groups   cablo 28913
      ​18.2  Complex vector spaces
            18.2.1  Definition and basic properties   cvc 28927
            18.2.2  Examples of complex vector spaces   cnaddabloOLD 28950
      ​18.3  Normed complex vector spaces
            18.3.1  Definition and basic properties   cnv 28953
            18.3.2  Examples of normed complex vector spaces   cnnv 29046
            18.3.3  Induced metric of a normed complex vector space   imsval 29054
            18.3.4  Inner product   cdip 29069
            18.3.5  Subspaces   css 29090
      ​18.4  Operators on complex vector spaces
            18.4.1  Definitions and basic properties   clno 29109
      ​18.5  Inner product (pre-Hilbert) spaces
            18.5.1  Definition and basic properties   ccphlo 29181
            18.5.2  Examples of pre-Hilbert spaces   cncph 29188
            18.5.3  Properties of pre-Hilbert spaces   isph 29191
      ​18.6  Complex Banach spaces
            18.6.1  Definition and basic properties   ccbn 29231
            18.6.2  Examples of complex Banach spaces   cnbn 29238
            18.6.3  Uniform Boundedness Theorem   ubthlem1 29239
            18.6.4  Minimizing Vector Theorem   minvecolem1 29243
      ​18.7  Complex Hilbert spaces
            18.7.1  Definition and basic properties   chlo 29254
            18.7.2  Standard axioms for a complex Hilbert space   hlex 29267
            18.7.3  Examples of complex Hilbert spaces   cnchl 29285
            18.7.4  Hellinger-Toeplitz Theorem   htthlem 29286
​​*PART 19  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      ​19.1  Axiomatization of complex pre-Hilbert spaces
            19.1.1  Basic Hilbert space definitions   chba 29288
            19.1.2  Preliminary ZFC lemmas   df-hnorm 29337
            ​*19.1.3  Derive the Hilbert space axioms from ZFC set theory   axhilex-zf 29350
            ​*19.1.4  Introduce the vector space axioms for a Hilbert space   ax-hilex 29368
            19.1.5  Vector operations   hvmulex 29380
            19.1.6  Inner product postulates for a Hilbert space   ax-hfi 29448
      ​19.2  Inner product and norms
            19.2.1  Inner product   his5 29455
            19.2.2  Norms   dfhnorm2 29491
            19.2.3  Relate Hilbert space to normed complex vector spaces   hilablo 29529
            19.2.4  Bunjakovaskij-Cauchy-Schwarz inequality   bcsiALT 29548
      ​19.3  Cauchy sequences and completeness axiom
            19.3.1  Cauchy sequences and limits   hcau 29553
            19.3.2  Derivation of the completeness axiom from ZF set theory   hilmet 29563
            19.3.3  Completeness postulate for a Hilbert space   ax-hcompl 29571
            19.3.4  Relate Hilbert space to ZFC pre-Hilbert and Hilbert spaces   hhcms 29572
      ​19.4  Subspaces and projections
            19.4.1  Subspaces   df-sh 29576
            19.4.2  Closed subspaces   df-ch 29590
            19.4.3  Orthocomplements   df-oc 29621
            19.4.4  Subspace sum, span, lattice join, lattice supremum   df-shs 29677
            19.4.5  Projection theorem   pjhthlem1 29760
            19.4.6  Projectors   df-pjh 29764
      ​19.5  Properties of Hilbert subspaces
            19.5.1  Orthomodular law   omlsilem 29771
            19.5.2  Projectors (cont.)   pjhtheu2 29785
            19.5.3  Hilbert lattice operations   sh0le 29809
            19.5.4  Span (cont.) and one-dimensional subspaces   spansn0 29910
            19.5.5  Commutes relation for Hilbert lattice elements   df-cm 29952
            19.5.6  Foulis-Holland theorem   fh1 29987
            19.5.7  Quantum Logic Explorer axioms   qlax1i 29996
            19.5.8  Orthogonal subspaces   chscllem1 30006
            19.5.9  Orthoarguesian laws 5OA and 3OA   5oalem1 30023
            19.5.10  Projectors (cont.)   pjorthi 30038
            19.5.11  Mayet's equation E_3   mayete3i 30097
      ​19.6  Operators on Hilbert spaces
            ​*19.6.1  Operator sum, difference, and scalar multiplication   df-hosum 30099
            19.6.2  Zero and identity operators   df-h0op 30117
            19.6.3  Operations on Hilbert space operators   hoaddcl 30127
            19.6.4  Linear, continuous, bounded, Hermitian, unitary operators and norms   df-nmop 30208
            19.6.5  Linear and continuous functionals and norms   df-nmfn 30214
            19.6.6  Adjoint   df-adjh 30218
            19.6.7  Dirac bra-ket notation   df-bra 30219
            19.6.8  Positive operators   df-leop 30221
            19.6.9  Eigenvectors, eigenvalues, spectrum   df-eigvec 30222
            19.6.10  Theorems about operators and functionals   nmopval 30225
            19.6.11  Riesz lemma   riesz3i 30431
            19.6.12  Adjoints (cont.)   cnlnadjlem1 30436
            19.6.13  Quantum computation error bound theorem   unierri 30473
            19.6.14  Dirac bra-ket notation (cont.)   branmfn 30474
            19.6.15  Positive operators (cont.)   leopg 30491
            19.6.16  Projectors as operators   pjhmopi 30515
      ​19.7  States on a Hilbert lattice and Godowski's equation
            19.7.1  States on a Hilbert lattice   df-st 30580
            19.7.2  Godowski's equation   golem1 30640
      ​19.8  Cover relation, atoms, exchange axiom, and modular symmetry
            19.8.1  Covers relation; modular pairs   df-cv 30648
            19.8.2  Atoms   df-at 30707
            19.8.3  Superposition principle   superpos 30723
            19.8.4  Atoms, exchange and covering properties, atomicity   chcv1 30724
            19.8.5  Irreducibility   chirredlem1 30759
            19.8.6  Atoms (cont.)   atcvat3i 30765
            19.8.7  Modular symmetry   mdsymlem1 30772
​PART 20  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      ​20.1  Mathboxes for user contributions
            20.1.1  Mathbox guidelines   mathbox 30811
      ​20.2  Mathbox for Stefan Allan
      ​20.3  Mathbox for Thierry Arnoux
            20.3.1  Propositional Calculus - misc additions   bian1d 30816
            20.3.2  Predicate Calculus   sbc2iedf 30822
                  20.3.2.1  Predicate Calculus - misc additions   sbc2iedf 30822
                  20.3.2.2  Restricted quantification - misc additions   ralcom4f 30825
                  20.3.2.3  Equality   eqtrb 30830
                  20.3.2.4  Double restricted existential uniqueness quantification   opsbc2ie 30831
                  20.3.2.5  Double restricted existential uniqueness quantification syntax   w2reu 30833
                  20.3.2.6  Substitution (without distinct variables) - misc additions   sbceqbidf 30842
                  20.3.2.7  Existential "at most one" - misc additions   mo5f 30844
                  20.3.2.8  Existential uniqueness - misc additions   reuxfrdf 30846
                  20.3.2.9  Restricted "at most one" - misc additions   rmoxfrd 30848
            20.3.3  General Set Theory   dmrab 30851
                  20.3.3.1  Class abstractions (a.k.a. class builders)   dmrab 30851
                  20.3.3.2  Image Sets   abrexdomjm 30859
                  20.3.3.3  Set relations and operations - misc additions   elunsn 30865
                  20.3.3.4  Unordered pairs   eqsnd 30884
                  20.3.3.5  Conditional operator - misc additions   ifeqeqx 30892
                  20.3.3.6  Set union   uniinn0 30897
                  20.3.3.7  Indexed union - misc additions   cbviunf 30902
                  20.3.3.8  Indexed intersection - misc additions   iinabrex 30915
                  20.3.3.9  Disjointness - misc additions   disjnf 30916
            20.3.4  Relations and Functions   xpdisjres 30944
                  20.3.4.1  Relations - misc additions   xpdisjres 30944
                  20.3.4.2  Functions - misc additions   ac6sf2 30967
                  20.3.4.3  Operations - misc additions   mpomptxf 31023
                  20.3.4.4  Explicit Functions with one or two points as a domain   cosnopne 31034
                  20.3.4.5  Isomorphisms - misc. add.   gtiso 31040
                  20.3.4.6  Disjointness (additional proof requiring functions)   disjdsct 31042
                  20.3.4.7  First and second members of an ordered pair - misc additions   df1stres 31043
                  20.3.4.8  Supremum - misc additions   supssd 31051
                  20.3.4.9  Finite Sets   imafi2 31053
                  20.3.4.10  Countable Sets   snct 31055
            20.3.5  Real and Complex Numbers   creq0 31077
                  20.3.5.1  Complex operations - misc. additions   creq0 31077
                  20.3.5.2  Ordering on reals - misc additions   lt2addrd 31081
                  20.3.5.3  Extended reals - misc additions   xrlelttric 31082
                  20.3.5.4  Extended nonnegative integers - misc additions   xnn0gt0 31099
                  20.3.5.5  Real number intervals - misc additions   joiniooico 31102
                  20.3.5.6  Finite intervals of integers - misc additions   uzssico 31112
                  20.3.5.7  Half-open integer ranges - misc additions   iundisjfi 31124
                  20.3.5.8  The ` # ` (set size) function - misc additions   hashunif 31133
                  20.3.5.9  The greatest common divisor operator - misc. add   dvdszzq 31136
                  20.3.5.10  Integers   nnindf 31140
                  20.3.5.11  Decimal numbers   dfdec100 31151
            ​*20.3.6  Decimal expansion   cdp2 31152
                  ​*20.3.6.1  Decimal point   cdp 31169
                  20.3.6.2  Division in the extended real number system   cxdiv 31198
            20.3.7  Words over a set - misc additions   wrdfd 31217
                  20.3.7.1  Splicing words (substring replacement)   splfv3 31237
                  20.3.7.2  Cyclic shift of words   1cshid 31238
            20.3.8  Extensible Structures   ressplusf 31242
                  20.3.8.1  Structure restriction operator   ressplusf 31242
                  20.3.8.2  The opposite group   oppgle 31245
                  20.3.8.3  Posets   ressprs 31248
                  20.3.8.4  Complete lattices   clatp0cl 31261
                  20.3.8.5  Order Theory   cmnt 31263
                  20.3.8.6  Extended reals Structure - misc additions   ax-xrssca 31289
                  20.3.8.7  The extended nonnegative real numbers commutative monoid   xrge0base 31301
            20.3.9  Algebra   abliso 31312
                  20.3.9.1  Monoids Homomorphisms   abliso 31312
                  20.3.9.2  Finitely supported group sums - misc additions   gsumsubg 31313
                  20.3.9.3  Centralizers and centers - misc additions   cntzun 31327
                  20.3.9.4  Totally ordered monoids and groups   comnd 31330
                  20.3.9.5  The symmetric group   symgfcoeu 31358
                  20.3.9.6  Transpositions   pmtridf1o 31368
                  20.3.9.7  Permutation Signs   psgnid 31371
                  20.3.9.8  Permutation cycles   ctocyc 31380
                  20.3.9.9  The Alternating Group   evpmval 31419
                  20.3.9.10  Signum in an ordered monoid   csgns 31432
                  20.3.9.11  The Archimedean property for generic ordered algebraic structures   cinftm 31437
                  20.3.9.12  Semiring left modules   cslmd 31460
                  20.3.9.13  Simple groups   prmsimpcyc 31488
                  20.3.9.14  Rings - misc additions   rngurd 31489
                  20.3.9.15  Subfields   primefldchr 31500
                  20.3.9.16  Totally ordered rings and fields   corng 31501
                  20.3.9.17  Ring homomorphisms - misc additions   rhmdvdsr 31524
                  20.3.9.18  Scalar restriction operation   cresv 31530
                  20.3.9.19  The commutative ring of gaussian integers   gzcrng 31550
                  20.3.9.20  The archimedean ordered field of real numbers   reofld 31551
                  20.3.9.21  The quotient map and quotient modules   qusker 31556
                  20.3.9.22  The ring of integers modulo ` N `   znfermltl 31569
                  20.3.9.23  Independent sets and families   islinds5 31570
                  ​*20.3.9.24  Subgroup sum / Sumset / Minkowski sum   elgrplsmsn 31585
                  20.3.9.25  The quotient map   quslsm 31600
                  20.3.9.26  Ideals   intlidl 31609
                  20.3.9.27  Prime Ideals   cprmidl 31617
                  20.3.9.28  Maximal Ideals   cmxidl 31638
                  20.3.9.29  The semiring of ideals of a ring   cidlsrg 31652
                  20.3.9.30  Unique factorization domains   cufd 31668
                  20.3.9.31  Associative algebras   asclmulg 31673
                  20.3.9.32  Univariate Polynomials   fply1 31674
                  20.3.9.33  The subring algebra   sra1r 31678
                  20.3.9.34  Division Ring Extensions   drgext0g 31684
                  20.3.9.35  Vector Spaces   lvecdimfi 31690
                  20.3.9.36  Vector Space Dimension   cldim 31691
            20.3.10  Field Extensions   cfldext 31720
            20.3.11  Matrices   csmat 31750
                  20.3.11.1  Submatrices   csmat 31750
                  20.3.11.2  Matrix literals   clmat 31768
                  20.3.11.3  Laplace expansion of determinants   mdetpmtr1 31780
            20.3.12  Topology   ist0cld 31790
                  20.3.12.1  Open maps   txomap 31791
                  20.3.12.2  Topology of the unit circle   qtopt1 31792
                  20.3.12.3  Refinements   reff 31796
                  20.3.12.4  Open cover refinement property   ccref 31799
                  20.3.12.5  Lindelöf spaces   cldlf 31809
                  20.3.12.6  Paracompact spaces   cpcmp 31812
                  ​*20.3.12.7  Spectrum of a ring   crspec 31819
                  20.3.12.8  Pseudometrics   cmetid 31843
                  20.3.12.9  Continuity - misc additions   hauseqcn 31855
                  20.3.12.10  Topology of the closed unit interval   elunitge0 31856
                  20.3.12.11  Topology of ` ( RR X. RR ) `   unicls 31860
                  20.3.12.12  Order topology - misc. additions   cnvordtrestixx 31870
                  20.3.12.13  Continuity in topological spaces - misc. additions   mndpluscn 31883
                  20.3.12.14  Topology of the extended nonnegative real numbers ordered monoid   xrge0hmph 31889
                  20.3.12.15  Limits - misc additions   lmlim 31904
                  20.3.12.16  Univariate polynomials   pl1cn 31912
            20.3.13  Uniform Stuctures and Spaces   chcmp 31913
                  20.3.13.1  Hausdorff uniform completion   chcmp 31913
            20.3.14  Topology and algebraic structures   zringnm 31915
                  20.3.14.1  The norm on the ring of the integer numbers   zringnm 31915
                  20.3.14.2  Topological ` ZZ ` -modules   zlm0 31917
                  20.3.14.3  Canonical embedding of the field of the rational numbers into a division ring   cqqh 31929
                  20.3.14.4  Canonical embedding of the real numbers into a complete ordered field   crrh 31950
                  20.3.14.5  Embedding from the extended real numbers into a complete lattice   cxrh 31973
                  20.3.14.6  Canonical embeddings into the ordered field of the real numbers   zrhre 31976
                  ​*20.3.14.7  Topological Manifolds   cmntop 31979
            20.3.15  Real and complex functions   nexple 31984
                  20.3.15.1  Integer powers - misc. additions   nexple 31984
                  20.3.15.2  Indicator Functions   cind 31985
                  20.3.15.3  Extended sum   cesum 32002
            20.3.16  Mixed Function/Constant operation   cofc 32070
            20.3.17  Abstract measure   csiga 32083
                  20.3.17.1  Sigma-Algebra   csiga 32083
                  20.3.17.2  Generated sigma-Algebra   csigagen 32113
                  ​*20.3.17.3  lambda and pi-Systems, Rings of Sets   ispisys 32127
                  20.3.17.4  The Borel algebra on the real numbers   cbrsiga 32156
                  20.3.17.5  Product Sigma-Algebra   csx 32163
                  20.3.17.6  Measures   cmeas 32170
                  20.3.17.7  The counting measure   cntmeas 32201
                  20.3.17.8  The Lebesgue measure - misc additions   voliune 32204
                  20.3.17.9  The Dirac delta measure   cdde 32207
                  20.3.17.10  The 'almost everywhere' relation   cae 32212
                  20.3.17.11  Measurable functions   cmbfm 32224
                  20.3.17.12  Borel Algebra on ` ( RR X. RR ) `   br2base 32243
                  ​*20.3.17.13  Caratheodory's extension theorem   coms 32265
            20.3.18  Integration   itgeq12dv 32300
                  20.3.18.1  Lebesgue integral - misc additions   itgeq12dv 32300
                  20.3.18.2  Bochner integral   citgm 32301
            20.3.19  Euler's partition theorem   oddpwdc 32328
            20.3.20  Sequences defined by strong recursion   csseq 32357
            20.3.21  Fibonacci Numbers   cfib 32370
            20.3.22  Probability   cprb 32381
                  20.3.22.1  Probability Theory   cprb 32381
                  20.3.22.2  Conditional Probabilities   ccprob 32405
                  20.3.22.3  Real-valued Random Variables   crrv 32414
                  20.3.22.4  Preimage set mapping operator   corvc 32429
                  20.3.22.5  Distribution Functions   orvcelval 32442
                  20.3.22.6  Cumulative Distribution Functions   orvclteel 32446
                  20.3.22.7  Probabilities - example   coinfliplem 32452
                  20.3.22.8  Bertrand's Ballot Problem   ballotlemoex 32459
            20.3.23  Signum (sgn or sign) function - misc. additions   sgncl 32512
                  20.3.23.1  Operations on words   ccatmulgnn0dir 32528
            20.3.24  Polynomials with real coefficients - misc additions   plymul02 32532
            20.3.25  Descartes's rule of signs   signspval 32538
                  20.3.25.1  Sign changes in a word over real numbers   signspval 32538
                  20.3.25.2  Counting sign changes in a word over real numbers   signslema 32548
            20.3.26  Number Theory   efcld 32578
                  20.3.26.1  Representations of a number as sums of integers   crepr 32595
                  20.3.26.2  Vinogradov Trigonometric Sums and the Circle Method   cvts 32622
                  20.3.26.3  The Ternary Goldbach Conjecture: Final Statement   ax-hgt749 32631
            20.3.27  Elementary Geometry   cstrkg2d 32651
                  ​*20.3.27.1  Two-dimensional geometry   cstrkg2d 32651
                  20.3.27.2  Outer Five Segment (not used, no need to move to main)   cafs 32656
            ​*20.3.28  LeftPad Project   clpad 32661
      ​​*20.4  Mathbox for Jonathan Ben-Naim
            20.4.1  First-order logic and set theory   bnj170 32684
            20.4.2  Well founded induction and recursion   bnj110 32845
            20.4.3  The existence of a minimal element in certain classes   bnj69 32997
            20.4.4  Well-founded induction   bnj1204 32999
            20.4.5  Well-founded recursion, part 1 of 3   bnj60 33049
            20.4.6  Well-founded recursion, part 2 of 3   bnj1500 33055
            20.4.7  Well-founded recursion, part 3 of 3   bnj1522 33059
      ​20.5  Mathbox for BTernaryTau
            20.5.1  ZF set theory   exdifsn 33060
                  20.5.1.1  Finitism   fineqvrep 33071
            20.5.2  Real and complex numbers   zltp1ne 33075
            20.5.3  Graph theory   lfuhgr 33086
                  20.5.3.1  Acyclic graphs   cacycgr 33111
      ​20.6  Mathbox for Mario Carneiro
            20.6.1  Predicate calculus with all distinct variables   ax-7d 33128
            20.6.2  Miscellaneous stuff   quartfull 33134
            20.6.3  Derangements and the Subfactorial   deranglem 33135
            20.6.4  The Erdős-Szekeres theorem   erdszelem1 33160
            20.6.5  The Kuratowski closure-complement theorem   kur14lem1 33175
            20.6.6  Retracts and sections   cretr 33186
            20.6.7  Path-connected and simply connected spaces   cpconn 33188
            20.6.8  Covering maps   ccvm 33224
            20.6.9  Normal numbers   snmlff 33298
            20.6.10  Godel-sets of formulas - part 1   cgoe 33302
            20.6.11  Godel-sets of formulas - part 2   cgon 33401
            20.6.12  Models of ZF   cgze 33415
            ​*20.6.13  Metamath formal systems   cmcn 33429
            20.6.14  Grammatical formal systems   cm0s 33554
            20.6.15  Models of formal systems   cmuv 33574
            20.6.16  Splitting fields   citr 33596
            20.6.17  p-adic number fields   czr 33612
      ​​*20.7  Mathbox for Filip Cernatescu
      ​20.8  Mathbox for Paul Chapman
            20.8.1  Real and complex numbers (cont.)   climuzcnv 33636
            20.8.2  Miscellaneous theorems   elfzm12 33640
      ​20.9  Mathbox for Scott Fenton
            20.9.1  ZFC Axioms in primitive form   axextprim 33649
            20.9.2  Untangled classes   untelirr 33656
            20.9.3  Extra propositional calculus theorems   3pm3.2ni 33663
            20.9.4  Misc. Useful Theorems   nepss 33669
            20.9.5  Properties of real and complex numbers   sqdivzi 33700
            20.9.6  Infinite products   iprodefisumlem 33713
            20.9.7  Factorial limits   faclimlem1 33716
            20.9.8  Greatest common divisor and divisibility   gcd32 33722
            20.9.9  Properties of relationships   brtp 33724
            20.9.10  Properties of functions and mappings   funpsstri 33746
            20.9.11  Set induction (or epsilon induction)   setinds 33761
            20.9.12  Ordinal numbers   elpotr 33764
            20.9.13  Defined equality axioms   axextdfeq 33780
            20.9.14  Hypothesis builders   hbntg 33788
            20.9.15  (Trans)finite Recursion Theorems   tfisg 33793
            20.9.16  Well-Founded Induction   frpoins3xpg 33794
            20.9.17  Ordering Cross Products, Part 2   xpord2lem 33796
            20.9.18  Ordering Ordinal Sequences   orderseqlem 33808
            20.9.19  Well-founded zero, successor, and limits   cwsuc 33811
            20.9.20  Natural operations on ordinals   cnadd 33831
            20.9.21  Surreal Numbers   csur 33850
            20.9.22  Surreal Numbers: Ordering   sltsolem1 33885
            20.9.23  Surreal Numbers: Birthday Function   bdayfo 33887
            20.9.24  Surreal Numbers: Density   fvnobday 33888
            ​*20.9.25  Surreal Numbers: Full-Eta Property   bdayimaon 33903
            20.9.26  Surreal numbers - ordering theorems   csle 33954
            20.9.27  Surreal numbers - birthday theorems   bdayfun 33974
            20.9.28  Surreal numbers: Conway cuts   csslt 33982
            20.9.29  Surreal numbers - zero and one   c0s 34023
            20.9.30  Surreal numbers - cuts and options   cmade 34033
            20.9.31  Surreal numbers: Cofinality and coinitiality   cofsslt 34095
            20.9.32  Surreal numbers: Induction and recursion on one variable   cnorec 34101
            20.9.33  Surreal numbers: Induction and recursion on two variables   cnorec2 34112
            20.9.34  Surreal numbers - addition, negation, and subtraction   cadds 34123
            20.9.35  Quantifier-free definitions   ctxp 34139
            20.9.36  Alternate ordered pairs   caltop 34265
            20.9.37  Geometry in the Euclidean space   cofs 34291
                  20.9.37.1  Congruence properties   cofs 34291
                  20.9.37.2  Betweenness properties   btwntriv2 34321
                  20.9.37.3  Segment Transportation   ctransport 34338
                  20.9.37.4  Properties relating betweenness and congruence   cifs 34344
                  20.9.37.5  Connectivity of betweenness   btwnconn1lem1 34396
                  20.9.37.6  Segment less than or equal to   csegle 34415
                  20.9.37.7  Outside-of relationship   coutsideof 34428
                  20.9.37.8  Lines and Rays   cline2 34443
            20.9.38  Forward difference   cfwddif 34467
            20.9.39  Rank theorems   rankung 34475
            20.9.40  Hereditarily Finite Sets   chf 34481
      ​20.10  Mathbox for Jeff Hankins
            20.10.1  Miscellany   a1i14 34496
            20.10.2  Basic topological facts   topbnd 34520
            20.10.3  Topology of the real numbers   ivthALT 34531
            20.10.4  Refinements   cfne 34532
            20.10.5  Neighborhood bases determine topologies   neibastop1 34555
            20.10.6  Lattice structure of topologies   topmtcl 34559
            20.10.7  Filter bases   fgmin 34566
            20.10.8  Directed sets, nets   tailfval 34568
      ​20.11  Mathbox for Anthony Hart
            20.11.1  Propositional Calculus   tb-ax1 34579
            20.11.2  Predicate Calculus   nalfal 34599
            20.11.3  Miscellaneous single axioms   meran1 34607
            20.11.4  Connective Symmetry   negsym1 34613
      ​20.12  Mathbox for Chen-Pang He
            20.12.1  Ordinal topology   ontopbas 34624
      ​20.13  Mathbox for Jeff Hoffman
            20.13.1  Inferences for finite induction on generic function values   fveleq 34647
            20.13.2  gdc.mm   nnssi2 34651
      ​20.14  Mathbox for Asger C. Ipsen
            20.14.1  Continuous nowhere differentiable functions   dnival 34658
      ​​*20.15  Mathbox for BJ
            *20.15.1  Propositional calculus   bj-mp2c 34727
                  *20.15.1.1  Derived rules of inference   bj-mp2c 34727
                  ​*20.15.1.2  A syntactic theorem   bj-0 34729
                  20.15.1.3  Minimal implicational calculus   bj-a1k 34731
                  ​*20.15.1.4  Positive calculus   bj-syl66ib 34742
                  20.15.1.5  Implication and negation   bj-con2com 34748
                  ​*20.15.1.6  Disjunction   bj-jaoi1 34759
                  ​*20.15.1.7  Logical equivalence   bj-dfbi4 34761
                  20.15.1.8  The conditional operator for propositions   bj-consensus 34766
                  ​*20.15.1.9  Propositional calculus: miscellaneous   bj-imbi12 34771
            ​*20.15.2  Modal logic   bj-axdd2 34781
            ​*20.15.3  Provability logic   cprvb 34786
            ​*20.15.4  First-order logic   bj-genr 34795
                  20.15.4.1  Adding ax-gen   bj-genr 34795
                  20.15.4.2  Adding ax-4   bj-2alim 34799
                  20.15.4.3  Adding ax-5   bj-ax12wlem 34832
                  20.15.4.4  Equality and substitution   bj-ssbeq 34841
                  20.15.4.5  Adding ax-6   bj-spimvwt 34857
                  20.15.4.6  Adding ax-7   bj-cbvexw 34864
                  20.15.4.7  Membership predicate, ax-8 and ax-9   bj-ax89 34866
                  20.15.4.8  Adding ax-11   bj-alcomexcom 34869
                  20.15.4.9  Adding ax-12   axc11n11 34871
                  20.15.4.10  Nonfreeness   wnnf 34912
                  20.15.4.11  Adding ax-13   bj-axc10 34972
                  ​*20.15.4.12  Removing dependencies on ax-13 (and ax-11)   bj-axc10v 34982
                  ​*20.15.4.13  Distinct var metavariables   bj-hbaeb2 35008
                  ​*20.15.4.14  Around ~ equsal   bj-equsal1t 35012
                  ​*20.15.4.15  Some Principia Mathematica proofs   stdpc5t 35017
                  20.15.4.16  Alternate definition of substitution   bj-sbsb 35027
                  20.15.4.17  Lemmas for substitution   bj-sbf3 35029
                  20.15.4.18  Existential uniqueness   bj-eu3f 35032
                  ​*20.15.4.19  First-order logic: miscellaneous   bj-sblem1 35033
            20.15.5  Set theory   eliminable1 35050
                  ​*20.15.5.1  Eliminability of class terms   eliminable1 35050
                  ​*20.15.5.2  Classes without the axiom of extensionality   bj-denoteslem 35062
                  20.15.5.3  Characterization among sets versus among classes   elelb 35089
                  ​*20.15.5.4  The nonfreeness quantifier for classes   bj-nfcsym 35091
                  ​*20.15.5.5  Lemmas for class substitution   bj-sbeqALT 35092
                  20.15.5.6  Removing some axiom requirements and disjoint variable conditions   bj-exlimvmpi 35103
                  ​*20.15.5.7  Class abstractions   bj-elabd2ALT 35120
                  20.15.5.8  Generalized class abstractions   bj-cgab 35128
                  ​*20.15.5.9  Restricted nonfreeness   wrnf 35136
                  ​*20.15.5.10  Russell's paradox   bj-ru0 35138
                  20.15.5.11  Curry's paradox in set theory   currysetlem 35141
                  ​*20.15.5.12  Some disjointness results   bj-n0i 35147
                  ​*20.15.5.13  Complements on direct products   bj-xpimasn 35152
                  ​*20.15.5.14  "Singletonization" and tagging   bj-snsetex 35160
                  ​*20.15.5.15  Tuples of classes   bj-cproj 35187
                  ​*20.15.5.16  Set theory: elementary operations relative to a universe   bj-rcleqf 35222
                  ​*20.15.5.17  Set theory: miscellaneous   eleq2w2ALT 35227
                  ​*20.15.5.18  Evaluation at a class   bj-evaleq 35250
                  20.15.5.19  Elementwise operations   celwise 35257
                  ​*20.15.5.20  Elementwise intersection (families of sets induced on a subset)   bj-rest00 35259
                  20.15.5.21  Moore collections (complements)   bj-raldifsn 35278
                  20.15.5.22  Maps-to notation for functions with three arguments   bj-0nelmpt 35294
                  ​*20.15.5.23  Currying   csethom 35300
                  ​*20.15.5.24  Setting components of extensible structures   cstrset 35312
            ​*20.15.6  Extended real and complex numbers, real and complex projective lines   bj-nfald 35315
                  20.15.6.1  Complements on class abstractions of ordered pairs and binary relations   bj-nfald 35315
                  ​*20.15.6.2  Identity relation (complements)   bj-opabssvv 35328
                  ​*20.15.6.3  Functionalized identity (diagonal in a Cartesian square)   cdiag2 35350
                  ​*20.15.6.4  Direct image and inverse image   cimdir 35356
                  ​*20.15.6.5  Extended numbers and projective lines as sets   cfractemp 35374
                  ​*20.15.6.6  Addition and opposite   caddcc 35415
                  ​*20.15.6.7  Order relation on the extended reals   cltxr 35419
                  ​*20.15.6.8  Argument, multiplication and inverse   carg 35421
                  20.15.6.9  The canonical bijection from the finite ordinals   ciomnn 35427
                  20.15.6.10  Divisibility   cnnbar 35438
            ​*20.15.7  Monoids   bj-smgrpssmgm 35446
                  ​*20.15.7.1  Finite sums in monoids   cfinsum 35461
            ​*20.15.8  Affine, Euclidean, and Cartesian geometry   bj-fvimacnv0 35464
                  *20.15.8.1  Real vector spaces   bj-fvimacnv0 35464
                  ​*20.15.8.2  Complex numbers (supplements)   bj-subcom 35486
                  ​*20.15.8.3  Barycentric coordinates   bj-bary1lem 35488
            20.15.9  Monoid of endomorphisms   cend 35491
      ​20.16  Mathbox for Jim Kingdon
                  20.16.0.1  Circle constant   taupilem3 35497
                  20.16.0.2  Number theory   dfgcd3 35502
                  20.16.0.3  Real numbers   irrdifflemf 35503
      ​20.17  Mathbox for ML
            20.17.1  Miscellaneous   csbrecsg 35506
            20.17.2  Cartesian exponentiation   cfinxp 35561
            20.17.3  Topology   iunctb2 35581
                  ​*20.17.3.1  Pi-base theorems   pibp16 35591
      ​20.18  Mathbox for Wolf Lammen
            20.18.1  1. Bootstrapping   wl-section-boot 35600
            20.18.2  Implication chains   wl-section-impchain 35624
            20.18.3  Theorems around the conditional operator   wl-ifp-ncond1 35642
            20.18.4  Alternative development of hadd, cadd   wl-df-3xor 35646
            20.18.5  An alternative axiom ~ ax-13   ax-wl-13v 35671
            20.18.6  Other stuff   wl-mps 35673
      ​20.19  Mathbox for Brendan Leahy
      ​20.20  Mathbox for Jeff Madsen
            20.20.1  Logic and set theory   unirep 35878
            20.20.2  Real and complex numbers; integers   filbcmb 35905
            20.20.3  Sequences and sums   sdclem2 35907
            20.20.4  Topology   subspopn 35917
            20.20.5  Metric spaces   metf1o 35920
            20.20.6  Continuous maps and homeomorphisms   constcncf 35927
            20.20.7  Boundedness   ctotbnd 35931
            20.20.8  Isometries   cismty 35963
            20.20.9  Heine-Borel Theorem   heibor1lem 35974
            20.20.10  Banach Fixed Point Theorem   bfplem1 35987
            20.20.11  Euclidean space   crrn 35990
            20.20.12  Intervals (continued)   ismrer1 36003
            20.20.13  Operation properties   cass 36007
            20.20.14  Groups and related structures   cmagm 36013
            20.20.15  Group homomorphism and isomorphism   cghomOLD 36048
            20.20.16  Rings   crngo 36059
            20.20.17  Division Rings   cdrng 36113
            20.20.18  Ring homomorphisms   crnghom 36125
            20.20.19  Commutative rings   ccm2 36154
            20.20.20  Ideals   cidl 36172
            20.20.21  Prime rings and integral domains   cprrng 36211
            20.20.22  Ideal generators   cigen 36224
      ​20.21  Mathbox for Giovanni Mascellani
            ​*20.21.1  Tools for automatic proof building   efald2 36243
            ​*20.21.2  Tseitin axioms   fald 36294
            ​*20.21.3  Equality deductions   iuneq2f 36321
            ​*20.21.4  Miscellanea   orcomdd 36332
      ​20.22  Mathbox for Peter Mazsa
            20.22.1  Notations   cxrn 36339
            20.22.2  Preparatory theorems   el2v1 36377
            20.22.3  Range Cartesian product   df-xrn 36506
            20.22.4  Cosets by ` R `   df-coss 36542
            20.22.5  Relations   df-rels 36608
            20.22.6  Subset relations   df-ssr 36621
            20.22.7  Reflexivity   df-refs 36633
            20.22.8  Converse reflexivity   df-cnvrefs 36646
            20.22.9  Symmetry   df-syms 36661
            20.22.10  Reflexivity and symmetry   symrefref2 36682
            20.22.11  Transitivity   df-trs 36691
            20.22.12  Equivalence relations   df-eqvrels 36702
            20.22.13  Redundancy   df-redunds 36741
            20.22.14  Domain quotients   df-dmqss 36756
            20.22.15  Equivalence relations on domain quotients   df-ers 36780
            20.22.16  Functions   df-funss 36796
            20.22.17  Disjoints vs. converse functions   df-disjss 36819
      ​20.23  Mathbox for Rodolfo Medina
            20.23.1  Partitions   prtlem60 36872
      ​​*20.24  Mathbox for Norm Megill
            *20.24.1  Obsolete schemes ax-c4,c5,c7,c10,c11,c11n,c15,c9,c14,c16   ax-c5 36902
            ​*20.24.2  Rederive new axioms ax-4, ax-10, ax-6, ax-12, ax-13 from old   axc5 36912
            ​*20.24.3  Legacy theorems using obsolete axioms   ax5ALT 36926
            20.24.4  Experiments with weak deduction theorem   elimhyps 36980
            20.24.5  Miscellanea   cnaddcom 36991
            20.24.6  Atoms, hyperplanes, and covering in a left vector space (or module)   clsa 36993
            20.24.7  Functionals and kernels of a left vector space (or module)   clfn 37076
            20.24.8  Opposite rings and dual vector spaces   cld 37142
            20.24.9  Ortholattices and orthomodular lattices   cops 37191
            20.24.10  Atomic lattices with covering property   ccvr 37281
            20.24.11  Hilbert lattices   chlt 37369
            20.24.12  Projective geometries based on Hilbert lattices   clln 37510
            20.24.13  Construction of a vector space from a Hilbert lattice   cdlema1N 37810
            20.24.14  Construction of involution and inner product from a Hilbert lattice   clpoN 39499
      ​20.25  Mathbox for metakunt
            20.25.1  General helpful statements   leexp1ad 39985
            20.25.2  Some gcd and lcm results   12gcd5e1 40016
            20.25.3  Least common multiple inequality theorem   3factsumint1 40034
            20.25.4  Logarithm inequalities   3exp7 40066
            20.25.5  Miscellaneous results for AKS formalisation   intlewftc 40074
            20.25.6  Sticks and stones   sticksstones1 40107
            20.25.7  Permutation results   metakunt1 40130
            20.25.8  Unused lemmas scheduled for deletion   andiff 40164
      ​20.26  Mathbox for Steven Nguyen
            ​*20.26.1  Miscellaneous theorems   bicomdALT 40169
            20.26.2  Utility theorems   ioin9i8 40178
            20.26.3  Structures   nelsubginvcld 40225
            ​*20.26.4  Arithmetic theorems   c0exALT 40294
            20.26.5  Exponents and divisibility   oexpreposd 40326
            20.26.6  Real subtraction   cresub 40353
            ​*20.26.7  Projective spaces   cprjsp 40445
            20.26.8  Basic reductions for Fermat's Last Theorem   dffltz 40476
      ​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 40519
            20.29.2  Additional theory of functions   imaiinfv 40520
            20.29.3  Additional topology   elrfi 40521
            20.29.4  Characterization of closure operators. Kuratowski closure axioms   ismrcd1 40525
            20.29.5  Algebraic closure systems   cnacs 40529
            20.29.6  Miscellanea 1. Map utilities   constmap 40540
            20.29.7  Miscellanea for polynomials   mptfcl 40547
            20.29.8  Multivariate polynomials over the integers   cmzpcl 40548
            20.29.9  Miscellanea for Diophantine sets 1   coeq0i 40580
            20.29.10  Diophantine sets 1: definitions   cdioph 40582
            20.29.11  Diophantine sets 2 miscellanea   ellz1 40594
            20.29.12  Diophantine sets 2: union and intersection. Monotone Boolean algebra   diophin 40599
            20.29.13  Diophantine sets 3: construction   diophrex 40602
            20.29.14  Diophantine sets 4 miscellanea   2sbcrex 40611
            20.29.15  Diophantine sets 4: Quantification   rexrabdioph 40621
            20.29.16  Diophantine sets 5: Arithmetic sets   rabdiophlem1 40628
            20.29.17  Diophantine sets 6: reusability. renumbering of variables   eldioph4b 40638
            20.29.18  Pigeonhole Principle and cardinality helpers   fphpd 40643
            20.29.19  A non-closed set of reals is infinite   rencldnfilem 40647
            20.29.20  Lagrange's rational approximation theorem   irrapxlem1 40649
            20.29.21  Pell equations 1: A nontrivial solution always exists   pellexlem1 40656
            20.29.22  Pell equations 2: Algebraic number theory of the solution set   csquarenn 40663
            20.29.23  Pell equations 3: characterizing fundamental solution   infmrgelbi 40705
            ​*20.29.24  Logarithm laws generalized to an arbitrary base   reglogcl 40717
            20.29.25  Pell equations 4: the positive solution group is infinite cyclic   pellfund14 40725
            20.29.26  X and Y sequences 1: Definition and recurrence laws   crmx 40727
            20.29.27  Ordering and induction lemmas for the integers   monotuz 40768
            20.29.28  X and Y sequences 2: Order properties   rmxypos 40774
            20.29.29  Congruential equations   congtr 40792
            20.29.30  Alternating congruential equations   acongid 40802
            20.29.31  Additional theorems on integer divisibility   coprmdvdsb 40812
            20.29.32  X and Y sequences 3: Divisibility properties   jm2.18 40815
            20.29.33  X and Y sequences 4: Diophantine representability of Y   jm2.27a 40832
            20.29.34  X and Y sequences 5: Diophantine representability of X, ^, _C   rmxdiophlem 40842
            20.29.35  Uncategorized stuff not associated with a major project   setindtr 40851
            20.29.36  More equivalents of the Axiom of Choice   axac10 40860
            20.29.37  Finitely generated left modules   clfig 40897
            20.29.38  Noetherian left modules I   clnm 40905
            20.29.39  Addenda for structure powers   pwssplit4 40919
            20.29.40  Every set admits a group structure iff choice   unxpwdom3 40925
            20.29.41  Noetherian rings and left modules II   clnr 40939
            20.29.42  Hilbert's Basis Theorem   cldgis 40951
            20.29.43  Additional material on polynomials [DEPRECATED]   cmnc 40961
            20.29.44  Degree and minimal polynomial of algebraic numbers   cdgraa 40970
            20.29.45  Algebraic integers I   citgo 40987
            20.29.46  Endomorphism algebra   cmend 41005
            20.29.47  Cyclic groups and order   idomrootle 41025
            20.29.48  Cyclotomic polynomials   ccytp 41032
            20.29.49  Miscellaneous topology   fgraphopab 41040
      ​20.30  Mathbox for Jon Pennant
      ​20.31  Mathbox for Richard Penner
            20.31.1  Short Studies   nlimsuc 41053
                  20.31.1.1  Additional work on conditional logical operator   ifpan123g 41071
                  20.31.1.2  Sophisms   rp-fakeimass 41124
                  ​*20.31.1.3  Finite Sets   rp-isfinite5 41129
                  20.31.1.4  General Observations   intabssd 41131
                  20.31.1.5  Infinite Sets   pwelg 41172
                  ​*20.31.1.6  Finite intersection property   fipjust 41177
                  20.31.1.7  RP ADDTO: Subclasses and subsets   rababg 41186
                  20.31.1.8  RP ADDTO: The intersection of a class   elintabg 41187
                  20.31.1.9  RP ADDTO: Theorems requiring subset and intersection existence   elinintrab 41190
                  20.31.1.10  RP ADDTO: Relations   xpinintabd 41193
                  ​*20.31.1.11  RP ADDTO: Functions   elmapintab 41209
                  ​*20.31.1.12  RP ADDTO: Finite induction (for finite ordinals)   cnvcnvintabd 41213
                  20.31.1.13  RP ADDTO: First and second members of an ordered pair   elcnvlem 41214
                  20.31.1.14  RP ADDTO: The reflexive and transitive properties of relations   undmrnresiss 41217
                  20.31.1.15  RP ADDTO: Basic properties of closures   cleq2lem 41221
                  20.31.1.16  RP REPLACE: Definitions and basic properties of transitive closures   trcleq2lemRP 41243
                  ​*20.31.1.17  Additions for square root; absolute value   sqrtcvallem1 41244
            20.31.2  Additional statements on relations and subclasses   al3im 41260
                  20.31.2.1  Transitive relations (not to be confused with transitive classes).   trrelind 41278
                  20.31.2.2  Reflexive closures   crcl 41285
                  ​*20.31.2.3  Finite relationship composition.   relexp2 41290
                  20.31.2.4  Transitive closure of a relation   dftrcl3 41333
                  ​*20.31.2.5  Adapted from Frege   frege77d 41359
            ​*20.31.3  Propositions from _Begriffsschrift_   dfxor4 41379
                  *20.31.3.1  _Begriffsschrift_ Chapter I   dfxor4 41379
                  ​*20.31.3.2  _Begriffsschrift_ Notation hints   whe 41385
                  20.31.3.3  _Begriffsschrift_ Chapter II Implication   ax-frege1 41403
                  20.31.3.4  _Begriffsschrift_ Chapter II Implication and Negation   axfrege28 41442
                  ​*20.31.3.5  _Begriffsschrift_ Chapter II with logical equivalence   axfrege52a 41469
                  20.31.3.6  _Begriffsschrift_ Chapter II with equivalence of sets   axfrege52c 41500
                  ​*20.31.3.7  _Begriffsschrift_ Chapter II with equivalence of classes   frege53c 41527
                  ​*20.31.3.8  _Begriffsschrift_ Chapter III Properties hereditary in a sequence   dffrege69 41545
                  ​*20.31.3.9  _Begriffsschrift_ Chapter III Following in a sequence   dffrege76 41552
                  ​*20.31.3.10  _Begriffsschrift_ Chapter III Member of sequence   dffrege99 41575
                  ​*20.31.3.11  _Begriffsschrift_ Chapter III Single-valued procedures   dffrege115 41591
            ​*20.31.4  Exploring Topology via Seifert and Threlfall   enrelmap 41610
                  *20.31.4.1  Equinumerosity of sets of relations and maps   enrelmap 41610
                  ​*20.31.4.2  Generic Pseudoclosure Spaces, Pseudointerior Spaces, and Pseudoneighborhoods   or3or 41636
                  ​*20.31.4.3  Generic Neighborhood Spaces   gneispa 41745
            ​*20.31.5  Exploring Higher Homotopy via Kerodon   k0004lem1 41762
                  *20.31.5.1  Simplicial Sets   k0004lem1 41762
      ​20.32  Mathbox for Stanislas Polu
            20.32.1  IMO Problems   wwlemuld 41771
                  20.32.1.1  IMO 1972 B2   wwlemuld 41771
            ​*20.32.2  INT Inequalities Proof Generator   int-addcomd 41789
            ​*20.32.3  N-Digit Addition Proof Generator   unitadd 41811
            20.32.4  AM-GM (for k = 2,3,4)   gsumws3 41812
      ​20.33  Mathbox for Rohan Ridenour
            20.33.1  Misc   spALT 41817
            20.33.2  Monoid rings   cmnring 41829
            20.33.3  Shorter primitive equivalent of ax-groth   gru0eld 41852
                  20.33.3.1  Grothendieck universes are closed under collection   gru0eld 41852
                  20.33.3.2  Minimal universes   ismnu 41884
                  20.33.3.3  Primitive equivalent of ax-groth   expandan 41911
      ​20.34  Mathbox for Steve Rodriguez
            20.34.1  Miscellanea   nanorxor 41928
            20.34.2  Ratio test for infinite series convergence and divergence   dvgrat 41935
            20.34.3  Multiples   reldvds 41938
            20.34.4  Function operations   caofcan 41946
            20.34.5  Calculus   lhe4.4ex1a 41952
            20.34.6  The generalized binomial coefficient operation   cbcc 41959
            20.34.7  Binomial series   uzmptshftfval 41969
      ​20.35  Mathbox for Andrew Salmon
            20.35.1  Principia Mathematica * 10   pm10.12 41981
            20.35.2  Principia Mathematica * 11   2alanimi 41995
            20.35.3  Predicate Calculus   sbeqal1 42021
            20.35.4  Principia Mathematica * 13 and * 14   pm13.13a 42030
            20.35.5  Set Theory   elnev 42061
            20.35.6  Arithmetic   addcomgi 42079
            20.35.7  Geometry   cplusr 42080
      ​​*20.36  Mathbox for Alan Sare
            20.36.1  Auxiliary theorems for the Virtual Deduction tool   idiALT 42102
            20.36.2  Supplementary unification deductions   bi1imp 42106
            20.36.3  Conventional Metamath proofs, some derived from VD proofs   iidn3 42126
            20.36.4  What is Virtual Deduction?   wvd1 42194
            20.36.5  Virtual Deduction Theorems   df-vd1 42195
            20.36.6  Theorems proved using Virtual Deduction   trsspwALT 42443
            20.36.7  Theorems proved using Virtual Deduction with mmj2 assistance   simplbi2VD 42471
            20.36.8  Virtual Deduction transcriptions of textbook proofs   sb5ALTVD 42538
            20.36.9  Theorems proved using conjunction-form Virtual Deduction   elpwgdedVD 42542
            20.36.10  Theorems with a VD proof in conventional notation derived from a VD proof   suctrALT3 42549
            ​*20.36.11  Theorems with a proof in conventional notation derived from a VD proof   notnotrALT2 42552
      ​20.37  Mathbox for Glauco Siliprandi
            20.37.1  Miscellanea   evth2f 42563
            20.37.2  Functions   feq1dd 42714
            20.37.3  Ordering on real numbers - Real and complex numbers basic operations   sub2times 42828
            20.37.4  Real intervals   gtnelioc 43045
            20.37.5  Finite sums   fsummulc1f 43128
            20.37.6  Finite multiplication of numbers and finite multiplication of functions   fmul01 43137
            20.37.7  Limits   clim1fr1 43158
                  20.37.7.1  Inferior limit (lim inf)   clsi 43308
                  ​*20.37.7.2  Limits for sequences of extended real numbers   clsxlim 43375
            20.37.8  Trigonometry   coseq0 43421
            20.37.9  Continuous Functions   mulcncff 43427
            20.37.10  Derivatives   dvsinexp 43468
            20.37.11  Integrals   itgsin0pilem1 43507
            20.37.12  Stone Weierstrass theorem - real version   stoweidlem1 43558
            20.37.13  Wallis' product for π   wallispilem1 43622
            20.37.14  Stirling's approximation formula for ` n ` factorial   stirlinglem1 43631
            20.37.15  Dirichlet kernel   dirkerval 43648
            20.37.16  Fourier Series   fourierdlem1 43665
            20.37.17  e is transcendental   elaa2lem 43790
            20.37.18  n-dimensional Euclidean space   rrxtopn 43841
            20.37.19  Basic measure theory   csalg 43865
                  ​*20.37.19.1  σ-Algebras   csalg 43865
                  20.37.19.2  Sum of nonnegative extended reals   csumge0 43917
                  ​*20.37.19.3  Measures   cmea 44004
                  ​*20.37.19.4  Outer measures and Caratheodory's construction   come 44044
                  ​*20.37.19.5  Lebesgue measure on n-dimensional Real numbers   covoln 44091
                  ​*20.37.19.6  Measurable functions   csmblfn 44250
      ​20.38  Mathbox for Saveliy Skresanov
            20.38.1  Ceva's theorem   sigarval 44387
            20.38.2  Simple groups   simpcntrab 44407
      ​20.39  Mathbox for Jarvin Udandy
      ​20.40  Mathbox for Adhemar
            ​*20.40.1  Minimal implicational calculus   adh-minim 44517
      ​20.41  Mathbox for Alexander van der Vekens
            20.41.1  General auxiliary theorems (1)   eusnsn 44541
                  20.41.1.1  Unordered and ordered pairs - extension for singletons   eusnsn 44541
                  20.41.1.2  Unordered and ordered pairs - extension for unordered pairs   elprneb 44544
                  20.41.1.3  Unordered and ordered pairs - extension for ordered pairs   oppr 44545
                  20.41.1.4  Relations - extension   eubrv 44550
                  20.41.1.5  Definite description binder (inverted iota) - extension   iota0def 44553
                  20.41.1.6  Functions - extension   fveqvfvv 44555
            20.41.2  Alternative for Russell's definition of a description binder   caiota 44596
            20.41.3  Double restricted existential uniqueness   r19.32 44611
                  20.41.3.1  Restricted quantification (extension)   r19.32 44611
                  20.41.3.2  Restricted uniqueness and "at most one" quantification   reuf1odnf 44620
                  20.41.3.3  Analogs to Existential uniqueness (double quantification)   2reu3 44623
                  20.41.3.4  Additional theorems for double restricted existential uniqueness   2reu8i 44626
            ​*20.41.4  Alternative definitions of function and operation values   wdfat 44629
                  20.41.4.1  Restricted quantification (extension)   ralbinrald 44635
                  20.41.4.2  The universal class (extension)   nvelim 44636
                  20.41.4.3  Introduce the Axiom of Power Sets (extension)   alneu 44637
                  20.41.4.4  Predicate "defined at"   dfateq12d 44639
                  20.41.4.5  Alternative definition of the value of a function   dfafv2 44645
                  20.41.4.6  Alternative definition of the value of an operation   aoveq123d 44691
            ​*20.41.5  Alternative definitions of function values (2)   cafv2 44721
            20.41.6  General auxiliary theorems (2)   an4com24 44781
                  20.41.6.1  Logical conjunction - extension   an4com24 44781
                  20.41.6.2  Abbreviated conjunction and disjunction of three wff's - extension   3an4ancom24 44782
                  20.41.6.3  Negated membership (alternative)   cnelbr 44784
                  20.41.6.4  The empty set - extension   ralralimp 44791
                  20.41.6.5  Indexed union and intersection - extension   otiunsndisjX 44792
                  20.41.6.6  Functions - extension   fvifeq 44793
                  20.41.6.7  Maps-to notation - extension   fvmptrab 44805
                  20.41.6.8  Subtraction - extension   cnambpcma 44807
                  20.41.6.9  Ordering on reals (cont.) - extension   leaddsuble 44810
                  20.41.6.10  Imaginary and complex number properties - extension   readdcnnred 44816
                  20.41.6.11  Nonnegative integers (as a subset of complex numbers) - extension   nn0resubcl 44821
                  20.41.6.12  Integers (as a subset of complex numbers) - extension   zgeltp1eq 44822
                  20.41.6.13  Decimal arithmetic - extension   1t10e1p1e11 44823
                  20.41.6.14  Upper sets of integers - extension   eluzge0nn0 44825
                  20.41.6.15  Infinity and the extended real number system (cont.) - extension   nltle2tri 44826
                  20.41.6.16  Finite intervals of integers - extension   ssfz12 44827
                  20.41.6.17  Half-open integer ranges - extension   fzopred 44835
                  20.41.6.18  The modulo (remainder) operation - extension   m1mod0mod1 44842
                  20.41.6.19  The infinite sequence builder "seq"   smonoord 44844
                  20.41.6.20  Finite and infinite sums - extension   fsummsndifre 44845
                  20.41.6.21  Extensible structures - extension   setsidel 44849
            ​*20.41.7  Preimages of function values   preimafvsnel 44852
            ​*20.41.8  Partitions of real intervals   ciccp 44886
            20.41.9  Shifting functions with an integer range domain   fargshiftfv 44912
            20.41.10  Words over a set (extension)   lswn0 44917
                  20.41.10.1  Last symbol of a word - extension   lswn0 44917
            20.41.11  Unordered pairs   wich 44918
                  20.41.11.1  Interchangeable setvar variables   wich 44918
                  20.41.11.2  Set of unordered pairs   sprid 44947
                  ​*20.41.11.3  Proper (unordered) pairs   prpair 44974
                  20.41.11.4  Set of proper unordered pairs   cprpr 44985
            20.41.12  Number theory (extension)   cfmtno 45000
                  ​*20.41.12.1  Fermat numbers   cfmtno 45000
                  ​*20.41.12.2  Mersenne primes   m2prm 45064
                  20.41.12.3  Proth's theorem   modexp2m1d 45085
                  20.41.12.4  Solutions of quadratic equations   quad1 45093
            ​*20.41.13  Even and odd numbers   ceven 45097
                  20.41.13.1  Definitions and basic properties   ceven 45097
                  20.41.13.2  Alternate definitions using the "divides" relation   dfeven2 45122
                  20.41.13.3  Alternate definitions using the "modulo" operation   dfeven3 45131
                  20.41.13.4  Alternate definitions using the "gcd" operation   iseven5 45137
                  20.41.13.5  Theorems of part 5 revised   zneoALTV 45142
                  20.41.13.6  Theorems of part 6 revised   odd2np1ALTV 45147
                  20.41.13.7  Theorems of AV's mathbox revised   0evenALTV 45161
                  20.41.13.8  Additional theorems   epoo 45176
                  20.41.13.9  Perfect Number Theorem (revised)   perfectALTVlem1 45194
            20.41.14  Number theory (extension 2)   cfppr 45197
                  ​*20.41.14.1  Fermat pseudoprimes   cfppr 45197
                  ​*20.41.14.2  Goldbach's conjectures   cgbe 45218
            20.41.15  Graph theory (extension)   cgrisom 45291
                  ​*20.41.15.1  Isomorphic graphs   cgrisom 45291
                  20.41.15.2  Loop-free graphs - extension   1hegrlfgr 45315
                  20.41.15.3  Walks - extension   cupwlks 45316
                  20.41.15.4  Edges of graphs expressed as sets of unordered pairs   upgredgssspr 45326
            20.41.16  Monoids (extension)   ovn0dmfun 45339
                  20.41.16.1  Auxiliary theorems   ovn0dmfun 45339
                  20.41.16.2  Magmas, Semigroups and Monoids (extension)   plusfreseq 45347
                  20.41.16.3  Magma homomorphisms and submagmas   cmgmhm 45352
                  20.41.16.4  Examples and counterexamples for magmas, semigroups and monoids (extension)   opmpoismgm 45382
                  20.41.16.5  Group sum operation (extension 1)   gsumsplit2f 45395
            ​*20.41.17  Magmas and internal binary operations (alternate approach)   ccllaw 45398
                  *20.41.17.1  Laws for internal binary operations   ccllaw 45398
                  ​*20.41.17.2  Internal binary operations   cintop 45411
                  20.41.17.3  Alternative definitions for magmas and semigroups   cmgm2 45430
            20.41.18  Categories (extension)   idfusubc0 45444
                  20.41.18.1  Subcategories (extension)   idfusubc0 45444
            20.41.19  Rings (extension)   lmod0rng 45447
                  20.41.19.1  Nonzero rings (extension)   lmod0rng 45447
                  ​*20.41.19.2  Non-unital rings ("rngs")   crng 45453
                  20.41.19.3  Rng homomorphisms   crngh 45464
                  20.41.19.4  Ring homomorphisms (extension)   rhmfn 45497
                  20.41.19.5  Ideals as non-unital rings   lidldomn1 45500
                  20.41.19.6  The non-unital ring of even integers   0even 45510
                  20.41.19.7  A constructed not unital ring   cznrnglem 45532
                  ​*20.41.19.8  The category of non-unital rings   crngc 45536
                  ​*20.41.19.9  The category of (unital) rings   cringc 45582
                  20.41.19.10  Subcategories of the category of rings   srhmsubclem1 45652
            20.41.20  Basic algebraic structures (extension)   opeliun2xp 45689
                  20.41.20.1  Auxiliary theorems   opeliun2xp 45689
                  20.41.20.2  The binomial coefficient operation (extension)   bcpascm1 45708
                  20.41.20.3  The ` ZZ `-module ` ZZ X. ZZ `   zlmodzxzlmod 45711
                  20.41.20.4  Group sum operation (extension 2)   mgpsumunsn 45718
                  20.41.20.5  Symmetric groups (extension)   exple2lt6 45721
                  20.41.20.6  Divisibility (extension)   invginvrid 45724
                  20.41.20.7  The support of functions (extension)   rmsupp0 45725
                  20.41.20.8  Finitely supported functions (extension)   rmsuppfi 45730
                  20.41.20.9  Left modules (extension)   lmodvsmdi 45739
                  20.41.20.10  Associative algebras (extension)   assaascl0 45741
                  20.41.20.11  Univariate polynomials (extension)   ply1vr1smo 45743
                  20.41.20.12  Univariate polynomials (examples)   linply1 45755
            20.41.21  Linear algebra (extension)   cdmatalt 45758
                  ​*20.41.21.1  The subalgebras of diagonal and scalar matrices (extension)   cdmatalt 45758
                  ​*20.41.21.2  Linear combinations   clinc 45766
                  ​*20.41.21.3  Linear independence   clininds 45802
                  20.41.21.4  Simple left modules and the ` ZZ `-module   lmod1lem1 45849
                  20.41.21.5  Differences between (left) modules and (left) vector spaces   lvecpsslmod 45869
            20.41.22  Complexity theory   suppdm 45872
                  20.41.22.1  Auxiliary theorems   suppdm 45872
                  20.41.22.2  The modulo (remainder) operation (extension)   fldivmod 45885
                  20.41.22.3  Even and odd integers   nn0onn0ex 45890
                  20.41.22.4  The natural logarithm on complex numbers (extension)   logcxp0 45902
                  20.41.22.5  Division of functions   cfdiv 45904
                  20.41.22.6  Upper bounds   cbigo 45914
                  20.41.22.7  Logarithm to an arbitrary base (extension)   rege1logbrege0 45925
                  ​*20.41.22.8  The binary logarithm   fldivexpfllog2 45932
                  20.41.22.9  Binary length   cblen 45936
                  ​*20.41.22.10  Digits   cdig 45962
                  20.41.22.11  Nonnegative integer as sum of its shifted digits   dignn0flhalflem1 45982
                  20.41.22.12  Algorithms for the multiplication of nonnegative integers   nn0mulfsum 45991
                  ​*20.41.22.13  N-ary functions   cnaryf 45993
                  ​*20.41.22.14  The Ackermann function   citco 46024
            20.41.23  Elementary geometry (extension)   fv1prop 46066
                  20.41.23.1  Auxiliary theorems   fv1prop 46066
                  20.41.23.2  Real euclidean space of dimension 2   rrx2pxel 46078
                  20.41.23.3  Spheres and lines in real Euclidean spaces   cline 46094
      ​20.42  Mathbox for Zhi Wang
            20.42.1  Propositional calculus   pm4.71da 46156
            20.42.2  Predicate calculus with equality   dtrucor3 46165
                  20.42.2.1  Axiom scheme ax-5 (Distinctness)   dtrucor3 46165
            20.42.3  ZF Set Theory - start with the Axiom of Extensionality   ralbidb 46166
                  20.42.3.1  Restricted quantification   ralbidb 46166
                  20.42.3.2  The empty set   ssdisjd 46174
                  20.42.3.3  Unordered and ordered pairs   vsn 46178
                  20.42.3.4  The union of a class   unilbss 46184
            20.42.4  ZF Set Theory - add the Axiom of Replacement   inpw 46185
                  20.42.4.1  Theorems requiring subset and intersection existence   inpw 46185
            20.42.5  ZF Set Theory - add the Axiom of Power Sets   mof0 46186
                  20.42.5.1  Functions   mof0 46186
                  20.42.5.2  Operations   fvconstr 46204
            20.42.6  ZF Set Theory - add the Axiom of Union   fvconst0ci 46207
                  20.42.6.1  Equinumerosity   fvconst0ci 46207
            20.42.7  Order sets   iccin 46211
                  20.42.7.1  Real number intervals   iccin 46211
            20.42.8  Moore spaces   mreuniss 46214
            ​*20.42.9  Topology   clduni 46215
                  20.42.9.1  Closure and interior   clduni 46215
                  20.42.9.2  Neighborhoods   neircl 46219
                  20.42.9.3  Subspace topologies   restcls2lem 46227
                  20.42.9.4  Limits and continuity in topological spaces   cnneiima 46231
                  20.42.9.5  Topological definitions using the reals   iooii 46232
                  20.42.9.6  Separated sets   sepnsepolem1 46236
                  20.42.9.7  Separated spaces: T0, T1, T2 (Hausdorff) ...   isnrm4 46245
            20.42.10  Preordered sets and directed sets using extensible structures   isprsd 46270
            20.42.11  Posets and lattices using extensible structures   lubeldm2 46271
                  20.42.11.1  Posets   lubeldm2 46271
                  20.42.11.2  Lattices   toslat 46289
                  20.42.11.3  Subset order structures   intubeu 46291
            20.42.12  Categories   catprslem 46312
                  20.42.12.1  Categories   catprslem 46312
                  20.42.12.2  Monomorphisms and epimorphisms   idmon 46318
                  20.42.12.3  Functors   funcf2lem 46320
            20.42.13  Examples of categories   cthinc 46321
                  20.42.13.1  Thin categories   cthinc 46321
                  20.42.13.2  Preordered sets as thin categories   cprstc 46364
                  20.42.13.3  Monoids as categories   cmndtc 46385
      ​20.43  Mathbox for Emmett Weisz
            ​*20.43.1  Miscellaneous Theorems   nfintd 46400
            20.43.2  Set Recursion   csetrecs 46410
                  ​*20.43.2.1  Basic Properties of Set Recursion   csetrecs 46410
                  20.43.2.2  Examples and properties of set recursion   elsetrecslem 46425
            ​*20.43.3  Construction of Games and Surreal Numbers   cpg 46435
      ​​*20.44  Mathbox for David A. Wheeler
            20.44.1  Natural deduction   sbidd 46441
            ​*20.44.2  Greater than, greater than or equal to.   cge-real 46443
            ​*20.44.3  Hyperbolic trigonometric functions   csinh 46453
            ​*20.44.4  Reciprocal trigonometric functions (sec, csc, cot)   csec 46464
            ​*20.44.5  Identities for "if"   ifnmfalse 46486
            ​*20.44.6  Logarithms generalized to arbitrary base using ` logb `   logb2aval 46487
            ​*20.44.7  Logarithm laws generalized to an arbitrary base - log_   clog- 46488
            ​*20.44.8  Formally define notions such as reflexivity   wreflexive 46490
            ​*20.44.9  Algebra helpers   comraddi 46494
            ​*20.44.10  Algebra helper examples   i2linesi 46503
            ​*20.44.11  Formal methods "surprises"   alimp-surprise 46505
            ​*20.44.12  Allsome quantifier   walsi 46511
            ​*20.44.13  Miscellaneous   5m4e1 46522
            20.44.14  Theorems about algebraic numbers   aacllem 46526
      ​20.45  Mathbox for Kunhao Zheng
            20.45.1  Weighted AM-GM inequality   amgmwlem 46527
      ​20.46  Mathbox for Ender Ting
            20.46.1  Increasing sequences and subsequences   et-ltneverrefl 46531