TOC of Theorem List - Metamath Proof Explorer

Table of Contents Summary

PART 1  CLASSICAL FIRST-ORDER LOGIC WITH EQUALITY
      1.1  Pre-logic
      1.2  Propositional calculus
      1.3  Other axiomatizations related to classical propositional calculus
      1.4  Predicate calculus with equality: Tarski's system S2 (1 rule, 6 schemes)
      1.5  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes)
      1.6  Uniqueness and unique existence
      1.7  Other axiomatizations related to classical predicate calculus
PART 2  ZF (ZERMELO-FRAENKEL) SET THEORY
      2.1  ZF Set Theory - start with the Axiom of Extensionality
      2.2  ZF Set Theory - add the Axiom of Replacement
      2.3  ZF Set Theory - add the Axiom of Power Sets
      2.4  ZF Set Theory - add the Axiom of Union
      2.5  ZF Set Theory - add the Axiom of Regularity
      2.6  ZF Set Theory - add the Axiom of Infinity
PART 3  ZFC (ZERMELO-FRAENKEL WITH CHOICE) SET THEORY
      3.1  ZFC Set Theory - add Countable Choice and Dependent Choice
      3.2  ZFC Set Theory - add the Axiom of Choice
      3.3  ZFC Axioms with no distinct variable requirements
      3.4  The Generalized Continuum Hypothesis
PART 4  TG (TARSKI-GROTHENDIECK) SET THEORY
      4.1  Inaccessibles
      4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
PART 5  REAL AND COMPLEX NUMBERS
      5.1  Construction and axiomatization of real and complex numbers
      5.2  Derive the basic properties from the field axioms
      5.3  Real and complex numbers - basic operations
      5.4  Integer sets
      5.5  Order sets
      5.6  Elementary integer functions
      5.7  Words over a set
      5.8  Reflexive and transitive closures of relations
      5.9  Elementary real and complex functions
      5.10  Elementary limits and convergence
      5.11  Elementary trigonometry
      5.12  Cardinality of real and complex number subsets
PART 6  ELEMENTARY NUMBER THEORY
      6.1  Elementary properties of divisibility
      6.2  Elementary prime number theory
PART 7  BASIC STRUCTURES
      7.1  Extensible structures
      7.2  Moore spaces
PART 8  BASIC CATEGORY THEORY
      8.1  Categories
      8.2  Arrows (disjointified hom-sets)
      8.3  Examples of categories
      8.4  Categorical constructions
PART 9  BASIC ORDER THEORY
      9.1  Dual of an order structure
      9.2  Preordered sets and directed sets
      9.3  Partially ordered sets (posets)
      9.4  Totally ordered sets (tosets)
      9.5  Lattices
      9.6  Posets, directed sets, and lattices as relations
      9.7  Chains
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  Subring algebras and ideals
      10.8  The complex numbers as an algebraic extensible structure
      10.9  Generalized pre-Hilbert and Hilbert spaces
PART 11  BASIC LINEAR ALGEBRA
      11.1  Vectors and free modules
      11.2  Associative algebras
      11.3  Abstract multivariate polynomials
      11.4  Matrices
      11.5  The determinant
      11.6  Polynomial matrices
      11.7  The characteristic polynomial
PART 12  BASIC TOPOLOGY
      12.1  Topology
      12.2  Filters and filter bases
      12.3  Uniform Structures and Spaces
      12.4  Metric spaces
      12.5  Metric subcomplex vector spaces
PART 13  BASIC REAL AND COMPLEX ANALYSIS
      13.1  Continuity
      13.2  Integrals
      13.3  Derivatives
PART 14  BASIC REAL AND COMPLEX FUNCTIONS
      14.1  Polynomials
      14.2  Sequences and series
      14.3  Basic trigonometry
      14.4  Basic number theory
PART 15  SURREAL NUMBERS
      15.1  Sign sequence representation and Alling's axioms
      15.2  Initial consequences of Alling's axioms
      15.3  Conway cut representation
      15.4  Induction and recursion
      15.5  Surreal arithmetic
      15.6  Subsystems of surreals
PART 16  ELEMENTARY GEOMETRY
      16.1  Definition and Tarski's Axioms of Geometry
      16.2  Tarskian Geometry
      16.3  Properties of geometries
      16.4  Geometry in Hilbert spaces
PART 17  GRAPH THEORY
      17.1  Vertices and edges
      17.2  Undirected graphs
      17.3  Walks, paths and cycles
      17.4  Eulerian paths and the Konigsberg Bridge problem
      17.5  The Friendship Theorem
PART 18  GUIDES AND MISCELLANEA
      18.1  Guides (conventions, explanations, and examples)
      18.2  Humor
      18.3  (Future - to be reviewed and classified)
PART 19  COMPLEX TOPOLOGICAL VECTOR SPACES (DEPRECATED)
      19.1  Additional material on group theory (deprecated)
      19.2  Complex vector spaces
      19.3  Normed complex vector spaces
      19.4  Operators on complex vector spaces
      19.5  Inner product (pre-Hilbert) spaces
      19.6  Complex Banach spaces
      19.7  Complex Hilbert spaces
PART 20  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      20.1  Axiomatization of complex pre-Hilbert spaces
      20.2  Inner product and norms
      20.3  Cauchy sequences and completeness axiom
      20.4  Subspaces and projections
      20.5  Properties of Hilbert subspaces
      20.6  Operators on Hilbert spaces
      20.7  States on a Hilbert lattice and Godowski's equation
      20.8  Cover relation, atoms, exchange axiom, and modular symmetry
PART 21  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      ​21.1  Mathboxes for user contributions
      21.2  Mathbox for Stefan Allan
      21.3  Mathbox for Thierry Arnoux
      21.4  Mathbox for Jonathan Ben-Naim
      21.5  Mathbox for BTernaryTau
      21.6  Mathbox for Mario Carneiro
      21.7  Mathbox for Filip Cernatescu
      21.8  Mathbox for Paul Chapman
      21.9  Mathbox for Hongxiu Chen
      21.10  Mathbox for Adrian Ducourtial
      21.11  Mathbox for Scott Fenton
      21.12  Mathbox for Gino Giotto
      21.13  Mathbox for Jeff Hankins
      21.14  Mathbox for Anthony Hart
      21.15  Mathbox for Chen-Pang He
      21.16  Mathbox for Jeff Hoffman
      21.17  Mathbox for Matthew House
      21.18  Mathbox for Asger C. Ipsen
      21.19  Mathbox for BJ
      21.20  Mathbox for Jim Kingdon
      21.21  Mathbox for ML
      21.22  Mathbox for Wolf Lammen
      21.23  Mathbox for Brendan Leahy
      21.24  Mathbox for Thomas van Maaren
      21.25  Mathbox for Jeff Madsen
      21.26  Mathbox for Giovanni Mascellani
      21.27  Mathbox for Peter Mazsa
      21.28  Mathbox for Rodolfo Medina
      21.29  Mathbox for Norm Megill
      21.30  Mathbox for metakunt
      21.31  Mathbox for Luke Murphy
      21.32  Mathbox for Steven Nguyen
      21.33  Mathbox for Igor Ieskov
      21.34  Mathbox for OpenAI
      21.35  Mathbox for Stefan O'Rear
      21.36  Mathbox for Noam Pasman
      21.37  Mathbox for Jon Pennant
      21.38  Mathbox for Richard Penner
      21.39  Mathbox for Stanislas Polu
      21.40  Mathbox for Rohan Ridenour
      21.41  Mathbox for Steve Rodriguez
      21.42  Mathbox for Andrew Salmon
      21.43  Mathbox for Alan Sare
      21.44  Mathbox for Eric Schmidt
      21.45  Mathbox for Glauco Siliprandi
      21.46  Mathbox for Saveliy Skresanov
      21.47  Mathbox for Ender Ting
      21.48  Mathbox for Jarvin Udandy
      21.49  Mathbox for Adhemar
      21.50  Mathbox for Alexander van der Vekens
      21.51  Mathbox for Zhi Wang
      21.52  Mathbox for Emmett Weisz
      21.53  Mathbox for David A. Wheeler
      21.54  Mathbox for Mingli Yuan
      21.55  Mathbox for Jiamin Zhao
      21.56  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 114
            ​*1.2.5  Logical equivalence   wb 209
            ​*1.2.6  Logical conjunction   wa 401
            ​*1.2.7  Logical disjunction   wo 861
            ​*1.2.8  Mixed connectives   jaao 969
            ​*1.2.9  The conditional operator for propositions   wif 1078
            ​*1.2.10  The weak deduction theorem for propositional calculus   elimh 1099
            1.2.11  Abbreviated conjunction and disjunction of three wff's   w3o 1102
            1.2.12  Logical "nand" (Sheffer stroke)   wnan 1521
            1.2.13  Logical "xor"   wxo 1541
            1.2.14  Logical "nor"   wnor 1558
            1.2.15  True and false constants   wal 1568
                  ​*1.2.15.1  Universal quantifier for use by df-tru   wal 1568
                  ​*1.2.15.2  Equality predicate for use by df-tru   cv 1569
                  1.2.15.3  The true constant   wtru 1571
                  1.2.15.4  The false constant   wfal 1582
            ​*1.2.16  Truth tables   truimtru 1593
                  1.2.16.1  Implication   truimtru 1593
                  1.2.16.2  Negation   nottru 1597
                  1.2.16.3  Equivalence   trubitru 1599
                  1.2.16.4  Conjunction   truantru 1603
                  1.2.16.5  Disjunction   truortru 1607
                  1.2.16.6  Alternative denial   trunantru 1611
                  1.2.16.7  Exclusive disjunction   truxortru 1615
                  1.2.16.8  Joint denial   trunortru 1619
            ​*1.2.17  Half adder and full adder in propositional calculus   whad 1623
                  1.2.17.1  Full adder: sum   whad 1623
                  1.2.17.2  Full adder: carry   wcad 1639
      ​1.3  Other axiomatizations related to classical propositional calculus
            ​*1.3.1  Minimal implicational calculus   minimp 1654
            ​*1.3.2  Implicational Calculus   impsingle 1660
            1.3.3  Derive the Lukasiewicz axioms from Meredith's sole axiom   meredith 1674
            1.3.4  Derive the standard axioms from the Lukasiewicz axioms   luklem1 1691
            ​*1.3.5  Derive Nicod's axiom from the standard axioms   nic-dfim 1702
            1.3.6  Derive the Lukasiewicz axioms from Nicod's axiom   nic-imp 1708
            1.3.7  Derive Nicod's Axiom from Lukasiewicz's First Sheffer Stroke Axiom   lukshef-ax1 1727
            1.3.8  Derive the Lukasiewicz Axioms from the Tarski-Bernays-Wajsberg Axioms   tbw-bijust 1731
            1.3.9  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's First CO Axiom   merco1 1746
            1.3.10  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's Second CO Axiom   merco2 1769
            1.3.11  Derive the Lukasiewicz axioms from the Russell-Bernays Axioms   rb-bijust 1782
            ​*1.3.12  Stoic logic non-modal portion (Chrysippus of Soli)   mptnan 1801
      ​​*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 1812
                  1.4.1.1  Existential quantifier   wex 1812
                  1.4.1.2  Nonfreeness predicate   wnf 1816
            1.4.2  Rule scheme ax-gen (Generalization)   ax-gen 1828
            1.4.3  Axiom scheme ax-4 (Quantified Implication)   ax-4 1842
                  ​*1.4.3.1  The empty domain of discourse   empty 1939
            1.4.4  Axiom scheme ax-5 (Distinctness) - first use of $d   ax-5 1943
            ​*1.4.5  Equality predicate (continued)   weq 1995
            1.4.6  Axiom scheme ax-6 (Existence)   ax-6 2000
            1.4.7  Axiom scheme ax-7 (Equality)   ax-7 2041
            1.4.8  Define proper substitution   sbjust 2098
            1.4.9  Membership predicate   wcel 2145
            1.4.10  Axiom scheme ax-8 (Left Equality for Binary Predicate)   ax-8 2147
            1.4.11  Axiom scheme ax-9 (Right Equality for Binary Predicate)   ax-9 2155
            ​*1.4.12  Logical redundancy of ax-10 , ax-11 , ax-12 , ax-13   ax6dgen 2165
      ​​*1.5  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes)
            1.5.1  Axiom scheme ax-10 (Quantified Negation)   ax-10 2178
            1.5.2  Axiom scheme ax-11 (Quantifier Commutation)   ax-11 2194
            1.5.3  Axiom scheme ax-12 (Substitution)   ax-12 2213
            1.5.4  Axiom scheme ax-13 (Quantified Equality)   ax-13 2401
      ​1.6  Uniqueness and unique existence
            1.6.1  Uniqueness: the at-most-one quantifier   wmo 2562
            1.6.2  Unique existence: the unique existential quantifier   weu 2593
      ​1.7  Other axiomatizations related to classical predicate calculus
            ​*1.7.1  Aristotelian logic: Assertic syllogisms   barbara 2687
            ​*1.7.2  Intuitionistic logic   axia1 2717
​​*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 2732
            2.1.2  Classes   cab 2738
                  2.1.2.1  Class abstractions   cab 2738
                  ​*2.1.2.2  Class equality   df-cleq 2752
                  2.1.2.3  Class membership   df-clel 2835
                  2.1.2.4  Elementary properties of class abstractions   eqabdv 2893
            2.1.3  Class form not-free predicate   wnfc 2907
            2.1.4  Negated equality and membership   wne 2955
                  2.1.4.1  Negated equality   wne 2955
                  2.1.4.2  Negated membership   wnel 3061
            2.1.5  Restricted quantification   wral 3076
                  2.1.5.1  Restricted universal and existential quantification   wral 3076
                  2.1.5.2  Restricted existential uniqueness and at-most-one quantifier   wreu 3363
                  2.1.5.3  Restricted class abstraction   crab 3412
            2.1.6  The universal class   cvv 3450
            ​*2.1.7  Conditional equality (experimental)   wcdeq 3720
            2.1.8  Russell's Paradox   rru 3736
            2.1.9  Proper substitution of classes for sets   wsbc 3738
            2.1.10  Proper substitution of classes for sets into classes   csb 3846
            2.1.11  Define basic set operations and relations   cdif 3895
            2.1.12  Subclasses and subsets   df-ss 3915
            2.1.13  The difference, union, and intersection of two classes   dfdif3 4065
                  2.1.13.1  The difference of two classes   dfdif3 4065
                  2.1.13.2  The union of two classes   elun 4099
                  2.1.13.3  The intersection of two classes   elini 4144
                  2.1.13.4  The symmetric difference of two classes   csymdif 4197
                  2.1.13.5  Combinations of difference, union, and intersection of two classes   unabs 4210
                  2.1.13.6  Class abstractions with difference, union, and intersection of two classes   unabw 4252
                  2.1.13.7  Restricted uniqueness with difference, union, and intersection   reuun2 4270
            2.1.14  The empty set   c0 4278
            ​*2.1.15  The conditional operator for classes   cif 4481
            ​*2.1.16  The weak deduction theorem for set theory   dedth 4540
            2.1.17  Power classes   cpw 4556
            2.1.18  Unordered and ordered pairs   snjust 4582
            2.1.19  The union of a class   cuni 4866
            2.1.20  The intersection of a class   cint 4906
            2.1.21  Indexed union and intersection   ciun 4950
            2.1.22  Disjointness   wdisj 5069
            2.1.23  Binary relations   wbr 5102
            2.1.24  Ordered-pair class abstractions (class builders)   copab 5166
            2.1.25  Functions in maps-to notation   cmpt 5185
            2.1.26  Transitive classes   wtr 5211
      ​2.2  ZF Set Theory - add the Axiom of Replacement
            2.2.1  Introduce the Axiom of Replacement   ax-rep 5231
            2.2.2  Derive the Axiom of Separation   axsepgfromrep 5246
            2.2.3  Derive the Null Set Axiom   axnulALT 5257
            2.2.4  Theorems requiring subset and intersection existence   exnelv 5266
            2.2.5  Theorems requiring empty set existence   class2set 5315
      ​2.3  ZF Set Theory - add the Axiom of Power Sets
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 5326
            2.3.2  Derive the Axiom of Pairing   axprlem1 5384
            2.3.3  Ordered pair theorem   opnz 5441
            2.3.4  Ordered-pair class abstractions (cont.)   opabidw 5494
            2.3.5  Power class of union and intersection   pwin 5538
            2.3.6  The identity relation   cid 5541
            2.3.7  The membership relation (or epsilon relation)   cep 5546
            ​*2.3.8  Partial and total orderings   wpo 5553
            2.3.9  Founded and well-ordering relations   wfr 5597
            2.3.10  Relations   cxp 5645
            2.3.11  The Predecessor Class   cpred 6292
            2.3.12  Well-founded induction (variant)   frpomin 6332
            2.3.13  Well-ordered induction   tz6.26 6339
            2.3.14  Ordinals   word 6350
            2.3.15  Definite description binder (inverted iota)   cio 6481
            2.3.16  Functions   wfun 6521
            2.3.17  Cantor's Theorem   canth 7362
            2.3.18  Restricted iota (description binder)   crio 7364
            2.3.19  Operations   co 7408
                  2.3.19.1  Variable-to-class conversion for operations   caovclg 7601
            2.3.20  Operations in maps-to notation   mpondm0 7649
            2.3.21  Functions with three arguments in maps-to notation   cmpt3 7671
            2.3.22  Function operation   cof 7674
            2.3.23  Proper subset relation   crpss 7721
      ​2.4  ZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 7734
            2.4.2  Ordinals (continued)   epweon 7772
            2.4.3  Transfinite induction   tfi 7847
            2.4.4  The natural numbers (i.e., finite ordinals)   com 7860
            2.4.5  Peano's postulates   peano1 7883
            2.4.6  Finite induction (for finite ordinals)   find 7890
            2.4.7  Relations and functions (cont.)   dmexg 7896
            2.4.8  First and second members of an ordered pair   c1st 7982
            2.4.9  Induction on Cartesian products   frpoins3xpg 8135
            2.4.10  Ordering on Cartesian products   xpord2lem 8137
            2.4.11  Ordering Ordinal Sequences   orderseqlem 8152
            ​*2.4.12  The support of functions   csupp 8155
            ​*2.4.13  Special maps-to operations   opeliunxp2f 8205
            2.4.14  Function transposition   ctpos 8220
            2.4.15  Curry and uncurry   ccur 8260
            2.4.16  Undefined values   cund 8267
            2.4.17  Well-founded recursion   cfrecs 8276
            2.4.18  Well-ordered recursion   cwrecs 8307
            2.4.19  Functions on ordinals; strictly monotone ordinal functions   iunon 8325
            2.4.20  "Strong" transfinite recursion   crecs 8356
            2.4.21  Recursive definition generator   crdg 8395
            2.4.22  Finite recursion   frfnom 8421
            2.4.23  Ordinal arithmetic   c1o 8447
            2.4.24  Natural number arithmetic   nna0 8591
            2.4.25  Natural addition   cnadd 8652
            2.4.26  Equivalence relations and classes   wer 8692
            2.4.27  The mapping operation   cmap 8825
            2.4.28  Infinite Cartesian products   cixp 8903
            2.4.29  Equinumerosity   cen 8948
            2.4.30  Schroeder-Bernstein Theorem   sbthlem1 9084
            2.4.31  Equinumerosity (cont.)   xpf1o 9136
            2.4.32  Finite sets   dif1enlem 9153
            2.4.33  Pigeonhole Principle   phplem1 9197
            2.4.34  Finite sets (cont.)   onomeneq 9207
            2.4.35  Finitely supported functions   cfsupp 9331
            2.4.36  Finite intersections   cfi 9380
            2.4.37  Hall's marriage theorem   marypha1lem 9403
            2.4.38  Supremum and infimum   csup 9410
            2.4.39  Ordinal isomorphism, Hartogs's theorem   coi 9481
            2.4.40  Hartogs function   char 9528
            2.4.41  Weak dominance   cwdom 9536
      ​2.5  ZF Set Theory - add the Axiom of Regularity
            2.5.1  Introduce the Axiom of Regularity   ax-reg 9564
            2.5.2  Axiom of Infinity equivalents   inf0 9600
      ​2.6  ZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-inf 9617
            2.6.2  Existence of omega (the set of natural numbers)   omex 9622
            2.6.3  Cantor normal form   ccnf 9640
            2.6.4  Transitive closure of a relation   cttrcl 9686
            2.6.5  Transitive closure   trcl 9707
            2.6.6  Set induction (or epsilon induction)   setind 9726
            2.6.7  Well-Founded Induction   frmin 9731
            2.6.8  Well-Founded Recursion   frr3g 9738
            2.6.9  Rank   cr1 9744
            2.6.10  Hereditarily finite sets   chf 9877
            2.6.11  Scott's trick; collection principle; Hilbert's epsilon   cscott 9900
            2.6.12  Set Recursion   csetrecs 9936
                  ​*2.6.12.1  Basic Properties of Set Recursion   csetrecs 9936
            2.6.13  Disjoint union   cdju 9951
            2.6.14  Cardinal numbers   ccrd 9988
            2.6.15  Axiom of Choice equivalents   wac 10166
            ​*2.6.16  Cardinal number arithmetic   undjudom 10218
            2.6.17  The Ackermann bijection   ackbij2lem1 10268
            2.6.18  Cofinality (without Axiom of Choice)   cflem 10295
            2.6.19  Eight inequivalent definitions of finite set   sornom 10327
            2.6.20  Hereditarily size-limited sets without Choice   itunifval 10466
​​*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 10485
            3.1.2  Introduce the Axiom of Dependent Choice   ax-dc 10496
      ​3.2  ZFC Set Theory - add the Axiom of Choice
            3.2.1  Introduce the Axiom of Choice   ax-ac 10509
            3.2.2  AC equivalents: well-ordering, Zorn's lemma   numthcor 10544
            3.2.3  Cardinal number theorems using Axiom of Choice   cardval 10602
            3.2.4  Cardinal number arithmetic using Axiom of Choice   iunctb 10631
            3.2.5  Cofinality using the Axiom of Choice   alephreg 10639
      ​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 10677
            3.4.2  Derivation of the Axiom of Choice   gchaclem 10735
​​*PART 4  TG (TARSKI-GROTHENDIECK) SET THEORY
      ​4.1  Inaccessibles
            4.1.1  Weakly and strongly inaccessible cardinals   cwina 10739
            4.1.2  Weak universes   cwun 10757
            4.1.3  Tarski classes   ctsk 10805
            4.1.4  Grothendieck universes   cgru 10847
      ​4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
            4.2.1  Introduce the Tarski-Grothendieck Axiom   ax-groth 10880
            4.2.2  Derive the Power Set, Infinity and Choice Axioms   grothpw 10883
            4.2.3  Tarski map function   ctskm 10894
​​*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 10901
            5.1.2  Final derivation of real and complex number postulates   axaddf 11202
            5.1.3  Real and complex number postulates restated as axioms   ax-cnex 11228
      ​5.2  Derive the basic properties from the field axioms
            5.2.1  Some deductions from the field axioms for complex numbers   cnex 11253
            5.2.2  Infinity and the extended real number system   cpnf 11312
            5.2.3  Restate the ordering postulates with extended real "less than"   axlttri 11353
            5.2.4  Ordering on reals   lttr 11358
            5.2.5  Initial properties of the complex numbers   mul12 11447
      ​5.3  Real and complex numbers - basic operations
            5.3.1  Addition   add12 11500
            5.3.2  Subtraction   cmin 11513
            5.3.3  Multiplication   kcnktkm1cn 11717
            5.3.4  Ordering on reals (cont.)   gt0ne0 11751
            5.3.5  Reciprocals   ixi 11915
            5.3.6  Division   cdiv 11943
            5.3.7  Ordering on reals (cont.)   elimgt0 12125
            5.3.8  Completeness Axiom and Suprema   fimaxre 12231
            5.3.9  Imaginary and complex number properties   neg1cn 12275
            5.3.10  Function operation analogue theorems   ofsubeq0 12287
            ​*5.3.11  Indicator Functions   cind 12290
      ​5.4  Integer sets
            5.4.1  Positive integers (as a subset of complex numbers)   cn 12305
            5.4.2  Principle of mathematical induction   nnind 12323
            ​*5.4.3  Decimal representation of numbers   c2 12367
            ​*5.4.4  Some properties of specific numbers   1pneg1e0 12430
            5.4.5  Simple number properties   halfcl 12542
            5.4.6  The Archimedean property   nnunb 12572
            5.4.7  Nonnegative integers (as a subset of complex numbers)   cn0 12576
            ​*5.4.8  Extended nonnegative integers   cxnn0 12649
            5.4.9  Integers (as a subset of complex numbers)   cz 12663
            5.4.10  Decimal arithmetic   cdc 12784
            5.4.11  Upper sets of integers   cuz 12935
            5.4.12  Well-ordering principle for bounded-below sets of integers   uzwo3 13040
            5.4.13  Rational numbers (as a subset of complex numbers)   cq 13045
            5.4.14  Existence of the set of complex numbers   rpnnen1lem2 13075
      ​5.5  Order sets
            5.5.1  Positive reals (as a subset of complex numbers)   crp 13090
            5.5.2  Infinity and the extended real number system (cont.)   cxne 13208
            5.5.3  Supremum and infimum on the extended reals   xrsupexmnf 13405
            5.5.4  Real number intervals   cioo 13446
            5.5.5  Finite intervals of integers   cfz 13609
            ​*5.5.6  Finite intervals of nonnegative integers   elfz2nn0 13721
            5.5.7  Half-open integer ranges   cfzo 13757
      ​5.6  Elementary integer functions
            5.6.1  The floor and ceiling functions   cfl 13899
            5.6.2  The modulo (remainder) operation   cmo 13978
            5.6.3  Miscellaneous theorems about integers   om2uz0i 14059
            5.6.4  Strong induction over upper sets of integers   uzsinds 14099
            5.6.5  Finitely supported functions over the nonnegative integers   fsuppmapnn0fiublem 14102
            5.6.6  The infinite sequence builder "seq" - extension   cseq 14113
            5.6.7  Integer powers   cexp 14173
            5.6.8  Ordered pair theorem for nonnegative integers   nn0le2msqi 14379
            5.6.9  Factorial function   cfa 14385
            5.6.10  The binomial coefficient operation   cbc 14414
            5.6.11  The ` # ` (set size) function   chash 14442
                  5.6.11.1  Proper unordered pairs and triples (sets of size 2 and 3)   hashprlei 14581
                  5.6.11.2  Functions with a domain containing at least two different elements   fundmge2nop0 14615
                  5.6.11.3  Finite induction on the size of the first component of a binary relation   hashdifsnp1 14619
      ​​*5.7  Words over a set
            5.7.1  Definitions and basic theorems   cword 14626
            5.7.2  Last symbol of a word   clsw 14675
            5.7.3  Concatenations of words   cconcat 14683
            5.7.4  Singleton words   cs1 14710
            5.7.5  Concatenations with singleton words   ccatws1cl 14732
            5.7.6  Subwords/substrings   csubstr 14756
            5.7.7  Prefixes of a word   cpfx 14788
            5.7.8  Subwords of subwords   swrdswrdlem 14821
            5.7.9  Subwords and concatenations   pfxcctswrd 14827
            5.7.10  Subwords of concatenations   swrdccatfn 14841
            5.7.11  Splicing words (substring replacement)   csplice 14866
            5.7.12  Reversing words   creverse 14875
            5.7.13  Repeated symbol words   creps 14887
            ​*5.7.14  Cyclical shifts of words   ccsh 14907
            5.7.15  Mapping words by a function   wrdco 14950
            5.7.16  Longer string literals   cs2 14960
      ​​*5.8  Reflexive and transitive closures of relations
            5.8.1  The reflexive and transitive properties of relations   coss12d 15093
            5.8.2  Basic properties of closures   cleq1lem 15103
            5.8.3  Definitions and basic properties of transitive closures   ctcl 15106
            5.8.4  Exponentiation of relations   crelexp 15140
            5.8.5  Reflexive-transitive closure as an indexed union   crtrcl 15176
            ​*5.8.6  Principle of transitive induction   relexpindlem 15184
      ​5.9  Elementary real and complex functions
            5.9.1  The "shift" operation   cshi 15187
            5.9.2  Signum (sgn or sign) function   csgn 15207
            5.9.3  Real and imaginary parts; conjugate   ccj 15231
            5.9.4  Square root; absolute value   csqrt 15368
      ​5.10  Elementary limits and convergence
            5.10.1  Superior limit (lim sup)   clsp 15605
            5.10.2  Limits   cli 15619
            5.10.3  Finite and infinite sums   csu 15821
            5.10.4  The binomial theorem   binomlem 15966
            5.10.5  The inclusion/exclusion principle   incexclem 15973
            5.10.6  Infinite sums (cont.)   isumshft 15976
            5.10.7  Miscellaneous converging and diverging sequences   divrcnv 15989
            5.10.8  Arithmetic series   arisum 15997
            5.10.9  Geometric series   expcnv 16001
            5.10.10  Ratio test for infinite series convergence   cvgrat 16020
            5.10.11  Mertens' theorem   mertenslem1 16021
            5.10.12  Finite and infinite products   prodf 16024
                  5.10.12.1  Product sequences   prodf 16024
                  5.10.12.2  Non-trivial convergence   ntrivcvg 16034
                  5.10.12.3  Complex products   cprod 16040
                  5.10.12.4  Finite products   fprod 16076
                  5.10.12.5  Infinite products   iprodclim 16133
            5.10.13  Falling and Rising Factorial   cfallfac 16139
            5.10.14  Bernoulli polynomials and sums of k-th powers   cbp 16180
      ​5.11  Elementary trigonometry
            5.11.1  The exponential, sine, and cosine functions   ce 16195
                  5.11.1.1  The circle constant (tau = 2 pi)   ctau 16338
            5.11.2  _e is irrational   eirrlem 16340
      ​5.12  Cardinality of real and complex number subsets
            5.12.1  Countability of integers and rationals   xpnnen 16347
            5.12.2  The reals are uncountable   rpnnen2lem1 16350
​​*PART 6  ELEMENTARY NUMBER THEORY
      ​6.1  Elementary properties of divisibility
            6.1.1  Irrationality of square root of 2   sqrt2irrlem 16384
            6.1.2  Some Number sets are chains of proper subsets   nthruc 16388
            6.1.3  The divides relation   cdvds 16390
            ​*6.1.4  Even and odd numbers   evenelz 16474
            6.1.5  The division algorithm   divalglem0 16531
            6.1.6  Bit sequences   cbits 16557
            6.1.7  The greatest common divisor operator   cgcd 16632
            6.1.8  Bézout's identity   bezoutlem1 16677
            6.1.9  Algorithms   nn0seqcvgd 16708
            6.1.10  Euclid's Algorithm   eucalgval2 16719
            ​*6.1.11  The least common multiple   clcm 16726
            ​*6.1.12  Coprimality and Euclid's lemma   coprmgcdb 16787
            6.1.13  Cancellability of congruences   congr 16802
      ​6.2  Elementary prime number theory
            ​*6.2.1  Elementary properties   cprime 16809
            ​*6.2.2  Coprimality and Euclid's lemma (cont.)   coprm 16850
            6.2.3  Properties of the canonical representation of a rational   cnumer 16872
            6.2.4  Euler's theorem   codz 16902
            6.2.5  Arithmetic modulo a prime number   modprm1div 16937
            6.2.6  Pythagorean Triples   coprimeprodsq 16948
            6.2.7  The prime count function   cpc 16976
            6.2.8  Pocklington's theorem   prmpwdvds 17044
            6.2.9  Infinite primes theorem   unbenlem 17048
            6.2.10  Sum of prime reciprocals   prmreclem1 17056
            6.2.11  Fundamental theorem of arithmetic   1arithlem1 17063
            6.2.12  Lagrange's four-square theorem   cgz 17069
            6.2.13  Van der Waerden's theorem   cvdwa 17105
            6.2.14  Ramsey's theorem   cram 17139
            ​*6.2.15  Primorial function   cprmo 17171
            ​*6.2.16  Prime gaps   prmgaplem1 17189
            6.2.17  Decimal arithmetic (cont.)   dec2dvds 17203
            6.2.18  Cyclical shifts of words (cont.)   cshwsidrepsw 17233
            6.2.19  Specific prime numbers   prmlem0 17245
            6.2.20  Very large primes   1259lem1 17271
​PART 7  BASIC STRUCTURES
      ​7.1  Extensible structures
            ​*7.1.1  Basic definitions   cstr 17286
                  7.1.1.1  Extensible structures as structures with components   cstr 17286
                  7.1.1.2  Substitution of components   csts 17303
                  7.1.1.3  Slots   cslot 17321
                  ​*7.1.1.4  Structure component indices   cnx 17333
                  7.1.1.5  Base sets   cbs 17349
                  7.1.1.6  Base set restrictions   cress 17370
            7.1.2  Slot definitions   cplusg 17390
            7.1.3  Definition of the structure product   crest 17553
            7.1.4  Definition of the structure quotient   cordt 17633
      ​7.2  Moore spaces
            7.2.1  Moore closures   mrcflem 17742
            7.2.2  Independent sets in a Moore system   mrisval 17766
            7.2.3  Algebraic closure systems   isacs 17787
​PART 8  BASIC CATEGORY THEORY
      ​8.1  Categories
            8.1.1  Categories   ccat 17800
            8.1.2  Opposite category   coppc 17847
            8.1.3  Monomorphisms and epimorphisms   cmon 17865
            8.1.4  Sections, inverses, isomorphisms   csect 17881
            ​*8.1.5  Isomorphic objects   ccic 17932
            8.1.6  Subcategories   cssc 17944
            8.1.7  Functors   cfunc 17991
            8.1.8  Full & faithful functors   cful 18041
            8.1.9  Natural transformations and the functor category   cnat 18081
            8.1.10  Initial, terminal and zero objects of a category   cinito 18118
      ​8.2  Arrows (disjointified hom-sets)
            8.2.1  Identity and composition for arrows   cida 18190
      ​8.3  Examples of categories
            8.3.1  The category of sets   csetc 18212
            8.3.2  The category of categories   ccatc 18235
            ​*8.3.3  The category of extensible structures   fncnvimaeqv 18256
      ​8.4  Categorical constructions
            8.4.1  Product of categories   cxpc 18304
            8.4.2  Functor evaluation   cevlf 18345
            8.4.3  Hom functor   chof 18384
​PART 9  BASIC ORDER THEORY
      ​9.1  Dual of an order structure
      ​9.2  Preordered sets and directed sets
      ​9.3  Partially ordered sets (posets)
      ​9.4  Totally ordered sets (tosets)
      ​9.5  Lattices
            9.5.1  Lattices   clat 18567
            9.5.2  Complete lattices   ccla 18634
            9.5.3  Distributive lattices   cdlat 18656
            9.5.4  Subset order structures   cipo 18663
      ​9.6  Posets, directed sets, and lattices as relations
            ​*9.6.1  Posets and lattices as relations   cps 18700
            9.6.2  Directed sets, nets   cdir 18730
      ​9.7  Chains
​PART 10  BASIC ALGEBRAIC STRUCTURES
      ​10.1  Monoids
            ​*10.1.1  Magmas   cplusf 18775
            ​*10.1.2  Identity elements   mgmidmo 18800
            ​*10.1.3  Iterated sums in a magma   gsumvalx 18827
            10.1.4  Magma homomorphisms and submagmas   cmgmhm 18841
            ​*10.1.5  Semigroups   csgrp 18869
            ​*10.1.6  Definition and basic properties of monoids   cmnd 18885
            10.1.7  Monoid homomorphisms and submonoids   cmhm 18938
            ​*10.1.8  Iterated sums in a monoid   gsumvallem2 18992
            10.1.9  Free monoids   cfrmd 19005
                  ​*10.1.9.1  Monoid of endofunctions   cefmnd 19026
            10.1.10  Examples and counterexamples for magmas, semigroups and monoids   mgm2nsgrplem1 19079
      ​10.2  Groups
            10.2.1  Definition and basic properties   cgrp 19106
            ​*10.2.2  Group multiple operation   cmg 19239
            10.2.3  Subgroups and Quotient groups   csubg 19292
            ​*10.2.4  Cyclic monoids and groups   cycsubmel 19377
            10.2.5  Elementary theory of group homomorphisms   cghm 19389
            10.2.6  Isomorphisms of groups   cgim 19433
                  10.2.6.1  The first isomorphism theorem of groups   ghmqusnsglem1 19456
            10.2.7  Group actions   cga 19465
            10.2.8  Centralizers and centers   ccntz 19491
            10.2.9  The opposite group   coppg 19521
            10.2.10  Symmetric groups   csymg 19545
                  ​*10.2.10.1  Definition and basic properties   csymg 19545
                  10.2.10.2  Cayley's theorem   cayleylem1 19588
                  10.2.10.3  Permutations fixing one element   symgfix2 19592
                  ​*10.2.10.4  Transpositions in the symmetric group   cpmtr 19617
                  10.2.10.5  The sign of a permutation   cpsgn 19665
            10.2.11  p-Groups and Sylow groups; Sylow's theorems   cod 19700
            10.2.12  Direct products   clsm 19810
                  10.2.12.1  Direct products (extension)   smndlsmidm 19832
            10.2.13  Free groups   cefg 19882
            10.2.14  Abelian groups   ccmn 19956
                  10.2.14.1  Definition and basic properties   ccmn 19956
                  10.2.14.2  Cyclic groups   ccyg 20053
                  10.2.14.3  Group sum operation   gsumval3a 20079
                  10.2.14.4  Group sums over (ranges of) integers   fsfnn0gsumfsffz 20159
                  10.2.14.5  Internal direct products   cdprd 20171
                  10.2.14.6  The Fundamental Theorem of Abelian Groups   ablfacrplem 20243
            10.2.15  Simple groups   csimpg 20268
                  10.2.15.1  Definition and basic properties   csimpg 20268
                  10.2.15.2  Classification of abelian simple groups   ablsimpnosubgd 20282
            10.2.16  Totally ordered monoids and groups   comnd 20295
      ​10.3  Rings
            10.3.1  Multiplicative Group   cmgp 20322
            ​*10.3.2  Non-unital rings ("rngs")   crng 20336
            ​*10.3.3  Ring unity (multiplicative identity)   cur 20369
            10.3.4  Semirings   csrg 20374
                  ​*10.3.4.1  The binomial theorem for semirings   srgbinomlem1 20414
            10.3.5  Unital rings   crg 20421
            10.3.6  Opposite ring   coppr 20528
            10.3.7  Divisibility   cdsr 20546
            10.3.8  Ring primes   crpm 20624
            10.3.9  Homomorphisms of non-unital rings   crnghm 20626
            10.3.10  Ring homomorphisms   crh 20661
            10.3.11  Nonzero rings and zero rings   cnzr 20724
            10.3.12  Local rings   clring 20752
            10.3.13  Subrings   csubrng 20759
                  10.3.13.1  Subrings of non-unital rings   csubrng 20759
                  10.3.13.2  Subrings of unital rings   csubrg 20783
                  10.3.13.3  Subrings generated by a subset   crgspn 20824
            10.3.14  Categories of rings   crngc 20830
                  ​*10.3.14.1  The category of non-unital rings   crngc 20830
                  ​*10.3.14.2  The category of (unital) rings   cringc 20859
                  10.3.14.3  Subcategories of the category of rings   srhmsubclem1 20891
            10.3.15  Left regular elements and domains   crlreg 20905
      ​10.4  Division rings and fields
            10.4.1  Definition and basic properties   cdr 20942
            10.4.2  Sub-division rings   csdrg 21005
            10.4.3  Absolute value (abstract algebra)   cabv 21027
            10.4.4  Star rings   cstf 21056
            10.4.5  Totally ordered rings and fields   corng 21076
      ​10.5  Left modules
            10.5.1  Definition and basic properties   clmod 21097
            10.5.2  Subspaces and spans in a left module   clss 21168
            10.5.3  Homomorphisms and isomorphisms of left modules   clmhm 21256
            10.5.4  Subspace sum; bases for a left module   clbs 21311
      ​10.6  Vector spaces
            10.6.1  Definition and basic properties   clvec 21339
      ​10.7  Subring algebras and ideals
            10.7.1  Subring algebras   csra 21408
            ​*10.7.2  Left ideals and spans   clidl 21446
            10.7.3  Two-sided ideals and quotient rings   c2idl 21504
                  ​*10.7.3.1  Condition for a non-unital ring to be unital   rngqiprng1elbas 21544
                  10.7.3.2  Prime Ideals   cprmidl 21578
            10.7.4  Principal ideal rings. Divisibility in the integers   clpidl 21606
            10.7.5  Principal ideal domains   cpid 21622
      ​10.8  The complex numbers as an algebraic extensible structure
            10.8.1  Definition and basic properties   cpsmet 21624
            ​*10.8.2  Ring of integers   czring 21714
                  ​*10.8.2.1  Example for a condition for a non-unital ring to be unital   pzriprnglem1 21749
            10.8.3  Algebraic constructions based on the complex numbers   czrh 21767
            10.8.4  Signs as subgroup of the complex numbers   cnmsgnsubg 21845
            10.8.5  Embedding of permutation signs into a ring   zrhpsgnmhm 21852
            10.8.6  The ordered field of real numbers   crefld 21872
      ​10.9  Generalized pre-Hilbert and Hilbert spaces
            10.9.1  Definition and basic properties   cphl 21892
            10.9.2  Orthocomplements and closed subspaces   cocv 21928
            10.9.3  Orthogonal projection and orthonormal bases   cpj 21968
​​*PART 11  BASIC LINEAR ALGEBRA
      ​11.1  Vectors and free modules
            *11.1.1  Direct sum of left modules   cdsmm 21999
            ​*11.1.2  Free modules   cfrlm 22014
            ​*11.1.3  Standard basis (unit vectors)   cuvc 22050
            ​*11.1.4  Independent sets and families   clindf 22072
            11.1.5  Characterization of free modules   lmimlbs 22104
      ​11.2  Associative algebras
            11.2.1  Definition and basic properties   casa 22120
      ​11.3  Abstract multivariate polynomials
            11.3.1  Definition and basic properties   cmps 22174
            11.3.2  Polynomial evaluation   ces 22343
            11.3.3  The "variable selection" function   cslv 22387
            11.3.4  Additional definitions for (multivariate) polynomials   cmhp 22416
            ​*11.3.5  Univariate polynomials   cps1 22455
            11.3.6  Univariate polynomial evaluation   ces1 22593
                  11.3.6.1  Specialization of polynomial evaluation as a ring homomorphism   evls1scafv 22646
      ​​*11.4  Matrices
            *11.4.1  The matrix multiplication   cmmul 22667
            ​*11.4.2  Square matrices   cmat 22684
            ​*11.4.3  The matrix algebra   matmulr 22715
            ​*11.4.4  Matrices of dimension 0 and 1   mat0dimbas0 22743
            ​*11.4.5  The subalgebras of diagonal and scalar matrices   cdmat 22765
            ​*11.4.6  Multiplication of a matrix with a "column vector"   cmvmul 22817
            11.4.7  Replacement functions for a square matrix   cmarrep 22833
            11.4.8  Submatrices   csubma 22853
      ​11.5  The determinant
            11.5.1  Definition and basic properties   cmdat 22861
            11.5.2  Determinants of 2 x 2 -matrices   m2detleiblem1 22901
            11.5.3  The matrix adjugate/adjunct   cmadu 22909
            ​*11.5.4  Laplace expansion of determinants (special case)   symgmatr01lem 22930
            11.5.5  Inverse matrix   invrvald 22953
            ​*11.5.6  Cramer's rule   slesolvec 22959
      ​​*11.6  Polynomial matrices
            11.6.1  Basic properties   pmatring 22972
            ​*11.6.2  Constant polynomial matrices   ccpmat 22983
            ​*11.6.3  Collecting coefficients of polynomial matrices   cdecpmat 23042
            ​*11.6.4  Ring isomorphism between polynomial matrices and polynomials over matrices   cpm2mp 23072
      ​​*11.7  The characteristic polynomial
            *11.7.1  Definition and basic properties   cchpmat 23106
            ​*11.7.2  The characteristic factor function G   fvmptnn04if 23129
            ​*11.7.3  The Cayley-Hamilton theorem   cpmadurid 23147
​PART 12  BASIC TOPOLOGY
      ​12.1  Topology
            ​*12.1.1  Topological spaces   ctop 23173
                  12.1.1.1  Topologies   ctop 23173
                  12.1.1.2  Topologies on sets   ctopon 23190
                  12.1.1.3  Topological spaces   ctps 23212
            12.1.2  Topological bases   ctb 23225
            12.1.3  Examples of topologies   distop 23275
            12.1.4  Closure and interior   ccld 23296
            12.1.5  Neighborhoods   cnei 23377
            12.1.6  Limit points and perfect sets   clp 23414
            12.1.7  Subspace topologies   restrcl 23437
            12.1.8  Order topology   ordtbaslem 23468
            12.1.9  Limits and continuity in topological spaces   ccn 23504
            12.1.10  Separated spaces: T0, T1, T2 (Hausdorff) ...   ct0 23586
            12.1.11  Compactness   ccmp 23666
            12.1.12  Bolzano-Weierstrass theorem   bwth 23690
            12.1.13  Connectedness   cconn 23691
            12.1.14  First- and second-countability   c1stc 23717
            12.1.15  Local topological properties   clly 23745
            12.1.16  Refinements   cref 23783
            12.1.17  Compactly generated spaces   ckgen 23814
            12.1.18  Product topologies   ctx 23841
            12.1.19  Continuous function-builders   cnmptid 23942
            12.1.20  Quotient maps and quotient topology   ckq 23974
            12.1.21  Homeomorphisms   chmeo 24034
      ​12.2  Filters and filter bases
            12.2.1  Filter bases   elmptrab 24108
            12.2.2  Filters   cfil 24126
            12.2.3  Ultrafilters   cufil 24180
            12.2.4  Filter limits   cfm 24214
            12.2.5  Extension by continuity   ccnext 24340
            12.2.6  Topological groups   ctmd 24351
            12.2.7  Infinite group sum on topological groups   ctsu 24407
            12.2.8  Topological rings, fields, vector spaces   ctrg 24437
      ​12.3  Uniform Structures and Spaces
            12.3.1  Uniform structures   cust 24481
            12.3.2  The topology induced by an uniform structure   cutop 24511
            12.3.3  Uniform Spaces   cuss 24534
            12.3.4  Uniform continuity   cucn 24555
            12.3.5  Cauchy filters in uniform spaces   ccfilu 24566
            12.3.6  Complete uniform spaces   ccusp 24577
      ​12.4  Metric spaces
            12.4.1  Pseudometric spaces   ispsmet 24585
            12.4.2  Basic metric space properties   cxms 24598
            12.4.3  Metric space balls   blfvalps 24664
            12.4.4  Open sets of a metric space   mopnval 24719
            12.4.5  Continuity in metric spaces   metcnp3 24821
            12.4.6  The uniform structure generated by a metric   metuval 24830
            12.4.7  Examples of metric spaces   dscmet 24853
            ​*12.4.8  Normed algebraic structures   cnm 24857
            12.4.9  Normed space homomorphisms (bounded linear operators)   cnmo 24986
            12.4.10  Topology on the reals   qtopbaslem 25039
            12.4.11  Topological definitions using the reals   cii 25158
            12.4.12  Path homotopy   chtpy 25250
            12.4.13  The fundamental group   cpco 25283
      ​12.5  Metric subcomplex vector spaces
            12.5.1  Subcomplex modules   cclm 25345
            ​*12.5.2  Subcomplex vector spaces   ccvs 25406
            ​*12.5.3  Normed subcomplex vector spaces   isncvsngp 25432
            12.5.4  Subcomplex pre-Hilbert spaces   ccph 25449
            12.5.5  Convergence and completeness   ccfil 25535
            12.5.6  Baire's Category Theorem   bcthlem1 25607
            12.5.7  Banach spaces and subcomplex Hilbert spaces   ccms 25615
                  12.5.7.1  The complete ordered field of the real numbers   retopn 25662
            12.5.8  Euclidean spaces   crrx 25666
            12.5.9  Minimizing Vector Theorem   minveclem1 25707
            12.5.10  Projection Theorem   pjthlem1 25720
​PART 13  BASIC REAL AND COMPLEX ANALYSIS
      ​13.1  Continuity
            13.1.1  Intermediate value theorem   pmltpclem1 25731
      ​13.2  Integrals
            13.2.1  Lebesgue measure   covol 25745
            13.2.2  Lebesgue integration   cmbf 25897
                  13.2.2.1  Lesbesgue integral   cmbf 25897
                  13.2.2.2  Lesbesgue directed integral   cdit 26128
      ​13.3  Derivatives
            13.3.1  Real and complex differentiation   climc 26144
                  13.3.1.1  Derivatives of functions of one complex or real variable   climc 26144
                  13.3.1.2  Results on real differentiation   dvferm1lem 26266
​PART 14  BASIC REAL AND COMPLEX FUNCTIONS
      ​14.1  Polynomials
            14.1.1  Polynomial degrees   cmdg 26333
            14.1.2  The division algorithm for univariate polynomials   cmn1 26406
            14.1.3  Elementary properties of complex polynomials   cply 26464
            14.1.4  The division algorithm for polynomials   cquot 26575
            14.1.5  Algebraic numbers   caa 26601
            14.1.6  Liouville's approximation theorem   aalioulem1 26623
      ​14.2  Sequences and series
            14.2.1  Taylor polynomials and Taylor's theorem   ctayl 26644
            14.2.2  Uniform convergence   culm 26667
            14.2.3  Power series   pserval 26701
      ​14.3  Basic trigonometry
            14.3.1  The exponential, sine, and cosine functions (cont.)   efcn 26734
            14.3.2  Properties of pi = 3.14159...   pilem1 26742
            14.3.3  Mapping of the exponential function   efgh 26833
            14.3.4  The natural logarithm on complex numbers   clog 26846
            ​*14.3.5  Logarithms to an arbitrary base   clogb 27056
            14.3.6  Theorems of Pythagoras, isosceles triangles, and intersecting chords   angval 27093
            14.3.7  Solutions of quadratic, cubic, and quartic equations   quad2 27131
            14.3.8  Inverse trigonometric functions   casin 27154
            14.3.9  The Birthday Problem   log2ublem1 27238
            14.3.10  Areas in R^2   carea 27247
            14.3.11  More miscellaneous converging sequences   rlimcnp 27257
            14.3.12  Inequality of arithmetic and geometric means   cvxcl 27276
            14.3.13  Euler-Mascheroni constant   cem 27283
            14.3.14  Zeta function   czeta 27304
            14.3.15  Gamma function   clgam 27307
      ​14.4  Basic number theory
            14.4.1  Wilson's theorem   wilthlem1 27359
            14.4.2  The Fundamental Theorem of Algebra   ftalem1 27364
            14.4.3  The Basel problem (ζ(2) = π²/6)   basellem1 27372
            14.4.4  Number-theoretical functions   ccht 27382
            14.4.5  Perfect Number Theorem   mersenne 27518
            14.4.6  Characters of Z/nZ   cdchr 27523
            14.4.7  Bertrand's postulate   bcctr 27566
            ​*14.4.8  Quadratic residues and the Legendre symbol   clgs 27585
            ​*14.4.9  Gauss' Lemma   gausslemma2dlem0a 27647
            14.4.10  Quadratic reciprocity   lgseisenlem1 27666
            14.4.11  All primes 4n+1 are the sum of two squares   2sqlem1 27708
            14.4.12  Chebyshev's Weak Prime Number Theorem, Dirichlet's Theorem   chebbnd1lem1 27760
            14.4.13  The Prime Number Theorem   mudivsum 27821
            14.4.14  Ostrowski's theorem   abvcxp 27906
​PART 15  SURREAL NUMBERS
      ​​*15.1  Sign sequence representation and Alling's axioms
            15.1.1  Definitions and initial properties   csur 27931
            15.1.2  Ordering   ltssolem1 27966
            15.1.3  Birthday Function   bdayfo 27968
            15.1.4  Density   fvnobday 27969
            ​*15.1.5  Full-Eta Property   bdayimaon 27984
      ​15.2  Initial consequences of Alling's axioms
            15.2.1  Ordering Theorems   cles 28035
            15.2.2  Birthday Theorems   bdayfun 28067
      ​​*15.3  Conway cut representation
            15.3.1  Conway cuts   cslts 28077
            15.3.2  Zero and One   c0s 28125
            15.3.3  Cuts and Options   cmade 28142
            15.3.4  Cofinality and coinitiality   cofslts 28238
      ​15.4  Induction and recursion
            15.4.1  Induction and recursion on one variable   cnorec 28257
            15.4.2  Induction and recursion on two variables   cnorec2 28268
      ​15.5  Surreal arithmetic
            15.5.1  Addition   cadds 28279
            15.5.2  Negation and Subtraction   cnegs 28339
            15.5.3  Multiplication   cmuls 28426
            15.5.4  Division   cdivs 28507
            15.5.5  Absolute value   cabss 28557
      ​15.6  Subsystems of surreals
            15.6.1  Ordinal numbers   cons 28571
            15.6.2  Surreal recursive sequences   cseqs 28603
            15.6.3  Natural numbers   cn0s 28632
            15.6.4  Integers   czs 28698
            15.6.5  Dyadic fractions   c2s 28730
            15.6.6  Real numbers   creno 28809
​​*PART 16  ELEMENTARY GEOMETRY
      ​16.1  Definition and Tarski's Axioms of Geometry
            16.1.1  Justification for the congruence notation   tgjustf 28869
      ​16.2  Tarskian Geometry
            16.2.1  Congruence   tgcgrcomimp 28873
            16.2.2  Betweenness   tgbtwntriv2 28884
            16.2.3  Dimension   tglowdim1 28897
            16.2.4  Betweenness and Congruence   tgifscgr 28905
            16.2.5  Congruence of a series of points   ccgrg 28907
            16.2.6  Motions   cismt 28929
            16.2.7  Colinearity   tglng 28943
            16.2.8  Connectivity of betweenness   tgbtwnconn1lem1 28969
            16.2.9  Less-than relation in geometric congruences   cleg 28979
            16.2.10  Rays   chlg 28997
            16.2.11  Lines   btwnlng1 29021
            16.2.12  Point inversions   cmir 29058
            16.2.13  Right angles   crag 29102
            16.2.14  Half-planes   islnopp 29149
            16.2.15  Planes   cplng 29185
            16.2.16  Midpoints and Line Mirroring   cmid 29211
            16.2.17  Congruence of angles   ccgra 29248
            16.2.18  Angle Comparisons   cinag 29288
            16.2.19  Angle Addition   cangmgm 29307
            16.2.20  Congruence Theorems   tgsas1 29333
            16.2.21  Equilateral triangles   ceqlg 29344
            16.2.22  Parallel lines   cprlng 29348
      ​16.3  Properties of geometries
            16.3.1  Isomorphisms between geometries   f1otrgds 29380
      ​16.4  Geometry in Hilbert spaces
            16.4.1  Geometry in the complex plane   cchhllem 29398
            16.4.2  Geometry in Euclidean spaces   cee 29399
                  16.4.2.1  Definition of the Euclidean space   cee 29399
                  16.4.2.2  Tarski's axioms for geometry for the Euclidean space   axdimuniq 29425
                  16.4.2.3  EE^n fulfills Tarski's Axioms   ceeng 29489
​​*PART 17  GRAPH THEORY
      ​*17.1  Vertices and edges
            17.1.1  The edge function extractor for extensible structures   cedgf 29500
            ​*17.1.2  Vertices and indexed edges   cvtx 29508
                  17.1.2.1  Definitions and basic properties   cvtx 29508
                  17.1.2.2  The vertices and edges of a graph represented as ordered pair   opvtxval 29515
                  17.1.2.3  The vertices and edges of a graph represented as extensible structure   funvtxdmge2val 29523
                  17.1.2.4  Representations of graphs without edges   snstrvtxval 29549
                  17.1.2.5  Degenerated cases of representations of graphs   vtxval0 29551
            17.1.3  Edges as range of the edge function   cedg 29559
      ​​*17.2  Undirected graphs
            17.2.1  Undirected hypergraphs   cuhgr 29568
            17.2.2  Undirected pseudographs and multigraphs   cupgr 29592
            ​*17.2.3  Loop-free graphs   umgrislfupgrlem 29634
            17.2.4  Edges as subsets of vertices of graphs   uhgredgiedgb 29638
            ​*17.2.5  Undirected simple graphs   cuspgr 29663
            17.2.6  Examples for graphs   usgr0e 29751
            17.2.7  Subgraphs   csubgr 29782
            17.2.8  Finite undirected simple graphs   cfusgr 29831
            17.2.9  Neighbors, complete graphs and universal vertices   cnbgr 29847
                  17.2.9.1  Neighbors   cnbgr 29847
                  17.2.9.2  Universal vertices   cuvtx 29900
                  17.2.9.3  Complete graphs   ccplgr 29924
            17.2.10  Vertex degree   cvtxdg 29980
            ​*17.2.11  Regular graphs   crgr 30070
      ​​*17.3  Walks, paths and cycles
            *17.3.1  Walks   cewlks 30110
            17.3.2  Walks for loop-free graphs   lfgrwlkprop 30204
            17.3.3  Trails   ctrls 30207
            17.3.4  Paths and simple paths   cpths 30229
            17.3.5  Closed walks   cclwlks 30291
            17.3.6  Circuits and cycles   ccrcts 30305
            ​*17.3.7  Walks as words   cwwlks 30348
            17.3.8  Walks/paths of length 2 (as length 3 strings)   2wlkdlem1 30448
            17.3.9  Walks in regular graphs   rusgrnumwwlkl1 30494
            ​*17.3.10  Closed walks as words   cclwwlk 30506
                  17.3.10.1  Closed walks as words   cclwwlk 30506
                  17.3.10.2  Closed walks of a fixed length as words   cclwwlkn 30549
                  17.3.10.3  Closed walks on a vertex of a fixed length as words   cclwwlknon 30612
            17.3.11  Examples for walks, trails and paths   0ewlk 30639
            17.3.12  Acyclic graphs   cacycgr 30682
            17.3.13  Connected graphs   cconngr 30721
      ​17.4  Eulerian paths and the Konigsberg Bridge problem
            ​*17.4.1  Eulerian paths   ceupth 30732
            ​*17.4.2  The Königsberg Bridge problem   konigsbergvtx 30781
      ​17.5  The Friendship Theorem
            17.5.1  Friendship graphs - basics   cfrgr 30793
            17.5.2  The friendship theorem for small graphs   frgr1v 30806
            17.5.3  Theorems according to Mertzios and Unger   2pthfrgrrn 30817
            ​*17.5.4  Huneke's Proof of the Friendship Theorem   frgrncvvdeqlem1 30834
​PART 18  GUIDES AND MISCELLANEA
      ​18.1  Guides (conventions, explanations, and examples)
            ​*18.1.1  Conventions   conventions 30935
            18.1.2  Natural deduction   natded 30938
            ​*18.1.3  Natural deduction examples   ex-natded5.2 30939
            18.1.4  Definitional examples   ex-or 30956
            18.1.5  Other examples   aevdemo 30995
      ​18.2  Humor
            18.2.1  April Fool's theorem   avril1 30998
      ​18.3  (Future - to be reviewed and classified)
            18.3.1  Planar incidence geometry   cplig 31010
            ​*18.3.2  Aliases kept to prevent broken links   dummylink 31023
​​*PART 19  COMPLEX TOPOLOGICAL VECTOR SPACES (DEPRECATED)
      ​*19.1  Additional material on group theory (deprecated)
            19.1.1  Definitions and basic properties for groups   cgr 31025
            19.1.2  Abelian groups   cablo 31080
      ​19.2  Complex vector spaces
            19.2.1  Definition and basic properties   cvc 31094
            19.2.2  Examples of complex vector spaces   cnaddabloOLD 31117
      ​19.3  Normed complex vector spaces
            19.3.1  Definition and basic properties   cnv 31120
            19.3.2  Examples of normed complex vector spaces   cnnv 31213
            19.3.3  Induced metric of a normed complex vector space   imsval 31221
            19.3.4  Inner product   cdip 31236
            19.3.5  Subspaces   css 31257
      ​19.4  Operators on complex vector spaces
            19.4.1  Definitions and basic properties   clno 31276
      ​19.5  Inner product (pre-Hilbert) spaces
            19.5.1  Definition and basic properties   ccphlo 31348
            19.5.2  Examples of pre-Hilbert spaces   cncph 31355
            19.5.3  Properties of pre-Hilbert spaces   isph 31358
      ​19.6  Complex Banach spaces
            19.6.1  Definition and basic properties   ccbn 31398
            19.6.2  Examples of complex Banach spaces   cnbn 31405
            19.6.3  Uniform Boundedness Theorem   ubthlem1 31406
            19.6.4  Minimizing Vector Theorem   minvecolem1 31410
      ​19.7  Complex Hilbert spaces
            19.7.1  Definition and basic properties   chlo 31421
            19.7.2  Standard axioms for a complex Hilbert space   hlex 31434
            19.7.3  Examples of complex Hilbert spaces   cnchl 31452
            19.7.4  Hellinger-Toeplitz Theorem   htthlem 31453
​​*PART 20  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      ​20.1  Axiomatization of complex pre-Hilbert spaces
            20.1.1  Basic Hilbert space definitions   chba 31455
            20.1.2  Preliminary ZFC lemmas   df-hnorm 31504
            ​*20.1.3  Derive the Hilbert space axioms from ZFC set theory   axhilex-zf 31517
            ​*20.1.4  Introduce the vector space axioms for a Hilbert space   ax-hilex 31535
            20.1.5  Vector operations   hvmulex 31547
            20.1.6  Inner product postulates for a Hilbert space   ax-hfi 31615
      ​20.2  Inner product and norms
            20.2.1  Inner product   his5 31622
            20.2.2  Norms   dfhnorm2 31658
            20.2.3  Relate Hilbert space to normed complex vector spaces   hilablo 31696
            20.2.4  Bunjakovaskij-Cauchy-Schwarz inequality   bcsiALT 31715
      ​20.3  Cauchy sequences and completeness axiom
            20.3.1  Cauchy sequences and limits   hcau 31720
            20.3.2  Derivation of the completeness axiom from ZF set theory   hilmet 31730
            20.3.3  Completeness postulate for a Hilbert space   ax-hcompl 31738
            20.3.4  Relate Hilbert space to ZFC pre-Hilbert and Hilbert spaces   hhcms 31739
      ​20.4  Subspaces and projections
            20.4.1  Subspaces   df-sh 31743
            20.4.2  Closed subspaces   df-ch 31757
            20.4.3  Orthocomplements   df-oc 31788
            20.4.4  Subspace sum, span, lattice join, lattice supremum   df-shs 31844
            20.4.5  Projection theorem   pjhthlem1 31927
            20.4.6  Projectors   df-pjh 31931
      ​20.5  Properties of Hilbert subspaces
            20.5.1  Orthomodular law   omlsilem 31938
            20.5.2  Projectors (cont.)   pjhtheu2 31952
            20.5.3  Hilbert lattice operations   sh0le 31976
            20.5.4  Span (cont.) and one-dimensional subspaces   spansn0 32077
            20.5.5  Commutes relation for Hilbert lattice elements   df-cm 32119
            20.5.6  Foulis-Holland theorem   fh1 32154
            20.5.7  Quantum Logic Explorer axioms   qlax1i 32163
            20.5.8  Orthogonal subspaces   chscllem1 32173
            20.5.9  Orthoarguesian laws 5OA and 3OA   5oalem1 32190
            20.5.10  Projectors (cont.)   pjorthi 32205
            20.5.11  Mayet's equation E_3   mayete3i 32264
      ​20.6  Operators on Hilbert spaces
            ​*20.6.1  Operator sum, difference, and scalar multiplication   df-hosum 32266
            20.6.2  Zero and identity operators   df-h0op 32284
            20.6.3  Operations on Hilbert space operators   hoaddcl 32294
            20.6.4  Linear, continuous, bounded, Hermitian, unitary operators and norms   df-nmop 32375
            20.6.5  Linear and continuous functionals and norms   df-nmfn 32381
            20.6.6  Adjoint   df-adjh 32385
            20.6.7  Dirac bra-ket notation   df-bra 32386
            20.6.8  Positive operators   df-leop 32388
            20.6.9  Eigenvectors, eigenvalues, spectrum   df-eigvec 32389
            20.6.10  Theorems about operators and functionals   nmopval 32392
            20.6.11  Riesz lemma   riesz3i 32598
            20.6.12  Adjoints (cont.)   cnlnadjlem1 32603
            20.6.13  Quantum computation error bound theorem   unierri 32640
            20.6.14  Dirac bra-ket notation (cont.)   branmfn 32641
            20.6.15  Positive operators (cont.)   leopg 32658
            20.6.16  Projectors as operators   pjhmopi 32682
      ​20.7  States on a Hilbert lattice and Godowski's equation
            20.7.1  States on a Hilbert lattice   df-st 32747
            20.7.2  Godowski's equation   golem1 32807
      ​20.8  Cover relation, atoms, exchange axiom, and modular symmetry
            20.8.1  Covers relation; modular pairs   df-cv 32815
            20.8.2  Atoms   df-at 32874
            20.8.3  Superposition principle   superpos 32890
            20.8.4  Atoms, exchange and covering properties, atomicity   chcv1 32891
            20.8.5  Irreducibility   chirredlem1 32926
            20.8.6  Atoms (cont.)   atcvat3i 32932
            20.8.7  Modular symmetry   mdsymlem1 32939
​PART 21  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      ​21.1  Mathboxes for user contributions
            21.1.1  Mathbox guidelines   mathbox 32978
      ​21.2  Mathbox for Stefan Allan
      ​21.3  Mathbox for Thierry Arnoux
            21.3.1  Propositional Calculus - misc additions   ad11antr 32983
            21.3.2  Predicate Calculus   sbc2iedf 32996
                  21.3.2.1  Predicate Calculus - misc additions   sbc2iedf 32996
                  21.3.2.2  Restricted quantification - misc additions   ralcom4f 32998
                  21.3.2.3  Equality   eqtrb 33004
                  21.3.2.4  Double restricted existential uniqueness quantification   opsbc2ie 33006
                  21.3.2.5  Double restricted existential uniqueness quantification syntax   w2reu 33008
                  21.3.2.6  Substitution (without distinct variables) - misc additions   sbceqbidf 33017
                  21.3.2.7  Existential "at most one" - misc additions   mo5f 33019
                  21.3.2.8  Existential uniqueness - misc additions   reuxfrdf 33021
                  21.3.2.9  Restricted "at most one" - misc additions   rmoxfrd 33023
                  21.3.2.10  Restricted iota (description binder)   riotaeqbidva 33026
            21.3.3  General Set Theory   dmrab 33027
                  21.3.3.1  Class abstractions (a.k.a. class builders)   dmrab 33027
                  21.3.3.2  Image Sets   abrexdomjm 33037
                  21.3.3.3  Set relations and operations - misc additions   nelun 33043
                  21.3.3.4  Unordered pairs   elpreq 33058
                  21.3.3.5  Unordered triples   tpssg 33067
                  21.3.3.6  Conditional operator - misc additions   ifeqeqx 33072
                  21.3.3.7  Set union   uniinn0 33081
                  21.3.3.8  Indexed union - misc additions   cbviunf 33084
                  21.3.3.9  Indexed intersection - misc additions   iinabrex 33097
                  21.3.3.10  Disjointness - misc additions   disjnf 33098
            21.3.4  Relations and Functions   xpdisjres 33126
                  21.3.4.1  Relations - misc additions   xpdisjres 33126
                  21.3.4.2  Functions - misc additions   fconst7v 33148
                  21.3.4.3  Operations - misc additions   mpomptxf 33206
                  21.3.4.4  Support of a function   suppovss 33208
                  21.3.4.5  Explicit Functions with one or two points as a domain   cosnopne 33221
                  21.3.4.6  Isomorphisms - misc. additions   gtiso 33228
                  21.3.4.7  Disjointness (additional proof requiring functions)   disjdsct 33230
                  21.3.4.8  First and second members of an ordered pair - misc additions   df1stres 33231
                  21.3.4.9  Countable Sets   snct 33239
            21.3.5  Real and Complex Numbers   sgnval2 33261
                  21.3.5.1  Complex operations - misc. additions   creq0 33262
                  21.3.5.2  Ordering on reals - misc additions   lt2addrd 33276
                  21.3.5.3  Extended reals - misc additions   nn0mnfxrd 33277
                  21.3.5.4  Extended nonnegative integers - misc additions   xnn0gt0 33295
                  21.3.5.5  Real number intervals - misc additions   joiniooico 33300
                  21.3.5.6  Finite intervals of integers - misc additions   uzssico 33310
                  21.3.5.7  Half-open integer ranges - misc additions   iundisjfi 33322
                  21.3.5.8  The ` # ` (set size) function - misc additions   hashunif 33332
                  21.3.5.9  The greatest common divisor operator - misc. additions   elq2 33337
                  21.3.5.10  Integers   nn0split01 33343
                  21.3.5.11  Decimal numbers   dfdec100 33355
            21.3.6  Real and complex functions   sgnsgn 33356
                  21.3.6.1  Signum (sgn or sign) function - misc. additions   sgnsgn 33356
                  21.3.6.2  Integer powers - misc. additions   nexple 33358
                  21.3.6.3  Indicator Functions (continued)   indsumin 33362
            ​*21.3.7  Decimal expansion   cdp2 33371
                  ​*21.3.7.1  Decimal point   cdp 33388
                  21.3.7.2  Division in the extended real number system   cxdiv 33417
            21.3.8  Words over a set - misc additions   wrdres 33436
                  21.3.8.1  Splicing words (substring replacement)   splfv3 33453
                  21.3.8.2  Cyclic shift of words   1cshid 33454
            21.3.9  Extensible Structures   ressplusf 33458
                  21.3.9.1  Structure restriction operator   ressplusf 33458
                  21.3.9.2  Posets   ressprs 33461
                  21.3.9.3  Complete lattices   clatp0cl 33471
                  21.3.9.4  Order Theory   cmnt 33473
                  21.3.9.5  Extended reals Structure - misc additions   ax-xrssca 33499
                  21.3.9.6  The extended nonnegative real numbers commutative monoid   xrge00 33509
            21.3.10  Algebra   mndcld 33517
                  21.3.10.1  Monoids   mndcld 33517
                  21.3.10.2  Monoids Homomorphisms   abliso 33530
                  21.3.10.3  Groups - misc additions   grpidcld 33534
                  21.3.10.4  Abelian Groups - misc additions   ablcomd 33540
                  21.3.10.5  Finitely supported group sums - misc additions   gsumsubg 33541
                  21.3.10.6  Group or monoid sums over words   gsumwun 33571
                  21.3.10.7  Centralizers and centers - misc additions   cntzun 33574
                  21.3.10.8  The symmetric group   symgfcoeu 33577
                  21.3.10.9  Transpositions   pmtridf1o 33589
                  21.3.10.10  Permutation Signs   psgnid 33592
                  21.3.10.11  Permutation cycles   ctocyc 33601
                  21.3.10.12  The Alternating Group   evpmval 33640
                  21.3.10.13  Signum in an ordered monoid   csgns 33653
                  21.3.10.14  Fixed points   cfxp 33658
                  21.3.10.15  The Archimedean property for generic ordered algebraic structures   cinftm 33671
                  21.3.10.16  Semiring left modules   cslmd 33695
                  21.3.10.17  Simple groups   prmsimpcyc 33723
                  21.3.10.18  Rings - misc additions   ringrngd 33724
                  21.3.10.19  Subrings generated by a set   elrgspnlem1 33737
                  21.3.10.20  The zero ring   irrednzr 33745
                  21.3.10.21  Localization of rings   cerl 33748
                  21.3.10.22  Integral Domains   domnmuln0rd 33772
                  21.3.10.23  Euclidean Domains   ceuf 33786
                  21.3.10.24  Division Rings   rndrhmcl 33792
                  21.3.10.25  The field of rational numbers   qfld 33793
                  21.3.10.26  Subfields   subsdrg 33794
                  21.3.10.27  Field of fractions   cfrac 33798
                  21.3.10.28  Field extensions generated by a set   cfldgen 33806
                  21.3.10.29  Ring homomorphisms - misc additions   rhmdvd 33819
                  21.3.10.30  Scalar restriction operation   cresv 33821
                  21.3.10.31  The commutative ring of gaussian integers   gzcrng 33836
                  21.3.10.32  The archimedean ordered field of real numbers   cnfldfld 33837
                  21.3.10.33  The quotient map and quotient modules   qusker 33844
                  21.3.10.34  The ring of integers modulo ` N `   znfermltl 33856
                  21.3.10.35  Independent sets and families   islinds5 33857
                  21.3.10.36  Ring associates, ring units   dvdsruassoi 33873
                  ​*21.3.10.37  Subgroup sum / Sumset / Minkowski sum   elgrplsmsn 33879
                  21.3.10.38  The quotient map   quslsm 33890
                  21.3.10.39  Ideals   intlidl 33904
                  21.3.10.40  Maximal Ideals   cmxidl 33918
                  21.3.10.41  Local rings   drnglring 33958
                  21.3.10.42  The semiring of ideals of a ring   cidlsrg 33966
                  21.3.10.43  Prime Elements   rprmval 33982
                  21.3.10.44  Unique factorization domains   cufd 34004
                  21.3.10.45  The ring of integers   zringidom 34017
                  21.3.10.46  Associative Algebra   assaassd 34021
                  21.3.10.47  Univariate Polynomials   0ringmon1p 34023
                  21.3.10.48  Polynomial quotient and polynomial remainder   q1pdir 34069
                  21.3.10.49  Multivariate Polynomials   psrbasfsupp 34077
                  21.3.10.50  The ring of symmetric polynomials   csply 34121
                  21.3.10.51  The subring algebra   sra1r 34147
                  21.3.10.52  Division Ring Extensions   drgext0g 34156
                  21.3.10.53  Vector Spaces   lvecdimfi 34162
                  21.3.10.54  Vector Space Dimension   cldim 34165
            21.3.11  Field Extensions   cfldext 34204
                  21.3.11.1  Algebraic numbers   cirng 34249
                  21.3.11.2  Algebraic extensions   calgext 34261
                  21.3.11.3  Minimal polynomials   cminply 34265
                  21.3.11.4  Quadratic Field Extensions   rtelextdg2lem 34292
                  21.3.11.5  Towers of quadratic extentions   fldext2chn 34294
            ​*21.3.12  Constructible Numbers   cconstr 34295
                  21.3.12.1  Impossible constructions   2sqr3minply 34346
            21.3.13  Matrices   csmat 34359
                  21.3.13.1  Submatrices   csmat 34359
                  21.3.13.2  Matrix literals   clmat 34377
                  21.3.13.3  Laplace expansion of determinants   mdetpmtr1 34389
            21.3.14  Topology   ist0cld 34399
                  21.3.14.1  Open maps   txomap 34400
                  21.3.14.2  Topology of the unit circle   qtopt1 34401
                  21.3.14.3  Refinements   reff 34405
                  21.3.14.4  Open cover refinement property   ccref 34408
                  21.3.14.5  Lindelöf spaces   cldlf 34418
                  21.3.14.6  Paracompact spaces   cpcmp 34421
                  ​*21.3.14.7  Spectrum of a ring   crspec 34428
                  21.3.14.8  Pseudometrics   cmetid 34452
                  21.3.14.9  Continuity - misc additions   hauseqcn 34464
                  21.3.14.10  Topology of the closed unit interval   elunitge0 34465
                  21.3.14.11  Topology of ` ( RR X. RR ) `   unicls 34469
                  21.3.14.12  Order topology - misc. additions   cnvordtrestixx 34479
                  21.3.14.13  Continuity in topological spaces - misc. additions   mndpluscn 34492
                  21.3.14.14  Topology of the extended nonnegative real numbers ordered monoid   xrge0hmph 34498
                  21.3.14.15  Limits - misc additions   lmlim 34513
                  21.3.14.16  Univariate polynomials   pl1cn 34521
            21.3.15  Uniform Stuctures and Spaces   chcmp 34522
                  21.3.15.1  Hausdorff uniform completion   chcmp 34522
            21.3.16  Topology and algebraic structures   zringnm 34524
                  21.3.16.1  The norm on the ring of the integer numbers   zringnm 34524
                  21.3.16.2  Topological ` ZZ ` -modules   zlm0 34526
                  21.3.16.3  Canonical embedding of the field of the rational numbers into a division ring   cqqh 34536
                  21.3.16.4  Canonical embedding of the real numbers into a complete ordered field   crrh 34559
                  21.3.16.5  Embedding from the extended real numbers into a complete lattice   cxrh 34582
                  21.3.16.6  Canonical embeddings into the ordered field of the real numbers   zrhre 34585
                  ​*21.3.16.7  Topological Manifolds   cmntop 34588
                  21.3.16.8  Extended sum   cesum 34593
            21.3.17  Mixed Function/Constant operation   cofc 34661
            21.3.18  Abstract measure   csiga 34674
                  21.3.18.1  Sigma-Algebra   csiga 34674
                  21.3.18.2  Generated sigma-Algebra   csigagen 34705
                  ​*21.3.18.3  lambda and pi-Systems, Rings of Sets   ispisys 34719
                  21.3.18.4  The Borel algebra on the real numbers   cbrsiga 34748
                  21.3.18.5  Product Sigma-Algebra   csx 34755
                  21.3.18.6  Measures   cmeas 34762
                  21.3.18.7  The counting measure   cntmeas 34793
                  21.3.18.8  The Lebesgue measure - misc additions   voliune 34796
                  21.3.18.9  The Dirac delta measure   cdde 34799
                  21.3.18.10  The 'almost everywhere' relation   cae 34804
                  21.3.18.11  Measurable functions   cmbfm 34816
                  21.3.18.12  Borel Algebra on ` ( RR X. RR ) `   br2base 34836
                  ​*21.3.18.13  Caratheodory's extension theorem   coms 34858
            21.3.19  Integration   itgeq12dv 34893
                  21.3.19.1  Lebesgue integral - misc additions   itgeq12dv 34893
                  21.3.19.2  Bochner integral   citgm 34894
            21.3.20  Euler's partition theorem   oddpwdc 34921
            21.3.21  Sequences defined by strong recursion   csseq 34950
            21.3.22  Fibonacci Numbers   cfib 34963
            21.3.23  Probability   cprb 34974
                  21.3.23.1  Probability Theory   cprb 34974
                  21.3.23.2  Conditional Probabilities   ccprob 34998
                  21.3.23.3  Real-valued Random Variables   crrv 35007
                  21.3.23.4  Preimage set mapping operator   corvc 35023
                  21.3.23.5  Distribution Functions   orvcelval 35036
                  21.3.23.6  Cumulative Distribution Functions   orvclteel 35040
                  21.3.23.7  Probabilities - example   coinfliplem 35046
                  21.3.23.8  Bertrand's Ballot Problem   ballotlemoex 35053
            21.3.24  Signum (sgn or sign) function - misc. additions   fzssfzo 35106
                  21.3.24.1  Operations on words   ccatmulgnn0dir 35109
            21.3.25  Polynomials with real coefficients - misc additions   plyrecld 35113
            21.3.26  Descartes's rule of signs   signspval 35116
                  21.3.26.1  Sign changes in a word over real numbers   signspval 35116
                  21.3.26.2  Counting sign changes in a word over real numbers   signslema 35126
            21.3.27  Number Theory   iblidicc 35156
                  21.3.27.1  Representations of a number as sums of integers   crepr 35172
                  21.3.27.2  Vinogradov Trigonometric Sums and the Circle Method   cvts 35199
                  21.3.27.3  The Ternary Goldbach Conjecture: Final Statement   ax-hgt749 35208
            21.3.28  Elementary Geometry   cstrkg2d 35228
                  ​*21.3.28.1  Two-dimensional geometry   cstrkg2d 35228
                  21.3.28.2  Morley's Miracle   cgranbtwn 35233
                  21.3.28.3  Outer Five Segment (not used, no need to move to main)   cafs 35236
            ​*21.3.29  LeftPad Project   clpad 35241
      ​​*21.4  Mathbox for Jonathan Ben-Naim
            21.4.1  First-order logic and set theory   bnj170 35264
            21.4.2  Well founded induction and recursion   bnj110 35423
            21.4.3  The existence of a minimal element in certain classes   bnj69 35575
            21.4.4  Well-founded induction   bnj1204 35577
            21.4.5  Well-founded recursion, part 1 of 3   bnj60 35627
            21.4.6  Well-founded recursion, part 2 of 3   bnj1500 35633
            21.4.7  Well-founded recursion, part 3 of 3   bnj1522 35637
      ​21.5  Mathbox for BTernaryTau
            21.5.1  First-order logic   nfan1c 35638
                  21.5.1.1  Auxiliary axiom schemes   nfan1c 35638
            21.5.2  ZF set theory   inv2 35644
                  21.5.2.1  Ordinals 5 through 9   c5o 35686
                  21.5.2.2  Finitism   prcinf 35706
                  21.5.2.3  Introduce ax-regs   ax-regs 35719
                  21.5.2.4  Derive ax-regs   axregs 35732
                  21.5.2.5  ZFC axioms with reduced distinct variable conditions   axsepg2 35733
                  21.5.2.6  Cardinality without the Axiom of Choice   ckard 35742
                  21.5.2.7  Gödel operations   ccnv2 35766
                  21.5.2.8  The constructible universe   clexo 35788
                  21.5.2.9  Global choice   gblacfnacd 35806
            21.5.3  Real and complex numbers   zltp1ne 35821
            21.5.4  Graph theory   cplgredgex 35826
                  21.5.4.1  Acyclic graphs   acycgr0v 35834
      ​21.6  Mathbox for Mario Carneiro
            21.6.1  Predicate calculus with all distinct variables   ax-7d 35845
            21.6.2  Miscellaneous stuff   quartfull 35851
            21.6.3  Derangements and the Subfactorial   deranglem 35852
            21.6.4  The Erdős-Szekeres theorem   erdszelem1 35877
            21.6.5  The Kuratowski closure-complement theorem   kur14lem1 35892
            21.6.6  Retracts and sections   cretr 35903
            21.6.7  Path-connected and simply connected spaces   cpconn 35905
            21.6.8  Covering maps   ccvm 35941
            21.6.9  Normal numbers   snmlff 36015
            21.6.10  Godel-sets of formulas - part 1   cgoe 36019
            21.6.11  Godel-sets of formulas - part 2   cgon 36118
            21.6.12  Models of ZF   cgze 36132
            ​*21.6.13  Metamath formal systems   cmcn 36146
            21.6.14  Grammatical formal systems   cm0s 36271
            21.6.15  Models of formal systems   cmuv 36291
            21.6.16  Splitting fields   ccpms 36313
            21.6.17  p-adic number fields   czr 36333
      ​​*21.7  Mathbox for Filip Cernatescu
      ​21.8  Mathbox for Paul Chapman
            21.8.1  Real and complex numbers (cont.)   climuzcnv 36357
            21.8.2  Miscellaneous theorems   elfzm12 36361
      ​21.9  Mathbox for Hongxiu Chen
      ​21.10  Mathbox for Adrian Ducourtial
            21.10.1  Propositional calculus   currybi 36374
            21.10.2  Clone theory   ccloneop 36381
      ​21.11  Mathbox for Scott Fenton
            21.11.1  ZFC Axioms in primitive form   axextprim 36387
            21.11.2  Untangled classes   untelirr 36394
            21.11.3  Extra propositional calculus theorems   3jaodd 36401
            21.11.4  Misc. Useful Theorems   nepss 36404
            21.11.5  Properties of real and complex numbers   sqdivzi 36414
            21.11.6  Infinite products   iprodefisumlem 36426
            21.11.7  Factorial limits   faclimlem1 36429
            21.11.8  Greatest common divisor and divisibility   gcd32 36435
            21.11.9  Properties of relationships   dftr6 36437
            21.11.10  Properties of functions and mappings   funpsstri 36452
            21.11.11  Ordinal numbers   elpotr 36465
            21.11.12  Defined equality axioms   axextdfeq 36481
            21.11.13  Hypothesis builders   hbntg 36489
            21.11.14  Well-founded zero, successor, and limits   cwsuc 36494
            21.11.15  Quantifier-free definitions   ctxp 36514
            21.11.16  Alternate ordered pairs   caltop 36643
            21.11.17  Geometry in the Euclidean space   cofs 36669
                  21.11.17.1  Congruence properties   cofs 36669
                  21.11.17.2  Betweenness properties   btwntriv2 36699
                  21.11.17.3  Segment Transportation   ctransport 36716
                  21.11.17.4  Properties relating betweenness and congruence   cifs 36722
                  21.11.17.5  Connectivity of betweenness   btwnconn1lem1 36774
                  21.11.17.6  Segment less than or equal to   csegle 36793
                  21.11.17.7  Outside-of relationship   coutsideof 36806
                  21.11.17.8  Lines and Rays   cline2 36821
            21.11.18  Forward difference   cfwddif 36845
            21.11.19  Rank theorems   rank0 36853
            21.11.20  Hereditarily Finite Sets   hftr 36855
            21.11.21  Natural ordinal operations   cnmul 36858
      ​21.12  Mathbox for Gino Giotto
            21.12.1  Equality theorems   rmoeqi 36898
                  21.12.1.1  Inference versions   rmoeqi 36898
                  21.12.1.2  Deduction versions   rmoeqdv 36923
            21.12.2  Change bound variables   in-ax8 36935
                  21.12.2.1  Change bound variables and domains   cbvralvw2 36937
                  21.12.2.2  Change bound variables, deduction versions   cbvmodavw 36961
                  21.12.2.3  Change bound variables and domains, deduction versions   cbvrmodavw2 36994
            21.12.3  Study of ax-mulf usage   mpomulnzcnf 37010
      ​21.13  Mathbox for Jeff Hankins
            21.13.1  Miscellany   a1i14 37011
            21.13.2  Basic topological facts   topbnd 37034
            21.13.3  Topology of the real numbers   ivthALT 37045
            21.13.4  Refinements   cfne 37046
            21.13.5  Neighborhood bases determine topologies   neibastop1 37069
            21.13.6  Lattice structure of topologies   topmtcl 37073
            21.13.7  Filter bases   fgmin 37080
            21.13.8  Directed sets, nets   tailfval 37082
      ​21.14  Mathbox for Anthony Hart
            21.14.1  Propositional Calculus   tb-ax1 37093
            21.14.2  Predicate Calculus   nalfal 37113
            21.14.3  Miscellaneous single axioms   meran1 37121
            21.14.4  Connective Symmetry   negsym1 37127
      ​21.15  Mathbox for Chen-Pang He
            21.15.1  Ordinal topology   ontopbas 37138
      ​21.16  Mathbox for Jeff Hoffman
            21.16.1  Inferences for finite induction on generic function values   fveleq 37161
            21.16.2  gdc.mm   nnssi2 37165
      ​21.17  Mathbox for Matthew House
            21.17.1  Relations on well-ordered indexed unions   weiunval 37172
            21.17.2  Axiom of Transitive Containment   axtco 37181
            21.17.3  Transitive closure of a class   tr0elw 37194
            ​*21.17.4  Stronger axioms of regularity   mh-setind 37246
            21.17.5  Short axioms written in primitive symbols   mh-inf3f1 37251
      ​21.18  Mathbox for Asger C. Ipsen
            21.18.1  Continuous nowhere differentiable functions   dnival 37259
      ​​*21.19  Mathbox for BJ
            *21.19.1  Propositional calculus   bj-mp2c 37328
                  *21.19.1.1  Derived rules of inference   bj-mp2c 37328
                  ​*21.19.1.2  A syntactic theorem   bj-0 37330
                  ​*21.19.1.3  Minimal implicational calculus   bj-poni 37332
                  ​*21.19.1.4  Positive calculus   bj-bisimpl 37344
                  ​*21.19.1.5  Implication and negation   bj-con2com 37352
                  ​*21.19.1.6  Disjunction   bj-jaoi1 37363
                  ​*21.19.1.7  Logical equivalence   bj-dfbi4 37365
                  21.19.1.8  The conditional operator for propositions   bj-consensus 37370
                  ​*21.19.1.9  Propositional calculus: miscellaneous   bj-imbi12 37375
            ​*21.19.2  Modal logic   bj-axdd2 37384
            ​*21.19.3  Provability logic   cprvb 37389
            ​*21.19.4  First-order logic   bj-exexalal 37398
                  21.19.4.1  Universal and existential quantifiers, nonfreeness predicate   bj-exexalal 37398
                  21.19.4.2  Adding ax-gen   bj-genr 37399
                  21.19.4.3  Adding ax-4   bj-almp 37403
                  21.19.4.4  Adding ax-5   bj-spvw 37456
                  21.19.4.5  Equality and substitution   bj-df-sb 37471
                  21.19.4.6  Adding ax-6   bj-spim0 37490
                  21.19.4.7  Adding ax-7   bj-cbvexw 37498
                  21.19.4.8  Membership predicate, ax-8 and ax-9   bj-ax89 37500
                  21.19.4.9  Adding ax-11   bj-alcomexcom 37502
                  21.19.4.10  Adding ax-12   axc11n11 37506
                  ​*21.19.4.11  Really adding ax-12   bj-substax12 37548
                  21.19.4.12  Nonfreeness   wnnf 37550
                  21.19.4.13  Adding ax-13   bj-axc10 37617
                  ​*21.19.4.14  Removing dependencies on ax-13 (and ax-11)   bj-axc10v 37627
                  ​*21.19.4.15  Distinct var metavariables   bj-hbaeb2 37652
                  ​*21.19.4.16  Around ~ equsal   bj-equsal1t 37656
                  ​*21.19.4.17  Some Principia Mathematica proofs   stdpc5t 37661
                  21.19.4.18  Alternate definition of substitution   bj-sbsb 37671
                  21.19.4.19  Lemmas for substitution   bj-sbf3 37673
                  21.19.4.20  Existential uniqueness   bj-eu3f 37675
                  ​*21.19.4.21  First-order logic: miscellaneous   bj-sblem1 37676
            21.19.5  Set theory   eliminable1 37693
                  ​*21.19.5.1  Eliminability of class terms   eliminable1 37693
                  ​*21.19.5.2  Classes without the axiom of extensionality   bj-denoteslem 37705
                  21.19.5.3  Characterization among sets versus among classes   elelb 37731
                  ​*21.19.5.4  The nonfreeness quantifier for classes   bj-nfcsym 37733
                  ​*21.19.5.5  Lemmas for class substitution   bj-sbeqALT 37734
                  21.19.5.6  Removing some axiom requirements and disjoint variable conditions   bj-exlimvmpi 37745
                  ​*21.19.5.7  Class abstractions   bj-elabd2ALT 37760
                  21.19.5.8  Generalized class abstractions   bj-cgab 37768
                  ​*21.19.5.9  Restricted nonfreeness   wrnf 37776
                  ​*21.19.5.10  Russell's paradox   bj-ru1 37778
                  21.19.5.11  Curry's paradox in set theory   currysetlem 37780
                  ​*21.19.5.12  Some disjointness results   bj-n0i 37786
                  ​*21.19.5.13  Complements on direct products   bj-xpimasn 37790
                  ​*21.19.5.14  "Singletonization" and tagging   bj-snsetex 37798
                  ​*21.19.5.15  Tuples of classes   bj-cproj 37825
                  ​*21.19.5.16  Set theory: elementary operations relative to a universe   bj-rcleqf 37860
                  ​*21.19.5.17  Axioms for finite unions   bj-abex 37865
                  ​*21.19.5.18  Set theory: miscellaneous   eleq2w2ALT 37882
                  ​*21.19.5.19  Axioms of separation and replacement   bj-axnul 37908
                  ​*21.19.5.20  Evaluation at a class   bj-evaleq 37912
                  21.19.5.21  Elementwise operations   celwise 37920
                  ​*21.19.5.22  Elementwise intersection (families of sets induced on a subset)   bj-rest00 37922
                  21.19.5.23  Moore collections (complements)   bj-raldifsn 37941
                  21.19.5.24  Maps-to notation for functions with three arguments   bj-0nelmpt 37957
                  ​*21.19.5.25  Currying   csethom 37961
                  ​*21.19.5.26  Setting components of extensible structures   cstrset 37973
            ​*21.19.6  Extended real and complex numbers, real and complex projective lines   bj-nfald 37976
                  21.19.6.1  Complements on class abstractions of ordered pairs and binary relations   bj-nfald 37976
                  ​*21.19.6.2  Identity relation (complements)   bj-opabssvv 37991
                  ​*21.19.6.3  Functionalized identity (diagonal in a Cartesian square)   cdiag2 38013
                  ​*21.19.6.4  Direct image and inverse image   cimdir 38019
                  ​*21.19.6.5  Extended numbers and projective lines as sets   cfractemp 38037
                  ​*21.19.6.6  Addition and opposite   caddcc 38078
                  ​*21.19.6.7  Order relation on the extended reals   cltxr 38082
                  ​*21.19.6.8  Argument, multiplication and inverse   carg 38084
                  21.19.6.9  The canonical bijection from the finite ordinals   ciomnn 38090
                  21.19.6.10  Divisibility   cnnbar 38101
            ​*21.19.7  Monoids   bj-smgrpssmgm 38109
                  ​*21.19.7.1  Finite sums in monoids   cfinsum 38124
            ​*21.19.8  Affine, Euclidean, and Cartesian geometry   bj-fvimacnv0 38127
                  *21.19.8.1  Real vector spaces   bj-fvimacnv0 38127
                  ​*21.19.8.2  Complex numbers (supplements)   bj-subcom 38149
                  ​*21.19.8.3  Barycentric coordinates   bj-bary1lem 38151
            21.19.9  Monoid of endomorphisms   cend 38154
      ​21.20  Mathbox for Jim Kingdon
            21.20.1  Circle constant   taupilem3 38160
            21.20.2  Number theory   dfgcd3 38165
            21.20.3  Real numbers   irrdifflemf 38166
      ​21.21  Mathbox for ML
            21.21.1  Miscellaneous   csbrecsg 38171
            21.21.2  Cartesian exponentiation   cfinxp 38226
            21.21.3  Topology   iunctb2 38246
                  ​*21.21.3.1  Pi-base theorems   pibp16 38256
      ​21.22  Mathbox for Wolf Lammen
            21.22.1  1. Bootstrapping   wl-section-boot 38265
            21.22.2  Implication chains   wl-section-impchain 38289
            21.22.3  Theorems around the conditional operator   wl-ifp-ncond1 38307
            21.22.4  Alternative development of hadd, cadd   wl-df-3xor 38311
            21.22.5  An alternative axiom ~ ax-13   ax-wl-13v 38336
            21.22.6  Bootstrapping set theory with classes   wl-cleq-0 38338
            21.22.7  Other stuff   wl-mps 38359
      ​21.23  Mathbox for Brendan Leahy
      ​21.24  Mathbox for Thomas van Maaren
            ​*21.24.1  The language of Propositional Calculus.   cpropvar 38553
      ​21.25  Mathbox for Jeff Madsen
            21.25.1  Logic and set theory   unirep 38568
            21.25.2  Real and complex numbers; integers   filbcmb 38594
            21.25.3  Sequences and sums   sdclem2 38596
            21.25.4  Topology   subspopn 38606
            21.25.5  Metric spaces   metf1o 38609
            21.25.6  Continuous maps and homeomorphisms   constcncf 38616
            21.25.7  Boundedness   ctotbnd 38620
            21.25.8  Isometries   cismty 38652
            21.25.9  Heine-Borel Theorem   heibor1lem 38663
            21.25.10  Banach Fixed Point Theorem   bfplem1 38676
            21.25.11  Euclidean space   crrn 38679
            21.25.12  Intervals (continued)   ismrer1 38692
            ​*21.25.13  Operation properties   cass 38696
            21.25.14  Groups and related structures   cmagm 38702
            21.25.15  Group homomorphism and isomorphism   cghomOLD 38737
            21.25.16  Rings   crngo 38748
            21.25.17  Division Rings   cdrng 38802
            21.25.18  Ring homomorphisms   crngohom 38814
            21.25.19  Commutative rings   ccm2 38843
            21.25.20  Ideals   cidl 38861
            21.25.21  Prime rings and integral domains   cprrng 38900
            21.25.22  Ideal generators   cigen 38913
      ​21.26  Mathbox for Giovanni Mascellani
            ​*21.26.1  Tools for automatic proof building   efald2 38932
            ​*21.26.2  Tseitin axioms   fald 38981
            ​*21.26.3  Equality deductions   iuneq2f 39008
            ​*21.26.4  Miscellanea   orcomdd 39019
      ​21.27  Mathbox for Peter Mazsa
            21.27.1  Notations   cxrn 39026
            21.27.2  Preparatory theorems   el2v1 39081
            21.27.3  Range Cartesian product   df-xrn 39232
            21.27.4  Relations   df-rels 39292
            21.27.5  Quotient map (coset map)   df-qmap 39298
            21.27.6  Lifts, shifts, successor, and predecessor   df-adjliftmap 39307
            21.27.7  Cosets by ` R `   df-coss 39353
            21.27.8  Subset relations   df-ssr 39430
            21.27.9  Reflexivity   df-refs 39442
            21.27.10  Converse reflexivity   df-cnvrefs 39457
            21.27.11  Symmetry   df-syms 39474
            21.27.12  Reflexivity and symmetry   symrefref2 39499
            21.27.13  Transitivity   df-trs 39508
            21.27.14  Equivalence relations   df-eqvrels 39520
            21.27.15  Redundancy   df-redunds 39559
            21.27.16  Domain quotients   df-dmqss 39574
            21.27.17  Equivalence relations on domain quotients   df-ers 39600
            21.27.18  Functions   df-funss 39617
            21.27.19  Disjoints vs. converse functions   df-disjss 39640
            21.27.20  Antisymmetry   df-antisymrel 39715
            21.27.21  Partitions: disjoints on domain quotients   df-parts 39720
            21.27.22  Partition-Equivalence Theorems   disjim 39736
            21.27.23  Type-safe Partition-Equivalence: PetParts, PetErs, Pet2Parts, Pet2Ers   df-petparts 39820
      ​21.28  Mathbox for Rodolfo Medina
            21.28.1  Partitions   prtlem60 39830
      ​​*21.29  Mathbox for Norm Megill
            *21.29.1  Obsolete schemes ax-c4,c5,c7,c10,c11,c11n,c15,c9,c14,c16   ax-c5 39860
            ​*21.29.2  Rederive new axioms ax-4, ax-10, ax-6, ax-12, ax-13 from old   axc5 39870
            ​*21.29.3  Legacy theorems using obsolete axioms   ax5ALT 39884
            21.29.4  Experiments with weak deduction theorem   elimhyps 39938
            21.29.5  Miscellanea   cnaddcom 39949
            21.29.6  Atoms, hyperplanes, and covering in a left vector space (or module)   clsa 39951
            21.29.7  Functionals and kernels of a left vector space (or module)   clfn 40034
            21.29.8  Opposite rings and dual vector spaces   cld 40100
            21.29.9  Ortholattices and orthomodular lattices   cops 40149
            21.29.10  Atomic lattices with covering property   ccvr 40239
            21.29.11  Hilbert lattices   chlt 40327
            21.29.12  Projective geometries based on Hilbert lattices   clln 40468
            21.29.13  Construction of a vector space from a Hilbert lattice   cdlema1N 40768
            21.29.14  Construction of involution and inner product from a Hilbert lattice   clpoN 42457
      ​21.30  Mathbox for metakunt
            21.30.1  Commutative Semiring   ccsrg 42939
            21.30.2  General helpful statements   rhmzrhval 42942
            21.30.3  Some gcd and lcm results   12gcd5e1 42973
            21.30.4  Least common multiple inequality theorem   3factsumint1 42991
            21.30.5  Logarithm inequalities   3exp7 43023
            21.30.6  Miscellaneous results for AKS formalisation   intlewftc 43031
            21.30.7  Sticks and stones   sticksstones1 43116
            21.30.8  Continuation AKS   aks6d1c6lem1 43140
      ​21.31  Mathbox for Luke Murphy
            21.31.1  Solutions of quadratic equations   quadfac 43175
            21.31.2  April Fool's theorem   25or6to4 43176
      ​21.32  Mathbox for Steven Nguyen
            21.32.1  Utility theorems   jarrii 43177
            ​*21.32.2  Arithmetic theorems   c0exALT 43223
            21.32.3  Exponents and divisibility   oexpreposd 43301
            21.32.4  Trigonometry and Calculus   tanhalfpim 43328
            ​*21.32.5  Independence of ax-mulcom   cresub 43344
            21.32.6  Structures   sn-base0 43487
            ​*21.32.7  Projective spaces   cprjsp 43551
            21.32.8  Basic reductions for Fermat's Last Theorem   dffltz 43584
            ​*21.32.9  Exemplar theorems   iddii 43614
                  ​*21.32.9.1  Standard replacements of ax-10 , ax-11 , ax-12   nfa1w 43625
      ​21.33  Mathbox for Igor Ieskov
      ​21.34  Mathbox for OpenAI
      ​21.35  Mathbox for Stefan O'Rear
            21.35.1  Additional elementary logic and set theory   moxfr 43641
            21.35.2  Additional theory of functions   imaiinfv 43642
            21.35.3  Additional topology   elrfi 43643
            21.35.4  Characterization of closure operators. Kuratowski closure axioms   ismrcd1 43647
            21.35.5  Algebraic closure systems   cnacs 43651
            21.35.6  Miscellanea 1. Map utilities   constmap 43662
            21.35.7  Miscellanea for polynomials   mptfcl 43669
            21.35.8  Multivariate polynomials over the integers   cmzpcl 43670
            21.35.9  Miscellanea for Diophantine sets 1   coeq0i 43702
            21.35.10  Diophantine sets 1: definitions   cdioph 43704
            21.35.11  Diophantine sets 2 miscellanea   ellz1 43716
            21.35.12  Diophantine sets 2: union and intersection. Monotone Boolean algebra   diophin 43721
            21.35.13  Diophantine sets 3: construction   diophrex 43724
            21.35.14  Diophantine sets 4 miscellanea   2sbcrex 43733
            21.35.15  Diophantine sets 4: Quantification   rexrabdioph 43739
            21.35.16  Diophantine sets 5: Arithmetic sets   rabdiophlem1 43746
            21.35.17  Diophantine sets 6: reusability. renumbering of variables   eldioph4b 43756
            21.35.18  Pigeonhole Principle and cardinality helpers   fphpd 43761
            21.35.19  A non-closed set of reals is infinite   rencldnfilem 43765
            21.35.20  Lagrange's rational approximation theorem   irrapxlem1 43767
            21.35.21  Pell equations 1: A nontrivial solution always exists   pellexlem1 43774
            21.35.22  Pell equations 2: Algebraic number theory of the solution set   csquarenn 43781
            21.35.23  Pell equations 3: characterizing fundamental solution   infmrgelbi 43823
            ​*21.35.24  Logarithm laws generalized to an arbitrary base   reglogcl 43835
            21.35.25  Pell equations 4: the positive solution group is infinite cyclic   pellfund14 43843
            21.35.26  X and Y sequences 1: Definition and recurrence laws   crmx 43845
            21.35.27  Ordering and induction lemmas for the integers   monotuz 43886
            21.35.28  X and Y sequences 2: Order properties   rmxypos 43892
            21.35.29  Congruential equations   congtr 43910
            21.35.30  Alternating congruential equations   acongid 43920
            21.35.31  Additional theorems on integer divisibility   coprmdvdsb 43930
            21.35.32  X and Y sequences 3: Divisibility properties   jm2.18 43933
            21.35.33  X and Y sequences 4: Diophantine representability of Y   jm2.27a 43950
            21.35.34  X and Y sequences 5: Diophantine representability of X, ^, _C   rmxdiophlem 43960
            21.35.35  Uncategorized stuff not associated with a major project   setindtr 43969
            21.35.36  More equivalents of the Axiom of Choice   axac10 43978
            21.35.37  Finitely generated left modules   clfig 44012
            21.35.38  Noetherian left modules I   clnm 44020
            21.35.39  Addenda for structure powers   pwssplit4 44034
            21.35.40  Every set admits a group structure iff choice   unxpwdom3 44040
            21.35.41  Noetherian rings and left modules II   clnr 44054
            21.35.42  Hilbert's Basis Theorem   cldgis 44066
            21.35.43  Additional material on polynomials [DEPRECATED]   cmnc 44076
            21.35.44  Degree and minimal polynomial of algebraic numbers   cdgraa 44085
            21.35.45  Algebraic integers I   citgo 44102
            21.35.46  Endomorphism algebra   cmend 44116
            21.35.47  Cyclic groups and order   idomodle 44136
            21.35.48  Cyclotomic polynomials   ccytp 44142
            21.35.49  Miscellaneous topology   fgraphopab 44148
      ​21.36  Mathbox for Noam Pasman
      ​21.37  Mathbox for Jon Pennant
      ​21.38  Mathbox for Richard Penner
            21.38.1  Set Theory and Ordinal Numbers   uniel 44162
            21.38.2  Natural addition of Cantor normal forms   oawordex2 44271
            21.38.3  Surreal Contributions   abeqabi 44352
            21.38.4  Short Studies   nlimsuc 44385
                  21.38.4.1  Additional work on conditional logical operator   ifpan123g 44403
                  21.38.4.2  Sophisms   rp-fakeimass 44456
                  ​*21.38.4.3  Finite Sets   rp-isfinite5 44461
                  21.38.4.4  General Observations   intabssd 44463
                  21.38.4.5  Infinite Sets   pwelg 44504
                  ​*21.38.4.6  Finite intersection property   fipjust 44509
                  21.38.4.7  RP ADDTO: Subclasses and subsets   rababg 44518
                  21.38.4.8  RP ADDTO: The intersection of a class   elinintab 44519
                  21.38.4.9  RP ADDTO: Theorems requiring subset and intersection existence   elinintrab 44521
                  21.38.4.10  RP ADDTO: Relations   xpinintabd 44524
                  ​*21.38.4.11  RP ADDTO: Functions   elmapintab 44540
                  ​*21.38.4.12  RP ADDTO: Finite induction (for finite ordinals)   cnvcnvintabd 44544
                  21.38.4.13  RP ADDTO: First and second members of an ordered pair   elcnvlem 44545
                  21.38.4.14  RP ADDTO: The reflexive and transitive properties of relations   undmrnresiss 44548
                  21.38.4.15  RP ADDTO: Basic properties of closures   cleq2lem 44552
                  21.38.4.16  RP REPLACE: Definitions and basic properties of transitive closures   trcleq2lemRP 44574
                  ​*21.38.4.17  Additions for square root; absolute value   sqrtcvallem1 44575
            21.38.5  Additional statements on relations and subclasses   al3im 44591
                  21.38.5.1  Transitive relations (not to be confused with transitive classes)   trrelind 44609
                  21.38.5.2  Reflexive closures   crcl 44616
                  ​*21.38.5.3  Finite relationship composition   relexp2 44621
                  21.38.5.4  Transitive closure of a relation   dftrcl3 44664
                  ​*21.38.5.5  Adapted from Frege   frege77d 44690
            ​*21.38.6  Propositions from _Begriffsschrift_   dfxor4 44710
                  *21.38.6.1  _Begriffsschrift_ Chapter I   dfxor4 44710
                  ​*21.38.6.2  _Begriffsschrift_ Notation hints   whe 44716
                  21.38.6.3  _Begriffsschrift_ Chapter II Implication   ax-frege1 44734
                  21.38.6.4  _Begriffsschrift_ Chapter II Implication and Negation   axfrege28 44773
                  ​*21.38.6.5  _Begriffsschrift_ Chapter II with logical equivalence   axfrege52a 44800
                  21.38.6.6  _Begriffsschrift_ Chapter II with equivalence of sets   axfrege52c 44831
                  ​*21.38.6.7  _Begriffsschrift_ Chapter II with equivalence of classes   frege53c 44858
                  ​*21.38.6.8  _Begriffsschrift_ Chapter III Properties hereditary in a sequence   dffrege69 44876
                  ​*21.38.6.9  _Begriffsschrift_ Chapter III Following in a sequence   dffrege76 44883
                  ​*21.38.6.10  _Begriffsschrift_ Chapter III Member of sequence   dffrege99 44906
                  ​*21.38.6.11  _Begriffsschrift_ Chapter III Single-valued procedures   dffrege115 44922
            ​*21.38.7  Exploring Topology via Seifert and Threlfall   enrelmap 44941
                  *21.38.7.1  Equinumerosity of sets of relations and maps   enrelmap 44941
                  ​*21.38.7.2  Generic Pseudoclosure Spaces, Pseudointerior Spaces, and Pseudoneighborhoods   or3or 44967
                  ​*21.38.7.3  Generic Neighborhood Spaces   gneispa 45074
            ​*21.38.8  Exploring Higher Homotopy via Kerodon   k0004lem1 45091
                  *21.38.8.1  Simplicial Sets   k0004lem1 45091
      ​21.39  Mathbox for Stanislas Polu
            21.39.1  IMO Problems   wwlemuld 45100
                  21.39.1.1  IMO 1972 B2   wwlemuld 45100
            ​*21.39.2  INT Inequalities Proof Generator   int-addcomd 45117
            ​*21.39.3  N-Digit Addition Proof Generator   unitadd 45139
            21.39.4  AM-GM (for k = 2,3,4)   gsumws3 45140
      ​21.40  Mathbox for Rohan Ridenour
            21.40.1  Misc   spALT 45145
            21.40.2  Monoid rings   cmnring 45153
            21.40.3  Shorter primitive equivalent of ax-groth   gru0eld 45171
                  21.40.3.1  Grothendieck universes are closed under collection   gru0eld 45171
                  21.40.3.2  Minimal universes   ismnu 45189
                  21.40.3.3  Primitive equivalent of ax-groth   expandan 45216
      ​21.41  Mathbox for Steve Rodriguez
            21.41.1  Miscellanea   nanorxor 45233
            21.41.2  Ratio test for infinite series convergence and divergence   dvgrat 45240
            21.41.3  Multiples   reldvds 45243
            21.41.4  Function operations   caofcan 45251
            21.41.5  Calculus   lhe4.4ex1a 45257
            21.41.6  The generalized binomial coefficient operation   cbcc 45264
            21.41.7  Binomial series   uzmptshftfval 45274
      ​21.42  Mathbox for Andrew Salmon
            21.42.1  Principia Mathematica * 10   pm10.12 45286
            21.42.2  Principia Mathematica * 11   2alanimi 45300
            21.42.3  Predicate Calculus   sbeqal1 45326
            21.42.4  Principia Mathematica * 13 and * 14   pm13.13a 45335
            21.42.5  Set Theory   elnev 45365
            21.42.6  Arithmetic   addcomgi 45382
            21.42.7  Geometry   cplusr 45383
      ​​*21.43  Mathbox for Alan Sare
            21.43.1  Auxiliary theorems for the Virtual Deduction tool   idiALT 45405
            21.43.2  Supplementary unification deductions   bi1imp 45409
            21.43.3  Conventional Metamath proofs, some derived from VD proofs   iidn3 45428
            21.43.4  What is Virtual Deduction?   wvd1 45496
            21.43.5  Virtual Deduction Theorems   df-vd1 45497
            21.43.6  Theorems proved using Virtual Deduction   trsspwALT 45744
            21.43.7  Theorems proved using Virtual Deduction with mmj2 assistance   simplbi2VD 45772
            21.43.8  Virtual Deduction transcriptions of textbook proofs   sb5ALTVD 45839
            21.43.9  Theorems proved using conjunction-form Virtual Deduction   elpwgdedVD 45843
            21.43.10  Theorems with a VD proof in conventional notation derived from a VD proof   suctrALT3 45850
            ​*21.43.11  Theorems with a proof in conventional notation derived from a VD proof   notnotrALT2 45853
      ​21.44  Mathbox for Eric Schmidt
            21.44.1  Miscellany   rspesbcd 45864
            21.44.2  Study of dfbi1ALT   dfbi1ALTa 45866
            21.44.3  Relation-preserving functions   wrelp 45869
            21.44.4  Orbits   orbitex 45882
            21.44.5  Well-founded sets   trwf 45886
            21.44.6  Absoluteness in transitive models   ralabso 45895
            21.44.7  Lemmas for showing axioms hold in models   traxext 45904
            21.44.8  The class of well-founded sets is a model for ZFC   wfaxext 45920
            21.44.9  Permutation models   brpermmodel 45930
            21.44.10  Isomorphism of finite ordinals and non-negative integers   hashnna 45946
      ​21.45  Mathbox for Glauco Siliprandi
            21.45.1  Miscellanea   evth2f 45953
            21.45.2  Functions   fnresdmss 46104
            21.45.3  Ordering on real numbers - Real and complex numbers basic operations   sub2times 46210
            21.45.4  Real intervals   gtnelioc 46425
            21.45.5  Finite sums   fsummulc1f 46505
            21.45.6  Finite multiplication of numbers and finite multiplication of functions   fmul01 46514
            21.45.7  Limits   clim1fr1 46535
                  21.45.7.1  Inferior limit (lim inf)   clsi 46683
                  ​*21.45.7.2  Limits for sequences of extended real numbers   clsxlim 46750
            21.45.8  Trigonometry   coseq0 46796
            21.45.9  Continuous Functions   mulcncff 46802
            21.45.10  Derivatives   dvsinexp 46843
            21.45.11  Integrals   itgsin0pilem1 46882
            21.45.12  Stone Weierstrass theorem - real version   stoweidlem1 46933
            21.45.13  Wallis' product for π   wallispilem1 46997
            21.45.14  Stirling's approximation formula for ` n ` factorial   stirlinglem1 47006
            21.45.15  Dirichlet kernel   dirkerval 47023
            21.45.16  Fourier Series   fourierdlem1 47040
            21.45.17  e is transcendental   elaa2lem 47165
            21.45.18  n-dimensional Euclidean space   rrxtopn 47216
            21.45.19  Basic measure theory   csalg 47240
                  ​*21.45.19.1  σ-Algebras   csalg 47240
                  21.45.19.2  Sum of nonnegative extended reals   csumge0 47294
                  ​*21.45.19.3  Measures   cmea 47381
                  ​*21.45.19.4  Outer measures and Caratheodory's construction   come 47421
                  ​*21.45.19.5  Lebesgue measure on n-dimensional Real numbers   covoln 47468
                  ​*21.45.19.6  Measurable functions   csmblfn 47627
      ​21.46  Mathbox for Saveliy Skresanov
            21.46.1  Ceva's theorem   sigarval 47782
            21.46.2  Simple groups   simpcntrab 47802
      ​21.47  Mathbox for Ender Ting
            21.47.1  Interesting facts   et-ltneverrefl 47803
            21.47.2  Increasing sequences and subsequences   ormklocald 47808
            21.47.3  Scratchpad for number theory   evenwodadd 47833
            21.47.4  Scratchpad for math on real numbers   squeezedltsq 47834
            21.47.5  Turing Machine Finite Reach theorem   tmachlem-extapes 47866
      ​21.48  Mathbox for Jarvin Udandy
      ​21.49  Mathbox for Adhemar
            ​*21.49.1  Minimal implicational calculus   adh-minim 47993
      ​21.50  Mathbox for Alexander van der Vekens
            21.50.1  General auxiliary theorems (1)   n0nsn2el 48017
                  21.50.1.1  Unordered and ordered pairs - extension for singletons   n0nsn2el 48017
                  21.50.1.2  Unordered and ordered pairs - extension for unordered pairs   elprneb 48021
                  21.50.1.3  Unordered and ordered pairs - extension for ordered pairs   oppr 48022
                  21.50.1.4  Relations - extension   eubrv 48027
                  21.50.1.5  Definite description binder (inverted iota) - extension   iota0def 48030
                  21.50.1.6  Functions - extension   fveqvfvv 48032
            21.50.2  Alternative for Russell's definition of a description binder   caiota 48075
            21.50.3  Double restricted existential uniqueness   r19.32 48090
                  21.50.3.1  Restricted quantification (extension)   r19.32 48090
                  21.50.3.2  Restricted uniqueness and "at most one" quantification   reuf1odnf 48099
                  21.50.3.3  Analogs to Existential uniqueness (double quantification)   2reu3 48102
                  21.50.3.4  Additional theorems for double restricted existential uniqueness   2reu8i 48105
            ​*21.50.4  Alternative definitions of function and operation values   wdfat 48108
                  21.50.4.1  Restricted quantification (extension)   ralbinrald 48114
                  21.50.4.2  The universal class (extension)   nvelim 48115
                  21.50.4.3  Introduce the Axiom of Power Sets (extension)   alneu 48116
                  21.50.4.4  Predicate "defined at"   dfateq12d 48118
                  21.50.4.5  Alternative definition of the value of a function   dfafv2 48124
                  21.50.4.6  Alternative definition of the value of an operation   aoveq123d 48170
            ​*21.50.5  Alternative definitions of function values (2)   cafv2 48200
            21.50.6  General auxiliary theorems (2)   an4com24 48260
                  21.50.6.1  Logical conjunction - extension   an4com24 48260
                  21.50.6.2  Abbreviated conjunction and disjunction of three wff's - extension   3an4ancom24 48261
                  21.50.6.3  Negated membership (alternative)   cnelbr 48263
                  21.50.6.4  The empty set - extension   ralralimp 48270
                  21.50.6.5  Indexed union and intersection - extension   otiunsndisjX 48271
                  21.50.6.6  Functions - extension   fvifeq 48272
                  21.50.6.7  Maps-to notation - extension   fvmptrab 48284
                  21.50.6.8  Subtraction - extension   cnambpcma 48286
                  21.50.6.9  Ordering on reals (cont.) - extension   leaddsuble 48289
                  21.50.6.10  Imaginary and complex number properties - extension   readdcnnred 48295
                  21.50.6.11  Nonnegative integers (as a subset of complex numbers) - extension   nn0resubcl 48300
                  21.50.6.12  Integers (as a subset of complex numbers) - extension   zgeltp1eq 48301
                  21.50.6.13  Decimal arithmetic - extension   1t10e1p1e11 48302
                  21.50.6.14  Upper sets of integers - extension   eluzge0nn0 48304
                  21.50.6.15  Infinity and the extended real number system (cont.) - extension   nltle2tri 48305
                  21.50.6.16  Finite intervals of integers - extension   ssfz12 48306
                  21.50.6.17  Half-open integer ranges - extension   fzopred 48315
                  21.50.6.18  The floor and ceiling functions - extension   2ltceilhalf 48324
                  21.50.6.19  The modulo (remainder) operation - extension   fldivmod 48336
                  21.50.6.20  The infinite sequence builder "seq"   smonoord 48369
                  21.50.6.21  Integer powers - extension   2timesltsq 48370
                  21.50.6.22  Finite and infinite sums - extension   fsummsndifre 48372
                  21.50.6.23  The divides relation - extension   nndivides2 48376
                  21.50.6.24  Extensible structures - extension   setsidel 48380
            ​*21.50.7  Preimages of function values   preimafvsnel 48383
            ​*21.50.8  Partitions of real intervals   ciccp 48417
            21.50.9  Shifting functions with an integer range domain   fargshiftfv 48443
            21.50.10  Words over a set (extension)   lswn0 48448
                  21.50.10.1  Last symbol of a word - extension   lswn0 48448
            21.50.11  Unordered pairs   wich 48449
                  21.50.11.1  Interchangeable setvar variables   wich 48449
                  21.50.11.2  Set of unordered pairs   sprid 48478
                  ​*21.50.11.3  Proper (unordered) pairs   prpair 48505
                  21.50.11.4  Set of proper unordered pairs   cprpr 48516
            21.50.12  Number theory (extension)   nprmmul1 48531
                  21.50.12.1  Properties of non-prime numbers   nprmmul1 48531
                  ​*21.50.12.2  Fermat numbers   cfmtno 48534
                  ​*21.50.12.3  Mersenne primes   m2prm 48598
                  21.50.12.4  Proth's theorem   modexp2m1d 48619
                  21.50.12.5  The prime-counting function according to Ján Mináč   nprmdvdsfacm1lem1 48627
                  21.50.12.6  Solutions of quadratic equations   quad1 48640
            ​*21.50.13  Even and odd numbers   ceven 48644
                  21.50.13.1  Definitions and basic properties   ceven 48644
                  21.50.13.2  Alternate definitions using the "divides" relation   dfeven2 48669
                  21.50.13.3  Alternate definitions using the "modulo" operation   dfeven3 48678
                  21.50.13.4  Alternate definitions using the "gcd" operation   iseven5 48684
                  21.50.13.5  Theorems of part 5 revised   zneoALTV 48689
                  21.50.13.6  Theorems of part 6 revised   odd2np1ALTV 48694
                  21.50.13.7  Theorems of AV's mathbox revised   0evenALTV 48708
                  21.50.13.8  Additional theorems   epoo 48723
                  21.50.13.9  Perfect Number Theorem (revised)   perfectALTVlem1 48741
            21.50.14  Number theory (extension 2)   cfppr 48744
                  ​*21.50.14.1  Fermat pseudoprimes   cfppr 48744
                  ​*21.50.14.2  Goldbach's conjectures   cgbe 48765
            21.50.15  Graph theory (extension)   cclnbgr 48838
                  21.50.15.1  Closed neighborhood of a vertex   cclnbgr 48838
                  ​*21.50.15.2  Semiclosed and semiopen neighborhoods (experimental)   dfsclnbgr2 48866
                  21.50.15.3  Induced subgraphs   cisubgr 48880
                  ​*21.50.15.4  Isomorphisms of graphs   cgrisom 48894
                  ​*21.50.15.5  Triangles in graphs   cgrtri 48957
                  ​*21.50.15.6  Star graphs   cstgr 48971
                  ​*21.50.15.7  Local isomorphisms of graphs   cgrlim 48996
                  ​*21.50.15.8  Generalized Petersen graphs   cgpg 49060
                  21.50.15.9  Loop-free graphs - extension   1hegrlfgr 49152
                  21.50.15.10  Walks - extension   cupwlks 49153
                  21.50.15.11  Edges of graphs expressed as sets of unordered pairs   upgredgssspr 49163
            21.50.16  Monoids (extension)   ovn0dmfun 49176
                  21.50.16.1  Auxiliary theorems   ovn0dmfun 49176
                  21.50.16.2  Magmas, Semigroups and Monoids (extension)   plusfreseq 49183
                  21.50.16.3  Examples and counterexamples for magmas, semigroups and monoids (extension)   opmpoismgm 49186
                  21.50.16.4  Group sum operation (extension 1)   gsumsplit2f 49199
            ​*21.50.17  Magmas and internal binary operations (alternate approach)   ccllaw 49202
                  *21.50.17.1  Laws for internal binary operations   ccllaw 49202
                  ​*21.50.17.2  Internal binary operations   cintop 49215
                  21.50.17.3  Alternative definitions for magmas and semigroups   cmgm2 49234
            21.50.18  Rings (extension)   lmod0rng 49248
                  21.50.18.1  Nonzero rings (extension)   lmod0rng 49248
                  21.50.18.2  Ideals as non-unital rings   lidldomn1 49250
                  21.50.18.3  The non-unital ring of even integers   0even 49256
                  21.50.18.4  A constructed not unital ring   cznrnglem 49278
                  ​*21.50.18.5  The category of non-unital rings (alternate definition)   crngcALTV 49282
                  ​*21.50.18.6  The category of (unital) rings (alternate definition)   cringcALTV 49306
            ​*21.50.19  Prime rings (and integral domains)   cprmrng 49353
            21.50.20  Basic algebraic structures (extension)   eliunxp2 49368
                  21.50.20.1  Auxiliary theorems   eliunxp2 49368
                  21.50.20.2  The binomial coefficient operation (extension)   bcpascm1 49385
                  21.50.20.3  The ` ZZ `-module ` ZZ X. ZZ `   zlmodzxzlmod 49388
                  21.50.20.4  Group sum operation (extension 2)   mgpsumunsn 49395
                  21.50.20.5  Symmetric groups (extension)   exple2lt6 49398
                  21.50.20.6  Divisibility (extension)   invginvrid 49401
                  21.50.20.7  The support of functions (extension)   rmsupp0 49402
                  21.50.20.8  Finitely supported functions (extension)   rmsuppfi 49406
                  21.50.20.9  Left modules (extension)   lmodvsmdi 49413
                  21.50.20.10  Associative algebras (extension)   assaascl0 49415
                  21.50.20.11  Univariate polynomials (extension)   ply1vr1smo 49417
                  21.50.20.12  Univariate polynomials (examples)   linply1 49427
            21.50.21  Linear algebra (extension)   cdmatalt 49430
                  ​*21.50.21.1  The subalgebras of diagonal and scalar matrices (extension)   cdmatalt 49430
                  ​*21.50.21.2  Linear combinations   clinc 49438
                  ​*21.50.21.3  Linear independence   clininds 49474
                  21.50.21.4  Simple left modules and the ` ZZ `-module   lmod1lem1 49521
                  21.50.21.5  Differences between (left) modules and (left) vector spaces   lvecpsslmod 49541
            21.50.22  Complexity theory   suppdm 49544
                  21.50.22.1  Auxiliary theorems   suppdm 49544
                  21.50.22.2  Even and odd integers   nn0onn0ex 49557
                  21.50.22.3  The natural logarithm on complex numbers (extension)   logcxp0 49569
                  21.50.22.4  Division of functions   cfdiv 49571
                  21.50.22.5  Upper bounds   cbigo 49581
                  21.50.22.6  Logarithm to an arbitrary base (extension)   rege1logbrege0 49592
                  ​*21.50.22.7  The binary logarithm   fldivexpfllog2 49599
                  21.50.22.8  Binary length   cblen 49603
                  ​*21.50.22.9  Digits   cdig 49629
                  21.50.22.10  Nonnegative integer as sum of its shifted digits   dignn0flhalflem1 49649
                  21.50.22.11  Algorithms for the multiplication of nonnegative integers   nn0mulfsum 49658
                  ​*21.50.22.12  N-ary functions   cnaryf 49660
                  ​*21.50.22.13  The Ackermann function   citco 49691
            21.50.23  Elementary geometry (extension)   fv1prop 49733
                  21.50.23.1  Auxiliary theorems   fv1prop 49733
                  21.50.23.2  Real euclidean space of dimension 2   rrx2pxel 49745
                  21.50.23.3  Spheres and lines in real Euclidean spaces   cline 49761
      ​21.51  Mathbox for Zhi Wang
            21.51.1  Propositional calculus   imbi12d2 49823
            21.51.2  Predicate calculus with equality   dtrucor3 49831
                  21.51.2.1  Axiom scheme ax-5 (Distinctness)   dtrucor3 49831
            21.51.3  ZF Set Theory - start with the Axiom of Extensionality   ralbidb 49832
                  21.51.3.1  Restricted quantification   ralbidb 49832
                  21.51.3.2  The universal class   reuxfr1dd 49839
                  21.51.3.3  The empty set   ssdisjd 49840
                  21.51.3.4  Unordered and ordered pairs   vsn 49844
                  21.51.3.5  The union of a class   unilbss 49850
                  21.51.3.6  Indexed union and intersection   iuneq0 49851
            21.51.4  ZF Set Theory - add the Axiom of Replacement   inpw 49857
                  21.51.4.1  Theorems requiring subset and intersection existence   inpw 49857
            21.51.5  ZF Set Theory - add the Axiom of Power Sets   opth1neg 49858
                  21.51.5.1  Ordered pair theorem   opth1neg 49858
                  21.51.5.2  Ordered-pair class abstractions (cont.)   brab2dd 49860
                  21.51.5.3  Relations   iinxp 49863
                  21.51.5.4  Functions   mof0 49870
                  21.51.5.5  Operations   ovsng 49890
            21.51.6  ZF Set Theory - add the Axiom of Union   fonex 49899
                  21.51.6.1  Relations and functions (cont.)   fonex 49899
                  21.51.6.2  First and second members of an ordered pair   eloprab1st2nd 49900
                  21.51.6.3  Function transposition   resinsnlem 49901
                  21.51.6.4  Infinite Cartesian products   ixpv 49920
                  21.51.6.5  Equinumerosity   fvconst0ci 49921
            21.51.7  Order sets   iccin 49926
                  21.51.7.1  Real number intervals   iccin 49926
            21.51.8  Extensible structures   slotresfo 49929
                  21.51.8.1  Basic definitions   slotresfo 49929
            21.51.9  Moore spaces   mreuniss 49930
            ​*21.51.10  Topology   clduni 49931
                  21.51.10.1  Closure and interior   clduni 49931
                  21.51.10.2  Neighborhoods   neircl 49935
                  21.51.10.3  Subspace topologies   restcls2lem 49943
                  21.51.10.4  Limits and continuity in topological spaces   cnneiima 49947
                  21.51.10.5  Topological definitions using the reals   iooii 49948
                  21.51.10.6  Separated sets   sepnsepolem1 49952
                  21.51.10.7  Separated spaces: T0, T1, T2 (Hausdorff) ...   isnrm4 49961
            21.51.11  Preordered sets and directed sets using extensible structures   isprsd 49985
            21.51.12  Posets and lattices using extensible structures   lubeldm2 49986
                  21.51.12.1  Posets   lubeldm2 49986
                  21.51.12.2  Lattices   toslat 50012
                  21.51.12.3  Subset order structures   intubeu 50014
            21.51.13  Rings   elmgpcntrd 50035
                  21.51.13.1  Multiplicative Group   elmgpcntrd 50035
            21.51.14  Associative algebras   asclelbasALT 50036
                  21.51.14.1  Definition and basic properties   asclelbasALT 50036
            21.51.15  Categories   homf0 50039
                  21.51.15.1  Categories   homf0 50039
                  21.51.15.2  Opposite category   oppccatb 50046
                  21.51.15.3  Monomorphisms and epimorphisms   idmon 50050
                  21.51.15.4  Sections, inverses, isomorphisms   sectrcl 50052
                  21.51.15.5  Isomorphic objects   cicfn 50072
                  21.51.15.6  Subcategories   dmdm 50083
                  21.51.15.7  Functors   reldmfunc 50105
                  21.51.15.8  Opposite functors   coppf 50152
                  21.51.15.9  Full & faithful functors   imasubc 50181
                  21.51.15.10  Universal property   upciclem1 50196
                  21.51.15.11  Natural transformations and the functor category   isnatd 50253
                  21.51.15.12  Initial, terminal and zero objects of a category   initoo2 50262
                  21.51.15.13  Product of categories   reldmxpc 50276
                  21.51.15.14  Swap functors   cswapf 50289
                  21.51.15.15  Functor evaluation   oppc1stflem 50317
                  21.51.15.16  Transposed curry functors   cofuswapfcl 50323
                  21.51.15.17  Constant functors   diag1 50334
                  21.51.15.18  Functor composition bifunctors   fucofulem1 50340
                  21.51.15.19  Post-composition functors   postcofval 50394
                  21.51.15.20  Pre-composition functors   precofvallem 50396
            21.51.16  Examples of categories   catcrcl 50425
                  21.51.16.1  The category of categories   catcrcl 50425
                  21.51.16.2  Thin categories   cthinc 50447
                  21.51.16.3  Terminal categories   ctermc 50502
                  21.51.16.4  Preordered sets as thin categories   cprstc 50579
                  21.51.16.5  Monoids as categories   cmndtc 50607
                  21.51.16.6  Categories with at most one object and at most two morphisms   2arwcatlem1 50625
            21.51.17  Kan extensions and related concepts   clan 50635
                  21.51.17.1  Kan extensions   clan 50635
                  21.51.17.2  Limits and colimits   clmd 50673
      ​21.52  Mathbox for Emmett Weisz
            ​*21.52.1  Miscellaneous Theorems   nfintd 50703
            21.52.2  Examples and properties of set recursion   setrecseq 50710
            ​*21.52.3  Construction of Games and Surreal Numbers   cpg 50724
      ​​*21.53  Mathbox for David A. Wheeler
            21.53.1  Natural deduction   sbidd 50733
            ​*21.53.2  Greater than, greater than or equal to   cge-real 50735
            ​*21.53.3  Hyperbolic trigonometric functions   csinh 50745
            ​*21.53.4  Reciprocal trigonometric functions (sec, csc, cot)   csec 50756
            ​*21.53.5  Identities for "if"   ifnmfalse 50781
            ​*21.53.6  Logarithms generalized to arbitrary base using ` logb `   logb2aval 50782
            ​*21.53.7  Logarithm laws generalized to an arbitrary base - log_   clog- 50783
            ​*21.53.8  Formally define notions such as reflexivity   wreflexive 50785
            ​*21.53.9  Algebra helpers   mvlraddi 50789
            ​*21.53.10  Algebra helper examples   i2linesi 50796
            ​*21.53.11  Formal methods "surprises"   alimp-surprise 50798
            ​*21.53.12  Allsome quantifier   wals 50804
            ​*21.53.13  Allsome one quantifier   walseu 50837
            ​*21.53.14  Miscellaneous   5m4e1 50857
            21.53.15  Theorems about algebraic numbers   aacllem 50861
      ​21.54  Mathbox for Mingli Yuan
      ​21.55  Mathbox for Jiamin Zhao
            21.55.1  Cross product and scalar triple product in RR^3   1ne3 50863
            21.55.2  Veronese map and linear dependence   cveronese 50890
      ​21.56  Mathbox for Kunhao Zheng
            21.56.1  Weighted AM-GM inequality   amgmwlem 50909