TOC of Theorem List - Metamath Proof Explorer

Table of Contents Summary

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

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

​​*PART 1  CLASSICAL FIRST-ORDER LOGIC WITH EQUALITY
      ​*1.1  Pre-logic
            *1.1.1  Inferences for assisting proof development   idi 1
      ​​*1.2  Propositional calculus
            1.2.1  Recursively define primitive wffs for propositional calculus   wn 3
            ​*1.2.2  The axioms of propositional calculus   ax-mp 5
            ​*1.2.3  Logical implication   mp2 9
            ​*1.2.4  Logical negation   con4 114
            ​*1.2.5  Logical equivalence   wb 209
            ​*1.2.6  Logical conjunction   wa 400
            ​*1.2.7  Logical disjunction   wo 860
            ​*1.2.8  Mixed connectives   jaao 969
            ​*1.2.9  The conditional operator for propositions   wif 1076
            ​*1.2.10  The weak deduction theorem for propositional calculus   elimh 1097
            1.2.11  Abbreviated conjunction and disjunction of three wff's   w3o 1100
            1.2.12  Logical "nand" (Sheffer stroke)   wnan 1514
            1.2.13  Logical "xor"   wxo 1534
            1.2.14  Logical "nor"   wnor 1551
            1.2.15  True and false constants   wal 1561
                  ​*1.2.15.1  Universal quantifier for use by df-tru   wal 1561
                  ​*1.2.15.2  Equality predicate for use by df-tru   cv 1562
                  1.2.15.3  The true constant   wtru 1564
                  1.2.15.4  The false constant   wfal 1575
            ​*1.2.16  Truth tables   truimtru 1586
                  1.2.16.1  Implication   truimtru 1586
                  1.2.16.2  Negation   nottru 1590
                  1.2.16.3  Equivalence   trubitru 1592
                  1.2.16.4  Conjunction   truantru 1596
                  1.2.16.5  Disjunction   truortru 1600
                  1.2.16.6  Alternative denial   trunantru 1604
                  1.2.16.7  Exclusive disjunction   truxortru 1608
                  1.2.16.8  Joint denial   trunortru 1612
            ​*1.2.17  Half adder and full adder in propositional calculus   whad 1616
                  1.2.17.1  Full adder: sum   whad 1616
                  1.2.17.2  Full adder: carry   wcad 1629
      ​1.3  Other axiomatizations related to classical propositional calculus
            ​*1.3.1  Minimal implicational calculus   minimp 1644
            ​*1.3.2  Implicational Calculus   impsingle 1650
            1.3.3  Derive the Lukasiewicz axioms from Meredith's sole axiom   meredith 1664
            1.3.4  Derive the standard axioms from the Lukasiewicz axioms   luklem1 1681
            ​*1.3.5  Derive Nicod's axiom from the standard axioms   nic-dfim 1692
            1.3.6  Derive the Lukasiewicz axioms from Nicod's axiom   nic-imp 1698
            1.3.7  Derive Nicod's Axiom from Lukasiewicz's First Sheffer Stroke Axiom   lukshef-ax1 1717
            1.3.8  Derive the Lukasiewicz Axioms from the Tarski-Bernays-Wajsberg Axioms   tbw-bijust 1721
            1.3.9  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's First CO Axiom   merco1 1736
            1.3.10  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's Second CO Axiom   merco2 1759
            1.3.11  Derive the Lukasiewicz axioms from the Russell-Bernays Axioms   rb-bijust 1772
            ​*1.3.12  Stoic logic non-modal portion (Chrysippus of Soli)   mptnan 1791
      ​​*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 1802
                  1.4.1.1  Existential quantifier   wex 1802
                  1.4.1.2  Nonfreeness predicate   wnf 1806
            1.4.2  Rule scheme ax-gen (Generalization)   ax-gen 1818
            1.4.3  Axiom scheme ax-4 (Quantified Implication)   ax-4 1832
                  ​*1.4.3.1  The empty domain of discourse   empty 1929
            1.4.4  Axiom scheme ax-5 (Distinctness) - first use of $d   ax-5 1933
            ​*1.4.5  Equality predicate (continued)   weq 1985
            1.4.6  Axiom scheme ax-6 (Existence)   ax-6 1990
            1.4.7  Axiom scheme ax-7 (Equality)   ax-7 2031
            1.4.8  Define proper substitution   justify-df 2088
            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 2406
      ​1.6  Uniqueness and unique existence
            1.6.1  Uniqueness: the at-most-one quantifier   wmo 2567
            1.6.2  Unique existence: the unique existential quantifier   weu 2598
      ​1.7  Other axiomatizations related to classical predicate calculus
            ​*1.7.1  Aristotelian logic: Assertic syllogisms   barbara 2692
            ​*1.7.2  Intuitionistic logic   axia1 2722
​​*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 2737
            2.1.2  Classes   cab 2743
                  2.1.2.1  Class abstractions   cab 2743
                  ​*2.1.2.2  Class equality   df-cleq 2757
                  2.1.2.3  Class membership   df-clel 2840
                  2.1.2.4  Elementary properties of class abstractions   eqabdv 2898
            2.1.3  Class form not-free predicate   wnfc 2912
            2.1.4  Negated equality and membership   wne 2960
                  2.1.4.1  Negated equality   wne 2960
                  2.1.4.2  Negated membership   wnel 3064
            2.1.5  Restricted quantification   wral 3079
                  2.1.5.1  Restricted universal and existential quantification   wral 3079
                  2.1.5.2  Restricted existential uniqueness and at-most-one quantifier   wreu 3368
                  2.1.5.3  Restricted class abstraction   crab 3417
            2.1.6  The universal class   cvv 3457
            ​*2.1.7  Conditional equality (experimental)   wcdeq 3729
            2.1.8  Russell's Paradox   rru 3745
            2.1.9  Proper substitution of classes for sets   wsbc 3747
            2.1.10  Proper substitution of classes for sets into classes   csb 3855
            2.1.11  Define basic set operations and relations   cdif 3904
            2.1.12  Subclasses and subsets   df-ss 3924
            2.1.13  The difference, union, and intersection of two classes   dfdif3 4074
                  2.1.13.1  The difference of two classes   dfdif3 4074
                  2.1.13.2  The union of two classes   elun 4109
                  2.1.13.3  The intersection of two classes   elini 4154
                  2.1.13.4  The symmetric difference of two classes   csymdif 4207
                  2.1.13.5  Combinations of difference, union, and intersection of two classes   unabs 4220
                  2.1.13.6  Class abstractions with difference, union, and intersection of two classes   unabw 4262
                  2.1.13.7  Restricted uniqueness with difference, union, and intersection   reuun2 4280
            2.1.14  The empty set   c0 4288
            ​*2.1.15  The conditional operator for classes   cif 4483
            ​*2.1.16  The weak deduction theorem for set theory   dedth 4542
            2.1.17  Power classes   cpw 4558
            2.1.18  Unordered and ordered pairs   snjust 4584
            2.1.19  The union of a class   cuni 4867
            2.1.20  The intersection of a class   cint 4907
            2.1.21  Indexed union and intersection   ciun 4951
            2.1.22  Disjointness   wdisj 5071
            2.1.23  Binary relations   wbr 5104
            2.1.24  Ordered-pair class abstractions (class builders)   copab 5166
            2.1.25  Functions in maps-to notation   cmpt 5185
            2.1.26  Transitive classes   wtr 5211
      ​2.2  ZF Set Theory - add the Axiom of Replacement
            2.2.1  Introduce the Axiom of Replacement   ax-rep 5231
            2.2.2  Derive the Axiom of Separation   axsepgfromrep 5248
            2.2.3  Derive the Null Set Axiom   axnulALT 5258
            2.2.4  Theorems requiring subset and intersection existence   exnelv 5267
            2.2.5  Theorems requiring empty set existence   class2set 5315
      ​2.3  ZF Set Theory - add the Axiom of Power Sets
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 5326
            2.3.2  Derive the Axiom of Pairing   axprlem1 5384
            2.3.3  Ordered pair theorem   opnz 5445
            2.3.4  Ordered-pair class abstractions (cont.)   opabidw 5498
            2.3.5  Power class of union and intersection   pwin 5542
            2.3.6  The identity relation   cid 5545
            2.3.7  The membership relation (or epsilon relation)   cep 5550
            ​*2.3.8  Partial and total orderings   wpo 5557
            2.3.9  Founded and well-ordering relations   wfr 5601
            2.3.10  Relations   cxp 5649
            2.3.11  The Predecessor Class   cpred 6290
            2.3.12  Well-founded induction (variant)   frpomin 6330
            2.3.13  Well-ordered induction   tz6.26 6337
            2.3.14  Ordinals   word 6348
            2.3.15  Definite description binder (inverted iota)   cio 6479
            2.3.16  Functions   wfun 6519
            2.3.17  Cantor's Theorem   canth 7354
            2.3.18  Restricted iota (description binder)   crio 7356
            2.3.19  Operations   co 7400
                  2.3.19.1  Variable-to-class conversion for operations   caovclg 7592
            2.3.20  Maps-to notation   mpondm0 7640
            2.3.21  Function operation   cof 7662
            2.3.22  Proper subset relation   crpss 7709
      ​2.4  ZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 7722
            2.4.2  Ordinals (continued)   epweon 7762
            2.4.3  Transfinite induction   tfi 7837
            2.4.4  The natural numbers (i.e., finite ordinals)   com 7850
            2.4.5  Peano's postulates   peano1 7873
            2.4.6  Finite induction (for finite ordinals)   find 7880
            2.4.7  Relations and functions (cont.)   dmexg 7886
            2.4.8  First and second members of an ordered pair   c1st 7972
            2.4.9  Induction on Cartesian products   frpoins3xpg 8124
            2.4.10  Ordering on Cartesian products   xpord2lem 8126
            2.4.11  Ordering Ordinal Sequences   orderseqlem 8141
            ​*2.4.12  The support of functions   csupp 8144
            ​*2.4.13  Special maps-to operations   opeliunxp2f 8194
            2.4.14  Function transposition   ctpos 8209
            2.4.15  Curry and uncurry   ccur 8249
            2.4.16  Undefined values   cund 8256
            2.4.17  Well-founded recursion   cfrecs 8265
            2.4.18  Well-ordered recursion   cwrecs 8296
            2.4.19  Functions on ordinals; strictly monotone ordinal functions   iunon 8314
            2.4.20  "Strong" transfinite recursion   crecs 8345
            2.4.21  Recursive definition generator   crdg 8384
            2.4.22  Finite recursion   frfnom 8410
            2.4.23  Ordinal arithmetic   c1o 8434
            2.4.24  Natural number arithmetic   nna0 8578
            2.4.25  Natural addition   cnadd 8639
            2.4.26  Equivalence relations and classes   wer 8679
            2.4.27  The mapping operation   cmap 8812
            2.4.28  Infinite Cartesian products   cixp 8883
            2.4.29  Equinumerosity   cen 8928
            2.4.30  Schroeder-Bernstein Theorem   sbthlem1 9063
            2.4.31  Equinumerosity (cont.)   xpf1o 9115
            2.4.32  Finite sets   dif1enlem 9132
            2.4.33  Pigeonhole Principle   phplem1 9176
            2.4.34  Finite sets (cont.)   onomeneq 9186
            2.4.35  Finitely supported functions   cfsupp 9309
            2.4.36  Finite intersections   cfi 9358
            2.4.37  Hall's marriage theorem   marypha1lem 9381
            2.4.38  Supremum and infimum   csup 9388
            2.4.39  Ordinal isomorphism, Hartogs's theorem   coi 9459
            2.4.40  Hartogs function   char 9506
            2.4.41  Weak dominance   cwdom 9514
      ​2.5  ZF Set Theory - add the Axiom of Regularity
            2.5.1  Introduce the Axiom of Regularity   ax-reg 9542
            2.5.2  Axiom of Infinity equivalents   inf0 9578
      ​2.6  ZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-inf 9595
            2.6.2  Existence of omega (the set of natural numbers)   omex 9600
            2.6.3  Cantor normal form   ccnf 9618
            2.6.4  Transitive closure of a relation   cttrcl 9664
            2.6.5  Transitive closure   trcl 9685
            2.6.6  Set induction (or epsilon induction)   setind 9704
            2.6.7  Well-Founded Induction   frmin 9709
            2.6.8  Well-Founded Recursion   frr3g 9716
            2.6.9  Rank   cr1 9722
            2.6.10  Scott's trick; collection principle; Hilbert's epsilon   cscott 9845
            2.6.11  Disjoint union   cdju 9872
            2.6.12  Cardinal numbers   ccrd 9909
            2.6.13  Axiom of Choice equivalents   wac 10087
            ​*2.6.14  Cardinal number arithmetic   undjudom 10139
            2.6.15  The Ackermann bijection   ackbij2lem1 10189
            2.6.16  Cofinality (without Axiom of Choice)   cflem 10216
            2.6.17  Eight inequivalent definitions of finite set   sornom 10249
            2.6.18  Hereditarily size-limited sets without Choice   itunifval 10388
​​*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 10407
            3.1.2  Introduce the Axiom of Dependent Choice   ax-dc 10418
      ​3.2  ZFC Set Theory - add the Axiom of Choice
            3.2.1  Introduce the Axiom of Choice   ax-ac 10431
            3.2.2  AC equivalents: well-ordering, Zorn's lemma   numthcor 10466
            3.2.3  Cardinal number theorems using Axiom of Choice   cardval 10518
            3.2.4  Cardinal number arithmetic using Axiom of Choice   iunctb 10547
            3.2.5  Cofinality using the Axiom of Choice   alephreg 10555
      ​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 10593
            3.4.2  Derivation of the Axiom of Choice   gchaclem 10651
​​*PART 4  TG (TARSKI-GROTHENDIECK) SET THEORY
      ​4.1  Inaccessibles
            4.1.1  Weakly and strongly inaccessible cardinals   cwina 10655
            4.1.2  Weak universes   cwun 10673
            4.1.3  Tarski classes   ctsk 10721
            4.1.4  Grothendieck universes   cgru 10763
      ​4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
            4.2.1  Introduce the Tarski-Grothendieck Axiom   ax-groth 10796
            4.2.2  Derive the Power Set, Infinity and Choice Axioms   grothpw 10799
            4.2.3  Tarski map function   ctskm 10810
​​*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 10817
            5.1.2  Final derivation of real and complex number postulates   axaddf 11118
            5.1.3  Real and complex number postulates restated as axioms   ax-cnex 11144
      ​5.2  Derive the basic properties from the field axioms
            5.2.1  Some deductions from the field axioms for complex numbers   cnex 11169
            5.2.2  Infinity and the extended real number system   cpnf 11228
            5.2.3  Restate the ordering postulates with extended real "less than"   axlttri 11269
            5.2.4  Ordering on reals   lttr 11274
            5.2.5  Initial properties of the complex numbers   mul12 11363
      ​5.3  Real and complex numbers - basic operations
            5.3.1  Addition   add12 11416
            5.3.2  Subtraction   cmin 11429
            5.3.3  Multiplication   kcnktkm1cn 11633
            5.3.4  Ordering on reals (cont.)   gt0ne0 11667
            5.3.5  Reciprocals   ixi 11831
            5.3.6  Division   cdiv 11859
            5.3.7  Ordering on reals (cont.)   elimgt0 12041
            5.3.8  Completeness Axiom and Suprema   fimaxre 12147
            5.3.9  Imaginary and complex number properties   neg1cn 12191
            5.3.10  Function operation analogue theorems   ofsubeq0 12203
            ​*5.3.11  Indicator Functions   cind 12206
      ​5.4  Integer sets
            5.4.1  Positive integers (as a subset of complex numbers)   cn 12221
            5.4.2  Principle of mathematical induction   nnind 12239
            ​*5.4.3  Decimal representation of numbers   c2 12283
            ​*5.4.4  Some properties of specific numbers   1pneg1e0 12346
            5.4.5  Simple number properties   halfcl 12458
            5.4.6  The Archimedean property   nnunb 12488
            5.4.7  Nonnegative integers (as a subset of complex numbers)   cn0 12492
            ​*5.4.8  Extended nonnegative integers   cxnn0 12565
            5.4.9  Integers (as a subset of complex numbers)   cz 12579
            5.4.10  Decimal arithmetic   cdc 12699
            5.4.11  Upper sets of integers   cuz 12850
            5.4.12  Well-ordering principle for bounded-below sets of integers   uzwo3 12955
            5.4.13  Rational numbers (as a subset of complex numbers)   cq 12960
            5.4.14  Existence of the set of complex numbers   rpnnen1lem2 12989
      ​5.5  Order sets
            5.5.1  Positive reals (as a subset of complex numbers)   crp 13004
            5.5.2  Infinity and the extended real number system (cont.)   cxne 13122
            5.5.3  Supremum and infimum on the extended reals   xrsupexmnf 13319
            5.5.4  Real number intervals   cioo 13360
            5.5.5  Finite intervals of integers   cfz 13523
            ​*5.5.6  Finite intervals of nonnegative integers   elfz2nn0 13634
            5.5.7  Half-open integer ranges   cfzo 13670
      ​5.6  Elementary integer functions
            5.6.1  The floor and ceiling functions   cfl 13811
            5.6.2  The modulo (remainder) operation   cmo 13890
            5.6.3  Miscellaneous theorems about integers   om2uz0i 13971
            5.6.4  Strong induction over upper sets of integers   uzsinds 14011
            5.6.5  Finitely supported functions over the nonnegative integers   fsuppmapnn0fiublem 14014
            5.6.6  The infinite sequence builder "seq" - extension   cseq 14025
            5.6.7  Integer powers   cexp 14085
            5.6.8  Ordered pair theorem for nonnegative integers   nn0le2msqi 14291
            5.6.9  Factorial function   cfa 14297
            5.6.10  The binomial coefficient operation   cbc 14326
            5.6.11  The ` # ` (set size) function   chash 14354
                  5.6.11.1  Proper unordered pairs and triples (sets of size 2 and 3)   hashprlei 14493
                  5.6.11.2  Functions with a domain containing at least two different elements   fundmge2nop0 14527
                  5.6.11.3  Finite induction on the size of the first component of a binary relation   hashdifsnp1 14531
      ​​*5.7  Words over a set
            5.7.1  Definitions and basic theorems   cword 14538
            5.7.2  Last symbol of a word   clsw 14587
            5.7.3  Concatenations of words   cconcat 14595
            5.7.4  Singleton words   cs1 14621
            5.7.5  Concatenations with singleton words   ccatws1cl 14642
            5.7.6  Subwords/substrings   csubstr 14666
            5.7.7  Prefixes of a word   cpfx 14696
            5.7.8  Subwords of subwords   swrdswrdlem 14729
            5.7.9  Subwords and concatenations   pfxcctswrd 14735
            5.7.10  Subwords of concatenations   swrdccatfn 14749
            5.7.11  Splicing words (substring replacement)   csplice 14774
            5.7.12  Reversing words   creverse 14783
            5.7.13  Repeated symbol words   creps 14793
            ​*5.7.14  Cyclical shifts of words   ccsh 14813
            5.7.15  Mapping words by a function   wrdco 14856
            5.7.16  Longer string literals   cs2 14866
      ​​*5.8  Reflexive and transitive closures of relations
            5.8.1  The reflexive and transitive properties of relations   coss12d 14997
            5.8.2  Basic properties of closures   cleq1lem 15007
            5.8.3  Definitions and basic properties of transitive closures   ctcl 15010
            5.8.4  Exponentiation of relations   crelexp 15044
            5.8.5  Reflexive-transitive closure as an indexed union   crtrcl 15080
            ​*5.8.6  Principle of transitive induction   relexpindlem 15088
      ​5.9  Elementary real and complex functions
            5.9.1  The "shift" operation   cshi 15091
            5.9.2  Signum (sgn or sign) function   csgn 15111
            5.9.3  Real and imaginary parts; conjugate   ccj 15135
            5.9.4  Square root; absolute value   csqrt 15272
      ​5.10  Elementary limits and convergence
            5.10.1  Superior limit (lim sup)   clsp 15509
            5.10.2  Limits   cli 15523
            5.10.3  Finite and infinite sums   csu 15725
            5.10.4  The binomial theorem   binomlem 15871
            5.10.5  The inclusion/exclusion principle   incexclem 15878
            5.10.6  Infinite sums (cont.)   isumshft 15881
            5.10.7  Miscellaneous converging and diverging sequences   divrcnv 15894
            5.10.8  Arithmetic series   arisum 15902
            5.10.9  Geometric series   expcnv 15906
            5.10.10  Ratio test for infinite series convergence   cvgrat 15925
            5.10.11  Mertens' theorem   mertenslem1 15926
            5.10.12  Finite and infinite products   prodf 15929
                  5.10.12.1  Product sequences   prodf 15929
                  5.10.12.2  Non-trivial convergence   ntrivcvg 15939
                  5.10.12.3  Complex products   cprod 15945
                  5.10.12.4  Finite products   fprod 15983
                  5.10.12.5  Infinite products   iprodclim 16040
            5.10.13  Falling and Rising Factorial   cfallfac 16046
            5.10.14  Bernoulli polynomials and sums of k-th powers   cbp 16088
      ​5.11  Elementary trigonometry
            5.11.1  The exponential, sine, and cosine functions   ce 16103
                  5.11.1.1  The circle constant (tau = 2 pi)   ctau 16246
            5.11.2  _e is irrational   eirrlem 16248
      ​5.12  Cardinality of real and complex number subsets
            5.12.1  Countability of integers and rationals   xpnnen 16255
            5.12.2  The reals are uncountable   rpnnen2lem1 16258
​​*PART 6  ELEMENTARY NUMBER THEORY
      ​6.1  Elementary properties of divisibility
            6.1.1  Irrationality of square root of 2   sqrt2irrlem 16292
            6.1.2  Some Number sets are chains of proper subsets   nthruc 16296
            6.1.3  The divides relation   cdvds 16298
            ​*6.1.4  Even and odd numbers   evenelz 16382
            6.1.5  The division algorithm   divalglem0 16439
            6.1.6  Bit sequences   cbits 16465
            6.1.7  The greatest common divisor operator   cgcd 16540
            6.1.8  Bézout's identity   bezoutlem1 16585
            6.1.9  Algorithms   nn0seqcvgd 16616
            6.1.10  Euclid's Algorithm   eucalgval2 16627
            ​*6.1.11  The least common multiple   clcm 16634
            ​*6.1.12  Coprimality and Euclid's lemma   coprmgcdb 16695
            6.1.13  Cancellability of congruences   congr 16710
      ​6.2  Elementary prime number theory
            ​*6.2.1  Elementary properties   cprime 16717
            ​*6.2.2  Coprimality and Euclid's lemma (cont.)   coprm 16758
            6.2.3  Properties of the canonical representation of a rational   cnumer 16780
            6.2.4  Euler's theorem   codz 16810
            6.2.5  Arithmetic modulo a prime number   modprm1div 16845
            6.2.6  Pythagorean Triples   coprimeprodsq 16856
            6.2.7  The prime count function   cpc 16884
            6.2.8  Pocklington's theorem   prmpwdvds 16952
            6.2.9  Infinite primes theorem   unbenlem 16956
            6.2.10  Sum of prime reciprocals   prmreclem1 16964
            6.2.11  Fundamental theorem of arithmetic   1arithlem1 16971
            6.2.12  Lagrange's four-square theorem   cgz 16977
            6.2.13  Van der Waerden's theorem   cvdwa 17013
            6.2.14  Ramsey's theorem   cram 17047
            ​*6.2.15  Primorial function   cprmo 17079
            ​*6.2.16  Prime gaps   prmgaplem1 17097
            6.2.17  Decimal arithmetic (cont.)   dec2dvds 17111
            6.2.18  Cyclical shifts of words (cont.)   cshwsidrepsw 17141
            6.2.19  Specific prime numbers   prmlem0 17153
            6.2.20  Very large primes   1259lem1 17179
​PART 7  BASIC STRUCTURES
      ​7.1  Extensible structures
            ​*7.1.1  Basic definitions   cstr 17194
                  7.1.1.1  Extensible structures as structures with components   cstr 17194
                  7.1.1.2  Substitution of components   csts 17211
                  7.1.1.3  Slots   cslot 17229
                  ​*7.1.1.4  Structure component indices   cnx 17241
                  7.1.1.5  Base sets   cbs 17257
                  7.1.1.6  Base set restrictions   cress 17278
            7.1.2  Slot definitions   cplusg 17298
            7.1.3  Definition of the structure product   crest 17461
            7.1.4  Definition of the structure quotient   cordt 17541
      ​7.2  Moore spaces
            7.2.1  Moore closures   mrcflem 17650
            7.2.2  Independent sets in a Moore system   mrisval 17674
            7.2.3  Algebraic closure systems   isacs 17695
​PART 8  BASIC CATEGORY THEORY
      ​8.1  Categories
            8.1.1  Categories   ccat 17708
            8.1.2  Opposite category   coppc 17755
            8.1.3  Monomorphisms and epimorphisms   cmon 17773
            8.1.4  Sections, inverses, isomorphisms   csect 17789
            ​*8.1.5  Isomorphic objects   ccic 17840
            8.1.6  Subcategories   cssc 17852
            8.1.7  Functors   cfunc 17899
            8.1.8  Full & faithful functors   cful 17949
            8.1.9  Natural transformations and the functor category   cnat 17989
            8.1.10  Initial, terminal and zero objects of a category   cinito 18026
      ​8.2  Arrows (disjointified hom-sets)
            8.2.1  Identity and composition for arrows   cida 18098
      ​8.3  Examples of categories
            8.3.1  The category of sets   csetc 18120
            8.3.2  The category of categories   ccatc 18143
            ​*8.3.3  The category of extensible structures   fncnvimaeqv 18164
      ​8.4  Categorical constructions
            8.4.1  Product of categories   cxpc 18212
            8.4.2  Functor evaluation   cevlf 18253
            8.4.3  Hom functor   chof 18292
​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 18475
            9.5.2  Complete lattices   ccla 18542
            9.5.3  Distributive lattices   cdlat 18564
            9.5.4  Subset order structures   cipo 18571
      ​9.6  Posets, directed sets, and lattices as relations
            ​*9.6.1  Posets and lattices as relations   cps 18608
            9.6.2  Directed sets, nets   cdir 18638
      ​9.7  Chains
​PART 10  BASIC ALGEBRAIC STRUCTURES
      ​10.1  Monoids
            ​*10.1.1  Magmas   cplusf 18683
            ​*10.1.2  Identity elements   mgmidmo 18706
            ​*10.1.3  Iterated sums in a magma   gsumvalx 18722
            10.1.4  Magma homomorphisms and submagmas   cmgmhm 18736
            ​*10.1.5  Semigroups   csgrp 18764
            ​*10.1.6  Definition and basic properties of monoids   cmnd 18780
            10.1.7  Monoid homomorphisms and submonoids   cmhm 18827
            ​*10.1.8  Iterated sums in a monoid   gsumvallem2 18881
            10.1.9  Free monoids   cfrmd 18894
                  ​*10.1.9.1  Monoid of endofunctions   cefmnd 18915
            10.1.10  Examples and counterexamples for magmas, semigroups and monoids   mgm2nsgrplem1 18968
      ​10.2  Groups
            10.2.1  Definition and basic properties   cgrp 18988
            ​*10.2.2  Group multiple operation   cmg 19121
            10.2.3  Subgroups and Quotient groups   csubg 19174
            ​*10.2.4  Cyclic monoids and groups   cycsubmel 19259
            10.2.5  Elementary theory of group homomorphisms   cghm 19271
            10.2.6  Isomorphisms of groups   cgim 19315
                  10.2.6.1  The first isomorphism theorem of groups   ghmqusnsglem1 19338
            10.2.7  Group actions   cga 19347
            10.2.8  Centralizers and centers   ccntz 19373
            10.2.9  The opposite group   coppg 19403
            10.2.10  Symmetric groups   csymg 19427
                  ​*10.2.10.1  Definition and basic properties   csymg 19427
                  10.2.10.2  Cayley's theorem   cayleylem1 19470
                  10.2.10.3  Permutations fixing one element   symgfix2 19474
                  ​*10.2.10.4  Transpositions in the symmetric group   cpmtr 19499
                  10.2.10.5  The sign of a permutation   cpsgn 19547
            10.2.11  p-Groups and Sylow groups; Sylow's theorems   cod 19582
            10.2.12  Direct products   clsm 19692
                  10.2.12.1  Direct products (extension)   smndlsmidm 19714
            10.2.13  Free groups   cefg 19764
            10.2.14  Abelian groups   ccmn 19838
                  10.2.14.1  Definition and basic properties   ccmn 19838
                  10.2.14.2  Cyclic groups   ccyg 19935
                  10.2.14.3  Group sum operation   gsumval3a 19961
                  10.2.14.4  Group sums over (ranges of) integers   fsfnn0gsumfsffz 20041
                  10.2.14.5  Internal direct products   cdprd 20053
                  10.2.14.6  The Fundamental Theorem of Abelian Groups   ablfacrplem 20125
            10.2.15  Simple groups   csimpg 20150
                  10.2.15.1  Definition and basic properties   csimpg 20150
                  10.2.15.2  Classification of abelian simple groups   ablsimpnosubgd 20164
            10.2.16  Totally ordered monoids and groups   comnd 20177
      ​10.3  Rings
            10.3.1  Multiplicative Group   cmgp 20204
            ​*10.3.2  Non-unital rings ("rngs")   crng 20218
            ​*10.3.3  Ring unity (multiplicative identity)   cur 20251
            10.3.4  Semirings   csrg 20256
                  ​*10.3.4.1  The binomial theorem for semirings   srgbinomlem1 20296
            10.3.5  Unital rings   crg 20303
            10.3.6  Opposite ring   coppr 20406
            10.3.7  Divisibility   cdsr 20424
            10.3.8  Ring primes   crpm 20502
            10.3.9  Homomorphisms of non-unital rings   crnghm 20504
            10.3.10  Ring homomorphisms   crh 20539
            10.3.11  Nonzero rings and zero rings   cnzr 20583
            10.3.12  Local rings   clring 20611
            10.3.13  Subrings   csubrng 20618
                  10.3.13.1  Subrings of non-unital rings   csubrng 20618
                  10.3.13.2  Subrings of unital rings   csubrg 20642
                  10.3.13.3  Subrings generated by a subset   crgspn 20683
            10.3.14  Categories of rings   crngc 20689
                  ​*10.3.14.1  The category of non-unital rings   crngc 20689
                  ​*10.3.14.2  The category of (unital) rings   cringc 20718
                  10.3.14.3  Subcategories of the category of rings   srhmsubclem1 20750
            10.3.15  Left regular elements and domains   crlreg 20764
      ​10.4  Division rings and fields
            10.4.1  Definition and basic properties   cdr 20801
            10.4.2  Sub-division rings   csdrg 20855
            10.4.3  Absolute value (abstract algebra)   cabv 20877
            10.4.4  Star rings   cstf 20906
            10.4.5  Totally ordered rings and fields   corng 20926
      ​10.5  Left modules
            10.5.1  Definition and basic properties   clmod 20947
            10.5.2  Subspaces and spans in a left module   clss 21018
            10.5.3  Homomorphisms and isomorphisms of left modules   clmhm 21106
            10.5.4  Subspace sum; bases for a left module   clbs 21161
      ​10.6  Vector spaces
            10.6.1  Definition and basic properties   clvec 21189
      ​10.7  Subring algebras and ideals
            10.7.1  Subring algebras   csra 21258
            ​*10.7.2  Left ideals and spans   clidl 21296
            10.7.3  Two-sided ideals and quotient rings   c2idl 21347
                  ​*10.7.3.1  Condition for a non-unital ring to be unital   rngqiprng1elbas 21385
                  10.7.3.2  Prime Ideals   cprmidl 21419
            10.7.4  Principal ideal rings. Divisibility in the integers   clpidl 21445
            10.7.5  Principal ideal domains   cpid 21461
      ​10.8  The complex numbers as an algebraic extensible structure
            10.8.1  Definition and basic properties   cpsmet 21463
            ​*10.8.2  Ring of integers   czring 21553
                  ​*10.8.2.1  Example for a condition for a non-unital ring to be unital   pzriprnglem1 21588
            10.8.3  Algebraic constructions based on the complex numbers   czrh 21606
            10.8.4  Signs as subgroup of the complex numbers   cnmsgnsubg 21684
            10.8.5  Embedding of permutation signs into a ring   zrhpsgnmhm 21691
            10.8.6  The ordered field of real numbers   crefld 21711
      ​10.9  Generalized pre-Hilbert and Hilbert spaces
            10.9.1  Definition and basic properties   cphl 21731
            10.9.2  Orthocomplements and closed subspaces   cocv 21767
            10.9.3  Orthogonal projection and orthonormal bases   cpj 21807
​​*PART 11  BASIC LINEAR ALGEBRA
      ​11.1  Vectors and free modules
            *11.1.1  Direct sum of left modules   cdsmm 21838
            ​*11.1.2  Free modules   cfrlm 21853
            ​*11.1.3  Standard basis (unit vectors)   cuvc 21889
            ​*11.1.4  Independent sets and families   clindf 21911
            11.1.5  Characterization of free modules   lmimlbs 21943
      ​11.2  Associative algebras
            11.2.1  Definition and basic properties   casa 21957
      ​11.3  Abstract multivariate polynomials
            11.3.1  Definition and basic properties   cmps 22011
            11.3.2  Polynomial evaluation   ces 22180
            11.3.3  The "variable selection" function   cslv 22224
            11.3.4  Additional definitions for (multivariate) polynomials   cmhp 22253
            ​*11.3.5  Univariate polynomials   cps1 22292
            11.3.6  Univariate polynomial evaluation   ces1 22430
                  11.3.6.1  Specialization of polynomial evaluation as a ring homomorphism   evls1scafv 22483
      ​​*11.4  Matrices
            *11.4.1  The matrix multiplication   cmmul 22504
            ​*11.4.2  Square matrices   cmat 22521
            ​*11.4.3  The matrix algebra   matmulr 22552
            ​*11.4.4  Matrices of dimension 0 and 1   mat0dimbas0 22580
            ​*11.4.5  The subalgebras of diagonal and scalar matrices   cdmat 22602
            ​*11.4.6  Multiplication of a matrix with a "column vector"   cmvmul 22654
            11.4.7  Replacement functions for a square matrix   cmarrep 22670
            11.4.8  Submatrices   csubma 22690
      ​11.5  The determinant
            11.5.1  Definition and basic properties   cmdat 22698
            11.5.2  Determinants of 2 x 2 -matrices   m2detleiblem1 22738
            11.5.3  The matrix adjugate/adjunct   cmadu 22746
            ​*11.5.4  Laplace expansion of determinants (special case)   symgmatr01lem 22767
            11.5.5  Inverse matrix   invrvald 22790
            ​*11.5.6  Cramer's rule   slesolvec 22793
      ​​*11.6  Polynomial matrices
            11.6.1  Basic properties   pmatring 22806
            ​*11.6.2  Constant polynomial matrices   ccpmat 22817
            ​*11.6.3  Collecting coefficients of polynomial matrices   cdecpmat 22876
            ​*11.6.4  Ring isomorphism between polynomial matrices and polynomials over matrices   cpm2mp 22906
      ​​*11.7  The characteristic polynomial
            *11.7.1  Definition and basic properties   cchpmat 22940
            ​*11.7.2  The characteristic factor function G   fvmptnn04if 22963
            ​*11.7.3  The Cayley-Hamilton theorem   cpmadurid 22981
​PART 12  BASIC TOPOLOGY
      ​12.1  Topology
            ​*12.1.1  Topological spaces   ctop 23007
                  12.1.1.1  Topologies   ctop 23007
                  12.1.1.2  Topologies on sets   ctopon 23024
                  12.1.1.3  Topological spaces   ctps 23046
            12.1.2  Topological bases   ctb 23059
            12.1.3  Examples of topologies   distop 23109
            12.1.4  Closure and interior   ccld 23130
            12.1.5  Neighborhoods   cnei 23211
            12.1.6  Limit points and perfect sets   clp 23248
            12.1.7  Subspace topologies   restrcl 23271
            12.1.8  Order topology   ordtbaslem 23302
            12.1.9  Limits and continuity in topological spaces   ccn 23338
            12.1.10  Separated spaces: T0, T1, T2 (Hausdorff) ...   ct0 23420
            12.1.11  Compactness   ccmp 23500
            12.1.12  Bolzano-Weierstrass theorem   bwth 23524
            12.1.13  Connectedness   cconn 23525
            12.1.14  First- and second-countability   c1stc 23551
            12.1.15  Local topological properties   clly 23578
            12.1.16  Refinements   cref 23616
            12.1.17  Compactly generated spaces   ckgen 23647
            12.1.18  Product topologies   ctx 23674
            12.1.19  Continuous function-builders   cnmptid 23775
            12.1.20  Quotient maps and quotient topology   ckq 23807
            12.1.21  Homeomorphisms   chmeo 23867
      ​12.2  Filters and filter bases
            12.2.1  Filter bases   elmptrab 23941
            12.2.2  Filters   cfil 23959
            12.2.3  Ultrafilters   cufil 24013
            12.2.4  Filter limits   cfm 24047
            12.2.5  Extension by continuity   ccnext 24173
            12.2.6  Topological groups   ctmd 24184
            12.2.7  Infinite group sum on topological groups   ctsu 24240
            12.2.8  Topological rings, fields, vector spaces   ctrg 24270
      ​12.3  Uniform Structures and Spaces
            12.3.1  Uniform structures   cust 24314
            12.3.2  The topology induced by an uniform structure   cutop 24344
            12.3.3  Uniform Spaces   cuss 24367
            12.3.4  Uniform continuity   cucn 24388
            12.3.5  Cauchy filters in uniform spaces   ccfilu 24399
            12.3.6  Complete uniform spaces   ccusp 24410
      ​12.4  Metric spaces
            12.4.1  Pseudometric spaces   ispsmet 24418
            12.4.2  Basic metric space properties   cxms 24431
            12.4.3  Metric space balls   blfvalps 24497
            12.4.4  Open sets of a metric space   mopnval 24552
            12.4.5  Continuity in metric spaces   metcnp3 24654
            12.4.6  The uniform structure generated by a metric   metuval 24663
            12.4.7  Examples of metric spaces   dscmet 24686
            ​*12.4.8  Normed algebraic structures   cnm 24690
            12.4.9  Normed space homomorphisms (bounded linear operators)   cnmo 24819
            12.4.10  Topology on the reals   qtopbaslem 24872
            12.4.11  Topological definitions using the reals   cii 24991
            12.4.12  Path homotopy   chtpy 25083
            12.4.13  The fundamental group   cpco 25116
      ​12.5  Metric subcomplex vector spaces
            12.5.1  Subcomplex modules   cclm 25178
            ​*12.5.2  Subcomplex vector spaces   ccvs 25239
            ​*12.5.3  Normed subcomplex vector spaces   isncvsngp 25265
            12.5.4  Subcomplex pre-Hilbert spaces   ccph 25282
            12.5.5  Convergence and completeness   ccfil 25368
            12.5.6  Baire's Category Theorem   bcthlem1 25440
            12.5.7  Banach spaces and subcomplex Hilbert spaces   ccms 25448
                  12.5.7.1  The complete ordered field of the real numbers   retopn 25495
            12.5.8  Euclidean spaces   crrx 25499
            12.5.9  Minimizing Vector Theorem   minveclem1 25540
            12.5.10  Projection Theorem   pjthlem1 25553
​PART 13  BASIC REAL AND COMPLEX ANALYSIS
      ​13.1  Continuity
            13.1.1  Intermediate value theorem   pmltpclem1 25564
      ​13.2  Integrals
            13.2.1  Lebesgue measure   covol 25578
            13.2.2  Lebesgue integration   cmbf 25730
                  13.2.2.1  Lesbesgue integral   cmbf 25730
                  13.2.2.2  Lesbesgue directed integral   cdit 25962
      ​13.3  Derivatives
            13.3.1  Real and complex differentiation   climc 25978
                  13.3.1.1  Derivatives of functions of one complex or real variable   climc 25978
                  13.3.1.2  Results on real differentiation   dvferm1lem 26100
​PART 14  BASIC REAL AND COMPLEX FUNCTIONS
      ​14.1  Polynomials
            14.1.1  Polynomial degrees   cmdg 26167
            14.1.2  The division algorithm for univariate polynomials   cmn1 26240
            14.1.3  Elementary properties of complex polynomials   cply 26298
            14.1.4  The division algorithm for polynomials   cquot 26408
            14.1.5  Algebraic numbers   caa 26432
            14.1.6  Liouville's approximation theorem   aalioulem1 26450
      ​14.2  Sequences and series
            14.2.1  Taylor polynomials and Taylor's theorem   ctayl 26470
            14.2.2  Uniform convergence   culm 26493
            14.2.3  Power series   pserval 26527
      ​14.3  Basic trigonometry
            14.3.1  The exponential, sine, and cosine functions (cont.)   efcn 26560
            14.3.2  Properties of pi = 3.14159...   pilem1 26568
            14.3.3  Mapping of the exponential function   efgh 26660
            14.3.4  The natural logarithm on complex numbers   clog 26673
            ​*14.3.5  Logarithms to an arbitrary base   clogb 26883
            14.3.6  Theorems of Pythagoras, isosceles triangles, and intersecting chords   angval 26920
            14.3.7  Solutions of quadratic, cubic, and quartic equations   quad2 26958
            14.3.8  Inverse trigonometric functions   casin 26981
            14.3.9  The Birthday Problem   log2ublem1 27065
            14.3.10  Areas in R^2   carea 27074
            14.3.11  More miscellaneous converging sequences   rlimcnp 27084
            14.3.12  Inequality of arithmetic and geometric means   cvxcl 27103
            14.3.13  Euler-Mascheroni constant   cem 27110
            14.3.14  Zeta function   czeta 27131
            14.3.15  Gamma function   clgam 27134
      ​14.4  Basic number theory
            14.4.1  Wilson's theorem   wilthlem1 27186
            14.4.2  The Fundamental Theorem of Algebra   ftalem1 27191
            14.4.3  The Basel problem (ζ(2) = π²/6)   basellem1 27199
            14.4.4  Number-theoretical functions   ccht 27209
            14.4.5  Perfect Number Theorem   mersenne 27345
            14.4.6  Characters of Z/nZ   cdchr 27350
            14.4.7  Bertrand's postulate   bcctr 27393
            ​*14.4.8  Quadratic residues and the Legendre symbol   clgs 27412
            ​*14.4.9  Gauss' Lemma   gausslemma2dlem0a 27474
            14.4.10  Quadratic reciprocity   lgseisenlem1 27493
            14.4.11  All primes 4n+1 are the sum of two squares   2sqlem1 27535
            14.4.12  Chebyshev's Weak Prime Number Theorem, Dirichlet's Theorem   chebbnd1lem1 27587
            14.4.13  The Prime Number Theorem   mudivsum 27648
            14.4.14  Ostrowski's theorem   abvcxp 27733
​PART 15  SURREAL NUMBERS
      ​​*15.1  Sign sequence representation and Alling's axioms
            15.1.1  Definitions and initial properties   csur 27758
            15.1.2  Ordering   ltssolem1 27793
            15.1.3  Birthday Function   bdayfo 27795
            15.1.4  Density   fvnobday 27796
            ​*15.1.5  Full-Eta Property   bdayimaon 27811
      ​15.2  Initial consequences of Alling's axioms
            15.2.1  Ordering Theorems   cles 27862
            15.2.2  Birthday Theorems   bdayfun 27894
      ​​*15.3  Conway cut representation
            15.3.1  Conway cuts   cslts 27904
            15.3.2  Zero and One   c0s 27952
            15.3.3  Cuts and Options   cmade 27969
            15.3.4  Cofinality and coinitiality   cofslts 28065
      ​15.4  Induction and recursion
            15.4.1  Induction and recursion on one variable   cnorec 28084
            15.4.2  Induction and recursion on two variables   cnorec2 28095
      ​15.5  Surreal arithmetic
            15.5.1  Addition   cadds 28106
            15.5.2  Negation and Subtraction   cnegs 28166
            15.5.3  Multiplication   cmuls 28253
            15.5.4  Division   cdivs 28334
            15.5.5  Absolute value   cabss 28384
      ​15.6  Subsystems of surreals
            15.6.1  Ordinal numbers   cons 28398
            15.6.2  Surreal recursive sequences   cseqs 28430
            15.6.3  Natural numbers   cn0s 28459
            15.6.4  Integers   czs 28525
            15.6.5  Dyadic fractions   c2s 28557
            15.6.6  Real numbers   creno 28636
​​*PART 16  ELEMENTARY GEOMETRY
      ​16.1  Definition and Tarski's Axioms of Geometry
            16.1.1  Justification for the congruence notation   tgjustf 28696
      ​16.2  Tarskian Geometry
            16.2.1  Congruence   tgcgrcomimp 28700
            16.2.2  Betweenness   tgbtwntriv2 28710
            16.2.3  Dimension   tglowdim1 28723
            16.2.4  Betweenness and Congruence   tgifscgr 28731
            16.2.5  Congruence of a series of points   ccgrg 28733
            16.2.6  Motions   cismt 28755
            16.2.7  Colinearity   tglng 28769
            16.2.8  Connectivity of betweenness   tgbtwnconn1lem1 28795
            16.2.9  Less-than relation in geometric congruences   cleg 28805
            16.2.10  Rays   chlg 28823
            16.2.11  Lines   btwnlng1 28842
            16.2.12  Point inversions   cmir 28879
            16.2.13  Right angles   crag 28920
            16.2.14  Half-planes   islnopp 28966
            16.2.15  Planes   cplng 28999
            16.2.16  Midpoints and Line Mirroring   cmid 29020
            16.2.17  Congruence of angles   ccgra 29055
            16.2.18  Angle Comparisons   cinag 29083
            16.2.19  Congruence Theorems   tgsas1 29102
            16.2.20  Equilateral triangles   ceqlg 29113
            16.2.21  Parallel lines   cprlng 29117
      ​16.3  Properties of geometries
            16.3.1  Isomorphisms between geometries   f1otrgds 29123
      ​16.4  Geometry in Hilbert spaces
            16.4.1  Geometry in the complex plane   cchhllem 29141
            16.4.2  Geometry in Euclidean spaces   cee 29142
                  16.4.2.1  Definition of the Euclidean space   cee 29142
                  16.4.2.2  Tarski's axioms for geometry for the Euclidean space   axdimuniq 29168
                  16.4.2.3  EE^n fulfills Tarski's Axioms   ceeng 29232
​​*PART 17  GRAPH THEORY
      ​*17.1  Vertices and edges
            17.1.1  The edge function extractor for extensible structures   cedgf 29243
            ​*17.1.2  Vertices and indexed edges   cvtx 29251
                  17.1.2.1  Definitions and basic properties   cvtx 29251
                  17.1.2.2  The vertices and edges of a graph represented as ordered pair   opvtxval 29258
                  17.1.2.3  The vertices and edges of a graph represented as extensible structure   funvtxdmge2val 29266
                  17.1.2.4  Representations of graphs without edges   snstrvtxval 29292
                  17.1.2.5  Degenerated cases of representations of graphs   vtxval0 29294
            17.1.3  Edges as range of the edge function   cedg 29302
      ​​*17.2  Undirected graphs
            17.2.1  Undirected hypergraphs   cuhgr 29311
            17.2.2  Undirected pseudographs and multigraphs   cupgr 29335
            ​*17.2.3  Loop-free graphs   umgrislfupgrlem 29377
            17.2.4  Edges as subsets of vertices of graphs   uhgredgiedgb 29381
            ​*17.2.5  Undirected simple graphs   cuspgr 29403
            17.2.6  Examples for graphs   usgr0e 29491
            17.2.7  Subgraphs   csubgr 29522
            17.2.8  Finite undirected simple graphs   cfusgr 29571
            17.2.9  Neighbors, complete graphs and universal vertices   cnbgr 29587
                  17.2.9.1  Neighbors   cnbgr 29587
                  17.2.9.2  Universal vertices   cuvtx 29640
                  17.2.9.3  Complete graphs   ccplgr 29664
            17.2.10  Vertex degree   cvtxdg 29720
            ​*17.2.11  Regular graphs   crgr 29810
      ​​*17.3  Walks, paths and cycles
            *17.3.1  Walks   cewlks 29850
            17.3.2  Walks for loop-free graphs   lfgrwlkprop 29940
            17.3.3  Trails   ctrls 29943
            17.3.4  Paths and simple paths   cpths 29964
            17.3.5  Closed walks   cclwlks 30024
            17.3.6  Circuits and cycles   ccrcts 30038
            ​*17.3.7  Walks as words   cwwlks 30079
            17.3.8  Walks/paths of length 2 (as length 3 strings)   2wlkdlem1 30179
            17.3.9  Walks in regular graphs   rusgrnumwwlkl1 30225
            ​*17.3.10  Closed walks as words   cclwwlk 30237
                  17.3.10.1  Closed walks as words   cclwwlk 30237
                  17.3.10.2  Closed walks of a fixed length as words   cclwwlkn 30280
                  17.3.10.3  Closed walks on a vertex of a fixed length as words   cclwwlknon 30343
            17.3.11  Examples for walks, trails and paths   0ewlk 30370
            17.3.12  Connected graphs   cconngr 30442
      ​17.4  Eulerian paths and the Konigsberg Bridge problem
            ​*17.4.1  Eulerian paths   ceupth 30453
            ​*17.4.2  The Königsberg Bridge problem   konigsbergvtx 30502
      ​17.5  The Friendship Theorem
            17.5.1  Friendship graphs - basics   cfrgr 30514
            17.5.2  The friendship theorem for small graphs   frgr1v 30527
            17.5.3  Theorems according to Mertzios and Unger   2pthfrgrrn 30538
            ​*17.5.4  Huneke's Proof of the Friendship Theorem   frgrncvvdeqlem1 30555
​PART 18  GUIDES AND MISCELLANEA
      ​18.1  Guides (conventions, explanations, and examples)
            ​*18.1.1  Conventions   conventions 30656
            18.1.2  Natural deduction   natded 30659
            ​*18.1.3  Natural deduction examples   ex-natded5.2 30660
            18.1.4  Definitional examples   ex-or 30677
            18.1.5  Other examples   aevdemo 30716
      ​18.2  Humor
            18.2.1  April Fool's theorem   avril1 30719
      ​18.3  (Future - to be reviewed and classified)
            18.3.1  Planar incidence geometry   cplig 30731
            ​*18.3.2  Aliases kept to prevent broken links   dummylink 30744
​​*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 30746
            19.1.2  Abelian groups   cablo 30801
      ​19.2  Complex vector spaces
            19.2.1  Definition and basic properties   cvc 30815
            19.2.2  Examples of complex vector spaces   cnaddabloOLD 30838
      ​19.3  Normed complex vector spaces
            19.3.1  Definition and basic properties   cnv 30841
            19.3.2  Examples of normed complex vector spaces   cnnv 30934
            19.3.3  Induced metric of a normed complex vector space   imsval 30942
            19.3.4  Inner product   cdip 30957
            19.3.5  Subspaces   css 30978
      ​19.4  Operators on complex vector spaces
            19.4.1  Definitions and basic properties   clno 30997
      ​19.5  Inner product (pre-Hilbert) spaces
            19.5.1  Definition and basic properties   ccphlo 31069
            19.5.2  Examples of pre-Hilbert spaces   cncph 31076
            19.5.3  Properties of pre-Hilbert spaces   isph 31079
      ​19.6  Complex Banach spaces
            19.6.1  Definition and basic properties   ccbn 31119
            19.6.2  Examples of complex Banach spaces   cnbn 31126
            19.6.3  Uniform Boundedness Theorem   ubthlem1 31127
            19.6.4  Minimizing Vector Theorem   minvecolem1 31131
      ​19.7  Complex Hilbert spaces
            19.7.1  Definition and basic properties   chlo 31142
            19.7.2  Standard axioms for a complex Hilbert space   hlex 31155
            19.7.3  Examples of complex Hilbert spaces   cnchl 31173
            19.7.4  Hellinger-Toeplitz Theorem   htthlem 31174
​​*PART 20  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      ​20.1  Axiomatization of complex pre-Hilbert spaces
            20.1.1  Basic Hilbert space definitions   chba 31176
            20.1.2  Preliminary ZFC lemmas   df-hnorm 31225
            ​*20.1.3  Derive the Hilbert space axioms from ZFC set theory   axhilex-zf 31238
            ​*20.1.4  Introduce the vector space axioms for a Hilbert space   ax-hilex 31256
            20.1.5  Vector operations   hvmulex 31268
            20.1.6  Inner product postulates for a Hilbert space   ax-hfi 31336
      ​20.2  Inner product and norms
            20.2.1  Inner product   his5 31343
            20.2.2  Norms   dfhnorm2 31379
            20.2.3  Relate Hilbert space to normed complex vector spaces   hilablo 31417
            20.2.4  Bunjakovaskij-Cauchy-Schwarz inequality   bcsiALT 31436
      ​20.3  Cauchy sequences and completeness axiom
            20.3.1  Cauchy sequences and limits   hcau 31441
            20.3.2  Derivation of the completeness axiom from ZF set theory   hilmet 31451
            20.3.3  Completeness postulate for a Hilbert space   ax-hcompl 31459
            20.3.4  Relate Hilbert space to ZFC pre-Hilbert and Hilbert spaces   hhcms 31460
      ​20.4  Subspaces and projections
            20.4.1  Subspaces   df-sh 31464
            20.4.2  Closed subspaces   df-ch 31478
            20.4.3  Orthocomplements   df-oc 31509
            20.4.4  Subspace sum, span, lattice join, lattice supremum   df-shs 31565
            20.4.5  Projection theorem   pjhthlem1 31648
            20.4.6  Projectors   df-pjh 31652
      ​20.5  Properties of Hilbert subspaces
            20.5.1  Orthomodular law   omlsilem 31659
            20.5.2  Projectors (cont.)   pjhtheu2 31673
            20.5.3  Hilbert lattice operations   sh0le 31697
            20.5.4  Span (cont.) and one-dimensional subspaces   spansn0 31798
            20.5.5  Commutes relation for Hilbert lattice elements   df-cm 31840
            20.5.6  Foulis-Holland theorem   fh1 31875
            20.5.7  Quantum Logic Explorer axioms   qlax1i 31884
            20.5.8  Orthogonal subspaces   chscllem1 31894
            20.5.9  Orthoarguesian laws 5OA and 3OA   5oalem1 31911
            20.5.10  Projectors (cont.)   pjorthi 31926
            20.5.11  Mayet's equation E_3   mayete3i 31985
      ​20.6  Operators on Hilbert spaces
            ​*20.6.1  Operator sum, difference, and scalar multiplication   df-hosum 31987
            20.6.2  Zero and identity operators   df-h0op 32005
            20.6.3  Operations on Hilbert space operators   hoaddcl 32015
            20.6.4  Linear, continuous, bounded, Hermitian, unitary operators and norms   df-nmop 32096
            20.6.5  Linear and continuous functionals and norms   df-nmfn 32102
            20.6.6  Adjoint   df-adjh 32106
            20.6.7  Dirac bra-ket notation   df-bra 32107
            20.6.8  Positive operators   df-leop 32109
            20.6.9  Eigenvectors, eigenvalues, spectrum   df-eigvec 32110
            20.6.10  Theorems about operators and functionals   nmopval 32113
            20.6.11  Riesz lemma   riesz3i 32319
            20.6.12  Adjoints (cont.)   cnlnadjlem1 32324
            20.6.13  Quantum computation error bound theorem   unierri 32361
            20.6.14  Dirac bra-ket notation (cont.)   branmfn 32362
            20.6.15  Positive operators (cont.)   leopg 32379
            20.6.16  Projectors as operators   pjhmopi 32403
      ​20.7  States on a Hilbert lattice and Godowski's equation
            20.7.1  States on a Hilbert lattice   df-st 32468
            20.7.2  Godowski's equation   golem1 32528
      ​20.8  Cover relation, atoms, exchange axiom, and modular symmetry
            20.8.1  Covers relation; modular pairs   df-cv 32536
            20.8.2  Atoms   df-at 32595
            20.8.3  Superposition principle   superpos 32611
            20.8.4  Atoms, exchange and covering properties, atomicity   chcv1 32612
            20.8.5  Irreducibility   chirredlem1 32647
            20.8.6  Atoms (cont.)   atcvat3i 32653
            20.8.7  Modular symmetry   mdsymlem1 32660
​PART 21  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      ​21.1  Mathboxes for user contributions
            21.1.1  Mathbox guidelines   mathbox 32699
      ​21.2  Mathbox for Stefan Allan
      ​21.3  Mathbox for Thierry Arnoux
            21.3.1  Propositional Calculus - misc additions   ad11antr 32704
            21.3.2  Predicate Calculus   sbc2iedf 32718
                  21.3.2.1  Predicate Calculus - misc additions   sbc2iedf 32718
                  21.3.2.2  Restricted quantification - misc additions   ralcom4f 32720
                  21.3.2.3  Equality   eqtrb 32726
                  21.3.2.4  Double restricted existential uniqueness quantification   opsbc2ie 32728
                  21.3.2.5  Double restricted existential uniqueness quantification syntax   w2reu 32730
                  21.3.2.6  Substitution (without distinct variables) - misc additions   sbceqbidf 32739
                  21.3.2.7  Existential "at most one" - misc additions   mo5f 32741
                  21.3.2.8  Existential uniqueness - misc additions   reuxfrdf 32743
                  21.3.2.9  Restricted "at most one" - misc additions   rmoxfrd 32745
                  21.3.2.10  Restricted iota (description binder)   riotaeqbidva 32748
            21.3.3  General Set Theory   dmrab 32749
                  21.3.3.1  Class abstractions (a.k.a. class builders)   dmrab 32749
                  21.3.3.2  Image Sets   abrexdomjm 32759
                  21.3.3.3  Set relations and operations - misc additions   nelun 32765
                  21.3.3.4  Unordered pairs   elpreq 32780
                  21.3.3.5  Unordered triples   tpssg 32789
                  21.3.3.6  Conditional operator - misc additions   ifeqeqx 32794
                  21.3.3.7  Set union   uniinn0 32803
                  21.3.3.8  Indexed union - misc additions   cbviunf 32806
                  21.3.3.9  Indexed intersection - misc additions   iinabrex 32820
                  21.3.3.10  Disjointness - misc additions   disjnf 32821
            21.3.4  Relations and Functions   xpdisjres 32849
                  21.3.4.1  Relations - misc additions   xpdisjres 32849
                  21.3.4.2  Functions - misc additions   fconst7v 32873
                  21.3.4.3  Operations - misc additions   mpomptxf 32931
                  21.3.4.4  The mapping operation   elmaprd 32933
                  21.3.4.5  Support of a function   suppovss 32934
                  21.3.4.6  Explicit Functions with one or two points as a domain   cosnopne 32947
                  21.3.4.7  Isomorphisms - misc. additions   gtiso 32954
                  21.3.4.8  Disjointness (additional proof requiring functions)   disjdsct 32956
                  21.3.4.9  First and second members of an ordered pair - misc additions   df1stres 32957
                  21.3.4.10  Countable Sets   snct 32965
            21.3.5  Real and Complex Numbers   sgnval2 32988
                  21.3.5.1  Complex operations - misc. additions   creq0 32989
                  21.3.5.2  Ordering on reals - misc additions   lt2addrd 33003
                  21.3.5.3  Extended reals - misc additions   nn0mnfxrd 33004
                  21.3.5.4  Extended nonnegative integers - misc additions   xnn0gt0 33022
                  21.3.5.5  Real number intervals - misc additions   joiniooico 33027
                  21.3.5.6  Finite intervals of integers - misc additions   uzssico 33037
                  21.3.5.7  Half-open integer ranges - misc additions   iundisjfi 33049
                  21.3.5.8  The ` # ` (set size) function - misc additions   hashunif 33059
                  21.3.5.9  The greatest common divisor operator - misc. additions   elq2 33064
                  21.3.5.10  Integers   nn0split01 33070
                  21.3.5.11  Decimal numbers   dfdec100 33082
            21.3.6  Real and complex functions   sgnsgn 33083
                  21.3.6.1  Signum (sgn or sign) function - misc. additions   sgnsgn 33083
                  21.3.6.2  Integer powers - misc. additions   nexple 33085
                  21.3.6.3  Indicator Functions (continued)   indsumin 33089
            ​*21.3.7  Decimal expansion   cdp2 33098
                  ​*21.3.7.1  Decimal point   cdp 33115
                  21.3.7.2  Division in the extended real number system   cxdiv 33144
            21.3.8  Words over a set - misc additions   wrdres 33163
                  21.3.8.1  Splicing words (substring replacement)   splfv3 33186
                  21.3.8.2  Cyclic shift of words   1cshid 33187
            21.3.9  Extensible Structures   ressplusf 33191
                  21.3.9.1  Structure restriction operator   ressplusf 33191
                  21.3.9.2  Posets   ressprs 33194
                  21.3.9.3  Complete lattices   clatp0cl 33204
                  21.3.9.4  Order Theory   cmnt 33206
                  21.3.9.5  Extended reals Structure - misc additions   ax-xrssca 33232
                  21.3.9.6  The extended nonnegative real numbers commutative monoid   xrge00 33242
            21.3.10  Algebra   mndcld 33250
                  21.3.10.1  Monoids   mndcld 33250
                  21.3.10.2  Monoids Homomorphisms   abliso 33263
                  21.3.10.3  Groups - misc additions   grpidcld 33267
                  21.3.10.4  Abelian Groups - misc additions   ablcomd 33273
                  21.3.10.5  Finitely supported group sums - misc additions   gsumsubg 33274
                  21.3.10.6  Group or monoid sums over words   gsumwun 33304
                  21.3.10.7  Centralizers and centers - misc additions   cntzun 33307
                  21.3.10.8  The symmetric group   symgfcoeu 33310
                  21.3.10.9  Transpositions   pmtridf1o 33322
                  21.3.10.10  Permutation Signs   psgnid 33325
                  21.3.10.11  Permutation cycles   ctocyc 33334
                  21.3.10.12  The Alternating Group   evpmval 33373
                  21.3.10.13  Signum in an ordered monoid   csgns 33386
                  21.3.10.14  Fixed points   cfxp 33391
                  21.3.10.15  The Archimedean property for generic ordered algebraic structures   cinftm 33404
                  21.3.10.16  Semiring left modules   cslmd 33428
                  21.3.10.17  Simple groups   prmsimpcyc 33456
                  21.3.10.18  Rings - misc additions   ringrngd 33457
                  21.3.10.19  Subrings generated by a set   elrgspnlem1 33470
                  21.3.10.20  The zero ring   irrednzr 33478
                  21.3.10.21  Localization of rings   cerl 33481
                  21.3.10.22  Integral Domains   domnmuln0rd 33505
                  21.3.10.23  Euclidean Domains   ceuf 33519
                  21.3.10.24  Division Rings   ringinveu 33525
                  21.3.10.25  The field of rational numbers   qfld 33528
                  21.3.10.26  Subfields   subsdrg 33529
                  21.3.10.27  Field of fractions   cfrac 33533
                  21.3.10.28  Field extensions generated by a set   cfldgen 33541
                  21.3.10.29  Ring homomorphisms - misc additions   rhmdvd 33554
                  21.3.10.30  Scalar restriction operation   cresv 33556
                  21.3.10.31  The commutative ring of gaussian integers   gzcrng 33571
                  21.3.10.32  The archimedean ordered field of real numbers   cnfldfld 33572
                  21.3.10.33  The quotient map and quotient modules   qusker 33579
                  21.3.10.34  The ring of integers modulo ` N `   znfermltl 33591
                  21.3.10.35  Independent sets and families   islinds5 33592
                  21.3.10.36  Ring associates, ring units   dvdsruassoi 33608
                  ​*21.3.10.37  Subgroup sum / Sumset / Minkowski sum   elgrplsmsn 33614
                  21.3.10.38  The quotient map   quslsm 33625
                  21.3.10.39  Ideals   intlidl 33639
                  21.3.10.40  Maximal Ideals   cmxidl 33654
                  21.3.10.41  Local rings   drnglring 33694
                  21.3.10.42  The semiring of ideals of a ring   cidlsrg 33702
                  21.3.10.43  Prime Elements   rprmval 33718
                  21.3.10.44  Unique factorization domains   cufd 33740
                  21.3.10.45  The ring of integers   zringidom 33753
                  21.3.10.46  Associative Algebra   assaassd 33757
                  21.3.10.47  Univariate Polynomials   0ringmon1p 33759
                  21.3.10.48  Polynomial quotient and polynomial remainder   q1pdir 33805
                  21.3.10.49  Multivariate Polynomials   psrbasfsupp 33813
                  21.3.10.50  The ring of symmetric polynomials   csply 33857
                  21.3.10.51  The subring algebra   sra1r 33883
                  21.3.10.52  Division Ring Extensions   drgext0g 33892
                  21.3.10.53  Vector Spaces   lvecdimfi 33898
                  21.3.10.54  Vector Space Dimension   cldim 33901
            21.3.11  Field Extensions   cfldext 33940
                  21.3.11.1  Algebraic numbers   cirng 33985
                  21.3.11.2  Algebraic extensions   calgext 33997
                  21.3.11.3  Minimal polynomials   cminply 34001
                  21.3.11.4  Quadratic Field Extensions   rtelextdg2lem 34028
                  21.3.11.5  Towers of quadratic extentions   fldext2chn 34030
            ​*21.3.12  Constructible Numbers   cconstr 34031
                  21.3.12.1  Impossible constructions   2sqr3minply 34082
            21.3.13  Matrices   csmat 34095
                  21.3.13.1  Submatrices   csmat 34095
                  21.3.13.2  Matrix literals   clmat 34113
                  21.3.13.3  Laplace expansion of determinants   mdetpmtr1 34125
            21.3.14  Topology   ist0cld 34135
                  21.3.14.1  Open maps   txomap 34136
                  21.3.14.2  Topology of the unit circle   qtopt1 34137
                  21.3.14.3  Refinements   reff 34141
                  21.3.14.4  Open cover refinement property   ccref 34144
                  21.3.14.5  Lindelöf spaces   cldlf 34154
                  21.3.14.6  Paracompact spaces   cpcmp 34157
                  ​*21.3.14.7  Spectrum of a ring   crspec 34164
                  21.3.14.8  Pseudometrics   cmetid 34188
                  21.3.14.9  Continuity - misc additions   hauseqcn 34200
                  21.3.14.10  Topology of the closed unit interval   elunitge0 34201
                  21.3.14.11  Topology of ` ( RR X. RR ) `   unicls 34205
                  21.3.14.12  Order topology - misc. additions   cnvordtrestixx 34215
                  21.3.14.13  Continuity in topological spaces - misc. additions   mndpluscn 34228
                  21.3.14.14  Topology of the extended nonnegative real numbers ordered monoid   xrge0hmph 34234
                  21.3.14.15  Limits - misc additions   lmlim 34249
                  21.3.14.16  Univariate polynomials   pl1cn 34257
            21.3.15  Uniform Stuctures and Spaces   chcmp 34258
                  21.3.15.1  Hausdorff uniform completion   chcmp 34258
            21.3.16  Topology and algebraic structures   zringnm 34260
                  21.3.16.1  The norm on the ring of the integer numbers   zringnm 34260
                  21.3.16.2  Topological ` ZZ ` -modules   zlm0 34262
                  21.3.16.3  Canonical embedding of the field of the rational numbers into a division ring   cqqh 34272
                  21.3.16.4  Canonical embedding of the real numbers into a complete ordered field   crrh 34295
                  21.3.16.5  Embedding from the extended real numbers into a complete lattice   cxrh 34318
                  21.3.16.6  Canonical embeddings into the ordered field of the real numbers   zrhre 34321
                  ​*21.3.16.7  Topological Manifolds   cmntop 34324
                  21.3.16.8  Extended sum   cesum 34329
            21.3.17  Mixed Function/Constant operation   cofc 34397
            21.3.18  Abstract measure   csiga 34410
                  21.3.18.1  Sigma-Algebra   csiga 34410
                  21.3.18.2  Generated sigma-Algebra   csigagen 34440
                  ​*21.3.18.3  lambda and pi-Systems, Rings of Sets   ispisys 34454
                  21.3.18.4  The Borel algebra on the real numbers   cbrsiga 34483
                  21.3.18.5  Product Sigma-Algebra   csx 34490
                  21.3.18.6  Measures   cmeas 34497
                  21.3.18.7  The counting measure   cntmeas 34528
                  21.3.18.8  The Lebesgue measure - misc additions   voliune 34531
                  21.3.18.9  The Dirac delta measure   cdde 34534
                  21.3.18.10  The 'almost everywhere' relation   cae 34539
                  21.3.18.11  Measurable functions   cmbfm 34551
                  21.3.18.12  Borel Algebra on ` ( RR X. RR ) `   br2base 34571
                  ​*21.3.18.13  Caratheodory's extension theorem   coms 34593
            21.3.19  Integration   itgeq12dv 34628
                  21.3.19.1  Lebesgue integral - misc additions   itgeq12dv 34628
                  21.3.19.2  Bochner integral   citgm 34629
            21.3.20  Euler's partition theorem   oddpwdc 34656
            21.3.21  Sequences defined by strong recursion   csseq 34685
            21.3.22  Fibonacci Numbers   cfib 34698
            21.3.23  Probability   cprb 34709
                  21.3.23.1  Probability Theory   cprb 34709
                  21.3.23.2  Conditional Probabilities   ccprob 34733
                  21.3.23.3  Real-valued Random Variables   crrv 34742
                  21.3.23.4  Preimage set mapping operator   corvc 34758
                  21.3.23.5  Distribution Functions   orvcelval 34771
                  21.3.23.6  Cumulative Distribution Functions   orvclteel 34775
                  21.3.23.7  Probabilities - example   coinfliplem 34781
                  21.3.23.8  Bertrand's Ballot Problem   ballotlemoex 34788
            21.3.24  Signum (sgn or sign) function - misc. additions   fzssfzo 34841
                  21.3.24.1  Operations on words   ccatmulgnn0dir 34844
            21.3.25  Polynomials with real coefficients - misc additions   plyrecld 34848
            21.3.26  Descartes's rule of signs   signspval 34851
                  21.3.26.1  Sign changes in a word over real numbers   signspval 34851
                  21.3.26.2  Counting sign changes in a word over real numbers   signslema 34861
            21.3.27  Number Theory   iblidicc 34891
                  21.3.27.1  Representations of a number as sums of integers   crepr 34907
                  21.3.27.2  Vinogradov Trigonometric Sums and the Circle Method   cvts 34934
                  21.3.27.3  The Ternary Goldbach Conjecture: Final Statement   ax-hgt749 34943
            21.3.28  Elementary Geometry   cstrkg2d 34963
                  ​*21.3.28.1  Two-dimensional geometry   cstrkg2d 34963
                  21.3.28.2  Morley's Miracle   cgranbtwn 34968
                  21.3.28.3  Outer Five Segment (not used, no need to move to main)   cafs 34971
            ​*21.3.29  LeftPad Project   clpad 34976
      ​​*21.4  Mathbox for Jonathan Ben-Naim
            21.4.1  First-order logic and set theory   bnj170 34999
            21.4.2  Well founded induction and recursion   bnj110 35158
            21.4.3  The existence of a minimal element in certain classes   bnj69 35310
            21.4.4  Well-founded induction   bnj1204 35312
            21.4.5  Well-founded recursion, part 1 of 3   bnj60 35362
            21.4.6  Well-founded recursion, part 2 of 3   bnj1500 35368
            21.4.7  Well-founded recursion, part 3 of 3   bnj1522 35372
      ​21.5  Mathbox for BTernaryTau
            21.5.1  First-order logic   nfan1c 35373
                  21.5.1.1  Auxiliary axiom schemes   nfan1c 35373
            21.5.2  ZF set theory   exdifsn 35379
                  21.5.2.1  Finitism   prcinf 35416
                  21.5.2.2  Introduce ax-regs   ax-regs 35429
                  21.5.2.3  Derive ax-regs   axregs 35442
                  21.5.2.4  ZFC axioms with reduced distinct variable conditions   axsepg2 35443
                  21.5.2.5  Global choice   gblacfnacd 35452
            21.5.3  Real and complex numbers   zltp1ne 35467
            21.5.4  Graph theory   lfuhgr 35476
                  21.5.4.1  Acyclic graphs   cacycgr 35500
      ​21.6  Mathbox for Mario Carneiro
            21.6.1  Predicate calculus with all distinct variables   ax-7d 35517
            21.6.2  Miscellaneous stuff   quartfull 35523
            21.6.3  Derangements and the Subfactorial   deranglem 35524
            21.6.4  The Erdős-Szekeres theorem   erdszelem1 35549
            21.6.5  The Kuratowski closure-complement theorem   kur14lem1 35564
            21.6.6  Retracts and sections   cretr 35575
            21.6.7  Path-connected and simply connected spaces   cpconn 35577
            21.6.8  Covering maps   ccvm 35613
            21.6.9  Normal numbers   snmlff 35687
            21.6.10  Godel-sets of formulas - part 1   cgoe 35691
            21.6.11  Godel-sets of formulas - part 2   cgon 35790
            21.6.12  Models of ZF   cgze 35804
            ​*21.6.13  Metamath formal systems   cmcn 35818
            21.6.14  Grammatical formal systems   cm0s 35943
            21.6.15  Models of formal systems   cmuv 35963
            21.6.16  Splitting fields   ccpms 35985
            21.6.17  p-adic number fields   czr 36005
      ​​*21.7  Mathbox for Filip Cernatescu
      ​21.8  Mathbox for Paul Chapman
            21.8.1  Real and complex numbers (cont.)   climuzcnv 36029
            21.8.2  Miscellaneous theorems   elfzm12 36033
      ​21.9  Mathbox for Hongxiu Chen
      ​21.10  Mathbox for Adrian Ducourtial
            21.10.1  Propositional calculus   currybi 36046
            21.10.2  Clone theory   ccloneop 36053
      ​21.11  Mathbox for Scott Fenton
            21.11.1  ZFC Axioms in primitive form   axextprim 36059
            21.11.2  Untangled classes   untelirr 36066
            21.11.3  Extra propositional calculus theorems   3jaodd 36073
            21.11.4  Misc. Useful Theorems   nepss 36076
            21.11.5  Properties of real and complex numbers   sqdivzi 36086
            21.11.6  Infinite products   iprodefisumlem 36098
            21.11.7  Factorial limits   faclimlem1 36101
            21.11.8  Greatest common divisor and divisibility   gcd32 36107
            21.11.9  Properties of relationships   dftr6 36109
            21.11.10  Properties of functions and mappings   funpsstri 36124
            21.11.11  Ordinal numbers   elpotr 36137
            21.11.12  Defined equality axioms   axextdfeq 36153
            21.11.13  Hypothesis builders   hbntg 36161
            21.11.14  Well-founded zero, successor, and limits   cwsuc 36166
            21.11.15  Quantifier-free definitions   ctxp 36186
            21.11.16  Alternate ordered pairs   caltop 36314
            21.11.17  Geometry in the Euclidean space   cofs 36340
                  21.11.17.1  Congruence properties   cofs 36340
                  21.11.17.2  Betweenness properties   btwntriv2 36370
                  21.11.17.3  Segment Transportation   ctransport 36387
                  21.11.17.4  Properties relating betweenness and congruence   cifs 36393
                  21.11.17.5  Connectivity of betweenness   btwnconn1lem1 36445
                  21.11.17.6  Segment less than or equal to   csegle 36464
                  21.11.17.7  Outside-of relationship   coutsideof 36477
                  21.11.17.8  Lines and Rays   cline2 36492
            21.11.18  Forward difference   cfwddif 36516
            21.11.19  Rank theorems   rankung 36524
            21.11.20  Hereditarily Finite Sets   chf 36530
            21.11.21  Natural ordinal operations   cnmul 36545
      ​21.12  Mathbox for Gino Giotto
            21.12.1  Equality theorems   rmoeqi 36555
                  21.12.1.1  Inference versions   rmoeqi 36555
                  21.12.1.2  Deduction versions   rmoeqdv 36580
            21.12.2  Change bound variables   in-ax8 36592
                  21.12.2.1  Change bound variables and domains   cbvralvw2 36594
                  21.12.2.2  Change bound variables, deduction versions   cbvmodavw 36618
                  21.12.2.3  Change bound variables and domains, deduction versions   cbvrmodavw2 36651
            21.12.3  Study of ax-mulf usage   mpomulnzcnf 36667
      ​21.13  Mathbox for Jeff Hankins
            21.13.1  Miscellany   a1i14 36668
            21.13.2  Basic topological facts   topbnd 36692
            21.13.3  Topology of the real numbers   ivthALT 36703
            21.13.4  Refinements   cfne 36704
            21.13.5  Neighborhood bases determine topologies   neibastop1 36727
            21.13.6  Lattice structure of topologies   topmtcl 36731
            21.13.7  Filter bases   fgmin 36738
            21.13.8  Directed sets, nets   tailfval 36740
      ​21.14  Mathbox for Anthony Hart
            21.14.1  Propositional Calculus   tb-ax1 36751
            21.14.2  Predicate Calculus   nalfal 36771
            21.14.3  Miscellaneous single axioms   meran1 36779
            21.14.4  Connective Symmetry   negsym1 36785
      ​21.15  Mathbox for Chen-Pang He
            21.15.1  Ordinal topology   ontopbas 36796
      ​21.16  Mathbox for Jeff Hoffman
            21.16.1  Inferences for finite induction on generic function values   fveleq 36819
            21.16.2  gdc.mm   nnssi2 36823
      ​21.17  Mathbox for Matthew House
            21.17.1  Relations on well-ordered indexed unions   weiunval 36830
            21.17.2  Axiom of Transitive Containment   axtco 36839
            21.17.3  Transitive closure of a class   tr0elw 36852
            ​*21.17.4  Stronger axioms of regularity   mh-setind 36904
            21.17.5  Short axioms written in primitive symbols   mh-inf3f1 36909
      ​21.18  Mathbox for Asger C. Ipsen
            21.18.1  Continuous nowhere differentiable functions   dnival 36917
      ​​*21.19  Mathbox for BJ
            *21.19.1  Propositional calculus   bj-mp2c 36986
                  *21.19.1.1  Derived rules of inference   bj-mp2c 36986
                  ​*21.19.1.2  A syntactic theorem   bj-0 36988
                  ​*21.19.1.3  Minimal implicational calculus   bj-poni 36990
                  ​*21.19.1.4  Positive calculus   bj-bisimpl 37002
                  ​*21.19.1.5  Implication and negation   bj-con2com 37010
                  ​*21.19.1.6  Disjunction   bj-jaoi1 37021
                  ​*21.19.1.7  Logical equivalence   bj-dfbi4 37023
                  21.19.1.8  The conditional operator for propositions   bj-consensus 37028
                  ​*21.19.1.9  Propositional calculus: miscellaneous   bj-imbi12 37033
            ​*21.19.2  Modal logic   bj-axdd2 37042
            ​*21.19.3  Provability logic   cprvb 37047
            ​*21.19.4  First-order logic   bj-exexalal 37056
                  21.19.4.1  Universal and existential quantifiers, nonfreeness predicate   bj-exexalal 37056
                  21.19.4.2  Adding ax-gen   bj-genr 37057
                  21.19.4.3  Adding ax-4   bj-almp 37061
                  21.19.4.4  Adding ax-5   bj-spvw 37114
                  21.19.4.5  Equality and substitution   bj-df-sb 37129
                  21.19.4.6  Adding ax-6   bj-spim0 37148
                  21.19.4.7  Adding ax-7   bj-cbvexw 37156
                  21.19.4.8  Membership predicate, ax-8 and ax-9   bj-ax89 37158
                  21.19.4.9  Adding ax-11   bj-alcomexcom 37160
                  21.19.4.10  Adding ax-12   axc11n11 37164
                  ​*21.19.4.11  Really adding ax-12   bj-substax12 37206
                  21.19.4.12  Nonfreeness   wnnf 37208
                  21.19.4.13  Adding ax-13   bj-axc10 37275
                  ​*21.19.4.14  Removing dependencies on ax-13 (and ax-11)   bj-axc10v 37285
                  ​*21.19.4.15  Distinct var metavariables   bj-hbaeb2 37310
                  ​*21.19.4.16  Around ~ equsal   bj-equsal1t 37314
                  ​*21.19.4.17  Some Principia Mathematica proofs   stdpc5t 37319
                  21.19.4.18  Alternate definition of substitution   bj-sbsb 37329
                  21.19.4.19  Lemmas for substitution   bj-sbf3 37331
                  21.19.4.20  Existential uniqueness   bj-eu3f 37333
                  ​*21.19.4.21  First-order logic: miscellaneous   bj-sblem1 37334
            21.19.5  Set theory   eliminable1 37351
                  ​*21.19.5.1  Eliminability of class terms   eliminable1 37351
                  ​*21.19.5.2  Classes without the axiom of extensionality   bj-denoteslem 37363
                  21.19.5.3  Characterization among sets versus among classes   elelb 37389
                  ​*21.19.5.4  The nonfreeness quantifier for classes   bj-nfcsym 37391
                  ​*21.19.5.5  Lemmas for class substitution   bj-sbeqALT 37392
                  21.19.5.6  Removing some axiom requirements and disjoint variable conditions   bj-exlimvmpi 37403
                  ​*21.19.5.7  Class abstractions   bj-elabd2ALT 37417
                  21.19.5.8  Generalized class abstractions   bj-cgab 37425
                  ​*21.19.5.9  Restricted nonfreeness   wrnf 37433
                  ​*21.19.5.10  Russell's paradox   bj-ru1 37435
                  21.19.5.11  Curry's paradox in set theory   currysetlem 37437
                  ​*21.19.5.12  Some disjointness results   bj-n0i 37443
                  ​*21.19.5.13  Complements on direct products   bj-xpimasn 37447
                  ​*21.19.5.14  "Singletonization" and tagging   bj-snsetex 37455
                  ​*21.19.5.15  Tuples of classes   bj-cproj 37482
                  ​*21.19.5.16  Set theory: elementary operations relative to a universe   bj-rcleqf 37517
                  ​*21.19.5.17  Axioms for finite unions   bj-abex 37522
                  ​*21.19.5.18  Set theory: miscellaneous   eleq2w2ALT 37539
                  ​*21.19.5.19  Axioms of separation and replacement   bj-axnul 37564
                  ​*21.19.5.20  Evaluation at a class   bj-evaleq 37568
                  21.19.5.21  Elementwise operations   celwise 37576
                  ​*21.19.5.22  Elementwise intersection (families of sets induced on a subset)   bj-rest00 37578
                  21.19.5.23  Moore collections (complements)   bj-raldifsn 37597
                  21.19.5.24  Maps-to notation for functions with three arguments   bj-0nelmpt 37613
                  ​*21.19.5.25  Currying   csethom 37619
                  ​*21.19.5.26  Setting components of extensible structures   cstrset 37631
            ​*21.19.6  Extended real and complex numbers, real and complex projective lines   bj-nfald 37634
                  21.19.6.1  Complements on class abstractions of ordered pairs and binary relations   bj-nfald 37634
                  ​*21.19.6.2  Identity relation (complements)   bj-opabssvv 37649
                  ​*21.19.6.3  Functionalized identity (diagonal in a Cartesian square)   cdiag2 37671
                  ​*21.19.6.4  Direct image and inverse image   cimdir 37677
                  ​*21.19.6.5  Extended numbers and projective lines as sets   cfractemp 37695
                  ​*21.19.6.6  Addition and opposite   caddcc 37736
                  ​*21.19.6.7  Order relation on the extended reals   cltxr 37740
                  ​*21.19.6.8  Argument, multiplication and inverse   carg 37742
                  21.19.6.9  The canonical bijection from the finite ordinals   ciomnn 37748
                  21.19.6.10  Divisibility   cnnbar 37759
            ​*21.19.7  Monoids   bj-smgrpssmgm 37767
                  ​*21.19.7.1  Finite sums in monoids   cfinsum 37782
            ​*21.19.8  Affine, Euclidean, and Cartesian geometry   bj-fvimacnv0 37785
                  *21.19.8.1  Real vector spaces   bj-fvimacnv0 37785
                  ​*21.19.8.2  Complex numbers (supplements)   bj-subcom 37807
                  ​*21.19.8.3  Barycentric coordinates   bj-bary1lem 37809
            21.19.9  Monoid of endomorphisms   cend 37812
      ​21.20  Mathbox for Jim Kingdon
            21.20.1  Circle constant   taupilem3 37818
            21.20.2  Number theory   dfgcd3 37823
            21.20.3  Real numbers   irrdifflemf 37824
      ​21.21  Mathbox for ML
            21.21.1  Miscellaneous   csbrecsg 37829
            21.21.2  Cartesian exponentiation   cfinxp 37884
            21.21.3  Topology   iunctb2 37904
                  ​*21.21.3.1  Pi-base theorems   pibp16 37914
      ​21.22  Mathbox for Wolf Lammen
            21.22.1  1. Bootstrapping   wl-section-boot 37923
            21.22.2  Implication chains   wl-section-impchain 37947
            21.22.3  Theorems around the conditional operator   wl-ifp-ncond1 37965
            21.22.4  Alternative development of hadd, cadd   wl-df-3xor 37969
            21.22.5  An alternative axiom ~ ax-13   ax-wl-13v 37994
            21.22.6  Bootstrapping set theory with classes   wl-cleq-0 37996
            21.22.7  Other stuff   wl-mps 38017
      ​21.23  Mathbox for Brendan Leahy
      ​21.24  Mathbox for Jeff Madsen
            21.24.1  Logic and set theory   unirep 38220
            21.24.2  Real and complex numbers; integers   filbcmb 38246
            21.24.3  Sequences and sums   sdclem2 38248
            21.24.4  Topology   subspopn 38258
            21.24.5  Metric spaces   metf1o 38261
            21.24.6  Continuous maps and homeomorphisms   constcncf 38268
            21.24.7  Boundedness   ctotbnd 38272
            21.24.8  Isometries   cismty 38304
            21.24.9  Heine-Borel Theorem   heibor1lem 38315
            21.24.10  Banach Fixed Point Theorem   bfplem1 38328
            21.24.11  Euclidean space   crrn 38331
            21.24.12  Intervals (continued)   ismrer1 38344
            21.24.13  Operation properties   cass 38348
            21.24.14  Groups and related structures   cmagm 38354
            21.24.15  Group homomorphism and isomorphism   cghomOLD 38389
            21.24.16  Rings   crngo 38400
            21.24.17  Division Rings   cdrng 38454
            21.24.18  Ring homomorphisms   crngohom 38466
            21.24.19  Commutative rings   ccm2 38495
            21.24.20  Ideals   cidl 38513
            21.24.21  Prime rings and integral domains   cprrng 38552
            21.24.22  Ideal generators   cigen 38565
      ​21.25  Mathbox for Giovanni Mascellani
            ​*21.25.1  Tools for automatic proof building   efald2 38584
            ​*21.25.2  Tseitin axioms   fald 38635
            ​*21.25.3  Equality deductions   iuneq2f 38662
            ​*21.25.4  Miscellanea   orcomdd 38673
      ​21.26  Mathbox for Peter Mazsa
            21.26.1  Notations   cxrn 38680
            21.26.2  Preparatory theorems   el2v1 38735
            21.26.3  Range Cartesian product   df-xrn 38886
            21.26.4  Relations   df-rels 38946
            21.26.5  Quotient map (coset map)   df-qmap 38952
            21.26.6  Lifts, shifts, successor, and predecessor   df-adjliftmap 38961
            21.26.7  Cosets by ` R `   df-coss 39007
            21.26.8  Subset relations   df-ssr 39084
            21.26.9  Reflexivity   df-refs 39096
            21.26.10  Converse reflexivity   df-cnvrefs 39111
            21.26.11  Symmetry   df-syms 39128
            21.26.12  Reflexivity and symmetry   symrefref2 39153
            21.26.13  Transitivity   df-trs 39162
            21.26.14  Equivalence relations   df-eqvrels 39174
            21.26.15  Redundancy   df-redunds 39213
            21.26.16  Domain quotients   df-dmqss 39228
            21.26.17  Equivalence relations on domain quotients   df-ers 39254
            21.26.18  Functions   df-funss 39271
            21.26.19  Disjoints vs. converse functions   df-disjss 39294
            21.26.20  Antisymmetry   df-antisymrel 39369
            21.26.21  Partitions: disjoints on domain quotients   df-parts 39374
            21.26.22  Partition-Equivalence Theorems   disjim 39390
            21.26.23  Type-safe Partition-Equivalence: PetParts, PetErs, Pet2Parts, Pet2Ers   df-petparts 39474
      ​21.27  Mathbox for Rodolfo Medina
            21.27.1  Partitions   prtlem60 39484
      ​​*21.28  Mathbox for Norm Megill
            *21.28.1  Obsolete schemes ax-c4,c5,c7,c10,c11,c11n,c15,c9,c14,c16   ax-c5 39514
            ​*21.28.2  Rederive new axioms ax-4, ax-10, ax-6, ax-12, ax-13 from old   axc5 39524
            ​*21.28.3  Legacy theorems using obsolete axioms   ax5ALT 39538
            21.28.4  Experiments with weak deduction theorem   elimhyps 39592
            21.28.5  Miscellanea   cnaddcom 39603
            21.28.6  Atoms, hyperplanes, and covering in a left vector space (or module)   clsa 39605
            21.28.7  Functionals and kernels of a left vector space (or module)   clfn 39688
            21.28.8  Opposite rings and dual vector spaces   cld 39754
            21.28.9  Ortholattices and orthomodular lattices   cops 39803
            21.28.10  Atomic lattices with covering property   ccvr 39893
            21.28.11  Hilbert lattices   chlt 39981
            21.28.12  Projective geometries based on Hilbert lattices   clln 40122
            21.28.13  Construction of a vector space from a Hilbert lattice   cdlema1N 40422
            21.28.14  Construction of involution and inner product from a Hilbert lattice   clpoN 42111
      ​21.29  Mathbox for metakunt
            21.29.1  Commutative Semiring   ccsrg 42593
            21.29.2  General helpful statements   rhmzrhval 42596
            21.29.3  Some gcd and lcm results   12gcd5e1 42627
            21.29.4  Least common multiple inequality theorem   3factsumint1 42645
            21.29.5  Logarithm inequalities   3exp7 42677
            21.29.6  Miscellaneous results for AKS formalisation   intlewftc 42685
            21.29.7  Sticks and stones   sticksstones1 42770
            21.29.8  Continuation AKS   aks6d1c6lem1 42794
      ​21.30  Mathbox for Steven Nguyen
            21.30.1  Utility theorems   jarrii 42829
            ​*21.30.2  Arithmetic theorems   c0exALT 42875
            21.30.3  Exponents and divisibility   oexpreposd 42938
            21.30.4  Trigonometry and Calculus   tanhalfpim 42965
            ​*21.30.5  Independence of ax-mulcom   cresub 42981
            21.30.6  Structures   sn-base0 43124
            ​*21.30.7  Projective spaces   cprjsp 43190
            21.30.8  Basic reductions for Fermat's Last Theorem   dffltz 43223
            ​*21.30.9  Exemplar theorems   iddii 43253
                  ​*21.30.9.1  Standard replacements of ax-10 , ax-11 , ax-12   nfa1w 43264
      ​21.31  Mathbox for Igor Ieskov
      ​21.32  Mathbox for OpenAI
      ​21.33  Mathbox for Stefan O'Rear
            21.33.1  Additional elementary logic and set theory   moxfr 43280
            21.33.2  Additional theory of functions   imaiinfv 43281
            21.33.3  Additional topology   elrfi 43282
            21.33.4  Characterization of closure operators. Kuratowski closure axioms   ismrcd1 43286
            21.33.5  Algebraic closure systems   cnacs 43290
            21.33.6  Miscellanea 1. Map utilities   constmap 43301
            21.33.7  Miscellanea for polynomials   mptfcl 43308
            21.33.8  Multivariate polynomials over the integers   cmzpcl 43309
            21.33.9  Miscellanea for Diophantine sets 1   coeq0i 43341
            21.33.10  Diophantine sets 1: definitions   cdioph 43343
            21.33.11  Diophantine sets 2 miscellanea   ellz1 43355
            21.33.12  Diophantine sets 2: union and intersection. Monotone Boolean algebra   diophin 43360
            21.33.13  Diophantine sets 3: construction   diophrex 43363
            21.33.14  Diophantine sets 4 miscellanea   2sbcrex 43372
            21.33.15  Diophantine sets 4: Quantification   rexrabdioph 43378
            21.33.16  Diophantine sets 5: Arithmetic sets   rabdiophlem1 43385
            21.33.17  Diophantine sets 6: reusability. renumbering of variables   eldioph4b 43395
            21.33.18  Pigeonhole Principle and cardinality helpers   fphpd 43400
            21.33.19  A non-closed set of reals is infinite   rencldnfilem 43404
            21.33.20  Lagrange's rational approximation theorem   irrapxlem1 43406
            21.33.21  Pell equations 1: A nontrivial solution always exists   pellexlem1 43413
            21.33.22  Pell equations 2: Algebraic number theory of the solution set   csquarenn 43420
            21.33.23  Pell equations 3: characterizing fundamental solution   infmrgelbi 43462
            ​*21.33.24  Logarithm laws generalized to an arbitrary base   reglogcl 43474
            21.33.25  Pell equations 4: the positive solution group is infinite cyclic   pellfund14 43482
            21.33.26  X and Y sequences 1: Definition and recurrence laws   crmx 43484
            21.33.27  Ordering and induction lemmas for the integers   monotuz 43525
            21.33.28  X and Y sequences 2: Order properties   rmxypos 43531
            21.33.29  Congruential equations   congtr 43549
            21.33.30  Alternating congruential equations   acongid 43559
            21.33.31  Additional theorems on integer divisibility   coprmdvdsb 43569
            21.33.32  X and Y sequences 3: Divisibility properties   jm2.18 43572
            21.33.33  X and Y sequences 4: Diophantine representability of Y   jm2.27a 43589
            21.33.34  X and Y sequences 5: Diophantine representability of X, ^, _C   rmxdiophlem 43599
            21.33.35  Uncategorized stuff not associated with a major project   setindtr 43608
            21.33.36  More equivalents of the Axiom of Choice   axac10 43617
            21.33.37  Finitely generated left modules   clfig 43651
            21.33.38  Noetherian left modules I   clnm 43659
            21.33.39  Addenda for structure powers   pwssplit4 43673
            21.33.40  Every set admits a group structure iff choice   unxpwdom3 43679
            21.33.41  Noetherian rings and left modules II   clnr 43693
            21.33.42  Hilbert's Basis Theorem   cldgis 43705
            21.33.43  Additional material on polynomials [DEPRECATED]   cmnc 43715
            21.33.44  Degree and minimal polynomial of algebraic numbers   cdgraa 43724
            21.33.45  Algebraic integers I   citgo 43741
            21.33.46  Endomorphism algebra   cmend 43755
            21.33.47  Cyclic groups and order   idomodle 43775
            21.33.48  Cyclotomic polynomials   ccytp 43781
            21.33.49  Miscellaneous topology   fgraphopab 43787
      ​21.34  Mathbox for Noam Pasman
      ​21.35  Mathbox for Jon Pennant
      ​21.36  Mathbox for Richard Penner
            21.36.1  Set Theory and Ordinal Numbers   uniel 43801
            21.36.2  Natural addition of Cantor normal forms   oawordex2 43910
            21.36.3  Surreal Contributions   abeqabi 43991
            21.36.4  Short Studies   nlimsuc 44024
                  21.36.4.1  Additional work on conditional logical operator   ifpan123g 44042
                  21.36.4.2  Sophisms   rp-fakeimass 44095
                  ​*21.36.4.3  Finite Sets   rp-isfinite5 44100
                  21.36.4.4  General Observations   intabssd 44102
                  21.36.4.5  Infinite Sets   pwelg 44143
                  ​*21.36.4.6  Finite intersection property   fipjust 44148
                  21.36.4.7  RP ADDTO: Subclasses and subsets   rababg 44157
                  21.36.4.8  RP ADDTO: The intersection of a class   elinintab 44158
                  21.36.4.9  RP ADDTO: Theorems requiring subset and intersection existence   elinintrab 44160
                  21.36.4.10  RP ADDTO: Relations   xpinintabd 44163
                  ​*21.36.4.11  RP ADDTO: Functions   elmapintab 44179
                  ​*21.36.4.12  RP ADDTO: Finite induction (for finite ordinals)   cnvcnvintabd 44183
                  21.36.4.13  RP ADDTO: First and second members of an ordered pair   elcnvlem 44184
                  21.36.4.14  RP ADDTO: The reflexive and transitive properties of relations   undmrnresiss 44187
                  21.36.4.15  RP ADDTO: Basic properties of closures   cleq2lem 44191
                  21.36.4.16  RP REPLACE: Definitions and basic properties of transitive closures   trcleq2lemRP 44213
                  ​*21.36.4.17  Additions for square root; absolute value   sqrtcvallem1 44214
            21.36.5  Additional statements on relations and subclasses   al3im 44230
                  21.36.5.1  Transitive relations (not to be confused with transitive classes)   trrelind 44248
                  21.36.5.2  Reflexive closures   crcl 44255
                  ​*21.36.5.3  Finite relationship composition   relexp2 44260
                  21.36.5.4  Transitive closure of a relation   dftrcl3 44303
                  ​*21.36.5.5  Adapted from Frege   frege77d 44329
            ​*21.36.6  Propositions from _Begriffsschrift_   dfxor4 44349
                  *21.36.6.1  _Begriffsschrift_ Chapter I   dfxor4 44349
                  ​*21.36.6.2  _Begriffsschrift_ Notation hints   whe 44355
                  21.36.6.3  _Begriffsschrift_ Chapter II Implication   ax-frege1 44373
                  21.36.6.4  _Begriffsschrift_ Chapter II Implication and Negation   axfrege28 44412
                  ​*21.36.6.5  _Begriffsschrift_ Chapter II with logical equivalence   axfrege52a 44439
                  21.36.6.6  _Begriffsschrift_ Chapter II with equivalence of sets   axfrege52c 44470
                  ​*21.36.6.7  _Begriffsschrift_ Chapter II with equivalence of classes   frege53c 44497
                  ​*21.36.6.8  _Begriffsschrift_ Chapter III Properties hereditary in a sequence   dffrege69 44515
                  ​*21.36.6.9  _Begriffsschrift_ Chapter III Following in a sequence   dffrege76 44522
                  ​*21.36.6.10  _Begriffsschrift_ Chapter III Member of sequence   dffrege99 44545
                  ​*21.36.6.11  _Begriffsschrift_ Chapter III Single-valued procedures   dffrege115 44561
            ​*21.36.7  Exploring Topology via Seifert and Threlfall   enrelmap 44580
                  *21.36.7.1  Equinumerosity of sets of relations and maps   enrelmap 44580
                  ​*21.36.7.2  Generic Pseudoclosure Spaces, Pseudointerior Spaces, and Pseudoneighborhoods   or3or 44606
                  ​*21.36.7.3  Generic Neighborhood Spaces   gneispa 44713
            ​*21.36.8  Exploring Higher Homotopy via Kerodon   k0004lem1 44730
                  *21.36.8.1  Simplicial Sets   k0004lem1 44730
      ​21.37  Mathbox for Stanislas Polu
            21.37.1  IMO Problems   wwlemuld 44739
                  21.37.1.1  IMO 1972 B2   wwlemuld 44739
            ​*21.37.2  INT Inequalities Proof Generator   int-addcomd 44756
            ​*21.37.3  N-Digit Addition Proof Generator   unitadd 44778
            21.37.4  AM-GM (for k = 2,3,4)   gsumws3 44779
      ​21.38  Mathbox for Rohan Ridenour
            21.38.1  Misc   spALT 44784
            21.38.2  Monoid rings   cmnring 44794
            21.38.3  Shorter primitive equivalent of ax-groth   gru0eld 44812
                  21.38.3.1  Grothendieck universes are closed under collection   gru0eld 44812
                  21.38.3.2  Minimal universes   ismnu 44830
                  21.38.3.3  Primitive equivalent of ax-groth   expandan 44857
      ​21.39  Mathbox for Steve Rodriguez
            21.39.1  Miscellanea   nanorxor 44874
            21.39.2  Ratio test for infinite series convergence and divergence   dvgrat 44881
            21.39.3  Multiples   reldvds 44884
            21.39.4  Function operations   caofcan 44892
            21.39.5  Calculus   lhe4.4ex1a 44898
            21.39.6  The generalized binomial coefficient operation   cbcc 44905
            21.39.7  Binomial series   uzmptshftfval 44915
      ​21.40  Mathbox for Andrew Salmon
            21.40.1  Principia Mathematica * 10   pm10.12 44927
            21.40.2  Principia Mathematica * 11   2alanimi 44941
            21.40.3  Predicate Calculus   sbeqal1 44967
            21.40.4  Principia Mathematica * 13 and * 14   pm13.13a 44976
            21.40.5  Set Theory   elnev 45006
            21.40.6  Arithmetic   addcomgi 45023
            21.40.7  Geometry   cplusr 45024
      ​​*21.41  Mathbox for Alan Sare
            21.41.1  Auxiliary theorems for the Virtual Deduction tool   idiALT 45046
            21.41.2  Supplementary unification deductions   bi1imp 45050
            21.41.3  Conventional Metamath proofs, some derived from VD proofs   iidn3 45069
            21.41.4  What is Virtual Deduction?   wvd1 45137
            21.41.5  Virtual Deduction Theorems   df-vd1 45138
            21.41.6  Theorems proved using Virtual Deduction   trsspwALT 45385
            21.41.7  Theorems proved using Virtual Deduction with mmj2 assistance   simplbi2VD 45413
            21.41.8  Virtual Deduction transcriptions of textbook proofs   sb5ALTVD 45480
            21.41.9  Theorems proved using conjunction-form Virtual Deduction   elpwgdedVD 45484
            21.41.10  Theorems with a VD proof in conventional notation derived from a VD proof   suctrALT3 45491
            ​*21.41.11  Theorems with a proof in conventional notation derived from a VD proof   notnotrALT2 45494
      ​21.42  Mathbox for Eric Schmidt
            21.42.1  Miscellany   rspesbcd 45505
            21.42.2  Study of dfbi1ALT   dfbi1ALTa 45507
            21.42.3  Relation-preserving functions   wrelp 45510
            21.42.4  Orbits   orbitex 45523
            21.42.5  Well-founded sets   trwf 45527
            21.42.6  Absoluteness in transitive models   ralabso 45536
            21.42.7  Lemmas for showing axioms hold in models   traxext 45545
            21.42.8  The class of well-founded sets is a model for ZFC   wfaxext 45561
            21.42.9  Permutation models   brpermmodel 45571
            21.42.10  Isomorphism of finite ordinals and non-negative integers   hashnna 45587
      ​21.43  Mathbox for Glauco Siliprandi
            21.43.1  Miscellanea   evth2f 45594
            21.43.2  Functions   fnresdmss 45745
            21.43.3  Ordering on real numbers - Real and complex numbers basic operations   sub2times 45851
            21.43.4  Real intervals   gtnelioc 46066
            21.43.5  Finite sums   fsummulc1f 46146
            21.43.6  Finite multiplication of numbers and finite multiplication of functions   fmul01 46155
            21.43.7  Limits   clim1fr1 46176
                  21.43.7.1  Inferior limit (lim inf)   clsi 46324
                  ​*21.43.7.2  Limits for sequences of extended real numbers   clsxlim 46391
            21.43.8  Trigonometry   coseq0 46437
            21.43.9  Continuous Functions   mulcncff 46443
            21.43.10  Derivatives   dvsinexp 46484
            21.43.11  Integrals   itgsin0pilem1 46523
            21.43.12  Stone Weierstrass theorem - real version   stoweidlem1 46574
            21.43.13  Wallis' product for π   wallispilem1 46638
            21.43.14  Stirling's approximation formula for ` n ` factorial   stirlinglem1 46647
            21.43.15  Dirichlet kernel   dirkerval 46664
            21.43.16  Fourier Series   fourierdlem1 46681
            21.43.17  e is transcendental   elaa2lem 46806
            21.43.18  n-dimensional Euclidean space   rrxtopn 46857
            21.43.19  Basic measure theory   csalg 46881
                  ​*21.43.19.1  σ-Algebras   csalg 46881
                  21.43.19.2  Sum of nonnegative extended reals   csumge0 46935
                  ​*21.43.19.3  Measures   cmea 47022
                  ​*21.43.19.4  Outer measures and Caratheodory's construction   come 47062
                  ​*21.43.19.5  Lebesgue measure on n-dimensional Real numbers   covoln 47109
                  ​*21.43.19.6  Measurable functions   csmblfn 47268
      ​21.44  Mathbox for Saveliy Skresanov
            21.44.1  Ceva's theorem   sigarval 47423
            21.44.2  Simple groups   simpcntrab 47443
      ​21.45  Mathbox for Ender Ting
            21.45.1  Interesting facts   et-ltneverrefl 47444
            21.45.2  Increasing sequences and subsequences   ormklocald 47449
            21.45.3  Scratchpad for number theory   evenwodadd 47462
            21.45.4  Scratchpad for math on real numbers   squeezedltsq 47463
      ​21.46  Mathbox for Jarvin Udandy
      ​21.47  Mathbox for Adhemar
            ​*21.47.1  Minimal implicational calculus   adh-minim 47594
      ​21.48  Mathbox for Alexander van der Vekens
            21.48.1  General auxiliary theorems (1)   n0nsn2el 47618
                  21.48.1.1  Unordered and ordered pairs - extension for singletons   n0nsn2el 47618
                  21.48.1.2  Unordered and ordered pairs - extension for unordered pairs   elprneb 47622
                  21.48.1.3  Unordered and ordered pairs - extension for ordered pairs   oppr 47623
                  21.48.1.4  Relations - extension   eubrv 47628
                  21.48.1.5  Definite description binder (inverted iota) - extension   iota0def 47631
                  21.48.1.6  Functions - extension   fveqvfvv 47633
            21.48.2  Alternative for Russell's definition of a description binder   caiota 47676
            21.48.3  Double restricted existential uniqueness   r19.32 47691
                  21.48.3.1  Restricted quantification (extension)   r19.32 47691
                  21.48.3.2  Restricted uniqueness and "at most one" quantification   reuf1odnf 47700
                  21.48.3.3  Analogs to Existential uniqueness (double quantification)   2reu3 47703
                  21.48.3.4  Additional theorems for double restricted existential uniqueness   2reu8i 47706
            ​*21.48.4  Alternative definitions of function and operation values   wdfat 47709
                  21.48.4.1  Restricted quantification (extension)   ralbinrald 47715
                  21.48.4.2  The universal class (extension)   nvelim 47716
                  21.48.4.3  Introduce the Axiom of Power Sets (extension)   alneu 47717
                  21.48.4.4  Predicate "defined at"   dfateq12d 47719
                  21.48.4.5  Alternative definition of the value of a function   dfafv2 47725
                  21.48.4.6  Alternative definition of the value of an operation   aoveq123d 47771
            ​*21.48.5  Alternative definitions of function values (2)   cafv2 47801
            21.48.6  General auxiliary theorems (2)   an4com24 47861
                  21.48.6.1  Logical conjunction - extension   an4com24 47861
                  21.48.6.2  Abbreviated conjunction and disjunction of three wff's - extension   3an4ancom24 47862
                  21.48.6.3  Negated membership (alternative)   cnelbr 47864
                  21.48.6.4  The empty set - extension   ralralimp 47871
                  21.48.6.5  Indexed union and intersection - extension   otiunsndisjX 47872
                  21.48.6.6  Functions - extension   fvifeq 47873
                  21.48.6.7  Maps-to notation - extension   fvmptrab 47885
                  21.48.6.8  Subtraction - extension   cnambpcma 47887
                  21.48.6.9  Ordering on reals (cont.) - extension   leaddsuble 47890
                  21.48.6.10  Imaginary and complex number properties - extension   readdcnnred 47896
                  21.48.6.11  Nonnegative integers (as a subset of complex numbers) - extension   nn0resubcl 47901
                  21.48.6.12  Integers (as a subset of complex numbers) - extension   zgeltp1eq 47902
                  21.48.6.13  Decimal arithmetic - extension   1t10e1p1e11 47903
                  21.48.6.14  Upper sets of integers - extension   eluzge0nn0 47905
                  21.48.6.15  Infinity and the extended real number system (cont.) - extension   nltle2tri 47906
                  21.48.6.16  Finite intervals of integers - extension   ssfz12 47907
                  21.48.6.17  Half-open integer ranges - extension   fzopred 47916
                  21.48.6.18  The floor and ceiling functions - extension   2ltceilhalf 47925
                  21.48.6.19  The modulo (remainder) operation - extension   fldivmod 47937
                  21.48.6.20  The infinite sequence builder "seq"   smonoord 47970
                  21.48.6.21  Integer powers - extension   2timesltsq 47971
                  21.48.6.22  Finite and infinite sums - extension   fsummsndifre 47973
                  21.48.6.23  The divides relation - extension   nndivides2 47977
                  21.48.6.24  Extensible structures - extension   setsidel 47981
            ​*21.48.7  Preimages of function values   preimafvsnel 47984
            ​*21.48.8  Partitions of real intervals   ciccp 48018
            21.48.9  Shifting functions with an integer range domain   fargshiftfv 48044
            21.48.10  Words over a set (extension)   lswn0 48049
                  21.48.10.1  Last symbol of a word - extension   lswn0 48049
            21.48.11  Unordered pairs   wich 48050
                  21.48.11.1  Interchangeable setvar variables   wich 48050
                  21.48.11.2  Set of unordered pairs   sprid 48079
                  ​*21.48.11.3  Proper (unordered) pairs   prpair 48106
                  21.48.11.4  Set of proper unordered pairs   cprpr 48117
            21.48.12  Number theory (extension)   nprmmul1 48132
                  21.48.12.1  Properties of non-prime numbers   nprmmul1 48132
                  ​*21.48.12.2  Fermat numbers   cfmtno 48135
                  ​*21.48.12.3  Mersenne primes   m2prm 48199
                  21.48.12.4  Proth's theorem   modexp2m1d 48220
                  21.48.12.5  The prime-counting function according to Ján Mináč   nprmdvdsfacm1lem1 48228
                  21.48.12.6  Solutions of quadratic equations   quad1 48241
            ​*21.48.13  Even and odd numbers   ceven 48245
                  21.48.13.1  Definitions and basic properties   ceven 48245
                  21.48.13.2  Alternate definitions using the "divides" relation   dfeven2 48270
                  21.48.13.3  Alternate definitions using the "modulo" operation   dfeven3 48279
                  21.48.13.4  Alternate definitions using the "gcd" operation   iseven5 48285
                  21.48.13.5  Theorems of part 5 revised   zneoALTV 48290
                  21.48.13.6  Theorems of part 6 revised   odd2np1ALTV 48295
                  21.48.13.7  Theorems of AV's mathbox revised   0evenALTV 48309
                  21.48.13.8  Additional theorems   epoo 48324
                  21.48.13.9  Perfect Number Theorem (revised)   perfectALTVlem1 48342
            21.48.14  Number theory (extension 2)   cfppr 48345
                  ​*21.48.14.1  Fermat pseudoprimes   cfppr 48345
                  ​*21.48.14.2  Goldbach's conjectures   cgbe 48366
            21.48.15  Graph theory (extension)   cclnbgr 48439
                  21.48.15.1  Closed neighborhood of a vertex   cclnbgr 48439
                  ​*21.48.15.2  Semiclosed and semiopen neighborhoods (experimental)   dfsclnbgr2 48467
                  21.48.15.3  Induced subgraphs   cisubgr 48481
                  ​*21.48.15.4  Isomorphisms of graphs   cgrisom 48495
                  ​*21.48.15.5  Triangles in graphs   cgrtri 48558
                  ​*21.48.15.6  Star graphs   cstgr 48572
                  ​*21.48.15.7  Local isomorphisms of graphs   cgrlim 48597
                  ​*21.48.15.8  Generalized Petersen graphs   cgpg 48661
                  21.48.15.9  Loop-free graphs - extension   1hegrlfgr 48753
                  21.48.15.10  Walks - extension   cupwlks 48754
                  21.48.15.11  Edges of graphs expressed as sets of unordered pairs   upgredgssspr 48764
            21.48.16  Monoids (extension)   ovn0dmfun 48777
                  21.48.16.1  Auxiliary theorems   ovn0dmfun 48777
                  21.48.16.2  Magmas, Semigroups and Monoids (extension)   plusfreseq 48785
                  21.48.16.3  Examples and counterexamples for magmas, semigroups and monoids (extension)   opmpoismgm 48788
                  21.48.16.4  Group sum operation (extension 1)   gsumsplit2f 48801
            ​*21.48.17  Magmas and internal binary operations (alternate approach)   ccllaw 48804
                  *21.48.17.1  Laws for internal binary operations   ccllaw 48804
                  ​*21.48.17.2  Internal binary operations   cintop 48817
                  21.48.17.3  Alternative definitions for magmas and semigroups   cmgm2 48836
            21.48.18  Rings (extension)   lmod0rng 48850
                  21.48.18.1  Nonzero rings (extension)   lmod0rng 48850
                  21.48.18.2  Ideals as non-unital rings   lidldomn1 48852
                  21.48.18.3  The non-unital ring of even integers   0even 48858
                  21.48.18.4  A constructed not unital ring   cznrnglem 48880
                  ​*21.48.18.5  The category of non-unital rings (alternate definition)   crngcALTV 48884
                  ​*21.48.18.6  The category of (unital) rings (alternate definition)   cringcALTV 48908
            ​*21.48.19  Prime rings (and integral domains)   cprmrng 48955
            21.48.20  Basic algebraic structures (extension)   eliunxp2 48966
                  21.48.20.1  Auxiliary theorems   eliunxp2 48966
                  21.48.20.2  The binomial coefficient operation (extension)   bcpascm1 48983
                  21.48.20.3  The ` ZZ `-module ` ZZ X. ZZ `   zlmodzxzlmod 48986
                  21.48.20.4  Group sum operation (extension 2)   mgpsumunsn 48993
                  21.48.20.5  Symmetric groups (extension)   exple2lt6 48996
                  21.48.20.6  Divisibility (extension)   invginvrid 48999
                  21.48.20.7  The support of functions (extension)   rmsupp0 49000
                  21.48.20.8  Finitely supported functions (extension)   rmsuppfi 49004
                  21.48.20.9  Left modules (extension)   lmodvsmdi 49011
                  21.48.20.10  Associative algebras (extension)   assaascl0 49013
                  21.48.20.11  Univariate polynomials (extension)   ply1vr1smo 49015
                  21.48.20.12  Univariate polynomials (examples)   linply1 49025
            21.48.21  Linear algebra (extension)   cdmatalt 49028
                  ​*21.48.21.1  The subalgebras of diagonal and scalar matrices (extension)   cdmatalt 49028
                  ​*21.48.21.2  Linear combinations   clinc 49036
                  ​*21.48.21.3  Linear independence   clininds 49072
                  21.48.21.4  Simple left modules and the ` ZZ `-module   lmod1lem1 49119
                  21.48.21.5  Differences between (left) modules and (left) vector spaces   lvecpsslmod 49139
            21.48.22  Complexity theory   suppdm 49142
                  21.48.22.1  Auxiliary theorems   suppdm 49142
                  21.48.22.2  Even and odd integers   nn0onn0ex 49155
                  21.48.22.3  The natural logarithm on complex numbers (extension)   logcxp0 49167
                  21.48.22.4  Division of functions   cfdiv 49169
                  21.48.22.5  Upper bounds   cbigo 49179
                  21.48.22.6  Logarithm to an arbitrary base (extension)   rege1logbrege0 49190
                  ​*21.48.22.7  The binary logarithm   fldivexpfllog2 49197
                  21.48.22.8  Binary length   cblen 49201
                  ​*21.48.22.9  Digits   cdig 49227
                  21.48.22.10  Nonnegative integer as sum of its shifted digits   dignn0flhalflem1 49247
                  21.48.22.11  Algorithms for the multiplication of nonnegative integers   nn0mulfsum 49256
                  ​*21.48.22.12  N-ary functions   cnaryf 49258
                  ​*21.48.22.13  The Ackermann function   citco 49289
            21.48.23  Elementary geometry (extension)   fv1prop 49331
                  21.48.23.1  Auxiliary theorems   fv1prop 49331
                  21.48.23.2  Real euclidean space of dimension 2   rrx2pxel 49343
                  21.48.23.3  Spheres and lines in real Euclidean spaces   cline 49359
      ​21.49  Mathbox for Zhi Wang
            21.49.1  Propositional calculus   logic1 49421
            21.49.2  Predicate calculus with equality   dtrucor3 49429
                  21.49.2.1  Axiom scheme ax-5 (Distinctness)   dtrucor3 49429
            21.49.3  ZF Set Theory - start with the Axiom of Extensionality   ralbidb 49430
                  21.49.3.1  Restricted quantification   ralbidb 49430
                  21.49.3.2  The universal class   reuxfr1dd 49437
                  21.49.3.3  The empty set   ssdisjd 49438
                  21.49.3.4  Unordered and ordered pairs   vsn 49442
                  21.49.3.5  The union of a class   unilbss 49448
                  21.49.3.6  Indexed union and intersection   iuneq0 49449
            21.49.4  ZF Set Theory - add the Axiom of Replacement   inpw 49455
                  21.49.4.1  Theorems requiring subset and intersection existence   inpw 49455
            21.49.5  ZF Set Theory - add the Axiom of Power Sets   opth1neg 49456
                  21.49.5.1  Ordered pair theorem   opth1neg 49456
                  21.49.5.2  Ordered-pair class abstractions (cont.)   brab2dd 49458
                  21.49.5.3  Relations   iinxp 49461
                  21.49.5.4  Functions   mof0 49468
                  21.49.5.5  Operations   ovsng 49488
            21.49.6  ZF Set Theory - add the Axiom of Union   fonex 49497
                  21.49.6.1  Relations and functions (cont.)   fonex 49497
                  21.49.6.2  First and second members of an ordered pair   eloprab1st2nd 49498
                  21.49.6.3  Operations in maps-to notation (continued)   fmpodg 49499
                  21.49.6.4  Function transposition   resinsnlem 49501
                  21.49.6.5  Infinite Cartesian products   ixpv 49520
                  21.49.6.6  Equinumerosity   fvconst0ci 49521
            21.49.7  Order sets   iccin 49526
                  21.49.7.1  Real number intervals   iccin 49526
            21.49.8  Extensible structures   slotresfo 49529
                  21.49.8.1  Basic definitions   slotresfo 49529
            21.49.9  Moore spaces   mreuniss 49530
            ​*21.49.10  Topology   clduni 49531
                  21.49.10.1  Closure and interior   clduni 49531
                  21.49.10.2  Neighborhoods   neircl 49535
                  21.49.10.3  Subspace topologies   restcls2lem 49543
                  21.49.10.4  Limits and continuity in topological spaces   cnneiima 49547
                  21.49.10.5  Topological definitions using the reals   iooii 49548
                  21.49.10.6  Separated sets   sepnsepolem1 49552
                  21.49.10.7  Separated spaces: T0, T1, T2 (Hausdorff) ...   isnrm4 49561
            21.49.11  Preordered sets and directed sets using extensible structures   isprsd 49585
            21.49.12  Posets and lattices using extensible structures   lubeldm2 49586
                  21.49.12.1  Posets   lubeldm2 49586
                  21.49.12.2  Lattices   toslat 49612
                  21.49.12.3  Subset order structures   intubeu 49614
            21.49.13  Rings   elmgpcntrd 49635
                  21.49.13.1  Multiplicative Group   elmgpcntrd 49635
            21.49.14  Associative algebras   asclelbasALT 49636
                  21.49.14.1  Definition and basic properties   asclelbasALT 49636
            21.49.15  Categories   homf0 49639
                  21.49.15.1  Categories   homf0 49639
                  21.49.15.2  Opposite category   oppccatb 49646
                  21.49.15.3  Monomorphisms and epimorphisms   idmon 49650
                  21.49.15.4  Sections, inverses, isomorphisms   sectrcl 49652
                  21.49.15.5  Isomorphic objects   cicfn 49672
                  21.49.15.6  Subcategories   dmdm 49683
                  21.49.15.7  Functors   reldmfunc 49705
                  21.49.15.8  Opposite functors   coppf 49752
                  21.49.15.9  Full & faithful functors   imasubc 49781
                  21.49.15.10  Universal property   upciclem1 49796
                  21.49.15.11  Natural transformations and the functor category   isnatd 49853
                  21.49.15.12  Initial, terminal and zero objects of a category   initoo2 49862
                  21.49.15.13  Product of categories   reldmxpc 49876
                  21.49.15.14  Swap functors   cswapf 49889
                  21.49.15.15  Functor evaluation   oppc1stflem 49917
                  21.49.15.16  Transposed curry functors   cofuswapfcl 49923
                  21.49.15.17  Constant functors   diag1 49934
                  21.49.15.18  Functor composition bifunctors   fucofulem1 49940
                  21.49.15.19  Post-composition functors   postcofval 49994
                  21.49.15.20  Pre-composition functors   precofvallem 49996
            21.49.16  Examples of categories   catcrcl 50025
                  21.49.16.1  The category of categories   catcrcl 50025
                  21.49.16.2  Thin categories   cthinc 50047
                  21.49.16.3  Terminal categories   ctermc 50102
                  21.49.16.4  Preordered sets as thin categories   cprstc 50179
                  21.49.16.5  Monoids as categories   cmndtc 50207
                  21.49.16.6  Categories with at most one object and at most two morphisms   2arwcatlem1 50225
            21.49.17  Kan extensions and related concepts   clan 50235
                  21.49.17.1  Kan extensions   clan 50235
                  21.49.17.2  Limits and colimits   clmd 50273
      ​21.50  Mathbox for Emmett Weisz
            ​*21.50.1  Miscellaneous Theorems   nfintd 50303
            21.50.2  Set Recursion   csetrecs 50313
                  ​*21.50.2.1  Basic Properties of Set Recursion   csetrecs 50313
                  21.50.2.2  Examples and properties of set recursion   elsetrecslem 50329
            ​*21.50.3  Construction of Games and Surreal Numbers   cpg 50339
      ​​*21.51  Mathbox for David A. Wheeler
            21.51.1  Natural deduction   sbidd 50348
            ​*21.51.2  Greater than, greater than or equal to   cge-real 50350
            ​*21.51.3  Hyperbolic trigonometric functions   csinh 50360
            ​*21.51.4  Reciprocal trigonometric functions (sec, csc, cot)   csec 50371
            ​*21.51.5  Identities for "if"   ifnmfalse 50393
            ​*21.51.6  Logarithms generalized to arbitrary base using ` logb `   logb2aval 50394
            ​*21.51.7  Logarithm laws generalized to an arbitrary base - log_   clog- 50395
            ​*21.51.8  Formally define notions such as reflexivity   wreflexive 50397
            ​*21.51.9  Algebra helpers   mvlraddi 50401
            ​*21.51.10  Algebra helper examples   i2linesi 50408
            ​*21.51.11  Formal methods "surprises"   alimp-surprise 50410
            ​*21.51.12  Allsome quantifier   walsi 50416
            ​*21.51.13  Miscellaneous   5m4e1 50427
            21.51.14  Theorems about algebraic numbers   aacllem 50431
      ​21.52  Mathbox for Kunhao Zheng
            21.52.1  Weighted AM-GM inequality   amgmwlem 50432