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 Jorge Agra
      21.3  Mathbox for Stefan Allan
      21.4  Mathbox for Thierry Arnoux
      21.5  Mathbox for Jonathan Ben-Naim
      21.6  Mathbox for BTernaryTau
      21.7  Mathbox for Mario Carneiro
      21.8  Mathbox for Filip Cernatescu
      21.9  Mathbox for Paul Chapman
      21.10  Mathbox for Hongxiu Chen
      21.11  Mathbox for Adrian Ducourtial
      21.12  Mathbox for Scott Fenton
      21.13  Mathbox for Gino Giotto
      21.14  Mathbox for Jeff Hankins
      21.15  Mathbox for Anthony Hart
      21.16  Mathbox for Chen-Pang He
      21.17  Mathbox for Jeff Hoffman
      21.18  Mathbox for Matthew House
      21.19  Mathbox for Asger C. Ipsen
      21.20  Mathbox for BJ
      21.21  Mathbox for Jim Kingdon
      21.22  Mathbox for ML
      21.23  Mathbox for Wolf Lammen
      21.24  Mathbox for Brendan Leahy
      21.25  Mathbox for Thomas van Maaren
      21.26  Mathbox for Jeff Madsen
      21.27  Mathbox for Giovanni Mascellani
      21.28  Mathbox for Peter Mazsa
      21.29  Mathbox for Rodolfo Medina
      21.30  Mathbox for Norm Megill
      21.31  Mathbox for metakunt
      21.32  Mathbox for Luke Murphy
      21.33  Mathbox for Steven Nguyen
      21.34  Mathbox for Igor Ieskov
      21.35  Mathbox for Timothy J. O'Malley
      21.36  Mathbox for OpenAI
      21.37  Mathbox for Stefan O'Rear
      21.38  Mathbox for Noam Pasman
      21.39  Mathbox for Jon Pennant
      21.40  Mathbox for Richard Penner
      21.41  Mathbox for Stanislas Polu
      21.42  Mathbox for Rohan Ridenour
      21.43  Mathbox for Steve Rodriguez
      21.44  Mathbox for Andrew Salmon
      21.45  Mathbox for Alan Sare
      21.46  Mathbox for Eric Schmidt
      21.47  Mathbox for Glauco Siliprandi
      21.48  Mathbox for Saveliy Skresanov
      21.49  Mathbox for Ender Ting
      21.50  Mathbox for Jarvin Udandy
      21.51  Mathbox for Adhemar
      21.52  Mathbox for Alexander van der Vekens
      21.53  Mathbox for Zhi Wang
      21.54  Mathbox for Emmett Weisz
      21.55  Mathbox for David A. Wheeler
      21.56  Mathbox for Mingli Yuan
      21.57  Mathbox for Jiamin Zhao
      21.58  Mathbox for Kunhao Zheng

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

​​*PART 1  CLASSICAL FIRST-ORDER LOGIC WITH EQUALITY
      ​*1.1  Pre-logic
            *1.1.1  Inferences for assisting proof development   idi 1
      ​​*1.2  Propositional calculus
            1.2.1  Recursively define primitive wffs for propositional calculus   wn 3
            ​*1.2.2  The axioms of propositional calculus   ax-mp 5
            ​*1.2.3  Logical implication   mp2 9
            ​*1.2.4  Logical negation   con4 114
            ​*1.2.5  Logical equivalence   wb 209
            ​*1.2.6  Logical conjunction   wa 401
            ​*1.2.7  Logical disjunction   wo 861
            ​*1.2.8  Mixed connectives   jaao 969
            ​*1.2.9  The conditional operator for propositions   wif 1078
            ​*1.2.10  The weak deduction theorem for propositional calculus   elimh 1099
            1.2.11  Abbreviated conjunction and disjunction of three wff's   w3o 1102
            1.2.12  Logical "nand" (Sheffer stroke)   wnan 1520
            1.2.13  Logical "xor"   wxo 1540
            1.2.14  Logical "nor"   wnor 1557
            1.2.15  True and false constants   wal 1567
                  ​*1.2.15.1  Universal quantifier for use by df-tru   wal 1567
                  ​*1.2.15.2  Equality predicate for use by df-tru   cv 1568
                  1.2.15.3  The true constant   wtru 1570
                  1.2.15.4  The false constant   wfal 1581
            ​*1.2.16  Truth tables   truimtru 1592
                  1.2.16.1  Implication   truimtru 1592
                  1.2.16.2  Negation   nottru 1596
                  1.2.16.3  Equivalence   trubitru 1598
                  1.2.16.4  Conjunction   truantru 1602
                  1.2.16.5  Disjunction   truortru 1606
                  1.2.16.6  Alternative denial   trunantru 1610
                  1.2.16.7  Exclusive disjunction   truxortru 1614
                  1.2.16.8  Joint denial   trunortru 1618
            ​*1.2.17  Half adder and full adder in propositional calculus   whad 1622
                  1.2.17.1  Full adder: sum   whad 1622
                  1.2.17.2  Full adder: carry   wcad 1638
      ​1.3  Other axiomatizations related to classical propositional calculus
            ​*1.3.1  Minimal implicational calculus   minimp 1653
            ​*1.3.2  Implicational Calculus   impsingle 1659
            1.3.3  Derive the Lukasiewicz axioms from Meredith's sole axiom   meredith 1673
            1.3.4  Derive the standard axioms from the Lukasiewicz axioms   luklem1 1690
            ​*1.3.5  Derive Nicod's axiom from the standard axioms   nic-dfim 1701
            1.3.6  Derive the Lukasiewicz axioms from Nicod's axiom   nic-imp 1707
            1.3.7  Derive Nicod's Axiom from Lukasiewicz's First Sheffer Stroke Axiom   lukshef-ax1 1726
            1.3.8  Derive the Lukasiewicz Axioms from the Tarski-Bernays-Wajsberg Axioms   tbw-bijust 1730
            1.3.9  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's First CO Axiom   merco1 1745
            1.3.10  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's Second CO Axiom   merco2 1768
            1.3.11  Derive the Lukasiewicz axioms from the Russell-Bernays Axioms   rb-bijust 1781
            ​*1.3.12  Stoic logic non-modal portion (Chrysippus of Soli)   mptnan 1800
      ​​*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 1811
                  1.4.1.1  Existential quantifier   wex 1811
                  1.4.1.2  Nonfreeness predicate   wnf 1815
            1.4.2  Rule scheme ax-gen (Generalization)   ax-gen 1827
            1.4.3  Axiom scheme ax-4 (Quantified Implication)   ax-4 1841
                  ​*1.4.3.1  The empty domain of discourse   empty 1938
            1.4.4  Axiom scheme ax-5 (Distinctness) - first use of $d   ax-5 1942
            ​*1.4.5  Equality predicate (continued)   weq 1994
            1.4.6  Axiom scheme ax-6 (Existence)   ax-6 1999
            1.4.7  Axiom scheme ax-7 (Equality)   ax-7 2040
            1.4.8  Define proper substitution   sbjust 2097
            1.4.9  Membership predicate   wcel 2144
            1.4.10  Axiom scheme ax-8 (Left Equality for Binary Predicate)   ax-8 2146
            1.4.11  Axiom scheme ax-9 (Right Equality for Binary Predicate)   ax-9 2154
            ​*1.4.12  Logical redundancy of ax-10 , ax-11 , ax-12 , ax-13   ax6dgen 2164
      ​​*1.5  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes)
            1.5.1  Axiom scheme ax-10 (Quantified Negation)   ax-10 2177
            1.5.2  Axiom scheme ax-11 (Quantifier Commutation)   ax-11 2193
            1.5.3  Axiom scheme ax-12 (Substitution)   ax-12 2212
            1.5.4  Axiom scheme ax-13 (Quantified Equality)   ax-13 2401
      ​1.6  Uniqueness and unique existence
            1.6.1  Uniqueness: the at-most-one quantifier   wmo 2562
            1.6.2  Unique existence: the unique existential quantifier   weu 2593
      ​1.7  Other axiomatizations related to classical predicate calculus
            ​*1.7.1  Aristotelian logic: Assertic syllogisms   barbara 2687
            ​*1.7.2  Intuitionistic logic   axia1 2717
​​*PART 2  ZF (ZERMELO-FRAENKEL) SET THEORY
      ​2.1  ZF Set Theory - start with the Axiom of Extensionality
            2.1.1  Introduce the Axiom of Extensionality   ax-ext 2732
            2.1.2  Classes   cab 2738
                  2.1.2.1  Class abstractions   cab 2738
                  ​*2.1.2.2  Class equality   df-cleq 2752
                  2.1.2.3  Class membership   df-clel 2835
                  2.1.2.4  Elementary properties of class abstractions   eqabdv 2893
            2.1.3  Class form not-free predicate   wnfc 2907
            2.1.4  Negated equality and membership   wne 2955
                  2.1.4.1  Negated equality   wne 2955
                  2.1.4.2  Negated membership   wnel 3061
            2.1.5  Restricted quantification   wral 3076
                  2.1.5.1  Restricted universal and existential quantification   wral 3076
                  2.1.5.2  Restricted existential uniqueness and at-most-one quantifier   wreu 3363
                  2.1.5.3  Restricted class abstraction   crab 3412
            2.1.6  The universal class   cvv 3450
            ​*2.1.7  Conditional equality (experimental)   wcdeq 3720
            2.1.8  Russell's Paradox   rru 3736
            2.1.9  Proper substitution of classes for sets   wsbc 3738
            2.1.10  Proper substitution of classes for sets into classes   csb 3846
            2.1.11  Subclasses and subsets, proper subclasses and subsets   wss 3895
                  2.1.11.1  Subclasses and subsets   wss 3895
                  2.1.11.2  Proper subclasses and subsets   wpss 4013
            2.1.12  Basic operations on classes   cdif 4045
                  2.1.12.1  The difference of two classes   cdif 4045
                  2.1.12.2  The intersection of two classes   cin 4089
                  2.1.12.3  The union of two classes   cun 4149
                  2.1.12.4  The symmetric difference of two classes   csymdif 4197
                  2.1.12.5  Combinations of difference, union, and intersection of two classes   unabs 4210
                  2.1.12.6  Class abstractions with difference, union, and intersection of two classes   unabw 4252
                  2.1.12.7  Restricted uniqueness with difference, union, and intersection   reuun2 4270
            2.1.13  The empty set   c0 4278
            ​*2.1.14  The conditional operator for classes   cif 4481
            ​*2.1.15  The weak deduction theorem for set theory   dedth 4540
            2.1.16  Power classes   cpw 4556
            2.1.17  Unordered and ordered pairs   snjust 4582
            2.1.18  The union of a class   cuni 4866
            2.1.19  The intersection of a class   cint 4906
            2.1.20  Indexed union and intersection   ciun 4950
            2.1.21  Disjointness   wdisj 5069
            2.1.22  Binary relations   wbr 5102
            2.1.23  Ordered-pair class abstractions (class builders)   copab 5166
            2.1.24  Functions in maps-to notation   cmpt 5185
            2.1.25  Transitive classes   wtr 5211
      ​2.2  ZF Set Theory - add the Axiom of Replacement
            2.2.1  Introduce the Axiom of Replacement   ax-rep 5231
            2.2.2  Derive the Axiom of Separation   axsepgfromrep 5246
            2.2.3  Derive the Null Set Axiom   axnulALT 5257
            2.2.4  Theorems requiring subset and intersection existence   exnelv 5266
            2.2.5  Theorems requiring empty set existence   class2set 5315
      ​2.3  ZF Set Theory - add the Axiom of Power Sets
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 5326
            2.3.2  Derive the Axiom of Pairing   axprlem1 5384
            2.3.3  Ordered pair theorem   opnz 5441
            2.3.4  Ordered-pair class abstractions (cont.)   opabidw 5497
            2.3.5  Power class of union and intersection   pwin 5541
            2.3.6  The identity relation   cid 5544
            2.3.7  The membership relation (or epsilon relation)   cep 5549
            ​*2.3.8  Partial and total orderings   wpo 5556
            2.3.9  Founded and well-ordering relations   wfr 5600
            2.3.10  Relations   cxp 5648
            2.3.11  The Predecessor Class   cpred 6303
            2.3.12  Well-founded induction (variant)   frpomin 6343
            2.3.13  Well-ordered induction   tz6.26 6350
            2.3.14  Ordinals   word 6361
            2.3.15  Definite description binder (inverted iota)   cio 6492
            2.3.16  Functions   wfun 6532
            2.3.17  Cantor's Theorem   canth 7374
            2.3.18  Restricted iota (description binder)   crio 7376
            2.3.19  Operations   co 7420
                  2.3.19.1  Variable-to-class conversion for operations   caovclg 7613
            2.3.20  Operations in maps-to notation   mpondm0 7661
            2.3.21  Functions with three arguments in maps-to notation   cmpt3 7683
            2.3.22  Function operation   cof 7691
            2.3.23  Proper subset relation   crpss 7738
      ​2.4  ZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 7751
            2.4.2  Ordinals (continued)   epweon 7789
            2.4.3  Transfinite induction   tfi 7864
            2.4.4  The natural numbers (i.e., finite ordinals)   com 7877
            2.4.5  Peano's postulates   peano1 7900
            2.4.6  Finite induction (for finite ordinals)   find 7907
            2.4.7  Relations and functions (cont.)   dmexg 7913
            2.4.8  First and second members of an ordered pair   c1st 7999
            2.4.9  Induction on Cartesian products   frpoins3xpg 8157
            2.4.10  Ordering on Cartesian products   xpord2lem 8159
            2.4.11  Ordering Ordinal Sequences   orderseqlem 8174
            ​*2.4.12  The support of functions   csupp 8177
            ​*2.4.13  Special maps-to operations   opeliunxp2f 8228
            2.4.14  Function transposition   ctpos 8243
            2.4.15  Curry and uncurry   ccur 8283
            2.4.16  Undefined values   cund 8290
            2.4.17  Well-founded recursion   cfrecs 8299
            2.4.18  Well-ordered recursion   cwrecs 8330
            2.4.19  Functions on ordinals; strictly monotone ordinal functions   iunon 8348
            2.4.20  "Strong" transfinite recursion   crecs 8379
            2.4.21  Recursive definition generator   crdg 8418
            2.4.22  Finite recursion   frfnom 8444
            2.4.23  Ordinal arithmetic   c1o 8470
            2.4.24  Natural number arithmetic   nna0 8614
            2.4.25  Natural addition   cnadd 8675
            2.4.26  Equivalence relations and classes   wer 8715
            2.4.27  The mapping operation   cmap 8848
            2.4.28  Infinite Cartesian products   cixp 8926
            2.4.29  Equinumerosity   cen 8971
            2.4.30  Schroeder-Bernstein Theorem   sbthlem1 9107
            2.4.31  Equinumerosity (cont.)   xpf1o 9159
            2.4.32  Finite sets   dif1enlem 9176
            2.4.33  Pigeonhole Principle   phplem1 9220
            2.4.34  Finite sets (cont.)   onomeneq 9230
            2.4.35  Finitely supported functions   cfsupp 9354
            2.4.36  Finite intersections   cfi 9403
            2.4.37  Hall's marriage theorem   marypha1lem 9426
            2.4.38  Supremum and infimum   csup 9433
            2.4.39  Ordinal isomorphism, Hartogs's theorem   coi 9504
            2.4.40  Hartogs function   char 9551
            2.4.41  Weak dominance   cwdom 9559
      ​2.5  ZF Set Theory - add the Axiom of Regularity
            2.5.1  Introduce the Axiom of Regularity   ax-reg 9587
            2.5.2  Axiom of Infinity equivalents   inf0 9623
      ​2.6  ZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-inf 9640
            2.6.2  Existence of omega (the set of natural numbers)   omex 9645
            2.6.3  Cantor normal form   ccnf 9663
            2.6.4  Transitive closure of a relation   cttrcl 9709
            2.6.5  Transitive closure   trcl 9730
            2.6.6  Set induction (or epsilon induction)   setind 9749
            2.6.7  Well-Founded Induction   frmin 9754
            2.6.8  Well-Founded Recursion   frr3g 9761
            2.6.9  Rank   cr1 9767
            2.6.10  Hereditarily finite sets   chf 9905
            2.6.11  Scott's trick; collection principle; Hilbert's epsilon   cscott 9929
            2.6.12  Set Recursion   csetrecs 9965
                  ​*2.6.12.1  Basic Properties of Set Recursion   csetrecs 9965
            2.6.13  Disjoint union   cdju 9980
            2.6.14  Cardinal numbers   ccrd 10017
            2.6.15  Axiom of Choice equivalents   wac 10195
            ​*2.6.16  Cardinal number arithmetic   undjudom 10247
            2.6.17  The Ackermann bijection   ackbij2lem1 10297
            2.6.18  Cofinality (without Axiom of Choice)   cflem 10324
            2.6.19  Eight inequivalent definitions of finite set   sornom 10356
            2.6.20  Hereditarily size-limited sets without Choice   itunifval 10495
​​*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 10514
            3.1.2  Introduce the Axiom of Dependent Choice   ax-dc 10525
      ​3.2  ZFC Set Theory - add the Axiom of Choice
            3.2.1  Introduce the Axiom of Choice   ax-ac 10538
            3.2.2  AC equivalents: well-ordering, Zorn's lemma   numthcor 10573
            3.2.3  Cardinal number theorems using Axiom of Choice   cardval 10631
            3.2.4  Cardinal number arithmetic using Axiom of Choice   iunctb 10660
            3.2.5  Cofinality using the Axiom of Choice   alephreg 10668
      ​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 10706
            3.4.2  Derivation of the Axiom of Choice   gchaclem 10764
​​*PART 4  TG (TARSKI-GROTHENDIECK) SET THEORY
      ​4.1  Inaccessibles
            4.1.1  Weakly and strongly inaccessible cardinals   cwina 10768
            4.1.2  Weak universes   cwun 10786
            4.1.3  Tarski classes   ctsk 10834
            4.1.4  Grothendieck universes   cgru 10876
      ​4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
            4.2.1  Introduce the Tarski-Grothendieck Axiom   ax-groth 10909
            4.2.2  Derive the Power Set, Infinity and Choice Axioms   grothpw 10912
            4.2.3  Tarski map function   ctskm 10923
​​*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 10930
            5.1.2  Final derivation of real and complex number postulates   axaddf 11231
            5.1.3  Real and complex number postulates restated as axioms   ax-cnex 11257
      ​5.2  Derive the basic properties from the field axioms
            5.2.1  Some deductions from the field axioms for complex numbers   cnex 11282
            5.2.2  Infinity and the extended real number system   cpnf 11341
            5.2.3  Restate the ordering postulates with extended real "less than"   axlttri 11382
            5.2.4  Ordering on reals   lttr 11387
            5.2.5  Initial properties of the complex numbers   mul12 11476
      ​5.3  Real and complex numbers - basic operations
            5.3.1  Addition   add12 11529
            5.3.2  Subtraction   cmin 11542
            5.3.3  Multiplication   kcnktkm1cn 11748
            5.3.4  Ordering on reals (cont.)   gt0ne0 11782
            5.3.5  Reciprocals   ixi 11946
            5.3.6  Division   cdiv 11974
            5.3.7  Ordering on reals (cont.)   elimgt0 12156
            5.3.8  Completeness Axiom and Suprema   fimaxre 12262
            5.3.9  Imaginary and complex number properties   neg1cn 12306
            5.3.10  Function operation analogue theorems   ofsubeq0 12318
            ​*5.3.11  Indicator Functions   cind 12321
      ​5.4  Integer sets
            5.4.1  Positive integers (as a subset of complex numbers)   cn 12336
            5.4.2  Principle of mathematical induction   nnind 12354
            ​*5.4.3  Decimal representation of numbers   c2 12398
            ​*5.4.4  Some properties of specific numbers   1pneg1e0 12461
            5.4.5  Simple number properties   halfcl 12573
            5.4.6  The Archimedean property   nnunb 12603
            5.4.7  Nonnegative integers (as a subset of complex numbers)   cn0 12607
            ​*5.4.8  Extended nonnegative integers   cxnn0 12680
            5.4.9  Integers (as a subset of complex numbers)   cz 12694
            5.4.10  Decimal arithmetic   cdc 12815
            5.4.11  Upper sets of integers   cuz 12966
            5.4.12  Well-ordering principle for bounded-below sets of integers   uzwo3 13071
            5.4.13  Rational numbers (as a subset of complex numbers)   cq 13076
            5.4.14  Existence of the set of complex numbers   rpnnen1lem2 13106
      ​5.5  Order sets
            5.5.1  Positive reals (as a subset of complex numbers)   crp 13121
            5.5.2  Infinity and the extended real number system (cont.)   cxne 13239
            5.5.3  Supremum and infimum on the extended reals   xrsupexmnf 13436
            5.5.4  Real number intervals   cioo 13477
            5.5.5  Finite intervals of integers   cfz 13640
            ​*5.5.6  Finite intervals of nonnegative integers   elfz2nn0 13753
            5.5.7  Half-open integer ranges   cfzo 13789
      ​5.6  Elementary integer functions
            5.6.1  The floor and ceiling functions   cfl 13931
            5.6.2  The modulo (remainder) operation   cmo 14010
            5.6.3  Miscellaneous theorems about integers   om2uz0i 14091
            5.6.4  Strong induction over upper sets of integers   uzsinds 14131
            5.6.5  Finitely supported functions over the nonnegative integers   fsuppmapnn0fiublem 14134
            5.6.6  The infinite sequence builder "seq" - extension   cseq 14145
            5.6.7  Integer powers   cexp 14205
            5.6.8  Ordered pair theorem for nonnegative integers   nn0le2msqi 14412
            5.6.9  Factorial function   cfa 14418
            5.6.10  The binomial coefficient operation   cbc 14447
            5.6.11  The ` # ` (set size) function   chash 14475
                  5.6.11.1  Proper unordered pairs and triples (sets of size 2 and 3)   hashprlei 14614
                  5.6.11.2  Functions with a domain containing at least two different elements   fundmge2nop0 14648
                  5.6.11.3  Finite induction on the size of the first component of a binary relation   hashdifsnp1 14652
      ​​*5.7  Words over a set
            5.7.1  Definitions and basic theorems   cword 14659
            5.7.2  Last symbol of a word   clsw 14708
            5.7.3  Concatenations of words   cconcat 14716
            5.7.4  Singleton words   cs1 14743
            5.7.5  Concatenations with singleton words   ccatws1cl 14765
            5.7.6  Subwords/substrings   csubstr 14789
            5.7.7  Prefixes of a word   cpfx 14821
            5.7.8  Subwords of subwords   swrdswrdlem 14854
            5.7.9  Subwords and concatenations   pfxcctswrd 14860
            5.7.10  Subwords of concatenations   swrdccatfn 14874
            5.7.11  Splicing words (substring replacement)   csplice 14899
            5.7.12  Reversing words   creverse 14908
            5.7.13  Repeated symbol words   creps 14920
            ​*5.7.14  Cyclical shifts of words   ccsh 14940
            5.7.15  Mapping words by a function   wrdco 14983
            5.7.16  Longer string literals   cs2 14993
      ​​*5.8  Reflexive and transitive closures of relations
            5.8.1  The reflexive and transitive properties of relations   coss12d 15126
            5.8.2  Basic properties of closures   cleq1lem 15136
            5.8.3  Definitions and basic properties of transitive closures   ctcl 15139
            5.8.4  Exponentiation of relations   crelexp 15173
            5.8.5  Reflexive-transitive closure as an indexed union   crtrcl 15209
            ​*5.8.6  Principle of transitive induction   relexpindlem 15217
      ​5.9  Elementary real and complex functions
            5.9.1  The "shift" operation   cshi 15220
            5.9.2  Signum (sgn or sign) function   csgn 15240
            5.9.3  Real and imaginary parts; conjugate   ccj 15264
            5.9.4  Square root; absolute value   csqrt 15401
      ​5.10  Elementary limits and convergence
            5.10.1  Superior limit (lim sup)   clsp 15638
            5.10.2  Limits   cli 15652
            5.10.3  Finite and infinite sums   csu 15854
            5.10.4  The binomial theorem   binomlem 15999
            5.10.5  The inclusion/exclusion principle   incexclem 16006
            5.10.6  Infinite sums (cont.)   isumshft 16009
            5.10.7  Miscellaneous converging and diverging sequences   divrcnv 16022
            5.10.8  Arithmetic series   arisum 16030
            5.10.9  Geometric series   expcnv 16034
            5.10.10  Ratio test for infinite series convergence   cvgrat 16053
            5.10.11  Mertens' theorem   mertenslem1 16054
            5.10.12  Finite and infinite products   prodf 16057
                  5.10.12.1  Product sequences   prodf 16057
                  5.10.12.2  Non-trivial convergence   ntrivcvg 16067
                  5.10.12.3  Complex products   cprod 16073
                  5.10.12.4  Finite products   fprod 16109
                  5.10.12.5  Infinite products   iprodclim 16166
            5.10.13  Falling and Rising Factorial   cfallfac 16172
            5.10.14  Bernoulli polynomials and sums of k-th powers   cbp 16213
      ​5.11  Elementary trigonometry
            5.11.1  The exponential, sine, and cosine functions   ce 16228
                  5.11.1.1  The circle constant (tau = 2 pi)   ctau 16371
            5.11.2  _e is irrational   eirrlem 16373
      ​5.12  Cardinality of real and complex number subsets
            5.12.1  Countability of integers and rationals   xpnnen 16380
            5.12.2  The reals are uncountable   rpnnen2lem1 16383
​​*PART 6  ELEMENTARY NUMBER THEORY
      ​6.1  Elementary properties of divisibility
            6.1.1  Irrationality of square root of 2   sqrt2irrlem 16417
            6.1.2  Some Number sets are chains of proper subsets   nthruc 16421
            6.1.3  The divides relation   cdvds 16423
            ​*6.1.4  Even and odd numbers   evenelz 16507
            6.1.5  The division algorithm   divalglem0 16564
            6.1.6  Bit sequences   cbits 16590
            6.1.7  The greatest common divisor operator   cgcd 16665
            6.1.8  Bézout's identity   bezoutlem1 16713
            6.1.9  Algorithms   nn0seqcvgd 16746
            6.1.10  Euclid's Algorithm   eucalgval2 16757
            ​*6.1.11  The least common multiple   clcm 16764
            ​*6.1.12  Coprimality and Euclid's lemma   coprmgcdb 16825
            6.1.13  Cancellability of congruences   congr 16840
      ​6.2  Elementary prime number theory
            ​*6.2.1  Elementary properties   cprime 16847
            ​*6.2.2  Coprimality and Euclid's lemma (cont.)   coprm 16888
            6.2.3  Properties of the canonical representation of a rational   cnumer 16910
            6.2.4  Euler's theorem   codz 16941
            6.2.5  Arithmetic modulo a prime number   modprm1div 16976
            6.2.6  Pythagorean Triples   coprimeprodsq 16987
            6.2.7  The prime count function   cpc 17015
            6.2.8  Pocklington's theorem   prmpwdvds 17083
            6.2.9  Infinite primes theorem   unbenlem 17087
            6.2.10  Sum of prime reciprocals   prmreclem1 17095
            6.2.11  Fundamental theorem of arithmetic   1arithlem1 17102
            6.2.12  Lagrange's four-square theorem   cgz 17108
            6.2.13  Van der Waerden's theorem   cvdwa 17144
            6.2.14  Ramsey's theorem   cram 17178
            ​*6.2.15  Primorial function   cprmo 17210
            ​*6.2.16  Prime gaps   prmgaplem1 17228
            6.2.17  Decimal arithmetic (cont.)   dec2dvds 17242
            6.2.18  Cyclical shifts of words (cont.)   cshwsidrepsw 17272
            6.2.19  Specific prime numbers   prmlem0 17284
            6.2.20  Very large primes   1259lem1 17310
​PART 7  BASIC STRUCTURES
      ​7.1  Extensible structures
            ​*7.1.1  Basic definitions   cstr 17325
                  7.1.1.1  Extensible structures as structures with components   cstr 17325
                  7.1.1.2  Substitution of components   csts 17342
                  7.1.1.3  Slots   cslot 17360
                  ​*7.1.1.4  Structure component indices   cnx 17372
                  7.1.1.5  Base sets   cbs 17388
                  7.1.1.6  Base set restrictions   cress 17409
            7.1.2  Slot definitions   cplusg 17429
            7.1.3  Definition of the structure product   crest 17592
            7.1.4  Definition of the structure quotient   cordt 17672
      ​7.2  Moore spaces
            7.2.1  Moore closures   mrcflem 17781
            7.2.2  Independent sets in a Moore system   mrisval 17805
            7.2.3  Algebraic closure systems   isacs 17826
​PART 8  BASIC CATEGORY THEORY
      ​8.1  Categories
            8.1.1  Categories   ccat 17839
            8.1.2  Opposite category   coppc 17886
            8.1.3  Monomorphisms and epimorphisms   cmon 17904
            8.1.4  Sections, inverses, isomorphisms   csect 17920
            ​*8.1.5  Isomorphic objects   ccic 17971
            8.1.6  Subcategories   cssc 17983
            8.1.7  Functors   cfunc 18030
            8.1.8  Full & faithful functors   cful 18080
            8.1.9  Natural transformations and the functor category   cnat 18120
            8.1.10  Initial, terminal and zero objects of a category   cinito 18157
      ​8.2  Arrows (disjointified hom-sets)
            8.2.1  Identity and composition for arrows   cida 18229
      ​8.3  Examples of categories
            8.3.1  The category of sets   csetc 18251
            8.3.2  The category of categories   ccatc 18274
            ​*8.3.3  The category of extensible structures   fncnvimaeqv 18295
      ​8.4  Categorical constructions
            8.4.1  Product of categories   cxpc 18343
            8.4.2  Functor evaluation   cevlf 18384
            8.4.3  Hom functor   chof 18423
​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 18606
            9.5.2  Complete lattices   ccla 18673
            9.5.3  Distributive lattices   cdlat 18695
            9.5.4  Subset order structures   cipo 18702
      ​9.6  Posets, directed sets, and lattices as relations
            ​*9.6.1  Posets and lattices as relations   cps 18739
            9.6.2  Directed sets, nets   cdir 18769
      ​9.7  Chains
​PART 10  BASIC ALGEBRAIC STRUCTURES
      ​10.1  Monoids
            ​*10.1.1  Magmas   cplusf 18814
            ​*10.1.2  Identity elements   mgmidmo 18839
            ​*10.1.3  Iterated sums in a magma   gsumvalx 18866
            10.1.4  Magma homomorphisms and submagmas   cmgmhm 18880
            ​*10.1.5  Semigroups   csgrp 18908
            ​*10.1.6  Definition and basic properties of monoids   cmnd 18924
            10.1.7  Monoid homomorphisms and submonoids   cmhm 18977
            ​*10.1.8  Iterated sums in a monoid   gsumvallem2 19031
            10.1.9  Free monoids   cfrmd 19044
                  ​*10.1.9.1  Monoid of endofunctions   cefmnd 19065
            10.1.10  Examples and counterexamples for magmas, semigroups and monoids   mgm2nsgrplem1 19118
      ​10.2  Groups
            10.2.1  Definition and basic properties   cgrp 19145
            ​*10.2.2  Group multiple operation   cmg 19278
            10.2.3  Subgroups and Quotient groups   csubg 19331
            ​*10.2.4  Cyclic monoids and groups   cycsubmel 19416
            10.2.5  Elementary theory of group homomorphisms   cghm 19428
            10.2.6  Isomorphisms of groups   cgim 19472
                  10.2.6.1  The first isomorphism theorem of groups   ghmqusnsglem1 19495
            10.2.7  Group actions   cga 19504
            10.2.8  Centralizers and centers   ccntz 19530
            10.2.9  The opposite group   coppg 19560
            10.2.10  Symmetric groups   csymg 19584
                  ​*10.2.10.1  Definition and basic properties   csymg 19584
                  10.2.10.2  Cayley's theorem   cayleylem1 19627
                  10.2.10.3  Permutations fixing one element   symgfix2 19631
                  ​*10.2.10.4  Transpositions in the symmetric group   cpmtr 19656
                  10.2.10.5  The sign of a permutation   cpsgn 19704
            10.2.11  p-Groups and Sylow groups; Sylow's theorems   cod 19739
            10.2.12  Direct products   clsm 19849
                  10.2.12.1  Direct products (extension)   smndlsmidm 19871
            10.2.13  Free groups   cefg 19921
            10.2.14  Abelian groups   ccmn 19995
                  10.2.14.1  Definition and basic properties   ccmn 19995
                  10.2.14.2  Cyclic groups   ccyg 20092
                  10.2.14.3  Group sum operation   gsumval3a 20118
                  10.2.14.4  Group sums over (ranges of) integers   fsfnn0gsumfsffz 20198
                  10.2.14.5  Internal direct products   cdprd 20210
                  10.2.14.6  The Fundamental Theorem of Abelian Groups   ablfacrplem 20282
            10.2.15  Simple groups   csimpg 20307
                  10.2.15.1  Definition and basic properties   csimpg 20307
                  10.2.15.2  Classification of abelian simple groups   ablsimpnosubgd 20321
            10.2.16  Totally ordered monoids and groups   comnd 20334
      ​10.3  Rings
            10.3.1  Multiplicative Group   cmgp 20361
            ​*10.3.2  Non-unital rings ("rngs")   crng 20375
            ​*10.3.3  Ring unity (multiplicative identity)   cur 20408
            10.3.4  Semirings   csrg 20413
                  ​*10.3.4.1  The binomial theorem for semirings   srgbinomlem1 20453
            10.3.5  Unital rings   crg 20460
            10.3.6  Opposite ring   coppr 20567
            10.3.7  Divisibility   cdsr 20585
            10.3.8  Ring primes   crpm 20663
            10.3.9  Homomorphisms of non-unital rings   crnghm 20665
            10.3.10  Ring homomorphisms   crh 20700
            10.3.11  Nonzero rings and zero rings   cnzr 20763
            10.3.12  Local rings   clring 20791
            10.3.13  Subrings   csubrng 20798
                  10.3.13.1  Subrings of non-unital rings   csubrng 20798
                  10.3.13.2  Subrings of unital rings   csubrg 20822
                  10.3.13.3  Subrings generated by a subset   crgspn 20863
            10.3.14  Categories of rings   crngc 20869
                  ​*10.3.14.1  The category of non-unital rings   crngc 20869
                  ​*10.3.14.2  The category of (unital) rings   cringc 20898
                  10.3.14.3  Subcategories of the category of rings   srhmsubclem1 20930
            10.3.15  Left regular elements and domains   crlreg 20944
      ​10.4  Division rings and fields
            10.4.1  Definition and basic properties   cdr 20981
            10.4.2  Sub-division rings   csdrg 21044
            10.4.3  Absolute value (abstract algebra)   cabv 21066
            10.4.4  Star rings   cstf 21095
            10.4.5  Totally ordered rings and fields   corng 21115
      ​10.5  Left modules
            10.5.1  Definition and basic properties   clmod 21136
            10.5.2  Subspaces and spans in a left module   clss 21207
            10.5.3  Homomorphisms and isomorphisms of left modules   clmhm 21295
            10.5.4  Subspace sum; bases for a left module   clbs 21350
      ​10.6  Vector spaces
            10.6.1  Definition and basic properties   clvec 21378
      ​10.7  Subring algebras and ideals
            10.7.1  Subring algebras   csra 21447
            ​*10.7.2  Left ideals and spans   clidl 21485
            10.7.3  Two-sided ideals and quotient rings   c2idl 21543
                  ​*10.7.3.1  Condition for a non-unital ring to be unital   rngqiprng1elbas 21583
                  10.7.3.2  Prime Ideals   cprmidl 21617
            10.7.4  Principal ideal rings. Divisibility in the integers   clpidl 21645
            10.7.5  Principal ideal domains   cpid 21661
      ​10.8  The complex numbers as an algebraic extensible structure
            10.8.1  Definition and basic properties   cpsmet 21663
            ​*10.8.2  Ring of integers   czring 21753
                  ​*10.8.2.1  Example for a condition for a non-unital ring to be unital   pzriprnglem1 21788
            10.8.3  Algebraic constructions based on the complex numbers   czrh 21806
            10.8.4  Signs as subgroup of the complex numbers   cnmsgnsubg 21884
            10.8.5  Embedding of permutation signs into a ring   zrhpsgnmhm 21891
            10.8.6  The ordered field of real numbers   crefld 21911
      ​10.9  Generalized pre-Hilbert and Hilbert spaces
            10.9.1  Definition and basic properties   cphl 21931
            10.9.2  Orthocomplements and closed subspaces   cocv 21967
            10.9.3  Orthogonal projection and orthonormal bases   cpj 22007
​​*PART 11  BASIC LINEAR ALGEBRA
      ​11.1  Vectors and free modules
            *11.1.1  Direct sum of left modules   cdsmm 22038
            ​*11.1.2  Free modules   cfrlm 22053
            ​*11.1.3  Standard basis (unit vectors)   cuvc 22089
            ​*11.1.4  Independent sets and families   clindf 22111
            11.1.5  Characterization of free modules   lmimlbs 22143
      ​11.2  Associative algebras
            11.2.1  Definition and basic properties   casa 22159
      ​11.3  Abstract multivariate polynomials
            11.3.1  Definition and basic properties   cmps 22213
            11.3.2  Polynomial evaluation   ces 22382
            11.3.3  The "variable selection" function   cslv 22426
            11.3.4  Additional definitions for (multivariate) polynomials   cmhp 22455
            ​*11.3.5  Univariate polynomials   cps1 22494
            11.3.6  Univariate polynomial evaluation   ces1 22632
                  11.3.6.1  Specialization of polynomial evaluation as a ring homomorphism   evls1scafv 22685
      ​​*11.4  Matrices
            *11.4.1  The matrix multiplication   cmmul 22706
            ​*11.4.2  Square matrices   cmat 22723
            ​*11.4.3  The matrix algebra   matmulr 22754
            ​*11.4.4  Matrices of dimension 0 and 1   mat0dimbas0 22782
            ​*11.4.5  The subalgebras of diagonal and scalar matrices   cdmat 22804
            ​*11.4.6  Multiplication of a matrix with a "column vector"   cmvmul 22856
            11.4.7  Replacement functions for a square matrix   cmarrep 22872
            11.4.8  Submatrices   csubma 22892
      ​11.5  The determinant
            11.5.1  Definition and basic properties   cmdat 22900
            11.5.2  Determinants of 2 x 2 -matrices   m2detleiblem1 22940
            11.5.3  The matrix adjugate/adjunct   cmadu 22948
            ​*11.5.4  Laplace expansion of determinants (special case)   symgmatr01lem 22969
            11.5.5  Inverse matrix   invrvald 22992
            ​*11.5.6  Cramer's rule   slesolvec 22998
      ​​*11.6  Polynomial matrices
            11.6.1  Basic properties   pmatring 23011
            ​*11.6.2  Constant polynomial matrices   ccpmat 23022
            ​*11.6.3  Collecting coefficients of polynomial matrices   cdecpmat 23081
            ​*11.6.4  Ring isomorphism between polynomial matrices and polynomials over matrices   cpm2mp 23111
      ​​*11.7  The characteristic polynomial
            *11.7.1  Definition and basic properties   cchpmat 23145
            ​*11.7.2  The characteristic factor function G   fvmptnn04if 23168
            ​*11.7.3  The Cayley-Hamilton theorem   cpmadurid 23186
​PART 12  BASIC TOPOLOGY
      ​12.1  Topology
            ​*12.1.1  Topological spaces   ctop 23212
                  12.1.1.1  Topologies   ctop 23212
                  12.1.1.2  Topologies on sets   ctopon 23229
                  12.1.1.3  Topological spaces   ctps 23251
            12.1.2  Topological bases   ctb 23264
            12.1.3  Examples of topologies   distop 23314
            12.1.4  Closure and interior   ccld 23335
            12.1.5  Neighborhoods   cnei 23416
            12.1.6  Limit points and perfect sets   clp 23453
            12.1.7  Subspace topologies   restrcl 23476
            12.1.8  Order topology   ordtbaslem 23507
            12.1.9  Limits and continuity in topological spaces   ccn 23543
            12.1.10  Separated spaces: T0, T1, T2 (Hausdorff) ...   ct0 23625
            12.1.11  Compactness   ccmp 23705
            12.1.12  Bolzano-Weierstrass theorem   bwth 23729
            12.1.13  Connectedness   cconn 23730
            12.1.14  First- and second-countability   c1stc 23756
            12.1.15  Local topological properties   clly 23784
            12.1.16  Refinements   cref 23822
            12.1.17  Compactly generated spaces   ckgen 23853
            12.1.18  Product topologies   ctx 23880
            12.1.19  Continuous function-builders   cnmptid 23981
            12.1.20  Quotient maps and quotient topology   ckq 24013
            12.1.21  Homeomorphisms   chmeo 24073
      ​12.2  Filters and filter bases
            12.2.1  Filter bases   elmptrab 24147
            12.2.2  Filters   cfil 24165
            12.2.3  Ultrafilters   cufil 24219
            12.2.4  Filter limits   cfm 24253
            12.2.5  Extension by continuity   ccnext 24379
            12.2.6  Topological groups   ctmd 24390
            12.2.7  Infinite group sum on topological groups   ctsu 24446
            12.2.8  Topological rings, fields, vector spaces   ctrg 24476
      ​12.3  Uniform Structures and Spaces
            12.3.1  Uniform structures   cust 24520
            12.3.2  The topology induced by an uniform structure   cutop 24550
            12.3.3  Uniform Spaces   cuss 24573
            12.3.4  Uniform continuity   cucn 24594
            12.3.5  Cauchy filters in uniform spaces   ccfilu 24605
            12.3.6  Complete uniform spaces   ccusp 24616
      ​12.4  Metric spaces
            12.4.1  Pseudometric spaces   ispsmet 24624
            12.4.2  Basic metric space properties   cxms 24637
            12.4.3  Metric space balls   blfvalps 24703
            12.4.4  Open sets of a metric space   mopnval 24758
            12.4.5  Continuity in metric spaces   metcnp3 24860
            12.4.6  The uniform structure generated by a metric   metuval 24869
            12.4.7  Examples of metric spaces   dscmet 24892
            ​*12.4.8  Normed algebraic structures   cnm 24896
            12.4.9  Normed space homomorphisms (bounded linear operators)   cnmo 25025
            12.4.10  Topology on the reals   qtopbaslem 25078
            12.4.11  Topological definitions using the reals   cii 25197
            12.4.12  Path homotopy   chtpy 25289
            12.4.13  The fundamental group   cpco 25322
      ​12.5  Metric subcomplex vector spaces
            12.5.1  Subcomplex modules   cclm 25384
            ​*12.5.2  Subcomplex vector spaces   ccvs 25445
            ​*12.5.3  Normed subcomplex vector spaces   isncvsngp 25471
            12.5.4  Subcomplex pre-Hilbert spaces   ccph 25488
            12.5.5  Convergence and completeness   ccfil 25574
            12.5.6  Baire's Category Theorem   bcthlem1 25646
            12.5.7  Banach spaces and subcomplex Hilbert spaces   ccms 25654
                  12.5.7.1  The complete ordered field of the real numbers   retopn 25701
            12.5.8  Euclidean spaces   crrx 25705
            12.5.9  Minimizing Vector Theorem   minveclem1 25746
            12.5.10  Projection Theorem   pjthlem1 25759
​PART 13  BASIC REAL AND COMPLEX ANALYSIS
      ​13.1  Continuity
            13.1.1  Intermediate value theorem   pmltpclem1 25770
      ​13.2  Integrals
            13.2.1  Lebesgue measure   covol 25784
            13.2.2  Lebesgue integration   cmbf 25936
                  13.2.2.1  Lesbesgue integral   cmbf 25936
                  13.2.2.2  Lesbesgue directed integral   cdit 26167
      ​13.3  Derivatives
            13.3.1  Real and complex differentiation   climc 26183
                  13.3.1.1  Derivatives of functions of one complex or real variable   climc 26183
                  13.3.1.2  Results on real differentiation   dvferm1lem 26305
​PART 14  BASIC REAL AND COMPLEX FUNCTIONS
      ​14.1  Polynomials
            14.1.1  Polynomial degrees   cmdg 26372
            14.1.2  The division algorithm for univariate polynomials   cmn1 26445
            14.1.3  Elementary properties of complex polynomials   cply 26503
            14.1.4  The division algorithm for polynomials   cquot 26612
            14.1.5  Algebraic numbers   caa 26638
            14.1.6  Liouville's approximation theorem   aalioulem1 26660
      ​14.2  Sequences and series
            14.2.1  Taylor polynomials and Taylor's theorem   ctayl 26681
            14.2.2  Uniform convergence   culm 26704
            14.2.3  Power series   pserval 26738
      ​14.3  Basic trigonometry
            14.3.1  The exponential, sine, and cosine functions (cont.)   efcn 26771
            14.3.2  Properties of pi = 3.14159...   pilem1 26779
            14.3.3  Mapping of the exponential function   efgh 26870
            14.3.4  The natural logarithm on complex numbers   clog 26883
            ​*14.3.5  Logarithms to an arbitrary base   clogb 27093
            14.3.6  Theorems of Pythagoras, isosceles triangles, and intersecting chords   angval 27130
            14.3.7  Solutions of quadratic, cubic, and quartic equations   quad2 27168
            14.3.8  Inverse trigonometric functions   casin 27191
            14.3.9  The Birthday Problem   log2ublem1 27275
            14.3.10  Areas in R^2   carea 27284
            14.3.11  More miscellaneous converging sequences   rlimcnp 27294
            14.3.12  Inequality of arithmetic and geometric means   cvxcl 27313
            14.3.13  Euler-Mascheroni constant   cem 27320
            14.3.14  Zeta function   czeta 27341
            14.3.15  Gamma function   clgam 27344
      ​14.4  Basic number theory
            14.4.1  Wilson's theorem   wilthlem1 27396
            14.4.2  The Fundamental Theorem of Algebra   ftalem1 27401
            14.4.3  The Basel problem (ζ(2) = π²/6)   basellem1 27409
            14.4.4  Number-theoretical functions   ccht 27419
            14.4.5  Perfect Number Theorem   mersenne 27555
            14.4.6  Characters of Z/nZ   cdchr 27560
            14.4.7  Bertrand's postulate   bcctr 27603
            ​*14.4.8  Quadratic residues and the Legendre symbol   clgs 27622
            ​*14.4.9  Gauss' Lemma   gausslemma2dlem0a 27684
            14.4.10  Quadratic reciprocity   lgseisenlem1 27703
            14.4.11  All primes 4n+1 are the sum of two squares   2sqlem1 27745
            14.4.12  Chebyshev's Weak Prime Number Theorem, Dirichlet's Theorem   chebbnd1lem1 27797
            14.4.13  The Prime Number Theorem   mudivsum 27858
            14.4.14  Ostrowski's theorem   abvcxp 27943
            14.4.15  Fermat's Last Theorem   infdesc 27968
                  14.4.15.1  Basic reductions for Fermat's Last Theorem   infdesc 27968
                  ​*14.4.15.2  Proof of Fermat's Last Theorem (work in progress)   fltoprmlem1 27994
​PART 15  SURREAL NUMBERS
      ​​*15.1  Sign sequence representation and Alling's axioms
            15.1.1  Definitions and initial properties   csur 27998
            15.1.2  Ordering   ltssolem1 28033
            15.1.3  Birthday Function   bdayfo 28035
            15.1.4  Density   fvnobday 28036
            ​*15.1.5  Full-Eta Property   bdayimaon 28051
      ​15.2  Initial consequences of Alling's axioms
            15.2.1  Ordering Theorems   cles 28102
            15.2.2  Birthday Theorems   bdayfun 28134
      ​​*15.3  Conway cut representation
            15.3.1  Conway cuts   cslts 28144
            15.3.2  Zero and One   c0s 28192
            15.3.3  Cuts and Options   cmade 28209
            15.3.4  Cofinality and coinitiality   cofslts 28305
      ​15.4  Induction and recursion
            15.4.1  Induction and recursion on one variable   cnorec 28324
            15.4.2  Induction and recursion on two variables   cnorec2 28335
      ​15.5  Surreal arithmetic
            15.5.1  Addition   cadds 28346
            15.5.2  Negation and Subtraction   cnegs 28406
            15.5.3  Multiplication   cmuls 28493
            15.5.4  Division   cdivs 28574
            15.5.5  Absolute value   cabss 28624
      ​15.6  Subsystems of surreals
            15.6.1  Ordinal numbers   cons 28638
            15.6.2  Surreal recursive sequences   cseqs 28670
            15.6.3  Natural numbers   cn0s 28699
            15.6.4  Integers   czs 28765
            15.6.5  Dyadic fractions   c2s 28797
            15.6.6  Real numbers   creno 28876
​​*PART 16  ELEMENTARY GEOMETRY
      ​16.1  Definition and Tarski's Axioms of Geometry
            16.1.1  Justification for the congruence notation   tgjustf 28936
      ​16.2  Tarskian Geometry
            16.2.1  Congruence   tgcgrcomimp 28940
            16.2.2  Betweenness   tgbtwntriv2 28951
            16.2.3  Dimension   tglowdim1 28964
            16.2.4  Betweenness and Congruence   tgifscgr 28972
            16.2.5  Congruence of a series of points   ccgrg 28974
            16.2.6  Motions   cismt 28996
            16.2.7  Colinearity   tglng 29010
            16.2.8  Connectivity of betweenness   tgbtwnconn1lem1 29036
            16.2.9  Less-than relation in geometric congruences   cleg 29046
            16.2.10  Rays   chlg 29064
            16.2.11  Lines   btwnlng1 29088
            16.2.12  Point inversions   cmir 29125
            16.2.13  Right angles   crag 29169
            16.2.14  Half-planes   islnopp 29216
            16.2.15  Planes   cplng 29252
            16.2.16  Midpoints and Line Mirroring   cmid 29278
            16.2.17  Congruence of angles   ccgra 29315
            16.2.18  Angle Comparisons   cinag 29355
            16.2.19  Angle Addition   cangmgm 29374
            16.2.20  Congruence Theorems   tgsas1 29400
            16.2.21  Equilateral triangles   ceqlg 29411
            16.2.22  Parallel lines   cprlng 29415
      ​16.3  Properties of geometries
            16.3.1  Isomorphisms between geometries   f1otrgds 29447
      ​16.4  Geometry in Hilbert spaces
            16.4.1  Geometry in the complex plane   cchhllem 29465
            16.4.2  Geometry in Euclidean spaces   cee 29466
                  16.4.2.1  Definition of the Euclidean space   cee 29466
                  16.4.2.2  Tarski's axioms for geometry for the Euclidean space   axdimuniq 29492
                  16.4.2.3  EE^n fulfills Tarski's Axioms   ceeng 29556
​​*PART 17  GRAPH THEORY
      ​*17.1  Vertices and edges
            17.1.1  The edge function extractor for extensible structures   cedgf 29567
            ​*17.1.2  Vertices and indexed edges   cvtx 29575
                  17.1.2.1  Definitions and basic properties   cvtx 29575
                  17.1.2.2  The vertices and edges of a graph represented as ordered pair   opvtxval 29582
                  17.1.2.3  The vertices and edges of a graph represented as extensible structure   funvtxdmge2val 29590
                  17.1.2.4  Representations of graphs without edges   snstrvtxval 29616
                  17.1.2.5  Degenerated cases of representations of graphs   vtxval0 29618
            17.1.3  Edges as range of the edge function   cedg 29626
      ​​*17.2  Undirected graphs
            17.2.1  Undirected hypergraphs   cuhgr 29635
            17.2.2  Undirected pseudographs and multigraphs   cupgr 29659
            ​*17.2.3  Loop-free graphs   umgrislfupgrlem 29701
            17.2.4  Edges as subsets of vertices of graphs   uhgredgiedgb 29705
            ​*17.2.5  Undirected simple graphs   cuspgr 29730
            17.2.6  Examples for graphs   usgr0e 29818
            17.2.7  Subgraphs   csubgr 29849
            17.2.8  Finite undirected simple graphs   cfusgr 29898
            17.2.9  Neighbors, complete graphs and universal vertices   cnbgr 29914
                  17.2.9.1  Neighbors   cnbgr 29914
                  17.2.9.2  Universal vertices   cuvtx 29967
                  17.2.9.3  Complete graphs   ccplgr 29991
            17.2.10  Vertex degree   cvtxdg 30047
            ​*17.2.11  Regular graphs   crgr 30137
      ​​*17.3  Walks, paths and cycles
            *17.3.1  Walks   cewlks 30177
            17.3.2  Walks for loop-free graphs   lfgrwlkprop 30271
            17.3.3  Trails   ctrls 30274
            17.3.4  Paths and simple paths   cpths 30296
            17.3.5  Closed walks   cclwlks 30358
            17.3.6  Circuits and cycles   ccrcts 30372
            ​*17.3.7  Walks as words   cwwlks 30415
            17.3.8  Walks/paths of length 2 (as length 3 strings)   2wlkdlem1 30515
            17.3.9  Walks in regular graphs   rusgrnumwwlkl1 30561
            ​*17.3.10  Closed walks as words   cclwwlk 30573
                  17.3.10.1  Closed walks as words   cclwwlk 30573
                  17.3.10.2  Closed walks of a fixed length as words   cclwwlkn 30616
                  17.3.10.3  Closed walks on a vertex of a fixed length as words   cclwwlknon 30679
            17.3.11  Examples for walks, trails and paths   0ewlk 30706
            17.3.12  Acyclic graphs   cacycgr 30749
            17.3.13  Connected graphs   cconngr 30788
      ​17.4  Eulerian paths and the Konigsberg Bridge problem
            ​*17.4.1  Eulerian paths   ceupth 30799
            ​*17.4.2  The Königsberg Bridge problem   konigsbergvtx 30848
      ​17.5  The Friendship Theorem
            17.5.1  Friendship graphs - basics   cfrgr 30860
            17.5.2  The friendship theorem for small graphs   frgr1v 30873
            17.5.3  Theorems according to Mertzios and Unger   2pthfrgrrn 30884
            ​*17.5.4  Huneke's Proof of the Friendship Theorem   frgrncvvdeqlem1 30901
​PART 18  GUIDES AND MISCELLANEA
      ​18.1  Guides (conventions, explanations, and examples)
            ​*18.1.1  Conventions   conventions 31002
            18.1.2  Natural deduction   natded 31005
            ​*18.1.3  Natural deduction examples   ex-natded5.2 31006
            18.1.4  Definitional examples   ex-or 31023
            18.1.5  Other examples   aevdemo 31062
      ​18.2  Humor
            18.2.1  April Fool's theorem   avril1 31065
      ​18.3  (Future - to be reviewed and classified)
            18.3.1  Planar incidence geometry   cplig 31077
            ​*18.3.2  Aliases kept to prevent broken links   dummylink 31090
​​*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 31092
            19.1.2  Abelian groups   cablo 31147
      ​19.2  Complex vector spaces
            19.2.1  Definition and basic properties   cvc 31161
            19.2.2  Examples of complex vector spaces   cnaddabloOLD 31184
      ​19.3  Normed complex vector spaces
            19.3.1  Definition and basic properties   cnv 31187
            19.3.2  Examples of normed complex vector spaces   cnnv 31280
            19.3.3  Induced metric of a normed complex vector space   imsval 31288
            19.3.4  Inner product   cdip 31303
            19.3.5  Subspaces   css 31324
      ​19.4  Operators on complex vector spaces
            19.4.1  Definitions and basic properties   clno 31343
      ​19.5  Inner product (pre-Hilbert) spaces
            19.5.1  Definition and basic properties   ccphlo 31415
            19.5.2  Examples of pre-Hilbert spaces   cncph 31422
            19.5.3  Properties of pre-Hilbert spaces   isph 31425
      ​19.6  Complex Banach spaces
            19.6.1  Definition and basic properties   ccbn 31465
            19.6.2  Examples of complex Banach spaces   cnbn 31472
            19.6.3  Uniform Boundedness Theorem   ubthlem1 31473
            19.6.4  Minimizing Vector Theorem   minvecolem1 31477
      ​19.7  Complex Hilbert spaces
            19.7.1  Definition and basic properties   chlo 31488
            19.7.2  Standard axioms for a complex Hilbert space   hlex 31501
            19.7.3  Examples of complex Hilbert spaces   cnchl 31519
            19.7.4  Hellinger-Toeplitz Theorem   htthlem 31520
​​*PART 20  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      ​20.1  Axiomatization of complex pre-Hilbert spaces
            20.1.1  Basic Hilbert space definitions   chba 31522
            20.1.2  Preliminary ZFC lemmas   df-hnorm 31571
            ​*20.1.3  Derive the Hilbert space axioms from ZFC set theory   axhilex-zf 31584
            ​*20.1.4  Introduce the vector space axioms for a Hilbert space   ax-hilex 31602
            20.1.5  Vector operations   hvmulex 31614
            20.1.6  Inner product postulates for a Hilbert space   ax-hfi 31682
      ​20.2  Inner product and norms
            20.2.1  Inner product   his5 31689
            20.2.2  Norms   dfhnorm2 31725
            20.2.3  Relate Hilbert space to normed complex vector spaces   hilablo 31763
            20.2.4  Bunjakovaskij-Cauchy-Schwarz inequality   bcsiALT 31782
      ​20.3  Cauchy sequences and completeness axiom
            20.3.1  Cauchy sequences and limits   hcau 31787
            20.3.2  Derivation of the completeness axiom from ZF set theory   hilmet 31797
            20.3.3  Completeness postulate for a Hilbert space   ax-hcompl 31805
            20.3.4  Relate Hilbert space to ZFC pre-Hilbert and Hilbert spaces   hhcms 31806
      ​20.4  Subspaces and projections
            20.4.1  Subspaces   df-sh 31810
            20.4.2  Closed subspaces   df-ch 31824
            20.4.3  Orthocomplements   df-oc 31855
            20.4.4  Subspace sum, span, lattice join, lattice supremum   df-shs 31911
            20.4.5  Projection theorem   pjhthlem1 31994
            20.4.6  Projectors   df-pjh 31998
      ​20.5  Properties of Hilbert subspaces
            20.5.1  Orthomodular law   omlsilem 32005
            20.5.2  Projectors (cont.)   pjhtheu2 32019
            20.5.3  Hilbert lattice operations   sh0le 32043
            20.5.4  Span (cont.) and one-dimensional subspaces   spansn0 32144
            20.5.5  Commutes relation for Hilbert lattice elements   df-cm 32186
            20.5.6  Foulis-Holland theorem   fh1 32221
            20.5.7  Quantum Logic Explorer axioms   qlax1i 32230
            20.5.8  Orthogonal subspaces   chscllem1 32240
            20.5.9  Orthoarguesian laws 5OA and 3OA   5oalem1 32257
            20.5.10  Projectors (cont.)   pjorthi 32272
            20.5.11  Mayet's equation E_3   mayete3i 32331
      ​20.6  Operators on Hilbert spaces
            ​*20.6.1  Operator sum, difference, and scalar multiplication   df-hosum 32333
            20.6.2  Zero and identity operators   df-h0op 32351
            20.6.3  Operations on Hilbert space operators   hoaddcl 32361
            20.6.4  Linear, continuous, bounded, Hermitian, unitary operators and norms   df-nmop 32442
            20.6.5  Linear and continuous functionals and norms   df-nmfn 32448
            20.6.6  Adjoint   df-adjh 32452
            20.6.7  Dirac bra-ket notation   df-bra 32453
            20.6.8  Positive operators   df-leop 32455
            20.6.9  Eigenvectors, eigenvalues, spectrum   df-eigvec 32456
            20.6.10  Theorems about operators and functionals   nmopval 32459
            20.6.11  Riesz lemma   riesz3i 32665
            20.6.12  Adjoints (cont.)   cnlnadjlem1 32670
            20.6.13  Quantum computation error bound theorem   unierri 32707
            20.6.14  Dirac bra-ket notation (cont.)   branmfn 32708
            20.6.15  Positive operators (cont.)   leopg 32725
            20.6.16  Projectors as operators   pjhmopi 32749
      ​20.7  States on a Hilbert lattice and Godowski's equation
            20.7.1  States on a Hilbert lattice   df-st 32814
            20.7.2  Godowski's equation   golem1 32874
      ​20.8  Cover relation, atoms, exchange axiom, and modular symmetry
            20.8.1  Covers relation; modular pairs   df-cv 32882
            20.8.2  Atoms   df-at 32941
            20.8.3  Superposition principle   superpos 32957
            20.8.4  Atoms, exchange and covering properties, atomicity   chcv1 32958
            20.8.5  Irreducibility   chirredlem1 32993
            20.8.6  Atoms (cont.)   atcvat3i 32999
            20.8.7  Modular symmetry   mdsymlem1 33006
​PART 21  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      ​21.1  Mathboxes for user contributions
            21.1.1  Mathbox guidelines   mathbox 33045
      ​21.2  Mathbox for Jorge Agra
      ​21.3  Mathbox for Stefan Allan
      ​21.4  Mathbox for Thierry Arnoux
            21.4.1  Propositional Calculus - misc additions   ad11antr 33092
            21.4.2  Predicate Calculus   sbc2iedf 33105
                  21.4.2.1  Predicate Calculus - misc additions   sbc2iedf 33105
                  21.4.2.2  Restricted quantification - misc additions   ralcom4f 33107
                  21.4.2.3  Equality   eqtrb 33113
                  21.4.2.4  Double restricted existential uniqueness quantification   opsbc2ie 33115
                  21.4.2.5  Double restricted existential uniqueness quantification syntax   w2reu 33117
                  21.4.2.6  Substitution (without distinct variables) - misc additions   sbceqbidf 33126
                  21.4.2.7  Existential "at most one" - misc additions   mo5f 33128
                  21.4.2.8  Existential uniqueness - misc additions   reuxfrdf 33130
                  21.4.2.9  Restricted "at most one" - misc additions   rmoxfrd 33132
                  21.4.2.10  Restricted iota (description binder)   riotaeqbidva 33135
            21.4.3  General Set Theory   dmrab 33136
                  21.4.3.1  Class abstractions (a.k.a. class builders)   dmrab 33136
                  21.4.3.2  Image Sets   abrexdomjm 33146
                  21.4.3.3  Set relations and operations - misc additions   nelun 33152
                  21.4.3.4  Unordered pairs   elpreq 33167
                  21.4.3.5  Unordered triples   tpssg 33176
                  21.4.3.6  Conditional operator - misc additions   ifeqeqx 33181
                  21.4.3.7  Set union   uniinn0 33190
                  21.4.3.8  Indexed union - misc additions   cbviunf 33193
                  21.4.3.9  Indexed intersection - misc additions   iinabrex 33206
                  21.4.3.10  Disjointness - misc additions   disjnf 33207
            21.4.4  Relations and Functions   xpdisjres 33235
                  21.4.4.1  Relations - misc additions   xpdisjres 33235
                  21.4.4.2  Functions - misc additions   fconst7v 33257
                  21.4.4.3  Operations - misc additions   mpomptxf 33315
                  21.4.4.4  Support of a function   suppovss 33317
                  21.4.4.5  Explicit Functions with one or two points as a domain   cosnopne 33330
                  21.4.4.6  Isomorphisms - misc. additions   gtiso 33337
                  21.4.4.7  Disjointness (additional proof requiring functions)   disjdsct 33339
                  21.4.4.8  First and second members of an ordered pair - misc additions   df1stres 33340
                  21.4.4.9  Countable Sets   snct 33348
            21.4.5  Real and Complex Numbers   sgnval2 33370
                  21.4.5.1  Complex operations - misc. additions   creq0 33371
                  21.4.5.2  Ordering on reals - misc additions   lt2addrd 33385
                  21.4.5.3  Extended reals - misc additions   nn0mnfxrd 33386
                  21.4.5.4  Extended nonnegative integers - misc additions   xnn0gt0 33404
                  21.4.5.5  Real number intervals - misc additions   joiniooico 33409
                  21.4.5.6  Finite intervals of integers - misc additions   uzssico 33419
                  21.4.5.7  Half-open integer ranges - misc additions   iundisjfi 33431
                  21.4.5.8  The ` # ` (set size) function - misc additions   hashunif 33441
                  21.4.5.9  The greatest common divisor operator - misc. additions   elq2 33446
                  21.4.5.10  Integers   nn0split01 33452
                  21.4.5.11  Decimal numbers   dfdec100 33464
            21.4.6  Real and complex functions   sgnsgn 33465
                  21.4.6.1  Signum (sgn or sign) function - misc. additions   sgnsgn 33465
                  21.4.6.2  Integer powers - misc. additions   nexple 33467
                  21.4.6.3  Indicator Functions (continued)   indsumin 33471
            ​*21.4.7  Decimal expansion   cdp2 33480
                  ​*21.4.7.1  Decimal point   cdp 33497
                  21.4.7.2  Division in the extended real number system   cxdiv 33526
            21.4.8  Words over a set - misc additions   wrdres 33545
                  21.4.8.1  Splicing words (substring replacement)   splfv3 33562
                  21.4.8.2  Cyclic shift of words   1cshid 33563
            21.4.9  Extensible Structures   ressplusf 33567
                  21.4.9.1  Structure restriction operator   ressplusf 33567
                  21.4.9.2  Posets   ressprs 33570
                  21.4.9.3  Complete lattices   clatp0cl 33580
                  21.4.9.4  Order Theory   cmnt 33582
                  21.4.9.5  Extended reals Structure - misc additions   ax-xrssca 33608
                  21.4.9.6  The extended nonnegative real numbers commutative monoid   xrge00 33618
            21.4.10  Algebra   mndcld 33626
                  21.4.10.1  Monoids   mndcld 33626
                  21.4.10.2  Monoids Homomorphisms   abliso 33639
                  21.4.10.3  Groups - misc additions   grpidcld 33643
                  21.4.10.4  Abelian Groups - misc additions   ablcomd 33649
                  21.4.10.5  Finitely supported group sums - misc additions   gsumsubg 33650
                  21.4.10.6  Group or monoid sums over words   gsumwun 33680
                  21.4.10.7  Centralizers and centers - misc additions   cntzun 33683
                  21.4.10.8  The symmetric group   symgfcoeu 33686
                  21.4.10.9  Transpositions   pmtridf1o 33698
                  21.4.10.10  Permutation Signs   psgnid 33701
                  21.4.10.11  Permutation cycles   ctocyc 33710
                  21.4.10.12  The Alternating Group   evpmval 33749
                  21.4.10.13  Signum in an ordered monoid   csgns 33762
                  21.4.10.14  Fixed points   cfxp 33767
                  21.4.10.15  The Archimedean property for generic ordered algebraic structures   cinftm 33780
                  21.4.10.16  Semiring left modules   cslmd 33804
                  21.4.10.17  Simple groups   prmsimpcyc 33832
                  21.4.10.18  Rings - misc additions   ringrngd 33833
                  21.4.10.19  Subrings generated by a set   elrgspnlem1 33846
                  21.4.10.20  The zero ring   irrednzr 33854
                  21.4.10.21  Localization of rings   cerl 33857
                  21.4.10.22  Integral Domains   domnmuln0rd 33881
                  21.4.10.23  Euclidean Domains   ceuf 33895
                  21.4.10.24  Division Rings   rndrhmcl 33901
                  21.4.10.25  The field of rational numbers   qfld 33902
                  21.4.10.26  Subfields   subsdrg 33903
                  21.4.10.27  Field of fractions   cfrac 33907
                  21.4.10.28  Field extensions generated by a set   cfldgen 33915
                  21.4.10.29  Ring homomorphisms - misc additions   rhmdvd 33928
                  21.4.10.30  Scalar restriction operation   cresv 33930
                  21.4.10.31  The commutative ring of gaussian integers   gzcrng 33945
                  21.4.10.32  The archimedean ordered field of real numbers   cnfldfld 33946
                  21.4.10.33  The quotient map and quotient modules   qusker 33953
                  21.4.10.34  The ring of integers modulo ` N `   znfermltl 33965
                  21.4.10.35  Independent sets and families   islinds5 33966
                  21.4.10.36  Ring associates, ring units   dvdsruassoi 33982
                  ​*21.4.10.37  Subgroup sum / Sumset / Minkowski sum   elgrplsmsn 33988
                  21.4.10.38  The quotient map   quslsm 33999
                  21.4.10.39  Ideals   intlidl 34013
                  21.4.10.40  Maximal Ideals   cmxidl 34027
                  21.4.10.41  Local rings   drnglring 34067
                  21.4.10.42  The semiring of ideals of a ring   cidlsrg 34075
                  21.4.10.43  Prime Elements   rprmval 34091
                  21.4.10.44  Unique factorization domains   cufd 34113
                  21.4.10.45  The ring of integers   zringidom 34126
                  21.4.10.46  Associative Algebra   assaassd 34130
                  21.4.10.47  Univariate Polynomials   0ringmon1p 34132
                  21.4.10.48  Polynomial quotient and polynomial remainder   q1pdir 34178
                  21.4.10.49  Multivariate Polynomials   psrbasfsupp 34186
                  21.4.10.50  The ring of symmetric polynomials   csply 34230
                  21.4.10.51  The subring algebra   sra1r 34256
                  21.4.10.52  Division Ring Extensions   drgext0g 34265
                  21.4.10.53  Vector Spaces   lvecdimfi 34271
                  21.4.10.54  Vector Space Dimension   cldim 34274
            21.4.11  Field Extensions   cfldext 34313
                  21.4.11.1  Algebraic numbers   cirng 34358
                  21.4.11.2  Algebraic extensions   calgext 34370
                  21.4.11.3  Minimal polynomials   cminply 34374
                  21.4.11.4  Quadratic Field Extensions   rtelextdg2lem 34401
                  21.4.11.5  Towers of quadratic extentions   fldext2chn 34403
            ​*21.4.12  Constructible Numbers   cconstr 34404
                  21.4.12.1  Impossible constructions   2sqr3minply 34455
            21.4.13  Matrices   csmat 34468
                  21.4.13.1  Submatrices   csmat 34468
                  21.4.13.2  Matrix literals   clmat 34486
                  21.4.13.3  Laplace expansion of determinants   mdetpmtr1 34498
            21.4.14  Topology   ist0cld 34508
                  21.4.14.1  Open maps   txomap 34509
                  21.4.14.2  Topology of the unit circle   qtopt1 34510
                  21.4.14.3  Refinements   reff 34514
                  21.4.14.4  Open cover refinement property   ccref 34517
                  21.4.14.5  Lindelöf spaces   cldlf 34527
                  21.4.14.6  Paracompact spaces   cpcmp 34530
                  ​*21.4.14.7  Spectrum of a ring   crspec 34537
                  21.4.14.8  Pseudometrics   cmetid 34561
                  21.4.14.9  Continuity - misc additions   hauseqcn 34573
                  21.4.14.10  Topology of the closed unit interval   elunitge0 34574
                  21.4.14.11  Topology of ` ( RR X. RR ) `   unicls 34578
                  21.4.14.12  Order topology - misc. additions   cnvordtrestixx 34588
                  21.4.14.13  Continuity in topological spaces - misc. additions   mndpluscn 34601
                  21.4.14.14  Topology of the extended nonnegative real numbers ordered monoid   xrge0hmph 34607
                  21.4.14.15  Limits - misc additions   lmlim 34622
                  21.4.14.16  Univariate polynomials   pl1cn 34630
            21.4.15  Uniform Stuctures and Spaces   chcmp 34631
                  21.4.15.1  Hausdorff uniform completion   chcmp 34631
            21.4.16  Topology and algebraic structures   zringnm 34633
                  21.4.16.1  The norm on the ring of the integer numbers   zringnm 34633
                  21.4.16.2  Topological ` ZZ ` -modules   zlm0 34635
                  21.4.16.3  Canonical embedding of the field of the rational numbers into a division ring   cqqh 34645
                  21.4.16.4  Canonical embedding of the real numbers into a complete ordered field   crrh 34668
                  21.4.16.5  Embedding from the extended real numbers into a complete lattice   cxrh 34691
                  21.4.16.6  Canonical embeddings into the ordered field of the real numbers   zrhre 34694
                  ​*21.4.16.7  Topological Manifolds   cmntop 34697
                  21.4.16.8  Extended sum   cesum 34702
            21.4.17  Mixed Function/Constant operation   cofc 34770
            21.4.18  Abstract measure   csiga 34783
                  21.4.18.1  Sigma-Algebra   csiga 34783
                  21.4.18.2  Generated sigma-Algebra   csigagen 34814
                  ​*21.4.18.3  lambda and pi-Systems, Rings of Sets   ispisys 34828
                  21.4.18.4  The Borel algebra on the real numbers   cbrsiga 34857
                  21.4.18.5  Product Sigma-Algebra   csx 34864
                  21.4.18.6  Measures   cmeas 34871
                  21.4.18.7  The counting measure   cntmeas 34902
                  21.4.18.8  The Lebesgue measure - misc additions   voliune 34905
                  21.4.18.9  The Dirac delta measure   cdde 34908
                  21.4.18.10  The 'almost everywhere' relation   cae 34913
                  21.4.18.11  Measurable functions   cmbfm 34925
                  21.4.18.12  Borel Algebra on ` ( RR X. RR ) `   br2base 34944
                  ​*21.4.18.13  Caratheodory's extension theorem   coms 34966
            21.4.19  Integration   itgeq12dv 35001
                  21.4.19.1  Lebesgue integral - misc additions   itgeq12dv 35001
                  21.4.19.2  Bochner integral   citgm 35002
            21.4.20  Euler's partition theorem   oddpwdc 35029
            21.4.21  Sequences defined by strong recursion   csseq 35058
            21.4.22  Fibonacci Numbers   cfib 35071
            21.4.23  Probability   cprb 35082
                  21.4.23.1  Probability Theory   cprb 35082
                  21.4.23.2  Conditional Probabilities   ccprob 35106
                  21.4.23.3  Real-valued Random Variables   crrv 35115
                  21.4.23.4  Preimage set mapping operator   corvc 35131
                  21.4.23.5  Distribution Functions   orvcelval 35144
                  21.4.23.6  Cumulative Distribution Functions   orvclteel 35148
                  21.4.23.7  Probabilities - example   coinfliplem 35154
                  21.4.23.8  Bertrand's Ballot Problem   ballotlemoex 35161
            21.4.24  Signum (sgn or sign) function - misc. additions   fzssfzo 35214
                  21.4.24.1  Operations on words   ccatmulgnn0dir 35217
            21.4.25  Polynomials with real coefficients - misc additions   plyrecld 35221
            21.4.26  Descartes's rule of signs   signspval 35224
                  21.4.26.1  Sign changes in a word over real numbers   signspval 35224
                  21.4.26.2  Counting sign changes in a word over real numbers   signslema 35234
            21.4.27  Number Theory   iblidicc 35264
                  21.4.27.1  Representations of a number as sums of integers   crepr 35280
                  21.4.27.2  Vinogradov Trigonometric Sums and the Circle Method   cvts 35307
                  21.4.27.3  The Ternary Goldbach Conjecture: Final Statement   ax-hgt749 35316
            21.4.28  Elementary Geometry   cstrkg2d 35336
                  ​*21.4.28.1  Two-dimensional geometry   cstrkg2d 35336
                  21.4.28.2  Morley's Miracle   cgranbtwn 35341
                  21.4.28.3  Outer Five Segment (not used, no need to move to main)   cafs 35344
            ​*21.4.29  LeftPad Project   clpad 35349
      ​​*21.5  Mathbox for Jonathan Ben-Naim
            21.5.1  First-order logic and set theory   bnj170 35372
            21.5.2  Well founded induction and recursion   bnj110 35531
            21.5.3  The existence of a minimal element in certain classes   bnj69 35683
            21.5.4  Well-founded induction   bnj1204 35685
            21.5.5  Well-founded recursion, part 1 of 3   bnj60 35735
            21.5.6  Well-founded recursion, part 2 of 3   bnj1500 35741
            21.5.7  Well-founded recursion, part 3 of 3   bnj1522 35745
      ​21.6  Mathbox for BTernaryTau
            21.6.1  First-order logic   nfan1c 35746
                  21.6.1.1  Auxiliary axiom schemes   nfan1c 35746
            21.6.2  ZF set theory   inv2 35752
                  21.6.2.1  Ordinals 5 through 9   c5o 35810
                  21.6.2.2  Finitism   prcinf 35830
                  21.6.2.3  Introduce ax-regs   ax-regs 35843
                  21.6.2.4  Derive ax-regs   axregs 35856
                  21.6.2.5  ZFC axioms with reduced distinct variable conditions   axsepg2 35857
                  21.6.2.6  Cardinality without the Axiom of Choice   ckard 35866
                  21.6.2.7  Gödel operations   ccnv2 35890
                  21.6.2.8  The constructible universe   clexo 35912
                  21.6.2.9  Global choice   gblacfnacd 35941
            21.6.3  Real and complex numbers   zltp1ne 35969
            21.6.4  Graph theory   cplgredgex 35974
                  21.6.4.1  Acyclic graphs   acycgr0v 35982
      ​21.7  Mathbox for Mario Carneiro
            21.7.1  Predicate calculus with all distinct variables   ax-7d 35993
            21.7.2  Miscellaneous stuff   quartfull 35999
            21.7.3  Derangements and the Subfactorial   deranglem 36000
            21.7.4  The Erdős-Szekeres theorem   erdszelem1 36025
            21.7.5  The Kuratowski closure-complement theorem   kur14lem1 36040
            21.7.6  Retracts and sections   cretr 36051
            21.7.7  Path-connected and simply connected spaces   cpconn 36053
            21.7.8  Covering maps   ccvm 36089
            21.7.9  Normal numbers   snmlff 36163
            21.7.10  Godel-sets of formulas - part 1   cgoe 36167
            21.7.11  Godel-sets of formulas - part 2   cgon 36266
            21.7.12  Models of ZF   cgze 36280
            ​*21.7.13  Metamath formal systems   cmcn 36294
            21.7.14  Grammatical formal systems   cm0s 36419
            21.7.15  Models of formal systems   cmuv 36439
            21.7.16  Splitting fields   ccpms 36461
            21.7.17  p-adic number fields   czr 36481
      ​​*21.8  Mathbox for Filip Cernatescu
      ​21.9  Mathbox for Paul Chapman
            21.9.1  Real and complex numbers (cont.)   climuzcnv 36505
            21.9.2  Miscellaneous theorems   elfzm12 36509
      ​21.10  Mathbox for Hongxiu Chen
      ​21.11  Mathbox for Adrian Ducourtial
            21.11.1  Propositional calculus   currybi 36522
            21.11.2  Set theory   mirimanoffr 36529
            21.11.3  Clone theory   ccloneop 36535
      ​21.12  Mathbox for Scott Fenton
            21.12.1  ZFC Axioms in primitive form   axextprim 36541
            21.12.2  Untangled classes   untelirr 36548
            21.12.3  Extra propositional calculus theorems   3jaodd 36555
            21.12.4  Misc. Useful Theorems   nepss 36558
            21.12.5  Properties of real and complex numbers   sqdivzi 36568
            21.12.6  Infinite products   iprodefisumlem 36580
            21.12.7  Factorial limits   faclimlem1 36583
            21.12.8  Greatest common divisor and divisibility   gcd32 36589
            21.12.9  Properties of relationships   dftr6 36591
            21.12.10  Properties of functions and mappings   funpsstri 36606
            21.12.11  Ordinal numbers   elpotr 36618
            21.12.12  Defined equality axioms   axextdfeq 36634
            21.12.13  Hypothesis builders   hbntg 36642
            21.12.14  Well-founded zero, successor, and limits   cwsuc 36647
            21.12.15  Quantifier-free definitions   ctxp 36667
            21.12.16  Alternate ordered pairs   caltop 36796
            21.12.17  Geometry in the Euclidean space   cofs 36822
                  21.12.17.1  Congruence properties   cofs 36822
                  21.12.17.2  Betweenness properties   btwntriv2 36852
                  21.12.17.3  Segment Transportation   ctransport 36869
                  21.12.17.4  Properties relating betweenness and congruence   cifs 36875
                  21.12.17.5  Connectivity of betweenness   btwnconn1lem1 36927
                  21.12.17.6  Segment less than or equal to   csegle 36946
                  21.12.17.7  Outside-of relationship   coutsideof 36959
                  21.12.17.8  Lines and Rays   cline2 36974
            21.12.18  Forward difference   cfwddif 36998
            21.12.19  Rank theorems   rank0 37006
            21.12.20  Hereditarily Finite Sets   hftr 37008
            21.12.21  Natural ordinal operations   cnmul 37011
      ​21.13  Mathbox for Gino Giotto
            21.13.1  Equality theorems   rmoeqi 37051
                  21.13.1.1  Inference versions   rmoeqi 37051
                  21.13.1.2  Deduction versions   rmoeqdv 37076
            21.13.2  Change bound variables   in-ax8 37088
                  21.13.2.1  Change bound variables and domains   cbvralvw2 37090
                  21.13.2.2  Change bound variables, deduction versions   cbvmodavw 37114
                  21.13.2.3  Change bound variables and domains, deduction versions   cbvrmodavw2 37147
            21.13.3  Study of ax-mulf usage   mpomulnzcnf 37163
      ​21.14  Mathbox for Jeff Hankins
            21.14.1  Miscellany   a1i14 37164
            21.14.2  Basic topological facts   topbnd 37187
            21.14.3  Topology of the real numbers   ivthALT 37198
            21.14.4  Refinements   cfne 37199
            21.14.5  Neighborhood bases determine topologies   neibastop1 37222
            21.14.6  Lattice structure of topologies   topmtcl 37226
            21.14.7  Filter bases   fgmin 37233
            21.14.8  Directed sets, nets   tailfval 37235
      ​21.15  Mathbox for Anthony Hart
            21.15.1  Propositional Calculus   tb-ax1 37246
            21.15.2  Predicate Calculus   nalfal 37266
            21.15.3  Miscellaneous single axioms   meran1 37274
            21.15.4  Connective Symmetry   negsym1 37280
      ​21.16  Mathbox for Chen-Pang He
            21.16.1  Ordinal topology   ontopbas 37291
      ​21.17  Mathbox for Jeff Hoffman
            21.17.1  Inferences for finite induction on generic function values   fveleq 37314
            21.17.2  gdc.mm   nnssi2 37318
      ​21.18  Mathbox for Matthew House
            21.18.1  Relations on well-ordered indexed unions   weiunval 37325
            21.18.2  Axiom of Transitive Containment   axtco 37334
            21.18.3  Transitive closure of a class   tr0elw 37347
            ​*21.18.4  Stronger axioms of regularity   mh-setind 37399
            21.18.5  Short axioms written in primitive symbols   mh-inf3f1 37404
      ​21.19  Mathbox for Asger C. Ipsen
            21.19.1  Continuous nowhere differentiable functions   dnival 37412
      ​​*21.20  Mathbox for BJ
            *21.20.1  Propositional calculus   bj-mp2c 37481
                  *21.20.1.1  Derived rules of inference   bj-mp2c 37481
                  ​*21.20.1.2  A syntactic theorem   bj-0 37483
                  ​*21.20.1.3  Minimal implicational calculus   bj-poni 37485
                  ​*21.20.1.4  Positive calculus   bj-bisimpl 37497
                  ​*21.20.1.5  Implication and negation   bj-con2com 37505
                  ​*21.20.1.6  Disjunction   bj-jaoi1 37516
                  ​*21.20.1.7  Logical equivalence   bj-dfbi4 37518
                  21.20.1.8  The conditional operator for propositions   bj-consensus 37523
                  ​*21.20.1.9  Propositional calculus: miscellaneous   bj-imbi12 37528
            ​*21.20.2  Modal logic   bj-axdd2 37537
            ​*21.20.3  Provability logic   cprvb 37542
            ​*21.20.4  First-order logic   bj-exexalal 37551
                  21.20.4.1  Universal and existential quantifiers, nonfreeness predicate   bj-exexalal 37551
                  21.20.4.2  Adding ax-gen   bj-genr 37552
                  21.20.4.3  Adding ax-4   bj-almp 37556
                  21.20.4.4  Adding ax-5   bj-spvw 37609
                  21.20.4.5  Equality and substitution   bj-df-sb 37624
                  21.20.4.6  Adding ax-6   bj-spim0 37643
                  21.20.4.7  Adding ax-7   bj-cbvexw 37651
                  21.20.4.8  Membership predicate, ax-8 and ax-9   bj-ax89 37653
                  21.20.4.9  Adding ax-11   bj-alcomexcom 37655
                  21.20.4.10  Adding ax-12   axc11n11 37659
                  ​*21.20.4.11  Really adding ax-12   bj-substax12 37701
                  21.20.4.12  Nonfreeness   wnnf 37703
                  21.20.4.13  Adding ax-13   bj-axc10 37770
                  ​*21.20.4.14  Removing dependencies on ax-13 (and ax-11)   bj-axc10v 37780
                  ​*21.20.4.15  Distinct var metavariables   bj-hbaeb2 37805
                  ​*21.20.4.16  Around ~ equsal   bj-equsal1t 37809
                  ​*21.20.4.17  Some Principia Mathematica proofs   stdpc5t 37814
                  21.20.4.18  Alternate definition of substitution   bj-sbsb 37824
                  21.20.4.19  Lemmas for substitution   bj-sbf3 37826
                  21.20.4.20  Existential uniqueness   bj-eu3f 37828
                  ​*21.20.4.21  First-order logic: miscellaneous   bj-sblem1 37829
            21.20.5  Set theory   eliminable1 37846
                  ​*21.20.5.1  Eliminability of class terms   eliminable1 37846
                  ​*21.20.5.2  Classes without the axiom of extensionality   bj-denoteslem 37858
                  21.20.5.3  Characterization among sets versus among classes   elelb 37884
                  ​*21.20.5.4  The nonfreeness quantifier for classes   bj-nfcsym 37886
                  ​*21.20.5.5  Lemmas for class substitution   bj-sbeqALT 37887
                  21.20.5.6  Removing some axiom requirements and disjoint variable conditions   bj-exlimvmpi 37898
                  ​*21.20.5.7  Class abstractions   bj-elabd2ALT 37913
                  21.20.5.8  Generalized class abstractions   bj-cgab 37921
                  ​*21.20.5.9  Restricted nonfreeness   wrnf 37929
                  ​*21.20.5.10  Russell's paradox   bj-ru1 37931
                  21.20.5.11  Curry's paradox in set theory   currysetlem 37933
                  ​*21.20.5.12  Some disjointness results   bj-n0i 37939
                  ​*21.20.5.13  Complements on direct products   bj-xpimasn 37943
                  ​*21.20.5.14  "Singletonization" and tagging   bj-snsetex 37951
                  ​*21.20.5.15  Tuples of classes   bj-cproj 37978
                  ​*21.20.5.16  Set theory: elementary operations relative to a universe   bj-rcleqf 38013
                  ​*21.20.5.17  Axioms for finite unions   bj-abex 38018
                  21.20.5.18  Relations, functions, composition: supplements   elco 38035
                  ​*21.20.5.19  Set theory: miscellaneous   eleq2w2ALT 38037
                  ​*21.20.5.20  Axioms of separation and replacement   bj-axnul 38063
                  ​*21.20.5.21  Evaluation at a class   bj-evaleq 38067
                  21.20.5.22  Elementwise operations   celwise 38075
                  ​*21.20.5.23  Elementwise intersection (families of sets induced on a subset)   bj-rest00 38077
                  21.20.5.24  Moore collections (complements)   bj-raldifsn 38096
                  21.20.5.25  Maps-to notation for functions with three arguments   bj-0nelmpt 38113
                  ​*21.20.5.26  Currying   csethom 38117
                  ​*21.20.5.27  Setting components of extensible structures   cstrset 38129
            ​*21.20.6  Extended real and complex numbers, real and complex projective lines   bj-nfald 38132
                  21.20.6.1  Complements on class abstractions of ordered pairs and binary relations   bj-nfald 38132
                  ​*21.20.6.2  Identity relation (complements)   bj-opabssvv 38147
                  ​*21.20.6.3  Functionalized identity (diagonal in a Cartesian square)   cdiag2 38169
                  ​*21.20.6.4  Direct image and inverse image   cimdir 38175
                  ​*21.20.6.5  Extended numbers and projective lines as sets   cfractemp 38193
                  ​*21.20.6.6  Addition and opposite   caddcc 38234
                  ​*21.20.6.7  Order relation on the extended reals   cltxr 38238
                  ​*21.20.6.8  Argument, multiplication and inverse   carg 38240
                  21.20.6.9  The canonical bijection from the finite ordinals   ciomnn 38246
                  21.20.6.10  Divisibility   cnnbar 38257
            ​*21.20.7  Monoids   bj-smgrpssmgm 38265
                  ​*21.20.7.1  Finite sums in monoids   cfinsum 38280
            ​*21.20.8  Affine, Euclidean, and Cartesian geometry   bj-fvimacnv0 38283
                  *21.20.8.1  Real vector spaces   bj-fvimacnv0 38283
                  ​*21.20.8.2  Complex numbers (supplements)   bj-subcom 38305
                  ​*21.20.8.3  Barycentric coordinates   bj-bary1lem 38307
            21.20.9  Monoid of endomorphisms   cend 38310
      ​21.21  Mathbox for Jim Kingdon
            21.21.1  Circle constant   taupilem3 38316
            21.21.2  Number theory   dfgcd3 38321
            21.21.3  Real numbers   irrdifflemf 38322
      ​21.22  Mathbox for ML
            21.22.1  Miscellaneous   csbrecsg 38327
            21.22.2  Cartesian exponentiation   cfinxp 38382
            21.22.3  Topology   iunctb2 38402
                  ​*21.22.3.1  Pi-base theorems   pibp16 38412
      ​21.23  Mathbox for Wolf Lammen
            21.23.1  1. Bootstrapping   wl-section-boot 38421
            21.23.2  Implication chains   wl-section-impchain 38445
            21.23.3  Theorems around the conditional operator   wl-ifp-ncond1 38463
            21.23.4  Alternative development of hadd, cadd   wl-df-3xor 38467
            21.23.5  An alternative axiom ~ ax-13   ax-wl-13v 38492
            21.23.6  Bootstrapping set theory with classes   wl-cleq-0 38494
            21.23.7  Other stuff   wl-mps 38515
      ​21.24  Mathbox for Brendan Leahy
      ​21.25  Mathbox for Thomas van Maaren
            ​*21.25.1  The language of Propositional Calculus.   cpropvar 38709
      ​21.26  Mathbox for Jeff Madsen
            21.26.1  Logic and set theory   unirep 38724
            21.26.2  Real and complex numbers; integers   filbcmb 38750
            21.26.3  Sequences and sums   sdclem2 38752
            21.26.4  Topology   subspopn 38762
            21.26.5  Metric spaces   metf1o 38765
            21.26.6  Continuous maps and homeomorphisms   constcncf 38772
            21.26.7  Boundedness   ctotbnd 38776
            21.26.8  Isometries   cismty 38808
            21.26.9  Heine-Borel Theorem   heibor1lem 38819
            21.26.10  Banach Fixed Point Theorem   bfplem1 38832
            21.26.11  Euclidean space   crrn 38835
            21.26.12  Intervals (continued)   ismrer1 38848
            ​*21.26.13  Operation properties   cass 38852
            21.26.14  Groups and related structures   cmagm 38858
            21.26.15  Group homomorphism and isomorphism   cghomOLD 38893
            21.26.16  Rings   crngo 38904
            21.26.17  Division Rings   cdrng 38958
            21.26.18  Ring homomorphisms   crngohom 38970
            21.26.19  Commutative rings   ccm2 38999
            21.26.20  Ideals   cidl 39017
            21.26.21  Prime rings and integral domains   cprrng 39056
            21.26.22  Ideal generators   cigen 39069
      ​21.27  Mathbox for Giovanni Mascellani
            ​*21.27.1  Tools for automatic proof building   efald2 39088
            ​*21.27.2  Tseitin axioms   fald 39137
            ​*21.27.3  Equality deductions   iuneq2f 39164
            ​*21.27.4  Miscellanea   orcomdd 39175
      ​21.28  Mathbox for Peter Mazsa
            21.28.1  Notations   cxrn 39182
            21.28.2  Preparatory theorems   el2v1 39237
            21.28.3  Range Cartesian product   df-xrn 39388
            21.28.4  Relations   df-rels 39448
            21.28.5  Quotient map (coset map)   df-qmap 39454
            21.28.6  Lifts, shifts, successor, and predecessor   df-adjliftmap 39463
            21.28.7  Cosets by ` R `   df-coss 39509
            21.28.8  Subset relations   df-ssr 39586
            21.28.9  Reflexivity   df-refs 39598
            21.28.10  Converse reflexivity   df-cnvrefs 39613
            21.28.11  Symmetry   df-syms 39630
            21.28.12  Reflexivity and symmetry   symrefref2 39655
            21.28.13  Transitivity   df-trs 39664
            21.28.14  Equivalence relations   df-eqvrels 39676
            21.28.15  Redundancy   df-redunds 39715
            21.28.16  Domain quotients   df-dmqss 39730
            21.28.17  Equivalence relations on domain quotients   df-ers 39756
            21.28.18  Functions   df-funss 39773
            21.28.19  Disjoints vs. converse functions   df-disjss 39796
            21.28.20  Antisymmetry   df-antisymrel 39871
            21.28.21  Partitions: disjoints on domain quotients   df-parts 39876
            21.28.22  Partition-Equivalence Theorems   disjim 39892
            21.28.23  Type-safe Partition-Equivalence: PetParts, PetErs, Pet2Parts, Pet2Ers   df-petparts 39976
      ​21.29  Mathbox for Rodolfo Medina
            21.29.1  Partitions   prtlem60 39986
      ​​*21.30  Mathbox for Norm Megill
            *21.30.1  Obsolete schemes ax-c4,c5,c7,c10,c11,c11n,c15,c9,c14,c16   ax-c5 40016
            ​*21.30.2  Rederive new axioms ax-4, ax-10, ax-6, ax-12, ax-13 from old   axc5 40026
            ​*21.30.3  Legacy theorems using obsolete axioms   ax5ALT 40040
            21.30.4  Experiments with weak deduction theorem   elimhyps 40094
            21.30.5  Miscellanea   cnaddcom 40105
            21.30.6  Atoms, hyperplanes, and covering in a left vector space (or module)   clsa 40107
            21.30.7  Functionals and kernels of a left vector space (or module)   clfn 40190
            21.30.8  Opposite rings and dual vector spaces   cld 40256
            21.30.9  Ortholattices and orthomodular lattices   cops 40305
            21.30.10  Atomic lattices with covering property   ccvr 40395
            21.30.11  Hilbert lattices   chlt 40483
            21.30.12  Projective geometries based on Hilbert lattices   clln 40624
            21.30.13  Construction of a vector space from a Hilbert lattice   cdlema1N 40924
            21.30.14  Construction of involution and inner product from a Hilbert lattice   clpoN 42613
      ​21.31  Mathbox for metakunt
            21.31.1  Commutative Semiring   ccsrg 43095
            21.31.2  General helpful statements   rhmzrhval 43098
            21.31.3  Some gcd and lcm results   12gcd5e1 43129
            21.31.4  Least common multiple inequality theorem   3factsumint1 43147
            21.31.5  Logarithm inequalities   3exp7 43179
            21.31.6  Miscellaneous results for AKS formalisation   intlewftc 43187
            21.31.7  Sticks and stones   sticksstones1 43272
            21.31.8  Continuation AKS   aks6d1c6lem1 43296
      ​21.32  Mathbox for Luke Murphy
            21.32.1  Solutions of quadratic equations   quadfac 43331
            21.32.2  April Fool's theorem   25or6to4 43332
      ​21.33  Mathbox for Steven Nguyen
            21.33.1  Utility theorems   jarrii 43333
            ​*21.33.2  Arithmetic theorems   c0exALT 43379
            21.33.3  Exponents and divisibility   oexpreposd 43455
            21.33.4  Trigonometry and Calculus   tanhalfpim 43476
            ​*21.33.5  Independence of ax-mulcom   cresub 43492
            21.33.6  Structures   sn-base0 43635
            ​*21.33.7  Projective spaces   cprjsp 43705
            21.33.8  Basic reductions for Fermat's Last Theorem   dffltz 43746
            ​*21.33.9  Exemplar theorems   iddii 43751
                  ​*21.33.9.1  Standard replacements of ax-10 , ax-11 , ax-12   nfa1w 43762
      ​21.34  Mathbox for Igor Ieskov
      ​21.35  Mathbox for Timothy J. O'Malley
      ​21.36  Mathbox for OpenAI
      ​21.37  Mathbox for Stefan O'Rear
            21.37.1  Additional elementary logic and set theory   moxfr 43779
            21.37.2  Additional theory of functions   imaiinfv 43780
            21.37.3  Additional topology   elrfi 43781
            21.37.4  Characterization of closure operators. Kuratowski closure axioms   ismrcd1 43785
            21.37.5  Algebraic closure systems   cnacs 43789
            21.37.6  Miscellanea 1. Map utilities   constmap 43800
            21.37.7  Miscellanea for polynomials   mptfcl 43807
            21.37.8  Multivariate polynomials over the integers   cmzpcl 43808
            21.37.9  Miscellanea for Diophantine sets 1   coeq0i 43840
            21.37.10  Diophantine sets 1: definitions   cdioph 43842
            21.37.11  Diophantine sets 2 miscellanea   ellz1 43854
            21.37.12  Diophantine sets 2: union and intersection. Monotone Boolean algebra   diophin 43859
            21.37.13  Diophantine sets 3: construction   diophrex 43862
            21.37.14  Diophantine sets 4 miscellanea   2sbcrex 43871
            21.37.15  Diophantine sets 4: Quantification   rexrabdioph 43877
            21.37.16  Diophantine sets 5: Arithmetic sets   rabdiophlem1 43884
            21.37.17  Diophantine sets 6: reusability. renumbering of variables   eldioph4b 43894
            21.37.18  Pigeonhole Principle and cardinality helpers   fphpd 43899
            21.37.19  A non-closed set of reals is infinite   rencldnfilem 43903
            21.37.20  Lagrange's rational approximation theorem   irrapxlem1 43905
            21.37.21  Pell equations 1: A nontrivial solution always exists   pellexlem1 43912
            21.37.22  Pell equations 2: Algebraic number theory of the solution set   csquarenn 43919
            21.37.23  Pell equations 3: characterizing fundamental solution   infmrgelbi 43961
            ​*21.37.24  Logarithm laws generalized to an arbitrary base   reglogcl 43973
            21.37.25  Pell equations 4: the positive solution group is infinite cyclic   pellfund14 43981
            21.37.26  X and Y sequences 1: Definition and recurrence laws   crmx 43983
            21.37.27  Ordering and induction lemmas for the integers   monotuz 44024
            21.37.28  X and Y sequences 2: Order properties   rmxypos 44030
            21.37.29  Congruential equations   congtr 44048
            21.37.30  Alternating congruential equations   acongid 44058
            21.37.31  Additional theorems on integer divisibility   coprmdvdsb 44068
            21.37.32  X and Y sequences 3: Divisibility properties   jm2.18 44071
            21.37.33  X and Y sequences 4: Diophantine representability of Y   jm2.27a 44088
            21.37.34  X and Y sequences 5: Diophantine representability of X, ^, _C   rmxdiophlem 44098
            21.37.35  Uncategorized stuff not associated with a major project   setindtr 44107
            21.37.36  More equivalents of the Axiom of Choice   axac10 44116
            21.37.37  Finitely generated left modules   clfig 44145
            21.37.38  Noetherian left modules I   clnm 44153
            21.37.39  Addenda for structure powers   pwssplit4 44167
            21.37.40  Every set admits a group structure iff choice   unxpwdom3 44173
            21.37.41  Noetherian rings and left modules II   clnr 44187
            21.37.42  Hilbert's Basis Theorem   cldgis 44199
            21.37.43  Additional material on polynomials [DEPRECATED]   cmnc 44209
            21.37.44  Degree and minimal polynomial of algebraic numbers   cdgraa 44218
            21.37.45  Algebraic integers I   citgo 44235
            21.37.46  Endomorphism algebra   cmend 44249
            21.37.47  Cyclic groups and order   idomodle 44269
            21.37.48  Cyclotomic polynomials   ccytp 44275
            21.37.49  Miscellaneous topology   fgraphopab 44281
      ​21.38  Mathbox for Noam Pasman
      ​21.39  Mathbox for Jon Pennant
      ​21.40  Mathbox for Richard Penner
            21.40.1  Set Theory and Ordinal Numbers   uniel 44295
            21.40.2  Natural addition of Cantor normal forms   oawordex2 44404
            21.40.3  Surreal Contributions   abeqabi 44485
            21.40.4  Short Studies   nlimsuc 44518
                  21.40.4.1  Additional work on conditional logical operator   ifpan123g 44536
                  21.40.4.2  Sophisms   rp-fakeimass 44589
                  ​*21.40.4.3  Finite Sets   rp-isfinite5 44594
                  21.40.4.4  General Observations   intabssd 44596
                  21.40.4.5  Infinite Sets   pwelg 44637
                  ​*21.40.4.6  Finite intersection property   fipjust 44642
                  21.40.4.7  RP ADDTO: Subclasses and subsets   rababg 44651
                  21.40.4.8  RP ADDTO: The intersection of a class   elinintab 44652
                  21.40.4.9  RP ADDTO: Theorems requiring subset and intersection existence   elinintrab 44654
                  21.40.4.10  RP ADDTO: Relations   xpinintabd 44657
                  ​*21.40.4.11  RP ADDTO: Functions   elmapintab 44672
                  ​*21.40.4.12  RP ADDTO: Finite induction (for finite ordinals)   cnvcnvintabd 44676
                  21.40.4.13  RP ADDTO: First and second members of an ordered pair   elcnvlem 44677
                  21.40.4.14  RP ADDTO: The reflexive and transitive properties of relations   undmrnresiss 44680
                  21.40.4.15  RP ADDTO: Basic properties of closures   cleq2lem 44684
                  21.40.4.16  RP REPLACE: Definitions and basic properties of transitive closures   trcleq2lemRP 44706
                  ​*21.40.4.17  Additions for square root; absolute value   sqrtcvallem1 44707
            21.40.5  Additional statements on relations and subclasses   al3im 44723
                  21.40.5.1  Transitive relations (not to be confused with transitive classes)   trrelind 44741
                  21.40.5.2  Reflexive closures   crcl 44748
                  ​*21.40.5.3  Finite relationship composition   relexp2 44753
                  21.40.5.4  Transitive closure of a relation   dftrcl3 44796
                  ​*21.40.5.5  Adapted from Frege   frege77d 44822
            ​*21.40.6  Propositions from _Begriffsschrift_   dfxor4 44842
                  *21.40.6.1  _Begriffsschrift_ Chapter I   dfxor4 44842
                  ​*21.40.6.2  _Begriffsschrift_ Notation hints   whe 44848
                  21.40.6.3  _Begriffsschrift_ Chapter II Implication   ax-frege1 44866
                  21.40.6.4  _Begriffsschrift_ Chapter II Implication and Negation   axfrege28 44905
                  ​*21.40.6.5  _Begriffsschrift_ Chapter II with logical equivalence   axfrege52a 44932
                  21.40.6.6  _Begriffsschrift_ Chapter II with equivalence of sets   axfrege52c 44963
                  ​*21.40.6.7  _Begriffsschrift_ Chapter II with equivalence of classes   frege53c 44990
                  ​*21.40.6.8  _Begriffsschrift_ Chapter III Properties hereditary in a sequence   dffrege69 45008
                  ​*21.40.6.9  _Begriffsschrift_ Chapter III Following in a sequence   dffrege76 45015
                  ​*21.40.6.10  _Begriffsschrift_ Chapter III Member of sequence   dffrege99 45038
                  ​*21.40.6.11  _Begriffsschrift_ Chapter III Single-valued procedures   dffrege115 45054
            ​*21.40.7  Exploring Topology via Seifert and Threlfall   enrelmap 45073
                  *21.40.7.1  Equinumerosity of sets of relations and maps   enrelmap 45073
                  ​*21.40.7.2  Generic Pseudoclosure Spaces, Pseudointerior Spaces, and Pseudoneighborhoods   or3or 45099
                  ​*21.40.7.3  Generic Neighborhood Spaces   gneispa 45206
            ​*21.40.8  Exploring Higher Homotopy via Kerodon   k0004lem1 45223
                  *21.40.8.1  Simplicial Sets   k0004lem1 45223
      ​21.41  Mathbox for Stanislas Polu
            21.41.1  IMO Problems   wwlemuld 45232
                  21.41.1.1  IMO 1972 B2   wwlemuld 45232
            ​*21.41.2  INT Inequalities Proof Generator   int-addcomd 45249
            ​*21.41.3  N-Digit Addition Proof Generator   unitadd 45271
            21.41.4  AM-GM (for k = 2,3,4)   gsumws3 45272
      ​21.42  Mathbox for Rohan Ridenour
            21.42.1  Misc   spALT 45277
            21.42.2  Monoid rings   cmnring 45285
            21.42.3  Shorter primitive equivalent of ax-groth   gru0eld 45303
                  21.42.3.1  Grothendieck universes are closed under collection   gru0eld 45303
                  21.42.3.2  Minimal universes   ismnu 45321
                  21.42.3.3  Primitive equivalent of ax-groth   expandan 45348
      ​21.43  Mathbox for Steve Rodriguez
            21.43.1  Miscellanea   nanorxor 45365
            21.43.2  Ratio test for infinite series convergence and divergence   dvgrat 45372
            21.43.3  Multiples   reldvds 45375
            21.43.4  Function operations   caofcan 45383
            21.43.5  Calculus   lhe4.4ex1a 45389
            21.43.6  The generalized binomial coefficient operation   cbcc 45396
            21.43.7  Binomial series   uzmptshftfval 45406
      ​21.44  Mathbox for Andrew Salmon
            21.44.1  Principia Mathematica * 10   pm10.12 45418
            21.44.2  Principia Mathematica * 11   2alanimi 45432
            21.44.3  Predicate Calculus   sbeqal1 45458
            21.44.4  Principia Mathematica * 13 and * 14   pm13.13a 45467
            21.44.5  Set Theory   elnev 45497
            21.44.6  Arithmetic   addcomgi 45514
            21.44.7  Geometry   cplusr 45515
      ​​*21.45  Mathbox for Alan Sare
            21.45.1  Auxiliary theorems for the Virtual Deduction tool   idiALT 45537
            21.45.2  Supplementary unification deductions   bi1imp 45541
            21.45.3  Conventional Metamath proofs, some derived from VD proofs   iidn3 45560
            21.45.4  What is Virtual Deduction?   wvd1 45628
            21.45.5  Virtual Deduction Theorems   df-vd1 45629
            21.45.6  Theorems proved using Virtual Deduction   trsspwALT 45876
            21.45.7  Theorems proved using Virtual Deduction with mmj2 assistance   simplbi2VD 45904
            21.45.8  Virtual Deduction transcriptions of textbook proofs   sb5ALTVD 45971
            21.45.9  Theorems proved using conjunction-form Virtual Deduction   elpwgdedVD 45975
            21.45.10  Theorems with a VD proof in conventional notation derived from a VD proof   suctrALT3 45982
            ​*21.45.11  Theorems with a proof in conventional notation derived from a VD proof   notnotrALT2 45985
      ​21.46  Mathbox for Eric Schmidt
            21.46.1  Miscellany   rspesbcd 45996
            21.46.2  Study of dfbi1ALT   dfbi1ALTa 46005
            21.46.3  Relation-preserving functions   wrelp 46008
            21.46.4  Orbits   orbitex 46021
            21.46.5  Well-founded sets   trwf 46025
            21.46.6  Absoluteness in transitive models   ralabso 46034
            21.46.7  Lemmas for showing axioms hold in models   traxext 46043
            21.46.8  The class of well-founded sets is a model for ZFC   wfaxext 46059
            21.46.9  Permutation models   brpermmodel 46069
            21.46.10  Isomorphism of finite ordinals and non-negative integers   hashnna 46085
            21.46.11  Hereditarily finite sets   hfpr 46092
            21.46.12  Hereditarily finite structures   chfstruct 46099
      ​21.47  Mathbox for Glauco Siliprandi
            21.47.1  Miscellanea   evth2f 46108
            21.47.2  Functions   fnresdmss 46259
            21.47.3  Ordering on real numbers - Real and complex numbers basic operations   sub2times 46365
            21.47.4  Real intervals   gtnelioc 46579
            21.47.5  Finite sums   fsummulc1f 46659
            21.47.6  Finite multiplication of numbers and finite multiplication of functions   fmul01 46668
            21.47.7  Limits   clim1fr1 46689
                  21.47.7.1  Inferior limit (lim inf)   clsi 46837
                  ​*21.47.7.2  Limits for sequences of extended real numbers   clsxlim 46904
            21.47.8  Trigonometry   coseq0 46950
            21.47.9  Continuous Functions   mulcncff 46956
            21.47.10  Derivatives   dvsinexp 46997
            21.47.11  Integrals   itgsin0pilem1 47036
            21.47.12  Stone Weierstrass theorem - real version   stoweidlem1 47087
            21.47.13  Wallis' product for π   wallispilem1 47151
            21.47.14  Stirling's approximation formula for ` n ` factorial   stirlinglem1 47160
            21.47.15  Dirichlet kernel   dirkerval 47177
            21.47.16  Fourier Series   fourierdlem1 47194
            21.47.17  e is transcendental   elaa2lem 47319
            21.47.18  n-dimensional Euclidean space   rrxtopn 47370
            21.47.19  Basic measure theory   csalg 47394
                  ​*21.47.19.1  σ-Algebras   csalg 47394
                  21.47.19.2  Sum of nonnegative extended reals   csumge0 47448
                  ​*21.47.19.3  Measures   cmea 47535
                  ​*21.47.19.4  Outer measures and Caratheodory's construction   come 47575
                  ​*21.47.19.5  Lebesgue measure on n-dimensional Real numbers   covoln 47622
                  ​*21.47.19.6  Measurable functions   csmblfn 47781
      ​21.48  Mathbox for Saveliy Skresanov
            21.48.1  Ceva's theorem   sigarval 47936
            21.48.2  Simple groups   simpcntrab 47956
      ​21.49  Mathbox for Ender Ting
            21.49.1  Interesting facts   et-ltneverrefl 47957
            21.49.2  Increasing sequences and subsequences   ormklocald 47962
            21.49.3  Scratchpad for number theory   evenwodadd 47987
            21.49.4  Scratchpad for math on real numbers   squeezedltsq 47988
            21.49.5  Turing Machine Finite Reach theorem   tmachlem-extapes 48020
      ​21.50  Mathbox for Jarvin Udandy
      ​21.51  Mathbox for Adhemar
            ​*21.51.1  Minimal implicational calculus   adh-minim 48147
      ​21.52  Mathbox for Alexander van der Vekens
            21.52.1  General auxiliary theorems (1)   n0nsn2el 48171
                  21.52.1.1  Unordered and ordered pairs - extension for singletons   n0nsn2el 48171
                  21.52.1.2  Unordered and ordered pairs - extension for unordered pairs   elprneb 48175
                  21.52.1.3  Unordered and ordered pairs - extension for ordered pairs   oppr 48176
                  21.52.1.4  Relations - extension   eubrv 48181
                  21.52.1.5  Definite description binder (inverted iota) - extension   iota0def 48184
                  21.52.1.6  Functions - extension   fveqvfvv 48186
            21.52.2  Alternative for Russell's definition of a description binder   caiota 48229
            21.52.3  Double restricted existential uniqueness   r19.32 48244
                  21.52.3.1  Restricted quantification (extension)   r19.32 48244
                  21.52.3.2  Restricted uniqueness and "at most one" quantification   reuf1odnf 48253
                  21.52.3.3  Analogs to Existential uniqueness (double quantification)   2reu3 48256
                  21.52.3.4  Additional theorems for double restricted existential uniqueness   2reu8i 48259
            ​*21.52.4  Alternative definitions of function and operation values   wdfat 48262
                  21.52.4.1  Restricted quantification (extension)   ralbinrald 48268
                  21.52.4.2  The universal class (extension)   nvelim 48269
                  21.52.4.3  Introduce the Axiom of Power Sets (extension)   alneu 48270
                  21.52.4.4  Predicate "defined at"   dfateq12d 48272
                  21.52.4.5  Alternative definition of the value of a function   dfafv2 48278
                  21.52.4.6  Alternative definition of the value of an operation   aoveq123d 48324
            ​*21.52.5  Alternative definitions of function values (2)   cafv2 48354
            21.52.6  General auxiliary theorems (2)   an4com24 48414
                  21.52.6.1  Logical conjunction - extension   an4com24 48414
                  21.52.6.2  Abbreviated conjunction and disjunction of three wff's - extension   3an4ancom24 48415
                  21.52.6.3  Negated membership (alternative)   cnelbr 48417
                  21.52.6.4  The empty set - extension   ralralimp 48424
                  21.52.6.5  Indexed union and intersection - extension   otiunsndisjX 48425
                  21.52.6.6  Functions - extension   fvifeq 48426
                  21.52.6.7  Maps-to notation - extension   fvmptrab 48438
                  21.52.6.8  Subtraction - extension   cnambpcma 48440
                  21.52.6.9  Ordering on reals (cont.) - extension   leaddsuble 48443
                  21.52.6.10  Imaginary and complex number properties - extension   readdcnnred 48449
                  21.52.6.11  Nonnegative integers (as a subset of complex numbers) - extension   nn0resubcl 48454
                  21.52.6.12  Integers (as a subset of complex numbers) - extension   zgeltp1eq 48455
                  21.52.6.13  Decimal arithmetic - extension   1t10e1p1e11 48456
                  21.52.6.14  Upper sets of integers - extension   eluzge0nn0 48458
                  21.52.6.15  Infinity and the extended real number system (cont.) - extension   nltle2tri 48459
                  21.52.6.16  Finite intervals of integers - extension   ssfz12 48460
                  21.52.6.17  Half-open integer ranges - extension   fzopred 48469
                  21.52.6.18  The floor and ceiling functions - extension   2ltceilhalf 48478
                  21.52.6.19  The modulo (remainder) operation - extension   fldivmod 48490
                  21.52.6.20  The infinite sequence builder "seq"   smonoord 48523
                  21.52.6.21  Integer powers - extension   2timesltsq 48524
                  21.52.6.22  Finite and infinite sums - extension   fsummsndifre 48526
                  21.52.6.23  The divides relation - extension   nndivides2 48530
                  21.52.6.24  Extensible structures - extension   setsidel 48534
            ​*21.52.7  Preimages of function values   preimafvsnel 48537
            ​*21.52.8  Partitions of real intervals   ciccp 48571
            21.52.9  Shifting functions with an integer range domain   fargshiftfv 48597
            21.52.10  Words over a set (extension)   lswn0 48602
                  21.52.10.1  Last symbol of a word - extension   lswn0 48602
            21.52.11  Unordered pairs   wich 48603
                  21.52.11.1  Interchangeable setvar variables   wich 48603
                  21.52.11.2  Set of unordered pairs   sprid 48632
                  ​*21.52.11.3  Proper (unordered) pairs   prpair 48659
                  21.52.11.4  Set of proper unordered pairs   cprpr 48670
            21.52.12  Number theory (extension)   nprmmul1 48685
                  21.52.12.1  Properties of non-prime numbers   nprmmul1 48685
                  ​*21.52.12.2  Fermat numbers   cfmtno 48688
                  ​*21.52.12.3  Mersenne primes   m2prm 48752
                  21.52.12.4  Proth's theorem   modexp2m1d 48773
                  21.52.12.5  The prime-counting function according to Ján Mináč   nprmdvdsfacm1lem1 48781
                  21.52.12.6  Solutions of quadratic equations   quad1 48794
            ​*21.52.13  Even and odd numbers   ceven 48798
                  21.52.13.1  Definitions and basic properties   ceven 48798
                  21.52.13.2  Alternate definitions using the "divides" relation   dfeven2 48823
                  21.52.13.3  Alternate definitions using the "modulo" operation   dfeven3 48832
                  21.52.13.4  Alternate definitions using the "gcd" operation   iseven5 48838
                  21.52.13.5  Theorems of part 5 revised   zneoALTV 48843
                  21.52.13.6  Theorems of part 6 revised   odd2np1ALTV 48848
                  21.52.13.7  Theorems of AV's mathbox revised   0evenALTV 48862
                  21.52.13.8  Additional theorems   epoo 48877
                  21.52.13.9  Perfect Number Theorem (revised)   perfectALTVlem1 48895
            21.52.14  Number theory (extension 2)   cfppr 48898
                  ​*21.52.14.1  Fermat pseudoprimes   cfppr 48898
                  ​*21.52.14.2  Goldbach's conjectures   cgbe 48919
            21.52.15  Graph theory (extension)   cclnbgr 48992
                  21.52.15.1  Closed neighborhood of a vertex   cclnbgr 48992
                  ​*21.52.15.2  Semiclosed and semiopen neighborhoods (experimental)   dfsclnbgr2 49020
                  21.52.15.3  Induced subgraphs   cisubgr 49034
                  ​*21.52.15.4  Isomorphisms of graphs   cgrisom 49048
                  ​*21.52.15.5  Triangles in graphs   cgrtri 49111
                  ​*21.52.15.6  Star graphs   cstgr 49125
                  ​*21.52.15.7  Local isomorphisms of graphs   cgrlim 49150
                  ​*21.52.15.8  Generalized Petersen graphs   cgpg 49214
                  21.52.15.9  Loop-free graphs - extension   1hegrlfgr 49306
                  21.52.15.10  Walks - extension   cupwlks 49307
                  21.52.15.11  Edges of graphs expressed as sets of unordered pairs   upgredgssspr 49317
            21.52.16  Monoids (extension)   ovn0dmfun 49330
                  21.52.16.1  Auxiliary theorems   ovn0dmfun 49330
                  21.52.16.2  Magmas, Semigroups and Monoids (extension)   plusfreseq 49337
                  21.52.16.3  Examples and counterexamples for magmas, semigroups and monoids (extension)   opmpoismgm 49340
                  21.52.16.4  Group sum operation (extension 1)   gsumsplit2f 49353
            ​*21.52.17  Magmas and internal binary operations (alternate approach)   ccllaw 49356
                  *21.52.17.1  Laws for internal binary operations   ccllaw 49356
                  ​*21.52.17.2  Internal binary operations   cintop 49369
                  21.52.17.3  Alternative definitions for magmas and semigroups   cmgm2 49388
            21.52.18  Rings (extension)   lmod0rng 49402
                  21.52.18.1  Nonzero rings (extension)   lmod0rng 49402
                  21.52.18.2  Ideals as non-unital rings   lidldomn1 49404
                  21.52.18.3  The non-unital ring of even integers   0even 49410
                  21.52.18.4  A constructed not unital ring   cznrnglem 49432
                  ​*21.52.18.5  The category of non-unital rings (alternate definition)   crngcALTV 49436
                  ​*21.52.18.6  The category of (unital) rings (alternate definition)   cringcALTV 49460
            ​*21.52.19  Prime rings (and integral domains)   cprmrng 49507
            21.52.20  Basic algebraic structures (extension)   eliunxp2 49522
                  21.52.20.1  Auxiliary theorems   eliunxp2 49522
                  21.52.20.2  The binomial coefficient operation (extension)   bcpascm1 49539
                  21.52.20.3  The ` ZZ `-module ` ZZ X. ZZ `   zlmodzxzlmod 49542
                  21.52.20.4  Group sum operation (extension 2)   mgpsumunsn 49549
                  21.52.20.5  Symmetric groups (extension)   exple2lt6 49552
                  21.52.20.6  Divisibility (extension)   invginvrid 49555
                  21.52.20.7  The support of functions (extension)   rmsupp0 49556
                  21.52.20.8  Finitely supported functions (extension)   rmsuppfi 49560
                  21.52.20.9  Left modules (extension)   lmodvsmdi 49567
                  21.52.20.10  Associative algebras (extension)   assaascl0 49569
                  21.52.20.11  Univariate polynomials (extension)   ply1vr1smo 49571
                  21.52.20.12  Univariate polynomials (examples)   linply1 49581
            21.52.21  Linear algebra (extension)   cdmatalt 49584
                  ​*21.52.21.1  The subalgebras of diagonal and scalar matrices (extension)   cdmatalt 49584
                  ​*21.52.21.2  Linear combinations   clinc 49592
                  ​*21.52.21.3  Linear independence   clininds 49628
                  21.52.21.4  Simple left modules and the ` ZZ `-module   lmod1lem1 49675
                  21.52.21.5  Differences between (left) modules and (left) vector spaces   lvecpsslmod 49695
            21.52.22  Complexity theory   suppdm 49698
                  21.52.22.1  Auxiliary theorems   suppdm 49698
                  21.52.22.2  Even and odd integers   nn0onn0ex 49711
                  21.52.22.3  The natural logarithm on complex numbers (extension)   logcxp0 49723
                  21.52.22.4  Division of functions   cfdiv 49725
                  21.52.22.5  Upper bounds   cbigo 49735
                  21.52.22.6  Logarithm to an arbitrary base (extension)   rege1logbrege0 49746
                  ​*21.52.22.7  The binary logarithm   fldivexpfllog2 49753
                  21.52.22.8  Binary length   cblen 49757
                  ​*21.52.22.9  Digits   cdig 49783
                  21.52.22.10  Nonnegative integer as sum of its shifted digits   dignn0flhalflem1 49803
                  21.52.22.11  Algorithms for the multiplication of nonnegative integers   nn0mulfsum 49812
                  ​*21.52.22.12  N-ary functions   cnaryf 49814
                  ​*21.52.22.13  The Ackermann function   citco 49845
            21.52.23  Elementary geometry (extension)   fv1prop 49887
                  21.52.23.1  Auxiliary theorems   fv1prop 49887
                  21.52.23.2  Real euclidean space of dimension 2   rrx2pxel 49899
                  21.52.23.3  Spheres and lines in real Euclidean spaces   cline 49915
      ​21.53  Mathbox for Zhi Wang
            21.53.1  Propositional calculus   imbi12d2 49977
            21.53.2  Predicate calculus with equality   dtrucor3 49985
                  21.53.2.1  Axiom scheme ax-5 (Distinctness)   dtrucor3 49985
            21.53.3  ZF Set Theory - start with the Axiom of Extensionality   ralbidb 49986
                  21.53.3.1  Restricted quantification   ralbidb 49986
                  21.53.3.2  The universal class   reuxfr1dd 49993
                  21.53.3.3  The empty set   ssdisjd 49994
                  21.53.3.4  Unordered and ordered pairs   vsn 49998
                  21.53.3.5  The union of a class   unilbss 50004
                  21.53.3.6  Indexed union and intersection   iuneq0 50005
            21.53.4  ZF Set Theory - add the Axiom of Replacement   inpw 50011
                  21.53.4.1  Theorems requiring subset and intersection existence   inpw 50011
            21.53.5  ZF Set Theory - add the Axiom of Power Sets   opth1neg 50012
                  21.53.5.1  Ordered pair theorem   opth1neg 50012
                  21.53.5.2  Ordered-pair class abstractions (cont.)   brab2dd 50014
                  21.53.5.3  Relations   iinxp 50017
                  21.53.5.4  Functions   mof0 50024
                  21.53.5.5  Operations   ovsng 50044
            21.53.6  ZF Set Theory - add the Axiom of Union   fonex 50053
                  21.53.6.1  Relations and functions (cont.)   fonex 50053
                  21.53.6.2  First and second members of an ordered pair   eloprab1st2nd 50054
                  21.53.6.3  Function transposition   resinsnlem 50055
                  21.53.6.4  Infinite Cartesian products   ixpv 50074
                  21.53.6.5  Equinumerosity   fvconst0ci 50075
            21.53.7  Order sets   iccin 50080
                  21.53.7.1  Real number intervals   iccin 50080
            21.53.8  Extensible structures   slotresfo 50083
                  21.53.8.1  Basic definitions   slotresfo 50083
            21.53.9  Moore spaces   mreuniss 50084
            ​*21.53.10  Topology   clduni 50085
                  21.53.10.1  Closure and interior   clduni 50085
                  21.53.10.2  Neighborhoods   neircl 50089
                  21.53.10.3  Subspace topologies   restcls2lem 50097
                  21.53.10.4  Limits and continuity in topological spaces   cnneiima 50101
                  21.53.10.5  Topological definitions using the reals   iooii 50102
                  21.53.10.6  Separated sets   sepnsepolem1 50106
                  21.53.10.7  Separated spaces: T0, T1, T2 (Hausdorff) ...   isnrm4 50115
            21.53.11  Preordered sets and directed sets using extensible structures   isprsd 50139
            21.53.12  Posets and lattices using extensible structures   lubeldm2 50140
                  21.53.12.1  Posets   lubeldm2 50140
                  21.53.12.2  Lattices   toslat 50166
                  21.53.12.3  Subset order structures   intubeu 50168
            21.53.13  Rings   elmgpcntrd 50189
                  21.53.13.1  Multiplicative Group   elmgpcntrd 50189
            21.53.14  Associative algebras   asclelbasALT 50190
                  21.53.14.1  Definition and basic properties   asclelbasALT 50190
            21.53.15  Categories   homf0 50193
                  21.53.15.1  Categories   homf0 50193
                  21.53.15.2  Opposite category   oppccatb 50200
                  21.53.15.3  Monomorphisms and epimorphisms   idmon 50204
                  21.53.15.4  Sections, inverses, isomorphisms   sectrcl 50206
                  21.53.15.5  Isomorphic objects   cicfn 50226
                  21.53.15.6  Subcategories   dmdm 50237
                  21.53.15.7  Functors   reldmfunc 50259
                  21.53.15.8  Opposite functors   coppf 50306
                  21.53.15.9  Full & faithful functors   imasubc 50335
                  21.53.15.10  Universal property   upciclem1 50350
                  21.53.15.11  Natural transformations and the functor category   isnatd 50407
                  21.53.15.12  Initial, terminal and zero objects of a category   initoo2 50416
                  21.53.15.13  Product of categories   reldmxpc 50430
                  21.53.15.14  Swap functors   cswapf 50443
                  21.53.15.15  Functor evaluation   oppc1stflem 50471
                  21.53.15.16  Transposed curry functors   cofuswapfcl 50477
                  21.53.15.17  Constant functors   diag1 50488
                  21.53.15.18  Functor composition bifunctors   fucofulem1 50494
                  21.53.15.19  Post-composition functors   postcofval 50548
                  21.53.15.20  Pre-composition functors   precofvallem 50550
            21.53.16  Examples of categories   catcrcl 50579
                  21.53.16.1  The category of categories   catcrcl 50579
                  21.53.16.2  Thin categories   cthinc 50601
                  21.53.16.3  Terminal categories   ctermc 50656
                  21.53.16.4  Preordered sets as thin categories   cprstc 50733
                  21.53.16.5  Monoids as categories   cmndtc 50761
                  21.53.16.6  Categories with at most one object and at most two morphisms   2arwcatlem1 50779
            21.53.17  Kan extensions and related concepts   clan 50789
                  21.53.17.1  Kan extensions   clan 50789
                  21.53.17.2  Limits and colimits   clmd 50827
      ​21.54  Mathbox for Emmett Weisz
            ​*21.54.1  Miscellaneous Theorems   nfintd 50857
            21.54.2  Examples and properties of set recursion   setrecseq 50864
            ​*21.54.3  Construction of Games and Surreal Numbers   cpg 50878
      ​​*21.55  Mathbox for David A. Wheeler
            21.55.1  Natural deduction   sbidd 50887
            ​*21.55.2  Greater than, greater than or equal to   cge-real 50889
            ​*21.55.3  Hyperbolic trigonometric functions   csinh 50899
            ​*21.55.4  Reciprocal trigonometric functions (sec, csc, cot)   csec 50910
            ​*21.55.5  Identities for "if"   ifnmfalse 50935
            ​*21.55.6  Logarithms generalized to arbitrary base using ` logb `   logb2aval 50936
            ​*21.55.7  Logarithm laws generalized to an arbitrary base - log_   clog- 50937
            ​*21.55.8  Formally define notions such as reflexivity   wreflexive 50939
            ​*21.55.9  Algebra helpers   mvlraddi 50943
            ​*21.55.10  Algebra helper examples   i2linesi 50950
            ​*21.55.11  Formal methods "surprises"   alimp-surprise 50952
            ​*21.55.12  Allsome quantifier   wals 50958
            ​*21.55.13  Allsome one quantifier   walseu 50991
            ​*21.55.14  Miscellaneous   5m4e1 51011
            21.55.15  Theorems about algebraic numbers   aacllem 51015
      ​21.56  Mathbox for Mingli Yuan
      ​21.57  Mathbox for Jiamin Zhao
            21.57.1  Cross product and scalar triple product in RR^3   1ne3 51017
            21.57.2  Veronese map and linear dependence   cveronese 51044
      ​21.58  Mathbox for Kunhao Zheng
            21.58.1  Weighted AM-GM inequality   amgmwlem 51063