TOC of Theorem List - Metamath Proof Explorer

Table of Contents Summary

PART 1  CLASSICAL FIRST-ORDER LOGIC WITH EQUALITY
      1.1  Pre-logic
      1.2  Propositional calculus
      1.3  Other axiomatizations related to classical propositional calculus
      1.4  Predicate calculus with equality: Tarski's system S2 (1 rule, 6 schemes)
      1.5  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes)
      1.6  Uniqueness and unique existence
      1.7  Other axiomatizations related to classical predicate calculus
PART 2  ZF (ZERMELO-FRAENKEL) SET THEORY
      2.1  ZF Set Theory - start with the Axiom of Extensionality
      2.2  ZF Set Theory - add the Axiom of Replacement
      2.3  ZF Set Theory - add the Axiom of Power Sets
      2.4  ZF Set Theory - add the Axiom of Union
      2.5  ZF Set Theory - add the Axiom of Regularity
      2.6  ZF Set Theory - add the Axiom of Infinity
PART 3  ZFC (ZERMELO-FRAENKEL WITH CHOICE) SET THEORY
      3.1  ZFC Set Theory - add Countable Choice and Dependent Choice
      3.2  ZFC Set Theory - add the Axiom of Choice
      3.3  ZFC Axioms with no distinct variable requirements
      3.4  The Generalized Continuum Hypothesis
PART 4  TG (TARSKI-GROTHENDIECK) SET THEORY
      4.1  Inaccessibles
      4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
PART 5  REAL AND COMPLEX NUMBERS
      5.1  Construction and axiomatization of real and complex numbers
      5.2  Derive the basic properties from the field axioms
      5.3  Real and complex numbers - basic operations
      5.4  Integer sets
      5.5  Order sets
      5.6  Elementary integer functions
      5.7  Words over a set
      5.8  Reflexive and transitive closures of relations
      5.9  Elementary real and complex functions
      5.10  Elementary limits and convergence
      5.11  Elementary trigonometry
      5.12  Cardinality of real and complex number subsets
PART 6  ELEMENTARY NUMBER THEORY
      6.1  Elementary properties of divisibility
      6.2  Elementary prime number theory
PART 7  BASIC STRUCTURES
      7.1  Extensible structures
      7.2  Moore spaces
PART 8  BASIC CATEGORY THEORY
      8.1  Categories
      8.2  Arrows (disjointified hom-sets)
      8.3  Examples of categories
      8.4  Categorical constructions
PART 9  BASIC ORDER THEORY
      9.1  Dual of an order structure
      9.2  Preordered sets and directed sets
      9.3  Partially ordered sets (posets)
      9.4  Totally ordered sets (tosets)
      9.5  Lattices
      9.6  Posets, directed sets, and lattices as relations
      9.7  Chains
PART 10  BASIC ALGEBRAIC STRUCTURES
      10.1  Monoids
      10.2  Groups
      10.3  Rings
      10.4  Division rings and fields
      10.5  Left modules
      10.6  Vector spaces
      10.7  Subring algebras and ideals
      10.8  The complex numbers as an algebraic extensible structure
      10.9  Generalized pre-Hilbert and Hilbert spaces
PART 11  BASIC LINEAR ALGEBRA
      11.1  Vectors and free modules
      11.2  Associative algebras
      11.3  Abstract multivariate polynomials
      11.4  Matrices
      11.5  The determinant
      11.6  Polynomial matrices
      11.7  The characteristic polynomial
PART 12  BASIC TOPOLOGY
      12.1  Topology
      12.2  Filters and filter bases
      12.3  Uniform Structures and Spaces
      12.4  Metric spaces
      12.5  Metric subcomplex vector spaces
PART 13  BASIC REAL AND COMPLEX ANALYSIS
      13.1  Continuity
      13.2  Integrals
      13.3  Derivatives
PART 14  BASIC REAL AND COMPLEX FUNCTIONS
      14.1  Polynomials
      14.2  Sequences and series
      14.3  Basic trigonometry
      14.4  Basic number theory
PART 15  SURREAL NUMBERS
      15.1  Sign sequence representation and Alling's axioms
      15.2  Initial consequences of Alling's axioms
      15.3  Conway cut representation
      15.4  Induction and recursion
      15.5  Surreal arithmetic
      15.6  Subsystems of surreals
PART 16  ELEMENTARY GEOMETRY
      16.1  Definition and Tarski's Axioms of Geometry
      16.2  Tarskian Geometry
      16.3  Properties of geometries
      16.4  Geometry in Hilbert spaces
PART 17  GRAPH THEORY
      17.1  Vertices and edges
      17.2  Undirected graphs
      17.3  Walks, paths and cycles
      17.4  Eulerian paths and the Konigsberg Bridge problem
      17.5  The Friendship Theorem
PART 18  GUIDES AND MISCELLANEA
      18.1  Guides (conventions, explanations, and examples)
      18.2  Humor
      18.3  (Future - to be reviewed and classified)
PART 19  COMPLEX TOPOLOGICAL VECTOR SPACES (DEPRECATED)
      19.1  Additional material on group theory (deprecated)
      19.2  Complex vector spaces
      19.3  Normed complex vector spaces
      19.4  Operators on complex vector spaces
      19.5  Inner product (pre-Hilbert) spaces
      19.6  Complex Banach spaces
      19.7  Complex Hilbert spaces
PART 20  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      20.1  Axiomatization of complex pre-Hilbert spaces
      20.2  Inner product and norms
      20.3  Cauchy sequences and completeness axiom
      20.4  Subspaces and projections
      20.5  Properties of Hilbert subspaces
      20.6  Operators on Hilbert spaces
      20.7  States on a Hilbert lattice and Godowski's equation
      20.8  Cover relation, atoms, exchange axiom, and modular symmetry
PART 21  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      ​21.1  Mathboxes for user contributions
      21.2  Mathbox for Stefan Allan
      21.3  Mathbox for Thierry Arnoux
      21.4  Mathbox for Jonathan Ben-Naim
      21.5  Mathbox for BTernaryTau
      21.6  Mathbox for Mario Carneiro
      21.7  Mathbox for Filip Cernatescu
      21.8  Mathbox for Paul Chapman
      21.9  Mathbox for Hongxiu Chen
      21.10  Mathbox for Adrian Ducourtial
      21.11  Mathbox for Scott Fenton
      21.12  Mathbox for Gino Giotto
      21.13  Mathbox for Jeff Hankins
      21.14  Mathbox for Anthony Hart
      21.15  Mathbox for Chen-Pang He
      21.16  Mathbox for Jeff Hoffman
      21.17  Mathbox for Matthew House
      21.18  Mathbox for Asger C. Ipsen
      21.19  Mathbox for BJ
      21.20  Mathbox for Jim Kingdon
      21.21  Mathbox for ML
      21.22  Mathbox for Wolf Lammen
      21.23  Mathbox for Brendan Leahy
      21.24  Mathbox for Thomas van Maaren
      21.25  Mathbox for Jeff Madsen
      21.26  Mathbox for Giovanni Mascellani
      21.27  Mathbox for Peter Mazsa
      21.28  Mathbox for Rodolfo Medina
      21.29  Mathbox for Norm Megill
      21.30  Mathbox for metakunt
      21.31  Mathbox for Luke Murphy
      21.32  Mathbox for Steven Nguyen
      21.33  Mathbox for Igor Ieskov
      21.34  Mathbox for OpenAI
      21.35  Mathbox for Stefan O'Rear
      21.36  Mathbox for Noam Pasman
      21.37  Mathbox for Jon Pennant
      21.38  Mathbox for Richard Penner
      21.39  Mathbox for Stanislas Polu
      21.40  Mathbox for Rohan Ridenour
      21.41  Mathbox for Steve Rodriguez
      21.42  Mathbox for Andrew Salmon
      21.43  Mathbox for Alan Sare
      21.44  Mathbox for Eric Schmidt
      21.45  Mathbox for Glauco Siliprandi
      21.46  Mathbox for Saveliy Skresanov
      21.47  Mathbox for Ender Ting
      21.48  Mathbox for Jarvin Udandy
      21.49  Mathbox for Adhemar
      21.50  Mathbox for Alexander van der Vekens
      21.51  Mathbox for Zhi Wang
      21.52  Mathbox for Emmett Weisz
      21.53  Mathbox for David A. Wheeler
      21.54  Mathbox for Mingli Yuan
      21.55  Mathbox for Jiamin Zhao
      21.56  Mathbox for Kunhao Zheng

Detailed Table of Contents
(* means the section header has a description)

​​*PART 1  CLASSICAL FIRST-ORDER LOGIC WITH EQUALITY
      ​*1.1  Pre-logic
            *1.1.1  Inferences for assisting proof development   idi 1
      ​​*1.2  Propositional calculus
            1.2.1  Recursively define primitive wffs for propositional calculus   wn 3
            ​*1.2.2  The axioms of propositional calculus   ax-mp 5
            ​*1.2.3  Logical implication   mp2 9
            ​*1.2.4  Logical negation   con4 114
            ​*1.2.5  Logical equivalence   wb 209
            ​*1.2.6  Logical conjunction   wa 401
            ​*1.2.7  Logical disjunction   wo 861
            ​*1.2.8  Mixed connectives   jaao 969
            ​*1.2.9  The conditional operator for propositions   wif 1078
            ​*1.2.10  The weak deduction theorem for propositional calculus   elimh 1099
            1.2.11  Abbreviated conjunction and disjunction of three wff's   w3o 1102
            1.2.12  Logical "nand" (Sheffer stroke)   wnan 1521
            1.2.13  Logical "xor"   wxo 1541
            1.2.14  Logical "nor"   wnor 1558
            1.2.15  True and false constants   wal 1568
                  ​*1.2.15.1  Universal quantifier for use by df-tru   wal 1568
                  ​*1.2.15.2  Equality predicate for use by df-tru   cv 1569
                  1.2.15.3  The true constant   wtru 1571
                  1.2.15.4  The false constant   wfal 1582
            ​*1.2.16  Truth tables   truimtru 1593
                  1.2.16.1  Implication   truimtru 1593
                  1.2.16.2  Negation   nottru 1597
                  1.2.16.3  Equivalence   trubitru 1599
                  1.2.16.4  Conjunction   truantru 1603
                  1.2.16.5  Disjunction   truortru 1607
                  1.2.16.6  Alternative denial   trunantru 1611
                  1.2.16.7  Exclusive disjunction   truxortru 1615
                  1.2.16.8  Joint denial   trunortru 1619
            ​*1.2.17  Half adder and full adder in propositional calculus   whad 1623
                  1.2.17.1  Full adder: sum   whad 1623
                  1.2.17.2  Full adder: carry   wcad 1639
      ​1.3  Other axiomatizations related to classical propositional calculus
            ​*1.3.1  Minimal implicational calculus   minimp 1654
            ​*1.3.2  Implicational Calculus   impsingle 1660
            1.3.3  Derive the Lukasiewicz axioms from Meredith's sole axiom   meredith 1674
            1.3.4  Derive the standard axioms from the Lukasiewicz axioms   luklem1 1691
            ​*1.3.5  Derive Nicod's axiom from the standard axioms   nic-dfim 1702
            1.3.6  Derive the Lukasiewicz axioms from Nicod's axiom   nic-imp 1708
            1.3.7  Derive Nicod's Axiom from Lukasiewicz's First Sheffer Stroke Axiom   lukshef-ax1 1727
            1.3.8  Derive the Lukasiewicz Axioms from the Tarski-Bernays-Wajsberg Axioms   tbw-bijust 1731
            1.3.9  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's First CO Axiom   merco1 1746
            1.3.10  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's Second CO Axiom   merco2 1769
            1.3.11  Derive the Lukasiewicz axioms from the Russell-Bernays Axioms   rb-bijust 1782
            ​*1.3.12  Stoic logic non-modal portion (Chrysippus of Soli)   mptnan 1801
      ​​*1.4  Predicate calculus with equality: Tarski's system S2 (1 rule, 6 schemes)
            *1.4.1  Universal quantifier (continued); define "exists" and "not free"   wex 1812
                  1.4.1.1  Existential quantifier   wex 1812
                  1.4.1.2  Nonfreeness predicate   wnf 1816
            1.4.2  Rule scheme ax-gen (Generalization)   ax-gen 1828
            1.4.3  Axiom scheme ax-4 (Quantified Implication)   ax-4 1842
                  ​*1.4.3.1  The empty domain of discourse   empty 1939
            1.4.4  Axiom scheme ax-5 (Distinctness) - first use of $d   ax-5 1943
            ​*1.4.5  Equality predicate (continued)   weq 1995
            1.4.6  Axiom scheme ax-6 (Existence)   ax-6 2000
            1.4.7  Axiom scheme ax-7 (Equality)   ax-7 2041
            1.4.8  Define proper substitution   sbjust 2098
            1.4.9  Membership predicate   wcel 2145
            1.4.10  Axiom scheme ax-8 (Left Equality for Binary Predicate)   ax-8 2147
            1.4.11  Axiom scheme ax-9 (Right Equality for Binary Predicate)   ax-9 2155
            ​*1.4.12  Logical redundancy of ax-10 , ax-11 , ax-12 , ax-13   ax6dgen 2165
      ​​*1.5  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes)
            1.5.1  Axiom scheme ax-10 (Quantified Negation)   ax-10 2178
            1.5.2  Axiom scheme ax-11 (Quantifier Commutation)   ax-11 2194
            1.5.3  Axiom scheme ax-12 (Substitution)   ax-12 2215
            1.5.4  Axiom scheme ax-13 (Quantified Equality)   ax-13 2403
      ​1.6  Uniqueness and unique existence
            1.6.1  Uniqueness: the at-most-one quantifier   wmo 2564
            1.6.2  Unique existence: the unique existential quantifier   weu 2595
      ​1.7  Other axiomatizations related to classical predicate calculus
            ​*1.7.1  Aristotelian logic: Assertic syllogisms   barbara 2689
            ​*1.7.2  Intuitionistic logic   axia1 2719
​​*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 2734
            2.1.2  Classes   cab 2740
                  2.1.2.1  Class abstractions   cab 2740
                  ​*2.1.2.2  Class equality   df-cleq 2754
                  2.1.2.3  Class membership   df-clel 2837
                  2.1.2.4  Elementary properties of class abstractions   eqabdv 2895
            2.1.3  Class form not-free predicate   wnfc 2909
            2.1.4  Negated equality and membership   wne 2957
                  2.1.4.1  Negated equality   wne 2957
                  2.1.4.2  Negated membership   wnel 3063
            2.1.5  Restricted quantification   wral 3078
                  2.1.5.1  Restricted universal and existential quantification   wral 3078
                  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 3724
            2.1.8  Russell's Paradox   rru 3740
            2.1.9  Proper substitution of classes for sets   wsbc 3742
            2.1.10  Proper substitution of classes for sets into classes   csb 3850
            2.1.11  Define basic set operations and relations   cdif 3899
            2.1.12  Subclasses and subsets   df-ss 3919
            2.1.13  The difference, union, and intersection of two classes   dfdif3 4069
                  2.1.13.1  The difference of two classes   dfdif3 4069
                  2.1.13.2  The union of two classes   elun 4103
                  2.1.13.3  The intersection of two classes   elini 4148
                  2.1.13.4  The symmetric difference of two classes   csymdif 4201
                  2.1.13.5  Combinations of difference, union, and intersection of two classes   unabs 4214
                  2.1.13.6  Class abstractions with difference, union, and intersection of two classes   unabw 4256
                  2.1.13.7  Restricted uniqueness with difference, union, and intersection   reuun2 4274
            2.1.14  The empty set   c0 4282
            ​*2.1.15  The conditional operator for classes   cif 4485
            ​*2.1.16  The weak deduction theorem for set theory   dedth 4544
            2.1.17  Power classes   cpw 4560
            2.1.18  Unordered and ordered pairs   snjust 4586
            2.1.19  The union of a class   cuni 4870
            2.1.20  The intersection of a class   cint 4910
            2.1.21  Indexed union and intersection   ciun 4954
            2.1.22  Disjointness   wdisj 5074
            2.1.23  Binary relations   wbr 5107
            2.1.24  Ordered-pair class abstractions (class builders)   copab 5171
            2.1.25  Functions in maps-to notation   cmpt 5190
            2.1.26  Transitive classes   wtr 5216
      ​2.2  ZF Set Theory - add the Axiom of Replacement
            2.2.1  Introduce the Axiom of Replacement   ax-rep 5236
            2.2.2  Derive the Axiom of Separation   axsepgfromrep 5253
            2.2.3  Derive the Null Set Axiom   axnulALT 5265
            2.2.4  Theorems requiring subset and intersection existence   exnelv 5274
            2.2.5  Theorems requiring empty set existence   class2set 5323
      ​2.3  ZF Set Theory - add the Axiom of Power Sets
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 5334
            2.3.2  Derive the Axiom of Pairing   axprlem1 5392
            2.3.3  Ordered pair theorem   opnz 5453
            2.3.4  Ordered-pair class abstractions (cont.)   opabidw 5506
            2.3.5  Power class of union and intersection   pwin 5550
            2.3.6  The identity relation   cid 5553
            2.3.7  The membership relation (or epsilon relation)   cep 5558
            ​*2.3.8  Partial and total orderings   wpo 5565
            2.3.9  Founded and well-ordering relations   wfr 5609
            2.3.10  Relations   cxp 5657
            2.3.11  The Predecessor Class   cpred 6302
            2.3.12  Well-founded induction (variant)   frpomin 6342
            2.3.13  Well-ordered induction   tz6.26 6349
            2.3.14  Ordinals   word 6360
            2.3.15  Definite description binder (inverted iota)   cio 6491
            2.3.16  Functions   wfun 6531
            2.3.17  Cantor's Theorem   canth 7371
            2.3.18  Restricted iota (description binder)   crio 7373
            2.3.19  Operations   co 7417
                  2.3.19.1  Variable-to-class conversion for operations   caovclg 7610
            2.3.20  Maps-to notation   mpondm0 7658
            2.3.21  Function operation   cof 7680
            2.3.22  Proper subset relation   crpss 7727
      ​2.4  ZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 7740
            2.4.2  Ordinals (continued)   epweon 7778
            2.4.3  Transfinite induction   tfi 7853
            2.4.4  The natural numbers (i.e., finite ordinals)   com 7866
            2.4.5  Peano's postulates   peano1 7889
            2.4.6  Finite induction (for finite ordinals)   find 7896
            2.4.7  Relations and functions (cont.)   dmexg 7902
            2.4.8  First and second members of an ordered pair   c1st 7988
            2.4.9  Induction on Cartesian products   frpoins3xpg 8142
            2.4.10  Ordering on Cartesian products   xpord2lem 8144
            2.4.11  Ordering Ordinal Sequences   orderseqlem 8159
            ​*2.4.12  The support of functions   csupp 8162
            ​*2.4.13  Special maps-to operations   opeliunxp2f 8212
            2.4.14  Function transposition   ctpos 8227
            2.4.15  Curry and uncurry   ccur 8267
            2.4.16  Undefined values   cund 8274
            2.4.17  Well-founded recursion   cfrecs 8283
            2.4.18  Well-ordered recursion   cwrecs 8314
            2.4.19  Functions on ordinals; strictly monotone ordinal functions   iunon 8332
            2.4.20  "Strong" transfinite recursion   crecs 8363
            2.4.21  Recursive definition generator   crdg 8402
            2.4.22  Finite recursion   frfnom 8428
            2.4.23  Ordinal arithmetic   c1o 8452
            2.4.24  Natural number arithmetic   nna0 8596
            2.4.25  Natural addition   cnadd 8657
            2.4.26  Equivalence relations and classes   wer 8697
            2.4.27  The mapping operation   cmap 8830
            2.4.28  Infinite Cartesian products   cixp 8908
            2.4.29  Equinumerosity   cen 8953
            2.4.30  Schroeder-Bernstein Theorem   sbthlem1 9089
            2.4.31  Equinumerosity (cont.)   xpf1o 9141
            2.4.32  Finite sets   dif1enlem 9158
            2.4.33  Pigeonhole Principle   phplem1 9202
            2.4.34  Finite sets (cont.)   onomeneq 9212
            2.4.35  Finitely supported functions   cfsupp 9335
            2.4.36  Finite intersections   cfi 9384
            2.4.37  Hall's marriage theorem   marypha1lem 9407
            2.4.38  Supremum and infimum   csup 9414
            2.4.39  Ordinal isomorphism, Hartogs's theorem   coi 9485
            2.4.40  Hartogs function   char 9532
            2.4.41  Weak dominance   cwdom 9540
      ​2.5  ZF Set Theory - add the Axiom of Regularity
            2.5.1  Introduce the Axiom of Regularity   ax-reg 9568
            2.5.2  Axiom of Infinity equivalents   inf0 9604
      ​2.6  ZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-inf 9621
            2.6.2  Existence of omega (the set of natural numbers)   omex 9626
            2.6.3  Cantor normal form   ccnf 9644
            2.6.4  Transitive closure of a relation   cttrcl 9690
            2.6.5  Transitive closure   trcl 9711
            2.6.6  Set induction (or epsilon induction)   setind 9730
            2.6.7  Well-Founded Induction   frmin 9735
            2.6.8  Well-Founded Recursion   frr3g 9742
            2.6.9  Rank   cr1 9748
            2.6.10  Scott's trick; collection principle; Hilbert's epsilon   cscott 9871
            2.6.11  Disjoint union   cdju 9907
            2.6.12  Cardinal numbers   ccrd 9944
            2.6.13  Axiom of Choice equivalents   wac 10122
            ​*2.6.14  Cardinal number arithmetic   undjudom 10174
            2.6.15  The Ackermann bijection   ackbij2lem1 10224
            2.6.16  Cofinality (without Axiom of Choice)   cflem 10251
            2.6.17  Eight inequivalent definitions of finite set   sornom 10283
            2.6.18  Hereditarily size-limited sets without Choice   itunifval 10422
​​*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 10441
            3.1.2  Introduce the Axiom of Dependent Choice   ax-dc 10452
      ​3.2  ZFC Set Theory - add the Axiom of Choice
            3.2.1  Introduce the Axiom of Choice   ax-ac 10465
            3.2.2  AC equivalents: well-ordering, Zorn's lemma   numthcor 10500
            3.2.3  Cardinal number theorems using Axiom of Choice   cardval 10558
            3.2.4  Cardinal number arithmetic using Axiom of Choice   iunctb 10587
            3.2.5  Cofinality using the Axiom of Choice   alephreg 10595
      ​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 10633
            3.4.2  Derivation of the Axiom of Choice   gchaclem 10691
​​*PART 4  TG (TARSKI-GROTHENDIECK) SET THEORY
      ​4.1  Inaccessibles
            4.1.1  Weakly and strongly inaccessible cardinals   cwina 10695
            4.1.2  Weak universes   cwun 10713
            4.1.3  Tarski classes   ctsk 10761
            4.1.4  Grothendieck universes   cgru 10803
      ​4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
            4.2.1  Introduce the Tarski-Grothendieck Axiom   ax-groth 10836
            4.2.2  Derive the Power Set, Infinity and Choice Axioms   grothpw 10839
            4.2.3  Tarski map function   ctskm 10850
​​*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 10857
            5.1.2  Final derivation of real and complex number postulates   axaddf 11158
            5.1.3  Real and complex number postulates restated as axioms   ax-cnex 11184
      ​5.2  Derive the basic properties from the field axioms
            5.2.1  Some deductions from the field axioms for complex numbers   cnex 11209
            5.2.2  Infinity and the extended real number system   cpnf 11268
            5.2.3  Restate the ordering postulates with extended real "less than"   axlttri 11309
            5.2.4  Ordering on reals   lttr 11314
            5.2.5  Initial properties of the complex numbers   mul12 11403
      ​5.3  Real and complex numbers - basic operations
            5.3.1  Addition   add12 11456
            5.3.2  Subtraction   cmin 11469
            5.3.3  Multiplication   kcnktkm1cn 11673
            5.3.4  Ordering on reals (cont.)   gt0ne0 11707
            5.3.5  Reciprocals   ixi 11871
            5.3.6  Division   cdiv 11899
            5.3.7  Ordering on reals (cont.)   elimgt0 12081
            5.3.8  Completeness Axiom and Suprema   fimaxre 12187
            5.3.9  Imaginary and complex number properties   neg1cn 12231
            5.3.10  Function operation analogue theorems   ofsubeq0 12243
            ​*5.3.11  Indicator Functions   cind 12246
      ​5.4  Integer sets
            5.4.1  Positive integers (as a subset of complex numbers)   cn 12261
            5.4.2  Principle of mathematical induction   nnind 12279
            ​*5.4.3  Decimal representation of numbers   c2 12323
            ​*5.4.4  Some properties of specific numbers   1pneg1e0 12386
            5.4.5  Simple number properties   halfcl 12498
            5.4.6  The Archimedean property   nnunb 12528
            5.4.7  Nonnegative integers (as a subset of complex numbers)   cn0 12532
            ​*5.4.8  Extended nonnegative integers   cxnn0 12605
            5.4.9  Integers (as a subset of complex numbers)   cz 12619
            5.4.10  Decimal arithmetic   cdc 12740
            5.4.11  Upper sets of integers   cuz 12891
            5.4.12  Well-ordering principle for bounded-below sets of integers   uzwo3 12996
            5.4.13  Rational numbers (as a subset of complex numbers)   cq 13001
            5.4.14  Existence of the set of complex numbers   rpnnen1lem2 13031
      ​5.5  Order sets
            5.5.1  Positive reals (as a subset of complex numbers)   crp 13046
            5.5.2  Infinity and the extended real number system (cont.)   cxne 13164
            5.5.3  Supremum and infimum on the extended reals   xrsupexmnf 13361
            5.5.4  Real number intervals   cioo 13402
            5.5.5  Finite intervals of integers   cfz 13565
            ​*5.5.6  Finite intervals of nonnegative integers   elfz2nn0 13677
            5.5.7  Half-open integer ranges   cfzo 13713
      ​5.6  Elementary integer functions
            5.6.1  The floor and ceiling functions   cfl 13855
            5.6.2  The modulo (remainder) operation   cmo 13934
            5.6.3  Miscellaneous theorems about integers   om2uz0i 14015
            5.6.4  Strong induction over upper sets of integers   uzsinds 14055
            5.6.5  Finitely supported functions over the nonnegative integers   fsuppmapnn0fiublem 14058
            5.6.6  The infinite sequence builder "seq" - extension   cseq 14069
            5.6.7  Integer powers   cexp 14129
            5.6.8  Ordered pair theorem for nonnegative integers   nn0le2msqi 14335
            5.6.9  Factorial function   cfa 14341
            5.6.10  The binomial coefficient operation   cbc 14370
            5.6.11  The ` # ` (set size) function   chash 14398
                  5.6.11.1  Proper unordered pairs and triples (sets of size 2 and 3)   hashprlei 14537
                  5.6.11.2  Functions with a domain containing at least two different elements   fundmge2nop0 14571
                  5.6.11.3  Finite induction on the size of the first component of a binary relation   hashdifsnp1 14575
      ​​*5.7  Words over a set
            5.7.1  Definitions and basic theorems   cword 14582
            5.7.2  Last symbol of a word   clsw 14631
            5.7.3  Concatenations of words   cconcat 14639
            5.7.4  Singleton words   cs1 14666
            5.7.5  Concatenations with singleton words   ccatws1cl 14688
            5.7.6  Subwords/substrings   csubstr 14712
            5.7.7  Prefixes of a word   cpfx 14744
            5.7.8  Subwords of subwords   swrdswrdlem 14777
            5.7.9  Subwords and concatenations   pfxcctswrd 14783
            5.7.10  Subwords of concatenations   swrdccatfn 14797
            5.7.11  Splicing words (substring replacement)   csplice 14822
            5.7.12  Reversing words   creverse 14831
            5.7.13  Repeated symbol words   creps 14843
            ​*5.7.14  Cyclical shifts of words   ccsh 14863
            5.7.15  Mapping words by a function   wrdco 14906
            5.7.16  Longer string literals   cs2 14916
      ​​*5.8  Reflexive and transitive closures of relations
            5.8.1  The reflexive and transitive properties of relations   coss12d 15049
            5.8.2  Basic properties of closures   cleq1lem 15059
            5.8.3  Definitions and basic properties of transitive closures   ctcl 15062
            5.8.4  Exponentiation of relations   crelexp 15096
            5.8.5  Reflexive-transitive closure as an indexed union   crtrcl 15132
            ​*5.8.6  Principle of transitive induction   relexpindlem 15140
      ​5.9  Elementary real and complex functions
            5.9.1  The "shift" operation   cshi 15143
            5.9.2  Signum (sgn or sign) function   csgn 15163
            5.9.3  Real and imaginary parts; conjugate   ccj 15187
            5.9.4  Square root; absolute value   csqrt 15324
      ​5.10  Elementary limits and convergence
            5.10.1  Superior limit (lim sup)   clsp 15561
            5.10.2  Limits   cli 15575
            5.10.3  Finite and infinite sums   csu 15777
            5.10.4  The binomial theorem   binomlem 15922
            5.10.5  The inclusion/exclusion principle   incexclem 15929
            5.10.6  Infinite sums (cont.)   isumshft 15932
            5.10.7  Miscellaneous converging and diverging sequences   divrcnv 15945
            5.10.8  Arithmetic series   arisum 15953
            5.10.9  Geometric series   expcnv 15957
            5.10.10  Ratio test for infinite series convergence   cvgrat 15976
            5.10.11  Mertens' theorem   mertenslem1 15977
            5.10.12  Finite and infinite products   prodf 15980
                  5.10.12.1  Product sequences   prodf 15980
                  5.10.12.2  Non-trivial convergence   ntrivcvg 15990
                  5.10.12.3  Complex products   cprod 15996
                  5.10.12.4  Finite products   fprod 16034
                  5.10.12.5  Infinite products   iprodclim 16091
            5.10.13  Falling and Rising Factorial   cfallfac 16097
            5.10.14  Bernoulli polynomials and sums of k-th powers   cbp 16138
      ​5.11  Elementary trigonometry
            5.11.1  The exponential, sine, and cosine functions   ce 16153
                  5.11.1.1  The circle constant (tau = 2 pi)   ctau 16296
            5.11.2  _e is irrational   eirrlem 16298
      ​5.12  Cardinality of real and complex number subsets
            5.12.1  Countability of integers and rationals   xpnnen 16305
            5.12.2  The reals are uncountable   rpnnen2lem1 16308
​​*PART 6  ELEMENTARY NUMBER THEORY
      ​6.1  Elementary properties of divisibility
            6.1.1  Irrationality of square root of 2   sqrt2irrlem 16342
            6.1.2  Some Number sets are chains of proper subsets   nthruc 16346
            6.1.3  The divides relation   cdvds 16348
            ​*6.1.4  Even and odd numbers   evenelz 16432
            6.1.5  The division algorithm   divalglem0 16489
            6.1.6  Bit sequences   cbits 16515
            6.1.7  The greatest common divisor operator   cgcd 16590
            6.1.8  Bézout's identity   bezoutlem1 16635
            6.1.9  Algorithms   nn0seqcvgd 16666
            6.1.10  Euclid's Algorithm   eucalgval2 16677
            ​*6.1.11  The least common multiple   clcm 16684
            ​*6.1.12  Coprimality and Euclid's lemma   coprmgcdb 16745
            6.1.13  Cancellability of congruences   congr 16760
      ​6.2  Elementary prime number theory
            ​*6.2.1  Elementary properties   cprime 16767
            ​*6.2.2  Coprimality and Euclid's lemma (cont.)   coprm 16808
            6.2.3  Properties of the canonical representation of a rational   cnumer 16830
            6.2.4  Euler's theorem   codz 16860
            6.2.5  Arithmetic modulo a prime number   modprm1div 16895
            6.2.6  Pythagorean Triples   coprimeprodsq 16906
            6.2.7  The prime count function   cpc 16934
            6.2.8  Pocklington's theorem   prmpwdvds 17002
            6.2.9  Infinite primes theorem   unbenlem 17006
            6.2.10  Sum of prime reciprocals   prmreclem1 17014
            6.2.11  Fundamental theorem of arithmetic   1arithlem1 17021
            6.2.12  Lagrange's four-square theorem   cgz 17027
            6.2.13  Van der Waerden's theorem   cvdwa 17063
            6.2.14  Ramsey's theorem   cram 17097
            ​*6.2.15  Primorial function   cprmo 17129
            ​*6.2.16  Prime gaps   prmgaplem1 17147
            6.2.17  Decimal arithmetic (cont.)   dec2dvds 17161
            6.2.18  Cyclical shifts of words (cont.)   cshwsidrepsw 17191
            6.2.19  Specific prime numbers   prmlem0 17203
            6.2.20  Very large primes   1259lem1 17229
​PART 7  BASIC STRUCTURES
      ​7.1  Extensible structures
            ​*7.1.1  Basic definitions   cstr 17244
                  7.1.1.1  Extensible structures as structures with components   cstr 17244
                  7.1.1.2  Substitution of components   csts 17261
                  7.1.1.3  Slots   cslot 17279
                  ​*7.1.1.4  Structure component indices   cnx 17291
                  7.1.1.5  Base sets   cbs 17307
                  7.1.1.6  Base set restrictions   cress 17328
            7.1.2  Slot definitions   cplusg 17348
            7.1.3  Definition of the structure product   crest 17511
            7.1.4  Definition of the structure quotient   cordt 17591
      ​7.2  Moore spaces
            7.2.1  Moore closures   mrcflem 17700
            7.2.2  Independent sets in a Moore system   mrisval 17724
            7.2.3  Algebraic closure systems   isacs 17745
​PART 8  BASIC CATEGORY THEORY
      ​8.1  Categories
            8.1.1  Categories   ccat 17758
            8.1.2  Opposite category   coppc 17805
            8.1.3  Monomorphisms and epimorphisms   cmon 17823
            8.1.4  Sections, inverses, isomorphisms   csect 17839
            ​*8.1.5  Isomorphic objects   ccic 17890
            8.1.6  Subcategories   cssc 17902
            8.1.7  Functors   cfunc 17949
            8.1.8  Full & faithful functors   cful 17999
            8.1.9  Natural transformations and the functor category   cnat 18039
            8.1.10  Initial, terminal and zero objects of a category   cinito 18076
      ​8.2  Arrows (disjointified hom-sets)
            8.2.1  Identity and composition for arrows   cida 18148
      ​8.3  Examples of categories
            8.3.1  The category of sets   csetc 18170
            8.3.2  The category of categories   ccatc 18193
            ​*8.3.3  The category of extensible structures   fncnvimaeqv 18214
      ​8.4  Categorical constructions
            8.4.1  Product of categories   cxpc 18262
            8.4.2  Functor evaluation   cevlf 18303
            8.4.3  Hom functor   chof 18342
​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 18525
            9.5.2  Complete lattices   ccla 18592
            9.5.3  Distributive lattices   cdlat 18614
            9.5.4  Subset order structures   cipo 18621
      ​9.6  Posets, directed sets, and lattices as relations
            ​*9.6.1  Posets and lattices as relations   cps 18658
            9.6.2  Directed sets, nets   cdir 18688
      ​9.7  Chains
​PART 10  BASIC ALGEBRAIC STRUCTURES
      ​10.1  Monoids
            ​*10.1.1  Magmas   cplusf 18733
            ​*10.1.2  Identity elements   mgmidmo 18758
            ​*10.1.3  Iterated sums in a magma   gsumvalx 18784
            10.1.4  Magma homomorphisms and submagmas   cmgmhm 18798
            ​*10.1.5  Semigroups   csgrp 18826
            ​*10.1.6  Definition and basic properties of monoids   cmnd 18842
            10.1.7  Monoid homomorphisms and submonoids   cmhm 18895
            ​*10.1.8  Iterated sums in a monoid   gsumvallem2 18949
            10.1.9  Free monoids   cfrmd 18962
                  ​*10.1.9.1  Monoid of endofunctions   cefmnd 18983
            10.1.10  Examples and counterexamples for magmas, semigroups and monoids   mgm2nsgrplem1 19036
      ​10.2  Groups
            10.2.1  Definition and basic properties   cgrp 19063
            ​*10.2.2  Group multiple operation   cmg 19196
            10.2.3  Subgroups and Quotient groups   csubg 19249
            ​*10.2.4  Cyclic monoids and groups   cycsubmel 19334
            10.2.5  Elementary theory of group homomorphisms   cghm 19346
            10.2.6  Isomorphisms of groups   cgim 19390
                  10.2.6.1  The first isomorphism theorem of groups   ghmqusnsglem1 19413
            10.2.7  Group actions   cga 19422
            10.2.8  Centralizers and centers   ccntz 19448
            10.2.9  The opposite group   coppg 19478
            10.2.10  Symmetric groups   csymg 19502
                  ​*10.2.10.1  Definition and basic properties   csymg 19502
                  10.2.10.2  Cayley's theorem   cayleylem1 19545
                  10.2.10.3  Permutations fixing one element   symgfix2 19549
                  ​*10.2.10.4  Transpositions in the symmetric group   cpmtr 19574
                  10.2.10.5  The sign of a permutation   cpsgn 19622
            10.2.11  p-Groups and Sylow groups; Sylow's theorems   cod 19657
            10.2.12  Direct products   clsm 19767
                  10.2.12.1  Direct products (extension)   smndlsmidm 19789
            10.2.13  Free groups   cefg 19839
            10.2.14  Abelian groups   ccmn 19913
                  10.2.14.1  Definition and basic properties   ccmn 19913
                  10.2.14.2  Cyclic groups   ccyg 20010
                  10.2.14.3  Group sum operation   gsumval3a 20036
                  10.2.14.4  Group sums over (ranges of) integers   fsfnn0gsumfsffz 20116
                  10.2.14.5  Internal direct products   cdprd 20128
                  10.2.14.6  The Fundamental Theorem of Abelian Groups   ablfacrplem 20200
            10.2.15  Simple groups   csimpg 20225
                  10.2.15.1  Definition and basic properties   csimpg 20225
                  10.2.15.2  Classification of abelian simple groups   ablsimpnosubgd 20239
            10.2.16  Totally ordered monoids and groups   comnd 20252
      ​10.3  Rings
            10.3.1  Multiplicative Group   cmgp 20279
            ​*10.3.2  Non-unital rings ("rngs")   crng 20293
            ​*10.3.3  Ring unity (multiplicative identity)   cur 20326
            10.3.4  Semirings   csrg 20331
                  ​*10.3.4.1  The binomial theorem for semirings   srgbinomlem1 20371
            10.3.5  Unital rings   crg 20378
            10.3.6  Opposite ring   coppr 20483
            10.3.7  Divisibility   cdsr 20501
            10.3.8  Ring primes   crpm 20579
            10.3.9  Homomorphisms of non-unital rings   crnghm 20581
            10.3.10  Ring homomorphisms   crh 20616
            10.3.11  Nonzero rings and zero rings   cnzr 20678
            10.3.12  Local rings   clring 20706
            10.3.13  Subrings   csubrng 20713
                  10.3.13.1  Subrings of non-unital rings   csubrng 20713
                  10.3.13.2  Subrings of unital rings   csubrg 20737
                  10.3.13.3  Subrings generated by a subset   crgspn 20778
            10.3.14  Categories of rings   crngc 20784
                  ​*10.3.14.1  The category of non-unital rings   crngc 20784
                  ​*10.3.14.2  The category of (unital) rings   cringc 20813
                  10.3.14.3  Subcategories of the category of rings   srhmsubclem1 20845
            10.3.15  Left regular elements and domains   crlreg 20859
      ​10.4  Division rings and fields
            10.4.1  Definition and basic properties   cdr 20896
            10.4.2  Sub-division rings   csdrg 20958
            10.4.3  Absolute value (abstract algebra)   cabv 20980
            10.4.4  Star rings   cstf 21009
            10.4.5  Totally ordered rings and fields   corng 21029
      ​10.5  Left modules
            10.5.1  Definition and basic properties   clmod 21050
            10.5.2  Subspaces and spans in a left module   clss 21121
            10.5.3  Homomorphisms and isomorphisms of left modules   clmhm 21209
            10.5.4  Subspace sum; bases for a left module   clbs 21264
      ​10.6  Vector spaces
            10.6.1  Definition and basic properties   clvec 21292
      ​10.7  Subring algebras and ideals
            10.7.1  Subring algebras   csra 21361
            ​*10.7.2  Left ideals and spans   clidl 21399
            10.7.3  Two-sided ideals and quotient rings   c2idl 21457
                  ​*10.7.3.1  Condition for a non-unital ring to be unital   rngqiprng1elbas 21495
                  10.7.3.2  Prime Ideals   cprmidl 21529
            10.7.4  Principal ideal rings. Divisibility in the integers   clpidl 21557
            10.7.5  Principal ideal domains   cpid 21573
      ​10.8  The complex numbers as an algebraic extensible structure
            10.8.1  Definition and basic properties   cpsmet 21575
            ​*10.8.2  Ring of integers   czring 21665
                  ​*10.8.2.1  Example for a condition for a non-unital ring to be unital   pzriprnglem1 21700
            10.8.3  Algebraic constructions based on the complex numbers   czrh 21718
            10.8.4  Signs as subgroup of the complex numbers   cnmsgnsubg 21796
            10.8.5  Embedding of permutation signs into a ring   zrhpsgnmhm 21803
            10.8.6  The ordered field of real numbers   crefld 21823
      ​10.9  Generalized pre-Hilbert and Hilbert spaces
            10.9.1  Definition and basic properties   cphl 21843
            10.9.2  Orthocomplements and closed subspaces   cocv 21879
            10.9.3  Orthogonal projection and orthonormal bases   cpj 21919
​​*PART 11  BASIC LINEAR ALGEBRA
      ​11.1  Vectors and free modules
            *11.1.1  Direct sum of left modules   cdsmm 21950
            ​*11.1.2  Free modules   cfrlm 21965
            ​*11.1.3  Standard basis (unit vectors)   cuvc 22001
            ​*11.1.4  Independent sets and families   clindf 22023
            11.1.5  Characterization of free modules   lmimlbs 22055
      ​11.2  Associative algebras
            11.2.1  Definition and basic properties   casa 22071
      ​11.3  Abstract multivariate polynomials
            11.3.1  Definition and basic properties   cmps 22125
            11.3.2  Polynomial evaluation   ces 22294
            11.3.3  The "variable selection" function   cslv 22338
            11.3.4  Additional definitions for (multivariate) polynomials   cmhp 22367
            ​*11.3.5  Univariate polynomials   cps1 22406
            11.3.6  Univariate polynomial evaluation   ces1 22544
                  11.3.6.1  Specialization of polynomial evaluation as a ring homomorphism   evls1scafv 22597
      ​​*11.4  Matrices
            *11.4.1  The matrix multiplication   cmmul 22618
            ​*11.4.2  Square matrices   cmat 22635
            ​*11.4.3  The matrix algebra   matmulr 22666
            ​*11.4.4  Matrices of dimension 0 and 1   mat0dimbas0 22694
            ​*11.4.5  The subalgebras of diagonal and scalar matrices   cdmat 22716
            ​*11.4.6  Multiplication of a matrix with a "column vector"   cmvmul 22768
            11.4.7  Replacement functions for a square matrix   cmarrep 22784
            11.4.8  Submatrices   csubma 22804
      ​11.5  The determinant
            11.5.1  Definition and basic properties   cmdat 22812
            11.5.2  Determinants of 2 x 2 -matrices   m2detleiblem1 22852
            11.5.3  The matrix adjugate/adjunct   cmadu 22860
            ​*11.5.4  Laplace expansion of determinants (special case)   symgmatr01lem 22881
            11.5.5  Inverse matrix   invrvald 22904
            ​*11.5.6  Cramer's rule   slesolvec 22910
      ​​*11.6  Polynomial matrices
            11.6.1  Basic properties   pmatring 22923
            ​*11.6.2  Constant polynomial matrices   ccpmat 22934
            ​*11.6.3  Collecting coefficients of polynomial matrices   cdecpmat 22993
            ​*11.6.4  Ring isomorphism between polynomial matrices and polynomials over matrices   cpm2mp 23023
      ​​*11.7  The characteristic polynomial
            *11.7.1  Definition and basic properties   cchpmat 23057
            ​*11.7.2  The characteristic factor function G   fvmptnn04if 23080
            ​*11.7.3  The Cayley-Hamilton theorem   cpmadurid 23098
​PART 12  BASIC TOPOLOGY
      ​12.1  Topology
            ​*12.1.1  Topological spaces   ctop 23124
                  12.1.1.1  Topologies   ctop 23124
                  12.1.1.2  Topologies on sets   ctopon 23141
                  12.1.1.3  Topological spaces   ctps 23163
            12.1.2  Topological bases   ctb 23176
            12.1.3  Examples of topologies   distop 23226
            12.1.4  Closure and interior   ccld 23247
            12.1.5  Neighborhoods   cnei 23328
            12.1.6  Limit points and perfect sets   clp 23365
            12.1.7  Subspace topologies   restrcl 23388
            12.1.8  Order topology   ordtbaslem 23419
            12.1.9  Limits and continuity in topological spaces   ccn 23455
            12.1.10  Separated spaces: T0, T1, T2 (Hausdorff) ...   ct0 23537
            12.1.11  Compactness   ccmp 23617
            12.1.12  Bolzano-Weierstrass theorem   bwth 23641
            12.1.13  Connectedness   cconn 23642
            12.1.14  First- and second-countability   c1stc 23668
            12.1.15  Local topological properties   clly 23696
            12.1.16  Refinements   cref 23734
            12.1.17  Compactly generated spaces   ckgen 23765
            12.1.18  Product topologies   ctx 23792
            12.1.19  Continuous function-builders   cnmptid 23893
            12.1.20  Quotient maps and quotient topology   ckq 23925
            12.1.21  Homeomorphisms   chmeo 23985
      ​12.2  Filters and filter bases
            12.2.1  Filter bases   elmptrab 24059
            12.2.2  Filters   cfil 24077
            12.2.3  Ultrafilters   cufil 24131
            12.2.4  Filter limits   cfm 24165
            12.2.5  Extension by continuity   ccnext 24291
            12.2.6  Topological groups   ctmd 24302
            12.2.7  Infinite group sum on topological groups   ctsu 24358
            12.2.8  Topological rings, fields, vector spaces   ctrg 24388
      ​12.3  Uniform Structures and Spaces
            12.3.1  Uniform structures   cust 24432
            12.3.2  The topology induced by an uniform structure   cutop 24462
            12.3.3  Uniform Spaces   cuss 24485
            12.3.4  Uniform continuity   cucn 24506
            12.3.5  Cauchy filters in uniform spaces   ccfilu 24517
            12.3.6  Complete uniform spaces   ccusp 24528
      ​12.4  Metric spaces
            12.4.1  Pseudometric spaces   ispsmet 24536
            12.4.2  Basic metric space properties   cxms 24549
            12.4.3  Metric space balls   blfvalps 24615
            12.4.4  Open sets of a metric space   mopnval 24670
            12.4.5  Continuity in metric spaces   metcnp3 24772
            12.4.6  The uniform structure generated by a metric   metuval 24781
            12.4.7  Examples of metric spaces   dscmet 24804
            ​*12.4.8  Normed algebraic structures   cnm 24808
            12.4.9  Normed space homomorphisms (bounded linear operators)   cnmo 24937
            12.4.10  Topology on the reals   qtopbaslem 24990
            12.4.11  Topological definitions using the reals   cii 25109
            12.4.12  Path homotopy   chtpy 25201
            12.4.13  The fundamental group   cpco 25234
      ​12.5  Metric subcomplex vector spaces
            12.5.1  Subcomplex modules   cclm 25296
            ​*12.5.2  Subcomplex vector spaces   ccvs 25357
            ​*12.5.3  Normed subcomplex vector spaces   isncvsngp 25383
            12.5.4  Subcomplex pre-Hilbert spaces   ccph 25400
            12.5.5  Convergence and completeness   ccfil 25486
            12.5.6  Baire's Category Theorem   bcthlem1 25558
            12.5.7  Banach spaces and subcomplex Hilbert spaces   ccms 25566
                  12.5.7.1  The complete ordered field of the real numbers   retopn 25613
            12.5.8  Euclidean spaces   crrx 25617
            12.5.9  Minimizing Vector Theorem   minveclem1 25658
            12.5.10  Projection Theorem   pjthlem1 25671
​PART 13  BASIC REAL AND COMPLEX ANALYSIS
      ​13.1  Continuity
            13.1.1  Intermediate value theorem   pmltpclem1 25682
      ​13.2  Integrals
            13.2.1  Lebesgue measure   covol 25696
            13.2.2  Lebesgue integration   cmbf 25848
                  13.2.2.1  Lesbesgue integral   cmbf 25848
                  13.2.2.2  Lesbesgue directed integral   cdit 26080
      ​13.3  Derivatives
            13.3.1  Real and complex differentiation   climc 26096
                  13.3.1.1  Derivatives of functions of one complex or real variable   climc 26096
                  13.3.1.2  Results on real differentiation   dvferm1lem 26218
​PART 14  BASIC REAL AND COMPLEX FUNCTIONS
      ​14.1  Polynomials
            14.1.1  Polynomial degrees   cmdg 26285
            14.1.2  The division algorithm for univariate polynomials   cmn1 26358
            14.1.3  Elementary properties of complex polynomials   cply 26416
            14.1.4  The division algorithm for polynomials   cquot 26527
            14.1.5  Algebraic numbers   caa 26553
            14.1.6  Liouville's approximation theorem   aalioulem1 26575
      ​14.2  Sequences and series
            14.2.1  Taylor polynomials and Taylor's theorem   ctayl 26596
            14.2.2  Uniform convergence   culm 26619
            14.2.3  Power series   pserval 26653
      ​14.3  Basic trigonometry
            14.3.1  The exponential, sine, and cosine functions (cont.)   efcn 26686
            14.3.2  Properties of pi = 3.14159...   pilem1 26694
            14.3.3  Mapping of the exponential function   efgh 26786
            14.3.4  The natural logarithm on complex numbers   clog 26799
            ​*14.3.5  Logarithms to an arbitrary base   clogb 27009
            14.3.6  Theorems of Pythagoras, isosceles triangles, and intersecting chords   angval 27046
            14.3.7  Solutions of quadratic, cubic, and quartic equations   quad2 27084
            14.3.8  Inverse trigonometric functions   casin 27107
            14.3.9  The Birthday Problem   log2ublem1 27191
            14.3.10  Areas in R^2   carea 27200
            14.3.11  More miscellaneous converging sequences   rlimcnp 27210
            14.3.12  Inequality of arithmetic and geometric means   cvxcl 27229
            14.3.13  Euler-Mascheroni constant   cem 27236
            14.3.14  Zeta function   czeta 27257
            14.3.15  Gamma function   clgam 27260
      ​14.4  Basic number theory
            14.4.1  Wilson's theorem   wilthlem1 27312
            14.4.2  The Fundamental Theorem of Algebra   ftalem1 27317
            14.4.3  The Basel problem (ζ(2) = π²/6)   basellem1 27325
            14.4.4  Number-theoretical functions   ccht 27335
            14.4.5  Perfect Number Theorem   mersenne 27471
            14.4.6  Characters of Z/nZ   cdchr 27476
            14.4.7  Bertrand's postulate   bcctr 27519
            ​*14.4.8  Quadratic residues and the Legendre symbol   clgs 27538
            ​*14.4.9  Gauss' Lemma   gausslemma2dlem0a 27600
            14.4.10  Quadratic reciprocity   lgseisenlem1 27619
            14.4.11  All primes 4n+1 are the sum of two squares   2sqlem1 27661
            14.4.12  Chebyshev's Weak Prime Number Theorem, Dirichlet's Theorem   chebbnd1lem1 27713
            14.4.13  The Prime Number Theorem   mudivsum 27774
            14.4.14  Ostrowski's theorem   abvcxp 27859
​PART 15  SURREAL NUMBERS
      ​​*15.1  Sign sequence representation and Alling's axioms
            15.1.1  Definitions and initial properties   csur 27884
            15.1.2  Ordering   ltssolem1 27919
            15.1.3  Birthday Function   bdayfo 27921
            15.1.4  Density   fvnobday 27922
            ​*15.1.5  Full-Eta Property   bdayimaon 27937
      ​15.2  Initial consequences of Alling's axioms
            15.2.1  Ordering Theorems   cles 27988
            15.2.2  Birthday Theorems   bdayfun 28020
      ​​*15.3  Conway cut representation
            15.3.1  Conway cuts   cslts 28030
            15.3.2  Zero and One   c0s 28078
            15.3.3  Cuts and Options   cmade 28095
            15.3.4  Cofinality and coinitiality   cofslts 28191
      ​15.4  Induction and recursion
            15.4.1  Induction and recursion on one variable   cnorec 28210
            15.4.2  Induction and recursion on two variables   cnorec2 28221
      ​15.5  Surreal arithmetic
            15.5.1  Addition   cadds 28232
            15.5.2  Negation and Subtraction   cnegs 28292
            15.5.3  Multiplication   cmuls 28379
            15.5.4  Division   cdivs 28460
            15.5.5  Absolute value   cabss 28510
      ​15.6  Subsystems of surreals
            15.6.1  Ordinal numbers   cons 28524
            15.6.2  Surreal recursive sequences   cseqs 28556
            15.6.3  Natural numbers   cn0s 28585
            15.6.4  Integers   czs 28651
            15.6.5  Dyadic fractions   c2s 28683
            15.6.6  Real numbers   creno 28762
​​*PART 16  ELEMENTARY GEOMETRY
      ​16.1  Definition and Tarski's Axioms of Geometry
            16.1.1  Justification for the congruence notation   tgjustf 28822
      ​16.2  Tarskian Geometry
            16.2.1  Congruence   tgcgrcomimp 28826
            16.2.2  Betweenness   tgbtwntriv2 28837
            16.2.3  Dimension   tglowdim1 28850
            16.2.4  Betweenness and Congruence   tgifscgr 28858
            16.2.5  Congruence of a series of points   ccgrg 28860
            16.2.6  Motions   cismt 28882
            16.2.7  Colinearity   tglng 28896
            16.2.8  Connectivity of betweenness   tgbtwnconn1lem1 28922
            16.2.9  Less-than relation in geometric congruences   cleg 28932
            16.2.10  Rays   chlg 28950
            16.2.11  Lines   btwnlng1 28974
            16.2.12  Point inversions   cmir 29011
            16.2.13  Right angles   crag 29055
            16.2.14  Half-planes   islnopp 29102
            16.2.15  Planes   cplng 29138
            16.2.16  Midpoints and Line Mirroring   cmid 29164
            16.2.17  Congruence of angles   ccgra 29201
            16.2.18  Angle Comparisons   cinag 29241
            16.2.19  Angle Addition   cangmgm 29260
            16.2.20  Congruence Theorems   tgsas1 29286
            16.2.21  Equilateral triangles   ceqlg 29297
            16.2.22  Parallel lines   cprlng 29301
      ​16.3  Properties of geometries
            16.3.1  Isomorphisms between geometries   f1otrgds 29333
      ​16.4  Geometry in Hilbert spaces
            16.4.1  Geometry in the complex plane   cchhllem 29351
            16.4.2  Geometry in Euclidean spaces   cee 29352
                  16.4.2.1  Definition of the Euclidean space   cee 29352
                  16.4.2.2  Tarski's axioms for geometry for the Euclidean space   axdimuniq 29378
                  16.4.2.3  EE^n fulfills Tarski's Axioms   ceeng 29442
​​*PART 17  GRAPH THEORY
      ​*17.1  Vertices and edges
            17.1.1  The edge function extractor for extensible structures   cedgf 29453
            ​*17.1.2  Vertices and indexed edges   cvtx 29461
                  17.1.2.1  Definitions and basic properties   cvtx 29461
                  17.1.2.2  The vertices and edges of a graph represented as ordered pair   opvtxval 29468
                  17.1.2.3  The vertices and edges of a graph represented as extensible structure   funvtxdmge2val 29476
                  17.1.2.4  Representations of graphs without edges   snstrvtxval 29502
                  17.1.2.5  Degenerated cases of representations of graphs   vtxval0 29504
            17.1.3  Edges as range of the edge function   cedg 29512
      ​​*17.2  Undirected graphs
            17.2.1  Undirected hypergraphs   cuhgr 29521
            17.2.2  Undirected pseudographs and multigraphs   cupgr 29545
            ​*17.2.3  Loop-free graphs   umgrislfupgrlem 29587
            17.2.4  Edges as subsets of vertices of graphs   uhgredgiedgb 29591
            ​*17.2.5  Undirected simple graphs   cuspgr 29616
            17.2.6  Examples for graphs   usgr0e 29704
            17.2.7  Subgraphs   csubgr 29735
            17.2.8  Finite undirected simple graphs   cfusgr 29784
            17.2.9  Neighbors, complete graphs and universal vertices   cnbgr 29800
                  17.2.9.1  Neighbors   cnbgr 29800
                  17.2.9.2  Universal vertices   cuvtx 29853
                  17.2.9.3  Complete graphs   ccplgr 29877
            17.2.10  Vertex degree   cvtxdg 29933
            ​*17.2.11  Regular graphs   crgr 30023
      ​​*17.3  Walks, paths and cycles
            *17.3.1  Walks   cewlks 30063
            17.3.2  Walks for loop-free graphs   lfgrwlkprop 30157
            17.3.3  Trails   ctrls 30160
            17.3.4  Paths and simple paths   cpths 30182
            17.3.5  Closed walks   cclwlks 30244
            17.3.6  Circuits and cycles   ccrcts 30258
            ​*17.3.7  Walks as words   cwwlks 30301
            17.3.8  Walks/paths of length 2 (as length 3 strings)   2wlkdlem1 30401
            17.3.9  Walks in regular graphs   rusgrnumwwlkl1 30447
            ​*17.3.10  Closed walks as words   cclwwlk 30459
                  17.3.10.1  Closed walks as words   cclwwlk 30459
                  17.3.10.2  Closed walks of a fixed length as words   cclwwlkn 30502
                  17.3.10.3  Closed walks on a vertex of a fixed length as words   cclwwlknon 30565
            17.3.11  Examples for walks, trails and paths   0ewlk 30592
            17.3.12  Acyclic graphs   cacycgr 30635
            17.3.13  Connected graphs   cconngr 30674
      ​17.4  Eulerian paths and the Konigsberg Bridge problem
            ​*17.4.1  Eulerian paths   ceupth 30685
            ​*17.4.2  The Königsberg Bridge problem   konigsbergvtx 30734
      ​17.5  The Friendship Theorem
            17.5.1  Friendship graphs - basics   cfrgr 30746
            17.5.2  The friendship theorem for small graphs   frgr1v 30759
            17.5.3  Theorems according to Mertzios and Unger   2pthfrgrrn 30770
            ​*17.5.4  Huneke's Proof of the Friendship Theorem   frgrncvvdeqlem1 30787
​PART 18  GUIDES AND MISCELLANEA
      ​18.1  Guides (conventions, explanations, and examples)
            ​*18.1.1  Conventions   conventions 30888
            18.1.2  Natural deduction   natded 30891
            ​*18.1.3  Natural deduction examples   ex-natded5.2 30892
            18.1.4  Definitional examples   ex-or 30909
            18.1.5  Other examples   aevdemo 30948
      ​18.2  Humor
            18.2.1  April Fool's theorem   avril1 30951
      ​18.3  (Future - to be reviewed and classified)
            18.3.1  Planar incidence geometry   cplig 30963
            ​*18.3.2  Aliases kept to prevent broken links   dummylink 30976
​​*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 30978
            19.1.2  Abelian groups   cablo 31033
      ​19.2  Complex vector spaces
            19.2.1  Definition and basic properties   cvc 31047
            19.2.2  Examples of complex vector spaces   cnaddabloOLD 31070
      ​19.3  Normed complex vector spaces
            19.3.1  Definition and basic properties   cnv 31073
            19.3.2  Examples of normed complex vector spaces   cnnv 31166
            19.3.3  Induced metric of a normed complex vector space   imsval 31174
            19.3.4  Inner product   cdip 31189
            19.3.5  Subspaces   css 31210
      ​19.4  Operators on complex vector spaces
            19.4.1  Definitions and basic properties   clno 31229
      ​19.5  Inner product (pre-Hilbert) spaces
            19.5.1  Definition and basic properties   ccphlo 31301
            19.5.2  Examples of pre-Hilbert spaces   cncph 31308
            19.5.3  Properties of pre-Hilbert spaces   isph 31311
      ​19.6  Complex Banach spaces
            19.6.1  Definition and basic properties   ccbn 31351
            19.6.2  Examples of complex Banach spaces   cnbn 31358
            19.6.3  Uniform Boundedness Theorem   ubthlem1 31359
            19.6.4  Minimizing Vector Theorem   minvecolem1 31363
      ​19.7  Complex Hilbert spaces
            19.7.1  Definition and basic properties   chlo 31374
            19.7.2  Standard axioms for a complex Hilbert space   hlex 31387
            19.7.3  Examples of complex Hilbert spaces   cnchl 31405
            19.7.4  Hellinger-Toeplitz Theorem   htthlem 31406
​​*PART 20  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      ​20.1  Axiomatization of complex pre-Hilbert spaces
            20.1.1  Basic Hilbert space definitions   chba 31408
            20.1.2  Preliminary ZFC lemmas   df-hnorm 31457
            ​*20.1.3  Derive the Hilbert space axioms from ZFC set theory   axhilex-zf 31470
            ​*20.1.4  Introduce the vector space axioms for a Hilbert space   ax-hilex 31488
            20.1.5  Vector operations   hvmulex 31500
            20.1.6  Inner product postulates for a Hilbert space   ax-hfi 31568
      ​20.2  Inner product and norms
            20.2.1  Inner product   his5 31575
            20.2.2  Norms   dfhnorm2 31611
            20.2.3  Relate Hilbert space to normed complex vector spaces   hilablo 31649
            20.2.4  Bunjakovaskij-Cauchy-Schwarz inequality   bcsiALT 31668
      ​20.3  Cauchy sequences and completeness axiom
            20.3.1  Cauchy sequences and limits   hcau 31673
            20.3.2  Derivation of the completeness axiom from ZF set theory   hilmet 31683
            20.3.3  Completeness postulate for a Hilbert space   ax-hcompl 31691
            20.3.4  Relate Hilbert space to ZFC pre-Hilbert and Hilbert spaces   hhcms 31692
      ​20.4  Subspaces and projections
            20.4.1  Subspaces   df-sh 31696
            20.4.2  Closed subspaces   df-ch 31710
            20.4.3  Orthocomplements   df-oc 31741
            20.4.4  Subspace sum, span, lattice join, lattice supremum   df-shs 31797
            20.4.5  Projection theorem   pjhthlem1 31880
            20.4.6  Projectors   df-pjh 31884
      ​20.5  Properties of Hilbert subspaces
            20.5.1  Orthomodular law   omlsilem 31891
            20.5.2  Projectors (cont.)   pjhtheu2 31905
            20.5.3  Hilbert lattice operations   sh0le 31929
            20.5.4  Span (cont.) and one-dimensional subspaces   spansn0 32030
            20.5.5  Commutes relation for Hilbert lattice elements   df-cm 32072
            20.5.6  Foulis-Holland theorem   fh1 32107
            20.5.7  Quantum Logic Explorer axioms   qlax1i 32116
            20.5.8  Orthogonal subspaces   chscllem1 32126
            20.5.9  Orthoarguesian laws 5OA and 3OA   5oalem1 32143
            20.5.10  Projectors (cont.)   pjorthi 32158
            20.5.11  Mayet's equation E_3   mayete3i 32217
      ​20.6  Operators on Hilbert spaces
            ​*20.6.1  Operator sum, difference, and scalar multiplication   df-hosum 32219
            20.6.2  Zero and identity operators   df-h0op 32237
            20.6.3  Operations on Hilbert space operators   hoaddcl 32247
            20.6.4  Linear, continuous, bounded, Hermitian, unitary operators and norms   df-nmop 32328
            20.6.5  Linear and continuous functionals and norms   df-nmfn 32334
            20.6.6  Adjoint   df-adjh 32338
            20.6.7  Dirac bra-ket notation   df-bra 32339
            20.6.8  Positive operators   df-leop 32341
            20.6.9  Eigenvectors, eigenvalues, spectrum   df-eigvec 32342
            20.6.10  Theorems about operators and functionals   nmopval 32345
            20.6.11  Riesz lemma   riesz3i 32551
            20.6.12  Adjoints (cont.)   cnlnadjlem1 32556
            20.6.13  Quantum computation error bound theorem   unierri 32593
            20.6.14  Dirac bra-ket notation (cont.)   branmfn 32594
            20.6.15  Positive operators (cont.)   leopg 32611
            20.6.16  Projectors as operators   pjhmopi 32635
      ​20.7  States on a Hilbert lattice and Godowski's equation
            20.7.1  States on a Hilbert lattice   df-st 32700
            20.7.2  Godowski's equation   golem1 32760
      ​20.8  Cover relation, atoms, exchange axiom, and modular symmetry
            20.8.1  Covers relation; modular pairs   df-cv 32768
            20.8.2  Atoms   df-at 32827
            20.8.3  Superposition principle   superpos 32843
            20.8.4  Atoms, exchange and covering properties, atomicity   chcv1 32844
            20.8.5  Irreducibility   chirredlem1 32879
            20.8.6  Atoms (cont.)   atcvat3i 32885
            20.8.7  Modular symmetry   mdsymlem1 32892
​PART 21  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      ​21.1  Mathboxes for user contributions
            21.1.1  Mathbox guidelines   mathbox 32931
      ​21.2  Mathbox for Stefan Allan
      ​21.3  Mathbox for Thierry Arnoux
            21.3.1  Propositional Calculus - misc additions   ad11antr 32936
            21.3.2  Predicate Calculus   sbc2iedf 32949
                  21.3.2.1  Predicate Calculus - misc additions   sbc2iedf 32949
                  21.3.2.2  Restricted quantification - misc additions   ralcom4f 32951
                  21.3.2.3  Equality   eqtrb 32957
                  21.3.2.4  Double restricted existential uniqueness quantification   opsbc2ie 32959
                  21.3.2.5  Double restricted existential uniqueness quantification syntax   w2reu 32961
                  21.3.2.6  Substitution (without distinct variables) - misc additions   sbceqbidf 32970
                  21.3.2.7  Existential "at most one" - misc additions   mo5f 32972
                  21.3.2.8  Existential uniqueness - misc additions   reuxfrdf 32974
                  21.3.2.9  Restricted "at most one" - misc additions   rmoxfrd 32976
                  21.3.2.10  Restricted iota (description binder)   riotaeqbidva 32979
            21.3.3  General Set Theory   dmrab 32980
                  21.3.3.1  Class abstractions (a.k.a. class builders)   dmrab 32980
                  21.3.3.2  Image Sets   abrexdomjm 32990
                  21.3.3.3  Set relations and operations - misc additions   nelun 32996
                  21.3.3.4  Unordered pairs   elpreq 33011
                  21.3.3.5  Unordered triples   tpssg 33020
                  21.3.3.6  Conditional operator - misc additions   ifeqeqx 33025
                  21.3.3.7  Set union   uniinn0 33034
                  21.3.3.8  Indexed union - misc additions   cbviunf 33037
                  21.3.3.9  Indexed intersection - misc additions   iinabrex 33050
                  21.3.3.10  Disjointness - misc additions   disjnf 33051
            21.3.4  Relations and Functions   xpdisjres 33079
                  21.3.4.1  Relations - misc additions   xpdisjres 33079
                  21.3.4.2  Functions - misc additions   fconst7v 33101
                  21.3.4.3  Operations - misc additions   mpomptxf 33159
                  21.3.4.4  Support of a function   suppovss 33161
                  21.3.4.5  Explicit Functions with one or two points as a domain   cosnopne 33174
                  21.3.4.6  Isomorphisms - misc. additions   gtiso 33181
                  21.3.4.7  Disjointness (additional proof requiring functions)   disjdsct 33183
                  21.3.4.8  First and second members of an ordered pair - misc additions   df1stres 33184
                  21.3.4.9  Countable Sets   snct 33192
            21.3.5  Real and Complex Numbers   sgnval2 33214
                  21.3.5.1  Complex operations - misc. additions   creq0 33215
                  21.3.5.2  Ordering on reals - misc additions   lt2addrd 33229
                  21.3.5.3  Extended reals - misc additions   nn0mnfxrd 33230
                  21.3.5.4  Extended nonnegative integers - misc additions   xnn0gt0 33248
                  21.3.5.5  Real number intervals - misc additions   joiniooico 33253
                  21.3.5.6  Finite intervals of integers - misc additions   uzssico 33263
                  21.3.5.7  Half-open integer ranges - misc additions   iundisjfi 33275
                  21.3.5.8  The ` # ` (set size) function - misc additions   hashunif 33285
                  21.3.5.9  The greatest common divisor operator - misc. additions   elq2 33290
                  21.3.5.10  Integers   nn0split01 33296
                  21.3.5.11  Decimal numbers   dfdec100 33308
            21.3.6  Real and complex functions   sgnsgn 33309
                  21.3.6.1  Signum (sgn or sign) function - misc. additions   sgnsgn 33309
                  21.3.6.2  Integer powers - misc. additions   nexple 33311
                  21.3.6.3  Indicator Functions (continued)   indsumin 33315
            ​*21.3.7  Decimal expansion   cdp2 33324
                  ​*21.3.7.1  Decimal point   cdp 33341
                  21.3.7.2  Division in the extended real number system   cxdiv 33370
            21.3.8  Words over a set - misc additions   wrdres 33389
                  21.3.8.1  Splicing words (substring replacement)   splfv3 33406
                  21.3.8.2  Cyclic shift of words   1cshid 33407
            21.3.9  Extensible Structures   ressplusf 33411
                  21.3.9.1  Structure restriction operator   ressplusf 33411
                  21.3.9.2  Posets   ressprs 33414
                  21.3.9.3  Complete lattices   clatp0cl 33424
                  21.3.9.4  Order Theory   cmnt 33426
                  21.3.9.5  Extended reals Structure - misc additions   ax-xrssca 33452
                  21.3.9.6  The extended nonnegative real numbers commutative monoid   xrge00 33462
            21.3.10  Algebra   mndcld 33470
                  21.3.10.1  Monoids   mndcld 33470
                  21.3.10.2  Monoids Homomorphisms   abliso 33483
                  21.3.10.3  Groups - misc additions   grpidcld 33487
                  21.3.10.4  Abelian Groups - misc additions   ablcomd 33493
                  21.3.10.5  Finitely supported group sums - misc additions   gsumsubg 33494
                  21.3.10.6  Group or monoid sums over words   gsumwun 33524
                  21.3.10.7  Centralizers and centers - misc additions   cntzun 33527
                  21.3.10.8  The symmetric group   symgfcoeu 33530
                  21.3.10.9  Transpositions   pmtridf1o 33542
                  21.3.10.10  Permutation Signs   psgnid 33545
                  21.3.10.11  Permutation cycles   ctocyc 33554
                  21.3.10.12  The Alternating Group   evpmval 33593
                  21.3.10.13  Signum in an ordered monoid   csgns 33606
                  21.3.10.14  Fixed points   cfxp 33611
                  21.3.10.15  The Archimedean property for generic ordered algebraic structures   cinftm 33624
                  21.3.10.16  Semiring left modules   cslmd 33648
                  21.3.10.17  Simple groups   prmsimpcyc 33676
                  21.3.10.18  Rings - misc additions   ringrngd 33677
                  21.3.10.19  Subrings generated by a set   elrgspnlem1 33690
                  21.3.10.20  The zero ring   irrednzr 33698
                  21.3.10.21  Localization of rings   cerl 33701
                  21.3.10.22  Integral Domains   domnmuln0rd 33725
                  21.3.10.23  Euclidean Domains   ceuf 33739
                  21.3.10.24  Division Rings   rndrhmcl 33745
                  21.3.10.25  The field of rational numbers   qfld 33746
                  21.3.10.26  Subfields   subsdrg 33747
                  21.3.10.27  Field of fractions   cfrac 33751
                  21.3.10.28  Field extensions generated by a set   cfldgen 33759
                  21.3.10.29  Ring homomorphisms - misc additions   rhmdvd 33772
                  21.3.10.30  Scalar restriction operation   cresv 33774
                  21.3.10.31  The commutative ring of gaussian integers   gzcrng 33789
                  21.3.10.32  The archimedean ordered field of real numbers   cnfldfld 33790
                  21.3.10.33  The quotient map and quotient modules   qusker 33797
                  21.3.10.34  The ring of integers modulo ` N `   znfermltl 33809
                  21.3.10.35  Independent sets and families   islinds5 33810
                  21.3.10.36  Ring associates, ring units   dvdsruassoi 33825
                  ​*21.3.10.37  Subgroup sum / Sumset / Minkowski sum   elgrplsmsn 33831
                  21.3.10.38  The quotient map   quslsm 33842
                  21.3.10.39  Ideals   intlidl 33856
                  21.3.10.40  Maximal Ideals   cmxidl 33870
                  21.3.10.41  Local rings   drnglring 33910
                  21.3.10.42  The semiring of ideals of a ring   cidlsrg 33918
                  21.3.10.43  Prime Elements   rprmval 33934
                  21.3.10.44  Unique factorization domains   cufd 33956
                  21.3.10.45  The ring of integers   zringidom 33969
                  21.3.10.46  Associative Algebra   assaassd 33973
                  21.3.10.47  Univariate Polynomials   0ringmon1p 33975
                  21.3.10.48  Polynomial quotient and polynomial remainder   q1pdir 34021
                  21.3.10.49  Multivariate Polynomials   psrbasfsupp 34029
                  21.3.10.50  The ring of symmetric polynomials   csply 34073
                  21.3.10.51  The subring algebra   sra1r 34099
                  21.3.10.52  Division Ring Extensions   drgext0g 34108
                  21.3.10.53  Vector Spaces   lvecdimfi 34114
                  21.3.10.54  Vector Space Dimension   cldim 34117
            21.3.11  Field Extensions   cfldext 34156
                  21.3.11.1  Algebraic numbers   cirng 34201
                  21.3.11.2  Algebraic extensions   calgext 34213
                  21.3.11.3  Minimal polynomials   cminply 34217
                  21.3.11.4  Quadratic Field Extensions   rtelextdg2lem 34244
                  21.3.11.5  Towers of quadratic extentions   fldext2chn 34246
            ​*21.3.12  Constructible Numbers   cconstr 34247
                  21.3.12.1  Impossible constructions   2sqr3minply 34298
            21.3.13  Matrices   csmat 34311
                  21.3.13.1  Submatrices   csmat 34311
                  21.3.13.2  Matrix literals   clmat 34329
                  21.3.13.3  Laplace expansion of determinants   mdetpmtr1 34341
            21.3.14  Topology   ist0cld 34351
                  21.3.14.1  Open maps   txomap 34352
                  21.3.14.2  Topology of the unit circle   qtopt1 34353
                  21.3.14.3  Refinements   reff 34357
                  21.3.14.4  Open cover refinement property   ccref 34360
                  21.3.14.5  Lindelöf spaces   cldlf 34370
                  21.3.14.6  Paracompact spaces   cpcmp 34373
                  ​*21.3.14.7  Spectrum of a ring   crspec 34380
                  21.3.14.8  Pseudometrics   cmetid 34404
                  21.3.14.9  Continuity - misc additions   hauseqcn 34416
                  21.3.14.10  Topology of the closed unit interval   elunitge0 34417
                  21.3.14.11  Topology of ` ( RR X. RR ) `   unicls 34421
                  21.3.14.12  Order topology - misc. additions   cnvordtrestixx 34431
                  21.3.14.13  Continuity in topological spaces - misc. additions   mndpluscn 34444
                  21.3.14.14  Topology of the extended nonnegative real numbers ordered monoid   xrge0hmph 34450
                  21.3.14.15  Limits - misc additions   lmlim 34465
                  21.3.14.16  Univariate polynomials   pl1cn 34473
            21.3.15  Uniform Stuctures and Spaces   chcmp 34474
                  21.3.15.1  Hausdorff uniform completion   chcmp 34474
            21.3.16  Topology and algebraic structures   zringnm 34476
                  21.3.16.1  The norm on the ring of the integer numbers   zringnm 34476
                  21.3.16.2  Topological ` ZZ ` -modules   zlm0 34478
                  21.3.16.3  Canonical embedding of the field of the rational numbers into a division ring   cqqh 34488
                  21.3.16.4  Canonical embedding of the real numbers into a complete ordered field   crrh 34511
                  21.3.16.5  Embedding from the extended real numbers into a complete lattice   cxrh 34534
                  21.3.16.6  Canonical embeddings into the ordered field of the real numbers   zrhre 34537
                  ​*21.3.16.7  Topological Manifolds   cmntop 34540
                  21.3.16.8  Extended sum   cesum 34545
            21.3.17  Mixed Function/Constant operation   cofc 34613
            21.3.18  Abstract measure   csiga 34626
                  21.3.18.1  Sigma-Algebra   csiga 34626
                  21.3.18.2  Generated sigma-Algebra   csigagen 34657
                  ​*21.3.18.3  lambda and pi-Systems, Rings of Sets   ispisys 34671
                  21.3.18.4  The Borel algebra on the real numbers   cbrsiga 34700
                  21.3.18.5  Product Sigma-Algebra   csx 34707
                  21.3.18.6  Measures   cmeas 34714
                  21.3.18.7  The counting measure   cntmeas 34745
                  21.3.18.8  The Lebesgue measure - misc additions   voliune 34748
                  21.3.18.9  The Dirac delta measure   cdde 34751
                  21.3.18.10  The 'almost everywhere' relation   cae 34756
                  21.3.18.11  Measurable functions   cmbfm 34768
                  21.3.18.12  Borel Algebra on ` ( RR X. RR ) `   br2base 34788
                  ​*21.3.18.13  Caratheodory's extension theorem   coms 34810
            21.3.19  Integration   itgeq12dv 34845
                  21.3.19.1  Lebesgue integral - misc additions   itgeq12dv 34845
                  21.3.19.2  Bochner integral   citgm 34846
            21.3.20  Euler's partition theorem   oddpwdc 34873
            21.3.21  Sequences defined by strong recursion   csseq 34902
            21.3.22  Fibonacci Numbers   cfib 34915
            21.3.23  Probability   cprb 34926
                  21.3.23.1  Probability Theory   cprb 34926
                  21.3.23.2  Conditional Probabilities   ccprob 34950
                  21.3.23.3  Real-valued Random Variables   crrv 34959
                  21.3.23.4  Preimage set mapping operator   corvc 34975
                  21.3.23.5  Distribution Functions   orvcelval 34988
                  21.3.23.6  Cumulative Distribution Functions   orvclteel 34992
                  21.3.23.7  Probabilities - example   coinfliplem 34998
                  21.3.23.8  Bertrand's Ballot Problem   ballotlemoex 35005
            21.3.24  Signum (sgn or sign) function - misc. additions   fzssfzo 35058
                  21.3.24.1  Operations on words   ccatmulgnn0dir 35061
            21.3.25  Polynomials with real coefficients - misc additions   plyrecld 35065
            21.3.26  Descartes's rule of signs   signspval 35068
                  21.3.26.1  Sign changes in a word over real numbers   signspval 35068
                  21.3.26.2  Counting sign changes in a word over real numbers   signslema 35078
            21.3.27  Number Theory   iblidicc 35108
                  21.3.27.1  Representations of a number as sums of integers   crepr 35124
                  21.3.27.2  Vinogradov Trigonometric Sums and the Circle Method   cvts 35151
                  21.3.27.3  The Ternary Goldbach Conjecture: Final Statement   ax-hgt749 35160
            21.3.28  Elementary Geometry   cstrkg2d 35180
                  ​*21.3.28.1  Two-dimensional geometry   cstrkg2d 35180
                  21.3.28.2  Morley's Miracle   cgranbtwn 35185
                  21.3.28.3  Outer Five Segment (not used, no need to move to main)   cafs 35188
            ​*21.3.29  LeftPad Project   clpad 35193
      ​​*21.4  Mathbox for Jonathan Ben-Naim
            21.4.1  First-order logic and set theory   bnj170 35216
            21.4.2  Well founded induction and recursion   bnj110 35375
            21.4.3  The existence of a minimal element in certain classes   bnj69 35527
            21.4.4  Well-founded induction   bnj1204 35529
            21.4.5  Well-founded recursion, part 1 of 3   bnj60 35579
            21.4.6  Well-founded recursion, part 2 of 3   bnj1500 35585
            21.4.7  Well-founded recursion, part 3 of 3   bnj1522 35589
      ​21.5  Mathbox for BTernaryTau
            21.5.1  First-order logic   nfan1c 35590
                  21.5.1.1  Auxiliary axiom schemes   nfan1c 35590
            21.5.2  ZF set theory   inv2 35596
                  21.5.2.1  Finitism   prcinf 35647
                  21.5.2.2  Introduce ax-regs   ax-regs 35660
                  21.5.2.3  Derive ax-regs   axregs 35673
                  21.5.2.4  ZFC axioms with reduced distinct variable conditions   axsepg2 35674
                  21.5.2.5  Cardinality without the Axiom of Choice   ckard 35683
                  21.5.2.6  Global choice   gblacfnacd 35707
            21.5.3  Real and complex numbers   zltp1ne 35722
            21.5.4  Graph theory   cplgredgex 35727
                  21.5.4.1  Acyclic graphs   acycgr0v 35735
      ​21.6  Mathbox for Mario Carneiro
            21.6.1  Predicate calculus with all distinct variables   ax-7d 35746
            21.6.2  Miscellaneous stuff   quartfull 35752
            21.6.3  Derangements and the Subfactorial   deranglem 35753
            21.6.4  The Erdős-Szekeres theorem   erdszelem1 35778
            21.6.5  The Kuratowski closure-complement theorem   kur14lem1 35793
            21.6.6  Retracts and sections   cretr 35804
            21.6.7  Path-connected and simply connected spaces   cpconn 35806
            21.6.8  Covering maps   ccvm 35842
            21.6.9  Normal numbers   snmlff 35916
            21.6.10  Godel-sets of formulas - part 1   cgoe 35920
            21.6.11  Godel-sets of formulas - part 2   cgon 36019
            21.6.12  Models of ZF   cgze 36033
            ​*21.6.13  Metamath formal systems   cmcn 36047
            21.6.14  Grammatical formal systems   cm0s 36172
            21.6.15  Models of formal systems   cmuv 36192
            21.6.16  Splitting fields   ccpms 36214
            21.6.17  p-adic number fields   czr 36234
      ​​*21.7  Mathbox for Filip Cernatescu
      ​21.8  Mathbox for Paul Chapman
            21.8.1  Real and complex numbers (cont.)   climuzcnv 36258
            21.8.2  Miscellaneous theorems   elfzm12 36262
      ​21.9  Mathbox for Hongxiu Chen
      ​21.10  Mathbox for Adrian Ducourtial
            21.10.1  Propositional calculus   currybi 36275
            21.10.2  Clone theory   ccloneop 36282
      ​21.11  Mathbox for Scott Fenton
            21.11.1  ZFC Axioms in primitive form   axextprim 36288
            21.11.2  Untangled classes   untelirr 36295
            21.11.3  Extra propositional calculus theorems   3jaodd 36302
            21.11.4  Misc. Useful Theorems   nepss 36305
            21.11.5  Properties of real and complex numbers   sqdivzi 36315
            21.11.6  Infinite products   iprodefisumlem 36327
            21.11.7  Factorial limits   faclimlem1 36330
            21.11.8  Greatest common divisor and divisibility   gcd32 36336
            21.11.9  Properties of relationships   dftr6 36338
            21.11.10  Properties of functions and mappings   funpsstri 36353
            21.11.11  Ordinal numbers   elpotr 36366
            21.11.12  Defined equality axioms   axextdfeq 36382
            21.11.13  Hypothesis builders   hbntg 36390
            21.11.14  Well-founded zero, successor, and limits   cwsuc 36395
            21.11.15  Quantifier-free definitions   ctxp 36415
            21.11.16  Alternate ordered pairs   caltop 36544
            21.11.17  Geometry in the Euclidean space   cofs 36570
                  21.11.17.1  Congruence properties   cofs 36570
                  21.11.17.2  Betweenness properties   btwntriv2 36600
                  21.11.17.3  Segment Transportation   ctransport 36617
                  21.11.17.4  Properties relating betweenness and congruence   cifs 36623
                  21.11.17.5  Connectivity of betweenness   btwnconn1lem1 36675
                  21.11.17.6  Segment less than or equal to   csegle 36694
                  21.11.17.7  Outside-of relationship   coutsideof 36707
                  21.11.17.8  Lines and Rays   cline2 36722
            21.11.18  Forward difference   cfwddif 36746
            21.11.19  Rank theorems   rankung 36754
            21.11.20  Hereditarily Finite Sets   chf 36760
            21.11.21  Natural ordinal operations   cnmul 36775
      ​21.12  Mathbox for Gino Giotto
            21.12.1  Equality theorems   rmoeqi 36815
                  21.12.1.1  Inference versions   rmoeqi 36815
                  21.12.1.2  Deduction versions   rmoeqdv 36840
            21.12.2  Change bound variables   in-ax8 36852
                  21.12.2.1  Change bound variables and domains   cbvralvw2 36854
                  21.12.2.2  Change bound variables, deduction versions   cbvmodavw 36878
                  21.12.2.3  Change bound variables and domains, deduction versions   cbvrmodavw2 36911
            21.12.3  Study of ax-mulf usage   mpomulnzcnf 36927
      ​21.13  Mathbox for Jeff Hankins
            21.13.1  Miscellany   a1i14 36928
            21.13.2  Basic topological facts   topbnd 36951
            21.13.3  Topology of the real numbers   ivthALT 36962
            21.13.4  Refinements   cfne 36963
            21.13.5  Neighborhood bases determine topologies   neibastop1 36986
            21.13.6  Lattice structure of topologies   topmtcl 36990
            21.13.7  Filter bases   fgmin 36997
            21.13.8  Directed sets, nets   tailfval 36999
      ​21.14  Mathbox for Anthony Hart
            21.14.1  Propositional Calculus   tb-ax1 37010
            21.14.2  Predicate Calculus   nalfal 37030
            21.14.3  Miscellaneous single axioms   meran1 37038
            21.14.4  Connective Symmetry   negsym1 37044
      ​21.15  Mathbox for Chen-Pang He
            21.15.1  Ordinal topology   ontopbas 37055
      ​21.16  Mathbox for Jeff Hoffman
            21.16.1  Inferences for finite induction on generic function values   fveleq 37078
            21.16.2  gdc.mm   nnssi2 37082
      ​21.17  Mathbox for Matthew House
            21.17.1  Relations on well-ordered indexed unions   weiunval 37089
            21.17.2  Axiom of Transitive Containment   axtco 37098
            21.17.3  Transitive closure of a class   tr0elw 37111
            ​*21.17.4  Stronger axioms of regularity   mh-setind 37163
            21.17.5  Short axioms written in primitive symbols   mh-inf3f1 37168
      ​21.18  Mathbox for Asger C. Ipsen
            21.18.1  Continuous nowhere differentiable functions   dnival 37176
      ​​*21.19  Mathbox for BJ
            *21.19.1  Propositional calculus   bj-mp2c 37245
                  *21.19.1.1  Derived rules of inference   bj-mp2c 37245
                  ​*21.19.1.2  A syntactic theorem   bj-0 37247
                  ​*21.19.1.3  Minimal implicational calculus   bj-poni 37249
                  ​*21.19.1.4  Positive calculus   bj-bisimpl 37261
                  ​*21.19.1.5  Implication and negation   bj-con2com 37269
                  ​*21.19.1.6  Disjunction   bj-jaoi1 37280
                  ​*21.19.1.7  Logical equivalence   bj-dfbi4 37282
                  21.19.1.8  The conditional operator for propositions   bj-consensus 37287
                  ​*21.19.1.9  Propositional calculus: miscellaneous   bj-imbi12 37292
            ​*21.19.2  Modal logic   bj-axdd2 37301
            ​*21.19.3  Provability logic   cprvb 37306
            ​*21.19.4  First-order logic   bj-exexalal 37315
                  21.19.4.1  Universal and existential quantifiers, nonfreeness predicate   bj-exexalal 37315
                  21.19.4.2  Adding ax-gen   bj-genr 37316
                  21.19.4.3  Adding ax-4   bj-almp 37320
                  21.19.4.4  Adding ax-5   bj-spvw 37373
                  21.19.4.5  Equality and substitution   bj-df-sb 37388
                  21.19.4.6  Adding ax-6   bj-spim0 37407
                  21.19.4.7  Adding ax-7   bj-cbvexw 37415
                  21.19.4.8  Membership predicate, ax-8 and ax-9   bj-ax89 37417
                  21.19.4.9  Adding ax-11   bj-alcomexcom 37419
                  21.19.4.10  Adding ax-12   axc11n11 37423
                  ​*21.19.4.11  Really adding ax-12   bj-substax12 37465
                  21.19.4.12  Nonfreeness   wnnf 37467
                  21.19.4.13  Adding ax-13   bj-axc10 37534
                  ​*21.19.4.14  Removing dependencies on ax-13 (and ax-11)   bj-axc10v 37544
                  ​*21.19.4.15  Distinct var metavariables   bj-hbaeb2 37569
                  ​*21.19.4.16  Around ~ equsal   bj-equsal1t 37573
                  ​*21.19.4.17  Some Principia Mathematica proofs   stdpc5t 37578
                  21.19.4.18  Alternate definition of substitution   bj-sbsb 37588
                  21.19.4.19  Lemmas for substitution   bj-sbf3 37590
                  21.19.4.20  Existential uniqueness   bj-eu3f 37592
                  ​*21.19.4.21  First-order logic: miscellaneous   bj-sblem1 37593
            21.19.5  Set theory   eliminable1 37610
                  ​*21.19.5.1  Eliminability of class terms   eliminable1 37610
                  ​*21.19.5.2  Classes without the axiom of extensionality   bj-denoteslem 37622
                  21.19.5.3  Characterization among sets versus among classes   elelb 37648
                  ​*21.19.5.4  The nonfreeness quantifier for classes   bj-nfcsym 37650
                  ​*21.19.5.5  Lemmas for class substitution   bj-sbeqALT 37651
                  21.19.5.6  Removing some axiom requirements and disjoint variable conditions   bj-exlimvmpi 37662
                  ​*21.19.5.7  Class abstractions   bj-elabd2ALT 37677
                  21.19.5.8  Generalized class abstractions   bj-cgab 37685
                  ​*21.19.5.9  Restricted nonfreeness   wrnf 37693
                  ​*21.19.5.10  Russell's paradox   bj-ru1 37695
                  21.19.5.11  Curry's paradox in set theory   currysetlem 37697
                  ​*21.19.5.12  Some disjointness results   bj-n0i 37703
                  ​*21.19.5.13  Complements on direct products   bj-xpimasn 37707
                  ​*21.19.5.14  "Singletonization" and tagging   bj-snsetex 37715
                  ​*21.19.5.15  Tuples of classes   bj-cproj 37742
                  ​*21.19.5.16  Set theory: elementary operations relative to a universe   bj-rcleqf 37777
                  ​*21.19.5.17  Axioms for finite unions   bj-abex 37782
                  ​*21.19.5.18  Set theory: miscellaneous   eleq2w2ALT 37799
                  ​*21.19.5.19  Axioms of separation and replacement   bj-axnul 37825
                  ​*21.19.5.20  Evaluation at a class   bj-evaleq 37829
                  21.19.5.21  Elementwise operations   celwise 37837
                  ​*21.19.5.22  Elementwise intersection (families of sets induced on a subset)   bj-rest00 37839
                  21.19.5.23  Moore collections (complements)   bj-raldifsn 37858
                  21.19.5.24  Maps-to notation for functions with three arguments   bj-0nelmpt 37874
                  ​*21.19.5.25  Currying   csethom 37880
                  ​*21.19.5.26  Setting components of extensible structures   cstrset 37892
            ​*21.19.6  Extended real and complex numbers, real and complex projective lines   bj-nfald 37895
                  21.19.6.1  Complements on class abstractions of ordered pairs and binary relations   bj-nfald 37895
                  ​*21.19.6.2  Identity relation (complements)   bj-opabssvv 37910
                  ​*21.19.6.3  Functionalized identity (diagonal in a Cartesian square)   cdiag2 37932
                  ​*21.19.6.4  Direct image and inverse image   cimdir 37938
                  ​*21.19.6.5  Extended numbers and projective lines as sets   cfractemp 37956
                  ​*21.19.6.6  Addition and opposite   caddcc 37997
                  ​*21.19.6.7  Order relation on the extended reals   cltxr 38001
                  ​*21.19.6.8  Argument, multiplication and inverse   carg 38003
                  21.19.6.9  The canonical bijection from the finite ordinals   ciomnn 38009
                  21.19.6.10  Divisibility   cnnbar 38020
            ​*21.19.7  Monoids   bj-smgrpssmgm 38028
                  ​*21.19.7.1  Finite sums in monoids   cfinsum 38043
            ​*21.19.8  Affine, Euclidean, and Cartesian geometry   bj-fvimacnv0 38046
                  *21.19.8.1  Real vector spaces   bj-fvimacnv0 38046
                  ​*21.19.8.2  Complex numbers (supplements)   bj-subcom 38068
                  ​*21.19.8.3  Barycentric coordinates   bj-bary1lem 38070
            21.19.9  Monoid of endomorphisms   cend 38073
      ​21.20  Mathbox for Jim Kingdon
            21.20.1  Circle constant   taupilem3 38079
            21.20.2  Number theory   dfgcd3 38084
            21.20.3  Real numbers   irrdifflemf 38085
      ​21.21  Mathbox for ML
            21.21.1  Miscellaneous   csbrecsg 38090
            21.21.2  Cartesian exponentiation   cfinxp 38145
            21.21.3  Topology   iunctb2 38165
                  ​*21.21.3.1  Pi-base theorems   pibp16 38175
      ​21.22  Mathbox for Wolf Lammen
            21.22.1  1. Bootstrapping   wl-section-boot 38184
            21.22.2  Implication chains   wl-section-impchain 38208
            21.22.3  Theorems around the conditional operator   wl-ifp-ncond1 38226
            21.22.4  Alternative development of hadd, cadd   wl-df-3xor 38230
            21.22.5  An alternative axiom ~ ax-13   ax-wl-13v 38255
            21.22.6  Bootstrapping set theory with classes   wl-cleq-0 38257
            21.22.7  Other stuff   wl-mps 38278
      ​21.23  Mathbox for Brendan Leahy
      ​21.24  Mathbox for Thomas van Maaren
      ​21.25  Mathbox for Jeff Madsen
            21.25.1  Logic and set theory   unirep 38472
            21.25.2  Real and complex numbers; integers   filbcmb 38498
            21.25.3  Sequences and sums   sdclem2 38500
            21.25.4  Topology   subspopn 38510
            21.25.5  Metric spaces   metf1o 38513
            21.25.6  Continuous maps and homeomorphisms   constcncf 38520
            21.25.7  Boundedness   ctotbnd 38524
            21.25.8  Isometries   cismty 38556
            21.25.9  Heine-Borel Theorem   heibor1lem 38567
            21.25.10  Banach Fixed Point Theorem   bfplem1 38580
            21.25.11  Euclidean space   crrn 38583
            21.25.12  Intervals (continued)   ismrer1 38596
            ​*21.25.13  Operation properties   cass 38600
            21.25.14  Groups and related structures   cmagm 38606
            21.25.15  Group homomorphism and isomorphism   cghomOLD 38641
            21.25.16  Rings   crngo 38652
            21.25.17  Division Rings   cdrng 38706
            21.25.18  Ring homomorphisms   crngohom 38718
            21.25.19  Commutative rings   ccm2 38747
            21.25.20  Ideals   cidl 38765
            21.25.21  Prime rings and integral domains   cprrng 38804
            21.25.22  Ideal generators   cigen 38817
      ​21.26  Mathbox for Giovanni Mascellani
            ​*21.26.1  Tools for automatic proof building   efald2 38836
            ​*21.26.2  Tseitin axioms   fald 38885
            ​*21.26.3  Equality deductions   iuneq2f 38912
            ​*21.26.4  Miscellanea   orcomdd 38923
      ​21.27  Mathbox for Peter Mazsa
            21.27.1  Notations   cxrn 38930
            21.27.2  Preparatory theorems   el2v1 38985
            21.27.3  Range Cartesian product   df-xrn 39136
            21.27.4  Relations   df-rels 39196
            21.27.5  Quotient map (coset map)   df-qmap 39202
            21.27.6  Lifts, shifts, successor, and predecessor   df-adjliftmap 39211
            21.27.7  Cosets by ` R `   df-coss 39257
            21.27.8  Subset relations   df-ssr 39334
            21.27.9  Reflexivity   df-refs 39346
            21.27.10  Converse reflexivity   df-cnvrefs 39361
            21.27.11  Symmetry   df-syms 39378
            21.27.12  Reflexivity and symmetry   symrefref2 39403
            21.27.13  Transitivity   df-trs 39412
            21.27.14  Equivalence relations   df-eqvrels 39424
            21.27.15  Redundancy   df-redunds 39463
            21.27.16  Domain quotients   df-dmqss 39478
            21.27.17  Equivalence relations on domain quotients   df-ers 39504
            21.27.18  Functions   df-funss 39521
            21.27.19  Disjoints vs. converse functions   df-disjss 39544
            21.27.20  Antisymmetry   df-antisymrel 39619
            21.27.21  Partitions: disjoints on domain quotients   df-parts 39624
            21.27.22  Partition-Equivalence Theorems   disjim 39640
            21.27.23  Type-safe Partition-Equivalence: PetParts, PetErs, Pet2Parts, Pet2Ers   df-petparts 39724
      ​21.28  Mathbox for Rodolfo Medina
            21.28.1  Partitions   prtlem60 39734
      ​​*21.29  Mathbox for Norm Megill
            *21.29.1  Obsolete schemes ax-c4,c5,c7,c10,c11,c11n,c15,c9,c14,c16   ax-c5 39764
            ​*21.29.2  Rederive new axioms ax-4, ax-10, ax-6, ax-12, ax-13 from old   axc5 39774
            ​*21.29.3  Legacy theorems using obsolete axioms   ax5ALT 39788
            21.29.4  Experiments with weak deduction theorem   elimhyps 39842
            21.29.5  Miscellanea   cnaddcom 39853
            21.29.6  Atoms, hyperplanes, and covering in a left vector space (or module)   clsa 39855
            21.29.7  Functionals and kernels of a left vector space (or module)   clfn 39938
            21.29.8  Opposite rings and dual vector spaces   cld 40004
            21.29.9  Ortholattices and orthomodular lattices   cops 40053
            21.29.10  Atomic lattices with covering property   ccvr 40143
            21.29.11  Hilbert lattices   chlt 40231
            21.29.12  Projective geometries based on Hilbert lattices   clln 40372
            21.29.13  Construction of a vector space from a Hilbert lattice   cdlema1N 40672
            21.29.14  Construction of involution and inner product from a Hilbert lattice   clpoN 42361
      ​21.30  Mathbox for metakunt
            21.30.1  Commutative Semiring   ccsrg 42843
            21.30.2  General helpful statements   rhmzrhval 42846
            21.30.3  Some gcd and lcm results   12gcd5e1 42877
            21.30.4  Least common multiple inequality theorem   3factsumint1 42895
            21.30.5  Logarithm inequalities   3exp7 42927
            21.30.6  Miscellaneous results for AKS formalisation   intlewftc 42935
            21.30.7  Sticks and stones   sticksstones1 43020
            21.30.8  Continuation AKS   aks6d1c6lem1 43044
      ​21.31  Mathbox for Luke Murphy
            21.31.1  Solutions of quadratic equations   quadfac 43079
            21.31.2  April Fool's theorem   25or6to4 43080
      ​21.32  Mathbox for Steven Nguyen
            21.32.1  Utility theorems   jarrii 43081
            ​*21.32.2  Arithmetic theorems   c0exALT 43127
            21.32.3  Exponents and divisibility   oexpreposd 43205
            21.32.4  Trigonometry and Calculus   tanhalfpim 43232
            ​*21.32.5  Independence of ax-mulcom   cresub 43248
            21.32.6  Structures   sn-base0 43391
            ​*21.32.7  Projective spaces   cprjsp 43455
            21.32.8  Basic reductions for Fermat's Last Theorem   dffltz 43488
            ​*21.32.9  Exemplar theorems   iddii 43518
                  ​*21.32.9.1  Standard replacements of ax-10 , ax-11 , ax-12   nfa1w 43529
      ​21.33  Mathbox for Igor Ieskov
      ​21.34  Mathbox for OpenAI
      ​21.35  Mathbox for Stefan O'Rear
            21.35.1  Additional elementary logic and set theory   moxfr 43545
            21.35.2  Additional theory of functions   imaiinfv 43546
            21.35.3  Additional topology   elrfi 43547
            21.35.4  Characterization of closure operators. Kuratowski closure axioms   ismrcd1 43551
            21.35.5  Algebraic closure systems   cnacs 43555
            21.35.6  Miscellanea 1. Map utilities   constmap 43566
            21.35.7  Miscellanea for polynomials   mptfcl 43573
            21.35.8  Multivariate polynomials over the integers   cmzpcl 43574
            21.35.9  Miscellanea for Diophantine sets 1   coeq0i 43606
            21.35.10  Diophantine sets 1: definitions   cdioph 43608
            21.35.11  Diophantine sets 2 miscellanea   ellz1 43620
            21.35.12  Diophantine sets 2: union and intersection. Monotone Boolean algebra   diophin 43625
            21.35.13  Diophantine sets 3: construction   diophrex 43628
            21.35.14  Diophantine sets 4 miscellanea   2sbcrex 43637
            21.35.15  Diophantine sets 4: Quantification   rexrabdioph 43643
            21.35.16  Diophantine sets 5: Arithmetic sets   rabdiophlem1 43650
            21.35.17  Diophantine sets 6: reusability. renumbering of variables   eldioph4b 43660
            21.35.18  Pigeonhole Principle and cardinality helpers   fphpd 43665
            21.35.19  A non-closed set of reals is infinite   rencldnfilem 43669
            21.35.20  Lagrange's rational approximation theorem   irrapxlem1 43671
            21.35.21  Pell equations 1: A nontrivial solution always exists   pellexlem1 43678
            21.35.22  Pell equations 2: Algebraic number theory of the solution set   csquarenn 43685
            21.35.23  Pell equations 3: characterizing fundamental solution   infmrgelbi 43727
            ​*21.35.24  Logarithm laws generalized to an arbitrary base   reglogcl 43739
            21.35.25  Pell equations 4: the positive solution group is infinite cyclic   pellfund14 43747
            21.35.26  X and Y sequences 1: Definition and recurrence laws   crmx 43749
            21.35.27  Ordering and induction lemmas for the integers   monotuz 43790
            21.35.28  X and Y sequences 2: Order properties   rmxypos 43796
            21.35.29  Congruential equations   congtr 43814
            21.35.30  Alternating congruential equations   acongid 43824
            21.35.31  Additional theorems on integer divisibility   coprmdvdsb 43834
            21.35.32  X and Y sequences 3: Divisibility properties   jm2.18 43837
            21.35.33  X and Y sequences 4: Diophantine representability of Y   jm2.27a 43854
            21.35.34  X and Y sequences 5: Diophantine representability of X, ^, _C   rmxdiophlem 43864
            21.35.35  Uncategorized stuff not associated with a major project   setindtr 43873
            21.35.36  More equivalents of the Axiom of Choice   axac10 43882
            21.35.37  Finitely generated left modules   clfig 43916
            21.35.38  Noetherian left modules I   clnm 43924
            21.35.39  Addenda for structure powers   pwssplit4 43938
            21.35.40  Every set admits a group structure iff choice   unxpwdom3 43944
            21.35.41  Noetherian rings and left modules II   clnr 43958
            21.35.42  Hilbert's Basis Theorem   cldgis 43970
            21.35.43  Additional material on polynomials [DEPRECATED]   cmnc 43980
            21.35.44  Degree and minimal polynomial of algebraic numbers   cdgraa 43989
            21.35.45  Algebraic integers I   citgo 44006
            21.35.46  Endomorphism algebra   cmend 44020
            21.35.47  Cyclic groups and order   idomodle 44040
            21.35.48  Cyclotomic polynomials   ccytp 44046
            21.35.49  Miscellaneous topology   fgraphopab 44052
      ​21.36  Mathbox for Noam Pasman
      ​21.37  Mathbox for Jon Pennant
      ​21.38  Mathbox for Richard Penner
            21.38.1  Set Theory and Ordinal Numbers   uniel 44066
            21.38.2  Natural addition of Cantor normal forms   oawordex2 44175
            21.38.3  Surreal Contributions   abeqabi 44256
            21.38.4  Short Studies   nlimsuc 44289
                  21.38.4.1  Additional work on conditional logical operator   ifpan123g 44307
                  21.38.4.2  Sophisms   rp-fakeimass 44360
                  ​*21.38.4.3  Finite Sets   rp-isfinite5 44365
                  21.38.4.4  General Observations   intabssd 44367
                  21.38.4.5  Infinite Sets   pwelg 44408
                  ​*21.38.4.6  Finite intersection property   fipjust 44413
                  21.38.4.7  RP ADDTO: Subclasses and subsets   rababg 44422
                  21.38.4.8  RP ADDTO: The intersection of a class   elinintab 44423
                  21.38.4.9  RP ADDTO: Theorems requiring subset and intersection existence   elinintrab 44425
                  21.38.4.10  RP ADDTO: Relations   xpinintabd 44428
                  ​*21.38.4.11  RP ADDTO: Functions   elmapintab 44444
                  ​*21.38.4.12  RP ADDTO: Finite induction (for finite ordinals)   cnvcnvintabd 44448
                  21.38.4.13  RP ADDTO: First and second members of an ordered pair   elcnvlem 44449
                  21.38.4.14  RP ADDTO: The reflexive and transitive properties of relations   undmrnresiss 44452
                  21.38.4.15  RP ADDTO: Basic properties of closures   cleq2lem 44456
                  21.38.4.16  RP REPLACE: Definitions and basic properties of transitive closures   trcleq2lemRP 44478
                  ​*21.38.4.17  Additions for square root; absolute value   sqrtcvallem1 44479
            21.38.5  Additional statements on relations and subclasses   al3im 44495
                  21.38.5.1  Transitive relations (not to be confused with transitive classes)   trrelind 44513
                  21.38.5.2  Reflexive closures   crcl 44520
                  ​*21.38.5.3  Finite relationship composition   relexp2 44525
                  21.38.5.4  Transitive closure of a relation   dftrcl3 44568
                  ​*21.38.5.5  Adapted from Frege   frege77d 44594
            ​*21.38.6  Propositions from _Begriffsschrift_   dfxor4 44614
                  *21.38.6.1  _Begriffsschrift_ Chapter I   dfxor4 44614
                  ​*21.38.6.2  _Begriffsschrift_ Notation hints   whe 44620
                  21.38.6.3  _Begriffsschrift_ Chapter II Implication   ax-frege1 44638
                  21.38.6.4  _Begriffsschrift_ Chapter II Implication and Negation   axfrege28 44677
                  ​*21.38.6.5  _Begriffsschrift_ Chapter II with logical equivalence   axfrege52a 44704
                  21.38.6.6  _Begriffsschrift_ Chapter II with equivalence of sets   axfrege52c 44735
                  ​*21.38.6.7  _Begriffsschrift_ Chapter II with equivalence of classes   frege53c 44762
                  ​*21.38.6.8  _Begriffsschrift_ Chapter III Properties hereditary in a sequence   dffrege69 44780
                  ​*21.38.6.9  _Begriffsschrift_ Chapter III Following in a sequence   dffrege76 44787
                  ​*21.38.6.10  _Begriffsschrift_ Chapter III Member of sequence   dffrege99 44810
                  ​*21.38.6.11  _Begriffsschrift_ Chapter III Single-valued procedures   dffrege115 44826
            ​*21.38.7  Exploring Topology via Seifert and Threlfall   enrelmap 44845
                  *21.38.7.1  Equinumerosity of sets of relations and maps   enrelmap 44845
                  ​*21.38.7.2  Generic Pseudoclosure Spaces, Pseudointerior Spaces, and Pseudoneighborhoods   or3or 44871
                  ​*21.38.7.3  Generic Neighborhood Spaces   gneispa 44978
            ​*21.38.8  Exploring Higher Homotopy via Kerodon   k0004lem1 44995
                  *21.38.8.1  Simplicial Sets   k0004lem1 44995
      ​21.39  Mathbox for Stanislas Polu
            21.39.1  IMO Problems   wwlemuld 45004
                  21.39.1.1  IMO 1972 B2   wwlemuld 45004
            ​*21.39.2  INT Inequalities Proof Generator   int-addcomd 45021
            ​*21.39.3  N-Digit Addition Proof Generator   unitadd 45043
            21.39.4  AM-GM (for k = 2,3,4)   gsumws3 45044
      ​21.40  Mathbox for Rohan Ridenour
            21.40.1  Misc   spALT 45049
            21.40.2  Monoid rings   cmnring 45057
            21.40.3  Shorter primitive equivalent of ax-groth   gru0eld 45075
                  21.40.3.1  Grothendieck universes are closed under collection   gru0eld 45075
                  21.40.3.2  Minimal universes   ismnu 45093
                  21.40.3.3  Primitive equivalent of ax-groth   expandan 45120
      ​21.41  Mathbox for Steve Rodriguez
            21.41.1  Miscellanea   nanorxor 45137
            21.41.2  Ratio test for infinite series convergence and divergence   dvgrat 45144
            21.41.3  Multiples   reldvds 45147
            21.41.4  Function operations   caofcan 45155
            21.41.5  Calculus   lhe4.4ex1a 45161
            21.41.6  The generalized binomial coefficient operation   cbcc 45168
            21.41.7  Binomial series   uzmptshftfval 45178
      ​21.42  Mathbox for Andrew Salmon
            21.42.1  Principia Mathematica * 10   pm10.12 45190
            21.42.2  Principia Mathematica * 11   2alanimi 45204
            21.42.3  Predicate Calculus   sbeqal1 45230
            21.42.4  Principia Mathematica * 13 and * 14   pm13.13a 45239
            21.42.5  Set Theory   elnev 45269
            21.42.6  Arithmetic   addcomgi 45286
            21.42.7  Geometry   cplusr 45287
      ​​*21.43  Mathbox for Alan Sare
            21.43.1  Auxiliary theorems for the Virtual Deduction tool   idiALT 45309
            21.43.2  Supplementary unification deductions   bi1imp 45313
            21.43.3  Conventional Metamath proofs, some derived from VD proofs   iidn3 45332
            21.43.4  What is Virtual Deduction?   wvd1 45400
            21.43.5  Virtual Deduction Theorems   df-vd1 45401
            21.43.6  Theorems proved using Virtual Deduction   trsspwALT 45648
            21.43.7  Theorems proved using Virtual Deduction with mmj2 assistance   simplbi2VD 45676
            21.43.8  Virtual Deduction transcriptions of textbook proofs   sb5ALTVD 45743
            21.43.9  Theorems proved using conjunction-form Virtual Deduction   elpwgdedVD 45747
            21.43.10  Theorems with a VD proof in conventional notation derived from a VD proof   suctrALT3 45754
            ​*21.43.11  Theorems with a proof in conventional notation derived from a VD proof   notnotrALT2 45757
      ​21.44  Mathbox for Eric Schmidt
            21.44.1  Miscellany   rspesbcd 45768
            21.44.2  Study of dfbi1ALT   dfbi1ALTa 45770
            21.44.3  Relation-preserving functions   wrelp 45773
            21.44.4  Orbits   orbitex 45786
            21.44.5  Well-founded sets   trwf 45790
            21.44.6  Absoluteness in transitive models   ralabso 45799
            21.44.7  Lemmas for showing axioms hold in models   traxext 45808
            21.44.8  The class of well-founded sets is a model for ZFC   wfaxext 45824
            21.44.9  Permutation models   brpermmodel 45834
            21.44.10  Isomorphism of finite ordinals and non-negative integers   hashnna 45850
      ​21.45  Mathbox for Glauco Siliprandi
            21.45.1  Miscellanea   evth2f 45857
            21.45.2  Functions   fnresdmss 46008
            21.45.3  Ordering on real numbers - Real and complex numbers basic operations   sub2times 46114
            21.45.4  Real intervals   gtnelioc 46329
            21.45.5  Finite sums   fsummulc1f 46409
            21.45.6  Finite multiplication of numbers and finite multiplication of functions   fmul01 46418
            21.45.7  Limits   clim1fr1 46439
                  21.45.7.1  Inferior limit (lim inf)   clsi 46587
                  ​*21.45.7.2  Limits for sequences of extended real numbers   clsxlim 46654
            21.45.8  Trigonometry   coseq0 46700
            21.45.9  Continuous Functions   mulcncff 46706
            21.45.10  Derivatives   dvsinexp 46747
            21.45.11  Integrals   itgsin0pilem1 46786
            21.45.12  Stone Weierstrass theorem - real version   stoweidlem1 46837
            21.45.13  Wallis' product for π   wallispilem1 46901
            21.45.14  Stirling's approximation formula for ` n ` factorial   stirlinglem1 46910
            21.45.15  Dirichlet kernel   dirkerval 46927
            21.45.16  Fourier Series   fourierdlem1 46944
            21.45.17  e is transcendental   elaa2lem 47069
            21.45.18  n-dimensional Euclidean space   rrxtopn 47120
            21.45.19  Basic measure theory   csalg 47144
                  ​*21.45.19.1  σ-Algebras   csalg 47144
                  21.45.19.2  Sum of nonnegative extended reals   csumge0 47198
                  ​*21.45.19.3  Measures   cmea 47285
                  ​*21.45.19.4  Outer measures and Caratheodory's construction   come 47325
                  ​*21.45.19.5  Lebesgue measure on n-dimensional Real numbers   covoln 47372
                  ​*21.45.19.6  Measurable functions   csmblfn 47531
      ​21.46  Mathbox for Saveliy Skresanov
            21.46.1  Ceva's theorem   sigarval 47686
            21.46.2  Simple groups   simpcntrab 47706
      ​21.47  Mathbox for Ender Ting
            21.47.1  Interesting facts   et-ltneverrefl 47707
            21.47.2  Increasing sequences and subsequences   ormklocald 47712
            21.47.3  Scratchpad for number theory   evenwodadd 47737
            21.47.4  Scratchpad for math on real numbers   squeezedltsq 47738
            21.47.5  Turing Machine Finite Reach theorem   tmachlem-extapes 47770
      ​21.48  Mathbox for Jarvin Udandy
      ​21.49  Mathbox for Adhemar
            ​*21.49.1  Minimal implicational calculus   adh-minim 47897
      ​21.50  Mathbox for Alexander van der Vekens
            21.50.1  General auxiliary theorems (1)   n0nsn2el 47921
                  21.50.1.1  Unordered and ordered pairs - extension for singletons   n0nsn2el 47921
                  21.50.1.2  Unordered and ordered pairs - extension for unordered pairs   elprneb 47925
                  21.50.1.3  Unordered and ordered pairs - extension for ordered pairs   oppr 47926
                  21.50.1.4  Relations - extension   eubrv 47931
                  21.50.1.5  Definite description binder (inverted iota) - extension   iota0def 47934
                  21.50.1.6  Functions - extension   fveqvfvv 47936
            21.50.2  Alternative for Russell's definition of a description binder   caiota 47979
            21.50.3  Double restricted existential uniqueness   r19.32 47994
                  21.50.3.1  Restricted quantification (extension)   r19.32 47994
                  21.50.3.2  Restricted uniqueness and "at most one" quantification   reuf1odnf 48003
                  21.50.3.3  Analogs to Existential uniqueness (double quantification)   2reu3 48006
                  21.50.3.4  Additional theorems for double restricted existential uniqueness   2reu8i 48009
            ​*21.50.4  Alternative definitions of function and operation values   wdfat 48012
                  21.50.4.1  Restricted quantification (extension)   ralbinrald 48018
                  21.50.4.2  The universal class (extension)   nvelim 48019
                  21.50.4.3  Introduce the Axiom of Power Sets (extension)   alneu 48020
                  21.50.4.4  Predicate "defined at"   dfateq12d 48022
                  21.50.4.5  Alternative definition of the value of a function   dfafv2 48028
                  21.50.4.6  Alternative definition of the value of an operation   aoveq123d 48074
            ​*21.50.5  Alternative definitions of function values (2)   cafv2 48104
            21.50.6  General auxiliary theorems (2)   an4com24 48164
                  21.50.6.1  Logical conjunction - extension   an4com24 48164
                  21.50.6.2  Abbreviated conjunction and disjunction of three wff's - extension   3an4ancom24 48165
                  21.50.6.3  Negated membership (alternative)   cnelbr 48167
                  21.50.6.4  The empty set - extension   ralralimp 48174
                  21.50.6.5  Indexed union and intersection - extension   otiunsndisjX 48175
                  21.50.6.6  Functions - extension   fvifeq 48176
                  21.50.6.7  Maps-to notation - extension   fvmptrab 48188
                  21.50.6.8  Subtraction - extension   cnambpcma 48190
                  21.50.6.9  Ordering on reals (cont.) - extension   leaddsuble 48193
                  21.50.6.10  Imaginary and complex number properties - extension   readdcnnred 48199
                  21.50.6.11  Nonnegative integers (as a subset of complex numbers) - extension   nn0resubcl 48204
                  21.50.6.12  Integers (as a subset of complex numbers) - extension   zgeltp1eq 48205
                  21.50.6.13  Decimal arithmetic - extension   1t10e1p1e11 48206
                  21.50.6.14  Upper sets of integers - extension   eluzge0nn0 48208
                  21.50.6.15  Infinity and the extended real number system (cont.) - extension   nltle2tri 48209
                  21.50.6.16  Finite intervals of integers - extension   ssfz12 48210
                  21.50.6.17  Half-open integer ranges - extension   fzopred 48219
                  21.50.6.18  The floor and ceiling functions - extension   2ltceilhalf 48228
                  21.50.6.19  The modulo (remainder) operation - extension   fldivmod 48240
                  21.50.6.20  The infinite sequence builder "seq"   smonoord 48273
                  21.50.6.21  Integer powers - extension   2timesltsq 48274
                  21.50.6.22  Finite and infinite sums - extension   fsummsndifre 48276
                  21.50.6.23  The divides relation - extension   nndivides2 48280
                  21.50.6.24  Extensible structures - extension   setsidel 48284
            ​*21.50.7  Preimages of function values   preimafvsnel 48287
            ​*21.50.8  Partitions of real intervals   ciccp 48321
            21.50.9  Shifting functions with an integer range domain   fargshiftfv 48347
            21.50.10  Words over a set (extension)   lswn0 48352
                  21.50.10.1  Last symbol of a word - extension   lswn0 48352
            21.50.11  Unordered pairs   wich 48353
                  21.50.11.1  Interchangeable setvar variables   wich 48353
                  21.50.11.2  Set of unordered pairs   sprid 48382
                  ​*21.50.11.3  Proper (unordered) pairs   prpair 48409
                  21.50.11.4  Set of proper unordered pairs   cprpr 48420
            21.50.12  Number theory (extension)   nprmmul1 48435
                  21.50.12.1  Properties of non-prime numbers   nprmmul1 48435
                  ​*21.50.12.2  Fermat numbers   cfmtno 48438
                  ​*21.50.12.3  Mersenne primes   m2prm 48502
                  21.50.12.4  Proth's theorem   modexp2m1d 48523
                  21.50.12.5  The prime-counting function according to Ján Mináč   nprmdvdsfacm1lem1 48531
                  21.50.12.6  Solutions of quadratic equations   quad1 48544
            ​*21.50.13  Even and odd numbers   ceven 48548
                  21.50.13.1  Definitions and basic properties   ceven 48548
                  21.50.13.2  Alternate definitions using the "divides" relation   dfeven2 48573
                  21.50.13.3  Alternate definitions using the "modulo" operation   dfeven3 48582
                  21.50.13.4  Alternate definitions using the "gcd" operation   iseven5 48588
                  21.50.13.5  Theorems of part 5 revised   zneoALTV 48593
                  21.50.13.6  Theorems of part 6 revised   odd2np1ALTV 48598
                  21.50.13.7  Theorems of AV's mathbox revised   0evenALTV 48612
                  21.50.13.8  Additional theorems   epoo 48627
                  21.50.13.9  Perfect Number Theorem (revised)   perfectALTVlem1 48645
            21.50.14  Number theory (extension 2)   cfppr 48648
                  ​*21.50.14.1  Fermat pseudoprimes   cfppr 48648
                  ​*21.50.14.2  Goldbach's conjectures   cgbe 48669
            21.50.15  Graph theory (extension)   cclnbgr 48742
                  21.50.15.1  Closed neighborhood of a vertex   cclnbgr 48742
                  ​*21.50.15.2  Semiclosed and semiopen neighborhoods (experimental)   dfsclnbgr2 48770
                  21.50.15.3  Induced subgraphs   cisubgr 48784
                  ​*21.50.15.4  Isomorphisms of graphs   cgrisom 48798
                  ​*21.50.15.5  Triangles in graphs   cgrtri 48861
                  ​*21.50.15.6  Star graphs   cstgr 48875
                  ​*21.50.15.7  Local isomorphisms of graphs   cgrlim 48900
                  ​*21.50.15.8  Generalized Petersen graphs   cgpg 48964
                  21.50.15.9  Loop-free graphs - extension   1hegrlfgr 49056
                  21.50.15.10  Walks - extension   cupwlks 49057
                  21.50.15.11  Edges of graphs expressed as sets of unordered pairs   upgredgssspr 49067
            21.50.16  Monoids (extension)   ovn0dmfun 49080
                  21.50.16.1  Auxiliary theorems   ovn0dmfun 49080
                  21.50.16.2  Magmas, Semigroups and Monoids (extension)   plusfreseq 49087
                  21.50.16.3  Examples and counterexamples for magmas, semigroups and monoids (extension)   opmpoismgm 49090
                  21.50.16.4  Group sum operation (extension 1)   gsumsplit2f 49103
            ​*21.50.17  Magmas and internal binary operations (alternate approach)   ccllaw 49106
                  *21.50.17.1  Laws for internal binary operations   ccllaw 49106
                  ​*21.50.17.2  Internal binary operations   cintop 49119
                  21.50.17.3  Alternative definitions for magmas and semigroups   cmgm2 49138
            21.50.18  Rings (extension)   lmod0rng 49152
                  21.50.18.1  Nonzero rings (extension)   lmod0rng 49152
                  21.50.18.2  Ideals as non-unital rings   lidldomn1 49154
                  21.50.18.3  The non-unital ring of even integers   0even 49160
                  21.50.18.4  A constructed not unital ring   cznrnglem 49182
                  ​*21.50.18.5  The category of non-unital rings (alternate definition)   crngcALTV 49186
                  ​*21.50.18.6  The category of (unital) rings (alternate definition)   cringcALTV 49210
            ​*21.50.19  Prime rings (and integral domains)   cprmrng 49257
            21.50.20  Basic algebraic structures (extension)   eliunxp2 49272
                  21.50.20.1  Auxiliary theorems   eliunxp2 49272
                  21.50.20.2  The binomial coefficient operation (extension)   bcpascm1 49289
                  21.50.20.3  The ` ZZ `-module ` ZZ X. ZZ `   zlmodzxzlmod 49292
                  21.50.20.4  Group sum operation (extension 2)   mgpsumunsn 49299
                  21.50.20.5  Symmetric groups (extension)   exple2lt6 49302
                  21.50.20.6  Divisibility (extension)   invginvrid 49305
                  21.50.20.7  The support of functions (extension)   rmsupp0 49306
                  21.50.20.8  Finitely supported functions (extension)   rmsuppfi 49310
                  21.50.20.9  Left modules (extension)   lmodvsmdi 49317
                  21.50.20.10  Associative algebras (extension)   assaascl0 49319
                  21.50.20.11  Univariate polynomials (extension)   ply1vr1smo 49321
                  21.50.20.12  Univariate polynomials (examples)   linply1 49331
            21.50.21  Linear algebra (extension)   cdmatalt 49334
                  ​*21.50.21.1  The subalgebras of diagonal and scalar matrices (extension)   cdmatalt 49334
                  ​*21.50.21.2  Linear combinations   clinc 49342
                  ​*21.50.21.3  Linear independence   clininds 49378
                  21.50.21.4  Simple left modules and the ` ZZ `-module   lmod1lem1 49425
                  21.50.21.5  Differences between (left) modules and (left) vector spaces   lvecpsslmod 49445
            21.50.22  Complexity theory   suppdm 49448
                  21.50.22.1  Auxiliary theorems   suppdm 49448
                  21.50.22.2  Even and odd integers   nn0onn0ex 49461
                  21.50.22.3  The natural logarithm on complex numbers (extension)   logcxp0 49473
                  21.50.22.4  Division of functions   cfdiv 49475
                  21.50.22.5  Upper bounds   cbigo 49485
                  21.50.22.6  Logarithm to an arbitrary base (extension)   rege1logbrege0 49496
                  ​*21.50.22.7  The binary logarithm   fldivexpfllog2 49503
                  21.50.22.8  Binary length   cblen 49507
                  ​*21.50.22.9  Digits   cdig 49533
                  21.50.22.10  Nonnegative integer as sum of its shifted digits   dignn0flhalflem1 49553
                  21.50.22.11  Algorithms for the multiplication of nonnegative integers   nn0mulfsum 49562
                  ​*21.50.22.12  N-ary functions   cnaryf 49564
                  ​*21.50.22.13  The Ackermann function   citco 49595
            21.50.23  Elementary geometry (extension)   fv1prop 49637
                  21.50.23.1  Auxiliary theorems   fv1prop 49637
                  21.50.23.2  Real euclidean space of dimension 2   rrx2pxel 49649
                  21.50.23.3  Spheres and lines in real Euclidean spaces   cline 49665
      ​21.51  Mathbox for Zhi Wang
            21.51.1  Propositional calculus   logic1 49727
            21.51.2  Predicate calculus with equality   dtrucor3 49735
                  21.51.2.1  Axiom scheme ax-5 (Distinctness)   dtrucor3 49735
            21.51.3  ZF Set Theory - start with the Axiom of Extensionality   ralbidb 49736
                  21.51.3.1  Restricted quantification   ralbidb 49736
                  21.51.3.2  The universal class   reuxfr1dd 49743
                  21.51.3.3  The empty set   ssdisjd 49744
                  21.51.3.4  Unordered and ordered pairs   vsn 49748
                  21.51.3.5  The union of a class   unilbss 49754
                  21.51.3.6  Indexed union and intersection   iuneq0 49755
            21.51.4  ZF Set Theory - add the Axiom of Replacement   inpw 49761
                  21.51.4.1  Theorems requiring subset and intersection existence   inpw 49761
            21.51.5  ZF Set Theory - add the Axiom of Power Sets   opth1neg 49762
                  21.51.5.1  Ordered pair theorem   opth1neg 49762
                  21.51.5.2  Ordered-pair class abstractions (cont.)   brab2dd 49764
                  21.51.5.3  Relations   iinxp 49767
                  21.51.5.4  Functions   mof0 49774
                  21.51.5.5  Operations   ovsng 49794
            21.51.6  ZF Set Theory - add the Axiom of Union   fonex 49803
                  21.51.6.1  Relations and functions (cont.)   fonex 49803
                  21.51.6.2  First and second members of an ordered pair   eloprab1st2nd 49804
                  21.51.6.3  Function transposition   resinsnlem 49805
                  21.51.6.4  Infinite Cartesian products   ixpv 49824
                  21.51.6.5  Equinumerosity   fvconst0ci 49825
            21.51.7  Order sets   iccin 49830
                  21.51.7.1  Real number intervals   iccin 49830
            21.51.8  Extensible structures   slotresfo 49833
                  21.51.8.1  Basic definitions   slotresfo 49833
            21.51.9  Moore spaces   mreuniss 49834
            ​*21.51.10  Topology   clduni 49835
                  21.51.10.1  Closure and interior   clduni 49835
                  21.51.10.2  Neighborhoods   neircl 49839
                  21.51.10.3  Subspace topologies   restcls2lem 49847
                  21.51.10.4  Limits and continuity in topological spaces   cnneiima 49851
                  21.51.10.5  Topological definitions using the reals   iooii 49852
                  21.51.10.6  Separated sets   sepnsepolem1 49856
                  21.51.10.7  Separated spaces: T0, T1, T2 (Hausdorff) ...   isnrm4 49865
            21.51.11  Preordered sets and directed sets using extensible structures   isprsd 49889
            21.51.12  Posets and lattices using extensible structures   lubeldm2 49890
                  21.51.12.1  Posets   lubeldm2 49890
                  21.51.12.2  Lattices   toslat 49916
                  21.51.12.3  Subset order structures   intubeu 49918
            21.51.13  Rings   elmgpcntrd 49939
                  21.51.13.1  Multiplicative Group   elmgpcntrd 49939
            21.51.14  Associative algebras   asclelbasALT 49940
                  21.51.14.1  Definition and basic properties   asclelbasALT 49940
            21.51.15  Categories   homf0 49943
                  21.51.15.1  Categories   homf0 49943
                  21.51.15.2  Opposite category   oppccatb 49950
                  21.51.15.3  Monomorphisms and epimorphisms   idmon 49954
                  21.51.15.4  Sections, inverses, isomorphisms   sectrcl 49956
                  21.51.15.5  Isomorphic objects   cicfn 49976
                  21.51.15.6  Subcategories   dmdm 49987
                  21.51.15.7  Functors   reldmfunc 50009
                  21.51.15.8  Opposite functors   coppf 50056
                  21.51.15.9  Full & faithful functors   imasubc 50085
                  21.51.15.10  Universal property   upciclem1 50100
                  21.51.15.11  Natural transformations and the functor category   isnatd 50157
                  21.51.15.12  Initial, terminal and zero objects of a category   initoo2 50166
                  21.51.15.13  Product of categories   reldmxpc 50180
                  21.51.15.14  Swap functors   cswapf 50193
                  21.51.15.15  Functor evaluation   oppc1stflem 50221
                  21.51.15.16  Transposed curry functors   cofuswapfcl 50227
                  21.51.15.17  Constant functors   diag1 50238
                  21.51.15.18  Functor composition bifunctors   fucofulem1 50244
                  21.51.15.19  Post-composition functors   postcofval 50298
                  21.51.15.20  Pre-composition functors   precofvallem 50300
            21.51.16  Examples of categories   catcrcl 50329
                  21.51.16.1  The category of categories   catcrcl 50329
                  21.51.16.2  Thin categories   cthinc 50351
                  21.51.16.3  Terminal categories   ctermc 50406
                  21.51.16.4  Preordered sets as thin categories   cprstc 50483
                  21.51.16.5  Monoids as categories   cmndtc 50511
                  21.51.16.6  Categories with at most one object and at most two morphisms   2arwcatlem1 50529
            21.51.17  Kan extensions and related concepts   clan 50539
                  21.51.17.1  Kan extensions   clan 50539
                  21.51.17.2  Limits and colimits   clmd 50577
      ​21.52  Mathbox for Emmett Weisz
            ​*21.52.1  Miscellaneous Theorems   nfintd 50607
            21.52.2  Set Recursion   csetrecs 50617
                  ​*21.52.2.1  Basic Properties of Set Recursion   csetrecs 50617
                  21.52.2.2  Examples and properties of set recursion   elsetrecslem 50633
            ​*21.52.3  Construction of Games and Surreal Numbers   cpg 50643
      ​​*21.53  Mathbox for David A. Wheeler
            21.53.1  Natural deduction   sbidd 50652
            ​*21.53.2  Greater than, greater than or equal to   cge-real 50654
            ​*21.53.3  Hyperbolic trigonometric functions   csinh 50664
            ​*21.53.4  Reciprocal trigonometric functions (sec, csc, cot)   csec 50675
            ​*21.53.5  Identities for "if"   ifnmfalse 50700
            ​*21.53.6  Logarithms generalized to arbitrary base using ` logb `   logb2aval 50701
            ​*21.53.7  Logarithm laws generalized to an arbitrary base - log_   clog- 50702
            ​*21.53.8  Formally define notions such as reflexivity   wreflexive 50704
            ​*21.53.9  Algebra helpers   mvlraddi 50708
            ​*21.53.10  Algebra helper examples   i2linesi 50715
            ​*21.53.11  Formal methods "surprises"   alimp-surprise 50717
            ​*21.53.12  Allsome quantifier   wals 50723
            ​*21.53.13  Allsome one quantifier   walseu 50756
            ​*21.53.14  Miscellaneous   5m4e1 50776
            21.53.15  Theorems about algebraic numbers   aacllem 50780
      ​21.54  Mathbox for Mingli Yuan
      ​21.55  Mathbox for Jiamin Zhao
            21.55.1  Cross product and scalar triple product in RR^3   1ne3 50782
            21.55.2  Veronese map and linear dependence   cveronese 50809
      ​21.56  Mathbox for Kunhao Zheng
            21.56.1  Weighted AM-GM inequality   amgmwlem 50828