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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  Preordered sets and directed sets using extensible structures
      9.2  Posets and lattices using extensible structures
PART 10  BASIC ALGEBRAIC STRUCTURES
      10.1  Monoids
      10.2  Groups
      10.3  Rings
      10.4  Division rings and fields
      10.5  Left modules
      10.6  Vector spaces
      10.7  Ideals
      10.8  Associative algebras
      10.9  Abstract multivariate polynomials
      10.10  The complex numbers as an algebraic extensible structure
      10.11  Generalized pre-Hilbert and Hilbert spaces
PART 11  BASIC LINEAR ALGEBRA
      11.1  Vectors and free modules
      11.2  Matrices
      11.3  The determinant
      11.4  Polynomial matrices
      11.5  The characteristic polynomial
PART 12  BASIC TOPOLOGY
      12.1  Topology
      12.2  Filters and filter bases
      12.3  Uniform Structures and Spaces
      12.4  Metric spaces
      12.5  Metric subcomplex vector spaces
PART 13  BASIC REAL AND COMPLEX ANALYSIS
      13.1  Continuity
      13.2  Integrals
      13.3  Derivatives
PART 14  BASIC REAL AND COMPLEX FUNCTIONS
      14.1  Polynomials
      14.2  Sequences and series
      14.3  Basic trigonometry
      14.4  Basic number theory
PART 15  ELEMENTARY GEOMETRY
      15.1  Definition and Tarski's Axioms of Geometry
      15.2  Tarskian Geometry
      15.3  Properties of geometries
      15.4  Geometry in Hilbert spaces
PART 16  GRAPH THEORY
      16.1  Vertices and edges
      16.2  Undirected graphs
      16.3  Walks, paths and cycles
      16.4  Eulerian paths and the Konigsberg Bridge problem
      16.5  The Friendship Theorem
PART 17  GUIDES AND MISCELLANEA
      17.1  Guides (conventions, explanations, and examples)
      17.2  Humor
      17.3  (Future - to be reviewed and classified)
PART 18  COMPLEX TOPOLOGICAL VECTOR SPACES (DEPRECATED)
      18.1  Additional material on group theory (deprecated)
      18.2  Complex vector spaces
      18.3  Normed complex vector spaces
      18.4  Operators on complex vector spaces
      18.5  Inner product (pre-Hilbert) spaces
      18.6  Complex Banach spaces
      18.7  Complex Hilbert spaces
PART 19  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      19.1  Axiomatization of complex pre-Hilbert spaces
      19.2  Inner product and norms
      19.3  Cauchy sequences and completeness axiom
      19.4  Subspaces and projections
      19.5  Properties of Hilbert subspaces
      19.6  Operators on Hilbert spaces
      19.7  States on a Hilbert lattice and Godowski's equation
      19.8  Cover relation, atoms, exchange axiom, and modular symmetry
PART 20  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      20.1  Mathboxes for user contributions
      20.2  Mathbox for Stefan Allan
      20.3  Mathbox for Thierry Arnoux
      20.4  Mathbox for Jonathan Ben-Naim
      20.5  Mathbox for BTernaryTau
      20.6  Mathbox for Mario Carneiro
      20.7  Mathbox for Filip Cernatescu
      20.8  Mathbox for Paul Chapman
      20.9  Mathbox for Scott Fenton
      20.10  Mathbox for Jeff Hankins
      20.11  Mathbox for Anthony Hart
      20.12  Mathbox for Chen-Pang He
      20.13  Mathbox for Jeff Hoffman
      20.14  Mathbox for Asger C. Ipsen
      20.15  Mathbox for BJ
      20.16  Mathbox for Jim Kingdon
      20.17  Mathbox for ML
      20.18  Mathbox for Wolf Lammen
      20.19  Mathbox for Brendan Leahy
      20.20  Mathbox for Jeff Madsen
      20.21  Mathbox for Giovanni Mascellani
      20.22  Mathbox for Peter Mazsa
      20.23  Mathbox for Rodolfo Medina
      20.24  Mathbox for Norm Megill
      20.25  Mathbox for metakunt
      20.26  Mathbox for Steven Nguyen
      20.27  Mathbox for Igor Ieskov
      20.28  Mathbox for OpenAI
      20.29  Mathbox for Stefan O'Rear
      20.30  Mathbox for Jon Pennant
      20.31  Mathbox for Richard Penner
      20.32  Mathbox for Stanislas Polu
      20.33  Mathbox for Rohan Ridenour
      20.34  Mathbox for Steve Rodriguez
      20.35  Mathbox for Andrew Salmon
      20.36  Mathbox for Alan Sare
      20.37  Mathbox for Glauco Siliprandi
      20.38  Mathbox for Saveliy Skresanov
      20.39  Mathbox for Jarvin Udandy
      20.40  Mathbox for Adhemar
      20.41  Mathbox for Alexander van der Vekens
      20.42  Mathbox for Emmett Weisz
      20.43  Mathbox for David A. Wheeler
      20.44  Mathbox for Kunhao Zheng

Detailed Table of Contents
(* means the section header has a description)
*PART 1  CLASSICAL FIRST-ORDER LOGIC WITH EQUALITY
      *1.1  Pre-logic
            *1.1.1  Inferences for assisting proof development   idi 1
      *1.2  Propositional calculus
            1.2.1  Recursively define primitive wffs for propositional calculus   wn 3
            *1.2.2  The axioms of propositional calculus   ax-mp 5
            *1.2.3  Logical implication   mp2 9
            *1.2.4  Logical negation   con4 113
            *1.2.5  Logical equivalence   wb 208
            *1.2.6  Logical conjunction   wa 398
            *1.2.7  Logical disjunction   wo 843
            *1.2.8  Mixed connectives   jaao 951
            *1.2.9  The conditional operator for propositions   wif 1057
            *1.2.10  The weak deduction theorem for propositional calculus   elimh 1075
            1.2.11  Abbreviated conjunction and disjunction of three wff's   w3o 1081
            1.2.12  Logical "nand" (Sheffer stroke)   wnan 1478
            1.2.13  Logical "xor"   wxo 1498
            1.2.14  Logical "nor"   wnor 1515
            1.2.15  True and false constants   wal 1529
                  *1.2.15.1  Universal quantifier for use by df-tru   wal 1529
                  *1.2.15.2  Equality predicate for use by df-tru   cv 1530
                  1.2.15.3  The true constant   wtru 1532
                  1.2.15.4  The false constant   wfal 1543
            *1.2.16  Truth tables   truimtru 1554
                  1.2.16.1  Implication   truimtru 1554
                  1.2.16.2  Negation   nottru 1558
                  1.2.16.3  Equivalence   trubitru 1560
                  1.2.16.4  Conjunction   truantru 1564
                  1.2.16.5  Disjunction   truortru 1568
                  1.2.16.6  Alternative denial   trunantru 1572
                  1.2.16.7  Exclusive disjunction   truxortru 1576
                  1.2.16.8  Joint denial   trunortru 1580
            *1.2.17  Half adder and full adder in propositional calculus   whad 1587
                  1.2.17.1  Full adder: sum   whad 1587
                  1.2.17.2  Full adder: carry   wcad 1601
      1.3  Other axiomatizations related to classical propositional calculus
            *1.3.1  Minimal implicational calculus   minimp 1616
            *1.3.2  Implicational Calculus   impsingle 1622
            1.3.3  Derive the Lukasiewicz axioms from Meredith's sole axiom   meredith 1636
            1.3.4  Derive the standard axioms from the Lukasiewicz axioms   luklem1 1653
            *1.3.5  Derive Nicod's axiom from the standard axioms   nic-dfim 1664
            1.3.6  Derive the Lukasiewicz axioms from Nicod's axiom   nic-imp 1670
            1.3.7  Derive Nicod's Axiom from Lukasiewicz's First Sheffer Stroke Axiom   lukshef-ax1 1689
            1.3.8  Derive the Lukasiewicz Axioms from the Tarski-Bernays-Wajsberg Axioms   tbw-bijust 1693
            1.3.9  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's First CO Axiom   merco1 1708
            1.3.10  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's Second CO Axiom   merco2 1731
            1.3.11  Derive the Lukasiewicz axioms from the Russell-Bernays Axioms   rb-bijust 1744
            *1.3.12  Stoic logic non-modal portion (Chrysippus of Soli)   mptnan 1763
      *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 1774
                  1.4.1.1  Existential quantifier   wex 1774
                  1.4.1.2  Non-freeness predicate   wnf 1778
            1.4.2  Rule scheme ax-gen (Generalization)   ax-gen 1790
            1.4.3  Axiom scheme ax-4 (Quantified Implication)   ax-4 1804
                  *1.4.3.1  The empty domain of discourse   empty 1901
            1.4.4  Axiom scheme ax-5 (Distinctness) - first use of $d   ax-5 1905
            *1.4.5  Equality predicate (continued)   weq 1958
            1.4.6  Axiom scheme ax-6 (Existence)   ax-6 1964
            1.4.7  Axiom scheme ax-7 (Equality)   ax-7 2009
            1.4.8  Define proper substitution   sbjust 2062
            1.4.9  Membership predicate   wcel 2108
            1.4.10  Axiom scheme ax-8 (Left Equality for Binary Predicate)   ax-8 2110
            1.4.11  Axiom scheme ax-9 (Right Equality for Binary Predicate)   ax-9 2118
            *1.4.12  Logical redundancy of ax-10 , ax-11 , ax-12 , ax-13   ax6dgen 2126
      *1.5  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes)
            1.5.1  Axiom scheme ax-10 (Quantified Negation)   ax-10 2139
            1.5.2  Axiom scheme ax-11 (Quantifier Commutation)   ax-11 2154
            1.5.3  Axiom scheme ax-12 (Substitution)   ax-12 2170
            1.5.4  Axiom scheme ax-13 (Quantified Equality)   ax-13 2384
            *1.5.5  Alternate definition of substitution   sbimiALT 2571
      1.6  Uniqueness and unique existence
            1.6.1  Uniqueness: the at-most-one quantifier   wmo 2614
            1.6.2  Unique existence: the unique existential quantifier   weu 2647
      1.7  Other axiomatizations related to classical predicate calculus
            *1.7.1  Aristotelian logic: Assertic syllogisms   barbara 2744
            *1.7.2  Intuitionistic logic   axia1 2774
*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 2791
            2.1.2  Classes   cab 2797
                  2.1.2.1  Class abstractions   cab 2797
                  *2.1.2.2  Class equality   df-cleq 2812
                  2.1.2.3  Class membership   df-clel 2891
                  2.1.2.4  Elementary properties of class abstractions   abeq2 2943
            2.1.3  Class form not-free predicate   wnfc 2959
            2.1.4  Negated equality and membership   wne 3014
                  2.1.4.1  Negated equality   wne 3014
                  2.1.4.2  Negated membership   wnel 3121
            2.1.5  Restricted quantification   wral 3136
            2.1.6  The universal class   cvv 3493
            *2.1.7  Conditional equality (experimental)   wcdeq 3752
            2.1.8  Russell's Paradox   rru 3768
            2.1.9  Proper substitution of classes for sets   wsbc 3770
            2.1.10  Proper substitution of classes for sets into classes   csb 3881
            2.1.11  Define basic set operations and relations   cdif 3931
            2.1.12  Subclasses and subsets   df-ss 3950
            2.1.13  The difference, union, and intersection of two classes   dfdif3 4089
                  2.1.13.1  The difference of two classes   dfdif3 4089
                  2.1.13.2  The union of two classes   elun 4123
                  2.1.13.3  The intersection of two classes   elin 4167
                  2.1.13.4  The symmetric difference of two classes   csymdif 4216
                  2.1.13.5  Combinations of difference, union, and intersection of two classes   unabs 4229
                  2.1.13.6  Class abstractions with difference, union, and intersection of two classes   unab 4268
                  2.1.13.7  Restricted uniqueness with difference, union, and intersection   reuss2 4281
            2.1.14  The empty set   c0 4289
            *2.1.15  The conditional operator for classes   cif 4465
            *2.1.16  The weak deduction theorem for set theory   dedth 4521
            2.1.17  Power classes   cpw 4537
            2.1.18  Unordered and ordered pairs   snjust 4558
            2.1.19  The union of a class   cuni 4830
            2.1.20  The intersection of a class   cint 4867
            2.1.21  Indexed union and intersection   ciun 4910
            2.1.22  Disjointness   wdisj 5022
            2.1.23  Binary relations   wbr 5057
            2.1.24  Ordered-pair class abstractions (class builders)   copab 5119
            2.1.25  Functions in maps-to notation   cmpt 5137
            2.1.26  Transitive classes   wtr 5163
      2.2  ZF Set Theory - add the Axiom of Replacement
            2.2.1  Introduce the Axiom of Replacement   ax-rep 5181
            2.2.2  Derive the Axiom of Separation   axsepgfromrep 5192
            2.2.3  Derive the Null Set Axiom   axnulALT 5199
            2.2.4  Theorems requiring subset and intersection existence   nalset 5208
            2.2.5  Theorems requiring empty set existence   class2set 5245
      2.3  ZF Set Theory - add the Axiom of Power Sets
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 5257
            2.3.2  Derive the Axiom of Pairing   axprlem1 5314
            2.3.3  Ordered pair theorem   opnz 5356
            2.3.4  Ordered-pair class abstractions (cont.)   opabidw 5403
            2.3.5  Power class of union and intersection   pwin 5445
            2.3.6  The identity relation   cid 5452
            2.3.7  The membership relation (or epsilon relation)   cep 5457
            *2.3.8  Partial and total orderings   wpo 5465
            2.3.9  Founded and well-ordering relations   wfr 5504
            2.3.10  Relations   cxp 5546
            2.3.11  The Predecessor Class   cpred 6140
            2.3.12  Well-founded induction   tz6.26 6172
            2.3.13  Ordinals   word 6183
            2.3.14  Definite description binder (inverted iota)   cio 6305
            2.3.15  Functions   wfun 6342
            2.3.16  Cantor's Theorem   canth 7103
            2.3.17  Restricted iota (description binder)   crio 7105
            2.3.18  Operations   co 7148
                  2.3.18.1  Variable-to-class conversion for operations   caovclg 7332
            2.3.19  Maps-to notation   mpondm0 7378
            2.3.20  Function operation   cof 7399
            2.3.21  Proper subset relation   crpss 7440
      2.4  ZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 7453
            2.4.2  Ordinals (continued)   epweon 7489
            2.4.3  Transfinite induction   tfi 7560
            2.4.4  The natural numbers (i.e., finite ordinals)   com 7572
            2.4.5  Peano's postulates   peano1 7593
            2.4.6  Finite induction (for finite ordinals)   find 7599
            2.4.7  Relations and functions (cont.)   dmexg 7605
            2.4.8  First and second members of an ordered pair   c1st 7679
            *2.4.9  The support of functions   csupp 7822
            *2.4.10  Special maps-to operations   opeliunxp2f 7868
            2.4.11  Function transposition   ctpos 7883
            2.4.12  Curry and uncurry   ccur 7923
            2.4.13  Undefined values   cund 7930
            2.4.14  Well-founded recursion   cwrecs 7938
            2.4.15  Functions on ordinals; strictly monotone ordinal functions   iunon 7968
            2.4.16  "Strong" transfinite recursion   crecs 7999
            2.4.17  Recursive definition generator   crdg 8037
            2.4.18  Finite recursion   frfnom 8062
            2.4.19  Ordinal arithmetic   c1o 8087
            2.4.20  Natural number arithmetic   nna0 8222
            2.4.21  Equivalence relations and classes   wer 8278
            2.4.22  The mapping operation   cmap 8398
            2.4.23  Infinite Cartesian products   cixp 8453
            2.4.24  Equinumerosity   cen 8498
            2.4.25  Schroeder-Bernstein Theorem   sbthlem1 8619
            2.4.26  Equinumerosity (cont.)   xpf1o 8671
            2.4.27  Pigeonhole Principle   phplem1 8688
            2.4.28  Finite sets   onomeneq 8700
            2.4.29  Finitely supported functions   cfsupp 8825
            2.4.30  Finite intersections   cfi 8866
            2.4.31  Hall's marriage theorem   marypha1lem 8889
            2.4.32  Supremum and infimum   csup 8896
            2.4.33  Ordinal isomorphism, Hartogs's theorem   coi 8965
            2.4.34  Hartogs function, order types, weak dominance   char 9012
      2.5  ZF Set Theory - add the Axiom of Regularity
            2.5.1  Introduce the Axiom of Regularity   ax-reg 9048
            2.5.2  Axiom of Infinity equivalents   inf0 9076
      2.6  ZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-inf 9093
            2.6.2  Existence of omega (the set of natural numbers)   omex 9098
            2.6.3  Cantor normal form   ccnf 9116
            2.6.4  Transitive closure   trcl 9162
            2.6.5  Rank   cr1 9183
            2.6.6  Scott's trick; collection principle; Hilbert's epsilon   scottex 9306
            2.6.7  Disjoint union   cdju 9319
            2.6.8  Cardinal numbers   ccrd 9356
            2.6.9  Axiom of Choice equivalents   wac 9533
            *2.6.10  Cardinal number arithmetic   undjudom 9585
            2.6.11  The Ackermann bijection   ackbij2lem1 9633
            2.6.12  Cofinality (without Axiom of Choice)   cflem 9660
            2.6.13  Eight inequivalent definitions of finite set   sornom 9691
            2.6.14  Hereditarily size-limited sets without Choice   itunifval 9830
*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 9849
            3.1.2  Introduce the Axiom of Dependent Choice   ax-dc 9860
      3.2  ZFC Set Theory - add the Axiom of Choice
            3.2.1  Introduce the Axiom of Choice   ax-ac 9873
            3.2.2  AC equivalents: well-ordering, Zorn's lemma   numthcor 9908
            3.2.3  Cardinal number theorems using Axiom of Choice   cardval 9960
            3.2.4  Cardinal number arithmetic using Axiom of Choice   iunctb 9988
            3.2.5  Cofinality using the Axiom of Choice   alephreg 9996
      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 10034
            3.4.2  Derivation of the Axiom of Choice   gchaclem 10092
*PART 4  TG (TARSKI-GROTHENDIECK) SET THEORY
      4.1  Inaccessibles
            4.1.1  Weakly and strongly inaccessible cardinals   cwina 10096
            4.1.2  Weak universes   cwun 10114
            4.1.3  Tarski classes   ctsk 10162
            4.1.4  Grothendieck universes   cgru 10204
      4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
            4.2.1  Introduce the Tarski-Grothendieck Axiom   ax-groth 10237
            4.2.2  Derive the Power Set, Infinity and Choice Axioms   grothpw 10240
            4.2.3  Tarski map function   ctskm 10251
*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 10258
            5.1.2  Final derivation of real and complex number postulates   axaddf 10559
            5.1.3  Real and complex number postulates restated as axioms   ax-cnex 10585
      5.2  Derive the basic properties from the field axioms
            5.2.1  Some deductions from the field axioms for complex numbers   cnex 10610
            5.2.2  Infinity and the extended real number system   cpnf 10664
            5.2.3  Restate the ordering postulates with extended real "less than"   axlttri 10704
            5.2.4  Ordering on reals   lttr 10709
            5.2.5  Initial properties of the complex numbers   mul12 10797
      5.3  Real and complex numbers - basic operations
            5.3.1  Addition   add12 10849
            5.3.2  Subtraction   cmin 10862
            5.3.3  Multiplication   kcnktkm1cn 11063
            5.3.4  Ordering on reals (cont.)   gt0ne0 11097
            5.3.5  Reciprocals   ixi 11261
            5.3.6  Division   cdiv 11289
            5.3.7  Ordering on reals (cont.)   elimgt0 11470
            5.3.8  Completeness Axiom and Suprema   fimaxre 11576
            5.3.9  Imaginary and complex number properties   inelr 11620
            5.3.10  Function operation analogue theorems   ofsubeq0 11627
      5.4  Integer sets
            5.4.1  Positive integers (as a subset of complex numbers)   cn 11630
            5.4.2  Principle of mathematical induction   nnind 11648
            *5.4.3  Decimal representation of numbers   c2 11684
            *5.4.4  Some properties of specific numbers   neg1cn 11743
            5.4.5  Simple number properties   halfcl 11854
            5.4.6  The Archimedean property   nnunb 11885
            5.4.7  Nonnegative integers (as a subset of complex numbers)   cn0 11889
            *5.4.8  Extended nonnegative integers   cxnn0 11959
            5.4.9  Integers (as a subset of complex numbers)   cz 11973
            5.4.10  Decimal arithmetic   cdc 12090
            5.4.11  Upper sets of integers   cuz 12235
            5.4.12  Well-ordering principle for bounded-below sets of integers   uzwo3 12335
            5.4.13  Rational numbers (as a subset of complex numbers)   cq 12340
            5.4.14  Existence of the set of complex numbers   rpnnen1lem2 12368
      5.5  Order sets
            5.5.1  Positive reals (as a subset of complex numbers)   crp 12381
            5.5.2  Infinity and the extended real number system (cont.)   cxne 12496
            5.5.3  Supremum and infimum on the extended reals   xrsupexmnf 12690
            5.5.4  Real number intervals   cioo 12730
            5.5.5  Finite intervals of integers   cfz 12884
            *5.5.6  Finite intervals of nonnegative integers   elfz2nn0 12990
            5.5.7  Half-open integer ranges   cfzo 13025
      5.6  Elementary integer functions
            5.6.1  The floor and ceiling functions   cfl 13152
            5.6.2  The modulo (remainder) operation   cmo 13229
            5.6.3  Miscellaneous theorems about integers   om2uz0i 13307
            5.6.4  Strong induction over upper sets of integers   uzsinds 13347
            5.6.5  Finitely supported functions over the nonnegative integers   fsuppmapnn0fiublem 13350
            5.6.6  The infinite sequence builder "seq" - extension   cseq 13361
            5.6.7  Integer powers   cexp 13421
            5.6.8  Ordered pair theorem for nonnegative integers   nn0le2msqi 13619
            5.6.9  Factorial function   cfa 13625
            5.6.10  The binomial coefficient operation   cbc 13654
            5.6.11  The ` # ` (set size) function   chash 13682
                  5.6.11.1  Proper unordered pairs and triples (sets of size 2 and 3)   hashprlei 13818
                  5.6.11.2  Functions with a domain containing at least two different elements   fundmge2nop0 13842
                  5.6.11.3  Finite induction on the size of the first component of a binary relation   hashdifsnp1 13846
      *5.7  Words over a set
            5.7.1  Definitions and basic theorems   cword 13853
            5.7.2  Last symbol of a word   clsw 13906
            5.7.3  Concatenations of words   cconcat 13914
            5.7.4  Singleton words   cs1 13941
            5.7.5  Concatenations with singleton words   ccatws1cl 13962
            5.7.6  Subwords/substrings   csubstr 13994
            5.7.7  Prefixes of a word   cpfx 14024
            5.7.8  Subwords of subwords   swrdswrdlem 14058
            5.7.9  Subwords and concatenations   pfxcctswrd 14064
            5.7.10  Subwords of concatenations   swrdccatfn 14078
            5.7.11  Splicing words (substring replacement)   csplice 14103
            5.7.12  Reversing words   creverse 14112
            5.7.13  Repeated symbol words   creps 14122
            *5.7.14  Cyclical shifts of words   ccsh 14142
            5.7.15  Mapping words by a function   wrdco 14185
            5.7.16  Longer string literals   cs2 14195
      *5.8  Reflexive and transitive closures of relations
            5.8.1  The reflexive and transitive properties of relations   coss12d 14324
            5.8.2  Basic properties of closures   cleq1lem 14334
            5.8.3  Definitions and basic properties of transitive closures   ctcl 14337
            5.8.4  Exponentiation of relations   crelexp 14371
            5.8.5  Reflexive-transitive closure as an indexed union   crtrcl 14406
            *5.8.6  Principle of transitive induction.   relexpindlem 14414
      5.9  Elementary real and complex functions
            5.9.1  The "shift" operation   cshi 14417
            5.9.2  Signum (sgn or sign) function   csgn 14437
            5.9.3  Real and imaginary parts; conjugate   ccj 14447
            5.9.4  Square root; absolute value   csqrt 14584
      5.10  Elementary limits and convergence
            5.10.1  Superior limit (lim sup)   clsp 14819
            5.10.2  Limits   cli 14833
            5.10.3  Finite and infinite sums   csu 15034
            5.10.4  The binomial theorem   binomlem 15176
            5.10.5  The inclusion/exclusion principle   incexclem 15183
            5.10.6  Infinite sums (cont.)   isumshft 15186
            5.10.7  Miscellaneous converging and diverging sequences   divrcnv 15199
            5.10.8  Arithmetic series   arisum 15207
            5.10.9  Geometric series   expcnv 15211
            5.10.10  Ratio test for infinite series convergence   cvgrat 15231
            5.10.11  Mertens' theorem   mertenslem1 15232
            5.10.12  Finite and infinite products   prodf 15235
                  5.10.12.1  Product sequences   prodf 15235
                  5.10.12.2  Non-trivial convergence   ntrivcvg 15245
                  5.10.12.3  Complex products   cprod 15251
                  5.10.12.4  Finite products   fprod 15287
                  5.10.12.5  Infinite products   iprodclim 15344
            5.10.13  Falling and Rising Factorial   cfallfac 15350
            5.10.14  Bernoulli polynomials and sums of k-th powers   cbp 15392
      5.11  Elementary trigonometry
            5.11.1  The exponential, sine, and cosine functions   ce 15407
                  5.11.1.1  The circle constant (tau = 2 pi)   ctau 15547
            5.11.2  _e is irrational   eirrlem 15549
      5.12  Cardinality of real and complex number subsets
            5.12.1  Countability of integers and rationals   xpnnen 15556
            5.12.2  The reals are uncountable   rpnnen2lem1 15559
*PART 6  ELEMENTARY NUMBER THEORY
      6.1  Elementary properties of divisibility
            6.1.1  Irrationality of square root of 2   sqrt2irrlem 15593
            6.1.2  Some Number sets are chains of proper subsets   nthruc 15597
            6.1.3  The divides relation   cdvds 15599
            *6.1.4  Even and odd numbers   evenelz 15677
            6.1.5  The division algorithm   divalglem0 15736
            6.1.6  Bit sequences   cbits 15760
            6.1.7  The greatest common divisor operator   cgcd 15835
            6.1.8  Bézout's identity   bezoutlem1 15879
            6.1.9  Algorithms   nn0seqcvgd 15906
            6.1.10  Euclid's Algorithm   eucalgval2 15917
            *6.1.11  The least common multiple   clcm 15924
            *6.1.12  Coprimality and Euclid's lemma   coprmgcdb 15985
            6.1.13  Cancellability of congruences   congr 16000
      6.2  Elementary prime number theory
            *6.2.1  Elementary properties   cprime 16007
            *6.2.2  Coprimality and Euclid's lemma (cont.)   coprm 16047
            6.2.3  Properties of the canonical representation of a rational   cnumer 16065
            6.2.4  Euler's theorem   codz 16092
            6.2.5  Arithmetic modulo a prime number   modprm1div 16126
            6.2.6  Pythagorean Triples   coprimeprodsq 16137
            6.2.7  The prime count function   cpc 16165
            6.2.8  Pocklington's theorem   prmpwdvds 16232
            6.2.9  Infinite primes theorem   unbenlem 16236
            6.2.10  Sum of prime reciprocals   prmreclem1 16244
            6.2.11  Fundamental theorem of arithmetic   1arithlem1 16251
            6.2.12  Lagrange's four-square theorem   cgz 16257
            6.2.13  Van der Waerden's theorem   cvdwa 16293
            6.2.14  Ramsey's theorem   cram 16327
            *6.2.15  Primorial function   cprmo 16359
            *6.2.16  Prime gaps   prmgaplem1 16377
            6.2.17  Decimal arithmetic (cont.)   dec2dvds 16391
            6.2.18  Cyclical shifts of words (cont.)   cshwsidrepsw 16419
            6.2.19  Specific prime numbers   prmlem0 16431
            6.2.20  Very large primes   1259lem1 16456
PART 7  BASIC STRUCTURES
      7.1  Extensible structures
            *7.1.1  Basic definitions   cstr 16471
            7.1.2  Slot definitions   cplusg 16557
            7.1.3  Definition of the structure product   crest 16686
            7.1.4  Definition of the structure quotient   cordt 16764
      7.2  Moore spaces
            7.2.1  Moore closures   mrcflem 16869
            7.2.2  Independent sets in a Moore system   mrisval 16893
            7.2.3  Algebraic closure systems   isacs 16914
PART 8  BASIC CATEGORY THEORY
      8.1  Categories
            8.1.1  Categories   ccat 16927
            8.1.2  Opposite category   coppc 16973
            8.1.3  Monomorphisms and epimorphisms   cmon 16990
            8.1.4  Sections, inverses, isomorphisms   csect 17006
            *8.1.5  Isomorphic objects   ccic 17057
            8.1.6  Subcategories   cssc 17069
            8.1.7  Functors   cfunc 17116
            8.1.8  Full & faithful functors   cful 17164
            8.1.9  Natural transformations and the functor category   cnat 17203
            8.1.10  Initial, terminal and zero objects of a category   cinito 17240
      8.2  Arrows (disjointified hom-sets)
            8.2.1  Identity and composition for arrows   cida 17305
      8.3  Examples of categories
            8.3.1  The category of sets   csetc 17327
            8.3.2  The category of categories   ccatc 17346
            *8.3.3  The category of extensible structures   fncnvimaeqv 17362
      8.4  Categorical constructions
            8.4.1  Product of categories   cxpc 17410
            8.4.2  Functor evaluation   cevlf 17451
            8.4.3  Hom functor   chof 17490
PART 9  BASIC ORDER THEORY
      9.1  Preordered sets and directed sets using extensible structures
      9.2  Posets and lattices using extensible structures
            9.2.1  Posets   cpo 17542
            9.2.2  Lattices   clat 17647
            9.2.3  The dual of an ordered set   codu 17730
            9.2.4  Subset order structures   cipo 17753
            9.2.5  Distributive lattices   latmass 17790
            9.2.6  Posets and lattices as relations   cps 17800
            9.2.7  Directed sets, nets   cdir 17830
PART 10  BASIC ALGEBRAIC STRUCTURES
      10.1  Monoids
            *10.1.1  Magmas   cplusf 17841
            *10.1.2  Identity elements   mgmidmo 17862
            *10.1.3  Iterated sums in a magma   gsumvalx 17878
            *10.1.4  Semigroups   csgrp 17892
            *10.1.5  Definition and basic properties of monoids   cmnd 17903
            10.1.6  Monoid homomorphisms and submonoids   cmhm 17946
            *10.1.7  Iterated sums in a monoid   gsumvallem2 17990
            10.1.8  Free monoids   cfrmd 18004
                  *10.1.8.1  Monoid of endofunctions   cefmnd 18025
            10.1.9  Examples and counterexamples for magmas, semigroups and monoids   mgm2nsgrplem1 18075
      10.2  Groups
            10.2.1  Definition and basic properties   cgrp 18095
            *10.2.2  Group multiple operation   cmg 18216
            10.2.3  Subgroups and Quotient groups   csubg 18265
            *10.2.4  Cyclic monoids and groups   cycsubmel 18335
            10.2.5  Elementary theory of group homomorphisms   cghm 18347
            10.2.6  Isomorphisms of groups   cgim 18389
            10.2.7  Group actions   cga 18411
            10.2.8  Centralizers and centers   ccntz 18437
            10.2.9  The opposite group   coppg 18465
            10.2.10  Symmetric groups   csymg 18487
                  *10.2.10.1  Definition and basic properties   csymg 18487
                  10.2.10.2  Cayley's theorem   cayleylem1 18532
                  10.2.10.3  Permutations fixing one element   symgfix2 18536
                  *10.2.10.4  Transpositions in the symmetric group   cpmtr 18561
                  10.2.10.5  The sign of a permutation   cpsgn 18609
            10.2.11  p-Groups and Sylow groups; Sylow's theorems   cod 18644
            10.2.12  Direct products   clsm 18751
                  10.2.12.1  Direct products (extension)   smndlsmidm 18773
            10.2.13  Free groups   cefg 18824
            10.2.14  Abelian groups   ccmn 18898
                  10.2.14.1  Definition and basic properties   ccmn 18898
                  10.2.14.2  Cyclic groups   ccyg 18988
                  10.2.14.3  Group sum operation   gsumval3a 19015
                  10.2.14.4  Group sums over (ranges of) integers   fsfnn0gsumfsffz 19095
                  10.2.14.5  Internal direct products   cdprd 19107
                  10.2.14.6  The Fundamental Theorem of Abelian Groups   ablfacrplem 19179
            10.2.15  Simple groups   csimpg 19204
                  10.2.15.1  Definition and basic properties   csimpg 19204
                  10.2.15.2  Classification of abelian simple groups   ablsimpnosubgd 19218
      10.3  Rings
            10.3.1  Multiplicative Group   cmgp 19231
            10.3.2  Ring unit   cur 19243
                  10.3.2.1  Semirings   csrg 19247
                  *10.3.2.2  The binomial theorem for semirings   srgbinomlem1 19282
            10.3.3  Definition and basic properties of unital rings   crg 19289
            10.3.4  Opposite ring   coppr 19364
            10.3.5  Divisibility   cdsr 19380
            10.3.6  Ring primes   crpm 19454
            10.3.7  Ring homomorphisms   crh 19456
      10.4  Division rings and fields
            10.4.1  Definition and basic properties   cdr 19494
            10.4.2  Subrings of a ring   csubrg 19523
                  10.4.2.1  Sub-division rings   csdrg 19564
            10.4.3  Absolute value (abstract algebra)   cabv 19579
            10.4.4  Star rings   cstf 19606
      10.5  Left modules
            10.5.1  Definition and basic properties   clmod 19626
            10.5.2  Subspaces and spans in a left module   clss 19695
            10.5.3  Homomorphisms and isomorphisms of left modules   clmhm 19783
            10.5.4  Subspace sum; bases for a left module   clbs 19838
      10.6  Vector spaces
            10.6.1  Definition and basic properties   clvec 19866
      10.7  Ideals
            10.7.1  The subring algebra; ideals   csra 19932
            10.7.2  Two-sided ideals and quotient rings   c2idl 19996
            10.7.3  Principal ideal rings. Divisibility in the integers   clpidl 20006
            10.7.4  Nonzero rings and zero rings   cnzr 20022
            10.7.5  Left regular elements. More kinds of rings   crlreg 20044
      10.8  Associative algebras
            10.8.1  Definition and basic properties   casa 20074
      10.9  Abstract multivariate polynomials
            10.9.1  Definition and basic properties   cmps 20123
            10.9.2  Polynomial evaluation   ces 20276
            10.9.3  Additional definitions for (multivariate) polynomials   cslv 20313
            *10.9.4  Univariate polynomials   cps1 20335
            10.9.5  Univariate polynomial evaluation   ces1 20468
      10.10  The complex numbers as an algebraic extensible structure
            10.10.1  Definition and basic properties   cpsmet 20521
            *10.10.2  Ring of integers   zring 20609
            10.10.3  Algebraic constructions based on the complex numbers   czrh 20639
            10.10.4  Signs as subgroup of the complex numbers   cnmsgnsubg 20713
            10.10.5  Embedding of permutation signs into a ring   zrhpsgnmhm 20720
            10.10.6  The ordered field of real numbers   crefld 20740
      10.11  Generalized pre-Hilbert and Hilbert spaces
            10.11.1  Definition and basic properties   cphl 20760
            10.11.2  Orthocomplements and closed subspaces   cocv 20796
            10.11.3  Orthogonal projection and orthonormal bases   cpj 20836
*PART 11  BASIC LINEAR ALGEBRA
      11.1  Vectors and free modules
            *11.1.1  Direct sum of left modules   cdsmm 20867
            *11.1.2  Free modules   cfrlm 20882
            *11.1.3  Standard basis (unit vectors)   cuvc 20918
            *11.1.4  Independent sets and families   clindf 20940
            11.1.5  Characterization of free modules   lmimlbs 20972
      *11.2  Matrices
            *11.2.1  The matrix multiplication   cmmul 20986
            *11.2.2  Square matrices   cmat 21008
            *11.2.3  The matrix algebra   matmulr 21039
            *11.2.4  Matrices of dimension 0 and 1   mat0dimbas0 21067
            *11.2.5  The subalgebras of diagonal and scalar matrices   cdmat 21089
            *11.2.6  Multiplication of a matrix with a "column vector"   cmvmul 21141
            11.2.7  Replacement functions for a square matrix   cmarrep 21157
            11.2.8  Submatrices   csubma 21177
      11.3  The determinant
            11.3.1  Definition and basic properties   cmdat 21185
            11.3.2  Determinants of 2 x 2 -matrices   m2detleiblem1 21225
            11.3.3  The matrix adjugate/adjunct   cmadu 21233
            *11.3.4  Laplace expansion of determinants (special case)   symgmatr01lem 21254
            11.3.5  Inverse matrix   invrvald 21277
            *11.3.6  Cramer's rule   slesolvec 21280
      *11.4  Polynomial matrices
            11.4.1  Basic properties   pmatring 21293
            *11.4.2  Constant polynomial matrices   ccpmat 21303
            *11.4.3  Collecting coefficients of polynomial matrices   cdecpmat 21362
            *11.4.4  Ring isomorphism between polynomial matrices and polynomials over matrices   cpm2mp 21392
      *11.5  The characteristic polynomial
            *11.5.1  Definition and basic properties   cchpmat 21426
            *11.5.2  The characteristic factor function G   fvmptnn04if 21449
            *11.5.3  The Cayley-Hamilton theorem   cpmadurid 21467
PART 12  BASIC TOPOLOGY
      12.1  Topology
            *12.1.1  Topological spaces   ctop 21493
                  12.1.1.1  Topologies   ctop 21493
                  12.1.1.2  Topologies on sets   ctopon 21510
                  12.1.1.3  Topological spaces   ctps 21532
            12.1.2  Topological bases   ctb 21545
            12.1.3  Examples of topologies   distop 21595
            12.1.4  Closure and interior   ccld 21616
            12.1.5  Neighborhoods   cnei 21697
            12.1.6  Limit points and perfect sets   clp 21734
            12.1.7  Subspace topologies   restrcl 21757
            12.1.8  Order topology   ordtbaslem 21788
            12.1.9  Limits and continuity in topological spaces   ccn 21824
            12.1.10  Separated spaces: T0, T1, T2 (Hausdorff) ...   ct0 21906
            12.1.11  Compactness   ccmp 21986
            12.1.12  Bolzano-Weierstrass theorem   bwth 22010
            12.1.13  Connectedness   cconn 22011
            12.1.14  First- and second-countability   c1stc 22037
            12.1.15  Local topological properties   clly 22064
            12.1.16  Refinements   cref 22102
            12.1.17  Compactly generated spaces   ckgen 22133
            12.1.18  Product topologies   ctx 22160
            12.1.19  Continuous function-builders   cnmptid 22261
            12.1.20  Quotient maps and quotient topology   ckq 22293
            12.1.21  Homeomorphisms   chmeo 22353
      12.2  Filters and filter bases
            12.2.1  Filter bases   elmptrab 22427
            12.2.2  Filters   cfil 22445
            12.2.3  Ultrafilters   cufil 22499
            12.2.4  Filter limits   cfm 22533
            12.2.5  Extension by continuity   ccnext 22659
            12.2.6  Topological groups   ctmd 22670
            12.2.7  Infinite group sum on topological groups   ctsu 22726
            12.2.8  Topological rings, fields, vector spaces   ctrg 22756
      12.3  Uniform Structures and Spaces
            12.3.1  Uniform structures   cust 22800
            12.3.2  The topology induced by an uniform structure   cutop 22831
            12.3.3  Uniform Spaces   cuss 22854
            12.3.4  Uniform continuity   cucn 22876
            12.3.5  Cauchy filters in uniform spaces   ccfilu 22887
            12.3.6  Complete uniform spaces   ccusp 22898
      12.4  Metric spaces
            12.4.1  Pseudometric spaces   ispsmet 22906
            12.4.2  Basic metric space properties   cxms 22919
            12.4.3  Metric space balls   blfvalps 22985
            12.4.4  Open sets of a metric space   mopnval 23040
            12.4.5  Continuity in metric spaces   metcnp3 23142
            12.4.6  The uniform structure generated by a metric   metuval 23151
            12.4.7  Examples of metric spaces   dscmet 23174
            *12.4.8  Normed algebraic structures   cnm 23178
            12.4.9  Normed space homomorphisms (bounded linear operators)   cnmo 23306
            12.4.10  Topology on the reals   qtopbaslem 23359
            12.4.11  Topological definitions using the reals   cii 23475
            12.4.12  Path homotopy   chtpy 23563
            12.4.13  The fundamental group   cpco 23596
      12.5  Metric subcomplex vector spaces
            12.5.1  Subcomplex modules   cclm 23658
            *12.5.2  Subcomplex vector spaces   ccvs 23719
            *12.5.3  Normed subcomplex vector spaces   isncvsngp 23745
            12.5.4  Subcomplex pre-Hilbert space   ccph 23762
            12.5.5  Convergence and completeness   ccfil 23847
            12.5.6  Baire's Category Theorem   bcthlem1 23919
            12.5.7  Banach spaces and subcomplex Hilbert spaces   ccms 23927
                  12.5.7.1  The complete ordered field of the real numbers   retopn 23974
            12.5.8  Euclidean spaces   crrx 23978
            12.5.9  Minimizing Vector Theorem   minveclem1 24019
            12.5.10  Projection Theorem   pjthlem1 24032
PART 13  BASIC REAL AND COMPLEX ANALYSIS
      13.1  Continuity
            13.1.1  Intermediate value theorem   pmltpclem1 24041
      13.2  Integrals
            13.2.1  Lebesgue measure   covol 24055
            13.2.2  Lebesgue integration   cmbf 24207
                  13.2.2.1  Lesbesgue integral   cmbf 24207
                  13.2.2.2  Lesbesgue directed integral   cdit 24436
      13.3  Derivatives
            13.3.1  Real and complex differentiation   climc 24452
                  13.3.1.1  Derivatives of functions of one complex or real variable   climc 24452
                  13.3.1.2  Results on real differentiation   dvferm1lem 24573
PART 14  BASIC REAL AND COMPLEX FUNCTIONS
      14.1  Polynomials
            14.1.1  Polynomial degrees   cmdg 24639
            14.1.2  The division algorithm for univariate polynomials   cmn1 24711
            14.1.3  Elementary properties of complex polynomials   cply 24766
            14.1.4  The division algorithm for polynomials   cquot 24871
            14.1.5  Algebraic numbers   caa 24895
            14.1.6  Liouville's approximation theorem   aalioulem1 24913
      14.2  Sequences and series
            14.2.1  Taylor polynomials and Taylor's theorem   ctayl 24933
            14.2.2  Uniform convergence   culm 24956
            14.2.3  Power series   pserval 24990
      14.3  Basic trigonometry
            14.3.1  The exponential, sine, and cosine functions (cont.)   efcn 25023
            14.3.2  Properties of pi = 3.14159...   pilem1 25031
            14.3.3  Mapping of the exponential function   efgh 25117
            14.3.4  The natural logarithm on complex numbers   clog 25130
            *14.3.5  Logarithms to an arbitrary base   clogb 25334
            14.3.6  Theorems of Pythagoras, isosceles triangles, and intersecting chords   angval 25371
            14.3.7  Solutions of quadratic, cubic, and quartic equations   quad2 25409
            14.3.8  Inverse trigonometric functions   casin 25432
            14.3.9  The Birthday Problem   log2ublem1 25516
            14.3.10  Areas in R^2   carea 25525
            14.3.11  More miscellaneous converging sequences   rlimcnp 25535
            14.3.12  Inequality of arithmetic and geometric means   cvxcl 25554
            14.3.13  Euler-Mascheroni constant   cem 25561
            14.3.14  Zeta function   czeta 25582
            14.3.15  Gamma function   clgam 25585
      14.4  Basic number theory
            14.4.1  Wilson's theorem   wilthlem1 25637
            14.4.2  The Fundamental Theorem of Algebra   ftalem1 25642
            14.4.3  The Basel problem (ζ(2) = π2/6)   basellem1 25650
            14.4.4  Number-theoretical functions   ccht 25660
            14.4.5  Perfect Number Theorem   mersenne 25795
            14.4.6  Characters of Z/nZ   cdchr 25800
            14.4.7  Bertrand's postulate   bcctr 25843
            *14.4.8  Quadratic residues and the Legendre symbol   clgs 25862
            *14.4.9  Gauss' Lemma   gausslemma2dlem0a 25924
            14.4.10  Quadratic reciprocity   lgseisenlem1 25943
            14.4.11  All primes 4n+1 are the sum of two squares   2sqlem1 25985
            14.4.12  Chebyshev's Weak Prime Number Theorem, Dirichlet's Theorem   chebbnd1lem1 26037
            14.4.13  The Prime Number Theorem   mudivsum 26098
            14.4.14  Ostrowski's theorem   abvcxp 26183
*PART 15  ELEMENTARY GEOMETRY
      15.1  Definition and Tarski's Axioms of Geometry
            15.1.1  Justification for the congruence notation   tgjustf 26251
      15.2  Tarskian Geometry
            15.2.1  Congruence   tgcgrcomimp 26255
            15.2.2  Betweenness   tgbtwntriv2 26265
            15.2.3  Dimension   tglowdim1 26278
            15.2.4  Betweenness and Congruence   tgifscgr 26286
            15.2.5  Congruence of a series of points   ccgrg 26288
            15.2.6  Motions   cismt 26310
            15.2.7  Colinearity   tglng 26324
            15.2.8  Connectivity of betweenness   tgbtwnconn1lem1 26350
            15.2.9  Less-than relation in geometric congruences   cleg 26360
            15.2.10  Rays   chlg 26378
            15.2.11  Lines   btwnlng1 26397
            15.2.12  Point inversions   cmir 26430
            15.2.13  Right angles   crag 26471
            15.2.14  Half-planes   islnopp 26517
            15.2.15  Midpoints and Line Mirroring   cmid 26550
            15.2.16  Congruence of angles   ccgra 26585
            15.2.17  Angle Comparisons   cinag 26613
            15.2.18  Congruence Theorems   tgsas1 26632
            15.2.19  Equilateral triangles   ceqlg 26643
      15.3  Properties of geometries
            15.3.1  Isomorphisms between geometries   f1otrgds 26647
      15.4  Geometry in Hilbert spaces
            15.4.1  Geometry in the complex plane   cchhllem 26665
            15.4.2  Geometry in Euclidean spaces   cee 26666
                  15.4.2.1  Definition of the Euclidean space   cee 26666
                  15.4.2.2  Tarski's axioms for geometry for the Euclidean space   axdimuniq 26691
                  15.4.2.3  EE^n fulfills Tarski's Axioms   ceeng 26755
*PART 16  GRAPH THEORY
      *16.1  Vertices and edges
            16.1.1  The edge function extractor for extensible structures   cedgf 26766
            *16.1.2  Vertices and indexed edges   cvtx 26773
                  16.1.2.1  Definitions and basic properties   cvtx 26773
                  16.1.2.2  The vertices and edges of a graph represented as ordered pair   opvtxval 26780
                  16.1.2.3  The vertices and edges of a graph represented as extensible structure   funvtxdmge2val 26788
                  16.1.2.4  Representations of graphs without edges   snstrvtxval 26814
                  16.1.2.5  Degenerated cases of representations of graphs   vtxval0 26816
            16.1.3  Edges as range of the edge function   cedg 26824
      *16.2  Undirected graphs
            16.2.1  Undirected hypergraphs   cuhgr 26833
            16.2.2  Undirected pseudographs and multigraphs   cupgr 26857
            *16.2.3  Loop-free graphs   umgrislfupgrlem 26899
            16.2.4  Edges as subsets of vertices of graphs   uhgredgiedgb 26903
            *16.2.5  Undirected simple graphs   cuspgr 26925
            16.2.6  Examples for graphs   usgr0e 27010
            16.2.7  Subgraphs   csubgr 27041
            16.2.8  Finite undirected simple graphs   cfusgr 27090
            16.2.9  Neighbors, complete graphs and universal vertices   cnbgr 27106
                  16.2.9.1  Neighbors   cnbgr 27106
                  16.2.9.2  Universal vertices   cuvtx 27159
                  16.2.9.3  Complete graphs   ccplgr 27183
            16.2.10  Vertex degree   cvtxdg 27239
            *16.2.11  Regular graphs   crgr 27329
      *16.3  Walks, paths and cycles
            *16.3.1  Walks   cewlks 27369
            16.3.2  Walks for loop-free graphs   lfgrwlkprop 27461
            16.3.3  Trails   ctrls 27464
            16.3.4  Paths and simple paths   cpths 27485
            16.3.5  Closed walks   cclwlks 27543
            16.3.6  Circuits and cycles   ccrcts 27557
            *16.3.7  Walks as words   cwwlks 27595
            16.3.8  Walks/paths of length 2 (as length 3 strings)   2wlkdlem1 27696
            16.3.9  Walks in regular graphs   rusgrnumwwlkl1 27739
            *16.3.10  Closed walks as words   cclwwlk 27751
                  16.3.10.1  Closed walks as words   cclwwlk 27751
                  16.3.10.2  Closed walks of a fixed length as words   cclwwlkn 27794
                  16.3.10.3  Closed walks on a vertex of a fixed length as words   cclwwlknon 27858
            16.3.11  Examples for walks, trails and paths   0ewlk 27885
            16.3.12  Connected graphs   cconngr 27957
      16.4  Eulerian paths and the Konigsberg Bridge problem
            *16.4.1  Eulerian paths   ceupth 27968
            *16.4.2  The Königsberg Bridge problem   konigsbergvtx 28017
      16.5  The Friendship Theorem
            16.5.1  Friendship graphs - basics   cfrgr 28029
            16.5.2  The friendship theorem for small graphs   frgr1v 28042
            16.5.3  Theorems according to Mertzios and Unger   2pthfrgrrn 28053
            *16.5.4  Huneke's Proof of the Friendship Theorem   frgrncvvdeqlem1 28070
PART 17  GUIDES AND MISCELLANEA
      17.1  Guides (conventions, explanations, and examples)
            *17.1.1  Conventions   conventions 28171
            17.1.2  Natural deduction   natded 28174
            *17.1.3  Natural deduction examples   ex-natded5.2 28175
            17.1.4  Definitional examples   ex-or 28192
            17.1.5  Other examples   aevdemo 28231
      17.2  Humor
            17.2.1  April Fool's theorem   avril1 28234
      17.3  (Future - to be reviewed and classified)
            17.3.1  Planar incidence geometry   cplig 28243
            *17.3.2  Aliases kept to prevent broken links   dummylink 28256
*PART 18  COMPLEX TOPOLOGICAL VECTOR SPACES (DEPRECATED)
      *18.1  Additional material on group theory (deprecated)
            18.1.1  Definitions and basic properties for groups   cgr 28258
            18.1.2  Abelian groups   cablo 28313
      18.2  Complex vector spaces
            18.2.1  Definition and basic properties   cvc 28327
            18.2.2  Examples of complex vector spaces   cnaddabloOLD 28350
      18.3  Normed complex vector spaces
            18.3.1  Definition and basic properties   cnv 28353
            18.3.2  Examples of normed complex vector spaces   cnnv 28446
            18.3.3  Induced metric of a normed complex vector space   imsval 28454
            18.3.4  Inner product   cdip 28469
            18.3.5  Subspaces   css 28490
      18.4  Operators on complex vector spaces
            18.4.1  Definitions and basic properties   clno 28509
      18.5  Inner product (pre-Hilbert) spaces
            18.5.1  Definition and basic properties   ccphlo 28581
            18.5.2  Examples of pre-Hilbert spaces   cncph 28588
            18.5.3  Properties of pre-Hilbert spaces   isph 28591
      18.6  Complex Banach spaces
            18.6.1  Definition and basic properties   ccbn 28631
            18.6.2  Examples of complex Banach spaces   cnbn 28638
            18.6.3  Uniform Boundedness Theorem   ubthlem1 28639
            18.6.4  Minimizing Vector Theorem   minvecolem1 28643
      18.7  Complex Hilbert spaces
            18.7.1  Definition and basic properties   chlo 28654
            18.7.2  Standard axioms for a complex Hilbert space   hlex 28667
            18.7.3  Examples of complex Hilbert spaces   cnchl 28685
            18.7.4  Hellinger-Toeplitz Theorem   htthlem 28686
*PART 19  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      19.1  Axiomatization of complex pre-Hilbert spaces
            19.1.1  Basic Hilbert space definitions   chba 28688
            19.1.2  Preliminary ZFC lemmas   df-hnorm 28737
            *19.1.3  Derive the Hilbert space axioms from ZFC set theory   axhilex-zf 28750
            *19.1.4  Introduce the vector space axioms for a Hilbert space   ax-hilex 28768
            19.1.5  Vector operations   hvmulex 28780
            19.1.6  Inner product postulates for a Hilbert space   ax-hfi 28848
      19.2  Inner product and norms
            19.2.1  Inner product   his5 28855
            19.2.2  Norms   dfhnorm2 28891
            19.2.3  Relate Hilbert space to normed complex vector spaces   hilablo 28929
            19.2.4  Bunjakovaskij-Cauchy-Schwarz inequality   bcsiALT 28948
      19.3  Cauchy sequences and completeness axiom
            19.3.1  Cauchy sequences and limits   hcau 28953
            19.3.2  Derivation of the completeness axiom from ZF set theory   hilmet 28963
            19.3.3  Completeness postulate for a Hilbert space   ax-hcompl 28971
            19.3.4  Relate Hilbert space to ZFC pre-Hilbert and Hilbert spaces   hhcms 28972
      19.4  Subspaces and projections
            19.4.1  Subspaces   df-sh 28976
            19.4.2  Closed subspaces   df-ch 28990
            19.4.3  Orthocomplements   df-oc 29021
            19.4.4  Subspace sum, span, lattice join, lattice supremum   df-shs 29077
            19.4.5  Projection theorem   pjhthlem1 29160
            19.4.6  Projectors   df-pjh 29164
      19.5  Properties of Hilbert subspaces
            19.5.1  Orthomodular law   omlsilem 29171
            19.5.2  Projectors (cont.)   pjhtheu2 29185
            19.5.3  Hilbert lattice operations   sh0le 29209
            19.5.4  Span (cont.) and one-dimensional subspaces   spansn0 29310
            19.5.5  Commutes relation for Hilbert lattice elements   df-cm 29352
            19.5.6  Foulis-Holland theorem   fh1 29387
            19.5.7  Quantum Logic Explorer axioms   qlax1i 29396
            19.5.8  Orthogonal subspaces   chscllem1 29406
            19.5.9  Orthoarguesian laws 5OA and 3OA   5oalem1 29423
            19.5.10  Projectors (cont.)   pjorthi 29438
            19.5.11  Mayet's equation E_3   mayete3i 29497
      19.6  Operators on Hilbert spaces
            *19.6.1  Operator sum, difference, and scalar multiplication   df-hosum 29499
            19.6.2  Zero and identity operators   df-h0op 29517
            19.6.3  Operations on Hilbert space operators   hoaddcl 29527
            19.6.4  Linear, continuous, bounded, Hermitian, unitary operators and norms   df-nmop 29608
            19.6.5  Linear and continuous functionals and norms   df-nmfn 29614
            19.6.6  Adjoint   df-adjh 29618
            19.6.7  Dirac bra-ket notation   df-bra 29619
            19.6.8  Positive operators   df-leop 29621
            19.6.9  Eigenvectors, eigenvalues, spectrum   df-eigvec 29622
            19.6.10  Theorems about operators and functionals   nmopval 29625
            19.6.11  Riesz lemma   riesz3i 29831
            19.6.12  Adjoints (cont.)   cnlnadjlem1 29836
            19.6.13  Quantum computation error bound theorem   unierri 29873
            19.6.14  Dirac bra-ket notation (cont.)   branmfn 29874
            19.6.15  Positive operators (cont.)   leopg 29891
            19.6.16  Projectors as operators   pjhmopi 29915
      19.7  States on a Hilbert lattice and Godowski's equation
            19.7.1  States on a Hilbert lattice   df-st 29980
            19.7.2  Godowski's equation   golem1 30040
      19.8  Cover relation, atoms, exchange axiom, and modular symmetry
            19.8.1  Covers relation; modular pairs   df-cv 30048
            19.8.2  Atoms   df-at 30107
            19.8.3  Superposition principle   superpos 30123
            19.8.4  Atoms, exchange and covering properties, atomicity   chcv1 30124
            19.8.5  Irreducibility   chirredlem1 30159
            19.8.6  Atoms (cont.)   atcvat3i 30165
            19.8.7  Modular symmetry   mdsymlem1 30172
PART 20  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      20.1  Mathboxes for user contributions
            20.1.1  Mathbox guidelines   mathbox 30211
      20.2  Mathbox for Stefan Allan
      20.3  Mathbox for Thierry Arnoux
            20.3.1  Propositional Calculus - misc additions   bian1d 30216
            20.3.2  Predicate Calculus   sbc2iedf 30222
                  20.3.2.1  Predicate Calculus - misc additions   sbc2iedf 30222
                  20.3.2.2  Restricted quantification - misc additions   ralcom4f 30225
                  20.3.2.3  Equality   eqtrb 30230
                  20.3.2.4  Double restricted existential uniqueness quantification   opsbc2ie 30231
                  20.3.2.5  Double restricted existential uniqueness quantification syntax   w2reu 30233
                  20.3.2.6  Substitution (without distinct variables) - misc additions   sbceqbidf 30242
                  20.3.2.7  Existential "at most one" - misc additions   moel 30244
                  20.3.2.8  Existential uniqueness - misc additions   reuxfrdf 30247
                  20.3.2.9  Restricted "at most one" - misc additions   rmoxfrd 30249
            20.3.3  General Set Theory   dmrab 30252
                  20.3.3.1  Class abstractions (a.k.a. class builders)   dmrab 30252
                  20.3.3.2  Image Sets   abrexdomjm 30259
                  20.3.3.3  Set relations and operations - misc additions   elunsn 30265
                  20.3.3.4  Unordered pairs   eqsnd 30281
                  20.3.3.5  Conditional operator - misc additions   ifeqeqx 30289
                  20.3.3.6  Set union   uniinn0 30294
                  20.3.3.7  Indexed union - misc additions   cbviunf 30299
                  20.3.3.8  Disjointness - misc additions   disjnf 30312
            20.3.4  Relations and Functions   xpdisjres 30340
                  20.3.4.1  Relations - misc additions   xpdisjres 30340
                  20.3.4.2  Functions - misc additions   ac6sf2 30362
                  20.3.4.3  Operations - misc additions   mpomptxf 30417
                  20.3.4.4  Explicit Functions with one or two points as a domain   brsnop 30421
                  20.3.4.5  Isomorphisms - misc. add.   gtiso 30428
                  20.3.4.6  Disjointness (additional proof requiring functions)   disjdsct 30430
                  20.3.4.7  First and second members of an ordered pair - misc additions   df1stres 30431
                  20.3.4.8  Supremum - misc additions   supssd 30437
                  20.3.4.9  Finite Sets   imafi2 30439
                  20.3.4.10  Countable Sets   snct 30441
            20.3.5  Real and Complex Numbers   creq0 30463
                  20.3.5.1  Complex operations - misc. additions   creq0 30463
                  20.3.5.2  Ordering on reals - misc additions   lt2addrd 30467
                  20.3.5.3  Extended reals - misc additions   xrlelttric 30468
                  20.3.5.4  Extended nonnegative integers - misc additions   xnn0gt0 30486
                  20.3.5.5  Real number intervals - misc additions   joiniooico 30489
                  20.3.5.6  Finite intervals of integers - misc additions   uzssico 30499
                  20.3.5.7  Half-open integer ranges - misc additions   iundisjfi 30511
                  20.3.5.8  The ` # ` (set size) function - misc additions   hashunif 30520
                  20.3.5.9  The greatest common divisor operator - misc. add   dvdszzq 30523
                  20.3.5.10  Integers   nnindf 30527
                  20.3.5.11  Decimal numbers   dfdec100 30539
            *20.3.6  Decimal expansion   cdp2 30540
                  *20.3.6.1  Decimal point   cdp 30557
                  20.3.6.2  Division in the extended real number system   cxdiv 30586
            20.3.7  Words over a set - misc additions   wrdfd 30605
                  20.3.7.1  Splicing words (substring replacement)   splfv3 30625
                  20.3.7.2  Cyclic shift of words   1cshid 30626
            20.3.8  Extensible Structures   ressplusf 30630
                  20.3.8.1  Structure restriction operator   ressplusf 30630
                  20.3.8.2  The opposite group   oppgle 30633
                  20.3.8.3  Posets   ressprs 30635
                  20.3.8.4  Complete lattices   clatp0cl 30651
                  20.3.8.5  Extended reals Structure - misc additions   ax-xrssca 30653
                  20.3.8.6  The extended nonnegative real numbers commutative monoid   xrge0base 30665
            20.3.9  Algebra   abliso 30676
                  20.3.9.1  Monoids Homomorphisms   abliso 30676
                  20.3.9.2  Finitely supported group sums - misc additions   gsumsubg 30677
                  20.3.9.3  Centralizers and centers - misc additions   cntzun 30688
                  20.3.9.4  Totally ordered monoids and groups   comnd 30691
                  20.3.9.5  The symmetric group   symgfcoeu 30719
                  20.3.9.6  Transpositions   pmtridf1o 30729
                  20.3.9.7  Permutation Signs   psgnid 30732
                  20.3.9.8  Permutation cycles   ctocyc 30741
                  20.3.9.9  The Alternating Group   evpmval 30780
                  20.3.9.10  Signum in an ordered monoid   csgns 30793
                  20.3.9.11  The Archimedean property for generic ordered algebraic structures   cinftm 30798
                  20.3.9.12  Semiring left modules   cslmd 30821
                  20.3.9.13  Simple groups   prmsimpcyc 30849
                  20.3.9.14  Rings - misc additions   rngurd 30850
                  20.3.9.15  Subfields   primefldchr 30860
                  20.3.9.16  Totally ordered rings and fields   corng 30861
                  20.3.9.17  Ring homomorphisms - misc additions   rhmdvdsr 30884
                  20.3.9.18  Scalar restriction operation   cresv 30890
                  20.3.9.19  The commutative ring of gaussian integers   gzcrng 30905
                  20.3.9.20  The archimedean ordered field of real numbers   reofld 30906
                  20.3.9.21  The quotient map and quotient modules   qusker 30911
                  20.3.9.22  Univariate Polynomials   fply1 30924
                  20.3.9.23  Independent sets and families   islinds5 30925
                  *20.3.9.24  Subgroup sum / Sumset / Minkowski sum   elgrplsmsn 30937
                  20.3.9.25  Prime Ideals   cprmidl 30945
                  20.3.9.26  Maximal Ideals   cmxidl 30961
                  20.3.9.27  The semiring of ideals of a ring   cidlsrg 30975
                  20.3.9.28  The subring algebra   sra1r 30979
                  20.3.9.29  Division Ring Extensions   drgext0g 30985
                  20.3.9.30  Vector Spaces   lvecdimfi 30991
                  20.3.9.31  Vector Space Dimension   cldim 30992
            20.3.10  Field Extensions   cfldext 31021
            20.3.11  Matrices   csmat 31051
                  20.3.11.1  Submatrices   csmat 31051
                  20.3.11.2  Matrix literals   clmat 31069
                  20.3.11.3  Laplace expansion of determinants   mdetpmtr1 31081
            20.3.12  Topology   txomap 31091
                  20.3.12.1  Open maps   txomap 31091
                  20.3.12.2  Topology of the unit circle   qtopt1 31092
                  20.3.12.3  Refinements   reff 31096
                  20.3.12.4  Open cover refinement property   ccref 31099
                  20.3.12.5  Lindelöf spaces   cldlf 31109
                  20.3.12.6  Paracompact spaces   cpcmp 31112
                  20.3.12.7  Pseudometrics   cmetid 31119
                  20.3.12.8  Continuity - misc additions   hauseqcn 31131
                  20.3.12.9  Topology of the closed unit interval   unitsscn 31132
                  20.3.12.10  Topology of ` ( RR X. RR ) `   unicls 31139
                  20.3.12.11  Order topology - misc. additions   cnvordtrestixx 31149
                  20.3.12.12  Continuity in topological spaces - misc. additions   mndpluscn 31162
                  20.3.12.13  Topology of the extended nonnegative real numbers ordered monoid   xrge0hmph 31168
                  20.3.12.14  Limits - misc additions   lmlim 31183
                  20.3.12.15  Univariate polynomials   pl1cn 31191
            20.3.13  Uniform Stuctures and Spaces   chcmp 31192
                  20.3.13.1  Hausdorff uniform completion   chcmp 31192
            20.3.14  Topology and algebraic structures   zringnm 31194
                  20.3.14.1  The norm on the ring of the integer numbers   zringnm 31194
                  20.3.14.2  Topological ` ZZ ` -modules   zlm0 31196
                  20.3.14.3  Canonical embedding of the field of the rational numbers into a division ring   cqqh 31206
                  20.3.14.4  Canonical embedding of the real numbers into a complete ordered field   crrh 31227
                  20.3.14.5  Embedding from the extended real numbers into a complete lattice   cxrh 31250
                  20.3.14.6  Canonical embeddings into the ordered field of the real numbers   zrhre 31253
                  *20.3.14.7  Topological Manifolds   cmntop 31256
            20.3.15  Real and complex functions   nexple 31261
                  20.3.15.1  Integer powers - misc. additions   nexple 31261
                  20.3.15.2  Indicator Functions   cind 31262
                  20.3.15.3  Extended sum   cesum 31279
            20.3.16  Mixed Function/Constant operation   cofc 31347
            20.3.17  Abstract measure   csiga 31360
                  20.3.17.1  Sigma-Algebra   csiga 31360
                  20.3.17.2  Generated sigma-Algebra   csigagen 31390
                  *20.3.17.3  lambda and pi-Systems, Rings of Sets   ispisys 31404
                  20.3.17.4  The Borel algebra on the real numbers   cbrsiga 31433
                  20.3.17.5  Product Sigma-Algebra   csx 31440
                  20.3.17.6  Measures   cmeas 31447
                  20.3.17.7  The counting measure   cntmeas 31478
                  20.3.17.8  The Lebesgue measure - misc additions   voliune 31481
                  20.3.17.9  The Dirac delta measure   cdde 31484
                  20.3.17.10  The 'almost everywhere' relation   cae 31489
                  20.3.17.11  Measurable functions   cmbfm 31501
                  20.3.17.12  Borel Algebra on ` ( RR X. RR ) `   br2base 31520
                  *20.3.17.13  Caratheodory's extension theorem   coms 31542
            20.3.18  Integration   itgeq12dv 31577
                  20.3.18.1  Lebesgue integral - misc additions   itgeq12dv 31577
                  20.3.18.2  Bochner integral   citgm 31578
            20.3.19  Euler's partition theorem   oddpwdc 31605
            20.3.20  Sequences defined by strong recursion   csseq 31634
            20.3.21  Fibonacci Numbers   cfib 31647
            20.3.22  Probability   cprb 31658
                  20.3.22.1  Probability Theory   cprb 31658
                  20.3.22.2  Conditional Probabilities   ccprob 31682
                  20.3.22.3  Real-valued Random Variables   crrv 31691
                  20.3.22.4  Preimage set mapping operator   corvc 31706
                  20.3.22.5  Distribution Functions   orvcelval 31719
                  20.3.22.6  Cumulative Distribution Functions   orvclteel 31723
                  20.3.22.7  Probabilities - example   coinfliplem 31729
                  20.3.22.8  Bertrand's Ballot Problem   ballotlemoex 31736
            20.3.23  Signum (sgn or sign) function - misc. additions   sgncl 31789
                  20.3.23.1  Operations on words   ccatmulgnn0dir 31805
            20.3.24  Polynomials with real coefficients - misc additions   plymul02 31809
            20.3.25  Descartes's rule of signs   signspval 31815
                  20.3.25.1  Sign changes in a word over real numbers   signspval 31815
                  20.3.25.2  Counting sign changes in a word over real numbers   signslema 31825
            20.3.26  Number Theory   efcld 31855
                  20.3.26.1  Representations of a number as sums of integers   crepr 31872
                  20.3.26.2  Vinogradov Trigonometric Sums and the Circle Method   cvts 31899
                  20.3.26.3  The Ternary Goldbach Conjecture: Final Statement   ax-hgt749 31908
            20.3.27  Elementary Geometry   cstrkg2d 31928
                  *20.3.27.1  Two-dimensional geometry   cstrkg2d 31928
                  20.3.27.2  Outer Five Segment (not used, no need to move to main)   cafs 31933
            *20.3.28  LeftPad Project   clpad 31938
      *20.4  Mathbox for Jonathan Ben-Naim
            20.4.1  First-order logic and set theory   bnj170 31961
            20.4.2  Well founded induction and recursion   bnj110 32123
            20.4.3  The existence of a minimal element in certain classes   bnj69 32275
            20.4.4  Well-founded induction   bnj1204 32277
            20.4.5  Well-founded recursion, part 1 of 3   bnj60 32327
            20.4.6  Well-founded recursion, part 2 of 3   bnj1500 32333
            20.4.7  Well-founded recursion, part 3 of 3   bnj1522 32337
      20.5  Mathbox for BTernaryTau
            20.5.1  Acyclic graphs   cacycgr 32382
      20.6  Mathbox for Mario Carneiro
            20.6.1  Predicate calculus with all distinct variables   ax-7d 32399
            20.6.2  Miscellaneous stuff   quartfull 32405
            20.6.3  Derangements and the Subfactorial   deranglem 32406
            20.6.4  The Erdős-Szekeres theorem   erdszelem1 32431
            20.6.5  The Kuratowski closure-complement theorem   kur14lem1 32446
            20.6.6  Retracts and sections   cretr 32457
            20.6.7  Path-connected and simply connected spaces   cpconn 32459
            20.6.8  Covering maps   ccvm 32495
            20.6.9  Normal numbers   snmlff 32569
            20.6.10  Godel-sets of formulas - part 1   cgoe 32573
            20.6.11  Godel-sets of formulas - part 2   cgon 32672
            20.6.12  Models of ZF   cgze 32686
            *20.6.13  Metamath formal systems   cmcn 32700
            20.6.14  Grammatical formal systems   cm0s 32825
            20.6.15  Models of formal systems   cmuv 32845
            20.6.16  Splitting fields   citr 32867
            20.6.17  p-adic number fields   czr 32883
      *20.7  Mathbox for Filip Cernatescu
      20.8  Mathbox for Paul Chapman
            20.8.1  Real and complex numbers (cont.)   climuzcnv 32907
            20.8.2  Miscellaneous theorems   elfzm12 32911
      20.9  Mathbox for Scott Fenton
            20.9.1  ZFC Axioms in primitive form   axextprim 32920
            20.9.2  Untangled classes   untelirr 32927
            20.9.3  Extra propositional calculus theorems   3orel2 32934
            20.9.4  Misc. Useful Theorems   nepss 32941
            20.9.5  Properties of real and complex numbers   sqdivzi 32952
            20.9.6  Infinite products   iprodefisumlem 32965
            20.9.7  Factorial limits   faclimlem1 32968
            20.9.8  Greatest common divisor and divisibility   pdivsq 32974
            20.9.9  Properties of relationships   brtp 32978
            20.9.10  Properties of functions and mappings   funpsstri 33001
            20.9.11  Set induction (or epsilon induction)   setinds 33016
            20.9.12  Ordinal numbers   elpotr 33019
            20.9.13  Defined equality axioms   axextdfeq 33035
            20.9.14  Hypothesis builders   hbntg 33043
            20.9.15  (Trans)finite Recursion Theorems   tfisg 33048
            20.9.16  Transitive closure under a relationship   ctrpred 33049
            20.9.17  Founded Induction   frpomin 33071
            20.9.18  Ordering Ordinal Sequences   orderseqlem 33087
            20.9.19  Well-founded zero, successor, and limits   cwsuc 33090
            20.9.20  Founded Partial Recursion   cfrecs 33110
            20.9.21  Surreal Numbers   csur 33140
            20.9.22  Surreal Numbers: Ordering   sltsolem1 33173
            20.9.23  Surreal Numbers: Birthday Function   bdayfo 33175
            20.9.24  Surreal Numbers: Density   fvnobday 33176
            20.9.25  Surreal Numbers: Full-Eta Property   bdayimaon 33190
            20.9.26  Surreal numbers - ordering theorems   csle 33216
            20.9.27  Surreal numbers - birthday theorems   bdayfun 33235
            20.9.28  Surreal numbers: Conway cuts   csslt 33243
            20.9.29  Surreal numbers - cuts and options   cmade 33272
            20.9.30  Quantifier-free definitions   ctxp 33284
            20.9.31  Alternate ordered pairs   caltop 33410
            20.9.32  Geometry in the Euclidean space   cofs 33436
                  20.9.32.1  Congruence properties   cofs 33436
                  20.9.32.2  Betweenness properties   btwntriv2 33466
                  20.9.32.3  Segment Transportation   ctransport 33483
                  20.9.32.4  Properties relating betweenness and congruence   cifs 33489
                  20.9.32.5  Connectivity of betweenness   btwnconn1lem1 33541
                  20.9.32.6  Segment less than or equal to   csegle 33560
                  20.9.32.7  Outside-of relationship   coutsideof 33573
                  20.9.32.8  Lines and Rays   cline2 33588
            20.9.33  Forward difference   cfwddif 33612
            20.9.34  Rank theorems   rankung 33620
            20.9.35  Hereditarily Finite Sets   chf 33626
      20.10  Mathbox for Jeff Hankins
            20.10.1  Miscellany   a1i14 33641
            20.10.2  Basic topological facts   topbnd 33665
            20.10.3  Topology of the real numbers   ivthALT 33676
            20.10.4  Refinements   cfne 33677
            20.10.5  Neighborhood bases determine topologies   neibastop1 33700
            20.10.6  Lattice structure of topologies   topmtcl 33704
            20.10.7  Filter bases   fgmin 33711
            20.10.8  Directed sets, nets   tailfval 33713
      20.11  Mathbox for Anthony Hart
            20.11.1  Propositional Calculus   tb-ax1 33724
            20.11.2  Predicate Calculus   nalfal 33744
            20.11.3  Miscellaneous single axioms   meran1 33752
            20.11.4  Connective Symmetry   negsym1 33758
      20.12  Mathbox for Chen-Pang He
            20.12.1  Ordinal topology   ontopbas 33769
      20.13  Mathbox for Jeff Hoffman
            20.13.1  Inferences for finite induction on generic function values   fveleq 33792
            20.13.2  gdc.mm   nnssi2 33796
      20.14  Mathbox for Asger C. Ipsen
            20.14.1  Continuous nowhere differentiable functions   dnival 33803
      *20.15  Mathbox for BJ
            *20.15.1  Propositional calculus   bj-mp2c 33872
                  *20.15.1.1  Derived rules of inference   bj-mp2c 33872
                  *20.15.1.2  A syntactic theorem   bj-0 33874
                  20.15.1.3  Minimal implicational calculus   bj-a1k 33876
                  *20.15.1.4  Positive calculus   bj-syl66ib 33883
                  20.15.1.5  Implication and negation   bj-con2com 33889
                  *20.15.1.6  Disjunction   bj-jaoi1 33897
                  *20.15.1.7  Logical equivalence   bj-dfbi4 33899
                  20.15.1.8  The conditional operator for propositions   bj-consensus 33904
                  *20.15.1.9  Propositional calculus: miscellaneous   bj-imbi12 33909
            *20.15.2  Modal logic   bj-axdd2 33919
            *20.15.3  Provability logic   cprvb 33924
            *20.15.4  First-order logic   bj-genr 33933
                  20.15.4.1  Adding ax-gen   bj-genr 33933
                  20.15.4.2  Adding ax-4   bj-2alim 33937
                  20.15.4.3  Adding ax-5   bj-ax12wlem 33970
                  20.15.4.4  Equality and substitution   bj-ssbeq 33979
                  20.15.4.5  Adding ax-6   bj-spimvwt 33995
                  20.15.4.6  Adding ax-7   bj-cbvexw 34002
                  20.15.4.7  Membership predicate, ax-8 and ax-9   bj-ax89 34004
                  20.15.4.8  Adding ax-11   bj-alcomexcom 34007
                  20.15.4.9  Adding ax-12   axc11n11 34009
                  20.15.4.10  Nonfreeness   wnnf 34048
                  20.15.4.11  Adding ax-13   bj-axc10 34098
                  *20.15.4.12  Removing dependencies on ax-13 (and ax-11)   bj-axc10v 34108
                  *20.15.4.13  Distinct var metavariables   bj-hbaeb2 34134
                  *20.15.4.14  Around ~ equsal   bj-equsal1t 34138
                  *20.15.4.15  Some Principia Mathematica proofs   stdpc5t 34143
                  20.15.4.16  Alternate definition of substitution   bj-sbsb 34153
                  20.15.4.17  Lemmas for substitution   bj-sbf3 34155
                  20.15.4.18  Existential uniqueness   bj-eu3f 34158
                  *20.15.4.19  First-order logic: miscellaneous   bj-sblem1 34159
            20.15.5  Set theory   eliminable1 34175
                  *20.15.5.1  Eliminability of class terms   eliminable1 34175
                  *20.15.5.2  Classes without the axiom of extensionality   bj-denotes 34181
                  20.15.5.3  Characterization among sets versus among classes   elelb 34206
                  *20.15.5.4  The nonfreeness quantifier for classes   bj-nfcsym 34208
                  *20.15.5.5  Proposal for the definitions of class membership and class equality   bj-ax9 34209
                  *20.15.5.6  Lemmas for class substitution   bj-sbeqALT 34210
                  20.15.5.7  Removing some axiom requirements and disjoint variable conditions   bj-exlimvmpi 34220
                  *20.15.5.8  Class abstractions   bj-unrab 34237
                  *20.15.5.9  Restricted nonfreeness   wrnf 34244
                  *20.15.5.10  Russell's paradox   bj-ru0 34246
                  20.15.5.11  Curry's paradox in set theory   currysetlem 34249
                  *20.15.5.12  Some disjointness results   bj-n0i 34255
                  *20.15.5.13  Complements on direct products   bj-xpimasn 34260
                  *20.15.5.14  "Singletonization" and tagging   bj-snsetex 34268
                  *20.15.5.15  Tuples of classes   bj-cproj 34295
                  *20.15.5.16  Set theory: elementary operations relative to a universe   bj-rcleqf 34330
                  *20.15.5.17  Set theory: miscellaneous   bj-sselpwuni 34335
                  *20.15.5.18  Evaluation   bj-evaleq 34355
                  20.15.5.19  Elementwise operations   celwise 34362
                  *20.15.5.20  Elementwise intersection (families of sets induced on a subset)   bj-rest00 34364
                  20.15.5.21  Moore collections (complements)   bj-intss 34383
                  20.15.5.22  Maps-to notation for functions with three arguments   bj-0nelmpt 34400
                  *20.15.5.23  Currying   csethom 34406
                  *20.15.5.24  Setting components of extensible structures   cstrset 34418
            *20.15.6  Extended real and complex numbers, real and complex projective lines   bj-nfald 34421
                  20.15.6.1  Complements on class abstractions of ordered pairs and binary relations   bj-nfald 34421
                  *20.15.6.2  Identity relation (complements)   bj-opabssvv 34434
                  *20.15.6.3  Functionalized identity (diagonal in a Cartesian square)   cdiag2 34456
                  *20.15.6.4  Direct image and inverse image   cimdir 34462
                  *20.15.6.5  Extended numbers and projective lines as sets   cfractemp 34470
                  *20.15.6.6  Addition and opposite   caddcc 34511
                  *20.15.6.7  Order relation on the extended reals   cltxr 34515
                  *20.15.6.8  Argument, multiplication and inverse   carg 34517
                  20.15.6.9  The canonical bijection from the finite ordinals   ciomnn 34523
                  20.15.6.10  Divisibility   cnnbar 34534
            *20.15.7  Monoids   bj-smgrpssmgm 34542
                  *20.15.7.1  Finite sums in monoids   cfinsum 34557
            *20.15.8  Affine, Euclidean, and Cartesian geometry   bj-fvimacnv0 34560
                  *20.15.8.1  Real vector spaces   bj-fvimacnv0 34560
                  *20.15.8.2  Complex numbers (supplements)   bj-subcom 34581
                  *20.15.8.3  Barycentric coordinates   bj-bary1lem 34583
            20.15.9  Monoid of endomorphisms   cend 34586
      20.16  Mathbox for Jim Kingdon
                  20.16.0.1  Circle constant   taupilem3 34592
                  20.16.0.2  Number theory   dfgcd3 34597
      20.17  Mathbox for ML
            20.17.1  Miscellaneous   csbdif 34598
            20.17.2  Cartesian exponentiation   cfinxp 34656
            20.17.3  Topology   iunctb2 34676
                  *20.17.3.1  Pi-base theorems   pibp16 34686
      20.18  Mathbox for Wolf Lammen
            20.18.1  1. Bootstrapping   wl-section-boot 34695
            20.18.2  Implication chains   wl-section-impchain 34719
            20.18.3  An alternative axiom ~ ax-13   ax-wl-13v 34737
            20.18.4  Other stuff   wl-mps 34739
            20.18.5  1. Restricted Quantifiers   wl-ral 34823
      20.19  Mathbox for Brendan Leahy
      20.20  Mathbox for Jeff Madsen
            20.20.1  Logic and set theory   unirep 34980
            20.20.2  Real and complex numbers; integers   filbcmb 35007
            20.20.3  Sequences and sums   sdclem2 35009
            20.20.4  Topology   subspopn 35019
            20.20.5  Metric spaces   metf1o 35022
            20.20.6  Continuous maps and homeomorphisms   constcncf 35029
            20.20.7  Boundedness   ctotbnd 35036
            20.20.8  Isometries   cismty 35068
            20.20.9  Heine-Borel Theorem   heibor1lem 35079
            20.20.10  Banach Fixed Point Theorem   bfplem1 35092
            20.20.11  Euclidean space   crrn 35095
            20.20.12  Intervals (continued)   ismrer1 35108
            20.20.13  Operation properties   cass 35112
            20.20.14  Groups and related structures   cmagm 35118
            20.20.15  Group homomorphism and isomorphism   cghomOLD 35153
            20.20.16  Rings   crngo 35164
            20.20.17  Division Rings   cdrng 35218
            20.20.18  Ring homomorphisms   crnghom 35230
            20.20.19  Commutative rings   ccm2 35259
            20.20.20  Ideals   cidl 35277
            20.20.21  Prime rings and integral domains   cprrng 35316
            20.20.22  Ideal generators   cigen 35329
      20.21  Mathbox for Giovanni Mascellani
            *20.21.1  Tools for automatic proof building   efald2 35348
            *20.21.2  Tseitin axioms   fald 35399
            *20.21.3  Equality deductions   iuneq2f 35426
            *20.21.4  Miscellanea   orcomdd 35437
      20.22  Mathbox for Peter Mazsa
            20.22.1  Notations   cxrn 35444
            20.22.2  Preparatory theorems   el2v1 35482
            20.22.3  Range Cartesian product   df-xrn 35615
            20.22.4  Cosets by ` R `   df-coss 35651
            20.22.5  Relations   df-rels 35717
            20.22.6  Subset relations   df-ssr 35730
            20.22.7  Reflexivity   df-refs 35742
            20.22.8  Converse reflexivity   df-cnvrefs 35755
            20.22.9  Symmetry   df-syms 35770
            20.22.10  Reflexivity and symmetry   symrefref2 35791
            20.22.11  Transitivity   df-trs 35800
            20.22.12  Equivalence relations   df-eqvrels 35811
            20.22.13  Redundancy   df-redunds 35850
            20.22.14  Domain quotients   df-dmqss 35865
            20.22.15  Equivalence relations on domain quotients   df-ers 35889
            20.22.16  Functions   df-funss 35905
            20.22.17  Disjoints vs. converse functions   df-disjss 35928
      20.23  Mathbox for Rodolfo Medina
            20.23.1  Partitions   prtlem60 35981
      *20.24  Mathbox for Norm Megill
            *20.24.1  Obsolete schemes ax-c4,c5,c7,c10,c11,c11n,c15,c9,c14,c16   ax-c5 36011
            *20.24.2  Rederive new axioms ax-4, ax-10, ax-6, ax-12, ax-13 from old   axc5 36021
            *20.24.3  Legacy theorems using obsolete axioms   ax5ALT 36035
            20.24.4  Experiments with weak deduction theorem   elimhyps 36089
            20.24.5  Miscellanea   cnaddcom 36100
            20.24.6  Atoms, hyperplanes, and covering in a left vector space (or module)   clsa 36102
            20.24.7  Functionals and kernels of a left vector space (or module)   clfn 36185
            20.24.8  Opposite rings and dual vector spaces   cld 36251
            20.24.9  Ortholattices and orthomodular lattices   cops 36300
            20.24.10  Atomic lattices with covering property   ccvr 36390
            20.24.11  Hilbert lattices   chlt 36478
            20.24.12  Projective geometries based on Hilbert lattices   clln 36619
            20.24.13  Construction of a vector space from a Hilbert lattice   cdlema1N 36919
            20.24.14  Construction of involution and inner product from a Hilbert lattice   clpoN 38608
      20.25  Mathbox for metakunt
      20.26  Mathbox for Steven Nguyen
            20.26.1  Utility theorems   ioin9i8 39091
            *20.26.2  Arithmetic theorems   c0exALT 39143
            20.26.3  Exponents   oexpreposd 39170
            20.26.4  Real subtraction   cresub 39186
            *20.26.5  Projective spaces   cprjsp 39242
            20.26.6  Equivalent formulations of Fermat's Last Theorem   dffltz 39262
      20.27  Mathbox for Igor Ieskov
      20.28  Mathbox for OpenAI
      20.29  Mathbox for Stefan O'Rear
            20.29.1  Additional elementary logic and set theory   moxfr 39280
            20.29.2  Additional theory of functions   imaiinfv 39281
            20.29.3  Additional topology   elrfi 39282
            20.29.4  Characterization of closure operators. Kuratowski closure axioms   ismrcd1 39286
            20.29.5  Algebraic closure systems   cnacs 39290
            20.29.6  Miscellanea 1. Map utilities   constmap 39301
            20.29.7  Miscellanea for polynomials   mptfcl 39308
            20.29.8  Multivariate polynomials over the integers   cmzpcl 39309
            20.29.9  Miscellanea for Diophantine sets 1   coeq0i 39341
            20.29.10  Diophantine sets 1: definitions   cdioph 39343
            20.29.11  Diophantine sets 2 miscellanea   ellz1 39355
            20.29.12  Diophantine sets 2: union and intersection. Monotone Boolean algebra   diophin 39360
            20.29.13  Diophantine sets 3: construction   diophrex 39363
            20.29.14  Diophantine sets 4 miscellanea   2sbcrex 39372
            20.29.15  Diophantine sets 4: Quantification   rexrabdioph 39382
            20.29.16  Diophantine sets 5: Arithmetic sets   rabdiophlem1 39389
            20.29.17  Diophantine sets 6: reusability. renumbering of variables   eldioph4b 39399
            20.29.18  Pigeonhole Principle and cardinality helpers   fphpd 39404
            20.29.19  A non-closed set of reals is infinite   rencldnfilem 39408
            20.29.20  Lagrange's rational approximation theorem   irrapxlem1 39410
            20.29.21  Pell equations 1: A nontrivial solution always exists   pellexlem1 39417
            20.29.22  Pell equations 2: Algebraic number theory of the solution set   csquarenn 39424
            20.29.23  Pell equations 3: characterizing fundamental solution   infmrgelbi 39466
            *20.29.24  Logarithm laws generalized to an arbitrary base   reglogcl 39478
            20.29.25  Pell equations 4: the positive solution group is infinite cyclic   pellfund14 39486
            20.29.26  X and Y sequences 1: Definition and recurrence laws   crmx 39488
            20.29.27  Ordering and induction lemmas for the integers   monotuz 39529
            20.29.28  X and Y sequences 2: Order properties   rmxypos 39535
            20.29.29  Congruential equations   congtr 39553
            20.29.30  Alternating congruential equations   acongid 39563
            20.29.31  Additional theorems on integer divisibility   coprmdvdsb 39573
            20.29.32  X and Y sequences 3: Divisibility properties   jm2.18 39576
            20.29.33  X and Y sequences 4: Diophantine representability of Y   jm2.27a 39593
            20.29.34  X and Y sequences 5: Diophantine representability of X, ^, _C   rmxdiophlem 39603
            20.29.35  Uncategorized stuff not associated with a major project   setindtr 39612
            20.29.36  More equivalents of the Axiom of Choice   axac10 39621
            20.29.37  Finitely generated left modules   clfig 39658
            20.29.38  Noetherian left modules I   clnm 39666
            20.29.39  Addenda for structure powers   pwssplit4 39680
            20.29.40  Every set admits a group structure iff choice   unxpwdom3 39686
            20.29.41  Noetherian rings and left modules II   clnr 39700
            20.29.42  Hilbert's Basis Theorem   cldgis 39712
            20.29.43  Additional material on polynomials [DEPRECATED]   cmnc 39722
            20.29.44  Degree and minimal polynomial of algebraic numbers   cdgraa 39731
            20.29.45  Algebraic integers I   citgo 39748
            20.29.46  Endomorphism algebra   cmend 39766
            20.29.47  Cyclic groups and order   idomrootle 39786
            20.29.48  Cyclotomic polynomials   ccytp 39793
            20.29.49  Miscellaneous topology   fgraphopab 39801
      20.30  Mathbox for Jon Pennant
      20.31  Mathbox for Richard Penner
            20.31.1  Short Studies   ifpan123g 39815
                  20.31.1.1  Additional work on conditional logical operator   ifpan123g 39815
                  20.31.1.2  Sophisms   rp-fakeimass 39869
                  *20.31.1.3  Finite Sets   rp-isfinite5 39874
                  20.31.1.4  General Observations   intabssd 39876
                  20.31.1.5  Infinite Sets   pwelg 39910
                  *20.31.1.6  Finite intersection property   fipjust 39915
                  20.31.1.7  RP ADDTO: Subclasses and subsets   rababg 39924
                  20.31.1.8  RP ADDTO: The intersection of a class   elintabg 39925
                  20.31.1.9  RP ADDTO: Theorems requiring subset and intersection existence   elinintrab 39928
                  20.31.1.10  RP ADDTO: Relations   xpinintabd 39931
                  *20.31.1.11  RP ADDTO: Functions   elmapintab 39947
                  *20.31.1.12  RP ADDTO: Finite induction (for finite ordinals)   cnvcnvintabd 39951
                  20.31.1.13  RP ADDTO: First and second members of an ordered pair   elcnvlem 39952
                  20.31.1.14  RP ADDTO: The reflexive and transitive properties of relations   undmrnresiss 39955
                  20.31.1.15  RP ADDTO: Basic properties of closures   cleq2lem 39959
                  20.31.1.16  RP REPLACE: Definitions and basic properties of transitive closures   trcleq2lemRP 39981
            20.31.2  Additional statements on relations and subclasses   al3im 39982
                  20.31.2.1  Transitive relations (not to be confused with transitive classes).   trrelind 40001
                  20.31.2.2  Reflexive closures   crcl 40008
                  *20.31.2.3  Finite relationship composition.   relexp2 40013
                  20.31.2.4  Transitive closure of a relation   dftrcl3 40056
                  *20.31.2.5  Adapted from Frege   frege77d 40082
            *20.31.3  Propositions from _Begriffsschrift_   dfxor4 40102
                  *20.31.3.1  _Begriffsschrift_ Chapter I   dfxor4 40102
                  *20.31.3.2  _Begriffsschrift_ Notation hints   rp-imass 40108
                  20.31.3.3  _Begriffsschrift_ Chapter II Implication   ax-frege1 40127
                  20.31.3.4  _Begriffsschrift_ Chapter II Implication and Negation   axfrege28 40166
                  *20.31.3.5  _Begriffsschrift_ Chapter II with logical equivalence   axfrege52a 40193
                  20.31.3.6  _Begriffsschrift_ Chapter II with equivalence of sets   axfrege52c 40224
                  *20.31.3.7  _Begriffsschrift_ Chapter II with equivalence of classes   frege53c 40251
                  *20.31.3.8  _Begriffsschrift_ Chapter III Properties hereditary in a sequence   dffrege69 40269
                  *20.31.3.9  _Begriffsschrift_ Chapter III Following in a sequence   dffrege76 40276
                  *20.31.3.10  _Begriffsschrift_ Chapter III Member of sequence   dffrege99 40299
                  *20.31.3.11  _Begriffsschrift_ Chapter III Single-valued procedures   dffrege115 40315
            *20.31.4  Exploring Topology via Seifert and Threlfall   enrelmap 40334
                  *20.31.4.1  Equinumerosity of sets of relations and maps   enrelmap 40334
                  *20.31.4.2  Generic Pseudoclosure Spaces, Pseudointerior Spaces, and Pseudoneighborhoods   sscon34b 40360
                  *20.31.4.3  Generic Neighborhood Spaces   gneispa 40471
            *20.31.5  Exploring Higher Homotopy via Kerodon   k0004lem1 40488
                  *20.31.5.1  Simplicial Sets   k0004lem1 40488
      20.32  Mathbox for Stanislas Polu
            20.32.1  IMO Problems   wwlemuld 40497
                  20.32.1.1  IMO 1972 B2   wwlemuld 40497
            *20.32.2  INT Inequalities Proof Generator   int-addcomd 40517
            *20.32.3  N-Digit Addition Proof Generator   unitadd 40539
            20.32.4  AM-GM (for k = 2,3,4)   gsumws3 40540
      20.33  Mathbox for Rohan Ridenour
            20.33.1  Misc   spALT 40545
            20.33.2  Shorter primitive equivalent of ax-groth   gru0eld 40556
                  20.33.2.1  Grothendieck universes are closed under collection   gru0eld 40556
                  20.33.2.2  Minimal universes   ismnu 40588
                  20.33.2.3  Primitive equivalent of ax-groth   expandan 40615
      20.34  Mathbox for Steve Rodriguez
            20.34.1  Miscellanea   nanorxor 40628
            20.34.2  Ratio test for infinite series convergence and divergence   dvgrat 40635
            20.34.3  Multiples   reldvds 40638
            20.34.4  Function operations   caofcan 40646
            20.34.5  Calculus   lhe4.4ex1a 40652
            20.34.6  The generalized binomial coefficient operation   cbcc 40659
            20.34.7  Binomial series   uzmptshftfval 40669
      20.35  Mathbox for Andrew Salmon
            20.35.1  Principia Mathematica * 10   pm10.12 40681
            20.35.2  Principia Mathematica * 11   2alanimi 40695
            20.35.3  Predicate Calculus   sbeqal1 40721
            20.35.4  Principia Mathematica * 13 and * 14   pm13.13a 40730
            20.35.5  Set Theory   elnev 40761
            20.35.6  Arithmetic   addcomgi 40779
            20.35.7  Geometry   cplusr 40780
      *20.36  Mathbox for Alan Sare
            20.36.1  Auxiliary theorems for the Virtual Deduction tool   idiALT 40802
            20.36.2  Supplementary unification deductions   bi1imp 40806
            20.36.3  Conventional Metamath proofs, some derived from VD proofs   iidn3 40826
            20.36.4  What is Virtual Deduction?   wvd1 40894
            20.36.5  Virtual Deduction Theorems   df-vd1 40895
            20.36.6  Theorems proved using Virtual Deduction   trsspwALT 41143
            20.36.7  Theorems proved using Virtual Deduction with mmj2 assistance   simplbi2VD 41171
            20.36.8  Virtual Deduction transcriptions of textbook proofs   sb5ALTVD 41238
            20.36.9  Theorems proved using conjunction-form Virtual Deduction   elpwgdedVD 41242
            20.36.10  Theorems with a VD proof in conventional notation derived from a VD proof   suctrALT3 41249
            *20.36.11  Theorems with a proof in conventional notation derived from a VD proof   notnotrALT2 41252
      20.37  Mathbox for Glauco Siliprandi
            20.37.1  Miscellanea   evth2f 41263
            20.37.2  Functions   feq1dd 41413
            20.37.3  Ordering on real numbers - Real and complex numbers basic operations   sub2times 41530
            20.37.4  Real intervals   gtnelioc 41755
            20.37.5  Finite sums   fsumclf 41840
            20.37.6  Finite multiplication of numbers and finite multiplication of functions   fmul01 41851
            20.37.7  Limits   clim1fr1 41872
                  20.37.7.1  Inferior limit (lim inf)   clsi 42022
                  *20.37.7.2  Limits for sequences of extended real numbers   clsxlim 42089
            20.37.8  Trigonometry   coseq0 42135
            20.37.9  Continuous Functions   mulcncff 42141
            20.37.10  Derivatives   dvsinexp 42185
            20.37.11  Integrals   itgsin0pilem1 42225
            20.37.12  Stone Weierstrass theorem - real version   stoweidlem1 42277
            20.37.13  Wallis' product for π   wallispilem1 42341
            20.37.14  Stirling's approximation formula for ` n ` factorial   stirlinglem1 42350
            20.37.15  Dirichlet kernel   dirkerval 42367
            20.37.16  Fourier Series   fourierdlem1 42384
            20.37.17  e is transcendental   elaa2lem 42509
            20.37.18  n-dimensional Euclidean space   rrxtopn 42560
            20.37.19  Basic measure theory   csalg 42584
                  *20.37.19.1  σ-Algebras   csalg 42584
                  20.37.19.2  Sum of nonnegative extended reals   csumge0 42635
                  *20.37.19.3  Measures   cmea 42722
                  *20.37.19.4  Outer measures and Caratheodory's construction   come 42762
                  *20.37.19.5  Lebesgue measure on n-dimensional Real numbers   covoln 42809
                  *20.37.19.6  Measurable functions   csmblfn 42968
      20.38  Mathbox for Saveliy Skresanov
            20.38.1  Ceva's theorem   sigarval 43098
            20.38.2  Simple groups   simpcntrab 43118
      20.39  Mathbox for Jarvin Udandy
      20.40  Mathbox for Adhemar
            *20.40.1  Minimal implicational calculus   adh-minim 43228
      20.41  Mathbox for Alexander van der Vekens
            20.41.1  General auxiliary theorems (1)   eusnsn 43252
                  20.41.1.1  Unordered and ordered pairs - extension for singletons   eusnsn 43252
                  20.41.1.2  Unordered and ordered pairs - extension for unordered pairs   elprneb 43255
                  20.41.1.3  Unordered and ordered pairs - extension for ordered pairs   oppr 43256
                  20.41.1.4  Relations - extension   eubrv 43261
                  20.41.1.5  Definite description binder (inverted iota) - extension   iota0def 43264
                  20.41.1.6  Functions - extension   fveqvfvv 43266
            20.41.2  Alternative for Russell's definition of a description binder   caiota 43274
            20.41.3  Double restricted existential uniqueness   r19.32 43287
                  20.41.3.1  Restricted quantification (extension)   r19.32 43287
                  20.41.3.2  Restricted uniqueness and "at most one" quantification   reuf1odnf 43297
                  20.41.3.3  Analogs to Existential uniqueness (double quantification)   2reu3 43300
                  20.41.3.4  Additional theorems for double restricted existential uniqueness   2reu8i 43303
            *20.41.4  Alternative definitions of function and operation values   wdfat 43306
                  20.41.4.1  Restricted quantification (extension)   ralbinrald 43312
                  20.41.4.2  The universal class (extension)   nvelim 43313
                  20.41.4.3  Introduce the Axiom of Power Sets (extension)   alneu 43314
                  20.41.4.4  Predicate "defined at"   dfateq12d 43316
                  20.41.4.5  Alternative definition of the value of a function   dfafv2 43322
                  20.41.4.6  Alternative definition of the value of an operation   aoveq123d 43368
            *20.41.5  Alternative definitions of function values (2)   cafv2 43398
            20.41.6  General auxiliary theorems (2)   an4com24 43458
                  20.41.6.1  Logical conjunction - extension   an4com24 43458
                  20.41.6.2  Abbreviated conjunction and disjunction of three wff's - extension   3an4ancom24 43459
                  20.41.6.3  Negated membership (alternative)   cnelbr 43461
                  20.41.6.4  The empty set - extension   ralralimp 43468
                  20.41.6.5  Indexed union and intersection - extension   otiunsndisjX 43469
                  20.41.6.6  Functions - extension   fvifeq 43470
                  20.41.6.7  Maps-to notation - extension   fvmptrab 43482
                  20.41.6.8  Ordering on reals - extension   leltletr 43484
                  20.41.6.9  Subtraction - extension   cnambpcma 43485
                  20.41.6.10  Ordering on reals (cont.) - extension   leaddsuble 43488
                  20.41.6.11  Imaginary and complex number properties - extension   readdcnnred 43494
                  20.41.6.12  Nonnegative integers (as a subset of complex numbers) - extension   nn0resubcl 43499
                  20.41.6.13  Integers (as a subset of complex numbers) - extension   zgeltp1eq 43500
                  20.41.6.14  Decimal arithmetic - extension   1t10e1p1e11 43501
                  20.41.6.15  Upper sets of integers - extension   eluzge0nn0 43503
                  20.41.6.16  Infinity and the extended real number system (cont.) - extension   nltle2tri 43504
                  20.41.6.17  Finite intervals of integers - extension   ssfz12 43505
                  20.41.6.18  Half-open integer ranges - extension   fzopred 43513
                  20.41.6.19  The modulo (remainder) operation - extension   m1mod0mod1 43520
                  20.41.6.20  The infinite sequence builder "seq"   smonoord 43522
                  20.41.6.21  Finite and infinite sums - extension   fsummsndifre 43523
                  20.41.6.22  Extensible structures - extension   setsidel 43527
            *20.41.7  Preimages of function values   preimafvsnel 43530
            *20.41.8  Partitions of real intervals   ciccp 43564
            20.41.9  Shifting functions with an integer range domain   fargshiftfv 43590
            20.41.10  Words over a set (extension)   lswn0 43595
                  20.41.10.1  Last symbol of a word - extension   lswn0 43595
            20.41.11  Unordered pairs   wich 43596
                  20.41.11.1  Interchangeable setvar variables   wich 43596
                  20.41.11.2  Set of unordered pairs   sprid 43627
                  *20.41.11.3  Proper (unordered) pairs   prpair 43654
                  20.41.11.4  Set of proper unordered pairs   cprpr 43665
            20.41.12  Number theory (extension)   cfmtno 43680
                  *20.41.12.1  Fermat numbers   cfmtno 43680
                  *20.41.12.2  Mersenne primes   m2prm 43744
                  20.41.12.3  Proth's theorem   modexp2m1d 43768
                  20.41.12.4  Solutions of quadratic equations   quad1 43776
            *20.41.13  Even and odd numbers   ceven 43780
                  20.41.13.1  Definitions and basic properties   ceven 43780
                  20.41.13.2  Alternate definitions using the "divides" relation   dfeven2 43805
                  20.41.13.3  Alternate definitions using the "modulo" operation   dfeven3 43814
                  20.41.13.4  Alternate definitions using the "gcd" operation   iseven5 43820
                  20.41.13.5  Theorems of part 5 revised   zneoALTV 43825
                  20.41.13.6  Theorems of part 6 revised   odd2np1ALTV 43830
                  20.41.13.7  Theorems of AV's mathbox revised   0evenALTV 43844
                  20.41.13.8  Additional theorems   epoo 43859
                  20.41.13.9  Perfect Number Theorem (revised)   perfectALTVlem1 43877
            20.41.14  Number theory (extension 2)   cfppr 43880
                  *20.41.14.1  Fermat pseudoprimes   cfppr 43880
                  *20.41.14.2  Goldbach's conjectures   cgbe 43901
            20.41.15  Graph theory (extension)   cgrisom 43974
                  *20.41.15.1  Isomorphic graphs   cgrisom 43974
                  20.41.15.2  Loop-free graphs - extension   1hegrlfgr 43998
                  20.41.15.3  Walks - extension   cupwlks 43999
                  20.41.15.4  Edges of graphs expressed as sets of unordered pairs   upgredgssspr 44009
            20.41.16  Monoids (extension)   ovn0dmfun 44022
                  20.41.16.1  Auxiliary theorems   ovn0dmfun 44022
                  20.41.16.2  Magmas, Semigroups and Monoids (extension)   plusfreseq 44030
                  20.41.16.3  Magma homomorphisms and submagmas   cmgmhm 44035
                  20.41.16.4  Examples and counterexamples for magmas, semigroups and monoids (extension)   opmpoismgm 44065
                  20.41.16.5  Group sum operation (extension 1)   gsumsplit2f 44078
            *20.41.17  Magmas and internal binary operations (alternate approach)   ccllaw 44081
                  *20.41.17.1  Laws for internal binary operations   ccllaw 44081
                  *20.41.17.2  Internal binary operations   cintop 44094
                  20.41.17.3  Alternative definitions for magmas and semigroups   cmgm2 44113
            20.41.18  Categories (extension)   idfusubc0 44127
                  20.41.18.1  Subcategories (extension)   idfusubc0 44127
            20.41.19  Rings (extension)   lmod0rng 44130
                  20.41.19.1  Nonzero rings (extension)   lmod0rng 44130
                  *20.41.19.2  Non-unital rings ("rngs")   crng 44136
                  20.41.19.3  Rng homomorphisms   crngh 44147
                  20.41.19.4  Ring homomorphisms (extension)   rhmfn 44180
                  20.41.19.5  Ideals as non-unital rings   lidldomn1 44183
                  20.41.19.6  The non-unital ring of even integers   0even 44193
                  20.41.19.7  A constructed not unital ring   cznrnglem 44215
                  *20.41.19.8  The category of non-unital rings   crngc 44219
                  *20.41.19.9  The category of (unital) rings   cringc 44265
                  20.41.19.10  Subcategories of the category of rings   srhmsubclem1 44335
            20.41.20  Basic algebraic structures (extension)   opeliun2xp 44372
                  20.41.20.1  Auxiliary theorems   opeliun2xp 44372
                  20.41.20.2  The binomial coefficient operation (extension)   bcpascm1 44390
                  20.41.20.3  The ` ZZ `-module ` ZZ X. ZZ `   zlmodzxzlmod 44393
                  20.41.20.4  Group sum operation (extension 2)   mgpsumunsn 44400
                  20.41.20.5  Symmetric groups (extension)   exple2lt6 44403
                  20.41.20.6  Divisibility (extension)   invginvrid 44406
                  20.41.20.7  The support of functions (extension)   rmsupp0 44407
                  20.41.20.8  Finitely supported functions (extension)   rmsuppfi 44412
                  20.41.20.9  Left modules (extension)   lmodvsmdi 44421
                  20.41.20.10  Associative algebras (extension)   ascl1 44423
                  20.41.20.11  Univariate polynomials (extension)   ply1vr1smo 44426
                  20.41.20.12  Univariate polynomials (examples)   linply1 44438
            20.41.21  Linear algebra (extension)   cdmatalt 44442
                  *20.41.21.1  The subalgebras of diagonal and scalar matrices (extension)   cdmatalt 44442
                  *20.41.21.2  Linear combinations   clinc 44450
                  *20.41.21.3  Linear independence   clininds 44486
                  20.41.21.4  Simple left modules and the ` ZZ `-module   lmod1lem1 44533
                  20.41.21.5  Differences between (left) modules and (left) vector spaces   lvecpsslmod 44553
            20.41.22  Complexity theory   suppdm 44556
                  20.41.22.1  Auxiliary theorems   suppdm 44556
                  20.41.22.2  The modulo (remainder) operation (extension)   fldivmod 44569
                  20.41.22.3  Even and odd integers   nn0onn0ex 44574
                  20.41.22.4  The natural logarithm on complex numbers (extension)   logcxp0 44586
                  20.41.22.5  Division of functions   cfdiv 44588
                  20.41.22.6  Upper bounds   cbigo 44598
                  20.41.22.7  Logarithm to an arbitrary base (extension)   rege1logbrege0 44609
                  *20.41.22.8  The binary logarithm   fldivexpfllog2 44616
                  20.41.22.9  Binary length   cblen 44620
                  *20.41.22.10  Digits   cdig 44646
                  20.41.22.11  Nonnegative integer as sum of its shifted digits   dignn0flhalflem1 44666
                  20.41.22.12  Algorithms for the multiplication of nonnegative integers   nn0mulfsum 44675
            20.41.23  Elementary geometry (extension)   fv1prop 44677
                  20.41.23.1  Auxiliary theorems   fv1prop 44677
                  20.41.23.2  Real euclidean space of dimension 2   rrx2pxel 44689
                  20.41.23.3  Spheres and lines in real Euclidean spaces   cline 44705
      20.42  Mathbox for Emmett Weisz
            *20.42.1  Miscellaneous Theorems   nfintd 44767
            20.42.2  Set Recursion   csetrecs 44777
                  *20.42.2.1  Basic Properties of Set Recursion   csetrecs 44777
                  20.42.2.2  Examples and properties of set recursion   elsetrecslem 44792
            *20.42.3  Construction of Games and Surreal Numbers   cpg 44802
      *20.43  Mathbox for David A. Wheeler
            20.43.1  Natural deduction   sbidd 44808
            *20.43.2  Greater than, greater than or equal to.   cge-real 44810
            *20.43.3  Hyperbolic trigonometric functions   csinh 44820
            *20.43.4  Reciprocal trigonometric functions (sec, csc, cot)   csec 44831
            *20.43.5  Identities for "if"   ifnmfalse 44853
            *20.43.6  Logarithms generalized to arbitrary base using ` logb `   logb2aval 44854
            *20.43.7  Logarithm laws generalized to an arbitrary base - log_   clog- 44855
            *20.43.8  Formally define terms such as Reflexivity   wreflexive 44857
            *20.43.9  Algebra helpers   comraddi 44861
            *20.43.10  Algebra helper examples   i2linesi 44870
            *20.43.11  Formal methods "surprises"   alimp-surprise 44872
            *20.43.12  Allsome quantifier   walsi 44878
            *20.43.13  Miscellaneous   5m4e1 44889
            20.43.14  Theorems about algebraic numbers   aacllem 44893
      20.44  Mathbox for Kunhao Zheng
            20.44.1  Weighted AM-GM inequality   amgmwlem 44894

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