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 Jeff Madsen
      21.25  Mathbox for Giovanni Mascellani
      21.26  Mathbox for Peter Mazsa
      21.27  Mathbox for Rodolfo Medina
      21.28  Mathbox for Norm Megill
      21.29  Mathbox for metakunt
      21.30  Mathbox for Luke Murphy
      21.31  Mathbox for Steven Nguyen
      21.32  Mathbox for Igor Ieskov
      21.33  Mathbox for OpenAI
      21.34  Mathbox for Stefan O'Rear
      21.35  Mathbox for Noam Pasman
      21.36  Mathbox for Jon Pennant
      21.37  Mathbox for Richard Penner
      21.38  Mathbox for Stanislas Polu
      21.39  Mathbox for Rohan Ridenour
      21.40  Mathbox for Steve Rodriguez
      21.41  Mathbox for Andrew Salmon
      21.42  Mathbox for Alan Sare
      21.43  Mathbox for Eric Schmidt
      21.44  Mathbox for Glauco Siliprandi
      21.45  Mathbox for Saveliy Skresanov
      21.46  Mathbox for Ender Ting
      21.47  Mathbox for Jarvin Udandy
      21.48  Mathbox for Adhemar
      21.49  Mathbox for Alexander van der Vekens
      21.50  Mathbox for Zhi Wang
      21.51  Mathbox for Emmett Weisz
      21.52  Mathbox for David A. Wheeler
      21.53  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 400
            ​*1.2.7  Logical disjunction   wo 860
            ​*1.2.8  Mixed connectives   jaao 969
            ​*1.2.9  The conditional operator for propositions   wif 1076
            ​*1.2.10  The weak deduction theorem for propositional calculus   elimh 1097
            1.2.11  Abbreviated conjunction and disjunction of three wff's   w3o 1100
            1.2.12  Logical "nand" (Sheffer stroke)   wnan 1519
            1.2.13  Logical "xor"   wxo 1539
            1.2.14  Logical "nor"   wnor 1556
            1.2.15  True and false constants   wal 1566
                  ​*1.2.15.1  Universal quantifier for use by df-tru   wal 1566
                  ​*1.2.15.2  Equality predicate for use by df-tru   cv 1567
                  1.2.15.3  The true constant   wtru 1569
                  1.2.15.4  The false constant   wfal 1580
            ​*1.2.16  Truth tables   truimtru 1591
                  1.2.16.1  Implication   truimtru 1591
                  1.2.16.2  Negation   nottru 1595
                  1.2.16.3  Equivalence   trubitru 1597
                  1.2.16.4  Conjunction   truantru 1601
                  1.2.16.5  Disjunction   truortru 1605
                  1.2.16.6  Alternative denial   trunantru 1609
                  1.2.16.7  Exclusive disjunction   truxortru 1613
                  1.2.16.8  Joint denial   trunortru 1617
            ​*1.2.17  Half adder and full adder in propositional calculus   whad 1621
                  1.2.17.1  Full adder: sum   whad 1621
                  1.2.17.2  Full adder: carry   wcad 1634
      ​1.3  Other axiomatizations related to classical propositional calculus
            ​*1.3.1  Minimal implicational calculus   minimp 1649
            ​*1.3.2  Implicational Calculus   impsingle 1655
            1.3.3  Derive the Lukasiewicz axioms from Meredith's sole axiom   meredith 1669
            1.3.4  Derive the standard axioms from the Lukasiewicz axioms   luklem1 1686
            ​*1.3.5  Derive Nicod's axiom from the standard axioms   nic-dfim 1697
            1.3.6  Derive the Lukasiewicz axioms from Nicod's axiom   nic-imp 1703
            1.3.7  Derive Nicod's Axiom from Lukasiewicz's First Sheffer Stroke Axiom   lukshef-ax1 1722
            1.3.8  Derive the Lukasiewicz Axioms from the Tarski-Bernays-Wajsberg Axioms   tbw-bijust 1726
            1.3.9  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's First CO Axiom   merco1 1741
            1.3.10  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's Second CO Axiom   merco2 1764
            1.3.11  Derive the Lukasiewicz axioms from the Russell-Bernays Axioms   rb-bijust 1777
            ​*1.3.12  Stoic logic non-modal portion (Chrysippus of Soli)   mptnan 1796
      ​​*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 1807
                  1.4.1.1  Existential quantifier   wex 1807
                  1.4.1.2  Nonfreeness predicate   wnf 1811
            1.4.2  Rule scheme ax-gen (Generalization)   ax-gen 1823
            1.4.3  Axiom scheme ax-4 (Quantified Implication)   ax-4 1837
                  ​*1.4.3.1  The empty domain of discourse   empty 1934
            1.4.4  Axiom scheme ax-5 (Distinctness) - first use of $d   ax-5 1938
            ​*1.4.5  Equality predicate (continued)   weq 1990
            1.4.6  Axiom scheme ax-6 (Existence)   ax-6 1995
            1.4.7  Axiom scheme ax-7 (Equality)   ax-7 2036
            1.4.8  Define proper substitution   sbjust 2093
            1.4.9  Membership predicate   wcel 2141
            1.4.10  Axiom scheme ax-8 (Left Equality for Binary Predicate)   ax-8 2143
            1.4.11  Axiom scheme ax-9 (Right Equality for Binary Predicate)   ax-9 2151
            ​*1.4.12  Logical redundancy of ax-10 , ax-11 , ax-12 , ax-13   ax6dgen 2161
      ​​*1.5  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes)
            1.5.1  Axiom scheme ax-10 (Quantified Negation)   ax-10 2174
            1.5.2  Axiom scheme ax-11 (Quantifier Commutation)   ax-11 2190
            1.5.3  Axiom scheme ax-12 (Substitution)   ax-12 2211
            1.5.4  Axiom scheme ax-13 (Quantified Equality)   ax-13 2402
      ​1.6  Uniqueness and unique existence
            1.6.1  Uniqueness: the at-most-one quantifier   wmo 2563
            1.6.2  Unique existence: the unique existential quantifier   weu 2594
      ​1.7  Other axiomatizations related to classical predicate calculus
            ​*1.7.1  Aristotelian logic: Assertic syllogisms   barbara 2688
            ​*1.7.2  Intuitionistic logic   axia1 2718
​​*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 2733
            2.1.2  Classes   cab 2739
                  2.1.2.1  Class abstractions   cab 2739
                  ​*2.1.2.2  Class equality   df-cleq 2753
                  2.1.2.3  Class membership   df-clel 2836
                  2.1.2.4  Elementary properties of class abstractions   eqabdv 2894
            2.1.3  Class form not-free predicate   wnfc 2908
            2.1.4  Negated equality and membership   wne 2956
                  2.1.4.1  Negated equality   wne 2956
                  2.1.4.2  Negated membership   wnel 3062
            2.1.5  Restricted quantification   wral 3077
                  2.1.5.1  Restricted universal and existential quantification   wral 3077
                  2.1.5.2  Restricted existential uniqueness and at-most-one quantifier   wreu 3365
                  2.1.5.3  Restricted class abstraction   crab 3414
            2.1.6  The universal class   cvv 3453
            ​*2.1.7  Conditional equality (experimental)   wcdeq 3725
            2.1.8  Russell's Paradox   rru 3741
            2.1.9  Proper substitution of classes for sets   wsbc 3743
            2.1.10  Proper substitution of classes for sets into classes   csb 3852
            2.1.11  Define basic set operations and relations   cdif 3901
            2.1.12  Subclasses and subsets   df-ss 3921
            2.1.13  The difference, union, and intersection of two classes   dfdif3 4071
                  2.1.13.1  The difference of two classes   dfdif3 4071
                  2.1.13.2  The union of two classes   elun 4106
                  2.1.13.3  The intersection of two classes   elini 4151
                  2.1.13.4  The symmetric difference of two classes   csymdif 4204
                  2.1.13.5  Combinations of difference, union, and intersection of two classes   unabs 4217
                  2.1.13.6  Class abstractions with difference, union, and intersection of two classes   unabw 4259
                  2.1.13.7  Restricted uniqueness with difference, union, and intersection   reuun2 4277
            2.1.14  The empty set   c0 4285
            ​*2.1.15  The conditional operator for classes   cif 4486
            ​*2.1.16  The weak deduction theorem for set theory   dedth 4545
            2.1.17  Power classes   cpw 4561
            2.1.18  Unordered and ordered pairs   snjust 4587
            2.1.19  The union of a class   cuni 4871
            2.1.20  The intersection of a class   cint 4911
            2.1.21  Indexed union and intersection   ciun 4955
            2.1.22  Disjointness   wdisj 5075
            2.1.23  Binary relations   wbr 5108
            2.1.24  Ordered-pair class abstractions (class builders)   copab 5172
            2.1.25  Functions in maps-to notation   cmpt 5191
            2.1.26  Transitive classes   wtr 5217
      ​2.2  ZF Set Theory - add the Axiom of Replacement
            2.2.1  Introduce the Axiom of Replacement   ax-rep 5237
            2.2.2  Derive the Axiom of Separation   axsepgfromrep 5254
            2.2.3  Derive the Null Set Axiom   axnulALT 5266
            2.2.4  Theorems requiring subset and intersection existence   exnelv 5275
            2.2.5  Theorems requiring empty set existence   class2set 5325
      ​2.3  ZF Set Theory - add the Axiom of Power Sets
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 5336
            2.3.2  Derive the Axiom of Pairing   axprlem1 5394
            2.3.3  Ordered pair theorem   opnz 5455
            2.3.4  Ordered-pair class abstractions (cont.)   opabidw 5508
            2.3.5  Power class of union and intersection   pwin 5552
            2.3.6  The identity relation   cid 5555
            2.3.7  The membership relation (or epsilon relation)   cep 5560
            ​*2.3.8  Partial and total orderings   wpo 5567
            2.3.9  Founded and well-ordering relations   wfr 5611
            2.3.10  Relations   cxp 5659
            2.3.11  The Predecessor Class   cpred 6301
            2.3.12  Well-founded induction (variant)   frpomin 6341
            2.3.13  Well-ordered induction   tz6.26 6348
            2.3.14  Ordinals   word 6359
            2.3.15  Definite description binder (inverted iota)   cio 6490
            2.3.16  Functions   wfun 6530
            2.3.17  Cantor's Theorem   canth 7364
            2.3.18  Restricted iota (description binder)   crio 7366
            2.3.19  Operations   co 7410
                  2.3.19.1  Variable-to-class conversion for operations   caovclg 7602
            2.3.20  Maps-to notation   mpondm0 7650
            2.3.21  Function operation   cof 7672
            2.3.22  Proper subset relation   crpss 7719
      ​2.4  ZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 7732
            2.4.2  Ordinals (continued)   epweon 7773
            2.4.3  Transfinite induction   tfi 7848
            2.4.4  The natural numbers (i.e., finite ordinals)   com 7861
            2.4.5  Peano's postulates   peano1 7884
            2.4.6  Finite induction (for finite ordinals)   find 7891
            2.4.7  Relations and functions (cont.)   dmexg 7897
            2.4.8  First and second members of an ordered pair   c1st 7983
            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 8445
            2.4.24  Natural number arithmetic   nna0 8589
            2.4.25  Natural addition   cnadd 8650
            2.4.26  Equivalence relations and classes   wer 8690
            2.4.27  The mapping operation   cmap 8823
            2.4.28  Infinite Cartesian products   cixp 8894
            2.4.29  Equinumerosity   cen 8939
            2.4.30  Schroeder-Bernstein Theorem   sbthlem1 9074
            2.4.31  Equinumerosity (cont.)   xpf1o 9126
            2.4.32  Finite sets   dif1enlem 9143
            2.4.33  Pigeonhole Principle   phplem1 9187
            2.4.34  Finite sets (cont.)   onomeneq 9197
            2.4.35  Finitely supported functions   cfsupp 9320
            2.4.36  Finite intersections   cfi 9369
            2.4.37  Hall's marriage theorem   marypha1lem 9392
            2.4.38  Supremum and infimum   csup 9399
            2.4.39  Ordinal isomorphism, Hartogs's theorem   coi 9470
            2.4.40  Hartogs function   char 9517
            2.4.41  Weak dominance   cwdom 9525
      ​2.5  ZF Set Theory - add the Axiom of Regularity
            2.5.1  Introduce the Axiom of Regularity   ax-reg 9553
            2.5.2  Axiom of Infinity equivalents   inf0 9589
      ​2.6  ZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-inf 9606
            2.6.2  Existence of omega (the set of natural numbers)   omex 9611
            2.6.3  Cantor normal form   ccnf 9629
            2.6.4  Transitive closure of a relation   cttrcl 9675
            2.6.5  Transitive closure   trcl 9696
            2.6.6  Set induction (or epsilon induction)   setind 9715
            2.6.7  Well-Founded Induction   frmin 9720
            2.6.8  Well-Founded Recursion   frr3g 9727
            2.6.9  Rank   cr1 9733
            2.6.10  Scott's trick; collection principle; Hilbert's epsilon   cscott 9856
            2.6.11  Disjoint union   cdju 9883
            2.6.12  Cardinal numbers   ccrd 9920
            2.6.13  Axiom of Choice equivalents   wac 10098
            ​*2.6.14  Cardinal number arithmetic   undjudom 10150
            2.6.15  The Ackermann bijection   ackbij2lem1 10200
            2.6.16  Cofinality (without Axiom of Choice)   cflem 10227
            2.6.17  Eight inequivalent definitions of finite set   sornom 10260
            2.6.18  Hereditarily size-limited sets without Choice   itunifval 10399
​​*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 10418
            3.1.2  Introduce the Axiom of Dependent Choice   ax-dc 10429
      ​3.2  ZFC Set Theory - add the Axiom of Choice
            3.2.1  Introduce the Axiom of Choice   ax-ac 10442
            3.2.2  AC equivalents: well-ordering, Zorn's lemma   numthcor 10477
            3.2.3  Cardinal number theorems using Axiom of Choice   cardval 10529
            3.2.4  Cardinal number arithmetic using Axiom of Choice   iunctb 10558
            3.2.5  Cofinality using the Axiom of Choice   alephreg 10566
      ​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 10604
            3.4.2  Derivation of the Axiom of Choice   gchaclem 10662
​​*PART 4  TG (TARSKI-GROTHENDIECK) SET THEORY
      ​4.1  Inaccessibles
            4.1.1  Weakly and strongly inaccessible cardinals   cwina 10666
            4.1.2  Weak universes   cwun 10684
            4.1.3  Tarski classes   ctsk 10732
            4.1.4  Grothendieck universes   cgru 10774
      ​4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
            4.2.1  Introduce the Tarski-Grothendieck Axiom   ax-groth 10807
            4.2.2  Derive the Power Set, Infinity and Choice Axioms   grothpw 10810
            4.2.3  Tarski map function   ctskm 10821
​​*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 10828
            5.1.2  Final derivation of real and complex number postulates   axaddf 11129
            5.1.3  Real and complex number postulates restated as axioms   ax-cnex 11155
      ​5.2  Derive the basic properties from the field axioms
            5.2.1  Some deductions from the field axioms for complex numbers   cnex 11180
            5.2.2  Infinity and the extended real number system   cpnf 11239
            5.2.3  Restate the ordering postulates with extended real "less than"   axlttri 11280
            5.2.4  Ordering on reals   lttr 11285
            5.2.5  Initial properties of the complex numbers   mul12 11374
      ​5.3  Real and complex numbers - basic operations
            5.3.1  Addition   add12 11427
            5.3.2  Subtraction   cmin 11440
            5.3.3  Multiplication   kcnktkm1cn 11644
            5.3.4  Ordering on reals (cont.)   gt0ne0 11678
            5.3.5  Reciprocals   ixi 11842
            5.3.6  Division   cdiv 11870
            5.3.7  Ordering on reals (cont.)   elimgt0 12052
            5.3.8  Completeness Axiom and Suprema   fimaxre 12158
            5.3.9  Imaginary and complex number properties   neg1cn 12202
            5.3.10  Function operation analogue theorems   ofsubeq0 12214
            ​*5.3.11  Indicator Functions   cind 12217
      ​5.4  Integer sets
            5.4.1  Positive integers (as a subset of complex numbers)   cn 12232
            5.4.2  Principle of mathematical induction   nnind 12250
            ​*5.4.3  Decimal representation of numbers   c2 12294
            ​*5.4.4  Some properties of specific numbers   1pneg1e0 12357
            5.4.5  Simple number properties   halfcl 12469
            5.4.6  The Archimedean property   nnunb 12499
            5.4.7  Nonnegative integers (as a subset of complex numbers)   cn0 12503
            ​*5.4.8  Extended nonnegative integers   cxnn0 12576
            5.4.9  Integers (as a subset of complex numbers)   cz 12590
            5.4.10  Decimal arithmetic   cdc 12710
            5.4.11  Upper sets of integers   cuz 12861
            5.4.12  Well-ordering principle for bounded-below sets of integers   uzwo3 12966
            5.4.13  Rational numbers (as a subset of complex numbers)   cq 12971
            5.4.14  Existence of the set of complex numbers   rpnnen1lem2 13000
      ​5.5  Order sets
            5.5.1  Positive reals (as a subset of complex numbers)   crp 13015
            5.5.2  Infinity and the extended real number system (cont.)   cxne 13133
            5.5.3  Supremum and infimum on the extended reals   xrsupexmnf 13330
            5.5.4  Real number intervals   cioo 13371
            5.5.5  Finite intervals of integers   cfz 13534
            ​*5.5.6  Finite intervals of nonnegative integers   elfz2nn0 13646
            5.5.7  Half-open integer ranges   cfzo 13682
      ​5.6  Elementary integer functions
            5.6.1  The floor and ceiling functions   cfl 13823
            5.6.2  The modulo (remainder) operation   cmo 13902
            5.6.3  Miscellaneous theorems about integers   om2uz0i 13983
            5.6.4  Strong induction over upper sets of integers   uzsinds 14023
            5.6.5  Finitely supported functions over the nonnegative integers   fsuppmapnn0fiublem 14026
            5.6.6  The infinite sequence builder "seq" - extension   cseq 14037
            5.6.7  Integer powers   cexp 14097
            5.6.8  Ordered pair theorem for nonnegative integers   nn0le2msqi 14303
            5.6.9  Factorial function   cfa 14309
            5.6.10  The binomial coefficient operation   cbc 14338
            5.6.11  The ` # ` (set size) function   chash 14366
                  5.6.11.1  Proper unordered pairs and triples (sets of size 2 and 3)   hashprlei 14505
                  5.6.11.2  Functions with a domain containing at least two different elements   fundmge2nop0 14539
                  5.6.11.3  Finite induction on the size of the first component of a binary relation   hashdifsnp1 14543
      ​​*5.7  Words over a set
            5.7.1  Definitions and basic theorems   cword 14550
            5.7.2  Last symbol of a word   clsw 14599
            5.7.3  Concatenations of words   cconcat 14607
            5.7.4  Singleton words   cs1 14633
            5.7.5  Concatenations with singleton words   ccatws1cl 14654
            5.7.6  Subwords/substrings   csubstr 14678
            5.7.7  Prefixes of a word   cpfx 14708
            5.7.8  Subwords of subwords   swrdswrdlem 14741
            5.7.9  Subwords and concatenations   pfxcctswrd 14747
            5.7.10  Subwords of concatenations   swrdccatfn 14761
            5.7.11  Splicing words (substring replacement)   csplice 14786
            5.7.12  Reversing words   creverse 14795
            5.7.13  Repeated symbol words   creps 14805
            ​*5.7.14  Cyclical shifts of words   ccsh 14825
            5.7.15  Mapping words by a function   wrdco 14868
            5.7.16  Longer string literals   cs2 14878
      ​​*5.8  Reflexive and transitive closures of relations
            5.8.1  The reflexive and transitive properties of relations   coss12d 15009
            5.8.2  Basic properties of closures   cleq1lem 15019
            5.8.3  Definitions and basic properties of transitive closures   ctcl 15022
            5.8.4  Exponentiation of relations   crelexp 15056
            5.8.5  Reflexive-transitive closure as an indexed union   crtrcl 15092
            ​*5.8.6  Principle of transitive induction   relexpindlem 15100
      ​5.9  Elementary real and complex functions
            5.9.1  The "shift" operation   cshi 15103
            5.9.2  Signum (sgn or sign) function   csgn 15123
            5.9.3  Real and imaginary parts; conjugate   ccj 15147
            5.9.4  Square root; absolute value   csqrt 15284
      ​5.10  Elementary limits and convergence
            5.10.1  Superior limit (lim sup)   clsp 15521
            5.10.2  Limits   cli 15535
            5.10.3  Finite and infinite sums   csu 15737
            5.10.4  The binomial theorem   binomlem 15883
            5.10.5  The inclusion/exclusion principle   incexclem 15890
            5.10.6  Infinite sums (cont.)   isumshft 15893
            5.10.7  Miscellaneous converging and diverging sequences   divrcnv 15906
            5.10.8  Arithmetic series   arisum 15914
            5.10.9  Geometric series   expcnv 15918
            5.10.10  Ratio test for infinite series convergence   cvgrat 15937
            5.10.11  Mertens' theorem   mertenslem1 15938
            5.10.12  Finite and infinite products   prodf 15941
                  5.10.12.1  Product sequences   prodf 15941
                  5.10.12.2  Non-trivial convergence   ntrivcvg 15951
                  5.10.12.3  Complex products   cprod 15957
                  5.10.12.4  Finite products   fprod 15995
                  5.10.12.5  Infinite products   iprodclim 16052
            5.10.13  Falling and Rising Factorial   cfallfac 16058
            5.10.14  Bernoulli polynomials and sums of k-th powers   cbp 16099
      ​5.11  Elementary trigonometry
            5.11.1  The exponential, sine, and cosine functions   ce 16114
                  5.11.1.1  The circle constant (tau = 2 pi)   ctau 16257
            5.11.2  _e is irrational   eirrlem 16259
      ​5.12  Cardinality of real and complex number subsets
            5.12.1  Countability of integers and rationals   xpnnen 16266
            5.12.2  The reals are uncountable   rpnnen2lem1 16269
​​*PART 6  ELEMENTARY NUMBER THEORY
      ​6.1  Elementary properties of divisibility
            6.1.1  Irrationality of square root of 2   sqrt2irrlem 16303
            6.1.2  Some Number sets are chains of proper subsets   nthruc 16307
            6.1.3  The divides relation   cdvds 16309
            ​*6.1.4  Even and odd numbers   evenelz 16393
            6.1.5  The division algorithm   divalglem0 16450
            6.1.6  Bit sequences   cbits 16476
            6.1.7  The greatest common divisor operator   cgcd 16551
            6.1.8  Bézout's identity   bezoutlem1 16596
            6.1.9  Algorithms   nn0seqcvgd 16627
            6.1.10  Euclid's Algorithm   eucalgval2 16638
            ​*6.1.11  The least common multiple   clcm 16645
            ​*6.1.12  Coprimality and Euclid's lemma   coprmgcdb 16706
            6.1.13  Cancellability of congruences   congr 16721
      ​6.2  Elementary prime number theory
            ​*6.2.1  Elementary properties   cprime 16728
            ​*6.2.2  Coprimality and Euclid's lemma (cont.)   coprm 16769
            6.2.3  Properties of the canonical representation of a rational   cnumer 16791
            6.2.4  Euler's theorem   codz 16821
            6.2.5  Arithmetic modulo a prime number   modprm1div 16856
            6.2.6  Pythagorean Triples   coprimeprodsq 16867
            6.2.7  The prime count function   cpc 16895
            6.2.8  Pocklington's theorem   prmpwdvds 16963
            6.2.9  Infinite primes theorem   unbenlem 16967
            6.2.10  Sum of prime reciprocals   prmreclem1 16975
            6.2.11  Fundamental theorem of arithmetic   1arithlem1 16982
            6.2.12  Lagrange's four-square theorem   cgz 16988
            6.2.13  Van der Waerden's theorem   cvdwa 17024
            6.2.14  Ramsey's theorem   cram 17058
            ​*6.2.15  Primorial function   cprmo 17090
            ​*6.2.16  Prime gaps   prmgaplem1 17108
            6.2.17  Decimal arithmetic (cont.)   dec2dvds 17122
            6.2.18  Cyclical shifts of words (cont.)   cshwsidrepsw 17152
            6.2.19  Specific prime numbers   prmlem0 17164
            6.2.20  Very large primes   1259lem1 17190
​PART 7  BASIC STRUCTURES
      ​7.1  Extensible structures
            ​*7.1.1  Basic definitions   cstr 17205
                  7.1.1.1  Extensible structures as structures with components   cstr 17205
                  7.1.1.2  Substitution of components   csts 17222
                  7.1.1.3  Slots   cslot 17240
                  ​*7.1.1.4  Structure component indices   cnx 17252
                  7.1.1.5  Base sets   cbs 17268
                  7.1.1.6  Base set restrictions   cress 17289
            7.1.2  Slot definitions   cplusg 17309
            7.1.3  Definition of the structure product   crest 17472
            7.1.4  Definition of the structure quotient   cordt 17552
      ​7.2  Moore spaces
            7.2.1  Moore closures   mrcflem 17661
            7.2.2  Independent sets in a Moore system   mrisval 17685
            7.2.3  Algebraic closure systems   isacs 17706
​PART 8  BASIC CATEGORY THEORY
      ​8.1  Categories
            8.1.1  Categories   ccat 17719
            8.1.2  Opposite category   coppc 17766
            8.1.3  Monomorphisms and epimorphisms   cmon 17784
            8.1.4  Sections, inverses, isomorphisms   csect 17800
            ​*8.1.5  Isomorphic objects   ccic 17851
            8.1.6  Subcategories   cssc 17863
            8.1.7  Functors   cfunc 17910
            8.1.8  Full & faithful functors   cful 17960
            8.1.9  Natural transformations and the functor category   cnat 18000
            8.1.10  Initial, terminal and zero objects of a category   cinito 18037
      ​8.2  Arrows (disjointified hom-sets)
            8.2.1  Identity and composition for arrows   cida 18109
      ​8.3  Examples of categories
            8.3.1  The category of sets   csetc 18131
            8.3.2  The category of categories   ccatc 18154
            ​*8.3.3  The category of extensible structures   fncnvimaeqv 18175
      ​8.4  Categorical constructions
            8.4.1  Product of categories   cxpc 18223
            8.4.2  Functor evaluation   cevlf 18264
            8.4.3  Hom functor   chof 18303
​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 18486
            9.5.2  Complete lattices   ccla 18553
            9.5.3  Distributive lattices   cdlat 18575
            9.5.4  Subset order structures   cipo 18582
      ​9.6  Posets, directed sets, and lattices as relations
            ​*9.6.1  Posets and lattices as relations   cps 18619
            9.6.2  Directed sets, nets   cdir 18649
      ​9.7  Chains
​PART 10  BASIC ALGEBRAIC STRUCTURES
      ​10.1  Monoids
            ​*10.1.1  Magmas   cplusf 18694
            ​*10.1.2  Identity elements   mgmidmo 18717
            ​*10.1.3  Iterated sums in a magma   gsumvalx 18733
            10.1.4  Magma homomorphisms and submagmas   cmgmhm 18747
            ​*10.1.5  Semigroups   csgrp 18775
            ​*10.1.6  Definition and basic properties of monoids   cmnd 18791
            10.1.7  Monoid homomorphisms and submonoids   cmhm 18838
            ​*10.1.8  Iterated sums in a monoid   gsumvallem2 18892
            10.1.9  Free monoids   cfrmd 18905
                  ​*10.1.9.1  Monoid of endofunctions   cefmnd 18926
            10.1.10  Examples and counterexamples for magmas, semigroups and monoids   mgm2nsgrplem1 18979
      ​10.2  Groups
            10.2.1  Definition and basic properties   cgrp 18999
            ​*10.2.2  Group multiple operation   cmg 19132
            10.2.3  Subgroups and Quotient groups   csubg 19185
            ​*10.2.4  Cyclic monoids and groups   cycsubmel 19270
            10.2.5  Elementary theory of group homomorphisms   cghm 19282
            10.2.6  Isomorphisms of groups   cgim 19326
                  10.2.6.1  The first isomorphism theorem of groups   ghmqusnsglem1 19349
            10.2.7  Group actions   cga 19358
            10.2.8  Centralizers and centers   ccntz 19384
            10.2.9  The opposite group   coppg 19414
            10.2.10  Symmetric groups   csymg 19438
                  ​*10.2.10.1  Definition and basic properties   csymg 19438
                  10.2.10.2  Cayley's theorem   cayleylem1 19481
                  10.2.10.3  Permutations fixing one element   symgfix2 19485
                  ​*10.2.10.4  Transpositions in the symmetric group   cpmtr 19510
                  10.2.10.5  The sign of a permutation   cpsgn 19558
            10.2.11  p-Groups and Sylow groups; Sylow's theorems   cod 19593
            10.2.12  Direct products   clsm 19703
                  10.2.12.1  Direct products (extension)   smndlsmidm 19725
            10.2.13  Free groups   cefg 19775
            10.2.14  Abelian groups   ccmn 19849
                  10.2.14.1  Definition and basic properties   ccmn 19849
                  10.2.14.2  Cyclic groups   ccyg 19946
                  10.2.14.3  Group sum operation   gsumval3a 19972
                  10.2.14.4  Group sums over (ranges of) integers   fsfnn0gsumfsffz 20052
                  10.2.14.5  Internal direct products   cdprd 20064
                  10.2.14.6  The Fundamental Theorem of Abelian Groups   ablfacrplem 20136
            10.2.15  Simple groups   csimpg 20161
                  10.2.15.1  Definition and basic properties   csimpg 20161
                  10.2.15.2  Classification of abelian simple groups   ablsimpnosubgd 20175
            10.2.16  Totally ordered monoids and groups   comnd 20188
      ​10.3  Rings
            10.3.1  Multiplicative Group   cmgp 20215
            ​*10.3.2  Non-unital rings ("rngs")   crng 20229
            ​*10.3.3  Ring unity (multiplicative identity)   cur 20262
            10.3.4  Semirings   csrg 20267
                  ​*10.3.4.1  The binomial theorem for semirings   srgbinomlem1 20307
            10.3.5  Unital rings   crg 20314
            10.3.6  Opposite ring   coppr 20417
            10.3.7  Divisibility   cdsr 20435
            10.3.8  Ring primes   crpm 20513
            10.3.9  Homomorphisms of non-unital rings   crnghm 20515
            10.3.10  Ring homomorphisms   crh 20550
            10.3.11  Nonzero rings and zero rings   cnzr 20594
            10.3.12  Local rings   clring 20622
            10.3.13  Subrings   csubrng 20629
                  10.3.13.1  Subrings of non-unital rings   csubrng 20629
                  10.3.13.2  Subrings of unital rings   csubrg 20653
                  10.3.13.3  Subrings generated by a subset   crgspn 20694
            10.3.14  Categories of rings   crngc 20700
                  ​*10.3.14.1  The category of non-unital rings   crngc 20700
                  ​*10.3.14.2  The category of (unital) rings   cringc 20729
                  10.3.14.3  Subcategories of the category of rings   srhmsubclem1 20761
            10.3.15  Left regular elements and domains   crlreg 20775
      ​10.4  Division rings and fields
            10.4.1  Definition and basic properties   cdr 20812
            10.4.2  Sub-division rings   csdrg 20868
            10.4.3  Absolute value (abstract algebra)   cabv 20890
            10.4.4  Star rings   cstf 20919
            10.4.5  Totally ordered rings and fields   corng 20939
      ​10.5  Left modules
            10.5.1  Definition and basic properties   clmod 20960
            10.5.2  Subspaces and spans in a left module   clss 21031
            10.5.3  Homomorphisms and isomorphisms of left modules   clmhm 21119
            10.5.4  Subspace sum; bases for a left module   clbs 21174
      ​10.6  Vector spaces
            10.6.1  Definition and basic properties   clvec 21202
      ​10.7  Subring algebras and ideals
            10.7.1  Subring algebras   csra 21271
            ​*10.7.2  Left ideals and spans   clidl 21309
            10.7.3  Two-sided ideals and quotient rings   c2idl 21367
                  ​*10.7.3.1  Condition for a non-unital ring to be unital   rngqiprng1elbas 21405
                  10.7.3.2  Prime Ideals   cprmidl 21439
            10.7.4  Principal ideal rings. Divisibility in the integers   clpidl 21467
            10.7.5  Principal ideal domains   cpid 21483
      ​10.8  The complex numbers as an algebraic extensible structure
            10.8.1  Definition and basic properties   cpsmet 21485
            ​*10.8.2  Ring of integers   czring 21575
                  ​*10.8.2.1  Example for a condition for a non-unital ring to be unital   pzriprnglem1 21610
            10.8.3  Algebraic constructions based on the complex numbers   czrh 21628
            10.8.4  Signs as subgroup of the complex numbers   cnmsgnsubg 21706
            10.8.5  Embedding of permutation signs into a ring   zrhpsgnmhm 21713
            10.8.6  The ordered field of real numbers   crefld 21733
      ​10.9  Generalized pre-Hilbert and Hilbert spaces
            10.9.1  Definition and basic properties   cphl 21753
            10.9.2  Orthocomplements and closed subspaces   cocv 21789
            10.9.3  Orthogonal projection and orthonormal bases   cpj 21829
​​*PART 11  BASIC LINEAR ALGEBRA
      ​11.1  Vectors and free modules
            *11.1.1  Direct sum of left modules   cdsmm 21860
            ​*11.1.2  Free modules   cfrlm 21875
            ​*11.1.3  Standard basis (unit vectors)   cuvc 21911
            ​*11.1.4  Independent sets and families   clindf 21933
            11.1.5  Characterization of free modules   lmimlbs 21965
      ​11.2  Associative algebras
            11.2.1  Definition and basic properties   casa 21979
      ​11.3  Abstract multivariate polynomials
            11.3.1  Definition and basic properties   cmps 22033
            11.3.2  Polynomial evaluation   ces 22202
            11.3.3  The "variable selection" function   cslv 22246
            11.3.4  Additional definitions for (multivariate) polynomials   cmhp 22275
            ​*11.3.5  Univariate polynomials   cps1 22314
            11.3.6  Univariate polynomial evaluation   ces1 22452
                  11.3.6.1  Specialization of polynomial evaluation as a ring homomorphism   evls1scafv 22505
      ​​*11.4  Matrices
            *11.4.1  The matrix multiplication   cmmul 22526
            ​*11.4.2  Square matrices   cmat 22543
            ​*11.4.3  The matrix algebra   matmulr 22574
            ​*11.4.4  Matrices of dimension 0 and 1   mat0dimbas0 22602
            ​*11.4.5  The subalgebras of diagonal and scalar matrices   cdmat 22624
            ​*11.4.6  Multiplication of a matrix with a "column vector"   cmvmul 22676
            11.4.7  Replacement functions for a square matrix   cmarrep 22692
            11.4.8  Submatrices   csubma 22712
      ​11.5  The determinant
            11.5.1  Definition and basic properties   cmdat 22720
            11.5.2  Determinants of 2 x 2 -matrices   m2detleiblem1 22760
            11.5.3  The matrix adjugate/adjunct   cmadu 22768
            ​*11.5.4  Laplace expansion of determinants (special case)   symgmatr01lem 22789
            11.5.5  Inverse matrix   invrvald 22812
            ​*11.5.6  Cramer's rule   slesolvec 22815
      ​​*11.6  Polynomial matrices
            11.6.1  Basic properties   pmatring 22828
            ​*11.6.2  Constant polynomial matrices   ccpmat 22839
            ​*11.6.3  Collecting coefficients of polynomial matrices   cdecpmat 22898
            ​*11.6.4  Ring isomorphism between polynomial matrices and polynomials over matrices   cpm2mp 22928
      ​​*11.7  The characteristic polynomial
            *11.7.1  Definition and basic properties   cchpmat 22962
            ​*11.7.2  The characteristic factor function G   fvmptnn04if 22985
            ​*11.7.3  The Cayley-Hamilton theorem   cpmadurid 23003
​PART 12  BASIC TOPOLOGY
      ​12.1  Topology
            ​*12.1.1  Topological spaces   ctop 23029
                  12.1.1.1  Topologies   ctop 23029
                  12.1.1.2  Topologies on sets   ctopon 23046
                  12.1.1.3  Topological spaces   ctps 23068
            12.1.2  Topological bases   ctb 23081
            12.1.3  Examples of topologies   distop 23131
            12.1.4  Closure and interior   ccld 23152
            12.1.5  Neighborhoods   cnei 23233
            12.1.6  Limit points and perfect sets   clp 23270
            12.1.7  Subspace topologies   restrcl 23293
            12.1.8  Order topology   ordtbaslem 23324
            12.1.9  Limits and continuity in topological spaces   ccn 23360
            12.1.10  Separated spaces: T0, T1, T2 (Hausdorff) ...   ct0 23442
            12.1.11  Compactness   ccmp 23522
            12.1.12  Bolzano-Weierstrass theorem   bwth 23546
            12.1.13  Connectedness   cconn 23547
            12.1.14  First- and second-countability   c1stc 23573
            12.1.15  Local topological properties   clly 23600
            12.1.16  Refinements   cref 23638
            12.1.17  Compactly generated spaces   ckgen 23669
            12.1.18  Product topologies   ctx 23696
            12.1.19  Continuous function-builders   cnmptid 23797
            12.1.20  Quotient maps and quotient topology   ckq 23829
            12.1.21  Homeomorphisms   chmeo 23889
      ​12.2  Filters and filter bases
            12.2.1  Filter bases   elmptrab 23963
            12.2.2  Filters   cfil 23981
            12.2.3  Ultrafilters   cufil 24035
            12.2.4  Filter limits   cfm 24069
            12.2.5  Extension by continuity   ccnext 24195
            12.2.6  Topological groups   ctmd 24206
            12.2.7  Infinite group sum on topological groups   ctsu 24262
            12.2.8  Topological rings, fields, vector spaces   ctrg 24292
      ​12.3  Uniform Structures and Spaces
            12.3.1  Uniform structures   cust 24336
            12.3.2  The topology induced by an uniform structure   cutop 24366
            12.3.3  Uniform Spaces   cuss 24389
            12.3.4  Uniform continuity   cucn 24410
            12.3.5  Cauchy filters in uniform spaces   ccfilu 24421
            12.3.6  Complete uniform spaces   ccusp 24432
      ​12.4  Metric spaces
            12.4.1  Pseudometric spaces   ispsmet 24440
            12.4.2  Basic metric space properties   cxms 24453
            12.4.3  Metric space balls   blfvalps 24519
            12.4.4  Open sets of a metric space   mopnval 24574
            12.4.5  Continuity in metric spaces   metcnp3 24676
            12.4.6  The uniform structure generated by a metric   metuval 24685
            12.4.7  Examples of metric spaces   dscmet 24708
            ​*12.4.8  Normed algebraic structures   cnm 24712
            12.4.9  Normed space homomorphisms (bounded linear operators)   cnmo 24841
            12.4.10  Topology on the reals   qtopbaslem 24894
            12.4.11  Topological definitions using the reals   cii 25013
            12.4.12  Path homotopy   chtpy 25105
            12.4.13  The fundamental group   cpco 25138
      ​12.5  Metric subcomplex vector spaces
            12.5.1  Subcomplex modules   cclm 25200
            ​*12.5.2  Subcomplex vector spaces   ccvs 25261
            ​*12.5.3  Normed subcomplex vector spaces   isncvsngp 25287
            12.5.4  Subcomplex pre-Hilbert spaces   ccph 25304
            12.5.5  Convergence and completeness   ccfil 25390
            12.5.6  Baire's Category Theorem   bcthlem1 25462
            12.5.7  Banach spaces and subcomplex Hilbert spaces   ccms 25470
                  12.5.7.1  The complete ordered field of the real numbers   retopn 25517
            12.5.8  Euclidean spaces   crrx 25521
            12.5.9  Minimizing Vector Theorem   minveclem1 25562
            12.5.10  Projection Theorem   pjthlem1 25575
​PART 13  BASIC REAL AND COMPLEX ANALYSIS
      ​13.1  Continuity
            13.1.1  Intermediate value theorem   pmltpclem1 25586
      ​13.2  Integrals
            13.2.1  Lebesgue measure   covol 25600
            13.2.2  Lebesgue integration   cmbf 25752
                  13.2.2.1  Lesbesgue integral   cmbf 25752
                  13.2.2.2  Lesbesgue directed integral   cdit 25984
      ​13.3  Derivatives
            13.3.1  Real and complex differentiation   climc 26000
                  13.3.1.1  Derivatives of functions of one complex or real variable   climc 26000
                  13.3.1.2  Results on real differentiation   dvferm1lem 26122
​PART 14  BASIC REAL AND COMPLEX FUNCTIONS
      ​14.1  Polynomials
            14.1.1  Polynomial degrees   cmdg 26189
            14.1.2  The division algorithm for univariate polynomials   cmn1 26262
            14.1.3  Elementary properties of complex polynomials   cply 26320
            14.1.4  The division algorithm for polynomials   cquot 26430
            14.1.5  Algebraic numbers   caa 26454
            14.1.6  Liouville's approximation theorem   aalioulem1 26472
      ​14.2  Sequences and series
            14.2.1  Taylor polynomials and Taylor's theorem   ctayl 26492
            14.2.2  Uniform convergence   culm 26515
            14.2.3  Power series   pserval 26549
      ​14.3  Basic trigonometry
            14.3.1  The exponential, sine, and cosine functions (cont.)   efcn 26582
            14.3.2  Properties of pi = 3.14159...   pilem1 26590
            14.3.3  Mapping of the exponential function   efgh 26682
            14.3.4  The natural logarithm on complex numbers   clog 26695
            ​*14.3.5  Logarithms to an arbitrary base   clogb 26905
            14.3.6  Theorems of Pythagoras, isosceles triangles, and intersecting chords   angval 26942
            14.3.7  Solutions of quadratic, cubic, and quartic equations   quad2 26980
            14.3.8  Inverse trigonometric functions   casin 27003
            14.3.9  The Birthday Problem   log2ublem1 27087
            14.3.10  Areas in R^2   carea 27096
            14.3.11  More miscellaneous converging sequences   rlimcnp 27106
            14.3.12  Inequality of arithmetic and geometric means   cvxcl 27125
            14.3.13  Euler-Mascheroni constant   cem 27132
            14.3.14  Zeta function   czeta 27153
            14.3.15  Gamma function   clgam 27156
      ​14.4  Basic number theory
            14.4.1  Wilson's theorem   wilthlem1 27208
            14.4.2  The Fundamental Theorem of Algebra   ftalem1 27213
            14.4.3  The Basel problem (ζ(2) = π²/6)   basellem1 27221
            14.4.4  Number-theoretical functions   ccht 27231
            14.4.5  Perfect Number Theorem   mersenne 27367
            14.4.6  Characters of Z/nZ   cdchr 27372
            14.4.7  Bertrand's postulate   bcctr 27415
            ​*14.4.8  Quadratic residues and the Legendre symbol   clgs 27434
            ​*14.4.9  Gauss' Lemma   gausslemma2dlem0a 27496
            14.4.10  Quadratic reciprocity   lgseisenlem1 27515
            14.4.11  All primes 4n+1 are the sum of two squares   2sqlem1 27557
            14.4.12  Chebyshev's Weak Prime Number Theorem, Dirichlet's Theorem   chebbnd1lem1 27609
            14.4.13  The Prime Number Theorem   mudivsum 27670
            14.4.14  Ostrowski's theorem   abvcxp 27755
​PART 15  SURREAL NUMBERS
      ​​*15.1  Sign sequence representation and Alling's axioms
            15.1.1  Definitions and initial properties   csur 27780
            15.1.2  Ordering   ltssolem1 27815
            15.1.3  Birthday Function   bdayfo 27817
            15.1.4  Density   fvnobday 27818
            ​*15.1.5  Full-Eta Property   bdayimaon 27833
      ​15.2  Initial consequences of Alling's axioms
            15.2.1  Ordering Theorems   cles 27884
            15.2.2  Birthday Theorems   bdayfun 27916
      ​​*15.3  Conway cut representation
            15.3.1  Conway cuts   cslts 27926
            15.3.2  Zero and One   c0s 27974
            15.3.3  Cuts and Options   cmade 27991
            15.3.4  Cofinality and coinitiality   cofslts 28087
      ​15.4  Induction and recursion
            15.4.1  Induction and recursion on one variable   cnorec 28106
            15.4.2  Induction and recursion on two variables   cnorec2 28117
      ​15.5  Surreal arithmetic
            15.5.1  Addition   cadds 28128
            15.5.2  Negation and Subtraction   cnegs 28188
            15.5.3  Multiplication   cmuls 28275
            15.5.4  Division   cdivs 28356
            15.5.5  Absolute value   cabss 28406
      ​15.6  Subsystems of surreals
            15.6.1  Ordinal numbers   cons 28420
            15.6.2  Surreal recursive sequences   cseqs 28452
            15.6.3  Natural numbers   cn0s 28481
            15.6.4  Integers   czs 28547
            15.6.5  Dyadic fractions   c2s 28579
            15.6.6  Real numbers   creno 28658
​​*PART 16  ELEMENTARY GEOMETRY
      ​16.1  Definition and Tarski's Axioms of Geometry
            16.1.1  Justification for the congruence notation   tgjustf 28718
      ​16.2  Tarskian Geometry
            16.2.1  Congruence   tgcgrcomimp 28722
            16.2.2  Betweenness   tgbtwntriv2 28732
            16.2.3  Dimension   tglowdim1 28745
            16.2.4  Betweenness and Congruence   tgifscgr 28753
            16.2.5  Congruence of a series of points   ccgrg 28755
            16.2.6  Motions   cismt 28777
            16.2.7  Colinearity   tglng 28791
            16.2.8  Connectivity of betweenness   tgbtwnconn1lem1 28817
            16.2.9  Less-than relation in geometric congruences   cleg 28827
            16.2.10  Rays   chlg 28845
            16.2.11  Lines   btwnlng1 28868
            16.2.12  Point inversions   cmir 28905
            16.2.13  Right angles   crag 28948
            16.2.14  Half-planes   islnopp 28995
            16.2.15  Planes   cplng 29029
            16.2.16  Midpoints and Line Mirroring   cmid 29055
            16.2.17  Congruence of angles   ccgra 29091
            16.2.18  Angle Comparisons   cinag 29125
            16.2.19  Congruence Theorems   tgsas1 29144
            16.2.20  Equilateral triangles   ceqlg 29155
            16.2.21  Parallel lines   cprlng 29159
      ​16.3  Properties of geometries
            16.3.1  Isomorphisms between geometries   f1otrgds 29184
      ​16.4  Geometry in Hilbert spaces
            16.4.1  Geometry in the complex plane   cchhllem 29202
            16.4.2  Geometry in Euclidean spaces   cee 29203
                  16.4.2.1  Definition of the Euclidean space   cee 29203
                  16.4.2.2  Tarski's axioms for geometry for the Euclidean space   axdimuniq 29229
                  16.4.2.3  EE^n fulfills Tarski's Axioms   ceeng 29293
​​*PART 17  GRAPH THEORY
      ​*17.1  Vertices and edges
            17.1.1  The edge function extractor for extensible structures   cedgf 29304
            ​*17.1.2  Vertices and indexed edges   cvtx 29312
                  17.1.2.1  Definitions and basic properties   cvtx 29312
                  17.1.2.2  The vertices and edges of a graph represented as ordered pair   opvtxval 29319
                  17.1.2.3  The vertices and edges of a graph represented as extensible structure   funvtxdmge2val 29327
                  17.1.2.4  Representations of graphs without edges   snstrvtxval 29353
                  17.1.2.5  Degenerated cases of representations of graphs   vtxval0 29355
            17.1.3  Edges as range of the edge function   cedg 29363
      ​​*17.2  Undirected graphs
            17.2.1  Undirected hypergraphs   cuhgr 29372
            17.2.2  Undirected pseudographs and multigraphs   cupgr 29396
            ​*17.2.3  Loop-free graphs   umgrislfupgrlem 29438
            17.2.4  Edges as subsets of vertices of graphs   uhgredgiedgb 29442
            ​*17.2.5  Undirected simple graphs   cuspgr 29464
            17.2.6  Examples for graphs   usgr0e 29552
            17.2.7  Subgraphs   csubgr 29583
            17.2.8  Finite undirected simple graphs   cfusgr 29632
            17.2.9  Neighbors, complete graphs and universal vertices   cnbgr 29648
                  17.2.9.1  Neighbors   cnbgr 29648
                  17.2.9.2  Universal vertices   cuvtx 29701
                  17.2.9.3  Complete graphs   ccplgr 29725
            17.2.10  Vertex degree   cvtxdg 29781
            ​*17.2.11  Regular graphs   crgr 29871
      ​​*17.3  Walks, paths and cycles
            *17.3.1  Walks   cewlks 29911
            17.3.2  Walks for loop-free graphs   lfgrwlkprop 30001
            17.3.3  Trails   ctrls 30004
            17.3.4  Paths and simple paths   cpths 30025
            17.3.5  Closed walks   cclwlks 30085
            17.3.6  Circuits and cycles   ccrcts 30099
            ​*17.3.7  Walks as words   cwwlks 30140
            17.3.8  Walks/paths of length 2 (as length 3 strings)   2wlkdlem1 30240
            17.3.9  Walks in regular graphs   rusgrnumwwlkl1 30286
            ​*17.3.10  Closed walks as words   cclwwlk 30298
                  17.3.10.1  Closed walks as words   cclwwlk 30298
                  17.3.10.2  Closed walks of a fixed length as words   cclwwlkn 30341
                  17.3.10.3  Closed walks on a vertex of a fixed length as words   cclwwlknon 30404
            17.3.11  Examples for walks, trails and paths   0ewlk 30431
            17.3.12  Connected graphs   cconngr 30503
      ​17.4  Eulerian paths and the Konigsberg Bridge problem
            ​*17.4.1  Eulerian paths   ceupth 30514
            ​*17.4.2  The Königsberg Bridge problem   konigsbergvtx 30563
      ​17.5  The Friendship Theorem
            17.5.1  Friendship graphs - basics   cfrgr 30575
            17.5.2  The friendship theorem for small graphs   frgr1v 30588
            17.5.3  Theorems according to Mertzios and Unger   2pthfrgrrn 30599
            ​*17.5.4  Huneke's Proof of the Friendship Theorem   frgrncvvdeqlem1 30616
​PART 18  GUIDES AND MISCELLANEA
      ​18.1  Guides (conventions, explanations, and examples)
            ​*18.1.1  Conventions   conventions 30717
            18.1.2  Natural deduction   natded 30720
            ​*18.1.3  Natural deduction examples   ex-natded5.2 30721
            18.1.4  Definitional examples   ex-or 30738
            18.1.5  Other examples   aevdemo 30777
      ​18.2  Humor
            18.2.1  April Fool's theorem   avril1 30780
      ​18.3  (Future - to be reviewed and classified)
            18.3.1  Planar incidence geometry   cplig 30792
            ​*18.3.2  Aliases kept to prevent broken links   dummylink 30805
​​*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 30807
            19.1.2  Abelian groups   cablo 30862
      ​19.2  Complex vector spaces
            19.2.1  Definition and basic properties   cvc 30876
            19.2.2  Examples of complex vector spaces   cnaddabloOLD 30899
      ​19.3  Normed complex vector spaces
            19.3.1  Definition and basic properties   cnv 30902
            19.3.2  Examples of normed complex vector spaces   cnnv 30995
            19.3.3  Induced metric of a normed complex vector space   imsval 31003
            19.3.4  Inner product   cdip 31018
            19.3.5  Subspaces   css 31039
      ​19.4  Operators on complex vector spaces
            19.4.1  Definitions and basic properties   clno 31058
      ​19.5  Inner product (pre-Hilbert) spaces
            19.5.1  Definition and basic properties   ccphlo 31130
            19.5.2  Examples of pre-Hilbert spaces   cncph 31137
            19.5.3  Properties of pre-Hilbert spaces   isph 31140
      ​19.6  Complex Banach spaces
            19.6.1  Definition and basic properties   ccbn 31180
            19.6.2  Examples of complex Banach spaces   cnbn 31187
            19.6.3  Uniform Boundedness Theorem   ubthlem1 31188
            19.6.4  Minimizing Vector Theorem   minvecolem1 31192
      ​19.7  Complex Hilbert spaces
            19.7.1  Definition and basic properties   chlo 31203
            19.7.2  Standard axioms for a complex Hilbert space   hlex 31216
            19.7.3  Examples of complex Hilbert spaces   cnchl 31234
            19.7.4  Hellinger-Toeplitz Theorem   htthlem 31235
​​*PART 20  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      ​20.1  Axiomatization of complex pre-Hilbert spaces
            20.1.1  Basic Hilbert space definitions   chba 31237
            20.1.2  Preliminary ZFC lemmas   df-hnorm 31286
            ​*20.1.3  Derive the Hilbert space axioms from ZFC set theory   axhilex-zf 31299
            ​*20.1.4  Introduce the vector space axioms for a Hilbert space   ax-hilex 31317
            20.1.5  Vector operations   hvmulex 31329
            20.1.6  Inner product postulates for a Hilbert space   ax-hfi 31397
      ​20.2  Inner product and norms
            20.2.1  Inner product   his5 31404
            20.2.2  Norms   dfhnorm2 31440
            20.2.3  Relate Hilbert space to normed complex vector spaces   hilablo 31478
            20.2.4  Bunjakovaskij-Cauchy-Schwarz inequality   bcsiALT 31497
      ​20.3  Cauchy sequences and completeness axiom
            20.3.1  Cauchy sequences and limits   hcau 31502
            20.3.2  Derivation of the completeness axiom from ZF set theory   hilmet 31512
            20.3.3  Completeness postulate for a Hilbert space   ax-hcompl 31520
            20.3.4  Relate Hilbert space to ZFC pre-Hilbert and Hilbert spaces   hhcms 31521
      ​20.4  Subspaces and projections
            20.4.1  Subspaces   df-sh 31525
            20.4.2  Closed subspaces   df-ch 31539
            20.4.3  Orthocomplements   df-oc 31570
            20.4.4  Subspace sum, span, lattice join, lattice supremum   df-shs 31626
            20.4.5  Projection theorem   pjhthlem1 31709
            20.4.6  Projectors   df-pjh 31713
      ​20.5  Properties of Hilbert subspaces
            20.5.1  Orthomodular law   omlsilem 31720
            20.5.2  Projectors (cont.)   pjhtheu2 31734
            20.5.3  Hilbert lattice operations   sh0le 31758
            20.5.4  Span (cont.) and one-dimensional subspaces   spansn0 31859
            20.5.5  Commutes relation for Hilbert lattice elements   df-cm 31901
            20.5.6  Foulis-Holland theorem   fh1 31936
            20.5.7  Quantum Logic Explorer axioms   qlax1i 31945
            20.5.8  Orthogonal subspaces   chscllem1 31955
            20.5.9  Orthoarguesian laws 5OA and 3OA   5oalem1 31972
            20.5.10  Projectors (cont.)   pjorthi 31987
            20.5.11  Mayet's equation E_3   mayete3i 32046
      ​20.6  Operators on Hilbert spaces
            ​*20.6.1  Operator sum, difference, and scalar multiplication   df-hosum 32048
            20.6.2  Zero and identity operators   df-h0op 32066
            20.6.3  Operations on Hilbert space operators   hoaddcl 32076
            20.6.4  Linear, continuous, bounded, Hermitian, unitary operators and norms   df-nmop 32157
            20.6.5  Linear and continuous functionals and norms   df-nmfn 32163
            20.6.6  Adjoint   df-adjh 32167
            20.6.7  Dirac bra-ket notation   df-bra 32168
            20.6.8  Positive operators   df-leop 32170
            20.6.9  Eigenvectors, eigenvalues, spectrum   df-eigvec 32171
            20.6.10  Theorems about operators and functionals   nmopval 32174
            20.6.11  Riesz lemma   riesz3i 32380
            20.6.12  Adjoints (cont.)   cnlnadjlem1 32385
            20.6.13  Quantum computation error bound theorem   unierri 32422
            20.6.14  Dirac bra-ket notation (cont.)   branmfn 32423
            20.6.15  Positive operators (cont.)   leopg 32440
            20.6.16  Projectors as operators   pjhmopi 32464
      ​20.7  States on a Hilbert lattice and Godowski's equation
            20.7.1  States on a Hilbert lattice   df-st 32529
            20.7.2  Godowski's equation   golem1 32589
      ​20.8  Cover relation, atoms, exchange axiom, and modular symmetry
            20.8.1  Covers relation; modular pairs   df-cv 32597
            20.8.2  Atoms   df-at 32656
            20.8.3  Superposition principle   superpos 32672
            20.8.4  Atoms, exchange and covering properties, atomicity   chcv1 32673
            20.8.5  Irreducibility   chirredlem1 32708
            20.8.6  Atoms (cont.)   atcvat3i 32714
            20.8.7  Modular symmetry   mdsymlem1 32721
​PART 21  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      ​21.1  Mathboxes for user contributions
            21.1.1  Mathbox guidelines   mathbox 32760
      ​21.2  Mathbox for Stefan Allan
      ​21.3  Mathbox for Thierry Arnoux
            21.3.1  Propositional Calculus - misc additions   ad11antr 32765
            21.3.2  Predicate Calculus   sbc2iedf 32778
                  21.3.2.1  Predicate Calculus - misc additions   sbc2iedf 32778
                  21.3.2.2  Restricted quantification - misc additions   ralcom4f 32780
                  21.3.2.3  Equality   eqtrb 32786
                  21.3.2.4  Double restricted existential uniqueness quantification   opsbc2ie 32788
                  21.3.2.5  Double restricted existential uniqueness quantification syntax   w2reu 32790
                  21.3.2.6  Substitution (without distinct variables) - misc additions   sbceqbidf 32799
                  21.3.2.7  Existential "at most one" - misc additions   mo5f 32801
                  21.3.2.8  Existential uniqueness - misc additions   reuxfrdf 32803
                  21.3.2.9  Restricted "at most one" - misc additions   rmoxfrd 32805
                  21.3.2.10  Restricted iota (description binder)   riotaeqbidva 32808
            21.3.3  General Set Theory   dmrab 32809
                  21.3.3.1  Class abstractions (a.k.a. class builders)   dmrab 32809
                  21.3.3.2  Image Sets   abrexdomjm 32819
                  21.3.3.3  Set relations and operations - misc additions   nelun 32825
                  21.3.3.4  Unordered pairs   elpreq 32840
                  21.3.3.5  Unordered triples   tpssg 32849
                  21.3.3.6  Conditional operator - misc additions   ifeqeqx 32854
                  21.3.3.7  Set union   uniinn0 32863
                  21.3.3.8  Indexed union - misc additions   cbviunf 32866
                  21.3.3.9  Indexed intersection - misc additions   iinabrex 32880
                  21.3.3.10  Disjointness - misc additions   disjnf 32881
            21.3.4  Relations and Functions   xpdisjres 32909
                  21.3.4.1  Relations - misc additions   xpdisjres 32909
                  21.3.4.2  Functions - misc additions   fconst7v 32931
                  21.3.4.3  Operations - misc additions   mpomptxf 32989
                  21.3.4.4  The mapping operation   elmaprd 32991
                  21.3.4.5  Support of a function   suppovss 32992
                  21.3.4.6  Explicit Functions with one or two points as a domain   cosnopne 33005
                  21.3.4.7  Isomorphisms - misc. additions   gtiso 33012
                  21.3.4.8  Disjointness (additional proof requiring functions)   disjdsct 33014
                  21.3.4.9  First and second members of an ordered pair - misc additions   df1stres 33015
                  21.3.4.10  Countable Sets   snct 33023
            21.3.5  Real and Complex Numbers   sgnval2 33046
                  21.3.5.1  Complex operations - misc. additions   creq0 33047
                  21.3.5.2  Ordering on reals - misc additions   lt2addrd 33061
                  21.3.5.3  Extended reals - misc additions   nn0mnfxrd 33062
                  21.3.5.4  Extended nonnegative integers - misc additions   xnn0gt0 33080
                  21.3.5.5  Real number intervals - misc additions   joiniooico 33085
                  21.3.5.6  Finite intervals of integers - misc additions   uzssico 33095
                  21.3.5.7  Half-open integer ranges - misc additions   iundisjfi 33107
                  21.3.5.8  The ` # ` (set size) function - misc additions   hashunif 33117
                  21.3.5.9  The greatest common divisor operator - misc. additions   elq2 33122
                  21.3.5.10  Integers   nn0split01 33128
                  21.3.5.11  Decimal numbers   dfdec100 33140
            21.3.6  Real and complex functions   sgnsgn 33141
                  21.3.6.1  Signum (sgn or sign) function - misc. additions   sgnsgn 33141
                  21.3.6.2  Integer powers - misc. additions   nexple 33143
                  21.3.6.3  Indicator Functions (continued)   indsumin 33147
            ​*21.3.7  Decimal expansion   cdp2 33156
                  ​*21.3.7.1  Decimal point   cdp 33173
                  21.3.7.2  Division in the extended real number system   cxdiv 33202
            21.3.8  Words over a set - misc additions   wrdres 33221
                  21.3.8.1  Splicing words (substring replacement)   splfv3 33244
                  21.3.8.2  Cyclic shift of words   1cshid 33245
            21.3.9  Extensible Structures   ressplusf 33249
                  21.3.9.1  Structure restriction operator   ressplusf 33249
                  21.3.9.2  Posets   ressprs 33252
                  21.3.9.3  Complete lattices   clatp0cl 33262
                  21.3.9.4  Order Theory   cmnt 33264
                  21.3.9.5  Extended reals Structure - misc additions   ax-xrssca 33290
                  21.3.9.6  The extended nonnegative real numbers commutative monoid   xrge00 33300
            21.3.10  Algebra   mndcld 33308
                  21.3.10.1  Monoids   mndcld 33308
                  21.3.10.2  Monoids Homomorphisms   abliso 33321
                  21.3.10.3  Groups - misc additions   grpidcld 33325
                  21.3.10.4  Abelian Groups - misc additions   ablcomd 33331
                  21.3.10.5  Finitely supported group sums - misc additions   gsumsubg 33332
                  21.3.10.6  Group or monoid sums over words   gsumwun 33362
                  21.3.10.7  Centralizers and centers - misc additions   cntzun 33365
                  21.3.10.8  The symmetric group   symgfcoeu 33368
                  21.3.10.9  Transpositions   pmtridf1o 33380
                  21.3.10.10  Permutation Signs   psgnid 33383
                  21.3.10.11  Permutation cycles   ctocyc 33392
                  21.3.10.12  The Alternating Group   evpmval 33431
                  21.3.10.13  Signum in an ordered monoid   csgns 33444
                  21.3.10.14  Fixed points   cfxp 33449
                  21.3.10.15  The Archimedean property for generic ordered algebraic structures   cinftm 33462
                  21.3.10.16  Semiring left modules   cslmd 33486
                  21.3.10.17  Simple groups   prmsimpcyc 33514
                  21.3.10.18  Rings - misc additions   ringrngd 33515
                  21.3.10.19  Subrings generated by a set   elrgspnlem1 33528
                  21.3.10.20  The zero ring   irrednzr 33536
                  21.3.10.21  Localization of rings   cerl 33539
                  21.3.10.22  Integral Domains   domnmuln0rd 33563
                  21.3.10.23  Euclidean Domains   ceuf 33577
                  21.3.10.24  Division Rings   rndrhmcl 33583
                  21.3.10.25  The field of rational numbers   qfld 33584
                  21.3.10.26  Subfields   subsdrg 33585
                  21.3.10.27  Field of fractions   cfrac 33589
                  21.3.10.28  Field extensions generated by a set   cfldgen 33597
                  21.3.10.29  Ring homomorphisms - misc additions   rhmdvd 33610
                  21.3.10.30  Scalar restriction operation   cresv 33612
                  21.3.10.31  The commutative ring of gaussian integers   gzcrng 33627
                  21.3.10.32  The archimedean ordered field of real numbers   cnfldfld 33628
                  21.3.10.33  The quotient map and quotient modules   qusker 33635
                  21.3.10.34  The ring of integers modulo ` N `   znfermltl 33647
                  21.3.10.35  Independent sets and families   islinds5 33648
                  21.3.10.36  Ring associates, ring units   dvdsruassoi 33663
                  ​*21.3.10.37  Subgroup sum / Sumset / Minkowski sum   elgrplsmsn 33669
                  21.3.10.38  The quotient map   quslsm 33680
                  21.3.10.39  Ideals   intlidl 33694
                  21.3.10.40  Maximal Ideals   cmxidl 33708
                  21.3.10.41  Local rings   drnglring 33748
                  21.3.10.42  The semiring of ideals of a ring   cidlsrg 33756
                  21.3.10.43  Prime Elements   rprmval 33772
                  21.3.10.44  Unique factorization domains   cufd 33794
                  21.3.10.45  The ring of integers   zringidom 33807
                  21.3.10.46  Associative Algebra   assaassd 33811
                  21.3.10.47  Univariate Polynomials   0ringmon1p 33813
                  21.3.10.48  Polynomial quotient and polynomial remainder   q1pdir 33859
                  21.3.10.49  Multivariate Polynomials   psrbasfsupp 33867
                  21.3.10.50  The ring of symmetric polynomials   csply 33911
                  21.3.10.51  The subring algebra   sra1r 33937
                  21.3.10.52  Division Ring Extensions   drgext0g 33946
                  21.3.10.53  Vector Spaces   lvecdimfi 33952
                  21.3.10.54  Vector Space Dimension   cldim 33955
            21.3.11  Field Extensions   cfldext 33994
                  21.3.11.1  Algebraic numbers   cirng 34039
                  21.3.11.2  Algebraic extensions   calgext 34051
                  21.3.11.3  Minimal polynomials   cminply 34055
                  21.3.11.4  Quadratic Field Extensions   rtelextdg2lem 34082
                  21.3.11.5  Towers of quadratic extentions   fldext2chn 34084
            ​*21.3.12  Constructible Numbers   cconstr 34085
                  21.3.12.1  Impossible constructions   2sqr3minply 34136
            21.3.13  Matrices   csmat 34149
                  21.3.13.1  Submatrices   csmat 34149
                  21.3.13.2  Matrix literals   clmat 34167
                  21.3.13.3  Laplace expansion of determinants   mdetpmtr1 34179
            21.3.14  Topology   ist0cld 34189
                  21.3.14.1  Open maps   txomap 34190
                  21.3.14.2  Topology of the unit circle   qtopt1 34191
                  21.3.14.3  Refinements   reff 34195
                  21.3.14.4  Open cover refinement property   ccref 34198
                  21.3.14.5  Lindelöf spaces   cldlf 34208
                  21.3.14.6  Paracompact spaces   cpcmp 34211
                  ​*21.3.14.7  Spectrum of a ring   crspec 34218
                  21.3.14.8  Pseudometrics   cmetid 34242
                  21.3.14.9  Continuity - misc additions   hauseqcn 34254
                  21.3.14.10  Topology of the closed unit interval   elunitge0 34255
                  21.3.14.11  Topology of ` ( RR X. RR ) `   unicls 34259
                  21.3.14.12  Order topology - misc. additions   cnvordtrestixx 34269
                  21.3.14.13  Continuity in topological spaces - misc. additions   mndpluscn 34282
                  21.3.14.14  Topology of the extended nonnegative real numbers ordered monoid   xrge0hmph 34288
                  21.3.14.15  Limits - misc additions   lmlim 34303
                  21.3.14.16  Univariate polynomials   pl1cn 34311
            21.3.15  Uniform Stuctures and Spaces   chcmp 34312
                  21.3.15.1  Hausdorff uniform completion   chcmp 34312
            21.3.16  Topology and algebraic structures   zringnm 34314
                  21.3.16.1  The norm on the ring of the integer numbers   zringnm 34314
                  21.3.16.2  Topological ` ZZ ` -modules   zlm0 34316
                  21.3.16.3  Canonical embedding of the field of the rational numbers into a division ring   cqqh 34326
                  21.3.16.4  Canonical embedding of the real numbers into a complete ordered field   crrh 34349
                  21.3.16.5  Embedding from the extended real numbers into a complete lattice   cxrh 34372
                  21.3.16.6  Canonical embeddings into the ordered field of the real numbers   zrhre 34375
                  ​*21.3.16.7  Topological Manifolds   cmntop 34378
                  21.3.16.8  Extended sum   cesum 34383
            21.3.17  Mixed Function/Constant operation   cofc 34451
            21.3.18  Abstract measure   csiga 34464
                  21.3.18.1  Sigma-Algebra   csiga 34464
                  21.3.18.2  Generated sigma-Algebra   csigagen 34494
                  ​*21.3.18.3  lambda and pi-Systems, Rings of Sets   ispisys 34508
                  21.3.18.4  The Borel algebra on the real numbers   cbrsiga 34537
                  21.3.18.5  Product Sigma-Algebra   csx 34544
                  21.3.18.6  Measures   cmeas 34551
                  21.3.18.7  The counting measure   cntmeas 34582
                  21.3.18.8  The Lebesgue measure - misc additions   voliune 34585
                  21.3.18.9  The Dirac delta measure   cdde 34588
                  21.3.18.10  The 'almost everywhere' relation   cae 34593
                  21.3.18.11  Measurable functions   cmbfm 34605
                  21.3.18.12  Borel Algebra on ` ( RR X. RR ) `   br2base 34625
                  ​*21.3.18.13  Caratheodory's extension theorem   coms 34647
            21.3.19  Integration   itgeq12dv 34682
                  21.3.19.1  Lebesgue integral - misc additions   itgeq12dv 34682
                  21.3.19.2  Bochner integral   citgm 34683
            21.3.20  Euler's partition theorem   oddpwdc 34710
            21.3.21  Sequences defined by strong recursion   csseq 34739
            21.3.22  Fibonacci Numbers   cfib 34752
            21.3.23  Probability   cprb 34763
                  21.3.23.1  Probability Theory   cprb 34763
                  21.3.23.2  Conditional Probabilities   ccprob 34787
                  21.3.23.3  Real-valued Random Variables   crrv 34796
                  21.3.23.4  Preimage set mapping operator   corvc 34812
                  21.3.23.5  Distribution Functions   orvcelval 34825
                  21.3.23.6  Cumulative Distribution Functions   orvclteel 34829
                  21.3.23.7  Probabilities - example   coinfliplem 34835
                  21.3.23.8  Bertrand's Ballot Problem   ballotlemoex 34842
            21.3.24  Signum (sgn or sign) function - misc. additions   fzssfzo 34895
                  21.3.24.1  Operations on words   ccatmulgnn0dir 34898
            21.3.25  Polynomials with real coefficients - misc additions   plyrecld 34902
            21.3.26  Descartes's rule of signs   signspval 34905
                  21.3.26.1  Sign changes in a word over real numbers   signspval 34905
                  21.3.26.2  Counting sign changes in a word over real numbers   signslema 34915
            21.3.27  Number Theory   iblidicc 34945
                  21.3.27.1  Representations of a number as sums of integers   crepr 34961
                  21.3.27.2  Vinogradov Trigonometric Sums and the Circle Method   cvts 34988
                  21.3.27.3  The Ternary Goldbach Conjecture: Final Statement   ax-hgt749 34997
            21.3.28  Elementary Geometry   cstrkg2d 35017
                  ​*21.3.28.1  Two-dimensional geometry   cstrkg2d 35017
                  21.3.28.2  Morley's Miracle   cgranbtwn 35022
                  21.3.28.3  Outer Five Segment (not used, no need to move to main)   cafs 35025
            ​*21.3.29  LeftPad Project   clpad 35030
      ​​*21.4  Mathbox for Jonathan Ben-Naim
            21.4.1  First-order logic and set theory   bnj170 35053
            21.4.2  Well founded induction and recursion   bnj110 35212
            21.4.3  The existence of a minimal element in certain classes   bnj69 35364
            21.4.4  Well-founded induction   bnj1204 35366
            21.4.5  Well-founded recursion, part 1 of 3   bnj60 35416
            21.4.6  Well-founded recursion, part 2 of 3   bnj1500 35422
            21.4.7  Well-founded recursion, part 3 of 3   bnj1522 35426
      ​21.5  Mathbox for BTernaryTau
            21.5.1  First-order logic   nfan1c 35427
                  21.5.1.1  Auxiliary axiom schemes   nfan1c 35427
            21.5.2  ZF set theory   inv2 35433
                  21.5.2.1  Finitism   prcinf 35492
                  21.5.2.2  Introduce ax-regs   ax-regs 35505
                  21.5.2.3  Derive ax-regs   axregs 35518
                  21.5.2.4  ZFC axioms with reduced distinct variable conditions   axsepg2 35519
                  21.5.2.5  Cardinality without the Axiom of Choice   ckard 35528
                  21.5.2.6  Global choice   gblacfnacd 35552
            21.5.3  Real and complex numbers   zltp1ne 35567
            21.5.4  Graph theory   lfuhgr 35576
                  21.5.4.1  Acyclic graphs   cacycgr 35600
      ​21.6  Mathbox for Mario Carneiro
            21.6.1  Predicate calculus with all distinct variables   ax-7d 35617
            21.6.2  Miscellaneous stuff   quartfull 35623
            21.6.3  Derangements and the Subfactorial   deranglem 35624
            21.6.4  The Erdős-Szekeres theorem   erdszelem1 35649
            21.6.5  The Kuratowski closure-complement theorem   kur14lem1 35664
            21.6.6  Retracts and sections   cretr 35675
            21.6.7  Path-connected and simply connected spaces   cpconn 35677
            21.6.8  Covering maps   ccvm 35713
            21.6.9  Normal numbers   snmlff 35787
            21.6.10  Godel-sets of formulas - part 1   cgoe 35791
            21.6.11  Godel-sets of formulas - part 2   cgon 35890
            21.6.12  Models of ZF   cgze 35904
            ​*21.6.13  Metamath formal systems   cmcn 35918
            21.6.14  Grammatical formal systems   cm0s 36043
            21.6.15  Models of formal systems   cmuv 36063
            21.6.16  Splitting fields   ccpms 36085
            21.6.17  p-adic number fields   czr 36105
      ​​*21.7  Mathbox for Filip Cernatescu
      ​21.8  Mathbox for Paul Chapman
            21.8.1  Real and complex numbers (cont.)   climuzcnv 36129
            21.8.2  Miscellaneous theorems   elfzm12 36133
      ​21.9  Mathbox for Hongxiu Chen
      ​21.10  Mathbox for Adrian Ducourtial
            21.10.1  Propositional calculus   currybi 36146
            21.10.2  Clone theory   ccloneop 36153
      ​21.11  Mathbox for Scott Fenton
            21.11.1  ZFC Axioms in primitive form   axextprim 36159
            21.11.2  Untangled classes   untelirr 36166
            21.11.3  Extra propositional calculus theorems   3jaodd 36173
            21.11.4  Misc. Useful Theorems   nepss 36176
            21.11.5  Properties of real and complex numbers   sqdivzi 36186
            21.11.6  Infinite products   iprodefisumlem 36198
            21.11.7  Factorial limits   faclimlem1 36201
            21.11.8  Greatest common divisor and divisibility   gcd32 36207
            21.11.9  Properties of relationships   dftr6 36209
            21.11.10  Properties of functions and mappings   funpsstri 36224
            21.11.11  Ordinal numbers   elpotr 36237
            21.11.12  Defined equality axioms   axextdfeq 36253
            21.11.13  Hypothesis builders   hbntg 36261
            21.11.14  Well-founded zero, successor, and limits   cwsuc 36266
            21.11.15  Quantifier-free definitions   ctxp 36286
            21.11.16  Alternate ordered pairs   caltop 36414
            21.11.17  Geometry in the Euclidean space   cofs 36440
                  21.11.17.1  Congruence properties   cofs 36440
                  21.11.17.2  Betweenness properties   btwntriv2 36470
                  21.11.17.3  Segment Transportation   ctransport 36487
                  21.11.17.4  Properties relating betweenness and congruence   cifs 36493
                  21.11.17.5  Connectivity of betweenness   btwnconn1lem1 36545
                  21.11.17.6  Segment less than or equal to   csegle 36564
                  21.11.17.7  Outside-of relationship   coutsideof 36577
                  21.11.17.8  Lines and Rays   cline2 36592
            21.11.18  Forward difference   cfwddif 36616
            21.11.19  Rank theorems   rankung 36624
            21.11.20  Hereditarily Finite Sets   chf 36630
            21.11.21  Natural ordinal operations   cnmul 36645
      ​21.12  Mathbox for Gino Giotto
            21.12.1  Equality theorems   rmoeqi 36665
                  21.12.1.1  Inference versions   rmoeqi 36665
                  21.12.1.2  Deduction versions   rmoeqdv 36690
            21.12.2  Change bound variables   in-ax8 36702
                  21.12.2.1  Change bound variables and domains   cbvralvw2 36704
                  21.12.2.2  Change bound variables, deduction versions   cbvmodavw 36728
                  21.12.2.3  Change bound variables and domains, deduction versions   cbvrmodavw2 36761
            21.12.3  Study of ax-mulf usage   mpomulnzcnf 36777
      ​21.13  Mathbox for Jeff Hankins
            21.13.1  Miscellany   a1i14 36778
            21.13.2  Basic topological facts   topbnd 36801
            21.13.3  Topology of the real numbers   ivthALT 36812
            21.13.4  Refinements   cfne 36813
            21.13.5  Neighborhood bases determine topologies   neibastop1 36836
            21.13.6  Lattice structure of topologies   topmtcl 36840
            21.13.7  Filter bases   fgmin 36847
            21.13.8  Directed sets, nets   tailfval 36849
      ​21.14  Mathbox for Anthony Hart
            21.14.1  Propositional Calculus   tb-ax1 36860
            21.14.2  Predicate Calculus   nalfal 36880
            21.14.3  Miscellaneous single axioms   meran1 36888
            21.14.4  Connective Symmetry   negsym1 36894
      ​21.15  Mathbox for Chen-Pang He
            21.15.1  Ordinal topology   ontopbas 36905
      ​21.16  Mathbox for Jeff Hoffman
            21.16.1  Inferences for finite induction on generic function values   fveleq 36928
            21.16.2  gdc.mm   nnssi2 36932
      ​21.17  Mathbox for Matthew House
            21.17.1  Relations on well-ordered indexed unions   weiunval 36939
            21.17.2  Axiom of Transitive Containment   axtco 36948
            21.17.3  Transitive closure of a class   tr0elw 36961
            ​*21.17.4  Stronger axioms of regularity   mh-setind 37013
            21.17.5  Short axioms written in primitive symbols   mh-inf3f1 37018
      ​21.18  Mathbox for Asger C. Ipsen
            21.18.1  Continuous nowhere differentiable functions   dnival 37026
      ​​*21.19  Mathbox for BJ
            *21.19.1  Propositional calculus   bj-mp2c 37095
                  *21.19.1.1  Derived rules of inference   bj-mp2c 37095
                  ​*21.19.1.2  A syntactic theorem   bj-0 37097
                  ​*21.19.1.3  Minimal implicational calculus   bj-poni 37099
                  ​*21.19.1.4  Positive calculus   bj-bisimpl 37111
                  ​*21.19.1.5  Implication and negation   bj-con2com 37119
                  ​*21.19.1.6  Disjunction   bj-jaoi1 37130
                  ​*21.19.1.7  Logical equivalence   bj-dfbi4 37132
                  21.19.1.8  The conditional operator for propositions   bj-consensus 37137
                  ​*21.19.1.9  Propositional calculus: miscellaneous   bj-imbi12 37142
            ​*21.19.2  Modal logic   bj-axdd2 37151
            ​*21.19.3  Provability logic   cprvb 37156
            ​*21.19.4  First-order logic   bj-exexalal 37165
                  21.19.4.1  Universal and existential quantifiers, nonfreeness predicate   bj-exexalal 37165
                  21.19.4.2  Adding ax-gen   bj-genr 37166
                  21.19.4.3  Adding ax-4   bj-almp 37170
                  21.19.4.4  Adding ax-5   bj-spvw 37223
                  21.19.4.5  Equality and substitution   bj-df-sb 37238
                  21.19.4.6  Adding ax-6   bj-spim0 37257
                  21.19.4.7  Adding ax-7   bj-cbvexw 37265
                  21.19.4.8  Membership predicate, ax-8 and ax-9   bj-ax89 37267
                  21.19.4.9  Adding ax-11   bj-alcomexcom 37269
                  21.19.4.10  Adding ax-12   axc11n11 37273
                  ​*21.19.4.11  Really adding ax-12   bj-substax12 37315
                  21.19.4.12  Nonfreeness   wnnf 37317
                  21.19.4.13  Adding ax-13   bj-axc10 37384
                  ​*21.19.4.14  Removing dependencies on ax-13 (and ax-11)   bj-axc10v 37394
                  ​*21.19.4.15  Distinct var metavariables   bj-hbaeb2 37419
                  ​*21.19.4.16  Around ~ equsal   bj-equsal1t 37423
                  ​*21.19.4.17  Some Principia Mathematica proofs   stdpc5t 37428
                  21.19.4.18  Alternate definition of substitution   bj-sbsb 37438
                  21.19.4.19  Lemmas for substitution   bj-sbf3 37440
                  21.19.4.20  Existential uniqueness   bj-eu3f 37442
                  ​*21.19.4.21  First-order logic: miscellaneous   bj-sblem1 37443
            21.19.5  Set theory   eliminable1 37460
                  ​*21.19.5.1  Eliminability of class terms   eliminable1 37460
                  ​*21.19.5.2  Classes without the axiom of extensionality   bj-denoteslem 37472
                  21.19.5.3  Characterization among sets versus among classes   elelb 37498
                  ​*21.19.5.4  The nonfreeness quantifier for classes   bj-nfcsym 37500
                  ​*21.19.5.5  Lemmas for class substitution   bj-sbeqALT 37501
                  21.19.5.6  Removing some axiom requirements and disjoint variable conditions   bj-exlimvmpi 37512
                  ​*21.19.5.7  Class abstractions   bj-elabd2ALT 37527
                  21.19.5.8  Generalized class abstractions   bj-cgab 37535
                  ​*21.19.5.9  Restricted nonfreeness   wrnf 37543
                  ​*21.19.5.10  Russell's paradox   bj-ru1 37545
                  21.19.5.11  Curry's paradox in set theory   currysetlem 37547
                  ​*21.19.5.12  Some disjointness results   bj-n0i 37553
                  ​*21.19.5.13  Complements on direct products   bj-xpimasn 37557
                  ​*21.19.5.14  "Singletonization" and tagging   bj-snsetex 37565
                  ​*21.19.5.15  Tuples of classes   bj-cproj 37592
                  ​*21.19.5.16  Set theory: elementary operations relative to a universe   bj-rcleqf 37627
                  ​*21.19.5.17  Axioms for finite unions   bj-abex 37632
                  ​*21.19.5.18  Set theory: miscellaneous   eleq2w2ALT 37649
                  ​*21.19.5.19  Axioms of separation and replacement   bj-axnul 37675
                  ​*21.19.5.20  Evaluation at a class   bj-evaleq 37679
                  21.19.5.21  Elementwise operations   celwise 37687
                  ​*21.19.5.22  Elementwise intersection (families of sets induced on a subset)   bj-rest00 37689
                  21.19.5.23  Moore collections (complements)   bj-raldifsn 37708
                  21.19.5.24  Maps-to notation for functions with three arguments   bj-0nelmpt 37724
                  ​*21.19.5.25  Currying   csethom 37730
                  ​*21.19.5.26  Setting components of extensible structures   cstrset 37742
            ​*21.19.6  Extended real and complex numbers, real and complex projective lines   bj-nfald 37745
                  21.19.6.1  Complements on class abstractions of ordered pairs and binary relations   bj-nfald 37745
                  ​*21.19.6.2  Identity relation (complements)   bj-opabssvv 37760
                  ​*21.19.6.3  Functionalized identity (diagonal in a Cartesian square)   cdiag2 37782
                  ​*21.19.6.4  Direct image and inverse image   cimdir 37788
                  ​*21.19.6.5  Extended numbers and projective lines as sets   cfractemp 37806
                  ​*21.19.6.6  Addition and opposite   caddcc 37847
                  ​*21.19.6.7  Order relation on the extended reals   cltxr 37851
                  ​*21.19.6.8  Argument, multiplication and inverse   carg 37853
                  21.19.6.9  The canonical bijection from the finite ordinals   ciomnn 37859
                  21.19.6.10  Divisibility   cnnbar 37870
            ​*21.19.7  Monoids   bj-smgrpssmgm 37878
                  ​*21.19.7.1  Finite sums in monoids   cfinsum 37893
            ​*21.19.8  Affine, Euclidean, and Cartesian geometry   bj-fvimacnv0 37896
                  *21.19.8.1  Real vector spaces   bj-fvimacnv0 37896
                  ​*21.19.8.2  Complex numbers (supplements)   bj-subcom 37918
                  ​*21.19.8.3  Barycentric coordinates   bj-bary1lem 37920
            21.19.9  Monoid of endomorphisms   cend 37923
      ​21.20  Mathbox for Jim Kingdon
            21.20.1  Circle constant   taupilem3 37929
            21.20.2  Number theory   dfgcd3 37934
            21.20.3  Real numbers   irrdifflemf 37935
      ​21.21  Mathbox for ML
            21.21.1  Miscellaneous   csbrecsg 37940
            21.21.2  Cartesian exponentiation   cfinxp 37995
            21.21.3  Topology   iunctb2 38015
                  ​*21.21.3.1  Pi-base theorems   pibp16 38025
      ​21.22  Mathbox for Wolf Lammen
            21.22.1  1. Bootstrapping   wl-section-boot 38034
            21.22.2  Implication chains   wl-section-impchain 38058
            21.22.3  Theorems around the conditional operator   wl-ifp-ncond1 38076
            21.22.4  Alternative development of hadd, cadd   wl-df-3xor 38080
            21.22.5  An alternative axiom ~ ax-13   ax-wl-13v 38105
            21.22.6  Bootstrapping set theory with classes   wl-cleq-0 38107
            21.22.7  Other stuff   wl-mps 38128
      ​21.23  Mathbox for Brendan Leahy
      ​21.24  Mathbox for Jeff Madsen
            21.24.1  Logic and set theory   unirep 38331
            21.24.2  Real and complex numbers; integers   filbcmb 38357
            21.24.3  Sequences and sums   sdclem2 38359
            21.24.4  Topology   subspopn 38369
            21.24.5  Metric spaces   metf1o 38372
            21.24.6  Continuous maps and homeomorphisms   constcncf 38379
            21.24.7  Boundedness   ctotbnd 38383
            21.24.8  Isometries   cismty 38415
            21.24.9  Heine-Borel Theorem   heibor1lem 38426
            21.24.10  Banach Fixed Point Theorem   bfplem1 38439
            21.24.11  Euclidean space   crrn 38442
            21.24.12  Intervals (continued)   ismrer1 38455
            21.24.13  Operation properties   cass 38459
            21.24.14  Groups and related structures   cmagm 38465
            21.24.15  Group homomorphism and isomorphism   cghomOLD 38500
            21.24.16  Rings   crngo 38511
            21.24.17  Division Rings   cdrng 38565
            21.24.18  Ring homomorphisms   crngohom 38577
            21.24.19  Commutative rings   ccm2 38606
            21.24.20  Ideals   cidl 38624
            21.24.21  Prime rings and integral domains   cprrng 38663
            21.24.22  Ideal generators   cigen 38676
      ​21.25  Mathbox for Giovanni Mascellani
            ​*21.25.1  Tools for automatic proof building   efald2 38695
            ​*21.25.2  Tseitin axioms   fald 38746
            ​*21.25.3  Equality deductions   iuneq2f 38773
            ​*21.25.4  Miscellanea   orcomdd 38784
      ​21.26  Mathbox for Peter Mazsa
            21.26.1  Notations   cxrn 38791
            21.26.2  Preparatory theorems   el2v1 38846
            21.26.3  Range Cartesian product   df-xrn 38997
            21.26.4  Relations   df-rels 39057
            21.26.5  Quotient map (coset map)   df-qmap 39063
            21.26.6  Lifts, shifts, successor, and predecessor   df-adjliftmap 39072
            21.26.7  Cosets by ` R `   df-coss 39118
            21.26.8  Subset relations   df-ssr 39195
            21.26.9  Reflexivity   df-refs 39207
            21.26.10  Converse reflexivity   df-cnvrefs 39222
            21.26.11  Symmetry   df-syms 39239
            21.26.12  Reflexivity and symmetry   symrefref2 39264
            21.26.13  Transitivity   df-trs 39273
            21.26.14  Equivalence relations   df-eqvrels 39285
            21.26.15  Redundancy   df-redunds 39324
            21.26.16  Domain quotients   df-dmqss 39339
            21.26.17  Equivalence relations on domain quotients   df-ers 39365
            21.26.18  Functions   df-funss 39382
            21.26.19  Disjoints vs. converse functions   df-disjss 39405
            21.26.20  Antisymmetry   df-antisymrel 39480
            21.26.21  Partitions: disjoints on domain quotients   df-parts 39485
            21.26.22  Partition-Equivalence Theorems   disjim 39501
            21.26.23  Type-safe Partition-Equivalence: PetParts, PetErs, Pet2Parts, Pet2Ers   df-petparts 39585
      ​21.27  Mathbox for Rodolfo Medina
            21.27.1  Partitions   prtlem60 39595
      ​​*21.28  Mathbox for Norm Megill
            *21.28.1  Obsolete schemes ax-c4,c5,c7,c10,c11,c11n,c15,c9,c14,c16   ax-c5 39625
            ​*21.28.2  Rederive new axioms ax-4, ax-10, ax-6, ax-12, ax-13 from old   axc5 39635
            ​*21.28.3  Legacy theorems using obsolete axioms   ax5ALT 39649
            21.28.4  Experiments with weak deduction theorem   elimhyps 39703
            21.28.5  Miscellanea   cnaddcom 39714
            21.28.6  Atoms, hyperplanes, and covering in a left vector space (or module)   clsa 39716
            21.28.7  Functionals and kernels of a left vector space (or module)   clfn 39799
            21.28.8  Opposite rings and dual vector spaces   cld 39865
            21.28.9  Ortholattices and orthomodular lattices   cops 39914
            21.28.10  Atomic lattices with covering property   ccvr 40004
            21.28.11  Hilbert lattices   chlt 40092
            21.28.12  Projective geometries based on Hilbert lattices   clln 40233
            21.28.13  Construction of a vector space from a Hilbert lattice   cdlema1N 40533
            21.28.14  Construction of involution and inner product from a Hilbert lattice   clpoN 42222
      ​21.29  Mathbox for metakunt
            21.29.1  Commutative Semiring   ccsrg 42704
            21.29.2  General helpful statements   rhmzrhval 42707
            21.29.3  Some gcd and lcm results   12gcd5e1 42738
            21.29.4  Least common multiple inequality theorem   3factsumint1 42756
            21.29.5  Logarithm inequalities   3exp7 42788
            21.29.6  Miscellaneous results for AKS formalisation   intlewftc 42796
            21.29.7  Sticks and stones   sticksstones1 42881
            21.29.8  Continuation AKS   aks6d1c6lem1 42905
      ​21.30  Mathbox for Luke Murphy
            21.30.1  Solutions of quadratic equations   quadfac 42940
            21.30.2  April Fool's theorem   25or6to4 42941
      ​21.31  Mathbox for Steven Nguyen
            21.31.1  Utility theorems   jarrii 42942
            ​*21.31.2  Arithmetic theorems   c0exALT 42988
            21.31.3  Exponents and divisibility   oexpreposd 43051
            21.31.4  Trigonometry and Calculus   tanhalfpim 43078
            ​*21.31.5  Independence of ax-mulcom   cresub 43094
            21.31.6  Structures   sn-base0 43237
            ​*21.31.7  Projective spaces   cprjsp 43303
            21.31.8  Basic reductions for Fermat's Last Theorem   dffltz 43336
            ​*21.31.9  Exemplar theorems   iddii 43366
                  ​*21.31.9.1  Standard replacements of ax-10 , ax-11 , ax-12   nfa1w 43377
      ​21.32  Mathbox for Igor Ieskov
      ​21.33  Mathbox for OpenAI
      ​21.34  Mathbox for Stefan O'Rear
            21.34.1  Additional elementary logic and set theory   moxfr 43393
            21.34.2  Additional theory of functions   imaiinfv 43394
            21.34.3  Additional topology   elrfi 43395
            21.34.4  Characterization of closure operators. Kuratowski closure axioms   ismrcd1 43399
            21.34.5  Algebraic closure systems   cnacs 43403
            21.34.6  Miscellanea 1. Map utilities   constmap 43414
            21.34.7  Miscellanea for polynomials   mptfcl 43421
            21.34.8  Multivariate polynomials over the integers   cmzpcl 43422
            21.34.9  Miscellanea for Diophantine sets 1   coeq0i 43454
            21.34.10  Diophantine sets 1: definitions   cdioph 43456
            21.34.11  Diophantine sets 2 miscellanea   ellz1 43468
            21.34.12  Diophantine sets 2: union and intersection. Monotone Boolean algebra   diophin 43473
            21.34.13  Diophantine sets 3: construction   diophrex 43476
            21.34.14  Diophantine sets 4 miscellanea   2sbcrex 43485
            21.34.15  Diophantine sets 4: Quantification   rexrabdioph 43491
            21.34.16  Diophantine sets 5: Arithmetic sets   rabdiophlem1 43498
            21.34.17  Diophantine sets 6: reusability. renumbering of variables   eldioph4b 43508
            21.34.18  Pigeonhole Principle and cardinality helpers   fphpd 43513
            21.34.19  A non-closed set of reals is infinite   rencldnfilem 43517
            21.34.20  Lagrange's rational approximation theorem   irrapxlem1 43519
            21.34.21  Pell equations 1: A nontrivial solution always exists   pellexlem1 43526
            21.34.22  Pell equations 2: Algebraic number theory of the solution set   csquarenn 43533
            21.34.23  Pell equations 3: characterizing fundamental solution   infmrgelbi 43575
            ​*21.34.24  Logarithm laws generalized to an arbitrary base   reglogcl 43587
            21.34.25  Pell equations 4: the positive solution group is infinite cyclic   pellfund14 43595
            21.34.26  X and Y sequences 1: Definition and recurrence laws   crmx 43597
            21.34.27  Ordering and induction lemmas for the integers   monotuz 43638
            21.34.28  X and Y sequences 2: Order properties   rmxypos 43644
            21.34.29  Congruential equations   congtr 43662
            21.34.30  Alternating congruential equations   acongid 43672
            21.34.31  Additional theorems on integer divisibility   coprmdvdsb 43682
            21.34.32  X and Y sequences 3: Divisibility properties   jm2.18 43685
            21.34.33  X and Y sequences 4: Diophantine representability of Y   jm2.27a 43702
            21.34.34  X and Y sequences 5: Diophantine representability of X, ^, _C   rmxdiophlem 43712
            21.34.35  Uncategorized stuff not associated with a major project   setindtr 43721
            21.34.36  More equivalents of the Axiom of Choice   axac10 43730
            21.34.37  Finitely generated left modules   clfig 43764
            21.34.38  Noetherian left modules I   clnm 43772
            21.34.39  Addenda for structure powers   pwssplit4 43786
            21.34.40  Every set admits a group structure iff choice   unxpwdom3 43792
            21.34.41  Noetherian rings and left modules II   clnr 43806
            21.34.42  Hilbert's Basis Theorem   cldgis 43818
            21.34.43  Additional material on polynomials [DEPRECATED]   cmnc 43828
            21.34.44  Degree and minimal polynomial of algebraic numbers   cdgraa 43837
            21.34.45  Algebraic integers I   citgo 43854
            21.34.46  Endomorphism algebra   cmend 43868
            21.34.47  Cyclic groups and order   idomodle 43888
            21.34.48  Cyclotomic polynomials   ccytp 43894
            21.34.49  Miscellaneous topology   fgraphopab 43900
      ​21.35  Mathbox for Noam Pasman
      ​21.36  Mathbox for Jon Pennant
      ​21.37  Mathbox for Richard Penner
            21.37.1  Set Theory and Ordinal Numbers   uniel 43914
            21.37.2  Natural addition of Cantor normal forms   oawordex2 44023
            21.37.3  Surreal Contributions   abeqabi 44104
            21.37.4  Short Studies   nlimsuc 44137
                  21.37.4.1  Additional work on conditional logical operator   ifpan123g 44155
                  21.37.4.2  Sophisms   rp-fakeimass 44208
                  ​*21.37.4.3  Finite Sets   rp-isfinite5 44213
                  21.37.4.4  General Observations   intabssd 44215
                  21.37.4.5  Infinite Sets   pwelg 44256
                  ​*21.37.4.6  Finite intersection property   fipjust 44261
                  21.37.4.7  RP ADDTO: Subclasses and subsets   rababg 44270
                  21.37.4.8  RP ADDTO: The intersection of a class   elinintab 44271
                  21.37.4.9  RP ADDTO: Theorems requiring subset and intersection existence   elinintrab 44273
                  21.37.4.10  RP ADDTO: Relations   xpinintabd 44276
                  ​*21.37.4.11  RP ADDTO: Functions   elmapintab 44292
                  ​*21.37.4.12  RP ADDTO: Finite induction (for finite ordinals)   cnvcnvintabd 44296
                  21.37.4.13  RP ADDTO: First and second members of an ordered pair   elcnvlem 44297
                  21.37.4.14  RP ADDTO: The reflexive and transitive properties of relations   undmrnresiss 44300
                  21.37.4.15  RP ADDTO: Basic properties of closures   cleq2lem 44304
                  21.37.4.16  RP REPLACE: Definitions and basic properties of transitive closures   trcleq2lemRP 44326
                  ​*21.37.4.17  Additions for square root; absolute value   sqrtcvallem1 44327
            21.37.5  Additional statements on relations and subclasses   al3im 44343
                  21.37.5.1  Transitive relations (not to be confused with transitive classes)   trrelind 44361
                  21.37.5.2  Reflexive closures   crcl 44368
                  ​*21.37.5.3  Finite relationship composition   relexp2 44373
                  21.37.5.4  Transitive closure of a relation   dftrcl3 44416
                  ​*21.37.5.5  Adapted from Frege   frege77d 44442
            ​*21.37.6  Propositions from _Begriffsschrift_   dfxor4 44462
                  *21.37.6.1  _Begriffsschrift_ Chapter I   dfxor4 44462
                  ​*21.37.6.2  _Begriffsschrift_ Notation hints   whe 44468
                  21.37.6.3  _Begriffsschrift_ Chapter II Implication   ax-frege1 44486
                  21.37.6.4  _Begriffsschrift_ Chapter II Implication and Negation   axfrege28 44525
                  ​*21.37.6.5  _Begriffsschrift_ Chapter II with logical equivalence   axfrege52a 44552
                  21.37.6.6  _Begriffsschrift_ Chapter II with equivalence of sets   axfrege52c 44583
                  ​*21.37.6.7  _Begriffsschrift_ Chapter II with equivalence of classes   frege53c 44610
                  ​*21.37.6.8  _Begriffsschrift_ Chapter III Properties hereditary in a sequence   dffrege69 44628
                  ​*21.37.6.9  _Begriffsschrift_ Chapter III Following in a sequence   dffrege76 44635
                  ​*21.37.6.10  _Begriffsschrift_ Chapter III Member of sequence   dffrege99 44658
                  ​*21.37.6.11  _Begriffsschrift_ Chapter III Single-valued procedures   dffrege115 44674
            ​*21.37.7  Exploring Topology via Seifert and Threlfall   enrelmap 44693
                  *21.37.7.1  Equinumerosity of sets of relations and maps   enrelmap 44693
                  ​*21.37.7.2  Generic Pseudoclosure Spaces, Pseudointerior Spaces, and Pseudoneighborhoods   or3or 44719
                  ​*21.37.7.3  Generic Neighborhood Spaces   gneispa 44826
            ​*21.37.8  Exploring Higher Homotopy via Kerodon   k0004lem1 44843
                  *21.37.8.1  Simplicial Sets   k0004lem1 44843
      ​21.38  Mathbox for Stanislas Polu
            21.38.1  IMO Problems   wwlemuld 44852
                  21.38.1.1  IMO 1972 B2   wwlemuld 44852
            ​*21.38.2  INT Inequalities Proof Generator   int-addcomd 44869
            ​*21.38.3  N-Digit Addition Proof Generator   unitadd 44891
            21.38.4  AM-GM (for k = 2,3,4)   gsumws3 44892
      ​21.39  Mathbox for Rohan Ridenour
            21.39.1  Misc   spALT 44897
            21.39.2  Monoid rings   cmnring 44905
            21.39.3  Shorter primitive equivalent of ax-groth   gru0eld 44923
                  21.39.3.1  Grothendieck universes are closed under collection   gru0eld 44923
                  21.39.3.2  Minimal universes   ismnu 44941
                  21.39.3.3  Primitive equivalent of ax-groth   expandan 44968
      ​21.40  Mathbox for Steve Rodriguez
            21.40.1  Miscellanea   nanorxor 44985
            21.40.2  Ratio test for infinite series convergence and divergence   dvgrat 44992
            21.40.3  Multiples   reldvds 44995
            21.40.4  Function operations   caofcan 45003
            21.40.5  Calculus   lhe4.4ex1a 45009
            21.40.6  The generalized binomial coefficient operation   cbcc 45016
            21.40.7  Binomial series   uzmptshftfval 45026
      ​21.41  Mathbox for Andrew Salmon
            21.41.1  Principia Mathematica * 10   pm10.12 45038
            21.41.2  Principia Mathematica * 11   2alanimi 45052
            21.41.3  Predicate Calculus   sbeqal1 45078
            21.41.4  Principia Mathematica * 13 and * 14   pm13.13a 45087
            21.41.5  Set Theory   elnev 45117
            21.41.6  Arithmetic   addcomgi 45134
            21.41.7  Geometry   cplusr 45135
      ​​*21.42  Mathbox for Alan Sare
            21.42.1  Auxiliary theorems for the Virtual Deduction tool   idiALT 45157
            21.42.2  Supplementary unification deductions   bi1imp 45161
            21.42.3  Conventional Metamath proofs, some derived from VD proofs   iidn3 45180
            21.42.4  What is Virtual Deduction?   wvd1 45248
            21.42.5  Virtual Deduction Theorems   df-vd1 45249
            21.42.6  Theorems proved using Virtual Deduction   trsspwALT 45496
            21.42.7  Theorems proved using Virtual Deduction with mmj2 assistance   simplbi2VD 45524
            21.42.8  Virtual Deduction transcriptions of textbook proofs   sb5ALTVD 45591
            21.42.9  Theorems proved using conjunction-form Virtual Deduction   elpwgdedVD 45595
            21.42.10  Theorems with a VD proof in conventional notation derived from a VD proof   suctrALT3 45602
            ​*21.42.11  Theorems with a proof in conventional notation derived from a VD proof   notnotrALT2 45605
      ​21.43  Mathbox for Eric Schmidt
            21.43.1  Miscellany   rspesbcd 45616
            21.43.2  Study of dfbi1ALT   dfbi1ALTa 45618
            21.43.3  Relation-preserving functions   wrelp 45621
            21.43.4  Orbits   orbitex 45634
            21.43.5  Well-founded sets   trwf 45638
            21.43.6  Absoluteness in transitive models   ralabso 45647
            21.43.7  Lemmas for showing axioms hold in models   traxext 45656
            21.43.8  The class of well-founded sets is a model for ZFC   wfaxext 45672
            21.43.9  Permutation models   brpermmodel 45682
            21.43.10  Isomorphism of finite ordinals and non-negative integers   hashnna 45698
      ​21.44  Mathbox for Glauco Siliprandi
            21.44.1  Miscellanea   evth2f 45705
            21.44.2  Functions   fnresdmss 45856
            21.44.3  Ordering on real numbers - Real and complex numbers basic operations   sub2times 45962
            21.44.4  Real intervals   gtnelioc 46177
            21.44.5  Finite sums   fsummulc1f 46257
            21.44.6  Finite multiplication of numbers and finite multiplication of functions   fmul01 46266
            21.44.7  Limits   clim1fr1 46287
                  21.44.7.1  Inferior limit (lim inf)   clsi 46435
                  ​*21.44.7.2  Limits for sequences of extended real numbers   clsxlim 46502
            21.44.8  Trigonometry   coseq0 46548
            21.44.9  Continuous Functions   mulcncff 46554
            21.44.10  Derivatives   dvsinexp 46595
            21.44.11  Integrals   itgsin0pilem1 46634
            21.44.12  Stone Weierstrass theorem - real version   stoweidlem1 46685
            21.44.13  Wallis' product for π   wallispilem1 46749
            21.44.14  Stirling's approximation formula for ` n ` factorial   stirlinglem1 46758
            21.44.15  Dirichlet kernel   dirkerval 46775
            21.44.16  Fourier Series   fourierdlem1 46792
            21.44.17  e is transcendental   elaa2lem 46917
            21.44.18  n-dimensional Euclidean space   rrxtopn 46968
            21.44.19  Basic measure theory   csalg 46992
                  ​*21.44.19.1  σ-Algebras   csalg 46992
                  21.44.19.2  Sum of nonnegative extended reals   csumge0 47046
                  ​*21.44.19.3  Measures   cmea 47133
                  ​*21.44.19.4  Outer measures and Caratheodory's construction   come 47173
                  ​*21.44.19.5  Lebesgue measure on n-dimensional Real numbers   covoln 47220
                  ​*21.44.19.6  Measurable functions   csmblfn 47379
      ​21.45  Mathbox for Saveliy Skresanov
            21.45.1  Ceva's theorem   sigarval 47534
            21.45.2  Simple groups   simpcntrab 47554
      ​21.46  Mathbox for Ender Ting
            21.46.1  Interesting facts   et-ltneverrefl 47555
            21.46.2  Increasing sequences and subsequences   ormklocald 47560
            21.46.3  Scratchpad for number theory   evenwodadd 47573
            21.46.4  Scratchpad for math on real numbers   squeezedltsq 47574
      ​21.47  Mathbox for Jarvin Udandy
      ​21.48  Mathbox for Adhemar
            ​*21.48.1  Minimal implicational calculus   adh-minim 47705
      ​21.49  Mathbox for Alexander van der Vekens
            21.49.1  General auxiliary theorems (1)   n0nsn2el 47729
                  21.49.1.1  Unordered and ordered pairs - extension for singletons   n0nsn2el 47729
                  21.49.1.2  Unordered and ordered pairs - extension for unordered pairs   elprneb 47733
                  21.49.1.3  Unordered and ordered pairs - extension for ordered pairs   oppr 47734
                  21.49.1.4  Relations - extension   eubrv 47739
                  21.49.1.5  Definite description binder (inverted iota) - extension   iota0def 47742
                  21.49.1.6  Functions - extension   fveqvfvv 47744
            21.49.2  Alternative for Russell's definition of a description binder   caiota 47787
            21.49.3  Double restricted existential uniqueness   r19.32 47802
                  21.49.3.1  Restricted quantification (extension)   r19.32 47802
                  21.49.3.2  Restricted uniqueness and "at most one" quantification   reuf1odnf 47811
                  21.49.3.3  Analogs to Existential uniqueness (double quantification)   2reu3 47814
                  21.49.3.4  Additional theorems for double restricted existential uniqueness   2reu8i 47817
            ​*21.49.4  Alternative definitions of function and operation values   wdfat 47820
                  21.49.4.1  Restricted quantification (extension)   ralbinrald 47826
                  21.49.4.2  The universal class (extension)   nvelim 47827
                  21.49.4.3  Introduce the Axiom of Power Sets (extension)   alneu 47828
                  21.49.4.4  Predicate "defined at"   dfateq12d 47830
                  21.49.4.5  Alternative definition of the value of a function   dfafv2 47836
                  21.49.4.6  Alternative definition of the value of an operation   aoveq123d 47882
            ​*21.49.5  Alternative definitions of function values (2)   cafv2 47912
            21.49.6  General auxiliary theorems (2)   an4com24 47972
                  21.49.6.1  Logical conjunction - extension   an4com24 47972
                  21.49.6.2  Abbreviated conjunction and disjunction of three wff's - extension   3an4ancom24 47973
                  21.49.6.3  Negated membership (alternative)   cnelbr 47975
                  21.49.6.4  The empty set - extension   ralralimp 47982
                  21.49.6.5  Indexed union and intersection - extension   otiunsndisjX 47983
                  21.49.6.6  Functions - extension   fvifeq 47984
                  21.49.6.7  Maps-to notation - extension   fvmptrab 47996
                  21.49.6.8  Subtraction - extension   cnambpcma 47998
                  21.49.6.9  Ordering on reals (cont.) - extension   leaddsuble 48001
                  21.49.6.10  Imaginary and complex number properties - extension   readdcnnred 48007
                  21.49.6.11  Nonnegative integers (as a subset of complex numbers) - extension   nn0resubcl 48012
                  21.49.6.12  Integers (as a subset of complex numbers) - extension   zgeltp1eq 48013
                  21.49.6.13  Decimal arithmetic - extension   1t10e1p1e11 48014
                  21.49.6.14  Upper sets of integers - extension   eluzge0nn0 48016
                  21.49.6.15  Infinity and the extended real number system (cont.) - extension   nltle2tri 48017
                  21.49.6.16  Finite intervals of integers - extension   ssfz12 48018
                  21.49.6.17  Half-open integer ranges - extension   fzopred 48027
                  21.49.6.18  The floor and ceiling functions - extension   2ltceilhalf 48036
                  21.49.6.19  The modulo (remainder) operation - extension   fldivmod 48048
                  21.49.6.20  The infinite sequence builder "seq"   smonoord 48081
                  21.49.6.21  Integer powers - extension   2timesltsq 48082
                  21.49.6.22  Finite and infinite sums - extension   fsummsndifre 48084
                  21.49.6.23  The divides relation - extension   nndivides2 48088
                  21.49.6.24  Extensible structures - extension   setsidel 48092
            ​*21.49.7  Preimages of function values   preimafvsnel 48095
            ​*21.49.8  Partitions of real intervals   ciccp 48129
            21.49.9  Shifting functions with an integer range domain   fargshiftfv 48155
            21.49.10  Words over a set (extension)   lswn0 48160
                  21.49.10.1  Last symbol of a word - extension   lswn0 48160
            21.49.11  Unordered pairs   wich 48161
                  21.49.11.1  Interchangeable setvar variables   wich 48161
                  21.49.11.2  Set of unordered pairs   sprid 48190
                  ​*21.49.11.3  Proper (unordered) pairs   prpair 48217
                  21.49.11.4  Set of proper unordered pairs   cprpr 48228
            21.49.12  Number theory (extension)   nprmmul1 48243
                  21.49.12.1  Properties of non-prime numbers   nprmmul1 48243
                  ​*21.49.12.2  Fermat numbers   cfmtno 48246
                  ​*21.49.12.3  Mersenne primes   m2prm 48310
                  21.49.12.4  Proth's theorem   modexp2m1d 48331
                  21.49.12.5  The prime-counting function according to Ján Mináč   nprmdvdsfacm1lem1 48339
                  21.49.12.6  Solutions of quadratic equations   quad1 48352
            ​*21.49.13  Even and odd numbers   ceven 48356
                  21.49.13.1  Definitions and basic properties   ceven 48356
                  21.49.13.2  Alternate definitions using the "divides" relation   dfeven2 48381
                  21.49.13.3  Alternate definitions using the "modulo" operation   dfeven3 48390
                  21.49.13.4  Alternate definitions using the "gcd" operation   iseven5 48396
                  21.49.13.5  Theorems of part 5 revised   zneoALTV 48401
                  21.49.13.6  Theorems of part 6 revised   odd2np1ALTV 48406
                  21.49.13.7  Theorems of AV's mathbox revised   0evenALTV 48420
                  21.49.13.8  Additional theorems   epoo 48435
                  21.49.13.9  Perfect Number Theorem (revised)   perfectALTVlem1 48453
            21.49.14  Number theory (extension 2)   cfppr 48456
                  ​*21.49.14.1  Fermat pseudoprimes   cfppr 48456
                  ​*21.49.14.2  Goldbach's conjectures   cgbe 48477
            21.49.15  Graph theory (extension)   cclnbgr 48550
                  21.49.15.1  Closed neighborhood of a vertex   cclnbgr 48550
                  ​*21.49.15.2  Semiclosed and semiopen neighborhoods (experimental)   dfsclnbgr2 48578
                  21.49.15.3  Induced subgraphs   cisubgr 48592
                  ​*21.49.15.4  Isomorphisms of graphs   cgrisom 48606
                  ​*21.49.15.5  Triangles in graphs   cgrtri 48669
                  ​*21.49.15.6  Star graphs   cstgr 48683
                  ​*21.49.15.7  Local isomorphisms of graphs   cgrlim 48708
                  ​*21.49.15.8  Generalized Petersen graphs   cgpg 48772
                  21.49.15.9  Loop-free graphs - extension   1hegrlfgr 48864
                  21.49.15.10  Walks - extension   cupwlks 48865
                  21.49.15.11  Edges of graphs expressed as sets of unordered pairs   upgredgssspr 48875
            21.49.16  Monoids (extension)   ovn0dmfun 48888
                  21.49.16.1  Auxiliary theorems   ovn0dmfun 48888
                  21.49.16.2  Magmas, Semigroups and Monoids (extension)   plusfreseq 48896
                  21.49.16.3  Examples and counterexamples for magmas, semigroups and monoids (extension)   opmpoismgm 48899
                  21.49.16.4  Group sum operation (extension 1)   gsumsplit2f 48912
            ​*21.49.17  Magmas and internal binary operations (alternate approach)   ccllaw 48915
                  *21.49.17.1  Laws for internal binary operations   ccllaw 48915
                  ​*21.49.17.2  Internal binary operations   cintop 48928
                  21.49.17.3  Alternative definitions for magmas and semigroups   cmgm2 48947
            21.49.18  Rings (extension)   lmod0rng 48961
                  21.49.18.1  Nonzero rings (extension)   lmod0rng 48961
                  21.49.18.2  Ideals as non-unital rings   lidldomn1 48963
                  21.49.18.3  The non-unital ring of even integers   0even 48969
                  21.49.18.4  A constructed not unital ring   cznrnglem 48991
                  ​*21.49.18.5  The category of non-unital rings (alternate definition)   crngcALTV 48995
                  ​*21.49.18.6  The category of (unital) rings (alternate definition)   cringcALTV 49019
            ​*21.49.19  Prime rings (and integral domains)   cprmrng 49066
            21.49.20  Basic algebraic structures (extension)   eliunxp2 49081
                  21.49.20.1  Auxiliary theorems   eliunxp2 49081
                  21.49.20.2  The binomial coefficient operation (extension)   bcpascm1 49098
                  21.49.20.3  The ` ZZ `-module ` ZZ X. ZZ `   zlmodzxzlmod 49101
                  21.49.20.4  Group sum operation (extension 2)   mgpsumunsn 49108
                  21.49.20.5  Symmetric groups (extension)   exple2lt6 49111
                  21.49.20.6  Divisibility (extension)   invginvrid 49114
                  21.49.20.7  The support of functions (extension)   rmsupp0 49115
                  21.49.20.8  Finitely supported functions (extension)   rmsuppfi 49119
                  21.49.20.9  Left modules (extension)   lmodvsmdi 49126
                  21.49.20.10  Associative algebras (extension)   assaascl0 49128
                  21.49.20.11  Univariate polynomials (extension)   ply1vr1smo 49130
                  21.49.20.12  Univariate polynomials (examples)   linply1 49140
            21.49.21  Linear algebra (extension)   cdmatalt 49143
                  ​*21.49.21.1  The subalgebras of diagonal and scalar matrices (extension)   cdmatalt 49143
                  ​*21.49.21.2  Linear combinations   clinc 49151
                  ​*21.49.21.3  Linear independence   clininds 49187
                  21.49.21.4  Simple left modules and the ` ZZ `-module   lmod1lem1 49234
                  21.49.21.5  Differences between (left) modules and (left) vector spaces   lvecpsslmod 49254
            21.49.22  Complexity theory   suppdm 49257
                  21.49.22.1  Auxiliary theorems   suppdm 49257
                  21.49.22.2  Even and odd integers   nn0onn0ex 49270
                  21.49.22.3  The natural logarithm on complex numbers (extension)   logcxp0 49282
                  21.49.22.4  Division of functions   cfdiv 49284
                  21.49.22.5  Upper bounds   cbigo 49294
                  21.49.22.6  Logarithm to an arbitrary base (extension)   rege1logbrege0 49305
                  ​*21.49.22.7  The binary logarithm   fldivexpfllog2 49312
                  21.49.22.8  Binary length   cblen 49316
                  ​*21.49.22.9  Digits   cdig 49342
                  21.49.22.10  Nonnegative integer as sum of its shifted digits   dignn0flhalflem1 49362
                  21.49.22.11  Algorithms for the multiplication of nonnegative integers   nn0mulfsum 49371
                  ​*21.49.22.12  N-ary functions   cnaryf 49373
                  ​*21.49.22.13  The Ackermann function   citco 49404
            21.49.23  Elementary geometry (extension)   fv1prop 49446
                  21.49.23.1  Auxiliary theorems   fv1prop 49446
                  21.49.23.2  Real euclidean space of dimension 2   rrx2pxel 49458
                  21.49.23.3  Spheres and lines in real Euclidean spaces   cline 49474
      ​21.50  Mathbox for Zhi Wang
            21.50.1  Propositional calculus   logic1 49536
            21.50.2  Predicate calculus with equality   dtrucor3 49544
                  21.50.2.1  Axiom scheme ax-5 (Distinctness)   dtrucor3 49544
            21.50.3  ZF Set Theory - start with the Axiom of Extensionality   ralbidb 49545
                  21.50.3.1  Restricted quantification   ralbidb 49545
                  21.50.3.2  The universal class   reuxfr1dd 49552
                  21.50.3.3  The empty set   ssdisjd 49553
                  21.50.3.4  Unordered and ordered pairs   vsn 49557
                  21.50.3.5  The union of a class   unilbss 49563
                  21.50.3.6  Indexed union and intersection   iuneq0 49564
            21.50.4  ZF Set Theory - add the Axiom of Replacement   inpw 49570
                  21.50.4.1  Theorems requiring subset and intersection existence   inpw 49570
            21.50.5  ZF Set Theory - add the Axiom of Power Sets   opth1neg 49571
                  21.50.5.1  Ordered pair theorem   opth1neg 49571
                  21.50.5.2  Ordered-pair class abstractions (cont.)   brab2dd 49573
                  21.50.5.3  Relations   iinxp 49576
                  21.50.5.4  Functions   mof0 49583
                  21.50.5.5  Operations   ovsng 49603
            21.50.6  ZF Set Theory - add the Axiom of Union   fonex 49612
                  21.50.6.1  Relations and functions (cont.)   fonex 49612
                  21.50.6.2  First and second members of an ordered pair   eloprab1st2nd 49613
                  21.50.6.3  Operations in maps-to notation (continued)   fmpodg 49614
                  21.50.6.4  Function transposition   resinsnlem 49616
                  21.50.6.5  Infinite Cartesian products   ixpv 49635
                  21.50.6.6  Equinumerosity   fvconst0ci 49636
            21.50.7  Order sets   iccin 49641
                  21.50.7.1  Real number intervals   iccin 49641
            21.50.8  Extensible structures   slotresfo 49644
                  21.50.8.1  Basic definitions   slotresfo 49644
            21.50.9  Moore spaces   mreuniss 49645
            ​*21.50.10  Topology   clduni 49646
                  21.50.10.1  Closure and interior   clduni 49646
                  21.50.10.2  Neighborhoods   neircl 49650
                  21.50.10.3  Subspace topologies   restcls2lem 49658
                  21.50.10.4  Limits and continuity in topological spaces   cnneiima 49662
                  21.50.10.5  Topological definitions using the reals   iooii 49663
                  21.50.10.6  Separated sets   sepnsepolem1 49667
                  21.50.10.7  Separated spaces: T0, T1, T2 (Hausdorff) ...   isnrm4 49676
            21.50.11  Preordered sets and directed sets using extensible structures   isprsd 49700
            21.50.12  Posets and lattices using extensible structures   lubeldm2 49701
                  21.50.12.1  Posets   lubeldm2 49701
                  21.50.12.2  Lattices   toslat 49727
                  21.50.12.3  Subset order structures   intubeu 49729
            21.50.13  Rings   elmgpcntrd 49750
                  21.50.13.1  Multiplicative Group   elmgpcntrd 49750
            21.50.14  Associative algebras   asclelbasALT 49751
                  21.50.14.1  Definition and basic properties   asclelbasALT 49751
            21.50.15  Categories   homf0 49754
                  21.50.15.1  Categories   homf0 49754
                  21.50.15.2  Opposite category   oppccatb 49761
                  21.50.15.3  Monomorphisms and epimorphisms   idmon 49765
                  21.50.15.4  Sections, inverses, isomorphisms   sectrcl 49767
                  21.50.15.5  Isomorphic objects   cicfn 49787
                  21.50.15.6  Subcategories   dmdm 49798
                  21.50.15.7  Functors   reldmfunc 49820
                  21.50.15.8  Opposite functors   coppf 49867
                  21.50.15.9  Full & faithful functors   imasubc 49896
                  21.50.15.10  Universal property   upciclem1 49911
                  21.50.15.11  Natural transformations and the functor category   isnatd 49968
                  21.50.15.12  Initial, terminal and zero objects of a category   initoo2 49977
                  21.50.15.13  Product of categories   reldmxpc 49991
                  21.50.15.14  Swap functors   cswapf 50004
                  21.50.15.15  Functor evaluation   oppc1stflem 50032
                  21.50.15.16  Transposed curry functors   cofuswapfcl 50038
                  21.50.15.17  Constant functors   diag1 50049
                  21.50.15.18  Functor composition bifunctors   fucofulem1 50055
                  21.50.15.19  Post-composition functors   postcofval 50109
                  21.50.15.20  Pre-composition functors   precofvallem 50111
            21.50.16  Examples of categories   catcrcl 50140
                  21.50.16.1  The category of categories   catcrcl 50140
                  21.50.16.2  Thin categories   cthinc 50162
                  21.50.16.3  Terminal categories   ctermc 50217
                  21.50.16.4  Preordered sets as thin categories   cprstc 50294
                  21.50.16.5  Monoids as categories   cmndtc 50322
                  21.50.16.6  Categories with at most one object and at most two morphisms   2arwcatlem1 50340
            21.50.17  Kan extensions and related concepts   clan 50350
                  21.50.17.1  Kan extensions   clan 50350
                  21.50.17.2  Limits and colimits   clmd 50388
      ​21.51  Mathbox for Emmett Weisz
            ​*21.51.1  Miscellaneous Theorems   nfintd 50418
            21.51.2  Set Recursion   csetrecs 50428
                  ​*21.51.2.1  Basic Properties of Set Recursion   csetrecs 50428
                  21.51.2.2  Examples and properties of set recursion   elsetrecslem 50444
            ​*21.51.3  Construction of Games and Surreal Numbers   cpg 50454
      ​​*21.52  Mathbox for David A. Wheeler
            21.52.1  Natural deduction   sbidd 50463
            ​*21.52.2  Greater than, greater than or equal to   cge-real 50465
            ​*21.52.3  Hyperbolic trigonometric functions   csinh 50475
            ​*21.52.4  Reciprocal trigonometric functions (sec, csc, cot)   csec 50486
            ​*21.52.5  Identities for "if"   ifnmfalse 50508
            ​*21.52.6  Logarithms generalized to arbitrary base using ` logb `   logb2aval 50509
            ​*21.52.7  Logarithm laws generalized to an arbitrary base - log_   clog- 50510
            ​*21.52.8  Formally define notions such as reflexivity   wreflexive 50512
            ​*21.52.9  Algebra helpers   mvlraddi 50516
            ​*21.52.10  Algebra helper examples   i2linesi 50523
            ​*21.52.11  Formal methods "surprises"   alimp-surprise 50525
            ​*21.52.12  Allsome quantifier   wals 50531
            ​*21.52.13  Miscellaneous   5m4e1 50564
            21.52.14  Theorems about algebraic numbers   aacllem 50568
      ​21.53  Mathbox for Kunhao Zheng
            21.53.1  Weighted AM-GM inequality   amgmwlem 50569