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
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  Ideals
      10.8  The complex numbers as an algebraic extensible structure
      10.9  Generalized pre-Hilbert and Hilbert spaces
PART 11  BASIC LINEAR ALGEBRA
      11.1  Vectors and free modules
      11.2  Associative algebras
      11.3  Abstract multivariate polynomials
      11.4  Matrices
      11.5  The determinant
      11.6  Polynomial matrices
      11.7  The characteristic polynomial
PART 12  BASIC TOPOLOGY
      12.1  Topology
      12.2  Filters and filter bases
      12.3  Uniform Structures and Spaces
      12.4  Metric spaces
      12.5  Metric subcomplex vector spaces
PART 13  BASIC REAL AND COMPLEX ANALYSIS
      13.1  Continuity
      13.2  Integrals
      13.3  Derivatives
PART 14  BASIC REAL AND COMPLEX FUNCTIONS
      14.1  Polynomials
      14.2  Sequences and series
      14.3  Basic trigonometry
      14.4  Basic number theory
PART 15  SURREAL NUMBERS
      15.1  Sign sequence representation and Alling's axioms
      15.2  Initial consequences of Alling's axioms
      15.3  Conway cut representation
      15.4  Induction and recursion
      15.5  Surreal arithmetic
      15.6  Subsystems of surreals
PART 16  ELEMENTARY GEOMETRY
      16.1  Definition and Tarski's Axioms of Geometry
      16.2  Tarskian Geometry
      16.3  Properties of geometries
      16.4  Geometry in Hilbert spaces
PART 17  GRAPH THEORY
      17.1  Vertices and edges
      17.2  Undirected graphs
      17.3  Walks, paths and cycles
      17.4  Eulerian paths and the Konigsberg Bridge problem
      17.5  The Friendship Theorem
PART 18  GUIDES AND MISCELLANEA
      18.1  Guides (conventions, explanations, and examples)
      18.2  Humor
      18.3  (Future - to be reviewed and classified)
PART 19  COMPLEX TOPOLOGICAL VECTOR SPACES (DEPRECATED)
      19.1  Additional material on group theory (deprecated)
      19.2  Complex vector spaces
      19.3  Normed complex vector spaces
      19.4  Operators on complex vector spaces
      19.5  Inner product (pre-Hilbert) spaces
      19.6  Complex Banach spaces
      19.7  Complex Hilbert spaces
PART 20  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      20.1  Axiomatization of complex pre-Hilbert spaces
      20.2  Inner product and norms
      20.3  Cauchy sequences and completeness axiom
      20.4  Subspaces and projections
      20.5  Properties of Hilbert subspaces
      20.6  Operators on Hilbert spaces
      20.7  States on a Hilbert lattice and Godowski's equation
      20.8  Cover relation, atoms, exchange axiom, and modular symmetry
PART 21  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      ​21.1  Mathboxes for user contributions
      21.2  Mathbox for Stefan Allan
      21.3  Mathbox for Thierry Arnoux
      21.4  Mathbox for Jonathan Ben-Naim
      21.5  Mathbox for BTernaryTau
      21.6  Mathbox for Mario Carneiro
      21.7  Mathbox for Filip Cernatescu
      21.8  Mathbox for Paul Chapman
      21.9  Mathbox for Adrian Ducourtial
      21.10  Mathbox for Scott Fenton
      21.11  Mathbox for Gino Giotto
      21.12  Mathbox for Jeff Hankins
      21.13  Mathbox for Anthony Hart
      21.14  Mathbox for Chen-Pang He
      21.15  Mathbox for Jeff Hoffman
      21.16  Mathbox for Asger C. Ipsen
      21.17  Mathbox for BJ
      21.18  Mathbox for Jim Kingdon
      21.19  Mathbox for ML
      21.20  Mathbox for Wolf Lammen
      21.21  Mathbox for Brendan Leahy
      21.22  Mathbox for Jeff Madsen
      21.23  Mathbox for Giovanni Mascellani
      21.24  Mathbox for Peter Mazsa
      21.25  Mathbox for Rodolfo Medina
      21.26  Mathbox for Norm Megill
      21.27  Mathbox for metakunt
      21.28  Mathbox for Steven Nguyen
      21.29  Mathbox for Igor Ieskov
      21.30  Mathbox for OpenAI
      21.31  Mathbox for Stefan O'Rear
      21.32  Mathbox for Noam Pasman
      21.33  Mathbox for Jon Pennant
      21.34  Mathbox for Richard Penner
      21.35  Mathbox for Stanislas Polu
      21.36  Mathbox for Rohan Ridenour
      21.37  Mathbox for Steve Rodriguez
      21.38  Mathbox for Andrew Salmon
      21.39  Mathbox for Alan Sare
      21.40  Mathbox for Glauco Siliprandi
      21.41  Mathbox for Saveliy Skresanov
      21.42  Mathbox for Ender Ting
      21.43  Mathbox for Jarvin Udandy
      21.44  Mathbox for Adhemar
      21.45  Mathbox for Alexander van der Vekens
      21.46  Mathbox for Zhi Wang
      21.47  Mathbox for Emmett Weisz
      21.48  Mathbox for David A. Wheeler
      21.49  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 113
            ​*1.2.5  Logical equivalence   wb 205
            ​*1.2.6  Logical conjunction   wa 395
            ​*1.2.7  Logical disjunction   wo 844
            ​*1.2.8  Mixed connectives   jaao 952
            ​*1.2.9  The conditional operator for propositions   wif 1060
            ​*1.2.10  The weak deduction theorem for propositional calculus   elimh 1082
            1.2.11  Abbreviated conjunction and disjunction of three wff's   w3o 1085
            1.2.12  Logical "nand" (Sheffer stroke)   wnan 1488
            1.2.13  Logical "xor"   wxo 1508
            1.2.14  Logical "nor"   wnor 1527
            1.2.15  True and false constants   wal 1538
                  ​*1.2.15.1  Universal quantifier for use by df-tru   wal 1538
                  ​*1.2.15.2  Equality predicate for use by df-tru   cv 1539
                  1.2.15.3  The true constant   wtru 1541
                  1.2.15.4  The false constant   wfal 1552
            ​*1.2.16  Truth tables   truimtru 1563
                  1.2.16.1  Implication   truimtru 1563
                  1.2.16.2  Negation   nottru 1567
                  1.2.16.3  Equivalence   trubitru 1569
                  1.2.16.4  Conjunction   truantru 1573
                  1.2.16.5  Disjunction   truortru 1577
                  1.2.16.6  Alternative denial   trunantru 1581
                  1.2.16.7  Exclusive disjunction   truxortru 1585
                  1.2.16.8  Joint denial   trunortru 1589
            ​*1.2.17  Half adder and full adder in propositional calculus   whad 1593
                  1.2.17.1  Full adder: sum   whad 1593
                  1.2.17.2  Full adder: carry   wcad 1606
      ​1.3  Other axiomatizations related to classical propositional calculus
            ​*1.3.1  Minimal implicational calculus   minimp 1622
            ​*1.3.2  Implicational Calculus   impsingle 1628
            1.3.3  Derive the Lukasiewicz axioms from Meredith's sole axiom   meredith 1642
            1.3.4  Derive the standard axioms from the Lukasiewicz axioms   luklem1 1659
            ​*1.3.5  Derive Nicod's axiom from the standard axioms   nic-dfim 1670
            1.3.6  Derive the Lukasiewicz axioms from Nicod's axiom   nic-imp 1676
            1.3.7  Derive Nicod's Axiom from Lukasiewicz's First Sheffer Stroke Axiom   lukshef-ax1 1695
            1.3.8  Derive the Lukasiewicz Axioms from the Tarski-Bernays-Wajsberg Axioms   tbw-bijust 1699
            1.3.9  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's First CO Axiom   merco1 1714
            1.3.10  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's Second CO Axiom   merco2 1737
            1.3.11  Derive the Lukasiewicz axioms from the Russell-Bernays Axioms   rb-bijust 1750
            ​*1.3.12  Stoic logic non-modal portion (Chrysippus of Soli)   mptnan 1769
      ​​*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 1780
                  1.4.1.1  Existential quantifier   wex 1780
                  1.4.1.2  Nonfreeness predicate   wnf 1784
            1.4.2  Rule scheme ax-gen (Generalization)   ax-gen 1796
            1.4.3  Axiom scheme ax-4 (Quantified Implication)   ax-4 1810
                  ​*1.4.3.1  The empty domain of discourse   empty 1908
            1.4.4  Axiom scheme ax-5 (Distinctness) - first use of $d   ax-5 1912
            ​*1.4.5  Equality predicate (continued)   weq 1965
            1.4.6  Axiom scheme ax-6 (Existence)   ax-6 1970
            1.4.7  Axiom scheme ax-7 (Equality)   ax-7 2010
            1.4.8  Define proper substitution   sbjust 2065
            1.4.9  Membership predicate   wcel 2105
            1.4.10  Axiom scheme ax-8 (Left Equality for Binary Predicate)   ax-8 2107
            1.4.11  Axiom scheme ax-9 (Right Equality for Binary Predicate)   ax-9 2115
            ​*1.4.12  Logical redundancy of ax-10 , ax-11 , ax-12 , ax-13   ax6dgen 2123
      ​​*1.5  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes)
            1.5.1  Axiom scheme ax-10 (Quantified Negation)   ax-10 2136
            1.5.2  Axiom scheme ax-11 (Quantifier Commutation)   ax-11 2153
            1.5.3  Axiom scheme ax-12 (Substitution)   ax-12 2170
            1.5.4  Axiom scheme ax-13 (Quantified Equality)   ax-13 2370
      ​1.6  Uniqueness and unique existence
            1.6.1  Uniqueness: the at-most-one quantifier   wmo 2531
            1.6.2  Unique existence: the unique existential quantifier   weu 2561
      ​1.7  Other axiomatizations related to classical predicate calculus
            ​*1.7.1  Aristotelian logic: Assertic syllogisms   barbara 2657
            ​*1.7.2  Intuitionistic logic   axia1 2687
​​*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 2702
            2.1.2  Classes   cab 2708
                  2.1.2.1  Class abstractions   cab 2708
                  ​*2.1.2.2  Class equality   df-cleq 2723
                  2.1.2.3  Class membership   df-clel 2809
                  2.1.2.4  Elementary properties of class abstractions   eqabdv 2866
            2.1.3  Class form not-free predicate   wnfc 2882
            2.1.4  Negated equality and membership   wne 2939
                  2.1.4.1  Negated equality   wne 2939
                  2.1.4.2  Negated membership   wnel 3045
            2.1.5  Restricted quantification   wral 3060
                  2.1.5.1  Restricted universal and existential quantification   wral 3060
                  2.1.5.2  Restricted existential uniqueness and at-most-one quantifier   wreu 3373
                  2.1.5.3  Restricted class abstraction   crab 3431
            2.1.6  The universal class   cvv 3473
            ​*2.1.7  Conditional equality (experimental)   wcdeq 3760
            2.1.8  Russell's Paradox   rru 3776
            2.1.9  Proper substitution of classes for sets   wsbc 3778
            2.1.10  Proper substitution of classes for sets into classes   csb 3894
            2.1.11  Define basic set operations and relations   cdif 3946
            2.1.12  Subclasses and subsets   df-ss 3966
            2.1.13  The difference, union, and intersection of two classes   dfdif3 4115
                  2.1.13.1  The difference of two classes   dfdif3 4115
                  2.1.13.2  The union of two classes   elun 4149
                  2.1.13.3  The intersection of two classes   elini 4194
                  2.1.13.4  The symmetric difference of two classes   csymdif 4242
                  2.1.13.5  Combinations of difference, union, and intersection of two classes   unabs 4255
                  2.1.13.6  Class abstractions with difference, union, and intersection of two classes   unabw 4298
                  2.1.13.7  Restricted uniqueness with difference, union, and intersection   reuun2 4315
            2.1.14  The empty set   c0 4323
            ​*2.1.15  The conditional operator for classes   cif 4529
            ​*2.1.16  The weak deduction theorem for set theory   dedth 4587
            2.1.17  Power classes   cpw 4603
            2.1.18  Unordered and ordered pairs   snjust 4628
            2.1.19  The union of a class   cuni 4909
            2.1.20  The intersection of a class   cint 4951
            2.1.21  Indexed union and intersection   ciun 4998
            2.1.22  Disjointness   wdisj 5114
            2.1.23  Binary relations   wbr 5149
            2.1.24  Ordered-pair class abstractions (class builders)   copab 5211
            2.1.25  Functions in maps-to notation   cmpt 5232
            2.1.26  Transitive classes   wtr 5266
      ​2.2  ZF Set Theory - add the Axiom of Replacement
            2.2.1  Introduce the Axiom of Replacement   ax-rep 5286
            2.2.2  Derive the Axiom of Separation   axsepgfromrep 5298
            2.2.3  Derive the Null Set Axiom   axnulALT 5305
            2.2.4  Theorems requiring subset and intersection existence   nalset 5314
            2.2.5  Theorems requiring empty set existence   class2set 5354
      ​2.3  ZF Set Theory - add the Axiom of Power Sets
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 5364
            2.3.2  Derive the Axiom of Pairing   axprlem1 5422
            2.3.3  Ordered pair theorem   opnz 5474
            2.3.4  Ordered-pair class abstractions (cont.)   opabidw 5525
            2.3.5  Power class of union and intersection   pwin 5571
            2.3.6  The identity relation   cid 5574
            2.3.7  The membership relation (or epsilon relation)   cep 5580
            ​*2.3.8  Partial and total orderings   wpo 5587
            2.3.9  Founded and well-ordering relations   wfr 5629
            2.3.10  Relations   cxp 5675
            2.3.11  The Predecessor Class   cpred 6300
            2.3.12  Well-founded induction (variant)   frpomin 6342
            2.3.13  Well-ordered induction   tz6.26 6349
            2.3.14  Ordinals   word 6364
            2.3.15  Definite description binder (inverted iota)   cio 6494
            2.3.16  Functions   wfun 6538
            2.3.17  Cantor's Theorem   canth 7365
            2.3.18  Restricted iota (description binder)   crio 7367
            2.3.19  Operations   co 7412
                  2.3.19.1  Variable-to-class conversion for operations   caovclg 7602
            2.3.20  Maps-to notation   mpondm0 7650
            2.3.21  Function operation   cof 7671
            2.3.22  Proper subset relation   crpss 7715
      ​2.4  ZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 7728
            2.4.2  Ordinals (continued)   epweon 7765
            2.4.3  Transfinite induction   tfi 7845
            2.4.4  The natural numbers (i.e., finite ordinals)   com 7858
            2.4.5  Peano's postulates   peano1 7882
            2.4.6  Finite induction (for finite ordinals)   find 7890
            2.4.7  Relations and functions (cont.)   dmexg 7897
            2.4.8  First and second members of an ordered pair   c1st 7976
            2.4.9  Induction on Cartesian products   frpoins3xpg 8129
            2.4.10  Ordering on Cartesian products   xpord2lem 8131
            2.4.11  Ordering Ordinal Sequences   orderseqlem 8146
            ​*2.4.12  The support of functions   csupp 8149
            ​*2.4.13  Special maps-to operations   opeliunxp2f 8198
            2.4.14  Function transposition   ctpos 8213
            2.4.15  Curry and uncurry   ccur 8253
            2.4.16  Undefined values   cund 8260
            2.4.17  Well-founded recursion   cfrecs 8268
            2.4.18  Well-ordered recursion   cwrecs 8299
            2.4.19  Functions on ordinals; strictly monotone ordinal functions   iunon 8342
            2.4.20  "Strong" transfinite recursion   crecs 8373
            2.4.21  Recursive definition generator   crdg 8412
            2.4.22  Finite recursion   frfnom 8438
            2.4.23  Ordinal arithmetic   c1o 8462
            2.4.24  Natural number arithmetic   nna0 8607
            2.4.25  Natural addition   cnadd 8667
            2.4.26  Equivalence relations and classes   wer 8703
            2.4.27  The mapping operation   cmap 8823
            2.4.28  Infinite Cartesian products   cixp 8894
            2.4.29  Equinumerosity   cen 8939
            2.4.30  Schroeder-Bernstein Theorem   sbthlem1 9086
            2.4.31  Equinumerosity (cont.)   xpf1o 9142
            2.4.32  Finite sets   dif1enlem 9159
            2.4.33  Pigeonhole Principle   phplem1 9210
            2.4.34  Finite sets (cont.)   onomeneq 9231
            2.4.35  Finitely supported functions   cfsupp 9364
            2.4.36  Finite intersections   cfi 9408
            2.4.37  Hall's marriage theorem   marypha1lem 9431
            2.4.38  Supremum and infimum   csup 9438
            2.4.39  Ordinal isomorphism, Hartogs's theorem   coi 9507
            2.4.40  Hartogs function   char 9554
            2.4.41  Weak dominance   cwdom 9562
      ​2.5  ZF Set Theory - add the Axiom of Regularity
            2.5.1  Introduce the Axiom of Regularity   ax-reg 9590
            2.5.2  Axiom of Infinity equivalents   inf0 9619
      ​2.6  ZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-inf 9636
            2.6.2  Existence of omega (the set of natural numbers)   omex 9641
            2.6.3  Cantor normal form   ccnf 9659
            2.6.4  Transitive closure of a relation   cttrcl 9705
            2.6.5  Transitive closure   trcl 9726
            2.6.6  Well-Founded Induction   frmin 9747
            2.6.7  Well-Founded Recursion   frr3g 9754
            2.6.8  Rank   cr1 9760
            2.6.9  Scott's trick; collection principle; Hilbert's epsilon   scottex 9883
            2.6.10  Disjoint union   cdju 9896
            2.6.11  Cardinal numbers   ccrd 9933
            2.6.12  Axiom of Choice equivalents   wac 10113
            ​*2.6.13  Cardinal number arithmetic   undjudom 10165
            2.6.14  The Ackermann bijection   ackbij2lem1 10217
            2.6.15  Cofinality (without Axiom of Choice)   cflem 10244
            2.6.16  Eight inequivalent definitions of finite set   sornom 10275
            2.6.17  Hereditarily size-limited sets without Choice   itunifval 10414
​​*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 10433
            3.1.2  Introduce the Axiom of Dependent Choice   ax-dc 10444
      ​3.2  ZFC Set Theory - add the Axiom of Choice
            3.2.1  Introduce the Axiom of Choice   ax-ac 10457
            3.2.2  AC equivalents: well-ordering, Zorn's lemma   numthcor 10492
            3.2.3  Cardinal number theorems using Axiom of Choice   cardval 10544
            3.2.4  Cardinal number arithmetic using Axiom of Choice   iunctb 10572
            3.2.5  Cofinality using the Axiom of Choice   alephreg 10580
      ​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 10618
            3.4.2  Derivation of the Axiom of Choice   gchaclem 10676
​​*PART 4  TG (TARSKI-GROTHENDIECK) SET THEORY
      ​4.1  Inaccessibles
            4.1.1  Weakly and strongly inaccessible cardinals   cwina 10680
            4.1.2  Weak universes   cwun 10698
            4.1.3  Tarski classes   ctsk 10746
            4.1.4  Grothendieck universes   cgru 10788
      ​4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
            4.2.1  Introduce the Tarski-Grothendieck Axiom   ax-groth 10821
            4.2.2  Derive the Power Set, Infinity and Choice Axioms   grothpw 10824
            4.2.3  Tarski map function   ctskm 10835
​​*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 10842
            5.1.2  Final derivation of real and complex number postulates   axaddf 11143
            5.1.3  Real and complex number postulates restated as axioms   ax-cnex 11169
      ​5.2  Derive the basic properties from the field axioms
            5.2.1  Some deductions from the field axioms for complex numbers   cnex 11194
            5.2.2  Infinity and the extended real number system   cpnf 11250
            5.2.3  Restate the ordering postulates with extended real "less than"   axlttri 11290
            5.2.4  Ordering on reals   lttr 11295
            5.2.5  Initial properties of the complex numbers   mul12 11384
      ​5.3  Real and complex numbers - basic operations
            5.3.1  Addition   add12 11436
            5.3.2  Subtraction   cmin 11449
            5.3.3  Multiplication   kcnktkm1cn 11650
            5.3.4  Ordering on reals (cont.)   gt0ne0 11684
            5.3.5  Reciprocals   ixi 11848
            5.3.6  Division   cdiv 11876
            5.3.7  Ordering on reals (cont.)   elimgt0 12057
            5.3.8  Completeness Axiom and Suprema   fimaxre 12163
            5.3.9  Imaginary and complex number properties   inelr 12207
            5.3.10  Function operation analogue theorems   ofsubeq0 12214
      ​5.4  Integer sets
            5.4.1  Positive integers (as a subset of complex numbers)   cn 12217
            5.4.2  Principle of mathematical induction   nnind 12235
            ​*5.4.3  Decimal representation of numbers   c2 12272
            ​*5.4.4  Some properties of specific numbers   neg1cn 12331
            5.4.5  Simple number properties   halfcl 12442
            5.4.6  The Archimedean property   nnunb 12473
            5.4.7  Nonnegative integers (as a subset of complex numbers)   cn0 12477
            ​*5.4.8  Extended nonnegative integers   cxnn0 12549
            5.4.9  Integers (as a subset of complex numbers)   cz 12563
            5.4.10  Decimal arithmetic   cdc 12682
            5.4.11  Upper sets of integers   cuz 12827
            5.4.12  Well-ordering principle for bounded-below sets of integers   uzwo3 12932
            5.4.13  Rational numbers (as a subset of complex numbers)   cq 12937
            5.4.14  Existence of the set of complex numbers   rpnnen1lem2 12966
      ​5.5  Order sets
            5.5.1  Positive reals (as a subset of complex numbers)   crp 12979
            5.5.2  Infinity and the extended real number system (cont.)   cxne 13094
            5.5.3  Supremum and infimum on the extended reals   xrsupexmnf 13289
            5.5.4  Real number intervals   cioo 13329
            5.5.5  Finite intervals of integers   cfz 13489
            ​*5.5.6  Finite intervals of nonnegative integers   elfz2nn0 13597
            5.5.7  Half-open integer ranges   cfzo 13632
      ​5.6  Elementary integer functions
            5.6.1  The floor and ceiling functions   cfl 13760
            5.6.2  The modulo (remainder) operation   cmo 13839
            5.6.3  Miscellaneous theorems about integers   om2uz0i 13917
            5.6.4  Strong induction over upper sets of integers   uzsinds 13957
            5.6.5  Finitely supported functions over the nonnegative integers   fsuppmapnn0fiublem 13960
            5.6.6  The infinite sequence builder "seq" - extension   cseq 13971
            5.6.7  Integer powers   cexp 14032
            5.6.8  Ordered pair theorem for nonnegative integers   nn0le2msqi 14232
            5.6.9  Factorial function   cfa 14238
            5.6.10  The binomial coefficient operation   cbc 14267
            5.6.11  The ` # ` (set size) function   chash 14295
                  5.6.11.1  Proper unordered pairs and triples (sets of size 2 and 3)   hashprlei 14434
                  5.6.11.2  Functions with a domain containing at least two different elements   fundmge2nop0 14458
                  5.6.11.3  Finite induction on the size of the first component of a binary relation   hashdifsnp1 14462
      ​​*5.7  Words over a set
            5.7.1  Definitions and basic theorems   cword 14469
            5.7.2  Last symbol of a word   clsw 14517
            5.7.3  Concatenations of words   cconcat 14525
            5.7.4  Singleton words   cs1 14550
            5.7.5  Concatenations with singleton words   ccatws1cl 14571
            5.7.6  Subwords/substrings   csubstr 14595
            5.7.7  Prefixes of a word   cpfx 14625
            5.7.8  Subwords of subwords   swrdswrdlem 14659
            5.7.9  Subwords and concatenations   pfxcctswrd 14665
            5.7.10  Subwords of concatenations   swrdccatfn 14679
            5.7.11  Splicing words (substring replacement)   csplice 14704
            5.7.12  Reversing words   creverse 14713
            5.7.13  Repeated symbol words   creps 14723
            ​*5.7.14  Cyclical shifts of words   ccsh 14743
            5.7.15  Mapping words by a function   wrdco 14787
            5.7.16  Longer string literals   cs2 14797
      ​​*5.8  Reflexive and transitive closures of relations
            5.8.1  The reflexive and transitive properties of relations   coss12d 14924
            5.8.2  Basic properties of closures   cleq1lem 14934
            5.8.3  Definitions and basic properties of transitive closures   ctcl 14937
            5.8.4  Exponentiation of relations   crelexp 14971
            5.8.5  Reflexive-transitive closure as an indexed union   crtrcl 15007
            ​*5.8.6  Principle of transitive induction.   relexpindlem 15015
      ​5.9  Elementary real and complex functions
            5.9.1  The "shift" operation   cshi 15018
            5.9.2  Signum (sgn or sign) function   csgn 15038
            5.9.3  Real and imaginary parts; conjugate   ccj 15048
            5.9.4  Square root; absolute value   csqrt 15185
      ​5.10  Elementary limits and convergence
            5.10.1  Superior limit (lim sup)   clsp 15419
            5.10.2  Limits   cli 15433
            5.10.3  Finite and infinite sums   csu 15637
            5.10.4  The binomial theorem   binomlem 15780
            5.10.5  The inclusion/exclusion principle   incexclem 15787
            5.10.6  Infinite sums (cont.)   isumshft 15790
            5.10.7  Miscellaneous converging and diverging sequences   divrcnv 15803
            5.10.8  Arithmetic series   arisum 15811
            5.10.9  Geometric series   expcnv 15815
            5.10.10  Ratio test for infinite series convergence   cvgrat 15834
            5.10.11  Mertens' theorem   mertenslem1 15835
            5.10.12  Finite and infinite products   prodf 15838
                  5.10.12.1  Product sequences   prodf 15838
                  5.10.12.2  Non-trivial convergence   ntrivcvg 15848
                  5.10.12.3  Complex products   cprod 15854
                  5.10.12.4  Finite products   fprod 15890
                  5.10.12.5  Infinite products   iprodclim 15947
            5.10.13  Falling and Rising Factorial   cfallfac 15953
            5.10.14  Bernoulli polynomials and sums of k-th powers   cbp 15995
      ​5.11  Elementary trigonometry
            5.11.1  The exponential, sine, and cosine functions   ce 16010
                  5.11.1.1  The circle constant (tau = 2 pi)   ctau 16150
            5.11.2  _e is irrational   eirrlem 16152
      ​5.12  Cardinality of real and complex number subsets
            5.12.1  Countability of integers and rationals   xpnnen 16159
            5.12.2  The reals are uncountable   rpnnen2lem1 16162
​​*PART 6  ELEMENTARY NUMBER THEORY
      ​6.1  Elementary properties of divisibility
            6.1.1  Irrationality of square root of 2   sqrt2irrlem 16196
            6.1.2  Some Number sets are chains of proper subsets   nthruc 16200
            6.1.3  The divides relation   cdvds 16202
            ​*6.1.4  Even and odd numbers   evenelz 16284
            6.1.5  The division algorithm   divalglem0 16341
            6.1.6  Bit sequences   cbits 16365
            6.1.7  The greatest common divisor operator   cgcd 16440
            6.1.8  Bézout's identity   bezoutlem1 16486
            6.1.9  Algorithms   nn0seqcvgd 16512
            6.1.10  Euclid's Algorithm   eucalgval2 16523
            ​*6.1.11  The least common multiple   clcm 16530
            ​*6.1.12  Coprimality and Euclid's lemma   coprmgcdb 16591
            6.1.13  Cancellability of congruences   congr 16606
      ​6.2  Elementary prime number theory
            ​*6.2.1  Elementary properties   cprime 16613
            ​*6.2.2  Coprimality and Euclid's lemma (cont.)   coprm 16653
            6.2.3  Properties of the canonical representation of a rational   cnumer 16674
            6.2.4  Euler's theorem   codz 16701
            6.2.5  Arithmetic modulo a prime number   modprm1div 16735
            6.2.6  Pythagorean Triples   coprimeprodsq 16746
            6.2.7  The prime count function   cpc 16774
            6.2.8  Pocklington's theorem   prmpwdvds 16842
            6.2.9  Infinite primes theorem   unbenlem 16846
            6.2.10  Sum of prime reciprocals   prmreclem1 16854
            6.2.11  Fundamental theorem of arithmetic   1arithlem1 16861
            6.2.12  Lagrange's four-square theorem   cgz 16867
            6.2.13  Van der Waerden's theorem   cvdwa 16903
            6.2.14  Ramsey's theorem   cram 16937
            ​*6.2.15  Primorial function   cprmo 16969
            ​*6.2.16  Prime gaps   prmgaplem1 16987
            6.2.17  Decimal arithmetic (cont.)   dec2dvds 17001
            6.2.18  Cyclical shifts of words (cont.)   cshwsidrepsw 17032
            6.2.19  Specific prime numbers   prmlem0 17044
            6.2.20  Very large primes   1259lem1 17069
​PART 7  BASIC STRUCTURES
      ​7.1  Extensible structures
            ​*7.1.1  Basic definitions   cstr 17084
                  7.1.1.1  Extensible structures as structures with components   cstr 17084
                  7.1.1.2  Substitution of components   csts 17101
                  7.1.1.3  Slots   cslot 17119
                  ​*7.1.1.4  Structure component indices   cnx 17131
                  7.1.1.5  Base sets   cbs 17149
                  7.1.1.6  Base set restrictions   cress 17178
            7.1.2  Slot definitions   cplusg 17202
            7.1.3  Definition of the structure product   crest 17371
            7.1.4  Definition of the structure quotient   cordt 17450
      ​7.2  Moore spaces
            7.2.1  Moore closures   mrcflem 17555
            7.2.2  Independent sets in a Moore system   mrisval 17579
            7.2.3  Algebraic closure systems   isacs 17600
​PART 8  BASIC CATEGORY THEORY
      ​8.1  Categories
            8.1.1  Categories   ccat 17613
            8.1.2  Opposite category   coppc 17660
            8.1.3  Monomorphisms and epimorphisms   cmon 17680
            8.1.4  Sections, inverses, isomorphisms   csect 17696
            ​*8.1.5  Isomorphic objects   ccic 17747
            8.1.6  Subcategories   cssc 17759
            8.1.7  Functors   cfunc 17809
            8.1.8  Full & faithful functors   cful 17858
            8.1.9  Natural transformations and the functor category   cnat 17897
            8.1.10  Initial, terminal and zero objects of a category   cinito 17936
      ​8.2  Arrows (disjointified hom-sets)
            8.2.1  Identity and composition for arrows   cida 18008
      ​8.3  Examples of categories
            8.3.1  The category of sets   csetc 18030
            8.3.2  The category of categories   ccatc 18053
            ​*8.3.3  The category of extensible structures   fncnvimaeqv 18076
      ​8.4  Categorical constructions
            8.4.1  Product of categories   cxpc 18125
            8.4.2  Functor evaluation   cevlf 18167
            8.4.3  Hom functor   chof 18206
​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 18389
            9.5.2  Complete lattices   ccla 18456
            9.5.3  Distributive lattices   cdlat 18478
            9.5.4  Subset order structures   cipo 18485
      ​9.6  Posets, directed sets, and lattices as relations
            ​*9.6.1  Posets and lattices as relations   cps 18522
            9.6.2  Directed sets, nets   cdir 18552
​PART 10  BASIC ALGEBRAIC STRUCTURES
      ​10.1  Monoids
            ​*10.1.1  Magmas   cplusf 18563
            ​*10.1.2  Identity elements   mgmidmo 18586
            ​*10.1.3  Iterated sums in a magma   gsumvalx 18602
            10.1.4  Magma homomorphisms and submagmas   cmgmhm 18616
            ​*10.1.5  Semigroups   csgrp 18644
            ​*10.1.6  Definition and basic properties of monoids   cmnd 18660
            10.1.7  Monoid homomorphisms and submonoids   cmhm 18704
            ​*10.1.8  Iterated sums in a monoid   gsumvallem2 18752
            10.1.9  Free monoids   cfrmd 18765
                  ​*10.1.9.1  Monoid of endofunctions   cefmnd 18786
            10.1.10  Examples and counterexamples for magmas, semigroups and monoids   mgm2nsgrplem1 18836
      ​10.2  Groups
            10.2.1  Definition and basic properties   cgrp 18856
            ​*10.2.2  Group multiple operation   cmg 18987
            10.2.3  Subgroups and Quotient groups   csubg 19037
            ​*10.2.4  Cyclic monoids and groups   cycsubmel 19116
            10.2.5  Elementary theory of group homomorphisms   cghm 19128
            10.2.6  Isomorphisms of groups   cgim 19172
            10.2.7  Group actions   cga 19195
            10.2.8  Centralizers and centers   ccntz 19221
            10.2.9  The opposite group   coppg 19251
            10.2.10  Symmetric groups   csymg 19276
                  ​*10.2.10.1  Definition and basic properties   csymg 19276
                  10.2.10.2  Cayley's theorem   cayleylem1 19322
                  10.2.10.3  Permutations fixing one element   symgfix2 19326
                  ​*10.2.10.4  Transpositions in the symmetric group   cpmtr 19351
                  10.2.10.5  The sign of a permutation   cpsgn 19399
            10.2.11  p-Groups and Sylow groups; Sylow's theorems   cod 19434
            10.2.12  Direct products   clsm 19544
                  10.2.12.1  Direct products (extension)   smndlsmidm 19566
            10.2.13  Free groups   cefg 19616
            10.2.14  Abelian groups   ccmn 19690
                  10.2.14.1  Definition and basic properties   ccmn 19690
                  10.2.14.2  Cyclic groups   ccyg 19787
                  10.2.14.3  Group sum operation   gsumval3a 19813
                  10.2.14.4  Group sums over (ranges of) integers   fsfnn0gsumfsffz 19893
                  10.2.14.5  Internal direct products   cdprd 19905
                  10.2.14.6  The Fundamental Theorem of Abelian Groups   ablfacrplem 19977
            10.2.15  Simple groups   csimpg 20002
                  10.2.15.1  Definition and basic properties   csimpg 20002
                  10.2.15.2  Classification of abelian simple groups   ablsimpnosubgd 20016
      ​10.3  Rings
            10.3.1  Multiplicative Group   cmgp 20029
            ​*10.3.2  Non-unital rings ("rngs")   crng 20047
            ​*10.3.3  Ring unity (multiplicative identity)   cur 20076
            10.3.4  Semirings   csrg 20081
                  ​*10.3.4.1  The binomial theorem for semirings   srgbinomlem1 20121
            10.3.5  Unital rings   crg 20128
            10.3.6  Opposite ring   coppr 20225
            10.3.7  Divisibility   cdsr 20246
            10.3.8  Ring primes   crpm 20324
            10.3.9  Homomorphisms of non-unital rings   crnghm 20326
            10.3.10  Ring homomorphisms   crh 20361
            10.3.11  Nonzero rings and zero rings   cnzr 20404
            10.3.12  Local rings   clring 20427
            10.3.13  Subrings   csubrng 20434
                  10.3.13.1  Subrings of non-unital rings   csubrng 20434
                  10.3.13.2  Subrings of unital rings   csubrg 20458
      ​10.4  Division rings and fields
            10.4.1  Definition and basic properties   cdr 20501
            10.4.2  Sub-division rings   csdrg 20546
            10.4.3  Absolute value (abstract algebra)   cabv 20568
            10.4.4  Star rings   cstf 20595
      ​10.5  Left modules
            10.5.1  Definition and basic properties   clmod 20615
            10.5.2  Subspaces and spans in a left module   clss 20687
            10.5.3  Homomorphisms and isomorphisms of left modules   clmhm 20775
            10.5.4  Subspace sum; bases for a left module   clbs 20830
      ​10.6  Vector spaces
            10.6.1  Definition and basic properties   clvec 20858
      ​10.7  Ideals
            10.7.1  The subring algebra; ideals   csra 20927
            10.7.2  Two-sided ideals and quotient rings   c2idl 21006
            ​*10.7.3  Ideals of non-unital rings   dflidl2rng 21031
                  ​*10.7.3.1  Condition for a non-unital ring to be unital   rngqiprng1elbas 21046
            10.7.4  Principal ideal rings. Divisibility in the integers   clpidl 21080
            10.7.5  Left regular elements. More kinds of rings   crlreg 21096
      ​10.8  The complex numbers as an algebraic extensible structure
            10.8.1  Definition and basic properties   cpsmet 21129
            ​*10.8.2  Ring of integers   czring 21218
                  ​*10.8.2.1  Example for a condition for a non-unital ring to be unital   pzriprnglem1 21251
            10.8.3  Algebraic constructions based on the complex numbers   czrh 21269
            10.8.4  Signs as subgroup of the complex numbers   cnmsgnsubg 21350
            10.8.5  Embedding of permutation signs into a ring   zrhpsgnmhm 21357
            10.8.6  The ordered field of real numbers   crefld 21377
      ​10.9  Generalized pre-Hilbert and Hilbert spaces
            10.9.1  Definition and basic properties   cphl 21397
            10.9.2  Orthocomplements and closed subspaces   cocv 21433
            10.9.3  Orthogonal projection and orthonormal bases   cpj 21475
​​*PART 11  BASIC LINEAR ALGEBRA
      ​11.1  Vectors and free modules
            *11.1.1  Direct sum of left modules   cdsmm 21506
            ​*11.1.2  Free modules   cfrlm 21521
            ​*11.1.3  Standard basis (unit vectors)   cuvc 21557
            ​*11.1.4  Independent sets and families   clindf 21579
            11.1.5  Characterization of free modules   lmimlbs 21611
      ​11.2  Associative algebras
            11.2.1  Definition and basic properties   casa 21625
      ​11.3  Abstract multivariate polynomials
            11.3.1  Definition and basic properties   cmps 21677
            11.3.2  Polynomial evaluation   ces 21853
            11.3.3  Additional definitions for (multivariate) polynomials   cslv 21891
            ​*11.3.4  Univariate polynomials   cps1 21919
            11.3.5  Univariate polynomial evaluation   ces1 22053
      ​​*11.4  Matrices
            *11.4.1  The matrix multiplication   cmmul 22106
            ​*11.4.2  Square matrices   cmat 22128
            ​*11.4.3  The matrix algebra   matmulr 22161
            ​*11.4.4  Matrices of dimension 0 and 1   mat0dimbas0 22189
            ​*11.4.5  The subalgebras of diagonal and scalar matrices   cdmat 22211
            ​*11.4.6  Multiplication of a matrix with a "column vector"   cmvmul 22263
            11.4.7  Replacement functions for a square matrix   cmarrep 22279
            11.4.8  Submatrices   csubma 22299
      ​11.5  The determinant
            11.5.1  Definition and basic properties   cmdat 22307
            11.5.2  Determinants of 2 x 2 -matrices   m2detleiblem1 22347
            11.5.3  The matrix adjugate/adjunct   cmadu 22355
            ​*11.5.4  Laplace expansion of determinants (special case)   symgmatr01lem 22376
            11.5.5  Inverse matrix   invrvald 22399
            ​*11.5.6  Cramer's rule   slesolvec 22402
      ​​*11.6  Polynomial matrices
            11.6.1  Basic properties   pmatring 22415
            ​*11.6.2  Constant polynomial matrices   ccpmat 22426
            ​*11.6.3  Collecting coefficients of polynomial matrices   cdecpmat 22485
            ​*11.6.4  Ring isomorphism between polynomial matrices and polynomials over matrices   cpm2mp 22515
      ​​*11.7  The characteristic polynomial
            *11.7.1  Definition and basic properties   cchpmat 22549
            ​*11.7.2  The characteristic factor function G   fvmptnn04if 22572
            ​*11.7.3  The Cayley-Hamilton theorem   cpmadurid 22590
​PART 12  BASIC TOPOLOGY
      ​12.1  Topology
            ​*12.1.1  Topological spaces   ctop 22616
                  12.1.1.1  Topologies   ctop 22616
                  12.1.1.2  Topologies on sets   ctopon 22633
                  12.1.1.3  Topological spaces   ctps 22655
            12.1.2  Topological bases   ctb 22669
            12.1.3  Examples of topologies   distop 22719
            12.1.4  Closure and interior   ccld 22741
            12.1.5  Neighborhoods   cnei 22822
            12.1.6  Limit points and perfect sets   clp 22859
            12.1.7  Subspace topologies   restrcl 22882
            12.1.8  Order topology   ordtbaslem 22913
            12.1.9  Limits and continuity in topological spaces   ccn 22949
            12.1.10  Separated spaces: T0, T1, T2 (Hausdorff) ...   ct0 23031
            12.1.11  Compactness   ccmp 23111
            12.1.12  Bolzano-Weierstrass theorem   bwth 23135
            12.1.13  Connectedness   cconn 23136
            12.1.14  First- and second-countability   c1stc 23162
            12.1.15  Local topological properties   clly 23189
            12.1.16  Refinements   cref 23227
            12.1.17  Compactly generated spaces   ckgen 23258
            12.1.18  Product topologies   ctx 23285
            12.1.19  Continuous function-builders   cnmptid 23386
            12.1.20  Quotient maps and quotient topology   ckq 23418
            12.1.21  Homeomorphisms   chmeo 23478
      ​12.2  Filters and filter bases
            12.2.1  Filter bases   elmptrab 23552
            12.2.2  Filters   cfil 23570
            12.2.3  Ultrafilters   cufil 23624
            12.2.4  Filter limits   cfm 23658
            12.2.5  Extension by continuity   ccnext 23784
            12.2.6  Topological groups   ctmd 23795
            12.2.7  Infinite group sum on topological groups   ctsu 23851
            12.2.8  Topological rings, fields, vector spaces   ctrg 23881
      ​12.3  Uniform Structures and Spaces
            12.3.1  Uniform structures   cust 23925
            12.3.2  The topology induced by an uniform structure   cutop 23956
            12.3.3  Uniform Spaces   cuss 23979
            12.3.4  Uniform continuity   cucn 24001
            12.3.5  Cauchy filters in uniform spaces   ccfilu 24012
            12.3.6  Complete uniform spaces   ccusp 24023
      ​12.4  Metric spaces
            12.4.1  Pseudometric spaces   ispsmet 24031
            12.4.2  Basic metric space properties   cxms 24044
            12.4.3  Metric space balls   blfvalps 24110
            12.4.4  Open sets of a metric space   mopnval 24165
            12.4.5  Continuity in metric spaces   metcnp3 24270
            12.4.6  The uniform structure generated by a metric   metuval 24279
            12.4.7  Examples of metric spaces   dscmet 24302
            ​*12.4.8  Normed algebraic structures   cnm 24306
            12.4.9  Normed space homomorphisms (bounded linear operators)   cnmo 24443
            12.4.10  Topology on the reals   qtopbaslem 24496
            12.4.11  Topological definitions using the reals   cii 24616
            12.4.12  Path homotopy   chtpy 24714
            12.4.13  The fundamental group   cpco 24748
      ​12.5  Metric subcomplex vector spaces
            12.5.1  Subcomplex modules   cclm 24810
            ​*12.5.2  Subcomplex vector spaces   ccvs 24871
            ​*12.5.3  Normed subcomplex vector spaces   isncvsngp 24898
            12.5.4  Subcomplex pre-Hilbert spaces   ccph 24915
            12.5.5  Convergence and completeness   ccfil 25001
            12.5.6  Baire's Category Theorem   bcthlem1 25073
            12.5.7  Banach spaces and subcomplex Hilbert spaces   ccms 25081
                  12.5.7.1  The complete ordered field of the real numbers   retopn 25128
            12.5.8  Euclidean spaces   crrx 25132
            12.5.9  Minimizing Vector Theorem   minveclem1 25173
            12.5.10  Projection Theorem   pjthlem1 25186
​PART 13  BASIC REAL AND COMPLEX ANALYSIS
      ​13.1  Continuity
            13.1.1  Intermediate value theorem   pmltpclem1 25198
      ​13.2  Integrals
            13.2.1  Lebesgue measure   covol 25212
            13.2.2  Lebesgue integration   cmbf 25364
                  13.2.2.1  Lesbesgue integral   cmbf 25364
                  13.2.2.2  Lesbesgue directed integral   cdit 25596
      ​13.3  Derivatives
            13.3.1  Real and complex differentiation   climc 25612
                  13.3.1.1  Derivatives of functions of one complex or real variable   climc 25612
                  13.3.1.2  Results on real differentiation   dvferm1lem 25734
​PART 14  BASIC REAL AND COMPLEX FUNCTIONS
      ​14.1  Polynomials
            14.1.1  Polynomial degrees   cmdg 25801
            14.1.2  The division algorithm for univariate polynomials   cmn1 25876
            14.1.3  Elementary properties of complex polynomials   cply 25931
            14.1.4  The division algorithm for polynomials   cquot 26036
            14.1.5  Algebraic numbers   caa 26060
            14.1.6  Liouville's approximation theorem   aalioulem1 26078
      ​14.2  Sequences and series
            14.2.1  Taylor polynomials and Taylor's theorem   ctayl 26098
            14.2.2  Uniform convergence   culm 26121
            14.2.3  Power series   pserval 26155
      ​14.3  Basic trigonometry
            14.3.1  The exponential, sine, and cosine functions (cont.)   efcn 26188
            14.3.2  Properties of pi = 3.14159...   pilem1 26196
            14.3.3  Mapping of the exponential function   efgh 26283
            14.3.4  The natural logarithm on complex numbers   clog 26296
            ​*14.3.5  Logarithms to an arbitrary base   clogb 26502
            14.3.6  Theorems of Pythagoras, isosceles triangles, and intersecting chords   angval 26539
            14.3.7  Solutions of quadratic, cubic, and quartic equations   quad2 26577
            14.3.8  Inverse trigonometric functions   casin 26600
            14.3.9  The Birthday Problem   log2ublem1 26684
            14.3.10  Areas in R^2   carea 26693
            14.3.11  More miscellaneous converging sequences   rlimcnp 26703
            14.3.12  Inequality of arithmetic and geometric means   cvxcl 26722
            14.3.13  Euler-Mascheroni constant   cem 26729
            14.3.14  Zeta function   czeta 26750
            14.3.15  Gamma function   clgam 26753
      ​14.4  Basic number theory
            14.4.1  Wilson's theorem   wilthlem1 26805
            14.4.2  The Fundamental Theorem of Algebra   ftalem1 26810
            14.4.3  The Basel problem (ζ(2) = π²/6)   basellem1 26818
            14.4.4  Number-theoretical functions   ccht 26828
            14.4.5  Perfect Number Theorem   mersenne 26963
            14.4.6  Characters of Z/nZ   cdchr 26968
            14.4.7  Bertrand's postulate   bcctr 27011
            ​*14.4.8  Quadratic residues and the Legendre symbol   clgs 27030
            ​*14.4.9  Gauss' Lemma   gausslemma2dlem0a 27092
            14.4.10  Quadratic reciprocity   lgseisenlem1 27111
            14.4.11  All primes 4n+1 are the sum of two squares   2sqlem1 27153
            14.4.12  Chebyshev's Weak Prime Number Theorem, Dirichlet's Theorem   chebbnd1lem1 27205
            14.4.13  The Prime Number Theorem   mudivsum 27266
            14.4.14  Ostrowski's theorem   abvcxp 27351
​PART 15  SURREAL NUMBERS
      ​​*15.1  Sign sequence representation and Alling's axioms
            15.1.1  Definitions and initial properties   csur 27376
            15.1.2  Ordering   sltsolem1 27411
            15.1.3  Birthday Function   bdayfo 27413
            15.1.4  Density   fvnobday 27414
            ​*15.1.5  Full-Eta Property   bdayimaon 27429
      ​15.2  Initial consequences of Alling's axioms
            15.2.1  Ordering Theorems   csle 27480
            15.2.2  Birthday Theorems   bdayfun 27507
      ​​*15.3  Conway cut representation
            15.3.1  Conway cuts   csslt 27515
            15.3.2  Zero and One   c0s 27557
            15.3.3  Cuts and Options   cmade 27571
            15.3.4  Cofinality and coinitiality   cofsslt 27640
      ​15.4  Induction and recursion
            15.4.1  Induction and recursion on one variable   cnorec 27656
            15.4.2  Induction and recursion on two variables   cnorec2 27667
      ​15.5  Surreal arithmetic
            15.5.1  Addition   cadds 27678
            15.5.2  Negation and Subtraction   cnegs 27730
            15.5.3  Multiplication   cmuls 27798
            15.5.4  Division   cdivs 27871
      ​15.6  Subsystems of surreals
            15.6.1  Ordinal numbers   cons 27914
            15.6.2  Natural numbers   cnn0s 27926
​​*PART 16  ELEMENTARY GEOMETRY
      ​16.1  Definition and Tarski's Axioms of Geometry
            16.1.1  Justification for the congruence notation   tgjustf 27988
      ​16.2  Tarskian Geometry
            16.2.1  Congruence   tgcgrcomimp 27992
            16.2.2  Betweenness   tgbtwntriv2 28002
            16.2.3  Dimension   tglowdim1 28015
            16.2.4  Betweenness and Congruence   tgifscgr 28023
            16.2.5  Congruence of a series of points   ccgrg 28025
            16.2.6  Motions   cismt 28047
            16.2.7  Colinearity   tglng 28061
            16.2.8  Connectivity of betweenness   tgbtwnconn1lem1 28087
            16.2.9  Less-than relation in geometric congruences   cleg 28097
            16.2.10  Rays   chlg 28115
            16.2.11  Lines   btwnlng1 28134
            16.2.12  Point inversions   cmir 28167
            16.2.13  Right angles   crag 28208
            16.2.14  Half-planes   islnopp 28254
            16.2.15  Midpoints and Line Mirroring   cmid 28287
            16.2.16  Congruence of angles   ccgra 28322
            16.2.17  Angle Comparisons   cinag 28350
            16.2.18  Congruence Theorems   tgsas1 28369
            16.2.19  Equilateral triangles   ceqlg 28380
      ​16.3  Properties of geometries
            16.3.1  Isomorphisms between geometries   f1otrgds 28384
      ​16.4  Geometry in Hilbert spaces
            16.4.1  Geometry in the complex plane   cchhllem 28408
            16.4.2  Geometry in Euclidean spaces   cee 28410
                  16.4.2.1  Definition of the Euclidean space   cee 28410
                  16.4.2.2  Tarski's axioms for geometry for the Euclidean space   axdimuniq 28435
                  16.4.2.3  EE^n fulfills Tarski's Axioms   ceeng 28499
​​*PART 17  GRAPH THEORY
      ​*17.1  Vertices and edges
            17.1.1  The edge function extractor for extensible structures   cedgf 28510
            ​*17.1.2  Vertices and indexed edges   cvtx 28520
                  17.1.2.1  Definitions and basic properties   cvtx 28520
                  17.1.2.2  The vertices and edges of a graph represented as ordered pair   opvtxval 28527
                  17.1.2.3  The vertices and edges of a graph represented as extensible structure   funvtxdmge2val 28535
                  17.1.2.4  Representations of graphs without edges   snstrvtxval 28561
                  17.1.2.5  Degenerated cases of representations of graphs   vtxval0 28563
            17.1.3  Edges as range of the edge function   cedg 28571
      ​​*17.2  Undirected graphs
            17.2.1  Undirected hypergraphs   cuhgr 28580
            17.2.2  Undirected pseudographs and multigraphs   cupgr 28604
            ​*17.2.3  Loop-free graphs   umgrislfupgrlem 28646
            17.2.4  Edges as subsets of vertices of graphs   uhgredgiedgb 28650
            ​*17.2.5  Undirected simple graphs   cuspgr 28672
            17.2.6  Examples for graphs   usgr0e 28757
            17.2.7  Subgraphs   csubgr 28788
            17.2.8  Finite undirected simple graphs   cfusgr 28837
            17.2.9  Neighbors, complete graphs and universal vertices   cnbgr 28853
                  17.2.9.1  Neighbors   cnbgr 28853
                  17.2.9.2  Universal vertices   cuvtx 28906
                  17.2.9.3  Complete graphs   ccplgr 28930
            17.2.10  Vertex degree   cvtxdg 28986
            ​*17.2.11  Regular graphs   crgr 29076
      ​​*17.3  Walks, paths and cycles
            *17.3.1  Walks   cewlks 29116
            17.3.2  Walks for loop-free graphs   lfgrwlkprop 29208
            17.3.3  Trails   ctrls 29211
            17.3.4  Paths and simple paths   cpths 29233
            17.3.5  Closed walks   cclwlks 29291
            17.3.6  Circuits and cycles   ccrcts 29305
            ​*17.3.7  Walks as words   cwwlks 29343
            17.3.8  Walks/paths of length 2 (as length 3 strings)   2wlkdlem1 29443
            17.3.9  Walks in regular graphs   rusgrnumwwlkl1 29486
            ​*17.3.10  Closed walks as words   cclwwlk 29498
                  17.3.10.1  Closed walks as words   cclwwlk 29498
                  17.3.10.2  Closed walks of a fixed length as words   cclwwlkn 29541
                  17.3.10.3  Closed walks on a vertex of a fixed length as words   cclwwlknon 29604
            17.3.11  Examples for walks, trails and paths   0ewlk 29631
            17.3.12  Connected graphs   cconngr 29703
      ​17.4  Eulerian paths and the Konigsberg Bridge problem
            ​*17.4.1  Eulerian paths   ceupth 29714
            ​*17.4.2  The Königsberg Bridge problem   konigsbergvtx 29763
      ​17.5  The Friendship Theorem
            17.5.1  Friendship graphs - basics   cfrgr 29775
            17.5.2  The friendship theorem for small graphs   frgr1v 29788
            17.5.3  Theorems according to Mertzios and Unger   2pthfrgrrn 29799
            ​*17.5.4  Huneke's Proof of the Friendship Theorem   frgrncvvdeqlem1 29816
​PART 18  GUIDES AND MISCELLANEA
      ​18.1  Guides (conventions, explanations, and examples)
            ​*18.1.1  Conventions   conventions 29917
            18.1.2  Natural deduction   natded 29920
            ​*18.1.3  Natural deduction examples   ex-natded5.2 29921
            18.1.4  Definitional examples   ex-or 29938
            18.1.5  Other examples   aevdemo 29977
      ​18.2  Humor
            18.2.1  April Fool's theorem   avril1 29980
      ​18.3  (Future - to be reviewed and classified)
            18.3.1  Planar incidence geometry   cplig 29991
            ​*18.3.2  Aliases kept to prevent broken links   dummylink 30004
​​*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 30006
            19.1.2  Abelian groups   cablo 30061
      ​19.2  Complex vector spaces
            19.2.1  Definition and basic properties   cvc 30075
            19.2.2  Examples of complex vector spaces   cnaddabloOLD 30098
      ​19.3  Normed complex vector spaces
            19.3.1  Definition and basic properties   cnv 30101
            19.3.2  Examples of normed complex vector spaces   cnnv 30194
            19.3.3  Induced metric of a normed complex vector space   imsval 30202
            19.3.4  Inner product   cdip 30217
            19.3.5  Subspaces   css 30238
      ​19.4  Operators on complex vector spaces
            19.4.1  Definitions and basic properties   clno 30257
      ​19.5  Inner product (pre-Hilbert) spaces
            19.5.1  Definition and basic properties   ccphlo 30329
            19.5.2  Examples of pre-Hilbert spaces   cncph 30336
            19.5.3  Properties of pre-Hilbert spaces   isph 30339
      ​19.6  Complex Banach spaces
            19.6.1  Definition and basic properties   ccbn 30379
            19.6.2  Examples of complex Banach spaces   cnbn 30386
            19.6.3  Uniform Boundedness Theorem   ubthlem1 30387
            19.6.4  Minimizing Vector Theorem   minvecolem1 30391
      ​19.7  Complex Hilbert spaces
            19.7.1  Definition and basic properties   chlo 30402
            19.7.2  Standard axioms for a complex Hilbert space   hlex 30415
            19.7.3  Examples of complex Hilbert spaces   cnchl 30433
            19.7.4  Hellinger-Toeplitz Theorem   htthlem 30434
​​*PART 20  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      ​20.1  Axiomatization of complex pre-Hilbert spaces
            20.1.1  Basic Hilbert space definitions   chba 30436
            20.1.2  Preliminary ZFC lemmas   df-hnorm 30485
            ​*20.1.3  Derive the Hilbert space axioms from ZFC set theory   axhilex-zf 30498
            ​*20.1.4  Introduce the vector space axioms for a Hilbert space   ax-hilex 30516
            20.1.5  Vector operations   hvmulex 30528
            20.1.6  Inner product postulates for a Hilbert space   ax-hfi 30596
      ​20.2  Inner product and norms
            20.2.1  Inner product   his5 30603
            20.2.2  Norms   dfhnorm2 30639
            20.2.3  Relate Hilbert space to normed complex vector spaces   hilablo 30677
            20.2.4  Bunjakovaskij-Cauchy-Schwarz inequality   bcsiALT 30696
      ​20.3  Cauchy sequences and completeness axiom
            20.3.1  Cauchy sequences and limits   hcau 30701
            20.3.2  Derivation of the completeness axiom from ZF set theory   hilmet 30711
            20.3.3  Completeness postulate for a Hilbert space   ax-hcompl 30719
            20.3.4  Relate Hilbert space to ZFC pre-Hilbert and Hilbert spaces   hhcms 30720
      ​20.4  Subspaces and projections
            20.4.1  Subspaces   df-sh 30724
            20.4.2  Closed subspaces   df-ch 30738
            20.4.3  Orthocomplements   df-oc 30769
            20.4.4  Subspace sum, span, lattice join, lattice supremum   df-shs 30825
            20.4.5  Projection theorem   pjhthlem1 30908
            20.4.6  Projectors   df-pjh 30912
      ​20.5  Properties of Hilbert subspaces
            20.5.1  Orthomodular law   omlsilem 30919
            20.5.2  Projectors (cont.)   pjhtheu2 30933
            20.5.3  Hilbert lattice operations   sh0le 30957
            20.5.4  Span (cont.) and one-dimensional subspaces   spansn0 31058
            20.5.5  Commutes relation for Hilbert lattice elements   df-cm 31100
            20.5.6  Foulis-Holland theorem   fh1 31135
            20.5.7  Quantum Logic Explorer axioms   qlax1i 31144
            20.5.8  Orthogonal subspaces   chscllem1 31154
            20.5.9  Orthoarguesian laws 5OA and 3OA   5oalem1 31171
            20.5.10  Projectors (cont.)   pjorthi 31186
            20.5.11  Mayet's equation E_3   mayete3i 31245
      ​20.6  Operators on Hilbert spaces
            ​*20.6.1  Operator sum, difference, and scalar multiplication   df-hosum 31247
            20.6.2  Zero and identity operators   df-h0op 31265
            20.6.3  Operations on Hilbert space operators   hoaddcl 31275
            20.6.4  Linear, continuous, bounded, Hermitian, unitary operators and norms   df-nmop 31356
            20.6.5  Linear and continuous functionals and norms   df-nmfn 31362
            20.6.6  Adjoint   df-adjh 31366
            20.6.7  Dirac bra-ket notation   df-bra 31367
            20.6.8  Positive operators   df-leop 31369
            20.6.9  Eigenvectors, eigenvalues, spectrum   df-eigvec 31370
            20.6.10  Theorems about operators and functionals   nmopval 31373
            20.6.11  Riesz lemma   riesz3i 31579
            20.6.12  Adjoints (cont.)   cnlnadjlem1 31584
            20.6.13  Quantum computation error bound theorem   unierri 31621
            20.6.14  Dirac bra-ket notation (cont.)   branmfn 31622
            20.6.15  Positive operators (cont.)   leopg 31639
            20.6.16  Projectors as operators   pjhmopi 31663
      ​20.7  States on a Hilbert lattice and Godowski's equation
            20.7.1  States on a Hilbert lattice   df-st 31728
            20.7.2  Godowski's equation   golem1 31788
      ​20.8  Cover relation, atoms, exchange axiom, and modular symmetry
            20.8.1  Covers relation; modular pairs   df-cv 31796
            20.8.2  Atoms   df-at 31855
            20.8.3  Superposition principle   superpos 31871
            20.8.4  Atoms, exchange and covering properties, atomicity   chcv1 31872
            20.8.5  Irreducibility   chirredlem1 31907
            20.8.6  Atoms (cont.)   atcvat3i 31913
            20.8.7  Modular symmetry   mdsymlem1 31920
​PART 21  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      ​21.1  Mathboxes for user contributions
            21.1.1  Mathbox guidelines   mathbox 31959
      ​21.2  Mathbox for Stefan Allan
      ​21.3  Mathbox for Thierry Arnoux
            21.3.1  Propositional Calculus - misc additions   bian1d 31964
            21.3.2  Predicate Calculus   sbc2iedf 31971
                  21.3.2.1  Predicate Calculus - misc additions   sbc2iedf 31971
                  21.3.2.2  Restricted quantification - misc additions   ralcom4f 31973
                  21.3.2.3  Equality   eqtrb 31978
                  21.3.2.4  Double restricted existential uniqueness quantification   opsbc2ie 31980
                  21.3.2.5  Double restricted existential uniqueness quantification syntax   w2reu 31982
                  21.3.2.6  Substitution (without distinct variables) - misc additions   sbceqbidf 31991
                  21.3.2.7  Existential "at most one" - misc additions   mo5f 31993
                  21.3.2.8  Existential uniqueness - misc additions   reuxfrdf 31995
                  21.3.2.9  Restricted "at most one" - misc additions   rmoxfrd 31997
                  21.3.2.10  Restricted iota (description binder)   riotaeqbidva 32000
            21.3.3  General Set Theory   dmrab 32001
                  21.3.3.1  Class abstractions (a.k.a. class builders)   dmrab 32001
                  21.3.3.2  Image Sets   abrexdomjm 32008
                  21.3.3.3  Set relations and operations - misc additions   elunsn 32014
                  21.3.3.4  Unordered pairs   eqsnd 32030
                  21.3.3.5  Conditional operator - misc additions   ifeqeqx 32038
                  21.3.3.6  Set union   uniinn0 32046
                  21.3.3.7  Indexed union - misc additions   cbviunf 32051
                  21.3.3.8  Indexed intersection - misc additions   iinabrex 32064
                  21.3.3.9  Disjointness - misc additions   disjnf 32065
            21.3.4  Relations and Functions   xpdisjres 32093
                  21.3.4.1  Relations - misc additions   xpdisjres 32093
                  21.3.4.2  Functions - misc additions   ac6sf2 32113
                  21.3.4.3  Operations - misc additions   mpomptxf 32169
                  21.3.4.4  Support of a function   suppovss 32170
                  21.3.4.5  Explicit Functions with one or two points as a domain   cosnopne 32180
                  21.3.4.6  Isomorphisms - misc. additions   gtiso 32186
                  21.3.4.7  Disjointness (additional proof requiring functions)   disjdsct 32188
                  21.3.4.8  First and second members of an ordered pair - misc additions   df1stres 32189
                  21.3.4.9  Equivalence relations and classes   ecref 32197
                  21.3.4.10  Supremum - misc additions   supssd 32198
                  21.3.4.11  Finite Sets   imafi2 32200
                  21.3.4.12  Countable Sets   snct 32202
            21.3.5  Real and Complex Numbers   creq0 32224
                  21.3.5.1  Complex operations - misc. additions   creq0 32224
                  21.3.5.2  Ordering on reals - misc additions   lt2addrd 32228
                  21.3.5.3  Extended reals - misc additions   xrlelttric 32229
                  21.3.5.4  Extended nonnegative integers - misc additions   xnn0gt0 32246
                  21.3.5.5  Real number intervals - misc additions   joiniooico 32249
                  21.3.5.6  Finite intervals of integers - misc additions   uzssico 32259
                  21.3.5.7  Half-open integer ranges - misc additions   iundisjfi 32271
                  21.3.5.8  The ` # ` (set size) function - misc additions   hashunif 32282
                  21.3.5.9  The greatest common divisor operator - misc. additions   dvdszzq 32285
                  21.3.5.10  Integers   nnindf 32289
                  21.3.5.11  Decimal numbers   dfdec100 32300
            ​*21.3.6  Decimal expansion   cdp2 32301
                  ​*21.3.6.1  Decimal point   cdp 32318
                  21.3.6.2  Division in the extended real number system   cxdiv 32347
            21.3.7  Words over a set - misc additions   wrdfd 32366
                  21.3.7.1  Splicing words (substring replacement)   splfv3 32386
                  21.3.7.2  Cyclic shift of words   1cshid 32387
            21.3.8  Extensible Structures   ressplusf 32391
                  21.3.8.1  Structure restriction operator   ressplusf 32391
                  21.3.8.2  The opposite group   oppgle 32394
                  21.3.8.3  Posets   ressprs 32397
                  21.3.8.4  Complete lattices   clatp0cl 32410
                  21.3.8.5  Order Theory   cmnt 32412
                  21.3.8.6  Extended reals Structure - misc additions   ax-xrssca 32438
                  21.3.8.7  The extended nonnegative real numbers commutative monoid   xrge0base 32450
            21.3.9  Algebra   abliso 32461
                  21.3.9.1  Monoids Homomorphisms   abliso 32461
                  21.3.9.2  Finitely supported group sums - misc additions   gsumsubg 32465
                  21.3.9.3  Centralizers and centers - misc additions   cntzun 32479
                  21.3.9.4  Totally ordered monoids and groups   comnd 32482
                  21.3.9.5  The symmetric group   symgfcoeu 32510
                  21.3.9.6  Transpositions   pmtridf1o 32520
                  21.3.9.7  Permutation Signs   psgnid 32523
                  21.3.9.8  Permutation cycles   ctocyc 32532
                  21.3.9.9  The Alternating Group   evpmval 32571
                  21.3.9.10  Signum in an ordered monoid   csgns 32584
                  21.3.9.11  The Archimedean property for generic ordered algebraic structures   cinftm 32589
                  21.3.9.12  Semiring left modules   cslmd 32612
                  21.3.9.13  Simple groups   prmsimpcyc 32640
                  21.3.9.14  Rings - misc additions   idomdomd 32641
                  21.3.9.15  Euclidean Domains   ceuf 32655
                  21.3.9.16  Division Rings   ringinveu 32661
                  21.3.9.17  Subfields   sdrgdvcl 32664
                  21.3.9.18  Field extensions generated by a set   cfldgen 32667
                  21.3.9.19  Totally ordered rings and fields   corng 32680
                  21.3.9.20  Ring homomorphisms - misc additions   rhmdvd 32703
                  21.3.9.21  Scalar restriction operation   cresv 32705
                  21.3.9.22  The commutative ring of gaussian integers   gzcrng 32725
                  21.3.9.23  The archimedean ordered field of real numbers   reofld 32726
                  21.3.9.24  The quotient map and quotient modules   qusker 32731
                  21.3.9.25  The ring of integers modulo ` N `   fermltlchr 32749
                  21.3.9.26  Independent sets and families   islinds5 32751
                  ​*21.3.9.27  Subgroup sum / Sumset / Minkowski sum   elgrplsmsn 32771
                  21.3.9.28  The quotient map   qusmul 32786
                  21.3.9.29  Ideals   intlidl 32807
                  21.3.9.30  Prime Ideals   cprmidl 32824
                  21.3.9.31  Maximal Ideals   cmxidl 32846
                  21.3.9.32  The semiring of ideals of a ring   cidlsrg 32885
                  21.3.9.33  Unique factorization domains   cufd 32901
                  21.3.9.34  Associative algebras   asclmulg 32906
                  21.3.9.35  Univariate Polynomials   0ringmon1p 32907
                  21.3.9.36  Polynomial quotient and polynomial remainder   q1pdir 32945
                  21.3.9.37  The subring algebra   sra1r 32954
                  21.3.9.38  Division Ring Extensions   drgext0g 32961
                  21.3.9.39  Vector Spaces   lvecdimfi 32967
                  21.3.9.40  Vector Space Dimension   cldim 32968
            21.3.10  Field Extensions   cfldext 33002
                  21.3.10.1  Algebraic numbers   cirng 33033
                  21.3.10.2  Minimal polynomials   cminply 33042
            21.3.11  Matrices   csmat 33068
                  21.3.11.1  Submatrices   csmat 33068
                  21.3.11.2  Matrix literals   clmat 33086
                  21.3.11.3  Laplace expansion of determinants   mdetpmtr1 33098
            21.3.12  Topology   ist0cld 33108
                  21.3.12.1  Open maps   txomap 33109
                  21.3.12.2  Topology of the unit circle   qtopt1 33110
                  21.3.12.3  Refinements   reff 33114
                  21.3.12.4  Open cover refinement property   ccref 33117
                  21.3.12.5  Lindelöf spaces   cldlf 33127
                  21.3.12.6  Paracompact spaces   cpcmp 33130
                  ​*21.3.12.7  Spectrum of a ring   crspec 33137
                  21.3.12.8  Pseudometrics   cmetid 33161
                  21.3.12.9  Continuity - misc additions   hauseqcn 33173
                  21.3.12.10  Topology of the closed unit interval   elunitge0 33174
                  21.3.12.11  Topology of ` ( RR X. RR ) `   unicls 33178
                  21.3.12.12  Order topology - misc. additions   cnvordtrestixx 33188
                  21.3.12.13  Continuity in topological spaces - misc. additions   mndpluscn 33201
                  21.3.12.14  Topology of the extended nonnegative real numbers ordered monoid   xrge0hmph 33207
                  21.3.12.15  Limits - misc additions   lmlim 33222
                  21.3.12.16  Univariate polynomials   pl1cn 33230
            21.3.13  Uniform Stuctures and Spaces   chcmp 33231
                  21.3.13.1  Hausdorff uniform completion   chcmp 33231
            21.3.14  Topology and algebraic structures   zringnm 33233
                  21.3.14.1  The norm on the ring of the integer numbers   zringnm 33233
                  21.3.14.2  Topological ` ZZ ` -modules   zlm0 33235
                  21.3.14.3  Canonical embedding of the field of the rational numbers into a division ring   cqqh 33247
                  21.3.14.4  Canonical embedding of the real numbers into a complete ordered field   crrh 33268
                  21.3.14.5  Embedding from the extended real numbers into a complete lattice   cxrh 33291
                  21.3.14.6  Canonical embeddings into the ordered field of the real numbers   zrhre 33294
                  ​*21.3.14.7  Topological Manifolds   cmntop 33297
            21.3.15  Real and complex functions   nexple 33302
                  21.3.15.1  Integer powers - misc. additions   nexple 33302
                  21.3.15.2  Indicator Functions   cind 33303
                  21.3.15.3  Extended sum   cesum 33320
            21.3.16  Mixed Function/Constant operation   cofc 33388
            21.3.17  Abstract measure   csiga 33401
                  21.3.17.1  Sigma-Algebra   csiga 33401
                  21.3.17.2  Generated sigma-Algebra   csigagen 33431
                  ​*21.3.17.3  lambda and pi-Systems, Rings of Sets   ispisys 33445
                  21.3.17.4  The Borel algebra on the real numbers   cbrsiga 33474
                  21.3.17.5  Product Sigma-Algebra   csx 33481
                  21.3.17.6  Measures   cmeas 33488
                  21.3.17.7  The counting measure   cntmeas 33519
                  21.3.17.8  The Lebesgue measure - misc additions   voliune 33522
                  21.3.17.9  The Dirac delta measure   cdde 33525
                  21.3.17.10  The 'almost everywhere' relation   cae 33530
                  21.3.17.11  Measurable functions   cmbfm 33542
                  21.3.17.12  Borel Algebra on ` ( RR X. RR ) `   br2base 33563
                  ​*21.3.17.13  Caratheodory's extension theorem   coms 33585
            21.3.18  Integration   itgeq12dv 33620
                  21.3.18.1  Lebesgue integral - misc additions   itgeq12dv 33620
                  21.3.18.2  Bochner integral   citgm 33621
            21.3.19  Euler's partition theorem   oddpwdc 33648
            21.3.20  Sequences defined by strong recursion   csseq 33677
            21.3.21  Fibonacci Numbers   cfib 33690
            21.3.22  Probability   cprb 33701
                  21.3.22.1  Probability Theory   cprb 33701
                  21.3.22.2  Conditional Probabilities   ccprob 33725
                  21.3.22.3  Real-valued Random Variables   crrv 33734
                  21.3.22.4  Preimage set mapping operator   corvc 33749
                  21.3.22.5  Distribution Functions   orvcelval 33762
                  21.3.22.6  Cumulative Distribution Functions   orvclteel 33766
                  21.3.22.7  Probabilities - example   coinfliplem 33772
                  21.3.22.8  Bertrand's Ballot Problem   ballotlemoex 33779
            21.3.23  Signum (sgn or sign) function - misc. additions   sgncl 33832
                  21.3.23.1  Operations on words   ccatmulgnn0dir 33848
            21.3.24  Polynomials with real coefficients - misc additions   plymul02 33852
            21.3.25  Descartes's rule of signs   signspval 33858
                  21.3.25.1  Sign changes in a word over real numbers   signspval 33858
                  21.3.25.2  Counting sign changes in a word over real numbers   signslema 33868
            21.3.26  Number Theory   efcld 33898
                  21.3.26.1  Representations of a number as sums of integers   crepr 33915
                  21.3.26.2  Vinogradov Trigonometric Sums and the Circle Method   cvts 33942
                  21.3.26.3  The Ternary Goldbach Conjecture: Final Statement   ax-hgt749 33951
            21.3.27  Elementary Geometry   cstrkg2d 33971
                  ​*21.3.27.1  Two-dimensional geometry   cstrkg2d 33971
                  21.3.27.2  Outer Five Segment (not used, no need to move to main)   cafs 33976
            ​*21.3.28  LeftPad Project   clpad 33981
      ​​*21.4  Mathbox for Jonathan Ben-Naim
            21.4.1  First-order logic and set theory   bnj170 34004
            21.4.2  Well founded induction and recursion   bnj110 34164
            21.4.3  The existence of a minimal element in certain classes   bnj69 34316
            21.4.4  Well-founded induction   bnj1204 34318
            21.4.5  Well-founded recursion, part 1 of 3   bnj60 34368
            21.4.6  Well-founded recursion, part 2 of 3   bnj1500 34374
            21.4.7  Well-founded recursion, part 3 of 3   bnj1522 34378
      ​21.5  Mathbox for BTernaryTau
            21.5.1  ZF set theory   exdifsn 34379
                  21.5.1.1  Finitism   fineqvrep 34390
            21.5.2  Real and complex numbers   zltp1ne 34394
            21.5.3  Graph theory   lfuhgr 34403
                  21.5.3.1  Acyclic graphs   cacycgr 34428
      ​21.6  Mathbox for Mario Carneiro
            21.6.1  Predicate calculus with all distinct variables   ax-7d 34445
            21.6.2  Miscellaneous stuff   quartfull 34451
            21.6.3  Derangements and the Subfactorial   deranglem 34452
            21.6.4  The Erdős-Szekeres theorem   erdszelem1 34477
            21.6.5  The Kuratowski closure-complement theorem   kur14lem1 34492
            21.6.6  Retracts and sections   cretr 34503
            21.6.7  Path-connected and simply connected spaces   cpconn 34505
            21.6.8  Covering maps   ccvm 34541
            21.6.9  Normal numbers   snmlff 34615
            21.6.10  Godel-sets of formulas - part 1   cgoe 34619
            21.6.11  Godel-sets of formulas - part 2   cgon 34718
            21.6.12  Models of ZF   cgze 34732
            ​*21.6.13  Metamath formal systems   cmcn 34746
            21.6.14  Grammatical formal systems   cm0s 34871
            21.6.15  Models of formal systems   cmuv 34891
            21.6.16  Splitting fields   ccpms 34913
            21.6.17  p-adic number fields   czr 34927
      ​​*21.7  Mathbox for Filip Cernatescu
      ​21.8  Mathbox for Paul Chapman
            21.8.1  Real and complex numbers (cont.)   climuzcnv 34951
            21.8.2  Miscellaneous theorems   elfzm12 34955
      ​21.9  Mathbox for Adrian Ducourtial
            21.9.1  Propositional calculus   currybi 34964
            21.9.2  Clone theory   ccloneop 34965
      ​21.10  Mathbox for Scott Fenton
            21.10.1  ZFC Axioms in primitive form   axextprim 34971
            21.10.2  Untangled classes   untelirr 34978
            21.10.3  Extra propositional calculus theorems   3jaodd 34985
            21.10.4  Misc. Useful Theorems   nepss 34988
            21.10.5  Properties of real and complex numbers   sqdivzi 34998
            21.10.6  Infinite products   iprodefisumlem 35011
            21.10.7  Factorial limits   faclimlem1 35014
            21.10.8  Greatest common divisor and divisibility   gcd32 35020
            21.10.9  Properties of relationships   dftr6 35022
            21.10.10  Properties of functions and mappings   funpsstri 35038
            21.10.11  Set induction (or epsilon induction)   setinds 35051
            21.10.12  Ordinal numbers   elpotr 35054
            21.10.13  Defined equality axioms   axextdfeq 35070
            21.10.14  Hypothesis builders   hbntg 35078
            21.10.15  Well-founded zero, successor, and limits   cwsuc 35083
            21.10.16  Quantifier-free definitions   ctxp 35103
            21.10.17  Alternate ordered pairs   caltop 35229
            21.10.18  Geometry in the Euclidean space   cofs 35255
                  21.10.18.1  Congruence properties   cofs 35255
                  21.10.18.2  Betweenness properties   btwntriv2 35285
                  21.10.18.3  Segment Transportation   ctransport 35302
                  21.10.18.4  Properties relating betweenness and congruence   cifs 35308
                  21.10.18.5  Connectivity of betweenness   btwnconn1lem1 35360
                  21.10.18.6  Segment less than or equal to   csegle 35379
                  21.10.18.7  Outside-of relationship   coutsideof 35392
                  21.10.18.8  Lines and Rays   cline2 35407
            21.10.19  Forward difference   cfwddif 35431
            21.10.20  Rank theorems   rankung 35439
            21.10.21  Hereditarily Finite Sets   chf 35445
      ​21.11  Mathbox for Gino Giotto
            21.11.1  Study of ax-mulf usage.   mpomulex 35460
                  21.11.1.1  Miscellaneous   gg-cnfldex 35467
                  21.11.1.2  Theorems avoiding ax-addf   mpoaddf 35472
                  21.11.1.3  Revision of df-cnfld   gg-dfcnfld 35474
                  21.11.1.4  Replace cnfldmul with mpocnfldmul   gg-cncrng 35487
      ​21.12  Mathbox for Jeff Hankins
            21.12.1  Miscellany   a1i14 35489
            21.12.2  Basic topological facts   topbnd 35513
            21.12.3  Topology of the real numbers   ivthALT 35524
            21.12.4  Refinements   cfne 35525
            21.12.5  Neighborhood bases determine topologies   neibastop1 35548
            21.12.6  Lattice structure of topologies   topmtcl 35552
            21.12.7  Filter bases   fgmin 35559
            21.12.8  Directed sets, nets   tailfval 35561
      ​21.13  Mathbox for Anthony Hart
            21.13.1  Propositional Calculus   tb-ax1 35572
            21.13.2  Predicate Calculus   nalfal 35592
            21.13.3  Miscellaneous single axioms   meran1 35600
            21.13.4  Connective Symmetry   negsym1 35606
      ​21.14  Mathbox for Chen-Pang He
            21.14.1  Ordinal topology   ontopbas 35617
      ​21.15  Mathbox for Jeff Hoffman
            21.15.1  Inferences for finite induction on generic function values   fveleq 35640
            21.15.2  gdc.mm   nnssi2 35644
      ​21.16  Mathbox for Asger C. Ipsen
            21.16.1  Continuous nowhere differentiable functions   dnival 35651
      ​​*21.17  Mathbox for BJ
            *21.17.1  Propositional calculus   bj-mp2c 35720
                  *21.17.1.1  Derived rules of inference   bj-mp2c 35720
                  ​*21.17.1.2  A syntactic theorem   bj-0 35722
                  21.17.1.3  Minimal implicational calculus   bj-a1k 35724
                  ​*21.17.1.4  Positive calculus   bj-syl66ib 35735
                  21.17.1.5  Implication and negation   bj-con2com 35741
                  ​*21.17.1.6  Disjunction   bj-jaoi1 35752
                  ​*21.17.1.7  Logical equivalence   bj-dfbi4 35754
                  21.17.1.8  The conditional operator for propositions   bj-consensus 35759
                  ​*21.17.1.9  Propositional calculus: miscellaneous   bj-imbi12 35764
            ​*21.17.2  Modal logic   bj-axdd2 35774
            ​*21.17.3  Provability logic   cprvb 35779
            ​*21.17.4  First-order logic   bj-genr 35788
                  21.17.4.1  Adding ax-gen   bj-genr 35788
                  21.17.4.2  Adding ax-4   bj-2alim 35792
                  21.17.4.3  Adding ax-5   bj-ax12wlem 35825
                  21.17.4.4  Equality and substitution   bj-ssbeq 35834
                  21.17.4.5  Adding ax-6   bj-spimvwt 35850
                  21.17.4.6  Adding ax-7   bj-cbvexw 35857
                  21.17.4.7  Membership predicate, ax-8 and ax-9   bj-ax89 35859
                  21.17.4.8  Adding ax-11   bj-alcomexcom 35862
                  21.17.4.9  Adding ax-12   axc11n11 35864
                  21.17.4.10  Nonfreeness   wnnf 35905
                  21.17.4.11  Adding ax-13   bj-axc10 35965
                  ​*21.17.4.12  Removing dependencies on ax-13 (and ax-11)   bj-axc10v 35975
                  ​*21.17.4.13  Distinct var metavariables   bj-hbaeb2 36000
                  ​*21.17.4.14  Around ~ equsal   bj-equsal1t 36004
                  ​*21.17.4.15  Some Principia Mathematica proofs   stdpc5t 36009
                  21.17.4.16  Alternate definition of substitution   bj-sbsb 36019
                  21.17.4.17  Lemmas for substitution   bj-sbf3 36021
                  21.17.4.18  Existential uniqueness   bj-eu3f 36024
                  ​*21.17.4.19  First-order logic: miscellaneous   bj-sblem1 36025
            21.17.5  Set theory   eliminable1 36042
                  ​*21.17.5.1  Eliminability of class terms   eliminable1 36042
                  ​*21.17.5.2  Classes without the axiom of extensionality   bj-denoteslem 36054
                  21.17.5.3  Characterization among sets versus among classes   elelb 36081
                  ​*21.17.5.4  The nonfreeness quantifier for classes   bj-nfcsym 36083
                  ​*21.17.5.5  Lemmas for class substitution   bj-sbeqALT 36084
                  21.17.5.6  Removing some axiom requirements and disjoint variable conditions   bj-exlimvmpi 36095
                  ​*21.17.5.7  Class abstractions   bj-elabd2ALT 36109
                  21.17.5.8  Generalized class abstractions   bj-cgab 36117
                  ​*21.17.5.9  Restricted nonfreeness   wrnf 36125
                  ​*21.17.5.10  Russell's paradox   bj-ru0 36127
                  21.17.5.11  Curry's paradox in set theory   currysetlem 36130
                  ​*21.17.5.12  Some disjointness results   bj-n0i 36136
                  ​*21.17.5.13  Complements on direct products   bj-xpimasn 36140
                  ​*21.17.5.14  "Singletonization" and tagging   bj-snsetex 36148
                  ​*21.17.5.15  Tuples of classes   bj-cproj 36175
                  ​*21.17.5.16  Set theory: elementary operations relative to a universe   bj-rcleqf 36210
                  ​*21.17.5.17  Axioms for finite unions   bj-abex 36215
                  ​*21.17.5.18  Set theory: miscellaneous   eleq2w2ALT 36232
                  ​*21.17.5.19  Evaluation at a class   bj-evaleq 36257
                  21.17.5.20  Elementwise operations   celwise 36264
                  ​*21.17.5.21  Elementwise intersection (families of sets induced on a subset)   bj-rest00 36266
                  21.17.5.22  Moore collections (complements)   bj-raldifsn 36285
                  21.17.5.23  Maps-to notation for functions with three arguments   bj-0nelmpt 36301
                  ​*21.17.5.24  Currying   csethom 36307
                  ​*21.17.5.25  Setting components of extensible structures   cstrset 36319
            ​*21.17.6  Extended real and complex numbers, real and complex projective lines   bj-nfald 36322
                  21.17.6.1  Complements on class abstractions of ordered pairs and binary relations   bj-nfald 36322
                  ​*21.17.6.2  Identity relation (complements)   bj-opabssvv 36335
                  ​*21.17.6.3  Functionalized identity (diagonal in a Cartesian square)   cdiag2 36357
                  ​*21.17.6.4  Direct image and inverse image   cimdir 36363
                  ​*21.17.6.5  Extended numbers and projective lines as sets   cfractemp 36381
                  ​*21.17.6.6  Addition and opposite   caddcc 36422
                  ​*21.17.6.7  Order relation on the extended reals   cltxr 36426
                  ​*21.17.6.8  Argument, multiplication and inverse   carg 36428
                  21.17.6.9  The canonical bijection from the finite ordinals   ciomnn 36434
                  21.17.6.10  Divisibility   cnnbar 36445
            ​*21.17.7  Monoids   bj-smgrpssmgm 36453
                  ​*21.17.7.1  Finite sums in monoids   cfinsum 36468
            ​*21.17.8  Affine, Euclidean, and Cartesian geometry   bj-fvimacnv0 36471
                  *21.17.8.1  Real vector spaces   bj-fvimacnv0 36471
                  ​*21.17.8.2  Complex numbers (supplements)   bj-subcom 36493
                  ​*21.17.8.3  Barycentric coordinates   bj-bary1lem 36495
            21.17.9  Monoid of endomorphisms   cend 36498
      ​21.18  Mathbox for Jim Kingdon
                  21.18.0.1  Circle constant   taupilem3 36504
                  21.18.0.2  Number theory   dfgcd3 36509
                  21.18.0.3  Real numbers   irrdifflemf 36510
      ​21.19  Mathbox for ML
            21.19.1  Miscellaneous   csbrecsg 36513
            21.19.2  Cartesian exponentiation   cfinxp 36568
            21.19.3  Topology   iunctb2 36588
                  ​*21.19.3.1  Pi-base theorems   pibp16 36598
      ​21.20  Mathbox for Wolf Lammen
            21.20.1  1. Bootstrapping   wl-section-boot 36607
            21.20.2  Implication chains   wl-section-impchain 36631
            21.20.3  Theorems around the conditional operator   wl-ifp-ncond1 36649
            21.20.4  Alternative development of hadd, cadd   wl-df-3xor 36653
            21.20.5  An alternative axiom ~ ax-13   ax-wl-13v 36678
            21.20.6  Other stuff   wl-mps 36680
      ​21.21  Mathbox for Brendan Leahy
      ​21.22  Mathbox for Jeff Madsen
            21.22.1  Logic and set theory   unirep 36886
            21.22.2  Real and complex numbers; integers   filbcmb 36912
            21.22.3  Sequences and sums   sdclem2 36914
            21.22.4  Topology   subspopn 36924
            21.22.5  Metric spaces   metf1o 36927
            21.22.6  Continuous maps and homeomorphisms   constcncf 36934
            21.22.7  Boundedness   ctotbnd 36938
            21.22.8  Isometries   cismty 36970
            21.22.9  Heine-Borel Theorem   heibor1lem 36981
            21.22.10  Banach Fixed Point Theorem   bfplem1 36994
            21.22.11  Euclidean space   crrn 36997
            21.22.12  Intervals (continued)   ismrer1 37010
            21.22.13  Operation properties   cass 37014
            21.22.14  Groups and related structures   cmagm 37020
            21.22.15  Group homomorphism and isomorphism   cghomOLD 37055
            21.22.16  Rings   crngo 37066
            21.22.17  Division Rings   cdrng 37120
            21.22.18  Ring homomorphisms   crngohom 37132
            21.22.19  Commutative rings   ccm2 37161
            21.22.20  Ideals   cidl 37179
            21.22.21  Prime rings and integral domains   cprrng 37218
            21.22.22  Ideal generators   cigen 37231
      ​21.23  Mathbox for Giovanni Mascellani
            ​*21.23.1  Tools for automatic proof building   efald2 37250
            ​*21.23.2  Tseitin axioms   fald 37301
            ​*21.23.3  Equality deductions   iuneq2f 37328
            ​*21.23.4  Miscellanea   orcomdd 37339
      ​21.24  Mathbox for Peter Mazsa
            21.24.1  Notations   cxrn 37346
            21.24.2  Preparatory theorems   el2v1 37389
            21.24.3  Range Cartesian product   df-xrn 37545
            21.24.4  Cosets by ` R `   df-coss 37585
            21.24.5  Relations   df-rels 37659
            21.24.6  Subset relations   df-ssr 37672
            21.24.7  Reflexivity   df-refs 37684
            21.24.8  Converse reflexivity   df-cnvrefs 37699
            21.24.9  Symmetry   df-syms 37716
            21.24.10  Reflexivity and symmetry   symrefref2 37737
            21.24.11  Transitivity   df-trs 37746
            21.24.12  Equivalence relations   df-eqvrels 37758
            21.24.13  Redundancy   df-redunds 37797
            21.24.14  Domain quotients   df-dmqss 37812
            21.24.15  Equivalence relations on domain quotients   df-ers 37837
            21.24.16  Functions   df-funss 37854
            21.24.17  Disjoints vs. converse functions   df-disjss 37877
            21.24.18  Antisymmetry   df-antisymrel 37934
            21.24.19  Partitions: disjoints on domain quotients   df-parts 37939
            21.24.20  Partition-Equivalence Theorems   disjim 37955
      ​21.25  Mathbox for Rodolfo Medina
            21.25.1  Partitions   prtlem60 38027
      ​​*21.26  Mathbox for Norm Megill
            *21.26.1  Obsolete schemes ax-c4,c5,c7,c10,c11,c11n,c15,c9,c14,c16   ax-c5 38057
            ​*21.26.2  Rederive new axioms ax-4, ax-10, ax-6, ax-12, ax-13 from old   axc5 38067
            ​*21.26.3  Legacy theorems using obsolete axioms   ax5ALT 38081
            21.26.4  Experiments with weak deduction theorem   elimhyps 38135
            21.26.5  Miscellanea   cnaddcom 38146
            21.26.6  Atoms, hyperplanes, and covering in a left vector space (or module)   clsa 38148
            21.26.7  Functionals and kernels of a left vector space (or module)   clfn 38231
            21.26.8  Opposite rings and dual vector spaces   cld 38297
            21.26.9  Ortholattices and orthomodular lattices   cops 38346
            21.26.10  Atomic lattices with covering property   ccvr 38436
            21.26.11  Hilbert lattices   chlt 38524
            21.26.12  Projective geometries based on Hilbert lattices   clln 38666
            21.26.13  Construction of a vector space from a Hilbert lattice   cdlema1N 38966
            21.26.14  Construction of involution and inner product from a Hilbert lattice   clpoN 40655
      ​21.27  Mathbox for metakunt
            21.27.1  Commutative Semiring   ccsrg 41141
            21.27.2  General helpful statements   leexp1ad 41144
            21.27.3  Some gcd and lcm results   12gcd5e1 41175
            21.27.4  Least common multiple inequality theorem   3factsumint1 41193
            21.27.5  Logarithm inequalities   3exp7 41225
            21.27.6  Miscellaneous results for AKS formalisation   intlewftc 41233
            21.27.7  Sticks and stones   sticksstones1 41269
            21.27.8  Permutation results   metakunt1 41292
            21.27.9  Unused lemmas scheduled for deletion   andiff 41326
      ​21.28  Mathbox for Steven Nguyen
            21.28.1  Utility theorems   ioin9i8 41331
            21.28.2  Structures   nelsubginvcld 41377
            ​*21.28.3  Arithmetic theorems   c0exALT 41476
            21.28.4  Exponents and divisibility   oexpreposd 41515
            21.28.5  Real subtraction   cresub 41541
            ​*21.28.6  Projective spaces   cprjsp 41646
            21.28.7  Basic reductions for Fermat's Last Theorem   dffltz 41679
            ​*21.28.8  Exemplar theorems   iddii 41709
      ​21.29  Mathbox for Igor Ieskov
      ​21.30  Mathbox for OpenAI
      ​21.31  Mathbox for Stefan O'Rear
            21.31.1  Additional elementary logic and set theory   moxfr 41733
            21.31.2  Additional theory of functions   imaiinfv 41734
            21.31.3  Additional topology   elrfi 41735
            21.31.4  Characterization of closure operators. Kuratowski closure axioms   ismrcd1 41739
            21.31.5  Algebraic closure systems   cnacs 41743
            21.31.6  Miscellanea 1. Map utilities   constmap 41754
            21.31.7  Miscellanea for polynomials   mptfcl 41761
            21.31.8  Multivariate polynomials over the integers   cmzpcl 41762
            21.31.9  Miscellanea for Diophantine sets 1   coeq0i 41794
            21.31.10  Diophantine sets 1: definitions   cdioph 41796
            21.31.11  Diophantine sets 2 miscellanea   ellz1 41808
            21.31.12  Diophantine sets 2: union and intersection. Monotone Boolean algebra   diophin 41813
            21.31.13  Diophantine sets 3: construction   diophrex 41816
            21.31.14  Diophantine sets 4 miscellanea   2sbcrex 41825
            21.31.15  Diophantine sets 4: Quantification   rexrabdioph 41835
            21.31.16  Diophantine sets 5: Arithmetic sets   rabdiophlem1 41842
            21.31.17  Diophantine sets 6: reusability. renumbering of variables   eldioph4b 41852
            21.31.18  Pigeonhole Principle and cardinality helpers   fphpd 41857
            21.31.19  A non-closed set of reals is infinite   rencldnfilem 41861
            21.31.20  Lagrange's rational approximation theorem   irrapxlem1 41863
            21.31.21  Pell equations 1: A nontrivial solution always exists   pellexlem1 41870
            21.31.22  Pell equations 2: Algebraic number theory of the solution set   csquarenn 41877
            21.31.23  Pell equations 3: characterizing fundamental solution   infmrgelbi 41919
            ​*21.31.24  Logarithm laws generalized to an arbitrary base   reglogcl 41931
            21.31.25  Pell equations 4: the positive solution group is infinite cyclic   pellfund14 41939
            21.31.26  X and Y sequences 1: Definition and recurrence laws   crmx 41941
            21.31.27  Ordering and induction lemmas for the integers   monotuz 41983
            21.31.28  X and Y sequences 2: Order properties   rmxypos 41989
            21.31.29  Congruential equations   congtr 42007
            21.31.30  Alternating congruential equations   acongid 42017
            21.31.31  Additional theorems on integer divisibility   coprmdvdsb 42027
            21.31.32  X and Y sequences 3: Divisibility properties   jm2.18 42030
            21.31.33  X and Y sequences 4: Diophantine representability of Y   jm2.27a 42047
            21.31.34  X and Y sequences 5: Diophantine representability of X, ^, _C   rmxdiophlem 42057
            21.31.35  Uncategorized stuff not associated with a major project   setindtr 42066
            21.31.36  More equivalents of the Axiom of Choice   axac10 42075
            21.31.37  Finitely generated left modules   clfig 42112
            21.31.38  Noetherian left modules I   clnm 42120
            21.31.39  Addenda for structure powers   pwssplit4 42134
            21.31.40  Every set admits a group structure iff choice   unxpwdom3 42140
            21.31.41  Noetherian rings and left modules II   clnr 42154
            21.31.42  Hilbert's Basis Theorem   cldgis 42166
            21.31.43  Additional material on polynomials [DEPRECATED]   cmnc 42176
            21.31.44  Degree and minimal polynomial of algebraic numbers   cdgraa 42185
            21.31.45  Algebraic integers I   citgo 42202
            21.31.46  Endomorphism algebra   cmend 42220
            21.31.47  Cyclic groups and order   idomrootle 42240
            21.31.48  Cyclotomic polynomials   ccytp 42247
            21.31.49  Miscellaneous topology   fgraphopab 42255
      ​21.32  Mathbox for Noam Pasman
      ​21.33  Mathbox for Jon Pennant
      ​21.34  Mathbox for Richard Penner
            21.34.1  Set Theory and Ordinal Numbers   uniel 42269
            21.34.2  Natural addition of Cantor normal forms   oawordex2 42379
            21.34.3  Surreal Contributions   abeqabi 42462
            21.34.4  Short Studies   nlimsuc 42495
                  21.34.4.1  Additional work on conditional logical operator   ifpan123g 42513
                  21.34.4.2  Sophisms   rp-fakeimass 42566
                  ​*21.34.4.3  Finite Sets   rp-isfinite5 42571
                  21.34.4.4  General Observations   intabssd 42573
                  21.34.4.5  Infinite Sets   pwelg 42614
                  ​*21.34.4.6  Finite intersection property   fipjust 42619
                  21.34.4.7  RP ADDTO: Subclasses and subsets   rababg 42628
                  21.34.4.8  RP ADDTO: The intersection of a class   elinintab 42629
                  21.34.4.9  RP ADDTO: Theorems requiring subset and intersection existence   elinintrab 42631
                  21.34.4.10  RP ADDTO: Relations   xpinintabd 42634
                  ​*21.34.4.11  RP ADDTO: Functions   elmapintab 42650
                  ​*21.34.4.12  RP ADDTO: Finite induction (for finite ordinals)   cnvcnvintabd 42654
                  21.34.4.13  RP ADDTO: First and second members of an ordered pair   elcnvlem 42655
                  21.34.4.14  RP ADDTO: The reflexive and transitive properties of relations   undmrnresiss 42658
                  21.34.4.15  RP ADDTO: Basic properties of closures   cleq2lem 42662
                  21.34.4.16  RP REPLACE: Definitions and basic properties of transitive closures   trcleq2lemRP 42684
                  ​*21.34.4.17  Additions for square root; absolute value   sqrtcvallem1 42685
            21.34.5  Additional statements on relations and subclasses   al3im 42701
                  21.34.5.1  Transitive relations (not to be confused with transitive classes).   trrelind 42719
                  21.34.5.2  Reflexive closures   crcl 42726
                  ​*21.34.5.3  Finite relationship composition.   relexp2 42731
                  21.34.5.4  Transitive closure of a relation   dftrcl3 42774
                  ​*21.34.5.5  Adapted from Frege   frege77d 42800
            ​*21.34.6  Propositions from _Begriffsschrift_   dfxor4 42820
                  *21.34.6.1  _Begriffsschrift_ Chapter I   dfxor4 42820
                  ​*21.34.6.2  _Begriffsschrift_ Notation hints   whe 42826
                  21.34.6.3  _Begriffsschrift_ Chapter II Implication   ax-frege1 42844
                  21.34.6.4  _Begriffsschrift_ Chapter II Implication and Negation   axfrege28 42883
                  ​*21.34.6.5  _Begriffsschrift_ Chapter II with logical equivalence   axfrege52a 42910
                  21.34.6.6  _Begriffsschrift_ Chapter II with equivalence of sets   axfrege52c 42941
                  ​*21.34.6.7  _Begriffsschrift_ Chapter II with equivalence of classes   frege53c 42968
                  ​*21.34.6.8  _Begriffsschrift_ Chapter III Properties hereditary in a sequence   dffrege69 42986
                  ​*21.34.6.9  _Begriffsschrift_ Chapter III Following in a sequence   dffrege76 42993
                  ​*21.34.6.10  _Begriffsschrift_ Chapter III Member of sequence   dffrege99 43016
                  ​*21.34.6.11  _Begriffsschrift_ Chapter III Single-valued procedures   dffrege115 43032
            ​*21.34.7  Exploring Topology via Seifert and Threlfall   enrelmap 43051
                  *21.34.7.1  Equinumerosity of sets of relations and maps   enrelmap 43051
                  ​*21.34.7.2  Generic Pseudoclosure Spaces, Pseudointerior Spaces, and Pseudoneighborhoods   or3or 43077
                  ​*21.34.7.3  Generic Neighborhood Spaces   gneispa 43184
            ​*21.34.8  Exploring Higher Homotopy via Kerodon   k0004lem1 43201
                  *21.34.8.1  Simplicial Sets   k0004lem1 43201
      ​21.35  Mathbox for Stanislas Polu
            21.35.1  IMO Problems   wwlemuld 43210
                  21.35.1.1  IMO 1972 B2   wwlemuld 43210
            ​*21.35.2  INT Inequalities Proof Generator   int-addcomd 43228
            ​*21.35.3  N-Digit Addition Proof Generator   unitadd 43250
            21.35.4  AM-GM (for k = 2,3,4)   gsumws3 43251
      ​21.36  Mathbox for Rohan Ridenour
            21.36.1  Misc   spALT 43256
            21.36.2  Monoid rings   cmnring 43268
            21.36.3  Shorter primitive equivalent of ax-groth   gru0eld 43291
                  21.36.3.1  Grothendieck universes are closed under collection   gru0eld 43291
                  21.36.3.2  Minimal universes   ismnu 43323
                  21.36.3.3  Primitive equivalent of ax-groth   expandan 43350
      ​21.37  Mathbox for Steve Rodriguez
            21.37.1  Miscellanea   nanorxor 43367
            21.37.2  Ratio test for infinite series convergence and divergence   dvgrat 43374
            21.37.3  Multiples   reldvds 43377
            21.37.4  Function operations   caofcan 43385
            21.37.5  Calculus   lhe4.4ex1a 43391
            21.37.6  The generalized binomial coefficient operation   cbcc 43398
            21.37.7  Binomial series   uzmptshftfval 43408
      ​21.38  Mathbox for Andrew Salmon
            21.38.1  Principia Mathematica * 10   pm10.12 43420
            21.38.2  Principia Mathematica * 11   2alanimi 43434
            21.38.3  Predicate Calculus   sbeqal1 43460
            21.38.4  Principia Mathematica * 13 and * 14   pm13.13a 43469
            21.38.5  Set Theory   elnev 43500
            21.38.6  Arithmetic   addcomgi 43518
            21.38.7  Geometry   cplusr 43519
      ​​*21.39  Mathbox for Alan Sare
            21.39.1  Auxiliary theorems for the Virtual Deduction tool   idiALT 43541
            21.39.2  Supplementary unification deductions   bi1imp 43545
            21.39.3  Conventional Metamath proofs, some derived from VD proofs   iidn3 43565
            21.39.4  What is Virtual Deduction?   wvd1 43633
            21.39.5  Virtual Deduction Theorems   df-vd1 43634
            21.39.6  Theorems proved using Virtual Deduction   trsspwALT 43882
            21.39.7  Theorems proved using Virtual Deduction with mmj2 assistance   simplbi2VD 43910
            21.39.8  Virtual Deduction transcriptions of textbook proofs   sb5ALTVD 43977
            21.39.9  Theorems proved using conjunction-form Virtual Deduction   elpwgdedVD 43981
            21.39.10  Theorems with a VD proof in conventional notation derived from a VD proof   suctrALT3 43988
            ​*21.39.11  Theorems with a proof in conventional notation derived from a VD proof   notnotrALT2 43991
      ​21.40  Mathbox for Glauco Siliprandi
            21.40.1  Miscellanea   evth2f 44002
            21.40.2  Functions   feq1dd 44166
            21.40.3  Ordering on real numbers - Real and complex numbers basic operations   sub2times 44282
            21.40.4  Real intervals   gtnelioc 44504
            21.40.5  Finite sums   fsummulc1f 44587
            21.40.6  Finite multiplication of numbers and finite multiplication of functions   fmul01 44596
            21.40.7  Limits   clim1fr1 44617
                  21.40.7.1  Inferior limit (lim inf)   clsi 44767
                  ​*21.40.7.2  Limits for sequences of extended real numbers   clsxlim 44834
            21.40.8  Trigonometry   coseq0 44880
            21.40.9  Continuous Functions   mulcncff 44886
            21.40.10  Derivatives   dvsinexp 44927
            21.40.11  Integrals   itgsin0pilem1 44966
            21.40.12  Stone Weierstrass theorem - real version   stoweidlem1 45017
            21.40.13  Wallis' product for π   wallispilem1 45081
            21.40.14  Stirling's approximation formula for ` n ` factorial   stirlinglem1 45090
            21.40.15  Dirichlet kernel   dirkerval 45107
            21.40.16  Fourier Series   fourierdlem1 45124
            21.40.17  e is transcendental   elaa2lem 45249
            21.40.18  n-dimensional Euclidean space   rrxtopn 45300
            21.40.19  Basic measure theory   csalg 45324
                  ​*21.40.19.1  σ-Algebras   csalg 45324
                  21.40.19.2  Sum of nonnegative extended reals   csumge0 45378
                  ​*21.40.19.3  Measures   cmea 45465
                  ​*21.40.19.4  Outer measures and Caratheodory's construction   come 45505
                  ​*21.40.19.5  Lebesgue measure on n-dimensional Real numbers   covoln 45552
                  ​*21.40.19.6  Measurable functions   csmblfn 45711
      ​21.41  Mathbox for Saveliy Skresanov
            21.41.1  Ceva's theorem   sigarval 45866
            21.41.2  Simple groups   simpcntrab 45886
      ​21.42  Mathbox for Ender Ting
            21.42.1  Increasing sequences and subsequences   et-ltneverrefl 45887
      ​21.43  Mathbox for Jarvin Udandy
      ​21.44  Mathbox for Adhemar
            ​*21.44.1  Minimal implicational calculus   adh-minim 46011
      ​21.45  Mathbox for Alexander van der Vekens
            21.45.1  General auxiliary theorems (1)   n0nsn2el 46035
                  21.45.1.1  Unordered and ordered pairs - extension for singletons   n0nsn2el 46035
                  21.45.1.2  Unordered and ordered pairs - extension for unordered pairs   elprneb 46039
                  21.45.1.3  Unordered and ordered pairs - extension for ordered pairs   oppr 46040
                  21.45.1.4  Relations - extension   eubrv 46045
                  21.45.1.5  Definite description binder (inverted iota) - extension   iota0def 46048
                  21.45.1.6  Functions - extension   fveqvfvv 46050
            21.45.2  Alternative for Russell's definition of a description binder   caiota 46091
            21.45.3  Double restricted existential uniqueness   r19.32 46106
                  21.45.3.1  Restricted quantification (extension)   r19.32 46106
                  21.45.3.2  Restricted uniqueness and "at most one" quantification   reuf1odnf 46115
                  21.45.3.3  Analogs to Existential uniqueness (double quantification)   2reu3 46118
                  21.45.3.4  Additional theorems for double restricted existential uniqueness   2reu8i 46121
            ​*21.45.4  Alternative definitions of function and operation values   wdfat 46124
                  21.45.4.1  Restricted quantification (extension)   ralbinrald 46130
                  21.45.4.2  The universal class (extension)   nvelim 46131
                  21.45.4.3  Introduce the Axiom of Power Sets (extension)   alneu 46132
                  21.45.4.4  Predicate "defined at"   dfateq12d 46134
                  21.45.4.5  Alternative definition of the value of a function   dfafv2 46140
                  21.45.4.6  Alternative definition of the value of an operation   aoveq123d 46186
            ​*21.45.5  Alternative definitions of function values (2)   cafv2 46216
            21.45.6  General auxiliary theorems (2)   an4com24 46276
                  21.45.6.1  Logical conjunction - extension   an4com24 46276
                  21.45.6.2  Abbreviated conjunction and disjunction of three wff's - extension   3an4ancom24 46277
                  21.45.6.3  Negated membership (alternative)   cnelbr 46279
                  21.45.6.4  The empty set - extension   ralralimp 46286
                  21.45.6.5  Indexed union and intersection - extension   otiunsndisjX 46287
                  21.45.6.6  Functions - extension   fvifeq 46288
                  21.45.6.7  Maps-to notation - extension   fvmptrab 46300
                  21.45.6.8  Subtraction - extension   cnambpcma 46302
                  21.45.6.9  Ordering on reals (cont.) - extension   leaddsuble 46305
                  21.45.6.10  Imaginary and complex number properties - extension   readdcnnred 46311
                  21.45.6.11  Nonnegative integers (as a subset of complex numbers) - extension   nn0resubcl 46316
                  21.45.6.12  Integers (as a subset of complex numbers) - extension   zgeltp1eq 46317
                  21.45.6.13  Decimal arithmetic - extension   1t10e1p1e11 46318
                  21.45.6.14  Upper sets of integers - extension   eluzge0nn0 46320
                  21.45.6.15  Infinity and the extended real number system (cont.) - extension   nltle2tri 46321
                  21.45.6.16  Finite intervals of integers - extension   ssfz12 46322
                  21.45.6.17  Half-open integer ranges - extension   fzopred 46330
                  21.45.6.18  The modulo (remainder) operation - extension   m1mod0mod1 46337
                  21.45.6.19  The infinite sequence builder "seq"   smonoord 46339
                  21.45.6.20  Finite and infinite sums - extension   fsummsndifre 46340
                  21.45.6.21  Extensible structures - extension   setsidel 46344
            ​*21.45.7  Preimages of function values   preimafvsnel 46347
            ​*21.45.8  Partitions of real intervals   ciccp 46381
            21.45.9  Shifting functions with an integer range domain   fargshiftfv 46407
            21.45.10  Words over a set (extension)   lswn0 46412
                  21.45.10.1  Last symbol of a word - extension   lswn0 46412
            21.45.11  Unordered pairs   wich 46413
                  21.45.11.1  Interchangeable setvar variables   wich 46413
                  21.45.11.2  Set of unordered pairs   sprid 46442
                  ​*21.45.11.3  Proper (unordered) pairs   prpair 46469
                  21.45.11.4  Set of proper unordered pairs   cprpr 46480
            21.45.12  Number theory (extension)   cfmtno 46495
                  ​*21.45.12.1  Fermat numbers   cfmtno 46495
                  ​*21.45.12.2  Mersenne primes   m2prm 46559
                  21.45.12.3  Proth's theorem   modexp2m1d 46580
                  21.45.12.4  Solutions of quadratic equations   quad1 46588
            ​*21.45.13  Even and odd numbers   ceven 46592
                  21.45.13.1  Definitions and basic properties   ceven 46592
                  21.45.13.2  Alternate definitions using the "divides" relation   dfeven2 46617
                  21.45.13.3  Alternate definitions using the "modulo" operation   dfeven3 46626
                  21.45.13.4  Alternate definitions using the "gcd" operation   iseven5 46632
                  21.45.13.5  Theorems of part 5 revised   zneoALTV 46637
                  21.45.13.6  Theorems of part 6 revised   odd2np1ALTV 46642
                  21.45.13.7  Theorems of AV's mathbox revised   0evenALTV 46656
                  21.45.13.8  Additional theorems   epoo 46671
                  21.45.13.9  Perfect Number Theorem (revised)   perfectALTVlem1 46689
            21.45.14  Number theory (extension 2)   cfppr 46692
                  ​*21.45.14.1  Fermat pseudoprimes   cfppr 46692
                  ​*21.45.14.2  Goldbach's conjectures   cgbe 46713
            21.45.15  Graph theory (extension)   cgrisom 46786
                  ​*21.45.15.1  Isomorphic graphs   cgrisom 46786
                  21.45.15.2  Loop-free graphs - extension   1hegrlfgr 46810
                  21.45.15.3  Walks - extension   cupwlks 46811
                  21.45.15.4  Edges of graphs expressed as sets of unordered pairs   upgredgssspr 46821
            21.45.16  Monoids (extension)   ovn0dmfun 46834
                  21.45.16.1  Auxiliary theorems   ovn0dmfun 46834
                  21.45.16.2  Magmas, Semigroups and Monoids (extension)   plusfreseq 46842
                  21.45.16.3  Examples and counterexamples for magmas, semigroups and monoids (extension)   opmpoismgm 46845
                  21.45.16.4  Group sum operation (extension 1)   gsumsplit2f 46858
            ​*21.45.17  Magmas and internal binary operations (alternate approach)   ccllaw 46861
                  *21.45.17.1  Laws for internal binary operations   ccllaw 46861
                  ​*21.45.17.2  Internal binary operations   cintop 46874
                  21.45.17.3  Alternative definitions for magmas and semigroups   cmgm2 46893
            21.45.18  Categories (extension)   idfusubc0 46907
                  21.45.18.1  Subcategories (extension)   idfusubc0 46907
            21.45.19  Rings (extension)   lmod0rng 46910
                  21.45.19.1  Nonzero rings (extension)   lmod0rng 46910
                  21.45.19.2  Ideals as non-unital rings   lidldomn1 46913
                  21.45.19.3  The non-unital ring of even integers   0even 46919
                  21.45.19.4  A constructed not unital ring   cznrnglem 46941
                  ​*21.45.19.5  The category of non-unital rings   crngc 46945
                  ​*21.45.19.6  The category of (unital) rings   cringc 46991
                  21.45.19.7  Subcategories of the category of rings   srhmsubclem1 47061
            21.45.20  Basic algebraic structures (extension)   opeliun2xp 47098
                  21.45.20.1  Auxiliary theorems   opeliun2xp 47098
                  21.45.20.2  The binomial coefficient operation (extension)   bcpascm1 47117
                  21.45.20.3  The ` ZZ `-module ` ZZ X. ZZ `   zlmodzxzlmod 47120
                  21.45.20.4  Group sum operation (extension 2)   mgpsumunsn 47127
                  21.45.20.5  Symmetric groups (extension)   exple2lt6 47130
                  21.45.20.6  Divisibility (extension)   invginvrid 47133
                  21.45.20.7  The support of functions (extension)   rmsupp0 47134
                  21.45.20.8  Finitely supported functions (extension)   rmsuppfi 47139
                  21.45.20.9  Left modules (extension)   lmodvsmdi 47148
                  21.45.20.10  Associative algebras (extension)   assaascl0 47150
                  21.45.20.11  Univariate polynomials (extension)   ply1vr1smo 47152
                  21.45.20.12  Univariate polynomials (examples)   linply1 47163
            21.45.21  Linear algebra (extension)   cdmatalt 47166
                  ​*21.45.21.1  The subalgebras of diagonal and scalar matrices (extension)   cdmatalt 47166
                  ​*21.45.21.2  Linear combinations   clinc 47174
                  ​*21.45.21.3  Linear independence   clininds 47210
                  21.45.21.4  Simple left modules and the ` ZZ `-module   lmod1lem1 47257
                  21.45.21.5  Differences between (left) modules and (left) vector spaces   lvecpsslmod 47277
            21.45.22  Complexity theory   suppdm 47280
                  21.45.22.1  Auxiliary theorems   suppdm 47280
                  21.45.22.2  The modulo (remainder) operation (extension)   fldivmod 47293
                  21.45.22.3  Even and odd integers   nn0onn0ex 47298
                  21.45.22.4  The natural logarithm on complex numbers (extension)   logcxp0 47310
                  21.45.22.5  Division of functions   cfdiv 47312
                  21.45.22.6  Upper bounds   cbigo 47322
                  21.45.22.7  Logarithm to an arbitrary base (extension)   rege1logbrege0 47333
                  ​*21.45.22.8  The binary logarithm   fldivexpfllog2 47340
                  21.45.22.9  Binary length   cblen 47344
                  ​*21.45.22.10  Digits   cdig 47370
                  21.45.22.11  Nonnegative integer as sum of its shifted digits   dignn0flhalflem1 47390
                  21.45.22.12  Algorithms for the multiplication of nonnegative integers   nn0mulfsum 47399
                  ​*21.45.22.13  N-ary functions   cnaryf 47401
                  ​*21.45.22.14  The Ackermann function   citco 47432
            21.45.23  Elementary geometry (extension)   fv1prop 47474
                  21.45.23.1  Auxiliary theorems   fv1prop 47474
                  21.45.23.2  Real euclidean space of dimension 2   rrx2pxel 47486
                  21.45.23.3  Spheres and lines in real Euclidean spaces   cline 47502
      ​21.46  Mathbox for Zhi Wang
            21.46.1  Propositional calculus   pm4.71da 47564
            21.46.2  Predicate calculus with equality   dtrucor3 47573
                  21.46.2.1  Axiom scheme ax-5 (Distinctness)   dtrucor3 47573
            21.46.3  ZF Set Theory - start with the Axiom of Extensionality   ralbidb 47574
                  21.46.3.1  Restricted quantification   ralbidb 47574
                  21.46.3.2  The empty set   ssdisjd 47581
                  21.46.3.3  Unordered and ordered pairs   vsn 47585
                  21.46.3.4  The union of a class   unilbss 47591
            21.46.4  ZF Set Theory - add the Axiom of Replacement   inpw 47592
                  21.46.4.1  Theorems requiring subset and intersection existence   inpw 47592
            21.46.5  ZF Set Theory - add the Axiom of Power Sets   mof0 47593
                  21.46.5.1  Functions   mof0 47593
                  21.46.5.2  Operations   fvconstr 47611
            21.46.6  ZF Set Theory - add the Axiom of Union   fvconst0ci 47614
                  21.46.6.1  Equinumerosity   fvconst0ci 47614
            21.46.7  Order sets   iccin 47618
                  21.46.7.1  Real number intervals   iccin 47618
            21.46.8  Moore spaces   mreuniss 47621
            ​*21.46.9  Topology   clduni 47622
                  21.46.9.1  Closure and interior   clduni 47622
                  21.46.9.2  Neighborhoods   neircl 47626
                  21.46.9.3  Subspace topologies   restcls2lem 47634
                  21.46.9.4  Limits and continuity in topological spaces   cnneiima 47638
                  21.46.9.5  Topological definitions using the reals   iooii 47639
                  21.46.9.6  Separated sets   sepnsepolem1 47643
                  21.46.9.7  Separated spaces: T0, T1, T2 (Hausdorff) ...   isnrm4 47652
            21.46.10  Preordered sets and directed sets using extensible structures   isprsd 47677
            21.46.11  Posets and lattices using extensible structures   lubeldm2 47678
                  21.46.11.1  Posets   lubeldm2 47678
                  21.46.11.2  Lattices   toslat 47696
                  21.46.11.3  Subset order structures   intubeu 47698
            21.46.12  Categories   catprslem 47719
                  21.46.12.1  Categories   catprslem 47719
                  21.46.12.2  Monomorphisms and epimorphisms   idmon 47725
                  21.46.12.3  Functors   funcf2lem 47727
            21.46.13  Examples of categories   cthinc 47728
                  21.46.13.1  Thin categories   cthinc 47728
                  21.46.13.2  Preordered sets as thin categories   cprstc 47771
                  21.46.13.3  Monoids as categories   cmndtc 47792
      ​21.47  Mathbox for Emmett Weisz
            ​*21.47.1  Miscellaneous Theorems   nfintd 47807
            21.47.2  Set Recursion   csetrecs 47817
                  ​*21.47.2.1  Basic Properties of Set Recursion   csetrecs 47817
                  21.47.2.2  Examples and properties of set recursion   elsetrecslem 47833
            ​*21.47.3  Construction of Games and Surreal Numbers   cpg 47843
      ​​*21.48  Mathbox for David A. Wheeler
            21.48.1  Natural deduction   sbidd 47852
            ​*21.48.2  Greater than, greater than or equal to.   cge-real 47854
            ​*21.48.3  Hyperbolic trigonometric functions   csinh 47864
            ​*21.48.4  Reciprocal trigonometric functions (sec, csc, cot)   csec 47875
            ​*21.48.5  Identities for "if"   ifnmfalse 47897
            ​*21.48.6  Logarithms generalized to arbitrary base using ` logb `   logb2aval 47898
            ​*21.48.7  Logarithm laws generalized to an arbitrary base - log_   clog- 47899
            ​*21.48.8  Formally define notions such as reflexivity   wreflexive 47901
            ​*21.48.9  Algebra helpers   comraddi 47905
            ​*21.48.10  Algebra helper examples   i2linesi 47914
            ​*21.48.11  Formal methods "surprises"   alimp-surprise 47916
            ​*21.48.12  Allsome quantifier   walsi 47922
            ​*21.48.13  Miscellaneous   5m4e1 47933
            21.48.14  Theorems about algebraic numbers   aacllem 47937
      ​21.49  Mathbox for Kunhao Zheng
            21.49.1  Weighted AM-GM inequality   amgmwlem 47938