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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  SURREAL NUMBERS
PART 16  ELEMENTARY GEOMETRY
      16.1  Definition and Tarski's Axioms of Geometry
      16.2  Tarskian Geometry
      16.3  Properties of geometries
      16.4  Geometry in Hilbert spaces
PART 17  GRAPH THEORY
      17.1  Vertices and edges
      17.2  Undirected graphs
      17.3  Walks, paths and cycles
      17.4  Eulerian paths and the Konigsberg Bridge problem
      17.5  The Friendship Theorem
PART 18  GUIDES AND MISCELLANEA
      18.1  Guides (conventions, explanations, and examples)
      18.2  Humor
      18.3  (Future - to be reviewed and classified)
PART 19  COMPLEX TOPOLOGICAL VECTOR SPACES (DEPRECATED)
      19.1  Additional material on group theory (deprecated)
      19.2  Complex vector spaces
      19.3  Normed complex vector spaces
      19.4  Operators on complex vector spaces
      19.5  Inner product (pre-Hilbert) spaces
      19.6  Complex Banach spaces
      19.7  Complex Hilbert spaces
PART 20  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      20.1  Axiomatization of complex pre-Hilbert spaces
      20.2  Inner product and norms
      20.3  Cauchy sequences and completeness axiom
      20.4  Subspaces and projections
      20.5  Properties of Hilbert subspaces
      20.6  Operators on Hilbert spaces
      20.7  States on a Hilbert lattice and Godowski's equation
      20.8  Cover relation, atoms, exchange axiom, and modular symmetry
PART 21  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      21.1  Mathboxes for user contributions
      21.2  Mathbox for Stefan Allan
      21.3  Mathbox for Thierry Arnoux
      21.4  Mathbox for Jonathan Ben-Naim
      21.5  Mathbox for BTernaryTau
      21.6  Mathbox for Mario Carneiro
      21.7  Mathbox for Filip Cernatescu
      21.8  Mathbox for Paul Chapman
      21.9  Mathbox for Scott Fenton
      21.10  Mathbox for Jeff Hankins
      21.11  Mathbox for Anthony Hart
      21.12  Mathbox for Chen-Pang He
      21.13  Mathbox for Jeff Hoffman
      21.14  Mathbox for Asger C. Ipsen
      21.15  Mathbox for BJ
      21.16  Mathbox for Jim Kingdon
      21.17  Mathbox for ML
      21.18  Mathbox for Wolf Lammen
      21.19  Mathbox for Brendan Leahy
      21.20  Mathbox for Jeff Madsen
      21.21  Mathbox for Giovanni Mascellani
      21.22  Mathbox for Peter Mazsa
      21.23  Mathbox for Rodolfo Medina
      21.24  Mathbox for Norm Megill
      21.25  Mathbox for metakunt
      21.26  Mathbox for Steven Nguyen
      21.27  Mathbox for Igor Ieskov
      21.28  Mathbox for OpenAI
      21.29  Mathbox for Stefan O'Rear
      21.30  Mathbox for Noam Pasman
      21.31  Mathbox for Jon Pennant
      21.32  Mathbox for Richard Penner
      21.33  Mathbox for Stanislas Polu
      21.34  Mathbox for Rohan Ridenour
      21.35  Mathbox for Steve Rodriguez
      21.36  Mathbox for Andrew Salmon
      21.37  Mathbox for Alan Sare
      21.38  Mathbox for Glauco Siliprandi
      21.39  Mathbox for Saveliy Skresanov
      21.40  Mathbox for Ender Ting
      21.41  Mathbox for Jarvin Udandy
      21.42  Mathbox for Adhemar
      21.43  Mathbox for Alexander van der Vekens
      21.44  Mathbox for Zhi Wang
      21.45  Mathbox for Emmett Weisz
      21.46  Mathbox for David A. Wheeler
      21.47  Mathbox for Kunhao Zheng

Detailed Table of Contents
(* means the section header has a description)
*PART 1  CLASSICAL FIRST-ORDER LOGIC WITH EQUALITY
      *1.1  Pre-logic
            *1.1.1  Inferences for assisting proof development   idi 1
      *1.2  Propositional calculus
            1.2.1  Recursively define primitive wffs for propositional calculus   wn 3
            *1.2.2  The axioms of propositional calculus   ax-mp 5
            *1.2.3  Logical implication   mp2 9
            *1.2.4  Logical negation   con4 113
            *1.2.5  Logical equivalence   wb 205
            *1.2.6  Logical conjunction   wa 397
            *1.2.7  Logical disjunction   wo 846
            *1.2.8  Mixed connectives   jaao 954
            *1.2.9  The conditional operator for propositions   wif 1062
            *1.2.10  The weak deduction theorem for propositional calculus   elimh 1084
            1.2.11  Abbreviated conjunction and disjunction of three wff's   w3o 1087
            1.2.12  Logical "nand" (Sheffer stroke)   wnan 1490
            1.2.13  Logical "xor"   wxo 1510
            1.2.14  Logical "nor"   wnor 1529
            1.2.15  True and false constants   wal 1540
                  *1.2.15.1  Universal quantifier for use by df-tru   wal 1540
                  *1.2.15.2  Equality predicate for use by df-tru   cv 1541
                  1.2.15.3  The true constant   wtru 1543
                  1.2.15.4  The false constant   wfal 1554
            *1.2.16  Truth tables   truimtru 1565
                  1.2.16.1  Implication   truimtru 1565
                  1.2.16.2  Negation   nottru 1569
                  1.2.16.3  Equivalence   trubitru 1571
                  1.2.16.4  Conjunction   truantru 1575
                  1.2.16.5  Disjunction   truortru 1579
                  1.2.16.6  Alternative denial   trunantru 1583
                  1.2.16.7  Exclusive disjunction   truxortru 1587
                  1.2.16.8  Joint denial   trunortru 1591
            *1.2.17  Half adder and full adder in propositional calculus   whad 1595
                  1.2.17.1  Full adder: sum   whad 1595
                  1.2.17.2  Full adder: carry   wcad 1608
      1.3  Other axiomatizations related to classical propositional calculus
            *1.3.1  Minimal implicational calculus   minimp 1624
            *1.3.2  Implicational Calculus   impsingle 1630
            1.3.3  Derive the Lukasiewicz axioms from Meredith's sole axiom   meredith 1644
            1.3.4  Derive the standard axioms from the Lukasiewicz axioms   luklem1 1661
            *1.3.5  Derive Nicod's axiom from the standard axioms   nic-dfim 1672
            1.3.6  Derive the Lukasiewicz axioms from Nicod's axiom   nic-imp 1678
            1.3.7  Derive Nicod's Axiom from Lukasiewicz's First Sheffer Stroke Axiom   lukshef-ax1 1697
            1.3.8  Derive the Lukasiewicz Axioms from the Tarski-Bernays-Wajsberg Axioms   tbw-bijust 1701
            1.3.9  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's First CO Axiom   merco1 1716
            1.3.10  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's Second CO Axiom   merco2 1739
            1.3.11  Derive the Lukasiewicz axioms from the Russell-Bernays Axioms   rb-bijust 1752
            *1.3.12  Stoic logic non-modal portion (Chrysippus of Soli)   mptnan 1771
      *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 1782
                  1.4.1.1  Existential quantifier   wex 1782
                  1.4.1.2  Nonfreeness predicate   wnf 1786
            1.4.2  Rule scheme ax-gen (Generalization)   ax-gen 1798
            1.4.3  Axiom scheme ax-4 (Quantified Implication)   ax-4 1812
                  *1.4.3.1  The empty domain of discourse   empty 1910
            1.4.4  Axiom scheme ax-5 (Distinctness) - first use of $d   ax-5 1914
            *1.4.5  Equality predicate (continued)   weq 1967
            1.4.6  Axiom scheme ax-6 (Existence)   ax-6 1972
            1.4.7  Axiom scheme ax-7 (Equality)   ax-7 2012
            1.4.8  Define proper substitution   sbjust 2067
            1.4.9  Membership predicate   wcel 2107
            1.4.10  Axiom scheme ax-8 (Left Equality for Binary Predicate)   ax-8 2109
            1.4.11  Axiom scheme ax-9 (Right Equality for Binary Predicate)   ax-9 2117
            *1.4.12  Logical redundancy of ax-10 , ax-11 , ax-12 , ax-13   ax6dgen 2125
      *1.5  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes)
            1.5.1  Axiom scheme ax-10 (Quantified Negation)   ax-10 2138
            1.5.2  Axiom scheme ax-11 (Quantifier Commutation)   ax-11 2155
            1.5.3  Axiom scheme ax-12 (Substitution)   ax-12 2172
            1.5.4  Axiom scheme ax-13 (Quantified Equality)   ax-13 2372
      1.6  Uniqueness and unique existence
            1.6.1  Uniqueness: the at-most-one quantifier   wmo 2538
            1.6.2  Unique existence: the unique existential quantifier   weu 2568
      1.7  Other axiomatizations related to classical predicate calculus
            *1.7.1  Aristotelian logic: Assertic syllogisms   barbara 2664
            *1.7.2  Intuitionistic logic   axia1 2694
*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 2709
            2.1.2  Classes   cab 2715
                  2.1.2.1  Class abstractions   cab 2715
                  *2.1.2.2  Class equality   df-cleq 2730
                  2.1.2.3  Class membership   df-clel 2816
                  2.1.2.4  Elementary properties of class abstractions   abeq2 2872
            2.1.3  Class form not-free predicate   wnfc 2886
            2.1.4  Negated equality and membership   wne 2942
                  2.1.4.1  Negated equality   wne 2942
                  2.1.4.2  Negated membership   wnel 3048
            2.1.5  Restricted quantification   wral 3063
                  2.1.5.1  Restricted universal and existential quantification   wral 3063
                  2.1.5.2  Restricted existential uniqueness and at-most-one quantifier   wreu 3350
                  2.1.5.3  Restricted class abstraction   crab 3406
            2.1.6  The universal class   cvv 3444
            *2.1.7  Conditional equality (experimental)   wcdeq 3720
            2.1.8  Russell's Paradox   rru 3736
            2.1.9  Proper substitution of classes for sets   wsbc 3738
            2.1.10  Proper substitution of classes for sets into classes   csb 3854
            2.1.11  Define basic set operations and relations   cdif 3906
            2.1.12  Subclasses and subsets   df-ss 3926
            2.1.13  The difference, union, and intersection of two classes   dfdif3 4073
                  2.1.13.1  The difference of two classes   dfdif3 4073
                  2.1.13.2  The union of two classes   elun 4107
                  2.1.13.3  The intersection of two classes   elini 4152
                  2.1.13.4  The symmetric difference of two classes   csymdif 4200
                  2.1.13.5  Combinations of difference, union, and intersection of two classes   unabs 4213
                  2.1.13.6  Class abstractions with difference, union, and intersection of two classes   unabw 4256
                  2.1.13.7  Restricted uniqueness with difference, union, and intersection   reuun2 4273
            2.1.14  The empty set   c0 4281
            *2.1.15  The conditional operator for classes   cif 4485
            *2.1.16  The weak deduction theorem for set theory   dedth 4543
            2.1.17  Power classes   cpw 4559
            2.1.18  Unordered and ordered pairs   snjust 4584
            2.1.19  The union of a class   cuni 4864
            2.1.20  The intersection of a class   cint 4906
            2.1.21  Indexed union and intersection   ciun 4953
            2.1.22  Disjointness   wdisj 5069
            2.1.23  Binary relations   wbr 5104
            2.1.24  Ordered-pair class abstractions (class builders)   copab 5166
            2.1.25  Functions in maps-to notation   cmpt 5187
            2.1.26  Transitive classes   wtr 5221
      2.2  ZF Set Theory - add the Axiom of Replacement
            2.2.1  Introduce the Axiom of Replacement   ax-rep 5241
            2.2.2  Derive the Axiom of Separation   axsepgfromrep 5253
            2.2.3  Derive the Null Set Axiom   axnulALT 5260
            2.2.4  Theorems requiring subset and intersection existence   nalset 5269
            2.2.5  Theorems requiring empty set existence   class2set 5309
      2.3  ZF Set Theory - add the Axiom of Power Sets
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 5319
            2.3.2  Derive the Axiom of Pairing   axprlem1 5377
            2.3.3  Ordered pair theorem   opnz 5429
            2.3.4  Ordered-pair class abstractions (cont.)   opabidw 5480
            2.3.5  Power class of union and intersection   pwin 5526
            2.3.6  The identity relation   cid 5529
            2.3.7  The membership relation (or epsilon relation)   cep 5535
            *2.3.8  Partial and total orderings   wpo 5542
            2.3.9  Founded and well-ordering relations   wfr 5584
            2.3.10  Relations   cxp 5630
            2.3.11  The Predecessor Class   cpred 6251
            2.3.12  Well-founded induction (variant)   frpomin 6293
            2.3.13  Well-ordered induction   tz6.26 6300
            2.3.14  Ordinals   word 6315
            2.3.15  Definite description binder (inverted iota)   cio 6444
            2.3.16  Functions   wfun 6488
            2.3.17  Cantor's Theorem   canth 7305
            2.3.18  Restricted iota (description binder)   crio 7307
            2.3.19  Operations   co 7352
                  2.3.19.1  Variable-to-class conversion for operations   caovclg 7541
            2.3.20  Maps-to notation   mpondm0 7587
            2.3.21  Function operation   cof 7608
            2.3.22  Proper subset relation   crpss 7652
      2.4  ZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 7665
            2.4.2  Ordinals (continued)   epweon 7702
            2.4.3  Transfinite induction   tfi 7782
            2.4.4  The natural numbers (i.e., finite ordinals)   com 7795
            2.4.5  Peano's postulates   peano1 7818
            2.4.6  Finite induction (for finite ordinals)   find 7826
            2.4.7  Relations and functions (cont.)   dmexg 7833
            2.4.8  First and second members of an ordered pair   c1st 7912
            2.4.9  Induction on Cartesian products   frpoins3xpg 8065
            2.4.10  Ordering on Cartesian products   xpord2lem 8067
            2.4.11  Ordering Ordinal Sequences   orderseqlem 8082
            *2.4.12  The support of functions   csupp 8085
            *2.4.13  Special maps-to operations   opeliunxp2f 8134
            2.4.14  Function transposition   ctpos 8149
            2.4.15  Curry and uncurry   ccur 8189
            2.4.16  Undefined values   cund 8196
            2.4.17  Well-founded recursion   cfrecs 8204
            2.4.18  Well-ordered recursion   cwrecs 8235
            2.4.19  Functions on ordinals; strictly monotone ordinal functions   iunon 8278
            2.4.20  "Strong" transfinite recursion   crecs 8309
            2.4.21  Recursive definition generator   crdg 8348
            2.4.22  Finite recursion   frfnom 8374
            2.4.23  Ordinal arithmetic   c1o 8398
            2.4.24  Natural number arithmetic   nna0 8544
            2.4.25  Equivalence relations and classes   wer 8604
            2.4.26  The mapping operation   cmap 8724
            2.4.27  Infinite Cartesian products   cixp 8794
            2.4.28  Equinumerosity   cen 8839
            2.4.29  Schroeder-Bernstein Theorem   sbthlem1 8986
            2.4.30  Equinumerosity (cont.)   xpf1o 9042
            2.4.31  Finite sets   dif1enlem 9059
            2.4.32  Pigeonhole Principle   phplem1 9110
            2.4.33  Finite sets (cont.)   onomeneq 9131
            2.4.34  Finitely supported functions   cfsupp 9264
            2.4.35  Finite intersections   cfi 9305
            2.4.36  Hall's marriage theorem   marypha1lem 9328
            2.4.37  Supremum and infimum   csup 9335
            2.4.38  Ordinal isomorphism, Hartogs's theorem   coi 9404
            2.4.39  Hartogs function   char 9451
            2.4.40  Weak dominance   cwdom 9459
      2.5  ZF Set Theory - add the Axiom of Regularity
            2.5.1  Introduce the Axiom of Regularity   ax-reg 9487
            2.5.2  Axiom of Infinity equivalents   inf0 9516
      2.6  ZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-inf 9533
            2.6.2  Existence of omega (the set of natural numbers)   omex 9538
            2.6.3  Cantor normal form   ccnf 9556
            2.6.4  Transitive closure of a relation   cttrcl 9602
            2.6.5  Transitive closure   trcl 9623
            2.6.6  Well-Founded Induction   frmin 9644
            2.6.7  Well-Founded Recursion   frr3g 9651
            2.6.8  Rank   cr1 9657
            2.6.9  Scott's trick; collection principle; Hilbert's epsilon   scottex 9780
            2.6.10  Disjoint union   cdju 9793
            2.6.11  Cardinal numbers   ccrd 9830
            2.6.12  Axiom of Choice equivalents   wac 10010
            *2.6.13  Cardinal number arithmetic   undjudom 10062
            2.6.14  The Ackermann bijection   ackbij2lem1 10114
            2.6.15  Cofinality (without Axiom of Choice)   cflem 10141
            2.6.16  Eight inequivalent definitions of finite set   sornom 10172
            2.6.17  Hereditarily size-limited sets without Choice   itunifval 10311
*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 10330
            3.1.2  Introduce the Axiom of Dependent Choice   ax-dc 10341
      3.2  ZFC Set Theory - add the Axiom of Choice
            3.2.1  Introduce the Axiom of Choice   ax-ac 10354
            3.2.2  AC equivalents: well-ordering, Zorn's lemma   numthcor 10389
            3.2.3  Cardinal number theorems using Axiom of Choice   cardval 10441
            3.2.4  Cardinal number arithmetic using Axiom of Choice   iunctb 10469
            3.2.5  Cofinality using the Axiom of Choice   alephreg 10477
      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 10515
            3.4.2  Derivation of the Axiom of Choice   gchaclem 10573
*PART 4  TG (TARSKI-GROTHENDIECK) SET THEORY
      4.1  Inaccessibles
            4.1.1  Weakly and strongly inaccessible cardinals   cwina 10577
            4.1.2  Weak universes   cwun 10595
            4.1.3  Tarski classes   ctsk 10643
            4.1.4  Grothendieck universes   cgru 10685
      4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
            4.2.1  Introduce the Tarski-Grothendieck Axiom   ax-groth 10718
            4.2.2  Derive the Power Set, Infinity and Choice Axioms   grothpw 10721
            4.2.3  Tarski map function   ctskm 10732
*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 10739
            5.1.2  Final derivation of real and complex number postulates   axaddf 11040
            5.1.3  Real and complex number postulates restated as axioms   ax-cnex 11066
      5.2  Derive the basic properties from the field axioms
            5.2.1  Some deductions from the field axioms for complex numbers   cnex 11091
            5.2.2  Infinity and the extended real number system   cpnf 11145
            5.2.3  Restate the ordering postulates with extended real "less than"   axlttri 11185
            5.2.4  Ordering on reals   lttr 11190
            5.2.5  Initial properties of the complex numbers   mul12 11279
      5.3  Real and complex numbers - basic operations
            5.3.1  Addition   add12 11331
            5.3.2  Subtraction   cmin 11344
            5.3.3  Multiplication   kcnktkm1cn 11545
            5.3.4  Ordering on reals (cont.)   gt0ne0 11579
            5.3.5  Reciprocals   ixi 11743
            5.3.6  Division   cdiv 11771
            5.3.7  Ordering on reals (cont.)   elimgt0 11952
            5.3.8  Completeness Axiom and Suprema   fimaxre 12058
            5.3.9  Imaginary and complex number properties   inelr 12102
            5.3.10  Function operation analogue theorems   ofsubeq0 12109
      5.4  Integer sets
            5.4.1  Positive integers (as a subset of complex numbers)   cn 12112
            5.4.2  Principle of mathematical induction   nnind 12130
            *5.4.3  Decimal representation of numbers   c2 12167
            *5.4.4  Some properties of specific numbers   neg1cn 12226
            5.4.5  Simple number properties   halfcl 12337
            5.4.6  The Archimedean property   nnunb 12368
            5.4.7  Nonnegative integers (as a subset of complex numbers)   cn0 12372
            *5.4.8  Extended nonnegative integers   cxnn0 12444
            5.4.9  Integers (as a subset of complex numbers)   cz 12458
            5.4.10  Decimal arithmetic   cdc 12577
            5.4.11  Upper sets of integers   cuz 12722
            5.4.12  Well-ordering principle for bounded-below sets of integers   uzwo3 12823
            5.4.13  Rational numbers (as a subset of complex numbers)   cq 12828
            5.4.14  Existence of the set of complex numbers   rpnnen1lem2 12857
      5.5  Order sets
            5.5.1  Positive reals (as a subset of complex numbers)   crp 12870
            5.5.2  Infinity and the extended real number system (cont.)   cxne 12985
            5.5.3  Supremum and infimum on the extended reals   xrsupexmnf 13179
            5.5.4  Real number intervals   cioo 13219
            5.5.5  Finite intervals of integers   cfz 13379
            *5.5.6  Finite intervals of nonnegative integers   elfz2nn0 13487
            5.5.7  Half-open integer ranges   cfzo 13522
      5.6  Elementary integer functions
            5.6.1  The floor and ceiling functions   cfl 13650
            5.6.2  The modulo (remainder) operation   cmo 13729
            5.6.3  Miscellaneous theorems about integers   om2uz0i 13807
            5.6.4  Strong induction over upper sets of integers   uzsinds 13847
            5.6.5  Finitely supported functions over the nonnegative integers   fsuppmapnn0fiublem 13850
            5.6.6  The infinite sequence builder "seq" - extension   cseq 13861
            5.6.7  Integer powers   cexp 13922
            5.6.8  Ordered pair theorem for nonnegative integers   nn0le2msqi 14121
            5.6.9  Factorial function   cfa 14127
            5.6.10  The binomial coefficient operation   cbc 14156
            5.6.11  The ` # ` (set size) function   chash 14184
                  5.6.11.1  Proper unordered pairs and triples (sets of size 2 and 3)   hashprlei 14321
                  5.6.11.2  Functions with a domain containing at least two different elements   fundmge2nop0 14345
                  5.6.11.3  Finite induction on the size of the first component of a binary relation   hashdifsnp1 14349
      *5.7  Words over a set
            5.7.1  Definitions and basic theorems   cword 14356
            5.7.2  Last symbol of a word   clsw 14404
            5.7.3  Concatenations of words   cconcat 14412
            5.7.4  Singleton words   cs1 14437
            5.7.5  Concatenations with singleton words   ccatws1cl 14458
            5.7.6  Subwords/substrings   csubstr 14486
            5.7.7  Prefixes of a word   cpfx 14516
            5.7.8  Subwords of subwords   swrdswrdlem 14550
            5.7.9  Subwords and concatenations   pfxcctswrd 14556
            5.7.10  Subwords of concatenations   swrdccatfn 14570
            5.7.11  Splicing words (substring replacement)   csplice 14595
            5.7.12  Reversing words   creverse 14604
            5.7.13  Repeated symbol words   creps 14614
            *5.7.14  Cyclical shifts of words   ccsh 14634
            5.7.15  Mapping words by a function   wrdco 14678
            5.7.16  Longer string literals   cs2 14688
      *5.8  Reflexive and transitive closures of relations
            5.8.1  The reflexive and transitive properties of relations   coss12d 14817
            5.8.2  Basic properties of closures   cleq1lem 14827
            5.8.3  Definitions and basic properties of transitive closures   ctcl 14830
            5.8.4  Exponentiation of relations   crelexp 14864
            5.8.5  Reflexive-transitive closure as an indexed union   crtrcl 14900
            *5.8.6  Principle of transitive induction.   relexpindlem 14908
      5.9  Elementary real and complex functions
            5.9.1  The "shift" operation   cshi 14911
            5.9.2  Signum (sgn or sign) function   csgn 14931
            5.9.3  Real and imaginary parts; conjugate   ccj 14941
            5.9.4  Square root; absolute value   csqrt 15078
      5.10  Elementary limits and convergence
            5.10.1  Superior limit (lim sup)   clsp 15312
            5.10.2  Limits   cli 15326
            5.10.3  Finite and infinite sums   csu 15530
            5.10.4  The binomial theorem   binomlem 15674
            5.10.5  The inclusion/exclusion principle   incexclem 15681
            5.10.6  Infinite sums (cont.)   isumshft 15684
            5.10.7  Miscellaneous converging and diverging sequences   divrcnv 15697
            5.10.8  Arithmetic series   arisum 15705
            5.10.9  Geometric series   expcnv 15709
            5.10.10  Ratio test for infinite series convergence   cvgrat 15728
            5.10.11  Mertens' theorem   mertenslem1 15729
            5.10.12  Finite and infinite products   prodf 15732
                  5.10.12.1  Product sequences   prodf 15732
                  5.10.12.2  Non-trivial convergence   ntrivcvg 15742
                  5.10.12.3  Complex products   cprod 15748
                  5.10.12.4  Finite products   fprod 15784
                  5.10.12.5  Infinite products   iprodclim 15841
            5.10.13  Falling and Rising Factorial   cfallfac 15847
            5.10.14  Bernoulli polynomials and sums of k-th powers   cbp 15889
      5.11  Elementary trigonometry
            5.11.1  The exponential, sine, and cosine functions   ce 15904
                  5.11.1.1  The circle constant (tau = 2 pi)   ctau 16044
            5.11.2  _e is irrational   eirrlem 16046
      5.12  Cardinality of real and complex number subsets
            5.12.1  Countability of integers and rationals   xpnnen 16053
            5.12.2  The reals are uncountable   rpnnen2lem1 16056
*PART 6  ELEMENTARY NUMBER THEORY
      6.1  Elementary properties of divisibility
            6.1.1  Irrationality of square root of 2   sqrt2irrlem 16090
            6.1.2  Some Number sets are chains of proper subsets   nthruc 16094
            6.1.3  The divides relation   cdvds 16096
            *6.1.4  Even and odd numbers   evenelz 16178
            6.1.5  The division algorithm   divalglem0 16235
            6.1.6  Bit sequences   cbits 16259
            6.1.7  The greatest common divisor operator   cgcd 16334
            6.1.8  Bézout's identity   bezoutlem1 16380
            6.1.9  Algorithms   nn0seqcvgd 16406
            6.1.10  Euclid's Algorithm   eucalgval2 16417
            *6.1.11  The least common multiple   clcm 16424
            *6.1.12  Coprimality and Euclid's lemma   coprmgcdb 16485
            6.1.13  Cancellability of congruences   congr 16500
      6.2  Elementary prime number theory
            *6.2.1  Elementary properties   cprime 16507
            *6.2.2  Coprimality and Euclid's lemma (cont.)   coprm 16547
            6.2.3  Properties of the canonical representation of a rational   cnumer 16568
            6.2.4  Euler's theorem   codz 16595
            6.2.5  Arithmetic modulo a prime number   modprm1div 16629
            6.2.6  Pythagorean Triples   coprimeprodsq 16640
            6.2.7  The prime count function   cpc 16668
            6.2.8  Pocklington's theorem   prmpwdvds 16736
            6.2.9  Infinite primes theorem   unbenlem 16740
            6.2.10  Sum of prime reciprocals   prmreclem1 16748
            6.2.11  Fundamental theorem of arithmetic   1arithlem1 16755
            6.2.12  Lagrange's four-square theorem   cgz 16761
            6.2.13  Van der Waerden's theorem   cvdwa 16797
            6.2.14  Ramsey's theorem   cram 16831
            *6.2.15  Primorial function   cprmo 16863
            *6.2.16  Prime gaps   prmgaplem1 16881
            6.2.17  Decimal arithmetic (cont.)   dec2dvds 16895
            6.2.18  Cyclical shifts of words (cont.)   cshwsidrepsw 16926
            6.2.19  Specific prime numbers   prmlem0 16938
            6.2.20  Very large primes   1259lem1 16963
PART 7  BASIC STRUCTURES
      7.1  Extensible structures
            *7.1.1  Basic definitions   cstr 16978
                  7.1.1.1  Extensible structures as structures with components   cstr 16978
                  7.1.1.2  Substitution of components   csts 16995
                  7.1.1.3  Slots   cslot 17013
                  *7.1.1.4  Structure component indices   cnx 17025
                  7.1.1.5  Base sets   cbs 17043
                  7.1.1.6  Base set restrictions   cress 17072
            7.1.2  Slot definitions   cplusg 17093
            7.1.3  Definition of the structure product   crest 17262
            7.1.4  Definition of the structure quotient   cordt 17341
      7.2  Moore spaces
            7.2.1  Moore closures   mrcflem 17446
            7.2.2  Independent sets in a Moore system   mrisval 17470
            7.2.3  Algebraic closure systems   isacs 17491
PART 8  BASIC CATEGORY THEORY
      8.1  Categories
            8.1.1  Categories   ccat 17504
            8.1.2  Opposite category   coppc 17551
            8.1.3  Monomorphisms and epimorphisms   cmon 17571
            8.1.4  Sections, inverses, isomorphisms   csect 17587
            *8.1.5  Isomorphic objects   ccic 17638
            8.1.6  Subcategories   cssc 17650
            8.1.7  Functors   cfunc 17700
            8.1.8  Full & faithful functors   cful 17749
            8.1.9  Natural transformations and the functor category   cnat 17788
            8.1.10  Initial, terminal and zero objects of a category   cinito 17827
      8.2  Arrows (disjointified hom-sets)
            8.2.1  Identity and composition for arrows   cida 17899
      8.3  Examples of categories
            8.3.1  The category of sets   csetc 17921
            8.3.2  The category of categories   ccatc 17944
            *8.3.3  The category of extensible structures   fncnvimaeqv 17967
      8.4  Categorical constructions
            8.4.1  Product of categories   cxpc 18016
            8.4.2  Functor evaluation   cevlf 18058
            8.4.3  Hom functor   chof 18097
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 18280
            9.5.2  Complete lattices   ccla 18347
            9.5.3  Distributive lattices   cdlat 18369
            9.5.4  Subset order structures   cipo 18376
      9.6  Posets, directed sets, and lattices as relations
            *9.6.1  Posets and lattices as relations   cps 18413
            9.6.2  Directed sets, nets   cdir 18443
PART 10  BASIC ALGEBRAIC STRUCTURES
      10.1  Monoids
            *10.1.1  Magmas   cplusf 18454
            *10.1.2  Identity elements   mgmidmo 18475
            *10.1.3  Iterated sums in a magma   gsumvalx 18491
            *10.1.4  Semigroups   csgrp 18505
            *10.1.5  Definition and basic properties of monoids   cmnd 18516
            10.1.6  Monoid homomorphisms and submonoids   cmhm 18559
            *10.1.7  Iterated sums in a monoid   gsumvallem2 18604
            10.1.8  Free monoids   cfrmd 18617
                  *10.1.8.1  Monoid of endofunctions   cefmnd 18638
            10.1.9  Examples and counterexamples for magmas, semigroups and monoids   mgm2nsgrplem1 18688
      10.2  Groups
            10.2.1  Definition and basic properties   cgrp 18708
            *10.2.2  Group multiple operation   cmg 18831
            10.2.3  Subgroups and Quotient groups   csubg 18881
            *10.2.4  Cyclic monoids and groups   cycsubmel 18952
            10.2.5  Elementary theory of group homomorphisms   cghm 18964
            10.2.6  Isomorphisms of groups   cgim 19006
            10.2.7  Group actions   cga 19028
            10.2.8  Centralizers and centers   ccntz 19054
            10.2.9  The opposite group   coppg 19082
            10.2.10  Symmetric groups   csymg 19107
                  *10.2.10.1  Definition and basic properties   csymg 19107
                  10.2.10.2  Cayley's theorem   cayleylem1 19153
                  10.2.10.3  Permutations fixing one element   symgfix2 19157
                  *10.2.10.4  Transpositions in the symmetric group   cpmtr 19182
                  10.2.10.5  The sign of a permutation   cpsgn 19230
            10.2.11  p-Groups and Sylow groups; Sylow's theorems   cod 19265
            10.2.12  Direct products   clsm 19375
                  10.2.12.1  Direct products (extension)   smndlsmidm 19397
            10.2.13  Free groups   cefg 19447
            10.2.14  Abelian groups   ccmn 19521
                  10.2.14.1  Definition and basic properties   ccmn 19521
                  10.2.14.2  Cyclic groups   ccyg 19613
                  10.2.14.3  Group sum operation   gsumval3a 19639
                  10.2.14.4  Group sums over (ranges of) integers   fsfnn0gsumfsffz 19719
                  10.2.14.5  Internal direct products   cdprd 19731
                  10.2.14.6  The Fundamental Theorem of Abelian Groups   ablfacrplem 19803
            10.2.15  Simple groups   csimpg 19828
                  10.2.15.1  Definition and basic properties   csimpg 19828
                  10.2.15.2  Classification of abelian simple groups   ablsimpnosubgd 19842
      10.3  Rings
            10.3.1  Multiplicative Group   cmgp 19855
            *10.3.2  Ring unity (multiplicative identity)   cur 19872
            10.3.3  Semirings   csrg 19876
                  *10.3.3.1  The binomial theorem for semirings   srgbinomlem1 19911
            10.3.4  Definition and basic properties of unital rings   crg 19918
            10.3.5  Opposite ring   coppr 20001
            10.3.6  Divisibility   cdsr 20020
            10.3.7  Ring primes   crpm 20094
            10.3.8  Ring homomorphisms   crh 20096
      10.4  Division rings and fields
            10.4.1  Definition and basic properties   cdr 20138
            10.4.2  Subrings of a ring   csubrg 20171
                  10.4.2.1  Sub-division rings   csdrg 20212
            10.4.3  Absolute value (abstract algebra)   cabv 20228
            10.4.4  Star rings   cstf 20255
      10.5  Left modules
            10.5.1  Definition and basic properties   clmod 20275
            10.5.2  Subspaces and spans in a left module   clss 20345
            10.5.3  Homomorphisms and isomorphisms of left modules   clmhm 20433
            10.5.4  Subspace sum; bases for a left module   clbs 20488
      10.6  Vector spaces
            10.6.1  Definition and basic properties   clvec 20516
      10.7  Ideals
            10.7.1  The subring algebra; ideals   csra 20582
            10.7.2  Two-sided ideals and quotient rings   c2idl 20654
            10.7.3  Principal ideal rings. Divisibility in the integers   clpidl 20664
            10.7.4  Nonzero rings and zero rings   cnzr 20680
            10.7.5  Left regular elements. More kinds of rings   crlreg 20702
      10.8  The complex numbers as an algebraic extensible structure
            10.8.1  Definition and basic properties   cpsmet 20733
            *10.8.2  Ring of integers   czring 20822
            10.8.3  Algebraic constructions based on the complex numbers   czrh 20853
            10.8.4  Signs as subgroup of the complex numbers   cnmsgnsubg 20934
            10.8.5  Embedding of permutation signs into a ring   zrhpsgnmhm 20941
            10.8.6  The ordered field of real numbers   crefld 20961
      10.9  Generalized pre-Hilbert and Hilbert spaces
            10.9.1  Definition and basic properties   cphl 20981
            10.9.2  Orthocomplements and closed subspaces   cocv 21017
            10.9.3  Orthogonal projection and orthonormal bases   cpj 21059
*PART 11  BASIC LINEAR ALGEBRA
      11.1  Vectors and free modules
            *11.1.1  Direct sum of left modules   cdsmm 21090
            *11.1.2  Free modules   cfrlm 21105
            *11.1.3  Standard basis (unit vectors)   cuvc 21141
            *11.1.4  Independent sets and families   clindf 21163
            11.1.5  Characterization of free modules   lmimlbs 21195
      11.2  Associative algebras
            11.2.1  Definition and basic properties   casa 21209
      11.3  Abstract multivariate polynomials
            11.3.1  Definition and basic properties   cmps 21259
            11.3.2  Polynomial evaluation   ces 21432
            11.3.3  Additional definitions for (multivariate) polynomials   cslv 21470
            *11.3.4  Univariate polynomials   cps1 21498
            11.3.5  Univariate polynomial evaluation   ces1 21631
      *11.4  Matrices
            *11.4.1  The matrix multiplication   cmmul 21684
            *11.4.2  Square matrices   cmat 21706
            *11.4.3  The matrix algebra   matmulr 21739
            *11.4.4  Matrices of dimension 0 and 1   mat0dimbas0 21767
            *11.4.5  The subalgebras of diagonal and scalar matrices   cdmat 21789
            *11.4.6  Multiplication of a matrix with a "column vector"   cmvmul 21841
            11.4.7  Replacement functions for a square matrix   cmarrep 21857
            11.4.8  Submatrices   csubma 21877
      11.5  The determinant
            11.5.1  Definition and basic properties   cmdat 21885
            11.5.2  Determinants of 2 x 2 -matrices   m2detleiblem1 21925
            11.5.3  The matrix adjugate/adjunct   cmadu 21933
            *11.5.4  Laplace expansion of determinants (special case)   symgmatr01lem 21954
            11.5.5  Inverse matrix   invrvald 21977
            *11.5.6  Cramer's rule   slesolvec 21980
      *11.6  Polynomial matrices
            11.6.1  Basic properties   pmatring 21993
            *11.6.2  Constant polynomial matrices   ccpmat 22004
            *11.6.3  Collecting coefficients of polynomial matrices   cdecpmat 22063
            *11.6.4  Ring isomorphism between polynomial matrices and polynomials over matrices   cpm2mp 22093
      *11.7  The characteristic polynomial
            *11.7.1  Definition and basic properties   cchpmat 22127
            *11.7.2  The characteristic factor function G   fvmptnn04if 22150
            *11.7.3  The Cayley-Hamilton theorem   cpmadurid 22168
PART 12  BASIC TOPOLOGY
      12.1  Topology
            *12.1.1  Topological spaces   ctop 22194
                  12.1.1.1  Topologies   ctop 22194
                  12.1.1.2  Topologies on sets   ctopon 22211
                  12.1.1.3  Topological spaces   ctps 22233
            12.1.2  Topological bases   ctb 22247
            12.1.3  Examples of topologies   distop 22297
            12.1.4  Closure and interior   ccld 22319
            12.1.5  Neighborhoods   cnei 22400
            12.1.6  Limit points and perfect sets   clp 22437
            12.1.7  Subspace topologies   restrcl 22460
            12.1.8  Order topology   ordtbaslem 22491
            12.1.9  Limits and continuity in topological spaces   ccn 22527
            12.1.10  Separated spaces: T0, T1, T2 (Hausdorff) ...   ct0 22609
            12.1.11  Compactness   ccmp 22689
            12.1.12  Bolzano-Weierstrass theorem   bwth 22713
            12.1.13  Connectedness   cconn 22714
            12.1.14  First- and second-countability   c1stc 22740
            12.1.15  Local topological properties   clly 22767
            12.1.16  Refinements   cref 22805
            12.1.17  Compactly generated spaces   ckgen 22836
            12.1.18  Product topologies   ctx 22863
            12.1.19  Continuous function-builders   cnmptid 22964
            12.1.20  Quotient maps and quotient topology   ckq 22996
            12.1.21  Homeomorphisms   chmeo 23056
      12.2  Filters and filter bases
            12.2.1  Filter bases   elmptrab 23130
            12.2.2  Filters   cfil 23148
            12.2.3  Ultrafilters   cufil 23202
            12.2.4  Filter limits   cfm 23236
            12.2.5  Extension by continuity   ccnext 23362
            12.2.6  Topological groups   ctmd 23373
            12.2.7  Infinite group sum on topological groups   ctsu 23429
            12.2.8  Topological rings, fields, vector spaces   ctrg 23459
      12.3  Uniform Structures and Spaces
            12.3.1  Uniform structures   cust 23503
            12.3.2  The topology induced by an uniform structure   cutop 23534
            12.3.3  Uniform Spaces   cuss 23557
            12.3.4  Uniform continuity   cucn 23579
            12.3.5  Cauchy filters in uniform spaces   ccfilu 23590
            12.3.6  Complete uniform spaces   ccusp 23601
      12.4  Metric spaces
            12.4.1  Pseudometric spaces   ispsmet 23609
            12.4.2  Basic metric space properties   cxms 23622
            12.4.3  Metric space balls   blfvalps 23688
            12.4.4  Open sets of a metric space   mopnval 23743
            12.4.5  Continuity in metric spaces   metcnp3 23848
            12.4.6  The uniform structure generated by a metric   metuval 23857
            12.4.7  Examples of metric spaces   dscmet 23880
            *12.4.8  Normed algebraic structures   cnm 23884
            12.4.9  Normed space homomorphisms (bounded linear operators)   cnmo 24021
            12.4.10  Topology on the reals   qtopbaslem 24074
            12.4.11  Topological definitions using the reals   cii 24190
            12.4.12  Path homotopy   chtpy 24282
            12.4.13  The fundamental group   cpco 24315
      12.5  Metric subcomplex vector spaces
            12.5.1  Subcomplex modules   cclm 24377
            *12.5.2  Subcomplex vector spaces   ccvs 24438
            *12.5.3  Normed subcomplex vector spaces   isncvsngp 24465
            12.5.4  Subcomplex pre-Hilbert spaces   ccph 24482
            12.5.5  Convergence and completeness   ccfil 24568
            12.5.6  Baire's Category Theorem   bcthlem1 24640
            12.5.7  Banach spaces and subcomplex Hilbert spaces   ccms 24648
                  12.5.7.1  The complete ordered field of the real numbers   retopn 24695
            12.5.8  Euclidean spaces   crrx 24699
            12.5.9  Minimizing Vector Theorem   minveclem1 24740
            12.5.10  Projection Theorem   pjthlem1 24753
PART 13  BASIC REAL AND COMPLEX ANALYSIS
      13.1  Continuity
            13.1.1  Intermediate value theorem   pmltpclem1 24764
      13.2  Integrals
            13.2.1  Lebesgue measure   covol 24778
            13.2.2  Lebesgue integration   cmbf 24930
                  13.2.2.1  Lesbesgue integral   cmbf 24930
                  13.2.2.2  Lesbesgue directed integral   cdit 25162
      13.3  Derivatives
            13.3.1  Real and complex differentiation   climc 25178
                  13.3.1.1  Derivatives of functions of one complex or real variable   climc 25178
                  13.3.1.2  Results on real differentiation   dvferm1lem 25300
PART 14  BASIC REAL AND COMPLEX FUNCTIONS
      14.1  Polynomials
            14.1.1  Polynomial degrees   cmdg 25367
            14.1.2  The division algorithm for univariate polynomials   cmn1 25442
            14.1.3  Elementary properties of complex polynomials   cply 25497
            14.1.4  The division algorithm for polynomials   cquot 25602
            14.1.5  Algebraic numbers   caa 25626
            14.1.6  Liouville's approximation theorem   aalioulem1 25644
      14.2  Sequences and series
            14.2.1  Taylor polynomials and Taylor's theorem   ctayl 25664
            14.2.2  Uniform convergence   culm 25687
            14.2.3  Power series   pserval 25721
      14.3  Basic trigonometry
            14.3.1  The exponential, sine, and cosine functions (cont.)   efcn 25754
            14.3.2  Properties of pi = 3.14159...   pilem1 25762
            14.3.3  Mapping of the exponential function   efgh 25849
            14.3.4  The natural logarithm on complex numbers   clog 25862
            *14.3.5  Logarithms to an arbitrary base   clogb 26066
            14.3.6  Theorems of Pythagoras, isosceles triangles, and intersecting chords   angval 26103
            14.3.7  Solutions of quadratic, cubic, and quartic equations   quad2 26141
            14.3.8  Inverse trigonometric functions   casin 26164
            14.3.9  The Birthday Problem   log2ublem1 26248
            14.3.10  Areas in R^2   carea 26257
            14.3.11  More miscellaneous converging sequences   rlimcnp 26267
            14.3.12  Inequality of arithmetic and geometric means   cvxcl 26286
            14.3.13  Euler-Mascheroni constant   cem 26293
            14.3.14  Zeta function   czeta 26314
            14.3.15  Gamma function   clgam 26317
      14.4  Basic number theory
            14.4.1  Wilson's theorem   wilthlem1 26369
            14.4.2  The Fundamental Theorem of Algebra   ftalem1 26374
            14.4.3  The Basel problem (ζ(2) = π2/6)   basellem1 26382
            14.4.4  Number-theoretical functions   ccht 26392
            14.4.5  Perfect Number Theorem   mersenne 26527
            14.4.6  Characters of Z/nZ   cdchr 26532
            14.4.7  Bertrand's postulate   bcctr 26575
            *14.4.8  Quadratic residues and the Legendre symbol   clgs 26594
            *14.4.9  Gauss' Lemma   gausslemma2dlem0a 26656
            14.4.10  Quadratic reciprocity   lgseisenlem1 26675
            14.4.11  All primes 4n+1 are the sum of two squares   2sqlem1 26717
            14.4.12  Chebyshev's Weak Prime Number Theorem, Dirichlet's Theorem   chebbnd1lem1 26769
            14.4.13  The Prime Number Theorem   mudivsum 26830
            14.4.14  Ostrowski's theorem   abvcxp 26915
PART 15  SURREAL NUMBERS
            15.0.1  Definitions and initial properties   csur 26940
            15.0.2  Ordering   sltsolem1 26975
            15.0.3  Birthday Function   bdayfo 26977
            15.0.4  Density   fvnobday 26978
            *15.0.5  Full-Eta Property   bdayimaon 26993
            15.0.6  Ordering Theorems   csle 27044
            15.0.7  Birthday Theorems   bdayfun 27064
            15.0.8  Conway cuts   csslt 27072
            15.0.9  Zero and One   c0s 27113
            15.0.10  Cuts and Options   cmade 27124
            15.0.11  Cofinality and coinitiality   cofsslt 27186
*PART 16  ELEMENTARY GEOMETRY
      16.1  Definition and Tarski's Axioms of Geometry
            16.1.1  Justification for the congruence notation   tgjustf 27244
      16.2  Tarskian Geometry
            16.2.1  Congruence   tgcgrcomimp 27248
            16.2.2  Betweenness   tgbtwntriv2 27258
            16.2.3  Dimension   tglowdim1 27271
            16.2.4  Betweenness and Congruence   tgifscgr 27279
            16.2.5  Congruence of a series of points   ccgrg 27281
            16.2.6  Motions   cismt 27303
            16.2.7  Colinearity   tglng 27317
            16.2.8  Connectivity of betweenness   tgbtwnconn1lem1 27343
            16.2.9  Less-than relation in geometric congruences   cleg 27353
            16.2.10  Rays   chlg 27371
            16.2.11  Lines   btwnlng1 27390
            16.2.12  Point inversions   cmir 27423
            16.2.13  Right angles   crag 27464
            16.2.14  Half-planes   islnopp 27510
            16.2.15  Midpoints and Line Mirroring   cmid 27543
            16.2.16  Congruence of angles   ccgra 27578
            16.2.17  Angle Comparisons   cinag 27606
            16.2.18  Congruence Theorems   tgsas1 27625
            16.2.19  Equilateral triangles   ceqlg 27636
      16.3  Properties of geometries
            16.3.1  Isomorphisms between geometries   f1otrgds 27640
      16.4  Geometry in Hilbert spaces
            16.4.1  Geometry in the complex plane   cchhllem 27664
            16.4.2  Geometry in Euclidean spaces   cee 27666
                  16.4.2.1  Definition of the Euclidean space   cee 27666
                  16.4.2.2  Tarski's axioms for geometry for the Euclidean space   axdimuniq 27691
                  16.4.2.3  EE^n fulfills Tarski's Axioms   ceeng 27755
*PART 17  GRAPH THEORY
      *17.1  Vertices and edges
            17.1.1  The edge function extractor for extensible structures   cedgf 27766
            *17.1.2  Vertices and indexed edges   cvtx 27776
                  17.1.2.1  Definitions and basic properties   cvtx 27776
                  17.1.2.2  The vertices and edges of a graph represented as ordered pair   opvtxval 27783
                  17.1.2.3  The vertices and edges of a graph represented as extensible structure   funvtxdmge2val 27791
                  17.1.2.4  Representations of graphs without edges   snstrvtxval 27817
                  17.1.2.5  Degenerated cases of representations of graphs   vtxval0 27819
            17.1.3  Edges as range of the edge function   cedg 27827
      *17.2  Undirected graphs
            17.2.1  Undirected hypergraphs   cuhgr 27836
            17.2.2  Undirected pseudographs and multigraphs   cupgr 27860
            *17.2.3  Loop-free graphs   umgrislfupgrlem 27902
            17.2.4  Edges as subsets of vertices of graphs   uhgredgiedgb 27906
            *17.2.5  Undirected simple graphs   cuspgr 27928
            17.2.6  Examples for graphs   usgr0e 28013
            17.2.7  Subgraphs   csubgr 28044
            17.2.8  Finite undirected simple graphs   cfusgr 28093
            17.2.9  Neighbors, complete graphs and universal vertices   cnbgr 28109
                  17.2.9.1  Neighbors   cnbgr 28109
                  17.2.9.2  Universal vertices   cuvtx 28162
                  17.2.9.3  Complete graphs   ccplgr 28186
            17.2.10  Vertex degree   cvtxdg 28242
            *17.2.11  Regular graphs   crgr 28332
      *17.3  Walks, paths and cycles
            *17.3.1  Walks   cewlks 28372
            17.3.2  Walks for loop-free graphs   lfgrwlkprop 28464
            17.3.3  Trails   ctrls 28467
            17.3.4  Paths and simple paths   cpths 28489
            17.3.5  Closed walks   cclwlks 28547
            17.3.6  Circuits and cycles   ccrcts 28561
            *17.3.7  Walks as words   cwwlks 28599
            17.3.8  Walks/paths of length 2 (as length 3 strings)   2wlkdlem1 28699
            17.3.9  Walks in regular graphs   rusgrnumwwlkl1 28742
            *17.3.10  Closed walks as words   cclwwlk 28754
                  17.3.10.1  Closed walks as words   cclwwlk 28754
                  17.3.10.2  Closed walks of a fixed length as words   cclwwlkn 28797
                  17.3.10.3  Closed walks on a vertex of a fixed length as words   cclwwlknon 28860
            17.3.11  Examples for walks, trails and paths   0ewlk 28887
            17.3.12  Connected graphs   cconngr 28959
      17.4  Eulerian paths and the Konigsberg Bridge problem
            *17.4.1  Eulerian paths   ceupth 28970
            *17.4.2  The Königsberg Bridge problem   konigsbergvtx 29019
      17.5  The Friendship Theorem
            17.5.1  Friendship graphs - basics   cfrgr 29031
            17.5.2  The friendship theorem for small graphs   frgr1v 29044
            17.5.3  Theorems according to Mertzios and Unger   2pthfrgrrn 29055
            *17.5.4  Huneke's Proof of the Friendship Theorem   frgrncvvdeqlem1 29072
PART 18  GUIDES AND MISCELLANEA
      18.1  Guides (conventions, explanations, and examples)
            *18.1.1  Conventions   conventions 29173
            18.1.2  Natural deduction   natded 29176
            *18.1.3  Natural deduction examples   ex-natded5.2 29177
            18.1.4  Definitional examples   ex-or 29194
            18.1.5  Other examples   aevdemo 29233
      18.2  Humor
            18.2.1  April Fool's theorem   avril1 29236
      18.3  (Future - to be reviewed and classified)
            18.3.1  Planar incidence geometry   cplig 29245
            *18.3.2  Aliases kept to prevent broken links   dummylink 29258
*PART 19  COMPLEX TOPOLOGICAL VECTOR SPACES (DEPRECATED)
      *19.1  Additional material on group theory (deprecated)
            19.1.1  Definitions and basic properties for groups   cgr 29260
            19.1.2  Abelian groups   cablo 29315
      19.2  Complex vector spaces
            19.2.1  Definition and basic properties   cvc 29329
            19.2.2  Examples of complex vector spaces   cnaddabloOLD 29352
      19.3  Normed complex vector spaces
            19.3.1  Definition and basic properties   cnv 29355
            19.3.2  Examples of normed complex vector spaces   cnnv 29448
            19.3.3  Induced metric of a normed complex vector space   imsval 29456
            19.3.4  Inner product   cdip 29471
            19.3.5  Subspaces   css 29492
      19.4  Operators on complex vector spaces
            19.4.1  Definitions and basic properties   clno 29511
      19.5  Inner product (pre-Hilbert) spaces
            19.5.1  Definition and basic properties   ccphlo 29583
            19.5.2  Examples of pre-Hilbert spaces   cncph 29590
            19.5.3  Properties of pre-Hilbert spaces   isph 29593
      19.6  Complex Banach spaces
            19.6.1  Definition and basic properties   ccbn 29633
            19.6.2  Examples of complex Banach spaces   cnbn 29640
            19.6.3  Uniform Boundedness Theorem   ubthlem1 29641
            19.6.4  Minimizing Vector Theorem   minvecolem1 29645
      19.7  Complex Hilbert spaces
            19.7.1  Definition and basic properties   chlo 29656
            19.7.2  Standard axioms for a complex Hilbert space   hlex 29669
            19.7.3  Examples of complex Hilbert spaces   cnchl 29687
            19.7.4  Hellinger-Toeplitz Theorem   htthlem 29688
*PART 20  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      20.1  Axiomatization of complex pre-Hilbert spaces
            20.1.1  Basic Hilbert space definitions   chba 29690
            20.1.2  Preliminary ZFC lemmas   df-hnorm 29739
            *20.1.3  Derive the Hilbert space axioms from ZFC set theory   axhilex-zf 29752
            *20.1.4  Introduce the vector space axioms for a Hilbert space   ax-hilex 29770
            20.1.5  Vector operations   hvmulex 29782
            20.1.6  Inner product postulates for a Hilbert space   ax-hfi 29850
      20.2  Inner product and norms
            20.2.1  Inner product   his5 29857
            20.2.2  Norms   dfhnorm2 29893
            20.2.3  Relate Hilbert space to normed complex vector spaces   hilablo 29931
            20.2.4  Bunjakovaskij-Cauchy-Schwarz inequality   bcsiALT 29950
      20.3  Cauchy sequences and completeness axiom
            20.3.1  Cauchy sequences and limits   hcau 29955
            20.3.2  Derivation of the completeness axiom from ZF set theory   hilmet 29965
            20.3.3  Completeness postulate for a Hilbert space   ax-hcompl 29973
            20.3.4  Relate Hilbert space to ZFC pre-Hilbert and Hilbert spaces   hhcms 29974
      20.4  Subspaces and projections
            20.4.1  Subspaces   df-sh 29978
            20.4.2  Closed subspaces   df-ch 29992
            20.4.3  Orthocomplements   df-oc 30023
            20.4.4  Subspace sum, span, lattice join, lattice supremum   df-shs 30079
            20.4.5  Projection theorem   pjhthlem1 30162
            20.4.6  Projectors   df-pjh 30166
      20.5  Properties of Hilbert subspaces
            20.5.1  Orthomodular law   omlsilem 30173
            20.5.2  Projectors (cont.)   pjhtheu2 30187
            20.5.3  Hilbert lattice operations   sh0le 30211
            20.5.4  Span (cont.) and one-dimensional subspaces   spansn0 30312
            20.5.5  Commutes relation for Hilbert lattice elements   df-cm 30354
            20.5.6  Foulis-Holland theorem   fh1 30389
            20.5.7  Quantum Logic Explorer axioms   qlax1i 30398
            20.5.8  Orthogonal subspaces   chscllem1 30408
            20.5.9  Orthoarguesian laws 5OA and 3OA   5oalem1 30425
            20.5.10  Projectors (cont.)   pjorthi 30440
            20.5.11  Mayet's equation E_3   mayete3i 30499
      20.6  Operators on Hilbert spaces
            *20.6.1  Operator sum, difference, and scalar multiplication   df-hosum 30501
            20.6.2  Zero and identity operators   df-h0op 30519
            20.6.3  Operations on Hilbert space operators   hoaddcl 30529
            20.6.4  Linear, continuous, bounded, Hermitian, unitary operators and norms   df-nmop 30610
            20.6.5  Linear and continuous functionals and norms   df-nmfn 30616
            20.6.6  Adjoint   df-adjh 30620
            20.6.7  Dirac bra-ket notation   df-bra 30621
            20.6.8  Positive operators   df-leop 30623
            20.6.9  Eigenvectors, eigenvalues, spectrum   df-eigvec 30624
            20.6.10  Theorems about operators and functionals   nmopval 30627
            20.6.11  Riesz lemma   riesz3i 30833
            20.6.12  Adjoints (cont.)   cnlnadjlem1 30838
            20.6.13  Quantum computation error bound theorem   unierri 30875
            20.6.14  Dirac bra-ket notation (cont.)   branmfn 30876
            20.6.15  Positive operators (cont.)   leopg 30893
            20.6.16  Projectors as operators   pjhmopi 30917
      20.7  States on a Hilbert lattice and Godowski's equation
            20.7.1  States on a Hilbert lattice   df-st 30982
            20.7.2  Godowski's equation   golem1 31042
      20.8  Cover relation, atoms, exchange axiom, and modular symmetry
            20.8.1  Covers relation; modular pairs   df-cv 31050
            20.8.2  Atoms   df-at 31109
            20.8.3  Superposition principle   superpos 31125
            20.8.4  Atoms, exchange and covering properties, atomicity   chcv1 31126
            20.8.5  Irreducibility   chirredlem1 31161
            20.8.6  Atoms (cont.)   atcvat3i 31167
            20.8.7  Modular symmetry   mdsymlem1 31174
PART 21  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      21.1  Mathboxes for user contributions
            21.1.1  Mathbox guidelines   mathbox 31213
      21.2  Mathbox for Stefan Allan
      21.3  Mathbox for Thierry Arnoux
            21.3.1  Propositional Calculus - misc additions   bian1d 31218
            21.3.2  Predicate Calculus   sbc2iedf 31225
                  21.3.2.1  Predicate Calculus - misc additions   sbc2iedf 31225
                  21.3.2.2  Restricted quantification - misc additions   ralcom4f 31227
                  21.3.2.3  Equality   eqtrb 31232
                  21.3.2.4  Double restricted existential uniqueness quantification   opsbc2ie 31233
                  21.3.2.5  Double restricted existential uniqueness quantification syntax   w2reu 31235
                  21.3.2.6  Substitution (without distinct variables) - misc additions   sbceqbidf 31244
                  21.3.2.7  Existential "at most one" - misc additions   mo5f 31246
                  21.3.2.8  Existential uniqueness - misc additions   reuxfrdf 31248
                  21.3.2.9  Restricted "at most one" - misc additions   rmoxfrd 31250
            21.3.3  General Set Theory   dmrab 31253
                  21.3.3.1  Class abstractions (a.k.a. class builders)   dmrab 31253
                  21.3.3.2  Image Sets   abrexdomjm 31261
                  21.3.3.3  Set relations and operations - misc additions   elunsn 31267
                  21.3.3.4  Unordered pairs   eqsnd 31284
                  21.3.3.5  Conditional operator - misc additions   ifeqeqx 31292
                  21.3.3.6  Set union   uniinn0 31297
                  21.3.3.7  Indexed union - misc additions   cbviunf 31302
                  21.3.3.8  Indexed intersection - misc additions   iinabrex 31315
                  21.3.3.9  Disjointness - misc additions   disjnf 31316
            21.3.4  Relations and Functions   xpdisjres 31344
                  21.3.4.1  Relations - misc additions   xpdisjres 31344
                  21.3.4.2  Functions - misc additions   ac6sf2 31367
                  21.3.4.3  Operations - misc additions   mpomptxf 31423
                  21.3.4.4  Explicit Functions with one or two points as a domain   cosnopne 31434
                  21.3.4.5  Isomorphisms - misc. additions   gtiso 31440
                  21.3.4.6  Disjointness (additional proof requiring functions)   disjdsct 31442
                  21.3.4.7  First and second members of an ordered pair - misc additions   df1stres 31443
                  21.3.4.8  Supremum - misc additions   supssd 31451
                  21.3.4.9  Finite Sets   imafi2 31453
                  21.3.4.10  Countable Sets   snct 31455
            21.3.5  Real and Complex Numbers   creq0 31477
                  21.3.5.1  Complex operations - misc. additions   creq0 31477
                  21.3.5.2  Ordering on reals - misc additions   lt2addrd 31481
                  21.3.5.3  Extended reals - misc additions   xrlelttric 31482
                  21.3.5.4  Extended nonnegative integers - misc additions   xnn0gt0 31499
                  21.3.5.5  Real number intervals - misc additions   joiniooico 31502
                  21.3.5.6  Finite intervals of integers - misc additions   uzssico 31512
                  21.3.5.7  Half-open integer ranges - misc additions   iundisjfi 31524
                  21.3.5.8  The ` # ` (set size) function - misc additions   hashunif 31533
                  21.3.5.9  The greatest common divisor operator - misc. additions   dvdszzq 31536
                  21.3.5.10  Integers   nnindf 31540
                  21.3.5.11  Decimal numbers   dfdec100 31551
            *21.3.6  Decimal expansion   cdp2 31552
                  *21.3.6.1  Decimal point   cdp 31569
                  21.3.6.2  Division in the extended real number system   cxdiv 31598
            21.3.7  Words over a set - misc additions   wrdfd 31617
                  21.3.7.1  Splicing words (substring replacement)   splfv3 31637
                  21.3.7.2  Cyclic shift of words   1cshid 31638
            21.3.8  Extensible Structures   ressplusf 31642
                  21.3.8.1  Structure restriction operator   ressplusf 31642
                  21.3.8.2  The opposite group   oppgle 31645
                  21.3.8.3  Posets   ressprs 31648
                  21.3.8.4  Complete lattices   clatp0cl 31661
                  21.3.8.5  Order Theory   cmnt 31663
                  21.3.8.6  Extended reals Structure - misc additions   ax-xrssca 31689
                  21.3.8.7  The extended nonnegative real numbers commutative monoid   xrge0base 31701
            21.3.9  Algebra   abliso 31712
                  21.3.9.1  Monoids Homomorphisms   abliso 31712
                  21.3.9.2  Finitely supported group sums - misc additions   gsumsubg 31713
                  21.3.9.3  Centralizers and centers - misc additions   cntzun 31727
                  21.3.9.4  Totally ordered monoids and groups   comnd 31730
                  21.3.9.5  The symmetric group   symgfcoeu 31758
                  21.3.9.6  Transpositions   pmtridf1o 31768
                  21.3.9.7  Permutation Signs   psgnid 31771
                  21.3.9.8  Permutation cycles   ctocyc 31780
                  21.3.9.9  The Alternating Group   evpmval 31819
                  21.3.9.10  Signum in an ordered monoid   csgns 31832
                  21.3.9.11  The Archimedean property for generic ordered algebraic structures   cinftm 31837
                  21.3.9.12  Semiring left modules   cslmd 31860
                  21.3.9.13  Simple groups   prmsimpcyc 31888
                  21.3.9.14  Rings - misc additions   rngurd 31889
                  21.3.9.15  Subfields   sdrgdvcl 31900
                  21.3.9.16  Field extensions generated by a set   cfldgen 31903
                  21.3.9.17  Totally ordered rings and fields   corng 31914
                  21.3.9.18  Ring homomorphisms - misc additions   rhmdvd 31937
                  21.3.9.19  Scalar restriction operation   cresv 31939
                  21.3.9.20  The commutative ring of gaussian integers   gzcrng 31959
                  21.3.9.21  The archimedean ordered field of real numbers   reofld 31960
                  21.3.9.22  The quotient map and quotient modules   qusker 31965
                  21.3.9.23  The ring of integers modulo ` N `   fermltlchr 31978
                  21.3.9.24  Independent sets and families   islinds5 31980
                  *21.3.9.25  Subgroup sum / Sumset / Minkowski sum   elgrplsmsn 31995
                  21.3.9.26  The quotient map   quslsm 32010
                  21.3.9.27  Ideals   intlidl 32019
                  21.3.9.28  Prime Ideals   cprmidl 32027
                  21.3.9.29  Maximal Ideals   cmxidl 32048
                  21.3.9.30  The semiring of ideals of a ring   cidlsrg 32062
                  21.3.9.31  Unique factorization domains   cufd 32078
                  21.3.9.32  Associative algebras   asclmulg 32083
                  21.3.9.33  Univariate Polynomials   fply1 32084
                  21.3.9.34  The subring algebra   sra1r 32099
                  21.3.9.35  Division Ring Extensions   drgext0g 32105
                  21.3.9.36  Vector Spaces   lvecdimfi 32111
                  21.3.9.37  Vector Space Dimension   cldim 32112
            21.3.10  Field Extensions   cfldext 32141
                  21.3.10.1  Algebraic numbers and Minimal polynomials   calgnb 32171
            21.3.11  Matrices   csmat 32178
                  21.3.11.1  Submatrices   csmat 32178
                  21.3.11.2  Matrix literals   clmat 32196
                  21.3.11.3  Laplace expansion of determinants   mdetpmtr1 32208
            21.3.12  Topology   ist0cld 32218
                  21.3.12.1  Open maps   txomap 32219
                  21.3.12.2  Topology of the unit circle   qtopt1 32220
                  21.3.12.3  Refinements   reff 32224
                  21.3.12.4  Open cover refinement property   ccref 32227
                  21.3.12.5  Lindelöf spaces   cldlf 32237
                  21.3.12.6  Paracompact spaces   cpcmp 32240
                  *21.3.12.7  Spectrum of a ring   crspec 32247
                  21.3.12.8  Pseudometrics   cmetid 32271
                  21.3.12.9  Continuity - misc additions   hauseqcn 32283
                  21.3.12.10  Topology of the closed unit interval   elunitge0 32284
                  21.3.12.11  Topology of ` ( RR X. RR ) `   unicls 32288
                  21.3.12.12  Order topology - misc. additions   cnvordtrestixx 32298
                  21.3.12.13  Continuity in topological spaces - misc. additions   mndpluscn 32311
                  21.3.12.14  Topology of the extended nonnegative real numbers ordered monoid   xrge0hmph 32317
                  21.3.12.15  Limits - misc additions   lmlim 32332
                  21.3.12.16  Univariate polynomials   pl1cn 32340
            21.3.13  Uniform Stuctures and Spaces   chcmp 32341
                  21.3.13.1  Hausdorff uniform completion   chcmp 32341
            21.3.14  Topology and algebraic structures   zringnm 32343
                  21.3.14.1  The norm on the ring of the integer numbers   zringnm 32343
                  21.3.14.2  Topological ` ZZ ` -modules   zlm0 32345
                  21.3.14.3  Canonical embedding of the field of the rational numbers into a division ring   cqqh 32357
                  21.3.14.4  Canonical embedding of the real numbers into a complete ordered field   crrh 32378
                  21.3.14.5  Embedding from the extended real numbers into a complete lattice   cxrh 32401
                  21.3.14.6  Canonical embeddings into the ordered field of the real numbers   zrhre 32404
                  *21.3.14.7  Topological Manifolds   cmntop 32407
            21.3.15  Real and complex functions   nexple 32412
                  21.3.15.1  Integer powers - misc. additions   nexple 32412
                  21.3.15.2  Indicator Functions   cind 32413
                  21.3.15.3  Extended sum   cesum 32430
            21.3.16  Mixed Function/Constant operation   cofc 32498
            21.3.17  Abstract measure   csiga 32511
                  21.3.17.1  Sigma-Algebra   csiga 32511
                  21.3.17.2  Generated sigma-Algebra   csigagen 32541
                  *21.3.17.3  lambda and pi-Systems, Rings of Sets   ispisys 32555
                  21.3.17.4  The Borel algebra on the real numbers   cbrsiga 32584
                  21.3.17.5  Product Sigma-Algebra   csx 32591
                  21.3.17.6  Measures   cmeas 32598
                  21.3.17.7  The counting measure   cntmeas 32629
                  21.3.17.8  The Lebesgue measure - misc additions   voliune 32632
                  21.3.17.9  The Dirac delta measure   cdde 32635
                  21.3.17.10  The 'almost everywhere' relation   cae 32640
                  21.3.17.11  Measurable functions   cmbfm 32652
                  21.3.17.12  Borel Algebra on ` ( RR X. RR ) `   br2base 32673
                  *21.3.17.13  Caratheodory's extension theorem   coms 32695
            21.3.18  Integration   itgeq12dv 32730
                  21.3.18.1  Lebesgue integral - misc additions   itgeq12dv 32730
                  21.3.18.2  Bochner integral   citgm 32731
            21.3.19  Euler's partition theorem   oddpwdc 32758
            21.3.20  Sequences defined by strong recursion   csseq 32787
            21.3.21  Fibonacci Numbers   cfib 32800
            21.3.22  Probability   cprb 32811
                  21.3.22.1  Probability Theory   cprb 32811
                  21.3.22.2  Conditional Probabilities   ccprob 32835
                  21.3.22.3  Real-valued Random Variables   crrv 32844
                  21.3.22.4  Preimage set mapping operator   corvc 32859
                  21.3.22.5  Distribution Functions   orvcelval 32872
                  21.3.22.6  Cumulative Distribution Functions   orvclteel 32876
                  21.3.22.7  Probabilities - example   coinfliplem 32882
                  21.3.22.8  Bertrand's Ballot Problem   ballotlemoex 32889
            21.3.23  Signum (sgn or sign) function - misc. additions   sgncl 32942
                  21.3.23.1  Operations on words   ccatmulgnn0dir 32958
            21.3.24  Polynomials with real coefficients - misc additions   plymul02 32962
            21.3.25  Descartes's rule of signs   signspval 32968
                  21.3.25.1  Sign changes in a word over real numbers   signspval 32968
                  21.3.25.2  Counting sign changes in a word over real numbers   signslema 32978
            21.3.26  Number Theory   efcld 33008
                  21.3.26.1  Representations of a number as sums of integers   crepr 33025
                  21.3.26.2  Vinogradov Trigonometric Sums and the Circle Method   cvts 33052
                  21.3.26.3  The Ternary Goldbach Conjecture: Final Statement   ax-hgt749 33061
            21.3.27  Elementary Geometry   cstrkg2d 33081
                  *21.3.27.1  Two-dimensional geometry   cstrkg2d 33081
                  21.3.27.2  Outer Five Segment (not used, no need to move to main)   cafs 33086
            *21.3.28  LeftPad Project   clpad 33091
      *21.4  Mathbox for Jonathan Ben-Naim
            21.4.1  First-order logic and set theory   bnj170 33114
            21.4.2  Well founded induction and recursion   bnj110 33274
            21.4.3  The existence of a minimal element in certain classes   bnj69 33426
            21.4.4  Well-founded induction   bnj1204 33428
            21.4.5  Well-founded recursion, part 1 of 3   bnj60 33478
            21.4.6  Well-founded recursion, part 2 of 3   bnj1500 33484
            21.4.7  Well-founded recursion, part 3 of 3   bnj1522 33488
      21.5  Mathbox for BTernaryTau
            21.5.1  ZF set theory   exdifsn 33489
                  21.5.1.1  Finitism   fineqvrep 33500
            21.5.2  Real and complex numbers   zltp1ne 33504
            21.5.3  Graph theory   lfuhgr 33515
                  21.5.3.1  Acyclic graphs   cacycgr 33540
      21.6  Mathbox for Mario Carneiro
            21.6.1  Predicate calculus with all distinct variables   ax-7d 33557
            21.6.2  Miscellaneous stuff   quartfull 33563
            21.6.3  Derangements and the Subfactorial   deranglem 33564
            21.6.4  The Erdős-Szekeres theorem   erdszelem1 33589
            21.6.5  The Kuratowski closure-complement theorem   kur14lem1 33604
            21.6.6  Retracts and sections   cretr 33615
            21.6.7  Path-connected and simply connected spaces   cpconn 33617
            21.6.8  Covering maps   ccvm 33653
            21.6.9  Normal numbers   snmlff 33727
            21.6.10  Godel-sets of formulas - part 1   cgoe 33731
            21.6.11  Godel-sets of formulas - part 2   cgon 33830
            21.6.12  Models of ZF   cgze 33844
            *21.6.13  Metamath formal systems   cmcn 33858
            21.6.14  Grammatical formal systems   cm0s 33983
            21.6.15  Models of formal systems   cmuv 34003
            21.6.16  Splitting fields   citr 34025
            21.6.17  p-adic number fields   czr 34041
      *21.7  Mathbox for Filip Cernatescu
      21.8  Mathbox for Paul Chapman
            21.8.1  Real and complex numbers (cont.)   climuzcnv 34065
            21.8.2  Miscellaneous theorems   elfzm12 34069
      21.9  Mathbox for Scott Fenton
            21.9.1  ZFC Axioms in primitive form   axextprim 34078
            21.9.2  Untangled classes   untelirr 34085
            21.9.3  Extra propositional calculus theorems   3jaodd 34092
            21.9.4  Misc. Useful Theorems   nepss 34095
            21.9.5  Properties of real and complex numbers   sqdivzi 34110
            21.9.6  Infinite products   iprodefisumlem 34123
            21.9.7  Factorial limits   faclimlem1 34126
            21.9.8  Greatest common divisor and divisibility   gcd32 34132
            21.9.9  Properties of relationships   dftr6 34134
            21.9.10  Properties of functions and mappings   funpsstri 34150
            21.9.11  Set induction (or epsilon induction)   setinds 34163
            21.9.12  Ordinal numbers   elpotr 34166
            21.9.13  Defined equality axioms   axextdfeq 34182
            21.9.14  Hypothesis builders   hbntg 34190
            21.9.15  Well-founded zero, successor, and limits   cwsuc 34195
            21.9.16  Natural operations on ordinals   cnadd 34215
            21.9.17  Surreal numbers: Induction and recursion on one variable   cnorec 34246
            21.9.18  Surreal numbers: Induction and recursion on two variables   cnorec2 34257
            21.9.19  Surreal numbers - addition   cadds 34268
            21.9.20  Surreal numbers - negation and subtraction   cnegs 34307
            21.9.21  Quantifier-free definitions   ctxp 34347
            21.9.22  Alternate ordered pairs   caltop 34473
            21.9.23  Geometry in the Euclidean space   cofs 34499
                  21.9.23.1  Congruence properties   cofs 34499
                  21.9.23.2  Betweenness properties   btwntriv2 34529
                  21.9.23.3  Segment Transportation   ctransport 34546
                  21.9.23.4  Properties relating betweenness and congruence   cifs 34552
                  21.9.23.5  Connectivity of betweenness   btwnconn1lem1 34604
                  21.9.23.6  Segment less than or equal to   csegle 34623
                  21.9.23.7  Outside-of relationship   coutsideof 34636
                  21.9.23.8  Lines and Rays   cline2 34651
            21.9.24  Forward difference   cfwddif 34675
            21.9.25  Rank theorems   rankung 34683
            21.9.26  Hereditarily Finite Sets   chf 34689
      21.10  Mathbox for Jeff Hankins
            21.10.1  Miscellany   a1i14 34704
            21.10.2  Basic topological facts   topbnd 34728
            21.10.3  Topology of the real numbers   ivthALT 34739
            21.10.4  Refinements   cfne 34740
            21.10.5  Neighborhood bases determine topologies   neibastop1 34763
            21.10.6  Lattice structure of topologies   topmtcl 34767
            21.10.7  Filter bases   fgmin 34774
            21.10.8  Directed sets, nets   tailfval 34776
      21.11  Mathbox for Anthony Hart
            21.11.1  Propositional Calculus   tb-ax1 34787
            21.11.2  Predicate Calculus   nalfal 34807
            21.11.3  Miscellaneous single axioms   meran1 34815
            21.11.4  Connective Symmetry   negsym1 34821
      21.12  Mathbox for Chen-Pang He
            21.12.1  Ordinal topology   ontopbas 34832
      21.13  Mathbox for Jeff Hoffman
            21.13.1  Inferences for finite induction on generic function values   fveleq 34855
            21.13.2  gdc.mm   nnssi2 34859
      21.14  Mathbox for Asger C. Ipsen
            21.14.1  Continuous nowhere differentiable functions   dnival 34866
      *21.15  Mathbox for BJ
            *21.15.1  Propositional calculus   bj-mp2c 34935
                  *21.15.1.1  Derived rules of inference   bj-mp2c 34935
                  *21.15.1.2  A syntactic theorem   bj-0 34937
                  21.15.1.3  Minimal implicational calculus   bj-a1k 34939
                  *21.15.1.4  Positive calculus   bj-syl66ib 34950
                  21.15.1.5  Implication and negation   bj-con2com 34956
                  *21.15.1.6  Disjunction   bj-jaoi1 34967
                  *21.15.1.7  Logical equivalence   bj-dfbi4 34969
                  21.15.1.8  The conditional operator for propositions   bj-consensus 34974
                  *21.15.1.9  Propositional calculus: miscellaneous   bj-imbi12 34979
            *21.15.2  Modal logic   bj-axdd2 34989
            *21.15.3  Provability logic   cprvb 34994
            *21.15.4  First-order logic   bj-genr 35003
                  21.15.4.1  Adding ax-gen   bj-genr 35003
                  21.15.4.2  Adding ax-4   bj-2alim 35007
                  21.15.4.3  Adding ax-5   bj-ax12wlem 35040
                  21.15.4.4  Equality and substitution   bj-ssbeq 35049
                  21.15.4.5  Adding ax-6   bj-spimvwt 35065
                  21.15.4.6  Adding ax-7   bj-cbvexw 35072
                  21.15.4.7  Membership predicate, ax-8 and ax-9   bj-ax89 35074
                  21.15.4.8  Adding ax-11   bj-alcomexcom 35077
                  21.15.4.9  Adding ax-12   axc11n11 35079
                  21.15.4.10  Nonfreeness   wnnf 35120
                  21.15.4.11  Adding ax-13   bj-axc10 35180
                  *21.15.4.12  Removing dependencies on ax-13 (and ax-11)   bj-axc10v 35190
                  *21.15.4.13  Distinct var metavariables   bj-hbaeb2 35215
                  *21.15.4.14  Around ~ equsal   bj-equsal1t 35219
                  *21.15.4.15  Some Principia Mathematica proofs   stdpc5t 35224
                  21.15.4.16  Alternate definition of substitution   bj-sbsb 35234
                  21.15.4.17  Lemmas for substitution   bj-sbf3 35236
                  21.15.4.18  Existential uniqueness   bj-eu3f 35239
                  *21.15.4.19  First-order logic: miscellaneous   bj-sblem1 35240
            21.15.5  Set theory   eliminable1 35257
                  *21.15.5.1  Eliminability of class terms   eliminable1 35257
                  *21.15.5.2  Classes without the axiom of extensionality   bj-denoteslem 35269
                  21.15.5.3  Characterization among sets versus among classes   elelb 35296
                  *21.15.5.4  The nonfreeness quantifier for classes   bj-nfcsym 35298
                  *21.15.5.5  Lemmas for class substitution   bj-sbeqALT 35299
                  21.15.5.6  Removing some axiom requirements and disjoint variable conditions   bj-exlimvmpi 35310
                  *21.15.5.7  Class abstractions   bj-elabd2ALT 35327
                  21.15.5.8  Generalized class abstractions   bj-cgab 35335
                  *21.15.5.9  Restricted nonfreeness   wrnf 35343
                  *21.15.5.10  Russell's paradox   bj-ru0 35345
                  21.15.5.11  Curry's paradox in set theory   currysetlem 35348
                  *21.15.5.12  Some disjointness results   bj-n0i 35354
                  *21.15.5.13  Complements on direct products   bj-xpimasn 35358
                  *21.15.5.14  "Singletonization" and tagging   bj-snsetex 35366
                  *21.15.5.15  Tuples of classes   bj-cproj 35393
                  *21.15.5.16  Set theory: elementary operations relative to a universe   bj-rcleqf 35428
                  *21.15.5.17  Axioms for finite unions   bj-abex 35433
                  *21.15.5.18  Set theory: miscellaneous   eleq2w2ALT 35450
                  *21.15.5.19  Evaluation at a class   bj-evaleq 35475
                  21.15.5.20  Elementwise operations   celwise 35482
                  *21.15.5.21  Elementwise intersection (families of sets induced on a subset)   bj-rest00 35484
                  21.15.5.22  Moore collections (complements)   bj-raldifsn 35503
                  21.15.5.23  Maps-to notation for functions with three arguments   bj-0nelmpt 35519
                  *21.15.5.24  Currying   csethom 35525
                  *21.15.5.25  Setting components of extensible structures   cstrset 35537
            *21.15.6  Extended real and complex numbers, real and complex projective lines   bj-nfald 35540
                  21.15.6.1  Complements on class abstractions of ordered pairs and binary relations   bj-nfald 35540
                  *21.15.6.2  Identity relation (complements)   bj-opabssvv 35553
                  *21.15.6.3  Functionalized identity (diagonal in a Cartesian square)   cdiag2 35575
                  *21.15.6.4  Direct image and inverse image   cimdir 35581
                  *21.15.6.5  Extended numbers and projective lines as sets   cfractemp 35599
                  *21.15.6.6  Addition and opposite   caddcc 35640
                  *21.15.6.7  Order relation on the extended reals   cltxr 35644
                  *21.15.6.8  Argument, multiplication and inverse   carg 35646
                  21.15.6.9  The canonical bijection from the finite ordinals   ciomnn 35652
                  21.15.6.10  Divisibility   cnnbar 35663
            *21.15.7  Monoids   bj-smgrpssmgm 35671
                  *21.15.7.1  Finite sums in monoids   cfinsum 35686
            *21.15.8  Affine, Euclidean, and Cartesian geometry   bj-fvimacnv0 35689
                  *21.15.8.1  Real vector spaces   bj-fvimacnv0 35689
                  *21.15.8.2  Complex numbers (supplements)   bj-subcom 35711
                  *21.15.8.3  Barycentric coordinates   bj-bary1lem 35713
            21.15.9  Monoid of endomorphisms   cend 35716
      21.16  Mathbox for Jim Kingdon
                  21.16.0.1  Circle constant   taupilem3 35722
                  21.16.0.2  Number theory   dfgcd3 35727
                  21.16.0.3  Real numbers   irrdifflemf 35728
      21.17  Mathbox for ML
            21.17.1  Miscellaneous   csbrecsg 35731
            21.17.2  Cartesian exponentiation   cfinxp 35786
            21.17.3  Topology   iunctb2 35806
                  *21.17.3.1  Pi-base theorems   pibp16 35816
      21.18  Mathbox for Wolf Lammen
            21.18.1  1. Bootstrapping   wl-section-boot 35825
            21.18.2  Implication chains   wl-section-impchain 35849
            21.18.3  Theorems around the conditional operator   wl-ifp-ncond1 35867
            21.18.4  Alternative development of hadd, cadd   wl-df-3xor 35871
            21.18.5  An alternative axiom ~ ax-13   ax-wl-13v 35896
            21.18.6  Other stuff   wl-mps 35898
      21.19  Mathbox for Brendan Leahy
      21.20  Mathbox for Jeff Madsen
            21.20.1  Logic and set theory   unirep 36104
            21.20.2  Real and complex numbers; integers   filbcmb 36131
            21.20.3  Sequences and sums   sdclem2 36133
            21.20.4  Topology   subspopn 36143
            21.20.5  Metric spaces   metf1o 36146
            21.20.6  Continuous maps and homeomorphisms   constcncf 36153
            21.20.7  Boundedness   ctotbnd 36157
            21.20.8  Isometries   cismty 36189
            21.20.9  Heine-Borel Theorem   heibor1lem 36200
            21.20.10  Banach Fixed Point Theorem   bfplem1 36213
            21.20.11  Euclidean space   crrn 36216
            21.20.12  Intervals (continued)   ismrer1 36229
            21.20.13  Operation properties   cass 36233
            21.20.14  Groups and related structures   cmagm 36239
            21.20.15  Group homomorphism and isomorphism   cghomOLD 36274
            21.20.16  Rings   crngo 36285
            21.20.17  Division Rings   cdrng 36339
            21.20.18  Ring homomorphisms   crnghom 36351
            21.20.19  Commutative rings   ccm2 36380
            21.20.20  Ideals   cidl 36398
            21.20.21  Prime rings and integral domains   cprrng 36437
            21.20.22  Ideal generators   cigen 36450
      21.21  Mathbox for Giovanni Mascellani
            *21.21.1  Tools for automatic proof building   efald2 36469
            *21.21.2  Tseitin axioms   fald 36520
            *21.21.3  Equality deductions   iuneq2f 36547
            *21.21.4  Miscellanea   orcomdd 36558
      21.22  Mathbox for Peter Mazsa
            21.22.1  Notations   cxrn 36565
            21.22.2  Preparatory theorems   el2v1 36608
            21.22.3  Range Cartesian product   df-xrn 36765
            21.22.4  Cosets by ` R `   df-coss 36805
            21.22.5  Relations   df-rels 36879
            21.22.6  Subset relations   df-ssr 36892
            21.22.7  Reflexivity   df-refs 36904
            21.22.8  Converse reflexivity   df-cnvrefs 36919
            21.22.9  Symmetry   df-syms 36936
            21.22.10  Reflexivity and symmetry   symrefref2 36957
            21.22.11  Transitivity   df-trs 36966
            21.22.12  Equivalence relations   df-eqvrels 36978
            21.22.13  Redundancy   df-redunds 37017
            21.22.14  Domain quotients   df-dmqss 37032
            21.22.15  Equivalence relations on domain quotients   df-ers 37057
            21.22.16  Functions   df-funss 37074
            21.22.17  Disjoints vs. converse functions   df-disjss 37097
            21.22.18  Antisymmetry   df-antisymrel 37154
            21.22.19  Partitions: disjoints on domain quotients   df-parts 37159
            21.22.20  Partition-Equivalence Theorems   disjim 37175
      21.23  Mathbox for Rodolfo Medina
            21.23.1  Partitions   prtlem60 37247
      *21.24  Mathbox for Norm Megill
            *21.24.1  Obsolete schemes ax-c4,c5,c7,c10,c11,c11n,c15,c9,c14,c16   ax-c5 37277
            *21.24.2  Rederive new axioms ax-4, ax-10, ax-6, ax-12, ax-13 from old   axc5 37287
            *21.24.3  Legacy theorems using obsolete axioms   ax5ALT 37301
            21.24.4  Experiments with weak deduction theorem   elimhyps 37355
            21.24.5  Miscellanea   cnaddcom 37366
            21.24.6  Atoms, hyperplanes, and covering in a left vector space (or module)   clsa 37368
            21.24.7  Functionals and kernels of a left vector space (or module)   clfn 37451
            21.24.8  Opposite rings and dual vector spaces   cld 37517
            21.24.9  Ortholattices and orthomodular lattices   cops 37566
            21.24.10  Atomic lattices with covering property   ccvr 37656
            21.24.11  Hilbert lattices   chlt 37744
            21.24.12  Projective geometries based on Hilbert lattices   clln 37886
            21.24.13  Construction of a vector space from a Hilbert lattice   cdlema1N 38186
            21.24.14  Construction of involution and inner product from a Hilbert lattice   clpoN 39875
      21.25  Mathbox for metakunt
            21.25.1  General helpful statements   leexp1ad 40361
            21.25.2  Some gcd and lcm results   12gcd5e1 40392
            21.25.3  Least common multiple inequality theorem   3factsumint1 40410
            21.25.4  Logarithm inequalities   3exp7 40442
            21.25.5  Miscellaneous results for AKS formalisation   intlewftc 40450
            21.25.6  Sticks and stones   sticksstones1 40486
            21.25.7  Permutation results   metakunt1 40509
            21.25.8  Unused lemmas scheduled for deletion   andiff 40543
      21.26  Mathbox for Steven Nguyen
            *21.26.1  Miscellaneous theorems   bicomdALT 40548
            21.26.2  Utility theorems   ioin9i8 40557
            21.26.3  Structures   ressbasssg 40599
            *21.26.4  Arithmetic theorems   c0exALT 40678
            21.26.5  Exponents and divisibility   oexpreposd 40710
            21.26.6  Real subtraction   cresub 40737
            *21.26.7  Projective spaces   cprjsp 40842
            21.26.8  Basic reductions for Fermat's Last Theorem   dffltz 40875
      21.27  Mathbox for Igor Ieskov
      21.28  Mathbox for OpenAI
      21.29  Mathbox for Stefan O'Rear
            21.29.1  Additional elementary logic and set theory   moxfr 40918
            21.29.2  Additional theory of functions   imaiinfv 40919
            21.29.3  Additional topology   elrfi 40920
            21.29.4  Characterization of closure operators. Kuratowski closure axioms   ismrcd1 40924
            21.29.5  Algebraic closure systems   cnacs 40928
            21.29.6  Miscellanea 1. Map utilities   constmap 40939
            21.29.7  Miscellanea for polynomials   mptfcl 40946
            21.29.8  Multivariate polynomials over the integers   cmzpcl 40947
            21.29.9  Miscellanea for Diophantine sets 1   coeq0i 40979
            21.29.10  Diophantine sets 1: definitions   cdioph 40981
            21.29.11  Diophantine sets 2 miscellanea   ellz1 40993
            21.29.12  Diophantine sets 2: union and intersection. Monotone Boolean algebra   diophin 40998
            21.29.13  Diophantine sets 3: construction   diophrex 41001
            21.29.14  Diophantine sets 4 miscellanea   2sbcrex 41010
            21.29.15  Diophantine sets 4: Quantification   rexrabdioph 41020
            21.29.16  Diophantine sets 5: Arithmetic sets   rabdiophlem1 41027
            21.29.17  Diophantine sets 6: reusability. renumbering of variables   eldioph4b 41037
            21.29.18  Pigeonhole Principle and cardinality helpers   fphpd 41042
            21.29.19  A non-closed set of reals is infinite   rencldnfilem 41046
            21.29.20  Lagrange's rational approximation theorem   irrapxlem1 41048
            21.29.21  Pell equations 1: A nontrivial solution always exists   pellexlem1 41055
            21.29.22  Pell equations 2: Algebraic number theory of the solution set   csquarenn 41062
            21.29.23  Pell equations 3: characterizing fundamental solution   infmrgelbi 41104
            *21.29.24  Logarithm laws generalized to an arbitrary base   reglogcl 41116
            21.29.25  Pell equations 4: the positive solution group is infinite cyclic   pellfund14 41124
            21.29.26  X and Y sequences 1: Definition and recurrence laws   crmx 41126
            21.29.27  Ordering and induction lemmas for the integers   monotuz 41168
            21.29.28  X and Y sequences 2: Order properties   rmxypos 41174
            21.29.29  Congruential equations   congtr 41192
            21.29.30  Alternating congruential equations   acongid 41202
            21.29.31  Additional theorems on integer divisibility   coprmdvdsb 41212
            21.29.32  X and Y sequences 3: Divisibility properties   jm2.18 41215
            21.29.33  X and Y sequences 4: Diophantine representability of Y   jm2.27a 41232
            21.29.34  X and Y sequences 5: Diophantine representability of X, ^, _C   rmxdiophlem 41242
            21.29.35  Uncategorized stuff not associated with a major project   setindtr 41251
            21.29.36  More equivalents of the Axiom of Choice   axac10 41260
            21.29.37  Finitely generated left modules   clfig 41297
            21.29.38  Noetherian left modules I   clnm 41305
            21.29.39  Addenda for structure powers   pwssplit4 41319
            21.29.40  Every set admits a group structure iff choice   unxpwdom3 41325
            21.29.41  Noetherian rings and left modules II   clnr 41339
            21.29.42  Hilbert's Basis Theorem   cldgis 41351
            21.29.43  Additional material on polynomials [DEPRECATED]   cmnc 41361
            21.29.44  Degree and minimal polynomial of algebraic numbers   cdgraa 41370
            21.29.45  Algebraic integers I   citgo 41387
            21.29.46  Endomorphism algebra   cmend 41405
            21.29.47  Cyclic groups and order   idomrootle 41425
            21.29.48  Cyclotomic polynomials   ccytp 41432
            21.29.49  Miscellaneous topology   fgraphopab 41440
      21.30  Mathbox for Noam Pasman
      21.31  Mathbox for Jon Pennant
      21.32  Mathbox for Richard Penner
            21.32.1  Set Theory and Ordinal Numbers   uniel 41454
            21.32.2  Natural addition of Cantor normal forms   oawordex2 41561
            21.32.3  Surreal Contributions   abeqabi 41585
            21.32.4  Short Studies   nlimsuc 41618
                  21.32.4.1  Additional work on conditional logical operator   ifpan123g 41636
                  21.32.4.2  Sophisms   rp-fakeimass 41689
                  *21.32.4.3  Finite Sets   rp-isfinite5 41694
                  21.32.4.4  General Observations   intabssd 41696
                  21.32.4.5  Infinite Sets   pwelg 41737
                  *21.32.4.6  Finite intersection property   fipjust 41742
                  21.32.4.7  RP ADDTO: Subclasses and subsets   rababg 41751
                  21.32.4.8  RP ADDTO: The intersection of a class   elinintab 41752
                  21.32.4.9  RP ADDTO: Theorems requiring subset and intersection existence   elinintrab 41754
                  21.32.4.10  RP ADDTO: Relations   xpinintabd 41757
                  *21.32.4.11  RP ADDTO: Functions   elmapintab 41773
                  *21.32.4.12  RP ADDTO: Finite induction (for finite ordinals)   cnvcnvintabd 41777
                  21.32.4.13  RP ADDTO: First and second members of an ordered pair   elcnvlem 41778
                  21.32.4.14  RP ADDTO: The reflexive and transitive properties of relations   undmrnresiss 41781
                  21.32.4.15  RP ADDTO: Basic properties of closures   cleq2lem 41785
                  21.32.4.16  RP REPLACE: Definitions and basic properties of transitive closures   trcleq2lemRP 41807
                  *21.32.4.17  Additions for square root; absolute value   sqrtcvallem1 41808
            21.32.5  Additional statements on relations and subclasses   al3im 41824
                  21.32.5.1  Transitive relations (not to be confused with transitive classes).   trrelind 41842
                  21.32.5.2  Reflexive closures   crcl 41849
                  *21.32.5.3  Finite relationship composition.   relexp2 41854
                  21.32.5.4  Transitive closure of a relation   dftrcl3 41897
                  *21.32.5.5  Adapted from Frege   frege77d 41923
            *21.32.6  Propositions from _Begriffsschrift_   dfxor4 41943
                  *21.32.6.1  _Begriffsschrift_ Chapter I   dfxor4 41943
                  *21.32.6.2  _Begriffsschrift_ Notation hints   whe 41949
                  21.32.6.3  _Begriffsschrift_ Chapter II Implication   ax-frege1 41967
                  21.32.6.4  _Begriffsschrift_ Chapter II Implication and Negation   axfrege28 42006
                  *21.32.6.5  _Begriffsschrift_ Chapter II with logical equivalence   axfrege52a 42033
                  21.32.6.6  _Begriffsschrift_ Chapter II with equivalence of sets   axfrege52c 42064
                  *21.32.6.7  _Begriffsschrift_ Chapter II with equivalence of classes   frege53c 42091
                  *21.32.6.8  _Begriffsschrift_ Chapter III Properties hereditary in a sequence   dffrege69 42109
                  *21.32.6.9  _Begriffsschrift_ Chapter III Following in a sequence   dffrege76 42116
                  *21.32.6.10  _Begriffsschrift_ Chapter III Member of sequence   dffrege99 42139
                  *21.32.6.11  _Begriffsschrift_ Chapter III Single-valued procedures   dffrege115 42155
            *21.32.7  Exploring Topology via Seifert and Threlfall   enrelmap 42174
                  *21.32.7.1  Equinumerosity of sets of relations and maps   enrelmap 42174
                  *21.32.7.2  Generic Pseudoclosure Spaces, Pseudointerior Spaces, and Pseudoneighborhoods   or3or 42200
                  *21.32.7.3  Generic Neighborhood Spaces   gneispa 42307
            *21.32.8  Exploring Higher Homotopy via Kerodon   k0004lem1 42324
                  *21.32.8.1  Simplicial Sets   k0004lem1 42324
      21.33  Mathbox for Stanislas Polu
            21.33.1  IMO Problems   wwlemuld 42333
                  21.33.1.1  IMO 1972 B2   wwlemuld 42333
            *21.33.2  INT Inequalities Proof Generator   int-addcomd 42351
            *21.33.3  N-Digit Addition Proof Generator   unitadd 42373
            21.33.4  AM-GM (for k = 2,3,4)   gsumws3 42374
      21.34  Mathbox for Rohan Ridenour
            21.34.1  Misc   spALT 42379
            21.34.2  Monoid rings   cmnring 42391
            21.34.3  Shorter primitive equivalent of ax-groth   gru0eld 42414
                  21.34.3.1  Grothendieck universes are closed under collection   gru0eld 42414
                  21.34.3.2  Minimal universes   ismnu 42446
                  21.34.3.3  Primitive equivalent of ax-groth   expandan 42473
      21.35  Mathbox for Steve Rodriguez
            21.35.1  Miscellanea   nanorxor 42490
            21.35.2  Ratio test for infinite series convergence and divergence   dvgrat 42497
            21.35.3  Multiples   reldvds 42500
            21.35.4  Function operations   caofcan 42508
            21.35.5  Calculus   lhe4.4ex1a 42514
            21.35.6  The generalized binomial coefficient operation   cbcc 42521
            21.35.7  Binomial series   uzmptshftfval 42531
      21.36  Mathbox for Andrew Salmon
            21.36.1  Principia Mathematica * 10   pm10.12 42543
            21.36.2  Principia Mathematica * 11   2alanimi 42557
            21.36.3  Predicate Calculus   sbeqal1 42583
            21.36.4  Principia Mathematica * 13 and * 14   pm13.13a 42592
            21.36.5  Set Theory   elnev 42623
            21.36.6  Arithmetic   addcomgi 42641
            21.36.7  Geometry   cplusr 42642
      *21.37  Mathbox for Alan Sare
            21.37.1  Auxiliary theorems for the Virtual Deduction tool   idiALT 42664
            21.37.2  Supplementary unification deductions   bi1imp 42668
            21.37.3  Conventional Metamath proofs, some derived from VD proofs   iidn3 42688
            21.37.4  What is Virtual Deduction?   wvd1 42756
            21.37.5  Virtual Deduction Theorems   df-vd1 42757
            21.37.6  Theorems proved using Virtual Deduction   trsspwALT 43005
            21.37.7  Theorems proved using Virtual Deduction with mmj2 assistance   simplbi2VD 43033
            21.37.8  Virtual Deduction transcriptions of textbook proofs   sb5ALTVD 43100
            21.37.9  Theorems proved using conjunction-form Virtual Deduction   elpwgdedVD 43104
            21.37.10  Theorems with a VD proof in conventional notation derived from a VD proof   suctrALT3 43111
            *21.37.11  Theorems with a proof in conventional notation derived from a VD proof   notnotrALT2 43114
      21.38  Mathbox for Glauco Siliprandi
            21.38.1  Miscellanea   evth2f 43125
            21.38.2  Functions   feq1dd 43283
            21.38.3  Ordering on real numbers - Real and complex numbers basic operations   sub2times 43405
            21.38.4  Real intervals   gtnelioc 43624
            21.38.5  Finite sums   fsummulc1f 43707
            21.38.6  Finite multiplication of numbers and finite multiplication of functions   fmul01 43716
            21.38.7  Limits   clim1fr1 43737
                  21.38.7.1  Inferior limit (lim inf)   clsi 43887
                  *21.38.7.2  Limits for sequences of extended real numbers   clsxlim 43954
            21.38.8  Trigonometry   coseq0 44000
            21.38.9  Continuous Functions   mulcncff 44006
            21.38.10  Derivatives   dvsinexp 44047
            21.38.11  Integrals   itgsin0pilem1 44086
            21.38.12  Stone Weierstrass theorem - real version   stoweidlem1 44137
            21.38.13  Wallis' product for π   wallispilem1 44201
            21.38.14  Stirling's approximation formula for ` n ` factorial   stirlinglem1 44210
            21.38.15  Dirichlet kernel   dirkerval 44227
            21.38.16  Fourier Series   fourierdlem1 44244
            21.38.17  e is transcendental   elaa2lem 44369
            21.38.18  n-dimensional Euclidean space   rrxtopn 44420
            21.38.19  Basic measure theory   csalg 44444
                  *21.38.19.1  σ-Algebras   csalg 44444
                  21.38.19.2  Sum of nonnegative extended reals   csumge0 44498
                  *21.38.19.3  Measures   cmea 44585
                  *21.38.19.4  Outer measures and Caratheodory's construction   come 44625
                  *21.38.19.5  Lebesgue measure on n-dimensional Real numbers   covoln 44672
                  *21.38.19.6  Measurable functions   csmblfn 44831
      21.39  Mathbox for Saveliy Skresanov
            21.39.1  Ceva's theorem   sigarval 44986
            21.39.2  Simple groups   simpcntrab 45006
      21.40  Mathbox for Ender Ting
            21.40.1  Increasing sequences and subsequences   et-ltneverrefl 45007
      21.41  Mathbox for Jarvin Udandy
      21.42  Mathbox for Adhemar
            *21.42.1  Minimal implicational calculus   adh-minim 45131
      21.43  Mathbox for Alexander van der Vekens
            21.43.1  General auxiliary theorems (1)   eusnsn 45155
                  21.43.1.1  Unordered and ordered pairs - extension for singletons   eusnsn 45155
                  21.43.1.2  Unordered and ordered pairs - extension for unordered pairs   elprneb 45158
                  21.43.1.3  Unordered and ordered pairs - extension for ordered pairs   oppr 45159
                  21.43.1.4  Relations - extension   eubrv 45164
                  21.43.1.5  Definite description binder (inverted iota) - extension   iota0def 45167
                  21.43.1.6  Functions - extension   fveqvfvv 45169
            21.43.2  Alternative for Russell's definition of a description binder   caiota 45210
            21.43.3  Double restricted existential uniqueness   r19.32 45225
                  21.43.3.1  Restricted quantification (extension)   r19.32 45225
                  21.43.3.2  Restricted uniqueness and "at most one" quantification   reuf1odnf 45234
                  21.43.3.3  Analogs to Existential uniqueness (double quantification)   2reu3 45237
                  21.43.3.4  Additional theorems for double restricted existential uniqueness   2reu8i 45240
            *21.43.4  Alternative definitions of function and operation values   wdfat 45243
                  21.43.4.1  Restricted quantification (extension)   ralbinrald 45249
                  21.43.4.2  The universal class (extension)   nvelim 45250
                  21.43.4.3  Introduce the Axiom of Power Sets (extension)   alneu 45251
                  21.43.4.4  Predicate "defined at"   dfateq12d 45253
                  21.43.4.5  Alternative definition of the value of a function   dfafv2 45259
                  21.43.4.6  Alternative definition of the value of an operation   aoveq123d 45305
            *21.43.5  Alternative definitions of function values (2)   cafv2 45335
            21.43.6  General auxiliary theorems (2)   an4com24 45395
                  21.43.6.1  Logical conjunction - extension   an4com24 45395
                  21.43.6.2  Abbreviated conjunction and disjunction of three wff's - extension   3an4ancom24 45396
                  21.43.6.3  Negated membership (alternative)   cnelbr 45398
                  21.43.6.4  The empty set - extension   ralralimp 45405
                  21.43.6.5  Indexed union and intersection - extension   otiunsndisjX 45406
                  21.43.6.6  Functions - extension   fvifeq 45407
                  21.43.6.7  Maps-to notation - extension   fvmptrab 45419
                  21.43.6.8  Subtraction - extension   cnambpcma 45421
                  21.43.6.9  Ordering on reals (cont.) - extension   leaddsuble 45424
                  21.43.6.10  Imaginary and complex number properties - extension   readdcnnred 45430
                  21.43.6.11  Nonnegative integers (as a subset of complex numbers) - extension   nn0resubcl 45435
                  21.43.6.12  Integers (as a subset of complex numbers) - extension   zgeltp1eq 45436
                  21.43.6.13  Decimal arithmetic - extension   1t10e1p1e11 45437
                  21.43.6.14  Upper sets of integers - extension   eluzge0nn0 45439
                  21.43.6.15  Infinity and the extended real number system (cont.) - extension   nltle2tri 45440
                  21.43.6.16  Finite intervals of integers - extension   ssfz12 45441
                  21.43.6.17  Half-open integer ranges - extension   fzopred 45449
                  21.43.6.18  The modulo (remainder) operation - extension   m1mod0mod1 45456
                  21.43.6.19  The infinite sequence builder "seq"   smonoord 45458
                  21.43.6.20  Finite and infinite sums - extension   fsummsndifre 45459
                  21.43.6.21  Extensible structures - extension   setsidel 45463
            *21.43.7  Preimages of function values   preimafvsnel 45466
            *21.43.8  Partitions of real intervals   ciccp 45500
            21.43.9  Shifting functions with an integer range domain   fargshiftfv 45526
            21.43.10  Words over a set (extension)   lswn0 45531
                  21.43.10.1  Last symbol of a word - extension   lswn0 45531
            21.43.11  Unordered pairs   wich 45532
                  21.43.11.1  Interchangeable setvar variables   wich 45532
                  21.43.11.2  Set of unordered pairs   sprid 45561
                  *21.43.11.3  Proper (unordered) pairs   prpair 45588
                  21.43.11.4  Set of proper unordered pairs   cprpr 45599
            21.43.12  Number theory (extension)   cfmtno 45614
                  *21.43.12.1  Fermat numbers   cfmtno 45614
                  *21.43.12.2  Mersenne primes   m2prm 45678
                  21.43.12.3  Proth's theorem   modexp2m1d 45699
                  21.43.12.4  Solutions of quadratic equations   quad1 45707
            *21.43.13  Even and odd numbers   ceven 45711
                  21.43.13.1  Definitions and basic properties   ceven 45711
                  21.43.13.2  Alternate definitions using the "divides" relation   dfeven2 45736
                  21.43.13.3  Alternate definitions using the "modulo" operation   dfeven3 45745
                  21.43.13.4  Alternate definitions using the "gcd" operation   iseven5 45751
                  21.43.13.5  Theorems of part 5 revised   zneoALTV 45756
                  21.43.13.6  Theorems of part 6 revised   odd2np1ALTV 45761
                  21.43.13.7  Theorems of AV's mathbox revised   0evenALTV 45775
                  21.43.13.8  Additional theorems   epoo 45790
                  21.43.13.9  Perfect Number Theorem (revised)   perfectALTVlem1 45808
            21.43.14  Number theory (extension 2)   cfppr 45811
                  *21.43.14.1  Fermat pseudoprimes   cfppr 45811
                  *21.43.14.2  Goldbach's conjectures   cgbe 45832
            21.43.15  Graph theory (extension)   cgrisom 45905
                  *21.43.15.1  Isomorphic graphs   cgrisom 45905
                  21.43.15.2  Loop-free graphs - extension   1hegrlfgr 45929
                  21.43.15.3  Walks - extension   cupwlks 45930
                  21.43.15.4  Edges of graphs expressed as sets of unordered pairs   upgredgssspr 45940
            21.43.16  Monoids (extension)   ovn0dmfun 45953
                  21.43.16.1  Auxiliary theorems   ovn0dmfun 45953
                  21.43.16.2  Magmas, Semigroups and Monoids (extension)   plusfreseq 45961
                  21.43.16.3  Magma homomorphisms and submagmas   cmgmhm 45966
                  21.43.16.4  Examples and counterexamples for magmas, semigroups and monoids (extension)   opmpoismgm 45996
                  21.43.16.5  Group sum operation (extension 1)   gsumsplit2f 46009
            *21.43.17  Magmas and internal binary operations (alternate approach)   ccllaw 46012
                  *21.43.17.1  Laws for internal binary operations   ccllaw 46012
                  *21.43.17.2  Internal binary operations   cintop 46025
                  21.43.17.3  Alternative definitions for magmas and semigroups   cmgm2 46044
            21.43.18  Categories (extension)   idfusubc0 46058
                  21.43.18.1  Subcategories (extension)   idfusubc0 46058
            21.43.19  Rings (extension)   lmod0rng 46061
                  21.43.19.1  Nonzero rings (extension)   lmod0rng 46061
                  *21.43.19.2  Non-unital rings ("rngs")   crng 46067
                  21.43.19.3  Rng homomorphisms   crngh 46078
                  21.43.19.4  Ring homomorphisms (extension)   rhmfn 46111
                  21.43.19.5  Ideals as non-unital rings   lidldomn1 46114
                  21.43.19.6  The non-unital ring of even integers   0even 46124
                  21.43.19.7  A constructed not unital ring   cznrnglem 46146
                  *21.43.19.8  The category of non-unital rings   crngc 46150
                  *21.43.19.9  The category of (unital) rings   cringc 46196
                  21.43.19.10  Subcategories of the category of rings   srhmsubclem1 46266
            21.43.20  Basic algebraic structures (extension)   opeliun2xp 46303
                  21.43.20.1  Auxiliary theorems   opeliun2xp 46303
                  21.43.20.2  The binomial coefficient operation (extension)   bcpascm1 46322
                  21.43.20.3  The ` ZZ `-module ` ZZ X. ZZ `   zlmodzxzlmod 46325
                  21.43.20.4  Group sum operation (extension 2)   mgpsumunsn 46332
                  21.43.20.5  Symmetric groups (extension)   exple2lt6 46335
                  21.43.20.6  Divisibility (extension)   invginvrid 46338
                  21.43.20.7  The support of functions (extension)   rmsupp0 46339
                  21.43.20.8  Finitely supported functions (extension)   rmsuppfi 46344
                  21.43.20.9  Left modules (extension)   lmodvsmdi 46353
                  21.43.20.10  Associative algebras (extension)   assaascl0 46355
                  21.43.20.11  Univariate polynomials (extension)   ply1vr1smo 46357
                  21.43.20.12  Univariate polynomials (examples)   linply1 46369
            21.43.21  Linear algebra (extension)   cdmatalt 46372
                  *21.43.21.1  The subalgebras of diagonal and scalar matrices (extension)   cdmatalt 46372
                  *21.43.21.2  Linear combinations   clinc 46380
                  *21.43.21.3  Linear independence   clininds 46416
                  21.43.21.4  Simple left modules and the ` ZZ `-module   lmod1lem1 46463
                  21.43.21.5  Differences between (left) modules and (left) vector spaces   lvecpsslmod 46483
            21.43.22  Complexity theory   suppdm 46486
                  21.43.22.1  Auxiliary theorems   suppdm 46486
                  21.43.22.2  The modulo (remainder) operation (extension)   fldivmod 46499
                  21.43.22.3  Even and odd integers   nn0onn0ex 46504
                  21.43.22.4  The natural logarithm on complex numbers (extension)   logcxp0 46516
                  21.43.22.5  Division of functions   cfdiv 46518
                  21.43.22.6  Upper bounds   cbigo 46528
                  21.43.22.7  Logarithm to an arbitrary base (extension)   rege1logbrege0 46539
                  *21.43.22.8  The binary logarithm   fldivexpfllog2 46546
                  21.43.22.9  Binary length   cblen 46550
                  *21.43.22.10  Digits   cdig 46576
                  21.43.22.11  Nonnegative integer as sum of its shifted digits   dignn0flhalflem1 46596
                  21.43.22.12  Algorithms for the multiplication of nonnegative integers   nn0mulfsum 46605
                  *21.43.22.13  N-ary functions   cnaryf 46607
                  *21.43.22.14  The Ackermann function   citco 46638
            21.43.23  Elementary geometry (extension)   fv1prop 46680
                  21.43.23.1  Auxiliary theorems   fv1prop 46680
                  21.43.23.2  Real euclidean space of dimension 2   rrx2pxel 46692
                  21.43.23.3  Spheres and lines in real Euclidean spaces   cline 46708
      21.44  Mathbox for Zhi Wang
            21.44.1  Propositional calculus   pm4.71da 46770
            21.44.2  Predicate calculus with equality   dtrucor3 46779
                  21.44.2.1  Axiom scheme ax-5 (Distinctness)   dtrucor3 46779
            21.44.3  ZF Set Theory - start with the Axiom of Extensionality   ralbidb 46780
                  21.44.3.1  Restricted quantification   ralbidb 46780
                  21.44.3.2  The empty set   ssdisjd 46787
                  21.44.3.3  Unordered and ordered pairs   vsn 46791
                  21.44.3.4  The union of a class   unilbss 46797
            21.44.4  ZF Set Theory - add the Axiom of Replacement   inpw 46798
                  21.44.4.1  Theorems requiring subset and intersection existence   inpw 46798
            21.44.5  ZF Set Theory - add the Axiom of Power Sets   mof0 46799
                  21.44.5.1  Functions   mof0 46799
                  21.44.5.2  Operations   fvconstr 46817
            21.44.6  ZF Set Theory - add the Axiom of Union   fvconst0ci 46820
                  21.44.6.1  Equinumerosity   fvconst0ci 46820
            21.44.7  Order sets   iccin 46824
                  21.44.7.1  Real number intervals   iccin 46824
            21.44.8  Moore spaces   mreuniss 46827
            *21.44.9  Topology   clduni 46828
                  21.44.9.1  Closure and interior   clduni 46828
                  21.44.9.2  Neighborhoods   neircl 46832
                  21.44.9.3  Subspace topologies   restcls2lem 46840
                  21.44.9.4  Limits and continuity in topological spaces   cnneiima 46844
                  21.44.9.5  Topological definitions using the reals   iooii 46845
                  21.44.9.6  Separated sets   sepnsepolem1 46849
                  21.44.9.7  Separated spaces: T0, T1, T2 (Hausdorff) ...   isnrm4 46858
            21.44.10  Preordered sets and directed sets using extensible structures   isprsd 46883
            21.44.11  Posets and lattices using extensible structures   lubeldm2 46884
                  21.44.11.1  Posets   lubeldm2 46884
                  21.44.11.2  Lattices   toslat 46902
                  21.44.11.3  Subset order structures   intubeu 46904
            21.44.12  Categories   catprslem 46925
                  21.44.12.1  Categories   catprslem 46925
                  21.44.12.2  Monomorphisms and epimorphisms   idmon 46931
                  21.44.12.3  Functors   funcf2lem 46933
            21.44.13  Examples of categories   cthinc 46934
                  21.44.13.1  Thin categories   cthinc 46934
                  21.44.13.2  Preordered sets as thin categories   cprstc 46977
                  21.44.13.3  Monoids as categories   cmndtc 46998
      21.45  Mathbox for Emmett Weisz
            *21.45.1  Miscellaneous Theorems   nfintd 47013
            21.45.2  Set Recursion   csetrecs 47023
                  *21.45.2.1  Basic Properties of Set Recursion   csetrecs 47023
                  21.45.2.2  Examples and properties of set recursion   elsetrecslem 47039
            *21.45.3  Construction of Games and Surreal Numbers   cpg 47049
      *21.46  Mathbox for David A. Wheeler
            21.46.1  Natural deduction   sbidd 47058
            *21.46.2  Greater than, greater than or equal to.   cge-real 47060
            *21.46.3  Hyperbolic trigonometric functions   csinh 47070
            *21.46.4  Reciprocal trigonometric functions (sec, csc, cot)   csec 47081
            *21.46.5  Identities for "if"   ifnmfalse 47103
            *21.46.6  Logarithms generalized to arbitrary base using ` logb `   logb2aval 47104
            *21.46.7  Logarithm laws generalized to an arbitrary base - log_   clog- 47105
            *21.46.8  Formally define notions such as reflexivity   wreflexive 47107
            *21.46.9  Algebra helpers   comraddi 47111
            *21.46.10  Algebra helper examples   i2linesi 47120
            *21.46.11  Formal methods "surprises"   alimp-surprise 47122
            *21.46.12  Allsome quantifier   walsi 47128
            *21.46.13  Miscellaneous   5m4e1 47139
            21.46.14  Theorems about algebraic numbers   aacllem 47143
      21.47  Mathbox for Kunhao Zheng
            21.47.1  Weighted AM-GM inequality   amgmwlem 47144

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