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 4868
            2.1.20  The intersection of a class   cint 4908
            2.1.21  Indexed union and intersection   ciun 4952
            2.1.22  Disjointness   wdisj 5072
            2.1.23  Binary relations   wbr 5105
            2.1.24  Ordered-pair class abstractions (class builders)   copab 5167
            2.1.25  Functions in maps-to notation   cmpt 5186
            2.1.26  Transitive classes   wtr 5212
      ​2.2  ZF Set Theory - add the Axiom of Replacement
            2.2.1  Introduce the Axiom of Replacement   ax-rep 5232
            2.2.2  Derive the Axiom of Separation   axsepgfromrep 5249
            2.2.3  Derive the Null Set Axiom   axnulALT 5259
            2.2.4  Theorems requiring subset and intersection existence   exnelv 5268
            2.2.5  Theorems requiring empty set existence   class2set 5316
      ​2.3  ZF Set Theory - add the Axiom of Power Sets
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 5327
            2.3.2  Derive the Axiom of Pairing   axprlem1 5385
            2.3.3  Ordered pair theorem   opnz 5446
            2.3.4  Ordered-pair class abstractions (cont.)   opabidw 5499
            2.3.5  Power class of union and intersection   pwin 5543
            2.3.6  The identity relation   cid 5546
            2.3.7  The membership relation (or epsilon relation)   cep 5551
            ​*2.3.8  Partial and total orderings   wpo 5558
            2.3.9  Founded and well-ordering relations   wfr 5602
            2.3.10  Relations   cxp 5650
            2.3.11  The Predecessor Class   cpred 6291
            2.3.12  Well-founded induction (variant)   frpomin 6331
            2.3.13  Well-ordered induction   tz6.26 6338
            2.3.14  Ordinals   word 6349
            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 12044
            5.3.8  Completeness Axiom and Suprema   fimaxre 12150
            5.3.9  Imaginary and complex number properties   neg1cn 12194
            5.3.10  Function operation analogue theorems   ofsubeq0 12206
            ​*5.3.11  Indicator Functions   cind 12209
      ​5.4  Integer sets
            5.4.1  Positive integers (as a subset of complex numbers)   cn 12224
            5.4.2  Principle of mathematical induction   nnind 12242
            ​*5.4.3  Decimal representation of numbers   c2 12286
            ​*5.4.4  Some properties of specific numbers   1pneg1e0 12349
            5.4.5  Simple number properties   halfcl 12461
            5.4.6  The Archimedean property   nnunb 12491
            5.4.7  Nonnegative integers (as a subset of complex numbers)   cn0 12495
            ​*5.4.8  Extended nonnegative integers   cxnn0 12568
            5.4.9  Integers (as a subset of complex numbers)   cz 12582
            5.4.10  Decimal arithmetic   cdc 12702
            5.4.11  Upper sets of integers   cuz 12853
            5.4.12  Well-ordering principle for bounded-below sets of integers   uzwo3 12958
            5.4.13  Rational numbers (as a subset of complex numbers)   cq 12963
            5.4.14  Existence of the set of complex numbers   rpnnen1lem2 12992
      ​5.5  Order sets
            5.5.1  Positive reals (as a subset of complex numbers)   crp 13007
            5.5.2  Infinity and the extended real number system (cont.)   cxne 13125
            5.5.3  Supremum and infimum on the extended reals   xrsupexmnf 13322
            5.5.4  Real number intervals   cioo 13363
            5.5.5  Finite intervals of integers   cfz 13526
            ​*5.5.6  Finite intervals of nonnegative integers   elfz2nn0 13637
            5.5.7  Half-open integer ranges   cfzo 13673
      ​5.6  Elementary integer functions
            5.6.1  The floor and ceiling functions   cfl 13814
            5.6.2  The modulo (remainder) operation   cmo 13893
            5.6.3  Miscellaneous theorems about integers   om2uz0i 13974
            5.6.4  Strong induction over upper sets of integers   uzsinds 14014
            5.6.5  Finitely supported functions over the nonnegative integers   fsuppmapnn0fiublem 14017
            5.6.6  The infinite sequence builder "seq" - extension   cseq 14028
            5.6.7  Integer powers   cexp 14088
            5.6.8  Ordered pair theorem for nonnegative integers   nn0le2msqi 14294
            5.6.9  Factorial function   cfa 14300
            5.6.10  The binomial coefficient operation   cbc 14329
            5.6.11  The ` # ` (set size) function   chash 14357
                  5.6.11.1  Proper unordered pairs and triples (sets of size 2 and 3)   hashprlei 14495
                  5.6.11.2  Functions with a domain containing at least two different elements   fundmge2nop0 14529
                  5.6.11.3  Finite induction on the size of the first component of a binary relation   hashdifsnp1 14533
      ​​*5.7  Words over a set
            5.7.1  Definitions and basic theorems   cword 14540
            5.7.2  Last symbol of a word   clsw 14589
            5.7.3  Concatenations of words   cconcat 14597
            5.7.4  Singleton words   cs1 14623
            5.7.5  Concatenations with singleton words   ccatws1cl 14644
            5.7.6  Subwords/substrings   csubstr 14668
            5.7.7  Prefixes of a word   cpfx 14698
            5.7.8  Subwords of subwords   swrdswrdlem 14731
            5.7.9  Subwords and concatenations   pfxcctswrd 14737
            5.7.10  Subwords of concatenations   swrdccatfn 14751
            5.7.11  Splicing words (substring replacement)   csplice 14776
            5.7.12  Reversing words   creverse 14785
            5.7.13  Repeated symbol words   creps 14795
            ​*5.7.14  Cyclical shifts of words   ccsh 14815
            5.7.15  Mapping words by a function   wrdco 14858
            5.7.16  Longer string literals   cs2 14868
      ​​*5.8  Reflexive and transitive closures of relations
            5.8.1  The reflexive and transitive properties of relations   coss12d 14999
            5.8.2  Basic properties of closures   cleq1lem 15009
            5.8.3  Definitions and basic properties of transitive closures   ctcl 15012
            5.8.4  Exponentiation of relations   crelexp 15046
            5.8.5  Reflexive-transitive closure as an indexed union   crtrcl 15082
            ​*5.8.6  Principle of transitive induction   relexpindlem 15090
      ​5.9  Elementary real and complex functions
            5.9.1  The "shift" operation   cshi 15093
            5.9.2  Signum (sgn or sign) function   csgn 15113
            5.9.3  Real and imaginary parts; conjugate   ccj 15137
            5.9.4  Square root; absolute value   csqrt 15274
      ​5.10  Elementary limits and convergence
            5.10.1  Superior limit (lim sup)   clsp 15511
            5.10.2  Limits   cli 15525
            5.10.3  Finite and infinite sums   csu 15727
            5.10.4  The binomial theorem   binomlem 15873
            5.10.5  The inclusion/exclusion principle   incexclem 15880
            5.10.6  Infinite sums (cont.)   isumshft 15883
            5.10.7  Miscellaneous converging and diverging sequences   divrcnv 15896
            5.10.8  Arithmetic series   arisum 15904
            5.10.9  Geometric series   expcnv 15908
            5.10.10  Ratio test for infinite series convergence   cvgrat 15927
            5.10.11  Mertens' theorem   mertenslem1 15928
            5.10.12  Finite and infinite products   prodf 15931
                  5.10.12.1  Product sequences   prodf 15931
                  5.10.12.2  Non-trivial convergence   ntrivcvg 15941
                  5.10.12.3  Complex products   cprod 15947
                  5.10.12.4  Finite products   fprod 15985
                  5.10.12.5  Infinite products   iprodclim 16042
            5.10.13  Falling and Rising Factorial   cfallfac 16048
            5.10.14  Bernoulli polynomials and sums of k-th powers   cbp 16090
      ​5.11  Elementary trigonometry
            5.11.1  The exponential, sine, and cosine functions   ce 16105
                  5.11.1.1  The circle constant (tau = 2 pi)   ctau 16248
            5.11.2  _e is irrational   eirrlem 16250
      ​5.12  Cardinality of real and complex number subsets
            5.12.1  Countability of integers and rationals   xpnnen 16257
            5.12.2  The reals are uncountable   rpnnen2lem1 16260
​​*PART 6  ELEMENTARY NUMBER THEORY
      ​6.1  Elementary properties of divisibility
            6.1.1  Irrationality of square root of 2   sqrt2irrlem 16294
            6.1.2  Some Number sets are chains of proper subsets   nthruc 16298
            6.1.3  The divides relation   cdvds 16300
            ​*6.1.4  Even and odd numbers   evenelz 16384
            6.1.5  The division algorithm   divalglem0 16441
            6.1.6  Bit sequences   cbits 16467
            6.1.7  The greatest common divisor operator   cgcd 16542
            6.1.8  Bézout's identity   bezoutlem1 16587
            6.1.9  Algorithms   nn0seqcvgd 16618
            6.1.10  Euclid's Algorithm   eucalgval2 16629
            ​*6.1.11  The least common multiple   clcm 16636
            ​*6.1.12  Coprimality and Euclid's lemma   coprmgcdb 16697
            6.1.13  Cancellability of congruences   congr 16712
      ​6.2  Elementary prime number theory
            ​*6.2.1  Elementary properties   cprime 16719
            ​*6.2.2  Coprimality and Euclid's lemma (cont.)   coprm 16760
            6.2.3  Properties of the canonical representation of a rational   cnumer 16782
            6.2.4  Euler's theorem   codz 16812
            6.2.5  Arithmetic modulo a prime number   modprm1div 16847
            6.2.6  Pythagorean Triples   coprimeprodsq 16858
            6.2.7  The prime count function   cpc 16886
            6.2.8  Pocklington's theorem   prmpwdvds 16954
            6.2.9  Infinite primes theorem   unbenlem 16958
            6.2.10  Sum of prime reciprocals   prmreclem1 16966
            6.2.11  Fundamental theorem of arithmetic   1arithlem1 16973
            6.2.12  Lagrange's four-square theorem   cgz 16979
            6.2.13  Van der Waerden's theorem   cvdwa 17015
            6.2.14  Ramsey's theorem   cram 17049
            ​*6.2.15  Primorial function   cprmo 17081
            ​*6.2.16  Prime gaps   prmgaplem1 17099
            6.2.17  Decimal arithmetic (cont.)   dec2dvds 17113
            6.2.18  Cyclical shifts of words (cont.)   cshwsidrepsw 17143
            6.2.19  Specific prime numbers   prmlem0 17155
            6.2.20  Very large primes   1259lem1 17181
​PART 7  BASIC STRUCTURES
      ​7.1  Extensible structures
            ​*7.1.1  Basic definitions   cstr 17196
                  7.1.1.1  Extensible structures as structures with components   cstr 17196
                  7.1.1.2  Substitution of components   csts 17213
                  7.1.1.3  Slots   cslot 17231
                  ​*7.1.1.4  Structure component indices   cnx 17243
                  7.1.1.5  Base sets   cbs 17259
                  7.1.1.6  Base set restrictions   cress 17280
            7.1.2  Slot definitions   cplusg 17300
            7.1.3  Definition of the structure product   crest 17463
            7.1.4  Definition of the structure quotient   cordt 17543
      ​7.2  Moore spaces
            7.2.1  Moore closures   mrcflem 17652
            7.2.2  Independent sets in a Moore system   mrisval 17676
            7.2.3  Algebraic closure systems   isacs 17697
​PART 8  BASIC CATEGORY THEORY
      ​8.1  Categories
            8.1.1  Categories   ccat 17710
            8.1.2  Opposite category   coppc 17757
            8.1.3  Monomorphisms and epimorphisms   cmon 17775
            8.1.4  Sections, inverses, isomorphisms   csect 17791
            ​*8.1.5  Isomorphic objects   ccic 17842
            8.1.6  Subcategories   cssc 17854
            8.1.7  Functors   cfunc 17901
            8.1.8  Full & faithful functors   cful 17951
            8.1.9  Natural transformations and the functor category   cnat 17991
            8.1.10  Initial, terminal and zero objects of a category   cinito 18028
      ​8.2  Arrows (disjointified hom-sets)
            8.2.1  Identity and composition for arrows   cida 18100
      ​8.3  Examples of categories
            8.3.1  The category of sets   csetc 18122
            8.3.2  The category of categories   ccatc 18145
            ​*8.3.3  The category of extensible structures   fncnvimaeqv 18166
      ​8.4  Categorical constructions
            8.4.1  Product of categories   cxpc 18214
            8.4.2  Functor evaluation   cevlf 18255
            8.4.3  Hom functor   chof 18294
​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 18477
            9.5.2  Complete lattices   ccla 18544
            9.5.3  Distributive lattices   cdlat 18566
            9.5.4  Subset order structures   cipo 18573
      ​9.6  Posets, directed sets, and lattices as relations
            ​*9.6.1  Posets and lattices as relations   cps 18610
            9.6.2  Directed sets, nets   cdir 18640
      ​9.7  Chains
​PART 10  BASIC ALGEBRAIC STRUCTURES
      ​10.1  Monoids
            ​*10.1.1  Magmas   cplusf 18685
            ​*10.1.2  Identity elements   mgmidmo 18708
            ​*10.1.3  Iterated sums in a magma   gsumvalx 18724
            10.1.4  Magma homomorphisms and submagmas   cmgmhm 18738
            ​*10.1.5  Semigroups   csgrp 18766
            ​*10.1.6  Definition and basic properties of monoids   cmnd 18782
            10.1.7  Monoid homomorphisms and submonoids   cmhm 18829
            ​*10.1.8  Iterated sums in a monoid   gsumvallem2 18883
            10.1.9  Free monoids   cfrmd 18896
                  ​*10.1.9.1  Monoid of endofunctions   cefmnd 18917
            10.1.10  Examples and counterexamples for magmas, semigroups and monoids   mgm2nsgrplem1 18970
      ​10.2  Groups
            10.2.1  Definition and basic properties   cgrp 18990
            ​*10.2.2  Group multiple operation   cmg 19124
            10.2.3  Subgroups and Quotient groups   csubg 19177
            ​*10.2.4  Cyclic monoids and groups   cycsubmel 19262
            10.2.5  Elementary theory of group homomorphisms   cghm 19274
            10.2.6  Isomorphisms of groups   cgim 19318
                  10.2.6.1  The first isomorphism theorem of groups   ghmqusnsglem1 19341
            10.2.7  Group actions   cga 19350
            10.2.8  Centralizers and centers   ccntz 19376
            10.2.9  The opposite group   coppg 19406
            10.2.10  Symmetric groups   csymg 19430
                  ​*10.2.10.1  Definition and basic properties   csymg 19430
                  10.2.10.2  Cayley's theorem   cayleylem1 19473
                  10.2.10.3  Permutations fixing one element   symgfix2 19477
                  ​*10.2.10.4  Transpositions in the symmetric group   cpmtr 19502
                  10.2.10.5  The sign of a permutation   cpsgn 19550
            10.2.11  p-Groups and Sylow groups; Sylow's theorems   cod 19585
            10.2.12  Direct products   clsm 19695
                  10.2.12.1  Direct products (extension)   smndlsmidm 19717
            10.2.13  Free groups   cefg 19767
            10.2.14  Abelian groups   ccmn 19841
                  10.2.14.1  Definition and basic properties   ccmn 19841
                  10.2.14.2  Cyclic groups   ccyg 19938
                  10.2.14.3  Group sum operation   gsumval3a 19964
                  10.2.14.4  Group sums over (ranges of) integers   fsfnn0gsumfsffz 20044
                  10.2.14.5  Internal direct products   cdprd 20056
                  10.2.14.6  The Fundamental Theorem of Abelian Groups   ablfacrplem 20128
            10.2.15  Simple groups   csimpg 20153
                  10.2.15.1  Definition and basic properties   csimpg 20153
                  10.2.15.2  Classification of abelian simple groups   ablsimpnosubgd 20167
            10.2.16  Totally ordered monoids and groups   comnd 20180
      ​10.3  Rings
            10.3.1  Multiplicative Group   cmgp 20207
            ​*10.3.2  Non-unital rings ("rngs")   crng 20221
            ​*10.3.3  Ring unity (multiplicative identity)   cur 20254
            10.3.4  Semirings   csrg 20259
                  ​*10.3.4.1  The binomial theorem for semirings   srgbinomlem1 20299
            10.3.5  Unital rings   crg 20306
            10.3.6  Opposite ring   coppr 20409
            10.3.7  Divisibility   cdsr 20427
            10.3.8  Ring primes   crpm 20505
            10.3.9  Homomorphisms of non-unital rings   crnghm 20507
            10.3.10  Ring homomorphisms   crh 20542
            10.3.11  Nonzero rings and zero rings   cnzr 20586
            10.3.12  Local rings   clring 20614
            10.3.13  Subrings   csubrng 20621
                  10.3.13.1  Subrings of non-unital rings   csubrng 20621
                  10.3.13.2  Subrings of unital rings   csubrg 20645
                  10.3.13.3  Subrings generated by a subset   crgspn 20686
            10.3.14  Categories of rings   crngc 20692
                  ​*10.3.14.1  The category of non-unital rings   crngc 20692
                  ​*10.3.14.2  The category of (unital) rings   cringc 20721
                  10.3.14.3  Subcategories of the category of rings   srhmsubclem1 20753
            10.3.15  Left regular elements and domains   crlreg 20767
      ​10.4  Division rings and fields
            10.4.1  Definition and basic properties   cdr 20804
            10.4.2  Sub-division rings   csdrg 20858
            10.4.3  Absolute value (abstract algebra)   cabv 20880
            10.4.4  Star rings   cstf 20909
            10.4.5  Totally ordered rings and fields   corng 20929
      ​10.5  Left modules
            10.5.1  Definition and basic properties   clmod 20950
            10.5.2  Subspaces and spans in a left module   clss 21021
            10.5.3  Homomorphisms and isomorphisms of left modules   clmhm 21109
            10.5.4  Subspace sum; bases for a left module   clbs 21164
      ​10.6  Vector spaces
            10.6.1  Definition and basic properties   clvec 21192
      ​10.7  Subring algebras and ideals
            10.7.1  Subring algebras   csra 21261
            ​*10.7.2  Left ideals and spans   clidl 21299
            10.7.3  Two-sided ideals and quotient rings   c2idl 21350
                  ​*10.7.3.1  Condition for a non-unital ring to be unital   rngqiprng1elbas 21388
                  10.7.3.2  Prime Ideals   cprmidl 21422
            10.7.4  Principal ideal rings. Divisibility in the integers   clpidl 21448
            10.7.5  Principal ideal domains   cpid 21464
      ​10.8  The complex numbers as an algebraic extensible structure
            10.8.1  Definition and basic properties   cpsmet 21466
            ​*10.8.2  Ring of integers   czring 21556
                  ​*10.8.2.1  Example for a condition for a non-unital ring to be unital   pzriprnglem1 21591
            10.8.3  Algebraic constructions based on the complex numbers   czrh 21609
            10.8.4  Signs as subgroup of the complex numbers   cnmsgnsubg 21687
            10.8.5  Embedding of permutation signs into a ring   zrhpsgnmhm 21694
            10.8.6  The ordered field of real numbers   crefld 21714
      ​10.9  Generalized pre-Hilbert and Hilbert spaces
            10.9.1  Definition and basic properties   cphl 21734
            10.9.2  Orthocomplements and closed subspaces   cocv 21770
            10.9.3  Orthogonal projection and orthonormal bases   cpj 21810
​​*PART 11  BASIC LINEAR ALGEBRA
      ​11.1  Vectors and free modules
            *11.1.1  Direct sum of left modules   cdsmm 21841
            ​*11.1.2  Free modules   cfrlm 21856
            ​*11.1.3  Standard basis (unit vectors)   cuvc 21892
            ​*11.1.4  Independent sets and families   clindf 21914
            11.1.5  Characterization of free modules   lmimlbs 21946
      ​11.2  Associative algebras
            11.2.1  Definition and basic properties   casa 21960
      ​11.3  Abstract multivariate polynomials
            11.3.1  Definition and basic properties   cmps 22014
            11.3.2  Polynomial evaluation   ces 22183
            11.3.3  The "variable selection" function   cslv 22227
            11.3.4  Additional definitions for (multivariate) polynomials   cmhp 22256
            ​*11.3.5  Univariate polynomials   cps1 22295
            11.3.6  Univariate polynomial evaluation   ces1 22434
                  11.3.6.1  Specialization of polynomial evaluation as a ring homomorphism   evls1scafv 22487
      ​​*11.4  Matrices
            *11.4.1  The matrix multiplication   cmmul 22508
            ​*11.4.2  Square matrices   cmat 22525
            ​*11.4.3  The matrix algebra   matmulr 22556
            ​*11.4.4  Matrices of dimension 0 and 1   mat0dimbas0 22584
            ​*11.4.5  The subalgebras of diagonal and scalar matrices   cdmat 22606
            ​*11.4.6  Multiplication of a matrix with a "column vector"   cmvmul 22658
            11.4.7  Replacement functions for a square matrix   cmarrep 22674
            11.4.8  Submatrices   csubma 22694
      ​11.5  The determinant
            11.5.1  Definition and basic properties   cmdat 22702
            11.5.2  Determinants of 2 x 2 -matrices   m2detleiblem1 22742
            11.5.3  The matrix adjugate/adjunct   cmadu 22750
            ​*11.5.4  Laplace expansion of determinants (special case)   symgmatr01lem 22771
            11.5.5  Inverse matrix   invrvald 22794
            ​*11.5.6  Cramer's rule   slesolvec 22797
      ​​*11.6  Polynomial matrices
            11.6.1  Basic properties   pmatring 22810
            ​*11.6.2  Constant polynomial matrices   ccpmat 22821
            ​*11.6.3  Collecting coefficients of polynomial matrices   cdecpmat 22880
            ​*11.6.4  Ring isomorphism between polynomial matrices and polynomials over matrices   cpm2mp 22910
      ​​*11.7  The characteristic polynomial
            *11.7.1  Definition and basic properties   cchpmat 22944
            ​*11.7.2  The characteristic factor function G   fvmptnn04if 22967
            ​*11.7.3  The Cayley-Hamilton theorem   cpmadurid 22985
​PART 12  BASIC TOPOLOGY
      ​12.1  Topology
            ​*12.1.1  Topological spaces   ctop 23011
                  12.1.1.1  Topologies   ctop 23011
                  12.1.1.2  Topologies on sets   ctopon 23028
                  12.1.1.3  Topological spaces   ctps 23050
            12.1.2  Topological bases   ctb 23063
            12.1.3  Examples of topologies   distop 23113
            12.1.4  Closure and interior   ccld 23134
            12.1.5  Neighborhoods   cnei 23215
            12.1.6  Limit points and perfect sets   clp 23252
            12.1.7  Subspace topologies   restrcl 23275
            12.1.8  Order topology   ordtbaslem 23306
            12.1.9  Limits and continuity in topological spaces   ccn 23342
            12.1.10  Separated spaces: T0, T1, T2 (Hausdorff) ...   ct0 23424
            12.1.11  Compactness   ccmp 23504
            12.1.12  Bolzano-Weierstrass theorem   bwth 23528
            12.1.13  Connectedness   cconn 23529
            12.1.14  First- and second-countability   c1stc 23555
            12.1.15  Local topological properties   clly 23582
            12.1.16  Refinements   cref 23620
            12.1.17  Compactly generated spaces   ckgen 23651
            12.1.18  Product topologies   ctx 23678
            12.1.19  Continuous function-builders   cnmptid 23779
            12.1.20  Quotient maps and quotient topology   ckq 23811
            12.1.21  Homeomorphisms   chmeo 23871
      ​12.2  Filters and filter bases
            12.2.1  Filter bases   elmptrab 23945
            12.2.2  Filters   cfil 23963
            12.2.3  Ultrafilters   cufil 24017
            12.2.4  Filter limits   cfm 24051
            12.2.5  Extension by continuity   ccnext 24177
            12.2.6  Topological groups   ctmd 24188
            12.2.7  Infinite group sum on topological groups   ctsu 24244
            12.2.8  Topological rings, fields, vector spaces   ctrg 24274
      ​12.3  Uniform Structures and Spaces
            12.3.1  Uniform structures   cust 24318
            12.3.2  The topology induced by an uniform structure   cutop 24348
            12.3.3  Uniform Spaces   cuss 24371
            12.3.4  Uniform continuity   cucn 24392
            12.3.5  Cauchy filters in uniform spaces   ccfilu 24403
            12.3.6  Complete uniform spaces   ccusp 24414
      ​12.4  Metric spaces
            12.4.1  Pseudometric spaces   ispsmet 24422
            12.4.2  Basic metric space properties   cxms 24435
            12.4.3  Metric space balls   blfvalps 24501
            12.4.4  Open sets of a metric space   mopnval 24556
            12.4.5  Continuity in metric spaces   metcnp3 24658
            12.4.6  The uniform structure generated by a metric   metuval 24667
            12.4.7  Examples of metric spaces   dscmet 24690
            ​*12.4.8  Normed algebraic structures   cnm 24694
            12.4.9  Normed space homomorphisms (bounded linear operators)   cnmo 24823
            12.4.10  Topology on the reals   qtopbaslem 24876
            12.4.11  Topological definitions using the reals   cii 24995
            12.4.12  Path homotopy   chtpy 25087
            12.4.13  The fundamental group   cpco 25120
      ​12.5  Metric subcomplex vector spaces
            12.5.1  Subcomplex modules   cclm 25182
            ​*12.5.2  Subcomplex vector spaces   ccvs 25243
            ​*12.5.3  Normed subcomplex vector spaces   isncvsngp 25269
            12.5.4  Subcomplex pre-Hilbert spaces   ccph 25286
            12.5.5  Convergence and completeness   ccfil 25372
            12.5.6  Baire's Category Theorem   bcthlem1 25444
            12.5.7  Banach spaces and subcomplex Hilbert spaces   ccms 25452
                  12.5.7.1  The complete ordered field of the real numbers   retopn 25499
            12.5.8  Euclidean spaces   crrx 25503
            12.5.9  Minimizing Vector Theorem   minveclem1 25544
            12.5.10  Projection Theorem   pjthlem1 25557
​PART 13  BASIC REAL AND COMPLEX ANALYSIS
      ​13.1  Continuity
            13.1.1  Intermediate value theorem   pmltpclem1 25568
      ​13.2  Integrals
            13.2.1  Lebesgue measure   covol 25582
            13.2.2  Lebesgue integration   cmbf 25734
                  13.2.2.1  Lesbesgue integral   cmbf 25734
                  13.2.2.2  Lesbesgue directed integral   cdit 25966
      ​13.3  Derivatives
            13.3.1  Real and complex differentiation   climc 25982
                  13.3.1.1  Derivatives of functions of one complex or real variable   climc 25982
                  13.3.1.2  Results on real differentiation   dvferm1lem 26104
​PART 14  BASIC REAL AND COMPLEX FUNCTIONS
      ​14.1  Polynomials
            14.1.1  Polynomial degrees   cmdg 26171
            14.1.2  The division algorithm for univariate polynomials   cmn1 26244
            14.1.3  Elementary properties of complex polynomials   cply 26302
            14.1.4  The division algorithm for polynomials   cquot 26412
            14.1.5  Algebraic numbers   caa 26436
            14.1.6  Liouville's approximation theorem   aalioulem1 26454
      ​14.2  Sequences and series
            14.2.1  Taylor polynomials and Taylor's theorem   ctayl 26474
            14.2.2  Uniform convergence   culm 26497
            14.2.3  Power series   pserval 26531
      ​14.3  Basic trigonometry
            14.3.1  The exponential, sine, and cosine functions (cont.)   efcn 26564
            14.3.2  Properties of pi = 3.14159...   pilem1 26572
            14.3.3  Mapping of the exponential function   efgh 26664
            14.3.4  The natural logarithm on complex numbers   clog 26677
            ​*14.3.5  Logarithms to an arbitrary base   clogb 26887
            14.3.6  Theorems of Pythagoras, isosceles triangles, and intersecting chords   angval 26924
            14.3.7  Solutions of quadratic, cubic, and quartic equations   quad2 26962
            14.3.8  Inverse trigonometric functions   casin 26985
            14.3.9  The Birthday Problem   log2ublem1 27069
            14.3.10  Areas in R^2   carea 27078
            14.3.11  More miscellaneous converging sequences   rlimcnp 27088
            14.3.12  Inequality of arithmetic and geometric means   cvxcl 27107
            14.3.13  Euler-Mascheroni constant   cem 27114
            14.3.14  Zeta function   czeta 27135
            14.3.15  Gamma function   clgam 27138
      ​14.4  Basic number theory
            14.4.1  Wilson's theorem   wilthlem1 27190
            14.4.2  The Fundamental Theorem of Algebra   ftalem1 27195
            14.4.3  The Basel problem (ζ(2) = π²/6)   basellem1 27203
            14.4.4  Number-theoretical functions   ccht 27213
            14.4.5  Perfect Number Theorem   mersenne 27349
            14.4.6  Characters of Z/nZ   cdchr 27354
            14.4.7  Bertrand's postulate   bcctr 27397
            ​*14.4.8  Quadratic residues and the Legendre symbol   clgs 27416
            ​*14.4.9  Gauss' Lemma   gausslemma2dlem0a 27478
            14.4.10  Quadratic reciprocity   lgseisenlem1 27497
            14.4.11  All primes 4n+1 are the sum of two squares   2sqlem1 27539
            14.4.12  Chebyshev's Weak Prime Number Theorem, Dirichlet's Theorem   chebbnd1lem1 27591
            14.4.13  The Prime Number Theorem   mudivsum 27652
            14.4.14  Ostrowski's theorem   abvcxp 27737
​PART 15  SURREAL NUMBERS
      ​​*15.1  Sign sequence representation and Alling's axioms
            15.1.1  Definitions and initial properties   csur 27762
            15.1.2  Ordering   ltssolem1 27797
            15.1.3  Birthday Function   bdayfo 27799
            15.1.4  Density   fvnobday 27800
            ​*15.1.5  Full-Eta Property   bdayimaon 27815
      ​15.2  Initial consequences of Alling's axioms
            15.2.1  Ordering Theorems   cles 27866
            15.2.2  Birthday Theorems   bdayfun 27898
      ​​*15.3  Conway cut representation
            15.3.1  Conway cuts   cslts 27908
            15.3.2  Zero and One   c0s 27956
            15.3.3  Cuts and Options   cmade 27973
            15.3.4  Cofinality and coinitiality   cofslts 28069
      ​15.4  Induction and recursion
            15.4.1  Induction and recursion on one variable   cnorec 28088
            15.4.2  Induction and recursion on two variables   cnorec2 28099
      ​15.5  Surreal arithmetic
            15.5.1  Addition   cadds 28110
            15.5.2  Negation and Subtraction   cnegs 28170
            15.5.3  Multiplication   cmuls 28257
            15.5.4  Division   cdivs 28338
            15.5.5  Absolute value   cabss 28388
      ​15.6  Subsystems of surreals
            15.6.1  Ordinal numbers   cons 28402
            15.6.2  Surreal recursive sequences   cseqs 28434
            15.6.3  Natural numbers   cn0s 28463
            15.6.4  Integers   czs 28529
            15.6.5  Dyadic fractions   c2s 28561
            15.6.6  Real numbers   creno 28640
​​*PART 16  ELEMENTARY GEOMETRY
      ​16.1  Definition and Tarski's Axioms of Geometry
            16.1.1  Justification for the congruence notation   tgjustf 28700
      ​16.2  Tarskian Geometry
            16.2.1  Congruence   tgcgrcomimp 28704
            16.2.2  Betweenness   tgbtwntriv2 28714
            16.2.3  Dimension   tglowdim1 28727
            16.2.4  Betweenness and Congruence   tgifscgr 28735
            16.2.5  Congruence of a series of points   ccgrg 28737
            16.2.6  Motions   cismt 28759
            16.2.7  Colinearity   tglng 28773
            16.2.8  Connectivity of betweenness   tgbtwnconn1lem1 28799
            16.2.9  Less-than relation in geometric congruences   cleg 28809
            16.2.10  Rays   chlg 28827
            16.2.11  Lines   btwnlng1 28846
            16.2.12  Point inversions   cmir 28883
            16.2.13  Right angles   crag 28924
            16.2.14  Half-planes   islnopp 28970
            16.2.15  Planes   cplng 29003
            16.2.16  Midpoints and Line Mirroring   cmid 29024
            16.2.17  Congruence of angles   ccgra 29059
            16.2.18  Angle Comparisons   cinag 29087
            16.2.19  Congruence Theorems   tgsas1 29106
            16.2.20  Equilateral triangles   ceqlg 29117
            16.2.21  Parallel lines   cprlng 29121
      ​16.3  Properties of geometries
            16.3.1  Isomorphisms between geometries   f1otrgds 29127
      ​16.4  Geometry in Hilbert spaces
            16.4.1  Geometry in the complex plane   cchhllem 29145
            16.4.2  Geometry in Euclidean spaces   cee 29146
                  16.4.2.1  Definition of the Euclidean space   cee 29146
                  16.4.2.2  Tarski's axioms for geometry for the Euclidean space   axdimuniq 29172
                  16.4.2.3  EE^n fulfills Tarski's Axioms   ceeng 29236
​​*PART 17  GRAPH THEORY
      ​*17.1  Vertices and edges
            17.1.1  The edge function extractor for extensible structures   cedgf 29247
            ​*17.1.2  Vertices and indexed edges   cvtx 29255
                  17.1.2.1  Definitions and basic properties   cvtx 29255
                  17.1.2.2  The vertices and edges of a graph represented as ordered pair   opvtxval 29262
                  17.1.2.3  The vertices and edges of a graph represented as extensible structure   funvtxdmge2val 29270
                  17.1.2.4  Representations of graphs without edges   snstrvtxval 29296
                  17.1.2.5  Degenerated cases of representations of graphs   vtxval0 29298
            17.1.3  Edges as range of the edge function   cedg 29306
      ​​*17.2  Undirected graphs
            17.2.1  Undirected hypergraphs   cuhgr 29315
            17.2.2  Undirected pseudographs and multigraphs   cupgr 29339
            ​*17.2.3  Loop-free graphs   umgrislfupgrlem 29381
            17.2.4  Edges as subsets of vertices of graphs   uhgredgiedgb 29385
            ​*17.2.5  Undirected simple graphs   cuspgr 29407
            17.2.6  Examples for graphs   usgr0e 29495
            17.2.7  Subgraphs   csubgr 29526
            17.2.8  Finite undirected simple graphs   cfusgr 29575
            17.2.9  Neighbors, complete graphs and universal vertices   cnbgr 29591
                  17.2.9.1  Neighbors   cnbgr 29591
                  17.2.9.2  Universal vertices   cuvtx 29644
                  17.2.9.3  Complete graphs   ccplgr 29668
            17.2.10  Vertex degree   cvtxdg 29724
            ​*17.2.11  Regular graphs   crgr 29814
      ​​*17.3  Walks, paths and cycles
            *17.3.1  Walks   cewlks 29854
            17.3.2  Walks for loop-free graphs   lfgrwlkprop 29944
            17.3.3  Trails   ctrls 29947
            17.3.4  Paths and simple paths   cpths 29968
            17.3.5  Closed walks   cclwlks 30028
            17.3.6  Circuits and cycles   ccrcts 30042
            ​*17.3.7  Walks as words   cwwlks 30083
            17.3.8  Walks/paths of length 2 (as length 3 strings)   2wlkdlem1 30183
            17.3.9  Walks in regular graphs   rusgrnumwwlkl1 30229
            ​*17.3.10  Closed walks as words   cclwwlk 30241
                  17.3.10.1  Closed walks as words   cclwwlk 30241
                  17.3.10.2  Closed walks of a fixed length as words   cclwwlkn 30284
                  17.3.10.3  Closed walks on a vertex of a fixed length as words   cclwwlknon 30347
            17.3.11  Examples for walks, trails and paths   0ewlk 30374
            17.3.12  Connected graphs   cconngr 30446
      ​17.4  Eulerian paths and the Konigsberg Bridge problem
            ​*17.4.1  Eulerian paths   ceupth 30457
            ​*17.4.2  The Königsberg Bridge problem   konigsbergvtx 30506
      ​17.5  The Friendship Theorem
            17.5.1  Friendship graphs - basics   cfrgr 30518
            17.5.2  The friendship theorem for small graphs   frgr1v 30531
            17.5.3  Theorems according to Mertzios and Unger   2pthfrgrrn 30542
            ​*17.5.4  Huneke's Proof of the Friendship Theorem   frgrncvvdeqlem1 30559
​PART 18  GUIDES AND MISCELLANEA
      ​18.1  Guides (conventions, explanations, and examples)
            ​*18.1.1  Conventions   conventions 30660
            18.1.2  Natural deduction   natded 30663
            ​*18.1.3  Natural deduction examples   ex-natded5.2 30664
            18.1.4  Definitional examples   ex-or 30681
            18.1.5  Other examples   aevdemo 30720
      ​18.2  Humor
            18.2.1  April Fool's theorem   avril1 30723
      ​18.3  (Future - to be reviewed and classified)
            18.3.1  Planar incidence geometry   cplig 30735
            ​*18.3.2  Aliases kept to prevent broken links   dummylink 30748
​​*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 30750
            19.1.2  Abelian groups   cablo 30805
      ​19.2  Complex vector spaces
            19.2.1  Definition and basic properties   cvc 30819
            19.2.2  Examples of complex vector spaces   cnaddabloOLD 30842
      ​19.3  Normed complex vector spaces
            19.3.1  Definition and basic properties   cnv 30845
            19.3.2  Examples of normed complex vector spaces   cnnv 30938
            19.3.3  Induced metric of a normed complex vector space   imsval 30946
            19.3.4  Inner product   cdip 30961
            19.3.5  Subspaces   css 30982
      ​19.4  Operators on complex vector spaces
            19.4.1  Definitions and basic properties   clno 31001
      ​19.5  Inner product (pre-Hilbert) spaces
            19.5.1  Definition and basic properties   ccphlo 31073
            19.5.2  Examples of pre-Hilbert spaces   cncph 31080
            19.5.3  Properties of pre-Hilbert spaces   isph 31083
      ​19.6  Complex Banach spaces
            19.6.1  Definition and basic properties   ccbn 31123
            19.6.2  Examples of complex Banach spaces   cnbn 31130
            19.6.3  Uniform Boundedness Theorem   ubthlem1 31131
            19.6.4  Minimizing Vector Theorem   minvecolem1 31135
      ​19.7  Complex Hilbert spaces
            19.7.1  Definition and basic properties   chlo 31146
            19.7.2  Standard axioms for a complex Hilbert space   hlex 31159
            19.7.3  Examples of complex Hilbert spaces   cnchl 31177
            19.7.4  Hellinger-Toeplitz Theorem   htthlem 31178
​​*PART 20  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      ​20.1  Axiomatization of complex pre-Hilbert spaces
            20.1.1  Basic Hilbert space definitions   chba 31180
            20.1.2  Preliminary ZFC lemmas   df-hnorm 31229
            ​*20.1.3  Derive the Hilbert space axioms from ZFC set theory   axhilex-zf 31242
            ​*20.1.4  Introduce the vector space axioms for a Hilbert space   ax-hilex 31260
            20.1.5  Vector operations   hvmulex 31272
            20.1.6  Inner product postulates for a Hilbert space   ax-hfi 31340
      ​20.2  Inner product and norms
            20.2.1  Inner product   his5 31347
            20.2.2  Norms   dfhnorm2 31383
            20.2.3  Relate Hilbert space to normed complex vector spaces   hilablo 31421
            20.2.4  Bunjakovaskij-Cauchy-Schwarz inequality   bcsiALT 31440
      ​20.3  Cauchy sequences and completeness axiom
            20.3.1  Cauchy sequences and limits   hcau 31445
            20.3.2  Derivation of the completeness axiom from ZF set theory   hilmet 31455
            20.3.3  Completeness postulate for a Hilbert space   ax-hcompl 31463
            20.3.4  Relate Hilbert space to ZFC pre-Hilbert and Hilbert spaces   hhcms 31464
      ​20.4  Subspaces and projections
            20.4.1  Subspaces   df-sh 31468
            20.4.2  Closed subspaces   df-ch 31482
            20.4.3  Orthocomplements   df-oc 31513
            20.4.4  Subspace sum, span, lattice join, lattice supremum   df-shs 31569
            20.4.5  Projection theorem   pjhthlem1 31652
            20.4.6  Projectors   df-pjh 31656
      ​20.5  Properties of Hilbert subspaces
            20.5.1  Orthomodular law   omlsilem 31663
            20.5.2  Projectors (cont.)   pjhtheu2 31677
            20.5.3  Hilbert lattice operations   sh0le 31701
            20.5.4  Span (cont.) and one-dimensional subspaces   spansn0 31802
            20.5.5  Commutes relation for Hilbert lattice elements   df-cm 31844
            20.5.6  Foulis-Holland theorem   fh1 31879
            20.5.7  Quantum Logic Explorer axioms   qlax1i 31888
            20.5.8  Orthogonal subspaces   chscllem1 31898
            20.5.9  Orthoarguesian laws 5OA and 3OA   5oalem1 31915
            20.5.10  Projectors (cont.)   pjorthi 31930
            20.5.11  Mayet's equation E_3   mayete3i 31989
      ​20.6  Operators on Hilbert spaces
            ​*20.6.1  Operator sum, difference, and scalar multiplication   df-hosum 31991
            20.6.2  Zero and identity operators   df-h0op 32009
            20.6.3  Operations on Hilbert space operators   hoaddcl 32019
            20.6.4  Linear, continuous, bounded, Hermitian, unitary operators and norms   df-nmop 32100
            20.6.5  Linear and continuous functionals and norms   df-nmfn 32106
            20.6.6  Adjoint   df-adjh 32110
            20.6.7  Dirac bra-ket notation   df-bra 32111
            20.6.8  Positive operators   df-leop 32113
            20.6.9  Eigenvectors, eigenvalues, spectrum   df-eigvec 32114
            20.6.10  Theorems about operators and functionals   nmopval 32117
            20.6.11  Riesz lemma   riesz3i 32323
            20.6.12  Adjoints (cont.)   cnlnadjlem1 32328
            20.6.13  Quantum computation error bound theorem   unierri 32365
            20.6.14  Dirac bra-ket notation (cont.)   branmfn 32366
            20.6.15  Positive operators (cont.)   leopg 32383
            20.6.16  Projectors as operators   pjhmopi 32407
      ​20.7  States on a Hilbert lattice and Godowski's equation
            20.7.1  States on a Hilbert lattice   df-st 32472
            20.7.2  Godowski's equation   golem1 32532
      ​20.8  Cover relation, atoms, exchange axiom, and modular symmetry
            20.8.1  Covers relation; modular pairs   df-cv 32540
            20.8.2  Atoms   df-at 32599
            20.8.3  Superposition principle   superpos 32615
            20.8.4  Atoms, exchange and covering properties, atomicity   chcv1 32616
            20.8.5  Irreducibility   chirredlem1 32651
            20.8.6  Atoms (cont.)   atcvat3i 32657
            20.8.7  Modular symmetry   mdsymlem1 32664
​PART 21  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      ​21.1  Mathboxes for user contributions
            21.1.1  Mathbox guidelines   mathbox 32703
      ​21.2  Mathbox for Stefan Allan
      ​21.3  Mathbox for Thierry Arnoux
            21.3.1  Propositional Calculus - misc additions   ad11antr 32708
            21.3.2  Predicate Calculus   sbc2iedf 32722
                  21.3.2.1  Predicate Calculus - misc additions   sbc2iedf 32722
                  21.3.2.2  Restricted quantification - misc additions   ralcom4f 32724
                  21.3.2.3  Equality   eqtrb 32730
                  21.3.2.4  Double restricted existential uniqueness quantification   opsbc2ie 32732
                  21.3.2.5  Double restricted existential uniqueness quantification syntax   w2reu 32734
                  21.3.2.6  Substitution (without distinct variables) - misc additions   sbceqbidf 32743
                  21.3.2.7  Existential "at most one" - misc additions   mo5f 32745
                  21.3.2.8  Existential uniqueness - misc additions   reuxfrdf 32747
                  21.3.2.9  Restricted "at most one" - misc additions   rmoxfrd 32749
                  21.3.2.10  Restricted iota (description binder)   riotaeqbidva 32752
            21.3.3  General Set Theory   dmrab 32753
                  21.3.3.1  Class abstractions (a.k.a. class builders)   dmrab 32753
                  21.3.3.2  Image Sets   abrexdomjm 32763
                  21.3.3.3  Set relations and operations - misc additions   nelun 32769
                  21.3.3.4  Unordered pairs   elpreq 32784
                  21.3.3.5  Unordered triples   tpssg 32793
                  21.3.3.6  Conditional operator - misc additions   ifeqeqx 32798
                  21.3.3.7  Set union   uniinn0 32807
                  21.3.3.8  Indexed union - misc additions   cbviunf 32810
                  21.3.3.9  Indexed intersection - misc additions   iinabrex 32824
                  21.3.3.10  Disjointness - misc additions   disjnf 32825
            21.3.4  Relations and Functions   xpdisjres 32853
                  21.3.4.1  Relations - misc additions   xpdisjres 32853
                  21.3.4.2  Functions - misc additions   fconst7v 32877
                  21.3.4.3  Operations - misc additions   mpomptxf 32935
                  21.3.4.4  The mapping operation   elmaprd 32937
                  21.3.4.5  Support of a function   suppovss 32938
                  21.3.4.6  Explicit Functions with one or two points as a domain   cosnopne 32951
                  21.3.4.7  Isomorphisms - misc. additions   gtiso 32958
                  21.3.4.8  Disjointness (additional proof requiring functions)   disjdsct 32960
                  21.3.4.9  First and second members of an ordered pair - misc additions   df1stres 32961
                  21.3.4.10  Countable Sets   snct 32969
            21.3.5  Real and Complex Numbers   sgnval2 32992
                  21.3.5.1  Complex operations - misc. additions   creq0 32993
                  21.3.5.2  Ordering on reals - misc additions   lt2addrd 33007
                  21.3.5.3  Extended reals - misc additions   nn0mnfxrd 33008
                  21.3.5.4  Extended nonnegative integers - misc additions   xnn0gt0 33026
                  21.3.5.5  Real number intervals - misc additions   joiniooico 33031
                  21.3.5.6  Finite intervals of integers - misc additions   uzssico 33041
                  21.3.5.7  Half-open integer ranges - misc additions   iundisjfi 33053
                  21.3.5.8  The ` # ` (set size) function - misc additions   hashunif 33063
                  21.3.5.9  The greatest common divisor operator - misc. additions   elq2 33069
                  21.3.5.10  Integers   nn0split01 33075
                  21.3.5.11  Decimal numbers   dfdec100 33087
            21.3.6  Real and complex functions   sgnsgn 33088
                  21.3.6.1  Signum (sgn or sign) function - misc. additions   sgnsgn 33088
                  21.3.6.2  Integer powers - misc. additions   nexple 33090
                  21.3.6.3  Indicator Functions (continued)   indsumin 33094
            ​*21.3.7  Decimal expansion   cdp2 33103
                  ​*21.3.7.1  Decimal point   cdp 33120
                  21.3.7.2  Division in the extended real number system   cxdiv 33149
            21.3.8  Words over a set - misc additions   wrdres 33168
                  21.3.8.1  Splicing words (substring replacement)   splfv3 33191
                  21.3.8.2  Cyclic shift of words   1cshid 33192
            21.3.9  Extensible Structures   ressplusf 33196
                  21.3.9.1  Structure restriction operator   ressplusf 33196
                  21.3.9.2  Posets   ressprs 33199
                  21.3.9.3  Complete lattices   clatp0cl 33209
                  21.3.9.4  Order Theory   cmnt 33211
                  21.3.9.5  Extended reals Structure - misc additions   ax-xrssca 33237
                  21.3.9.6  The extended nonnegative real numbers commutative monoid   xrge00 33247
            21.3.10  Algebra   mndcld 33255
                  21.3.10.1  Monoids   mndcld 33255
                  21.3.10.2  Monoids Homomorphisms   abliso 33268
                  21.3.10.3  Groups - misc additions   grpidcld 33272
                  21.3.10.4  Abelian Groups - misc additions   ablcomd 33278
                  21.3.10.5  Finitely supported group sums - misc additions   gsumsubg 33279
                  21.3.10.6  Group or monoid sums over words   gsumwun 33309
                  21.3.10.7  Centralizers and centers - misc additions   cntzun 33312
                  21.3.10.8  The symmetric group   symgfcoeu 33315
                  21.3.10.9  Transpositions   pmtridf1o 33327
                  21.3.10.10  Permutation Signs   psgnid 33330
                  21.3.10.11  Permutation cycles   ctocyc 33339
                  21.3.10.12  The Alternating Group   evpmval 33378
                  21.3.10.13  Signum in an ordered monoid   csgns 33391
                  21.3.10.14  Fixed points   cfxp 33396
                  21.3.10.15  The Archimedean property for generic ordered algebraic structures   cinftm 33409
                  21.3.10.16  Semiring left modules   cslmd 33433
                  21.3.10.17  Simple groups   prmsimpcyc 33461
                  21.3.10.18  Rings - misc additions   ringrngd 33462
                  21.3.10.19  Subrings generated by a set   elrgspnlem1 33475
                  21.3.10.20  The zero ring   irrednzr 33483
                  21.3.10.21  Localization of rings   cerl 33486
                  21.3.10.22  Integral Domains   domnmuln0rd 33510
                  21.3.10.23  Euclidean Domains   ceuf 33524
                  21.3.10.24  Division Rings   ringinveu 33530
                  21.3.10.25  The field of rational numbers   qfld 33533
                  21.3.10.26  Subfields   subsdrg 33534
                  21.3.10.27  Field of fractions   cfrac 33538
                  21.3.10.28  Field extensions generated by a set   cfldgen 33546
                  21.3.10.29  Ring homomorphisms - misc additions   rhmdvd 33559
                  21.3.10.30  Scalar restriction operation   cresv 33561
                  21.3.10.31  The commutative ring of gaussian integers   gzcrng 33576
                  21.3.10.32  The archimedean ordered field of real numbers   cnfldfld 33577
                  21.3.10.33  The quotient map and quotient modules   qusker 33584
                  21.3.10.34  The ring of integers modulo ` N `   znfermltl 33596
                  21.3.10.35  Independent sets and families   islinds5 33597
                  21.3.10.36  Ring associates, ring units   dvdsruassoi 33613
                  ​*21.3.10.37  Subgroup sum / Sumset / Minkowski sum   elgrplsmsn 33619
                  21.3.10.38  The quotient map   quslsm 33630
                  21.3.10.39  Ideals   intlidl 33644
                  21.3.10.40  Maximal Ideals   cmxidl 33659
                  21.3.10.41  Local rings   drnglring 33699
                  21.3.10.42  The semiring of ideals of a ring   cidlsrg 33707
                  21.3.10.43  Prime Elements   rprmval 33723
                  21.3.10.44  Unique factorization domains   cufd 33745
                  21.3.10.45  The ring of integers   zringidom 33758
                  21.3.10.46  Associative Algebra   assaassd 33762
                  21.3.10.47  Univariate Polynomials   0ringmon1p 33764
                  21.3.10.48  Polynomial quotient and polynomial remainder   q1pdir 33810
                  21.3.10.49  Multivariate Polynomials   psrbasfsupp 33818
                  21.3.10.50  The ring of symmetric polynomials   csply 33862
                  21.3.10.51  The subring algebra   sra1r 33888
                  21.3.10.52  Division Ring Extensions   drgext0g 33897
                  21.3.10.53  Vector Spaces   lvecdimfi 33903
                  21.3.10.54  Vector Space Dimension   cldim 33906
            21.3.11  Field Extensions   cfldext 33945
                  21.3.11.1  Algebraic numbers   cirng 33990
                  21.3.11.2  Algebraic extensions   calgext 34002
                  21.3.11.3  Minimal polynomials   cminply 34006
                  21.3.11.4  Quadratic Field Extensions   rtelextdg2lem 34033
                  21.3.11.5  Towers of quadratic extentions   fldext2chn 34035
            ​*21.3.12  Constructible Numbers   cconstr 34036
                  21.3.12.1  Impossible constructions   2sqr3minply 34087
            21.3.13  Matrices   csmat 34100
                  21.3.13.1  Submatrices   csmat 34100
                  21.3.13.2  Matrix literals   clmat 34118
                  21.3.13.3  Laplace expansion of determinants   mdetpmtr1 34130
            21.3.14  Topology   ist0cld 34140
                  21.3.14.1  Open maps   txomap 34141
                  21.3.14.2  Topology of the unit circle   qtopt1 34142
                  21.3.14.3  Refinements   reff 34146
                  21.3.14.4  Open cover refinement property   ccref 34149
                  21.3.14.5  Lindelöf spaces   cldlf 34159
                  21.3.14.6  Paracompact spaces   cpcmp 34162
                  ​*21.3.14.7  Spectrum of a ring   crspec 34169
                  21.3.14.8  Pseudometrics   cmetid 34193
                  21.3.14.9  Continuity - misc additions   hauseqcn 34205
                  21.3.14.10  Topology of the closed unit interval   elunitge0 34206
                  21.3.14.11  Topology of ` ( RR X. RR ) `   unicls 34210
                  21.3.14.12  Order topology - misc. additions   cnvordtrestixx 34220
                  21.3.14.13  Continuity in topological spaces - misc. additions   mndpluscn 34233
                  21.3.14.14  Topology of the extended nonnegative real numbers ordered monoid   xrge0hmph 34239
                  21.3.14.15  Limits - misc additions   lmlim 34254
                  21.3.14.16  Univariate polynomials   pl1cn 34262
            21.3.15  Uniform Stuctures and Spaces   chcmp 34263
                  21.3.15.1  Hausdorff uniform completion   chcmp 34263
            21.3.16  Topology and algebraic structures   zringnm 34265
                  21.3.16.1  The norm on the ring of the integer numbers   zringnm 34265
                  21.3.16.2  Topological ` ZZ ` -modules   zlm0 34267
                  21.3.16.3  Canonical embedding of the field of the rational numbers into a division ring   cqqh 34277
                  21.3.16.4  Canonical embedding of the real numbers into a complete ordered field   crrh 34300
                  21.3.16.5  Embedding from the extended real numbers into a complete lattice   cxrh 34323
                  21.3.16.6  Canonical embeddings into the ordered field of the real numbers   zrhre 34326
                  ​*21.3.16.7  Topological Manifolds   cmntop 34329
                  21.3.16.8  Extended sum   cesum 34334
            21.3.17  Mixed Function/Constant operation   cofc 34402
            21.3.18  Abstract measure   csiga 34415
                  21.3.18.1  Sigma-Algebra   csiga 34415
                  21.3.18.2  Generated sigma-Algebra   csigagen 34445
                  ​*21.3.18.3  lambda and pi-Systems, Rings of Sets   ispisys 34459
                  21.3.18.4  The Borel algebra on the real numbers   cbrsiga 34488
                  21.3.18.5  Product Sigma-Algebra   csx 34495
                  21.3.18.6  Measures   cmeas 34502
                  21.3.18.7  The counting measure   cntmeas 34533
                  21.3.18.8  The Lebesgue measure - misc additions   voliune 34536
                  21.3.18.9  The Dirac delta measure   cdde 34539
                  21.3.18.10  The 'almost everywhere' relation   cae 34544
                  21.3.18.11  Measurable functions   cmbfm 34556
                  21.3.18.12  Borel Algebra on ` ( RR X. RR ) `   br2base 34576
                  ​*21.3.18.13  Caratheodory's extension theorem   coms 34598
            21.3.19  Integration   itgeq12dv 34633
                  21.3.19.1  Lebesgue integral - misc additions   itgeq12dv 34633
                  21.3.19.2  Bochner integral   citgm 34634
            21.3.20  Euler's partition theorem   oddpwdc 34661
            21.3.21  Sequences defined by strong recursion   csseq 34690
            21.3.22  Fibonacci Numbers   cfib 34703
            21.3.23  Probability   cprb 34714
                  21.3.23.1  Probability Theory   cprb 34714
                  21.3.23.2  Conditional Probabilities   ccprob 34738
                  21.3.23.3  Real-valued Random Variables   crrv 34747
                  21.3.23.4  Preimage set mapping operator   corvc 34763
                  21.3.23.5  Distribution Functions   orvcelval 34776
                  21.3.23.6  Cumulative Distribution Functions   orvclteel 34780
                  21.3.23.7  Probabilities - example   coinfliplem 34786
                  21.3.23.8  Bertrand's Ballot Problem   ballotlemoex 34793
            21.3.24  Signum (sgn or sign) function - misc. additions   fzssfzo 34846
                  21.3.24.1  Operations on words   ccatmulgnn0dir 34849
            21.3.25  Polynomials with real coefficients - misc additions   plyrecld 34853
            21.3.26  Descartes's rule of signs   signspval 34856
                  21.3.26.1  Sign changes in a word over real numbers   signspval 34856
                  21.3.26.2  Counting sign changes in a word over real numbers   signslema 34866
            21.3.27  Number Theory   iblidicc 34896
                  21.3.27.1  Representations of a number as sums of integers   crepr 34912
                  21.3.27.2  Vinogradov Trigonometric Sums and the Circle Method   cvts 34939
                  21.3.27.3  The Ternary Goldbach Conjecture: Final Statement   ax-hgt749 34948
            21.3.28  Elementary Geometry   cstrkg2d 34968
                  ​*21.3.28.1  Two-dimensional geometry   cstrkg2d 34968
                  21.3.28.2  Morley's Miracle   cgranbtwn 34973
                  21.3.28.3  Outer Five Segment (not used, no need to move to main)   cafs 34976
            ​*21.3.29  LeftPad Project   clpad 34981
      ​​*21.4  Mathbox for Jonathan Ben-Naim
            21.4.1  First-order logic and set theory   bnj170 35004
            21.4.2  Well founded induction and recursion   bnj110 35163
            21.4.3  The existence of a minimal element in certain classes   bnj69 35315
            21.4.4  Well-founded induction   bnj1204 35317
            21.4.5  Well-founded recursion, part 1 of 3   bnj60 35367
            21.4.6  Well-founded recursion, part 2 of 3   bnj1500 35373
            21.4.7  Well-founded recursion, part 3 of 3   bnj1522 35377
      ​21.5  Mathbox for BTernaryTau
            21.5.1  First-order logic   nfan1c 35378
                  21.5.1.1  Auxiliary axiom schemes   nfan1c 35378
            21.5.2  ZF set theory   exdifsn 35384
                  21.5.2.1  Finitism   prcinf 35421
                  21.5.2.2  Introduce ax-regs   ax-regs 35434
                  21.5.2.3  Derive ax-regs   axregs 35447
                  21.5.2.4  ZFC axioms with reduced distinct variable conditions   axsepg2 35448
                  21.5.2.5  Global choice   gblacfnacd 35457
            21.5.3  Real and complex numbers   zltp1ne 35472
            21.5.4  Graph theory   lfuhgr 35481
                  21.5.4.1  Acyclic graphs   cacycgr 35505
      ​21.6  Mathbox for Mario Carneiro
            21.6.1  Predicate calculus with all distinct variables   ax-7d 35522
            21.6.2  Miscellaneous stuff   quartfull 35528
            21.6.3  Derangements and the Subfactorial   deranglem 35529
            21.6.4  The Erdős-Szekeres theorem   erdszelem1 35554
            21.6.5  The Kuratowski closure-complement theorem   kur14lem1 35569
            21.6.6  Retracts and sections   cretr 35580
            21.6.7  Path-connected and simply connected spaces   cpconn 35582
            21.6.8  Covering maps   ccvm 35618
            21.6.9  Normal numbers   snmlff 35692
            21.6.10  Godel-sets of formulas - part 1   cgoe 35696
            21.6.11  Godel-sets of formulas - part 2   cgon 35795
            21.6.12  Models of ZF   cgze 35809
            ​*21.6.13  Metamath formal systems   cmcn 35823
            21.6.14  Grammatical formal systems   cm0s 35948
            21.6.15  Models of formal systems   cmuv 35968
            21.6.16  Splitting fields   ccpms 35990
            21.6.17  p-adic number fields   czr 36010
      ​​*21.7  Mathbox for Filip Cernatescu
      ​21.8  Mathbox for Paul Chapman
            21.8.1  Real and complex numbers (cont.)   climuzcnv 36034
            21.8.2  Miscellaneous theorems   elfzm12 36038
      ​21.9  Mathbox for Hongxiu Chen
      ​21.10  Mathbox for Adrian Ducourtial
            21.10.1  Propositional calculus   currybi 36051
            21.10.2  Clone theory   ccloneop 36058
      ​21.11  Mathbox for Scott Fenton
            21.11.1  ZFC Axioms in primitive form   axextprim 36064
            21.11.2  Untangled classes   untelirr 36071
            21.11.3  Extra propositional calculus theorems   3jaodd 36078
            21.11.4  Misc. Useful Theorems   nepss 36081
            21.11.5  Properties of real and complex numbers   sqdivzi 36091
            21.11.6  Infinite products   iprodefisumlem 36103
            21.11.7  Factorial limits   faclimlem1 36106
            21.11.8  Greatest common divisor and divisibility   gcd32 36112
            21.11.9  Properties of relationships   dftr6 36114
            21.11.10  Properties of functions and mappings   funpsstri 36129
            21.11.11  Ordinal numbers   elpotr 36142
            21.11.12  Defined equality axioms   axextdfeq 36158
            21.11.13  Hypothesis builders   hbntg 36166
            21.11.14  Well-founded zero, successor, and limits   cwsuc 36171
            21.11.15  Quantifier-free definitions   ctxp 36191
            21.11.16  Alternate ordered pairs   caltop 36319
            21.11.17  Geometry in the Euclidean space   cofs 36345
                  21.11.17.1  Congruence properties   cofs 36345
                  21.11.17.2  Betweenness properties   btwntriv2 36375
                  21.11.17.3  Segment Transportation   ctransport 36392
                  21.11.17.4  Properties relating betweenness and congruence   cifs 36398
                  21.11.17.5  Connectivity of betweenness   btwnconn1lem1 36450
                  21.11.17.6  Segment less than or equal to   csegle 36469
                  21.11.17.7  Outside-of relationship   coutsideof 36482
                  21.11.17.8  Lines and Rays   cline2 36497
            21.11.18  Forward difference   cfwddif 36521
            21.11.19  Rank theorems   rankung 36529
            21.11.20  Hereditarily Finite Sets   chf 36535
            21.11.21  Natural ordinal operations   cnmul 36550
      ​21.12  Mathbox for Gino Giotto
            21.12.1  Equality theorems   rmoeqi 36560
                  21.12.1.1  Inference versions   rmoeqi 36560
                  21.12.1.2  Deduction versions   rmoeqdv 36585
            21.12.2  Change bound variables   in-ax8 36597
                  21.12.2.1  Change bound variables and domains   cbvralvw2 36599
                  21.12.2.2  Change bound variables, deduction versions   cbvmodavw 36623
                  21.12.2.3  Change bound variables and domains, deduction versions   cbvrmodavw2 36656
            21.12.3  Study of ax-mulf usage   mpomulnzcnf 36672
      ​21.13  Mathbox for Jeff Hankins
            21.13.1  Miscellany   a1i14 36673
            21.13.2  Basic topological facts   topbnd 36697
            21.13.3  Topology of the real numbers   ivthALT 36708
            21.13.4  Refinements   cfne 36709
            21.13.5  Neighborhood bases determine topologies   neibastop1 36732
            21.13.6  Lattice structure of topologies   topmtcl 36736
            21.13.7  Filter bases   fgmin 36743
            21.13.8  Directed sets, nets   tailfval 36745
      ​21.14  Mathbox for Anthony Hart
            21.14.1  Propositional Calculus   tb-ax1 36756
            21.14.2  Predicate Calculus   nalfal 36776
            21.14.3  Miscellaneous single axioms   meran1 36784
            21.14.4  Connective Symmetry   negsym1 36790
      ​21.15  Mathbox for Chen-Pang He
            21.15.1  Ordinal topology   ontopbas 36801
      ​21.16  Mathbox for Jeff Hoffman
            21.16.1  Inferences for finite induction on generic function values   fveleq 36824
            21.16.2  gdc.mm   nnssi2 36828
      ​21.17  Mathbox for Matthew House
            21.17.1  Relations on well-ordered indexed unions   weiunval 36835
            21.17.2  Axiom of Transitive Containment   axtco 36844
            21.17.3  Transitive closure of a class   tr0elw 36857
            ​*21.17.4  Stronger axioms of regularity   mh-setind 36909
            21.17.5  Short axioms written in primitive symbols   mh-inf3f1 36914
      ​21.18  Mathbox for Asger C. Ipsen
            21.18.1  Continuous nowhere differentiable functions   dnival 36922
      ​​*21.19  Mathbox for BJ
            *21.19.1  Propositional calculus   bj-mp2c 36991
                  *21.19.1.1  Derived rules of inference   bj-mp2c 36991
                  ​*21.19.1.2  A syntactic theorem   bj-0 36993
                  ​*21.19.1.3  Minimal implicational calculus   bj-poni 36995
                  ​*21.19.1.4  Positive calculus   bj-bisimpl 37007
                  ​*21.19.1.5  Implication and negation   bj-con2com 37015
                  ​*21.19.1.6  Disjunction   bj-jaoi1 37026
                  ​*21.19.1.7  Logical equivalence   bj-dfbi4 37028
                  21.19.1.8  The conditional operator for propositions   bj-consensus 37033
                  ​*21.19.1.9  Propositional calculus: miscellaneous   bj-imbi12 37038
            ​*21.19.2  Modal logic   bj-axdd2 37047
            ​*21.19.3  Provability logic   cprvb 37052
            ​*21.19.4  First-order logic   bj-exexalal 37061
                  21.19.4.1  Universal and existential quantifiers, nonfreeness predicate   bj-exexalal 37061
                  21.19.4.2  Adding ax-gen   bj-genr 37062
                  21.19.4.3  Adding ax-4   bj-almp 37066
                  21.19.4.4  Adding ax-5   bj-spvw 37119
                  21.19.4.5  Equality and substitution   bj-df-sb 37134
                  21.19.4.6  Adding ax-6   bj-spim0 37153
                  21.19.4.7  Adding ax-7   bj-cbvexw 37161
                  21.19.4.8  Membership predicate, ax-8 and ax-9   bj-ax89 37163
                  21.19.4.9  Adding ax-11   bj-alcomexcom 37165
                  21.19.4.10  Adding ax-12   axc11n11 37169
                  ​*21.19.4.11  Really adding ax-12   bj-substax12 37211
                  21.19.4.12  Nonfreeness   wnnf 37213
                  21.19.4.13  Adding ax-13   bj-axc10 37280
                  ​*21.19.4.14  Removing dependencies on ax-13 (and ax-11)   bj-axc10v 37290
                  ​*21.19.4.15  Distinct var metavariables   bj-hbaeb2 37315
                  ​*21.19.4.16  Around ~ equsal   bj-equsal1t 37319
                  ​*21.19.4.17  Some Principia Mathematica proofs   stdpc5t 37324
                  21.19.4.18  Alternate definition of substitution   bj-sbsb 37334
                  21.19.4.19  Lemmas for substitution   bj-sbf3 37336
                  21.19.4.20  Existential uniqueness   bj-eu3f 37338
                  ​*21.19.4.21  First-order logic: miscellaneous   bj-sblem1 37339
            21.19.5  Set theory   eliminable1 37356
                  ​*21.19.5.1  Eliminability of class terms   eliminable1 37356
                  ​*21.19.5.2  Classes without the axiom of extensionality   bj-denoteslem 37368
                  21.19.5.3  Characterization among sets versus among classes   elelb 37394
                  ​*21.19.5.4  The nonfreeness quantifier for classes   bj-nfcsym 37396
                  ​*21.19.5.5  Lemmas for class substitution   bj-sbeqALT 37397
                  21.19.5.6  Removing some axiom requirements and disjoint variable conditions   bj-exlimvmpi 37408
                  ​*21.19.5.7  Class abstractions   bj-elabd2ALT 37422
                  21.19.5.8  Generalized class abstractions   bj-cgab 37430
                  ​*21.19.5.9  Restricted nonfreeness   wrnf 37438
                  ​*21.19.5.10  Russell's paradox   bj-ru1 37440
                  21.19.5.11  Curry's paradox in set theory   currysetlem 37442
                  ​*21.19.5.12  Some disjointness results   bj-n0i 37448
                  ​*21.19.5.13  Complements on direct products   bj-xpimasn 37452
                  ​*21.19.5.14  "Singletonization" and tagging   bj-snsetex 37460
                  ​*21.19.5.15  Tuples of classes   bj-cproj 37487
                  ​*21.19.5.16  Set theory: elementary operations relative to a universe   bj-rcleqf 37522
                  ​*21.19.5.17  Axioms for finite unions   bj-abex 37527
                  ​*21.19.5.18  Set theory: miscellaneous   eleq2w2ALT 37544
                  ​*21.19.5.19  Axioms of separation and replacement   bj-axnul 37569
                  ​*21.19.5.20  Evaluation at a class   bj-evaleq 37573
                  21.19.5.21  Elementwise operations   celwise 37581
                  ​*21.19.5.22  Elementwise intersection (families of sets induced on a subset)   bj-rest00 37583
                  21.19.5.23  Moore collections (complements)   bj-raldifsn 37602
                  21.19.5.24  Maps-to notation for functions with three arguments   bj-0nelmpt 37618
                  ​*21.19.5.25  Currying   csethom 37624
                  ​*21.19.5.26  Setting components of extensible structures   cstrset 37636
            ​*21.19.6  Extended real and complex numbers, real and complex projective lines   bj-nfald 37639
                  21.19.6.1  Complements on class abstractions of ordered pairs and binary relations   bj-nfald 37639
                  ​*21.19.6.2  Identity relation (complements)   bj-opabssvv 37654
                  ​*21.19.6.3  Functionalized identity (diagonal in a Cartesian square)   cdiag2 37676
                  ​*21.19.6.4  Direct image and inverse image   cimdir 37682
                  ​*21.19.6.5  Extended numbers and projective lines as sets   cfractemp 37700
                  ​*21.19.6.6  Addition and opposite   caddcc 37741
                  ​*21.19.6.7  Order relation on the extended reals   cltxr 37745
                  ​*21.19.6.8  Argument, multiplication and inverse   carg 37747
                  21.19.6.9  The canonical bijection from the finite ordinals   ciomnn 37753
                  21.19.6.10  Divisibility   cnnbar 37764
            ​*21.19.7  Monoids   bj-smgrpssmgm 37772
                  ​*21.19.7.1  Finite sums in monoids   cfinsum 37787
            ​*21.19.8  Affine, Euclidean, and Cartesian geometry   bj-fvimacnv0 37790
                  *21.19.8.1  Real vector spaces   bj-fvimacnv0 37790
                  ​*21.19.8.2  Complex numbers (supplements)   bj-subcom 37812
                  ​*21.19.8.3  Barycentric coordinates   bj-bary1lem 37814
            21.19.9  Monoid of endomorphisms   cend 37817
      ​21.20  Mathbox for Jim Kingdon
            21.20.1  Circle constant   taupilem3 37823
            21.20.2  Number theory   dfgcd3 37828
            21.20.3  Real numbers   irrdifflemf 37829
      ​21.21  Mathbox for ML
            21.21.1  Miscellaneous   csbrecsg 37834
            21.21.2  Cartesian exponentiation   cfinxp 37889
            21.21.3  Topology   iunctb2 37909
                  ​*21.21.3.1  Pi-base theorems   pibp16 37919
      ​21.22  Mathbox for Wolf Lammen
            21.22.1  1. Bootstrapping   wl-section-boot 37928
            21.22.2  Implication chains   wl-section-impchain 37952
            21.22.3  Theorems around the conditional operator   wl-ifp-ncond1 37970
            21.22.4  Alternative development of hadd, cadd   wl-df-3xor 37974
            21.22.5  An alternative axiom ~ ax-13   ax-wl-13v 37999
            21.22.6  Bootstrapping set theory with classes   wl-cleq-0 38001
            21.22.7  Other stuff   wl-mps 38022
      ​21.23  Mathbox for Brendan Leahy
      ​21.24  Mathbox for Jeff Madsen
            21.24.1  Logic and set theory   unirep 38225
            21.24.2  Real and complex numbers; integers   filbcmb 38251
            21.24.3  Sequences and sums   sdclem2 38253
            21.24.4  Topology   subspopn 38263
            21.24.5  Metric spaces   metf1o 38266
            21.24.6  Continuous maps and homeomorphisms   constcncf 38273
            21.24.7  Boundedness   ctotbnd 38277
            21.24.8  Isometries   cismty 38309
            21.24.9  Heine-Borel Theorem   heibor1lem 38320
            21.24.10  Banach Fixed Point Theorem   bfplem1 38333
            21.24.11  Euclidean space   crrn 38336
            21.24.12  Intervals (continued)   ismrer1 38349
            21.24.13  Operation properties   cass 38353
            21.24.14  Groups and related structures   cmagm 38359
            21.24.15  Group homomorphism and isomorphism   cghomOLD 38394
            21.24.16  Rings   crngo 38405
            21.24.17  Division Rings   cdrng 38459
            21.24.18  Ring homomorphisms   crngohom 38471
            21.24.19  Commutative rings   ccm2 38500
            21.24.20  Ideals   cidl 38518
            21.24.21  Prime rings and integral domains   cprrng 38557
            21.24.22  Ideal generators   cigen 38570
      ​21.25  Mathbox for Giovanni Mascellani
            ​*21.25.1  Tools for automatic proof building   efald2 38589
            ​*21.25.2  Tseitin axioms   fald 38640
            ​*21.25.3  Equality deductions   iuneq2f 38667
            ​*21.25.4  Miscellanea   orcomdd 38678
      ​21.26  Mathbox for Peter Mazsa
            21.26.1  Notations   cxrn 38685
            21.26.2  Preparatory theorems   el2v1 38740
            21.26.3  Range Cartesian product   df-xrn 38891
            21.26.4  Relations   df-rels 38951
            21.26.5  Quotient map (coset map)   df-qmap 38957
            21.26.6  Lifts, shifts, successor, and predecessor   df-adjliftmap 38966
            21.26.7  Cosets by ` R `   df-coss 39012
            21.26.8  Subset relations   df-ssr 39089
            21.26.9  Reflexivity   df-refs 39101
            21.26.10  Converse reflexivity   df-cnvrefs 39116
            21.26.11  Symmetry   df-syms 39133
            21.26.12  Reflexivity and symmetry   symrefref2 39158
            21.26.13  Transitivity   df-trs 39167
            21.26.14  Equivalence relations   df-eqvrels 39179
            21.26.15  Redundancy   df-redunds 39218
            21.26.16  Domain quotients   df-dmqss 39233
            21.26.17  Equivalence relations on domain quotients   df-ers 39259
            21.26.18  Functions   df-funss 39276
            21.26.19  Disjoints vs. converse functions   df-disjss 39299
            21.26.20  Antisymmetry   df-antisymrel 39374
            21.26.21  Partitions: disjoints on domain quotients   df-parts 39379
            21.26.22  Partition-Equivalence Theorems   disjim 39395
            21.26.23  Type-safe Partition-Equivalence: PetParts, PetErs, Pet2Parts, Pet2Ers   df-petparts 39479
      ​21.27  Mathbox for Rodolfo Medina
            21.27.1  Partitions   prtlem60 39489
      ​​*21.28  Mathbox for Norm Megill
            *21.28.1  Obsolete schemes ax-c4,c5,c7,c10,c11,c11n,c15,c9,c14,c16   ax-c5 39519
            ​*21.28.2  Rederive new axioms ax-4, ax-10, ax-6, ax-12, ax-13 from old   axc5 39529
            ​*21.28.3  Legacy theorems using obsolete axioms   ax5ALT 39543
            21.28.4  Experiments with weak deduction theorem   elimhyps 39597
            21.28.5  Miscellanea   cnaddcom 39608
            21.28.6  Atoms, hyperplanes, and covering in a left vector space (or module)   clsa 39610
            21.28.7  Functionals and kernels of a left vector space (or module)   clfn 39693
            21.28.8  Opposite rings and dual vector spaces   cld 39759
            21.28.9  Ortholattices and orthomodular lattices   cops 39808
            21.28.10  Atomic lattices with covering property   ccvr 39898
            21.28.11  Hilbert lattices   chlt 39986
            21.28.12  Projective geometries based on Hilbert lattices   clln 40127
            21.28.13  Construction of a vector space from a Hilbert lattice   cdlema1N 40427
            21.28.14  Construction of involution and inner product from a Hilbert lattice   clpoN 42116
      ​21.29  Mathbox for metakunt
            21.29.1  Commutative Semiring   ccsrg 42598
            21.29.2  General helpful statements   rhmzrhval 42601
            21.29.3  Some gcd and lcm results   12gcd5e1 42632
            21.29.4  Least common multiple inequality theorem   3factsumint1 42650
            21.29.5  Logarithm inequalities   3exp7 42682
            21.29.6  Miscellaneous results for AKS formalisation   intlewftc 42690
            21.29.7  Sticks and stones   sticksstones1 42775
            21.29.8  Continuation AKS   aks6d1c6lem1 42799
      ​21.30  Mathbox for Steven Nguyen
            21.30.1  Utility theorems   jarrii 42834
            ​*21.30.2  Arithmetic theorems   c0exALT 42880
            21.30.3  Exponents and divisibility   oexpreposd 42943
            21.30.4  Trigonometry and Calculus   tanhalfpim 42970
            ​*21.30.5  Independence of ax-mulcom   cresub 42986
            21.30.6  Structures   sn-base0 43129
            ​*21.30.7  Projective spaces   cprjsp 43195
            21.30.8  Basic reductions for Fermat's Last Theorem   dffltz 43228
            ​*21.30.9  Exemplar theorems   iddii 43258
                  ​*21.30.9.1  Standard replacements of ax-10 , ax-11 , ax-12   nfa1w 43269
      ​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 43285
            21.33.2  Additional theory of functions   imaiinfv 43286
            21.33.3  Additional topology   elrfi 43287
            21.33.4  Characterization of closure operators. Kuratowski closure axioms   ismrcd1 43291
            21.33.5  Algebraic closure systems   cnacs 43295
            21.33.6  Miscellanea 1. Map utilities   constmap 43306
            21.33.7  Miscellanea for polynomials   mptfcl 43313
            21.33.8  Multivariate polynomials over the integers   cmzpcl 43314
            21.33.9  Miscellanea for Diophantine sets 1   coeq0i 43346
            21.33.10  Diophantine sets 1: definitions   cdioph 43348
            21.33.11  Diophantine sets 2 miscellanea   ellz1 43360
            21.33.12  Diophantine sets 2: union and intersection. Monotone Boolean algebra   diophin 43365
            21.33.13  Diophantine sets 3: construction   diophrex 43368
            21.33.14  Diophantine sets 4 miscellanea   2sbcrex 43377
            21.33.15  Diophantine sets 4: Quantification   rexrabdioph 43383
            21.33.16  Diophantine sets 5: Arithmetic sets   rabdiophlem1 43390
            21.33.17  Diophantine sets 6: reusability. renumbering of variables   eldioph4b 43400
            21.33.18  Pigeonhole Principle and cardinality helpers   fphpd 43405
            21.33.19  A non-closed set of reals is infinite   rencldnfilem 43409
            21.33.20  Lagrange's rational approximation theorem   irrapxlem1 43411
            21.33.21  Pell equations 1: A nontrivial solution always exists   pellexlem1 43418
            21.33.22  Pell equations 2: Algebraic number theory of the solution set   csquarenn 43425
            21.33.23  Pell equations 3: characterizing fundamental solution   infmrgelbi 43467
            ​*21.33.24  Logarithm laws generalized to an arbitrary base   reglogcl 43479
            21.33.25  Pell equations 4: the positive solution group is infinite cyclic   pellfund14 43487
            21.33.26  X and Y sequences 1: Definition and recurrence laws   crmx 43489
            21.33.27  Ordering and induction lemmas for the integers   monotuz 43530
            21.33.28  X and Y sequences 2: Order properties   rmxypos 43536
            21.33.29  Congruential equations   congtr 43554
            21.33.30  Alternating congruential equations   acongid 43564
            21.33.31  Additional theorems on integer divisibility   coprmdvdsb 43574
            21.33.32  X and Y sequences 3: Divisibility properties   jm2.18 43577
            21.33.33  X and Y sequences 4: Diophantine representability of Y   jm2.27a 43594
            21.33.34  X and Y sequences 5: Diophantine representability of X, ^, _C   rmxdiophlem 43604
            21.33.35  Uncategorized stuff not associated with a major project   setindtr 43613
            21.33.36  More equivalents of the Axiom of Choice   axac10 43622
            21.33.37  Finitely generated left modules   clfig 43656
            21.33.38  Noetherian left modules I   clnm 43664
            21.33.39  Addenda for structure powers   pwssplit4 43678
            21.33.40  Every set admits a group structure iff choice   unxpwdom3 43684
            21.33.41  Noetherian rings and left modules II   clnr 43698
            21.33.42  Hilbert's Basis Theorem   cldgis 43710
            21.33.43  Additional material on polynomials [DEPRECATED]   cmnc 43720
            21.33.44  Degree and minimal polynomial of algebraic numbers   cdgraa 43729
            21.33.45  Algebraic integers I   citgo 43746
            21.33.46  Endomorphism algebra   cmend 43760
            21.33.47  Cyclic groups and order   idomodle 43780
            21.33.48  Cyclotomic polynomials   ccytp 43786
            21.33.49  Miscellaneous topology   fgraphopab 43792
      ​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 43806
            21.36.2  Natural addition of Cantor normal forms   oawordex2 43915
            21.36.3  Surreal Contributions   abeqabi 43996
            21.36.4  Short Studies   nlimsuc 44029
                  21.36.4.1  Additional work on conditional logical operator   ifpan123g 44047
                  21.36.4.2  Sophisms   rp-fakeimass 44100
                  ​*21.36.4.3  Finite Sets   rp-isfinite5 44105
                  21.36.4.4  General Observations   intabssd 44107
                  21.36.4.5  Infinite Sets   pwelg 44148
                  ​*21.36.4.6  Finite intersection property   fipjust 44153
                  21.36.4.7  RP ADDTO: Subclasses and subsets   rababg 44162
                  21.36.4.8  RP ADDTO: The intersection of a class   elinintab 44163
                  21.36.4.9  RP ADDTO: Theorems requiring subset and intersection existence   elinintrab 44165
                  21.36.4.10  RP ADDTO: Relations   xpinintabd 44168
                  ​*21.36.4.11  RP ADDTO: Functions   elmapintab 44184
                  ​*21.36.4.12  RP ADDTO: Finite induction (for finite ordinals)   cnvcnvintabd 44188
                  21.36.4.13  RP ADDTO: First and second members of an ordered pair   elcnvlem 44189
                  21.36.4.14  RP ADDTO: The reflexive and transitive properties of relations   undmrnresiss 44192
                  21.36.4.15  RP ADDTO: Basic properties of closures   cleq2lem 44196
                  21.36.4.16  RP REPLACE: Definitions and basic properties of transitive closures   trcleq2lemRP 44218
                  ​*21.36.4.17  Additions for square root; absolute value   sqrtcvallem1 44219
            21.36.5  Additional statements on relations and subclasses   al3im 44235
                  21.36.5.1  Transitive relations (not to be confused with transitive classes)   trrelind 44253
                  21.36.5.2  Reflexive closures   crcl 44260
                  ​*21.36.5.3  Finite relationship composition   relexp2 44265
                  21.36.5.4  Transitive closure of a relation   dftrcl3 44308
                  ​*21.36.5.5  Adapted from Frege   frege77d 44334
            ​*21.36.6  Propositions from _Begriffsschrift_   dfxor4 44354
                  *21.36.6.1  _Begriffsschrift_ Chapter I   dfxor4 44354
                  ​*21.36.6.2  _Begriffsschrift_ Notation hints   whe 44360
                  21.36.6.3  _Begriffsschrift_ Chapter II Implication   ax-frege1 44378
                  21.36.6.4  _Begriffsschrift_ Chapter II Implication and Negation   axfrege28 44417
                  ​*21.36.6.5  _Begriffsschrift_ Chapter II with logical equivalence   axfrege52a 44444
                  21.36.6.6  _Begriffsschrift_ Chapter II with equivalence of sets   axfrege52c 44475
                  ​*21.36.6.7  _Begriffsschrift_ Chapter II with equivalence of classes   frege53c 44502
                  ​*21.36.6.8  _Begriffsschrift_ Chapter III Properties hereditary in a sequence   dffrege69 44520
                  ​*21.36.6.9  _Begriffsschrift_ Chapter III Following in a sequence   dffrege76 44527
                  ​*21.36.6.10  _Begriffsschrift_ Chapter III Member of sequence   dffrege99 44550
                  ​*21.36.6.11  _Begriffsschrift_ Chapter III Single-valued procedures   dffrege115 44566
            ​*21.36.7  Exploring Topology via Seifert and Threlfall   enrelmap 44585
                  *21.36.7.1  Equinumerosity of sets of relations and maps   enrelmap 44585
                  ​*21.36.7.2  Generic Pseudoclosure Spaces, Pseudointerior Spaces, and Pseudoneighborhoods   or3or 44611
                  ​*21.36.7.3  Generic Neighborhood Spaces   gneispa 44718
            ​*21.36.8  Exploring Higher Homotopy via Kerodon   k0004lem1 44735
                  *21.36.8.1  Simplicial Sets   k0004lem1 44735
      ​21.37  Mathbox for Stanislas Polu
            21.37.1  IMO Problems   wwlemuld 44744
                  21.37.1.1  IMO 1972 B2   wwlemuld 44744
            ​*21.37.2  INT Inequalities Proof Generator   int-addcomd 44761
            ​*21.37.3  N-Digit Addition Proof Generator   unitadd 44783
            21.37.4  AM-GM (for k = 2,3,4)   gsumws3 44784
      ​21.38  Mathbox for Rohan Ridenour
            21.38.1  Misc   spALT 44789
            21.38.2  Monoid rings   cmnring 44799
            21.38.3  Shorter primitive equivalent of ax-groth   gru0eld 44817
                  21.38.3.1  Grothendieck universes are closed under collection   gru0eld 44817
                  21.38.3.2  Minimal universes   ismnu 44835
                  21.38.3.3  Primitive equivalent of ax-groth   expandan 44862
      ​21.39  Mathbox for Steve Rodriguez
            21.39.1  Miscellanea   nanorxor 44879
            21.39.2  Ratio test for infinite series convergence and divergence   dvgrat 44886
            21.39.3  Multiples   reldvds 44889
            21.39.4  Function operations   caofcan 44897
            21.39.5  Calculus   lhe4.4ex1a 44903
            21.39.6  The generalized binomial coefficient operation   cbcc 44910
            21.39.7  Binomial series   uzmptshftfval 44920
      ​21.40  Mathbox for Andrew Salmon
            21.40.1  Principia Mathematica * 10   pm10.12 44932
            21.40.2  Principia Mathematica * 11   2alanimi 44946
            21.40.3  Predicate Calculus   sbeqal1 44972
            21.40.4  Principia Mathematica * 13 and * 14   pm13.13a 44981
            21.40.5  Set Theory   elnev 45011
            21.40.6  Arithmetic   addcomgi 45029
            21.40.7  Geometry   cplusr 45030
      ​​*21.41  Mathbox for Alan Sare
            21.41.1  Auxiliary theorems for the Virtual Deduction tool   idiALT 45052
            21.41.2  Supplementary unification deductions   bi1imp 45056
            21.41.3  Conventional Metamath proofs, some derived from VD proofs   iidn3 45075
            21.41.4  What is Virtual Deduction?   wvd1 45143
            21.41.5  Virtual Deduction Theorems   df-vd1 45144
            21.41.6  Theorems proved using Virtual Deduction   trsspwALT 45391
            21.41.7  Theorems proved using Virtual Deduction with mmj2 assistance   simplbi2VD 45419
            21.41.8  Virtual Deduction transcriptions of textbook proofs   sb5ALTVD 45486
            21.41.9  Theorems proved using conjunction-form Virtual Deduction   elpwgdedVD 45490
            21.41.10  Theorems with a VD proof in conventional notation derived from a VD proof   suctrALT3 45497
            ​*21.41.11  Theorems with a proof in conventional notation derived from a VD proof   notnotrALT2 45500
      ​21.42  Mathbox for Eric Schmidt
            21.42.1  Miscellany   rspesbcd 45511
            21.42.2  Study of dfbi1ALT   dfbi1ALTa 45513
            21.42.3  Relation-preserving functions   wrelp 45516
            21.42.4  Orbits   orbitex 45529
            21.42.5  Well-founded sets   trwf 45533
            21.42.6  Absoluteness in transitive models   ralabso 45542
            21.42.7  Lemmas for showing axioms hold in models   traxext 45551
            21.42.8  The class of well-founded sets is a model for ZFC   wfaxext 45567
            21.42.9  Permutation models   brpermmodel 45577
      ​21.43  Mathbox for Glauco Siliprandi
            21.43.1  Miscellanea   evth2f 45593
            21.43.2  Functions   fnresdmss 45744
            21.43.3  Ordering on real numbers - Real and complex numbers basic operations   sub2times 45850
            21.43.4  Real intervals   gtnelioc 46065
            21.43.5  Finite sums   fsummulc1f 46145
            21.43.6  Finite multiplication of numbers and finite multiplication of functions   fmul01 46154
            21.43.7  Limits   clim1fr1 46175
                  21.43.7.1  Inferior limit (lim inf)   clsi 46323
                  ​*21.43.7.2  Limits for sequences of extended real numbers   clsxlim 46390
            21.43.8  Trigonometry   coseq0 46436
            21.43.9  Continuous Functions   mulcncff 46442
            21.43.10  Derivatives   dvsinexp 46483
            21.43.11  Integrals   itgsin0pilem1 46522
            21.43.12  Stone Weierstrass theorem - real version   stoweidlem1 46573
            21.43.13  Wallis' product for π   wallispilem1 46637
            21.43.14  Stirling's approximation formula for ` n ` factorial   stirlinglem1 46646
            21.43.15  Dirichlet kernel   dirkerval 46663
            21.43.16  Fourier Series   fourierdlem1 46680
            21.43.17  e is transcendental   elaa2lem 46805
            21.43.18  n-dimensional Euclidean space   rrxtopn 46856
            21.43.19  Basic measure theory   csalg 46880
                  ​*21.43.19.1  σ-Algebras   csalg 46880
                  21.43.19.2  Sum of nonnegative extended reals   csumge0 46934
                  ​*21.43.19.3  Measures   cmea 47021
                  ​*21.43.19.4  Outer measures and Caratheodory's construction   come 47061
                  ​*21.43.19.5  Lebesgue measure on n-dimensional Real numbers   covoln 47108
                  ​*21.43.19.6  Measurable functions   csmblfn 47267
      ​21.44  Mathbox for Saveliy Skresanov
            21.44.1  Ceva's theorem   sigarval 47422
            21.44.2  Simple groups   simpcntrab 47442
      ​21.45  Mathbox for Ender Ting
            21.45.1  Interesting facts   et-ltneverrefl 47443
            21.45.2  Increasing sequences and subsequences   ormklocald 47448
            21.45.3  Scratchpad for number theory   evenwodadd 47461
            21.45.4  Scratchpad for math on real numbers   squeezedltsq 47462
      ​21.46  Mathbox for Jarvin Udandy
      ​21.47  Mathbox for Adhemar
            ​*21.47.1  Minimal implicational calculus   adh-minim 47593
      ​21.48  Mathbox for Alexander van der Vekens
            21.48.1  General auxiliary theorems (1)   n0nsn2el 47617
                  21.48.1.1  Unordered and ordered pairs - extension for singletons   n0nsn2el 47617
                  21.48.1.2  Unordered and ordered pairs - extension for unordered pairs   elprneb 47621
                  21.48.1.3  Unordered and ordered pairs - extension for ordered pairs   oppr 47622
                  21.48.1.4  Relations - extension   eubrv 47627
                  21.48.1.5  Definite description binder (inverted iota) - extension   iota0def 47630
                  21.48.1.6  Functions - extension   fveqvfvv 47632
            21.48.2  Alternative for Russell's definition of a description binder   caiota 47675
            21.48.3  Double restricted existential uniqueness   r19.32 47690
                  21.48.3.1  Restricted quantification (extension)   r19.32 47690
                  21.48.3.2  Restricted uniqueness and "at most one" quantification   reuf1odnf 47699
                  21.48.3.3  Analogs to Existential uniqueness (double quantification)   2reu3 47702
                  21.48.3.4  Additional theorems for double restricted existential uniqueness   2reu8i 47705
            ​*21.48.4  Alternative definitions of function and operation values   wdfat 47708
                  21.48.4.1  Restricted quantification (extension)   ralbinrald 47714
                  21.48.4.2  The universal class (extension)   nvelim 47715
                  21.48.4.3  Introduce the Axiom of Power Sets (extension)   alneu 47716
                  21.48.4.4  Predicate "defined at"   dfateq12d 47718
                  21.48.4.5  Alternative definition of the value of a function   dfafv2 47724
                  21.48.4.6  Alternative definition of the value of an operation   aoveq123d 47770
            ​*21.48.5  Alternative definitions of function values (2)   cafv2 47800
            21.48.6  General auxiliary theorems (2)   an4com24 47860
                  21.48.6.1  Logical conjunction - extension   an4com24 47860
                  21.48.6.2  Abbreviated conjunction and disjunction of three wff's - extension   3an4ancom24 47861
                  21.48.6.3  Negated membership (alternative)   cnelbr 47863
                  21.48.6.4  The empty set - extension   ralralimp 47870
                  21.48.6.5  Indexed union and intersection - extension   otiunsndisjX 47871
                  21.48.6.6  Functions - extension   fvifeq 47872
                  21.48.6.7  Maps-to notation - extension   fvmptrab 47884
                  21.48.6.8  Subtraction - extension   cnambpcma 47886
                  21.48.6.9  Ordering on reals (cont.) - extension   leaddsuble 47889
                  21.48.6.10  Imaginary and complex number properties - extension   readdcnnred 47895
                  21.48.6.11  Nonnegative integers (as a subset of complex numbers) - extension   nn0resubcl 47900
                  21.48.6.12  Integers (as a subset of complex numbers) - extension   zgeltp1eq 47901
                  21.48.6.13  Decimal arithmetic - extension   1t10e1p1e11 47902
                  21.48.6.14  Upper sets of integers - extension   eluzge0nn0 47904
                  21.48.6.15  Infinity and the extended real number system (cont.) - extension   nltle2tri 47905
                  21.48.6.16  Finite intervals of integers - extension   ssfz12 47906
                  21.48.6.17  Half-open integer ranges - extension   fzopred 47915
                  21.48.6.18  The floor and ceiling functions - extension   2ltceilhalf 47924
                  21.48.6.19  The modulo (remainder) operation - extension   fldivmod 47936
                  21.48.6.20  The infinite sequence builder "seq"   smonoord 47969
                  21.48.6.21  Integer powers - extension   2timesltsq 47970
                  21.48.6.22  Finite and infinite sums - extension   fsummsndifre 47972
                  21.48.6.23  The divides relation - extension   nndivides2 47976
                  21.48.6.24  Extensible structures - extension   setsidel 47980
            ​*21.48.7  Preimages of function values   preimafvsnel 47983
            ​*21.48.8  Partitions of real intervals   ciccp 48017
            21.48.9  Shifting functions with an integer range domain   fargshiftfv 48043
            21.48.10  Words over a set (extension)   lswn0 48048
                  21.48.10.1  Last symbol of a word - extension   lswn0 48048
            21.48.11  Unordered pairs   wich 48049
                  21.48.11.1  Interchangeable setvar variables   wich 48049
                  21.48.11.2  Set of unordered pairs   sprid 48078
                  ​*21.48.11.3  Proper (unordered) pairs   prpair 48105
                  21.48.11.4  Set of proper unordered pairs   cprpr 48116
            21.48.12  Number theory (extension)   nprmmul1 48131
                  21.48.12.1  Properties of non-prime numbers   nprmmul1 48131
                  ​*21.48.12.2  Fermat numbers   cfmtno 48134
                  ​*21.48.12.3  Mersenne primes   m2prm 48198
                  21.48.12.4  Proth's theorem   modexp2m1d 48219
                  21.48.12.5  The prime-counting function according to Ján Mináč   nprmdvdsfacm1lem1 48227
                  21.48.12.6  Solutions of quadratic equations   quad1 48240
            ​*21.48.13  Even and odd numbers   ceven 48244
                  21.48.13.1  Definitions and basic properties   ceven 48244
                  21.48.13.2  Alternate definitions using the "divides" relation   dfeven2 48269
                  21.48.13.3  Alternate definitions using the "modulo" operation   dfeven3 48278
                  21.48.13.4  Alternate definitions using the "gcd" operation   iseven5 48284
                  21.48.13.5  Theorems of part 5 revised   zneoALTV 48289
                  21.48.13.6  Theorems of part 6 revised   odd2np1ALTV 48294
                  21.48.13.7  Theorems of AV's mathbox revised   0evenALTV 48308
                  21.48.13.8  Additional theorems   epoo 48323
                  21.48.13.9  Perfect Number Theorem (revised)   perfectALTVlem1 48341
            21.48.14  Number theory (extension 2)   cfppr 48344
                  ​*21.48.14.1  Fermat pseudoprimes   cfppr 48344
                  ​*21.48.14.2  Goldbach's conjectures   cgbe 48365
            21.48.15  Graph theory (extension)   cclnbgr 48438
                  21.48.15.1  Closed neighborhood of a vertex   cclnbgr 48438
                  ​*21.48.15.2  Semiclosed and semiopen neighborhoods (experimental)   dfsclnbgr2 48466
                  21.48.15.3  Induced subgraphs   cisubgr 48480
                  ​*21.48.15.4  Isomorphisms of graphs   cgrisom 48494
                  ​*21.48.15.5  Triangles in graphs   cgrtri 48557
                  ​*21.48.15.6  Star graphs   cstgr 48571
                  ​*21.48.15.7  Local isomorphisms of graphs   cgrlim 48596
                  ​*21.48.15.8  Generalized Petersen graphs   cgpg 48660
                  21.48.15.9  Loop-free graphs - extension   1hegrlfgr 48752
                  21.48.15.10  Walks - extension   cupwlks 48753
                  21.48.15.11  Edges of graphs expressed as sets of unordered pairs   upgredgssspr 48763
            21.48.16  Monoids (extension)   ovn0dmfun 48776
                  21.48.16.1  Auxiliary theorems   ovn0dmfun 48776
                  21.48.16.2  Magmas, Semigroups and Monoids (extension)   plusfreseq 48784
                  21.48.16.3  Examples and counterexamples for magmas, semigroups and monoids (extension)   opmpoismgm 48787
                  21.48.16.4  Group sum operation (extension 1)   gsumsplit2f 48800
            ​*21.48.17  Magmas and internal binary operations (alternate approach)   ccllaw 48803
                  *21.48.17.1  Laws for internal binary operations   ccllaw 48803
                  ​*21.48.17.2  Internal binary operations   cintop 48816
                  21.48.17.3  Alternative definitions for magmas and semigroups   cmgm2 48835
            21.48.18  Rings (extension)   lmod0rng 48849
                  21.48.18.1  Nonzero rings (extension)   lmod0rng 48849
                  21.48.18.2  Ideals as non-unital rings   lidldomn1 48851
                  21.48.18.3  The non-unital ring of even integers   0even 48857
                  21.48.18.4  A constructed not unital ring   cznrnglem 48879
                  ​*21.48.18.5  The category of non-unital rings (alternate definition)   crngcALTV 48883
                  ​*21.48.18.6  The category of (unital) rings (alternate definition)   cringcALTV 48907
            ​*21.48.19  Prime rings (and integral domains)   cprmrng 48954
            21.48.20  Basic algebraic structures (extension)   eliunxp2 48965
                  21.48.20.1  Auxiliary theorems   eliunxp2 48965
                  21.48.20.2  The binomial coefficient operation (extension)   bcpascm1 48982
                  21.48.20.3  The ` ZZ `-module ` ZZ X. ZZ `   zlmodzxzlmod 48985
                  21.48.20.4  Group sum operation (extension 2)   mgpsumunsn 48992
                  21.48.20.5  Symmetric groups (extension)   exple2lt6 48995
                  21.48.20.6  Divisibility (extension)   invginvrid 48998
                  21.48.20.7  The support of functions (extension)   rmsupp0 48999
                  21.48.20.8  Finitely supported functions (extension)   rmsuppfi 49003
                  21.48.20.9  Left modules (extension)   lmodvsmdi 49010
                  21.48.20.10  Associative algebras (extension)   assaascl0 49012
                  21.48.20.11  Univariate polynomials (extension)   ply1vr1smo 49014
                  21.48.20.12  Univariate polynomials (examples)   linply1 49024
            21.48.21  Linear algebra (extension)   cdmatalt 49027
                  ​*21.48.21.1  The subalgebras of diagonal and scalar matrices (extension)   cdmatalt 49027
                  ​*21.48.21.2  Linear combinations   clinc 49035
                  ​*21.48.21.3  Linear independence   clininds 49071
                  21.48.21.4  Simple left modules and the ` ZZ `-module   lmod1lem1 49118
                  21.48.21.5  Differences between (left) modules and (left) vector spaces   lvecpsslmod 49138
            21.48.22  Complexity theory   suppdm 49141
                  21.48.22.1  Auxiliary theorems   suppdm 49141
                  21.48.22.2  Even and odd integers   nn0onn0ex 49154
                  21.48.22.3  The natural logarithm on complex numbers (extension)   logcxp0 49166
                  21.48.22.4  Division of functions   cfdiv 49168
                  21.48.22.5  Upper bounds   cbigo 49178
                  21.48.22.6  Logarithm to an arbitrary base (extension)   rege1logbrege0 49189
                  ​*21.48.22.7  The binary logarithm   fldivexpfllog2 49196
                  21.48.22.8  Binary length   cblen 49200
                  ​*21.48.22.9  Digits   cdig 49226
                  21.48.22.10  Nonnegative integer as sum of its shifted digits   dignn0flhalflem1 49246
                  21.48.22.11  Algorithms for the multiplication of nonnegative integers   nn0mulfsum 49255
                  ​*21.48.22.12  N-ary functions   cnaryf 49257
                  ​*21.48.22.13  The Ackermann function   citco 49288
            21.48.23  Elementary geometry (extension)   fv1prop 49330
                  21.48.23.1  Auxiliary theorems   fv1prop 49330
                  21.48.23.2  Real euclidean space of dimension 2   rrx2pxel 49342
                  21.48.23.3  Spheres and lines in real Euclidean spaces   cline 49358
      ​21.49  Mathbox for Zhi Wang
            21.49.1  Propositional calculus   logic1 49420
            21.49.2  Predicate calculus with equality   dtrucor3 49428
                  21.49.2.1  Axiom scheme ax-5 (Distinctness)   dtrucor3 49428
            21.49.3  ZF Set Theory - start with the Axiom of Extensionality   ralbidb 49429
                  21.49.3.1  Restricted quantification   ralbidb 49429
                  21.49.3.2  The universal class   reuxfr1dd 49436
                  21.49.3.3  The empty set   ssdisjd 49437
                  21.49.3.4  Unordered and ordered pairs   vsn 49441
                  21.49.3.5  The union of a class   unilbss 49447
                  21.49.3.6  Indexed union and intersection   iuneq0 49448
            21.49.4  ZF Set Theory - add the Axiom of Replacement   inpw 49454
                  21.49.4.1  Theorems requiring subset and intersection existence   inpw 49454
            21.49.5  ZF Set Theory - add the Axiom of Power Sets   opth1neg 49455
                  21.49.5.1  Ordered pair theorem   opth1neg 49455
                  21.49.5.2  Ordered-pair class abstractions (cont.)   brab2dd 49457
                  21.49.5.3  Relations   iinxp 49460
                  21.49.5.4  Functions   mof0 49467
                  21.49.5.5  Operations   ovsng 49487
            21.49.6  ZF Set Theory - add the Axiom of Union   fonex 49496
                  21.49.6.1  Relations and functions (cont.)   fonex 49496
                  21.49.6.2  First and second members of an ordered pair   eloprab1st2nd 49497
                  21.49.6.3  Operations in maps-to notation (continued)   fmpodg 49498
                  21.49.6.4  Function transposition   resinsnlem 49500
                  21.49.6.5  Infinite Cartesian products   ixpv 49519
                  21.49.6.6  Equinumerosity   fvconst0ci 49520
            21.49.7  Order sets   iccin 49525
                  21.49.7.1  Real number intervals   iccin 49525
            21.49.8  Extensible structures   slotresfo 49528
                  21.49.8.1  Basic definitions   slotresfo 49528
            21.49.9  Moore spaces   mreuniss 49529
            ​*21.49.10  Topology   clduni 49530
                  21.49.10.1  Closure and interior   clduni 49530
                  21.49.10.2  Neighborhoods   neircl 49534
                  21.49.10.3  Subspace topologies   restcls2lem 49542
                  21.49.10.4  Limits and continuity in topological spaces   cnneiima 49546
                  21.49.10.5  Topological definitions using the reals   iooii 49547
                  21.49.10.6  Separated sets   sepnsepolem1 49551
                  21.49.10.7  Separated spaces: T0, T1, T2 (Hausdorff) ...   isnrm4 49560
            21.49.11  Preordered sets and directed sets using extensible structures   isprsd 49584
            21.49.12  Posets and lattices using extensible structures   lubeldm2 49585
                  21.49.12.1  Posets   lubeldm2 49585
                  21.49.12.2  Lattices   toslat 49611
                  21.49.12.3  Subset order structures   intubeu 49613
            21.49.13  Rings   elmgpcntrd 49634
                  21.49.13.1  Multiplicative Group   elmgpcntrd 49634
            21.49.14  Associative algebras   asclelbasALT 49635
                  21.49.14.1  Definition and basic properties   asclelbasALT 49635
            21.49.15  Categories   homf0 49638
                  21.49.15.1  Categories   homf0 49638
                  21.49.15.2  Opposite category   oppccatb 49645
                  21.49.15.3  Monomorphisms and epimorphisms   idmon 49649
                  21.49.15.4  Sections, inverses, isomorphisms   sectrcl 49651
                  21.49.15.5  Isomorphic objects   cicfn 49671
                  21.49.15.6  Subcategories   dmdm 49682
                  21.49.15.7  Functors   reldmfunc 49704
                  21.49.15.8  Opposite functors   coppf 49751
                  21.49.15.9  Full & faithful functors   imasubc 49780
                  21.49.15.10  Universal property   upciclem1 49795
                  21.49.15.11  Natural transformations and the functor category   isnatd 49852
                  21.49.15.12  Initial, terminal and zero objects of a category   initoo2 49861
                  21.49.15.13  Product of categories   reldmxpc 49875
                  21.49.15.14  Swap functors   cswapf 49888
                  21.49.15.15  Functor evaluation   oppc1stflem 49916
                  21.49.15.16  Transposed curry functors   cofuswapfcl 49922
                  21.49.15.17  Constant functors   diag1 49933
                  21.49.15.18  Functor composition bifunctors   fucofulem1 49939
                  21.49.15.19  Post-composition functors   postcofval 49993
                  21.49.15.20  Pre-composition functors   precofvallem 49995
            21.49.16  Examples of categories   catcrcl 50024
                  21.49.16.1  The category of categories   catcrcl 50024
                  21.49.16.2  Thin categories   cthinc 50046
                  21.49.16.3  Terminal categories   ctermc 50101
                  21.49.16.4  Preordered sets as thin categories   cprstc 50178
                  21.49.16.5  Monoids as categories   cmndtc 50206
                  21.49.16.6  Categories with at most one object and at most two morphisms   2arwcatlem1 50224
            21.49.17  Kan extensions and related concepts   clan 50234
                  21.49.17.1  Kan extensions   clan 50234
                  21.49.17.2  Limits and colimits   clmd 50272
      ​21.50  Mathbox for Emmett Weisz
            ​*21.50.1  Miscellaneous Theorems   nfintd 50302
            21.50.2  Set Recursion   csetrecs 50312
                  ​*21.50.2.1  Basic Properties of Set Recursion   csetrecs 50312
                  21.50.2.2  Examples and properties of set recursion   elsetrecslem 50328
            ​*21.50.3  Construction of Games and Surreal Numbers   cpg 50338
      ​​*21.51  Mathbox for David A. Wheeler
            21.51.1  Natural deduction   sbidd 50347
            ​*21.51.2  Greater than, greater than or equal to   cge-real 50349
            ​*21.51.3  Hyperbolic trigonometric functions   csinh 50359
            ​*21.51.4  Reciprocal trigonometric functions (sec, csc, cot)   csec 50370
            ​*21.51.5  Identities for "if"   ifnmfalse 50392
            ​*21.51.6  Logarithms generalized to arbitrary base using ` logb `   logb2aval 50393
            ​*21.51.7  Logarithm laws generalized to an arbitrary base - log_   clog- 50394
            ​*21.51.8  Formally define notions such as reflexivity   wreflexive 50396
            ​*21.51.9  Algebra helpers   mvlraddi 50400
            ​*21.51.10  Algebra helper examples   i2linesi 50407
            ​*21.51.11  Formal methods "surprises"   alimp-surprise 50409
            ​*21.51.12  Allsome quantifier   walsi 50415
            ​*21.51.13  Miscellaneous   5m4e1 50426
            21.51.14  Theorems about algebraic numbers   aacllem 50430
      ​21.52  Mathbox for Kunhao Zheng
            21.52.1  Weighted AM-GM inequality   amgmwlem 50431