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  Subring algebras and ideals
      10.8  The complex numbers as an algebraic extensible structure
      10.9  Generalized pre-Hilbert and Hilbert spaces
PART 11  BASIC LINEAR ALGEBRA
      11.1  Vectors and free modules
      11.2  Associative algebras
      11.3  Abstract multivariate polynomials
      11.4  Matrices
      11.5  The determinant
      11.6  Polynomial matrices
      11.7  The characteristic polynomial
PART 12  BASIC TOPOLOGY
      12.1  Topology
      12.2  Filters and filter bases
      12.3  Uniform Structures and Spaces
      12.4  Metric spaces
      12.5  Metric subcomplex vector spaces
PART 13  BASIC REAL AND COMPLEX ANALYSIS
      13.1  Continuity
      13.2  Integrals
      13.3  Derivatives
PART 14  BASIC REAL AND COMPLEX FUNCTIONS
      14.1  Polynomials
      14.2  Sequences and series
      14.3  Basic trigonometry
      14.4  Basic number theory
PART 15  SURREAL NUMBERS
      15.1  Sign sequence representation and Alling's axioms
      15.2  Initial consequences of Alling's axioms
      15.3  Conway cut representation
      15.4  Induction and recursion
      15.5  Surreal arithmetic
      15.6  Subsystems of surreals
PART 16  ELEMENTARY GEOMETRY
      16.1  Definition and Tarski's Axioms of Geometry
      16.2  Tarskian Geometry
      16.3  Properties of geometries
      16.4  Geometry in Hilbert spaces
PART 17  GRAPH THEORY
      17.1  Vertices and edges
      17.2  Undirected graphs
      17.3  Walks, paths and cycles
      17.4  Eulerian paths and the Konigsberg Bridge problem
      17.5  The Friendship Theorem
PART 18  GUIDES AND MISCELLANEA
      18.1  Guides (conventions, explanations, and examples)
      18.2  Humor
      18.3  (Future - to be reviewed and classified)
PART 19  COMPLEX TOPOLOGICAL VECTOR SPACES (DEPRECATED)
      19.1  Additional material on group theory (deprecated)
      19.2  Complex vector spaces
      19.3  Normed complex vector spaces
      19.4  Operators on complex vector spaces
      19.5  Inner product (pre-Hilbert) spaces
      19.6  Complex Banach spaces
      19.7  Complex Hilbert spaces
PART 20  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      20.1  Axiomatization of complex pre-Hilbert spaces
      20.2  Inner product and norms
      20.3  Cauchy sequences and completeness axiom
      20.4  Subspaces and projections
      20.5  Properties of Hilbert subspaces
      20.6  Operators on Hilbert spaces
      20.7  States on a Hilbert lattice and Godowski's equation
      20.8  Cover relation, atoms, exchange axiom, and modular symmetry
PART 21  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      ​21.1  Mathboxes for user contributions
      21.2  Mathbox for Stefan Allan
      21.3  Mathbox for Thierry Arnoux
      21.4  Mathbox for Jonathan Ben-Naim
      21.5  Mathbox for BTernaryTau
      21.6  Mathbox for Mario Carneiro
      21.7  Mathbox for Filip Cernatescu
      21.8  Mathbox for Paul Chapman
      21.9  Mathbox for 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 846
            ​*1.2.8  Mixed connectives   jaao 953
            ​*1.2.9  The conditional operator for propositions   wif 1061
            ​*1.2.10  The weak deduction theorem for propositional calculus   elimh 1081
            1.2.11  Abbreviated conjunction and disjunction of three wff's   w3o 1084
            1.2.12  Logical "nand" (Sheffer stroke)   wnan 1485
            1.2.13  Logical "xor"   wxo 1505
            1.2.14  Logical "nor"   wnor 1522
            1.2.15  True and false constants   wal 1532
                  ​*1.2.15.1  Universal quantifier for use by df-tru   wal 1532
                  ​*1.2.15.2  Equality predicate for use by df-tru   cv 1533
                  1.2.15.3  The true constant   wtru 1535
                  1.2.15.4  The false constant   wfal 1546
            ​*1.2.16  Truth tables   truimtru 1557
                  1.2.16.1  Implication   truimtru 1557
                  1.2.16.2  Negation   nottru 1561
                  1.2.16.3  Equivalence   trubitru 1563
                  1.2.16.4  Conjunction   truantru 1567
                  1.2.16.5  Disjunction   truortru 1571
                  1.2.16.6  Alternative denial   trunantru 1575
                  1.2.16.7  Exclusive disjunction   truxortru 1579
                  1.2.16.8  Joint denial   trunortru 1583
            ​*1.2.17  Half adder and full adder in propositional calculus   whad 1587
                  1.2.17.1  Full adder: sum   whad 1587
                  1.2.17.2  Full adder: carry   wcad 1600
      ​1.3  Other axiomatizations related to classical propositional calculus
            ​*1.3.1  Minimal implicational calculus   minimp 1616
            ​*1.3.2  Implicational Calculus   impsingle 1622
            1.3.3  Derive the Lukasiewicz axioms from Meredith's sole axiom   meredith 1636
            1.3.4  Derive the standard axioms from the Lukasiewicz axioms   luklem1 1653
            ​*1.3.5  Derive Nicod's axiom from the standard axioms   nic-dfim 1664
            1.3.6  Derive the Lukasiewicz axioms from Nicod's axiom   nic-imp 1670
            1.3.7  Derive Nicod's Axiom from Lukasiewicz's First Sheffer Stroke Axiom   lukshef-ax1 1689
            1.3.8  Derive the Lukasiewicz Axioms from the Tarski-Bernays-Wajsberg Axioms   tbw-bijust 1693
            1.3.9  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's First CO Axiom   merco1 1708
            1.3.10  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's Second CO Axiom   merco2 1731
            1.3.11  Derive the Lukasiewicz axioms from the Russell-Bernays Axioms   rb-bijust 1744
            ​*1.3.12  Stoic logic non-modal portion (Chrysippus of Soli)   mptnan 1763
      ​​*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 1774
                  1.4.1.1  Existential quantifier   wex 1774
                  1.4.1.2  Nonfreeness predicate   wnf 1778
            1.4.2  Rule scheme ax-gen (Generalization)   ax-gen 1790
            1.4.3  Axiom scheme ax-4 (Quantified Implication)   ax-4 1804
                  ​*1.4.3.1  The empty domain of discourse   empty 1902
            1.4.4  Axiom scheme ax-5 (Distinctness) - first use of $d   ax-5 1906
            ​*1.4.5  Equality predicate (continued)   weq 1959
            1.4.6  Axiom scheme ax-6 (Existence)   ax-6 1964
            1.4.7  Axiom scheme ax-7 (Equality)   ax-7 2004
            1.4.8  Define proper substitution   sbjust 2059
            1.4.9  Membership predicate   wcel 2099
            1.4.10  Axiom scheme ax-8 (Left Equality for Binary Predicate)   ax-8 2101
            1.4.11  Axiom scheme ax-9 (Right Equality for Binary Predicate)   ax-9 2109
            ​*1.4.12  Logical redundancy of ax-10 , ax-11 , ax-12 , ax-13   ax6dgen 2117
      ​​*1.5  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes)
            1.5.1  Axiom scheme ax-10 (Quantified Negation)   ax-10 2130
            1.5.2  Axiom scheme ax-11 (Quantifier Commutation)   ax-11 2147
            1.5.3  Axiom scheme ax-12 (Substitution)   ax-12 2164
            1.5.4  Axiom scheme ax-13 (Quantified Equality)   ax-13 2366
      ​1.6  Uniqueness and unique existence
            1.6.1  Uniqueness: the at-most-one quantifier   wmo 2527
            1.6.2  Unique existence: the unique existential quantifier   weu 2557
      ​1.7  Other axiomatizations related to classical predicate calculus
            ​*1.7.1  Aristotelian logic: Assertic syllogisms   barbara 2653
            ​*1.7.2  Intuitionistic logic   axia1 2683
​​*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 2698
            2.1.2  Classes   cab 2704
                  2.1.2.1  Class abstractions   cab 2704
                  ​*2.1.2.2  Class equality   df-cleq 2719
                  2.1.2.3  Class membership   df-clel 2805
                  2.1.2.4  Elementary properties of class abstractions   eqabdv 2862
            2.1.3  Class form not-free predicate   wnfc 2878
            2.1.4  Negated equality and membership   wne 2935
                  2.1.4.1  Negated equality   wne 2935
                  2.1.4.2  Negated membership   wnel 3041
            2.1.5  Restricted quantification   wral 3056
                  2.1.5.1  Restricted universal and existential quantification   wral 3056
                  2.1.5.2  Restricted existential uniqueness and at-most-one quantifier   wreu 3369
                  2.1.5.3  Restricted class abstraction   crab 3427
            2.1.6  The universal class   cvv 3469
            ​*2.1.7  Conditional equality (experimental)   wcdeq 3756
            2.1.8  Russell's Paradox   rru 3772
            2.1.9  Proper substitution of classes for sets   wsbc 3774
            2.1.10  Proper substitution of classes for sets into classes   csb 3889
            2.1.11  Define basic set operations and relations   cdif 3941
            2.1.12  Subclasses and subsets   df-ss 3961
            2.1.13  The difference, union, and intersection of two classes   dfdif3 4110
                  2.1.13.1  The difference of two classes   dfdif3 4110
                  2.1.13.2  The union of two classes   elun 4144
                  2.1.13.3  The intersection of two classes   elini 4189
                  2.1.13.4  The symmetric difference of two classes   csymdif 4237
                  2.1.13.5  Combinations of difference, union, and intersection of two classes   unabs 4250
                  2.1.13.6  Class abstractions with difference, union, and intersection of two classes   unabw 4293
                  2.1.13.7  Restricted uniqueness with difference, union, and intersection   reuun2 4310
            2.1.14  The empty set   c0 4318
            ​*2.1.15  The conditional operator for classes   cif 4524
            ​*2.1.16  The weak deduction theorem for set theory   dedth 4582
            2.1.17  Power classes   cpw 4598
            2.1.18  Unordered and ordered pairs   snjust 4623
            2.1.19  The union of a class   cuni 4903
            2.1.20  The intersection of a class   cint 4944
            2.1.21  Indexed union and intersection   ciun 4991
            2.1.22  Disjointness   wdisj 5107
            2.1.23  Binary relations   wbr 5142
            2.1.24  Ordered-pair class abstractions (class builders)   copab 5204
            2.1.25  Functions in maps-to notation   cmpt 5225
            2.1.26  Transitive classes   wtr 5259
      ​2.2  ZF Set Theory - add the Axiom of Replacement
            2.2.1  Introduce the Axiom of Replacement   ax-rep 5279
            2.2.2  Derive the Axiom of Separation   axsepgfromrep 5291
            2.2.3  Derive the Null Set Axiom   axnulALT 5298
            2.2.4  Theorems requiring subset and intersection existence   nalset 5307
            2.2.5  Theorems requiring empty set existence   class2set 5349
      ​2.3  ZF Set Theory - add the Axiom of Power Sets
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 5359
            2.3.2  Derive the Axiom of Pairing   axprlem1 5417
            2.3.3  Ordered pair theorem   opnz 5469
            2.3.4  Ordered-pair class abstractions (cont.)   opabidw 5520
            2.3.5  Power class of union and intersection   pwin 5566
            2.3.6  The identity relation   cid 5569
            2.3.7  The membership relation (or epsilon relation)   cep 5575
            ​*2.3.8  Partial and total orderings   wpo 5582
            2.3.9  Founded and well-ordering relations   wfr 5624
            2.3.10  Relations   cxp 5670
            2.3.11  The Predecessor Class   cpred 6298
            2.3.12  Well-founded induction (variant)   frpomin 6340
            2.3.13  Well-ordered induction   tz6.26 6347
            2.3.14  Ordinals   word 6362
            2.3.15  Definite description binder (inverted iota)   cio 6492
            2.3.16  Functions   wfun 6536
            2.3.17  Cantor's Theorem   canth 7367
            2.3.18  Restricted iota (description binder)   crio 7369
            2.3.19  Operations   co 7414
                  2.3.19.1  Variable-to-class conversion for operations   caovclg 7607
            2.3.20  Maps-to notation   mpondm0 7655
            2.3.21  Function operation   cof 7677
            2.3.22  Proper subset relation   crpss 7721
      ​2.4  ZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 7734
            2.4.2  Ordinals (continued)   epweon 7771
            2.4.3  Transfinite induction   tfi 7851
            2.4.4  The natural numbers (i.e., finite ordinals)   com 7864
            2.4.5  Peano's postulates   peano1 7888
            2.4.6  Finite induction (for finite ordinals)   find 7896
            2.4.7  Relations and functions (cont.)   dmexg 7903
            2.4.8  First and second members of an ordered pair   c1st 7985
            2.4.9  Induction on Cartesian products   frpoins3xpg 8139
            2.4.10  Ordering on Cartesian products   xpord2lem 8141
            2.4.11  Ordering Ordinal Sequences   orderseqlem 8156
            ​*2.4.12  The support of functions   csupp 8159
            ​*2.4.13  Special maps-to operations   opeliunxp2f 8209
            2.4.14  Function transposition   ctpos 8224
            2.4.15  Curry and uncurry   ccur 8264
            2.4.16  Undefined values   cund 8271
            2.4.17  Well-founded recursion   cfrecs 8279
            2.4.18  Well-ordered recursion   cwrecs 8310
            2.4.19  Functions on ordinals; strictly monotone ordinal functions   iunon 8353
            2.4.20  "Strong" transfinite recursion   crecs 8384
            2.4.21  Recursive definition generator   crdg 8423
            2.4.22  Finite recursion   frfnom 8449
            2.4.23  Ordinal arithmetic   c1o 8473
            2.4.24  Natural number arithmetic   nna0 8618
            2.4.25  Natural addition   cnadd 8679
            2.4.26  Equivalence relations and classes   wer 8715
            2.4.27  The mapping operation   cmap 8836
            2.4.28  Infinite Cartesian products   cixp 8907
            2.4.29  Equinumerosity   cen 8952
            2.4.30  Schroeder-Bernstein Theorem   sbthlem1 9099
            2.4.31  Equinumerosity (cont.)   xpf1o 9155
            2.4.32  Finite sets   dif1enlem 9172
            2.4.33  Pigeonhole Principle   phplem1 9223
            2.4.34  Finite sets (cont.)   onomeneq 9244
            2.4.35  Finitely supported functions   cfsupp 9377
            2.4.36  Finite intersections   cfi 9425
            2.4.37  Hall's marriage theorem   marypha1lem 9448
            2.4.38  Supremum and infimum   csup 9455
            2.4.39  Ordinal isomorphism, Hartogs's theorem   coi 9524
            2.4.40  Hartogs function   char 9571
            2.4.41  Weak dominance   cwdom 9579
      ​2.5  ZF Set Theory - add the Axiom of Regularity
            2.5.1  Introduce the Axiom of Regularity   ax-reg 9607
            2.5.2  Axiom of Infinity equivalents   inf0 9636
      ​2.6  ZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-inf 9653
            2.6.2  Existence of omega (the set of natural numbers)   omex 9658
            2.6.3  Cantor normal form   ccnf 9676
            2.6.4  Transitive closure of a relation   cttrcl 9722
            2.6.5  Transitive closure   trcl 9743
            2.6.6  Well-Founded Induction   frmin 9764
            2.6.7  Well-Founded Recursion   frr3g 9771
            2.6.8  Rank   cr1 9777
            2.6.9  Scott's trick; collection principle; Hilbert's epsilon   scottex 9900
            2.6.10  Disjoint union   cdju 9913
            2.6.11  Cardinal numbers   ccrd 9950
            2.6.12  Axiom of Choice equivalents   wac 10130
            ​*2.6.13  Cardinal number arithmetic   undjudom 10182
            2.6.14  The Ackermann bijection   ackbij2lem1 10234
            2.6.15  Cofinality (without Axiom of Choice)   cflem 10261
            2.6.16  Eight inequivalent definitions of finite set   sornom 10292
            2.6.17  Hereditarily size-limited sets without Choice   itunifval 10431
​​*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 10450
            3.1.2  Introduce the Axiom of Dependent Choice   ax-dc 10461
      ​3.2  ZFC Set Theory - add the Axiom of Choice
            3.2.1  Introduce the Axiom of Choice   ax-ac 10474
            3.2.2  AC equivalents: well-ordering, Zorn's lemma   numthcor 10509
            3.2.3  Cardinal number theorems using Axiom of Choice   cardval 10561
            3.2.4  Cardinal number arithmetic using Axiom of Choice   iunctb 10589
            3.2.5  Cofinality using the Axiom of Choice   alephreg 10597
      ​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 10635
            3.4.2  Derivation of the Axiom of Choice   gchaclem 10693
​​*PART 4  TG (TARSKI-GROTHENDIECK) SET THEORY
      ​4.1  Inaccessibles
            4.1.1  Weakly and strongly inaccessible cardinals   cwina 10697
            4.1.2  Weak universes   cwun 10715
            4.1.3  Tarski classes   ctsk 10763
            4.1.4  Grothendieck universes   cgru 10805
      ​4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
            4.2.1  Introduce the Tarski-Grothendieck Axiom   ax-groth 10838
            4.2.2  Derive the Power Set, Infinity and Choice Axioms   grothpw 10841
            4.2.3  Tarski map function   ctskm 10852
​​*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 10859
            5.1.2  Final derivation of real and complex number postulates   axaddf 11160
            5.1.3  Real and complex number postulates restated as axioms   ax-cnex 11186
      ​5.2  Derive the basic properties from the field axioms
            5.2.1  Some deductions from the field axioms for complex numbers   cnex 11211
            5.2.2  Infinity and the extended real number system   cpnf 11267
            5.2.3  Restate the ordering postulates with extended real "less than"   axlttri 11307
            5.2.4  Ordering on reals   lttr 11312
            5.2.5  Initial properties of the complex numbers   mul12 11401
      ​5.3  Real and complex numbers - basic operations
            5.3.1  Addition   add12 11453
            5.3.2  Subtraction   cmin 11466
            5.3.3  Multiplication   kcnktkm1cn 11667
            5.3.4  Ordering on reals (cont.)   gt0ne0 11701
            5.3.5  Reciprocals   ixi 11865
            5.3.6  Division   cdiv 11893
            5.3.7  Ordering on reals (cont.)   elimgt0 12074
            5.3.8  Completeness Axiom and Suprema   fimaxre 12180
            5.3.9  Imaginary and complex number properties   inelr 12224
            5.3.10  Function operation analogue theorems   ofsubeq0 12231
      ​5.4  Integer sets
            5.4.1  Positive integers (as a subset of complex numbers)   cn 12234
            5.4.2  Principle of mathematical induction   nnind 12252
            ​*5.4.3  Decimal representation of numbers   c2 12289
            ​*5.4.4  Some properties of specific numbers   neg1cn 12348
            5.4.5  Simple number properties   halfcl 12459
            5.4.6  The Archimedean property   nnunb 12490
            5.4.7  Nonnegative integers (as a subset of complex numbers)   cn0 12494
            ​*5.4.8  Extended nonnegative integers   cxnn0 12566
            5.4.9  Integers (as a subset of complex numbers)   cz 12580
            5.4.10  Decimal arithmetic   cdc 12699
            5.4.11  Upper sets of integers   cuz 12844
            5.4.12  Well-ordering principle for bounded-below sets of integers   uzwo3 12949
            5.4.13  Rational numbers (as a subset of complex numbers)   cq 12954
            5.4.14  Existence of the set of complex numbers   rpnnen1lem2 12983
      ​5.5  Order sets
            5.5.1  Positive reals (as a subset of complex numbers)   crp 12998
            5.5.2  Infinity and the extended real number system (cont.)   cxne 13113
            5.5.3  Supremum and infimum on the extended reals   xrsupexmnf 13308
            5.5.4  Real number intervals   cioo 13348
            5.5.5  Finite intervals of integers   cfz 13508
            ​*5.5.6  Finite intervals of nonnegative integers   elfz2nn0 13616
            5.5.7  Half-open integer ranges   cfzo 13651
      ​5.6  Elementary integer functions
            5.6.1  The floor and ceiling functions   cfl 13779
            5.6.2  The modulo (remainder) operation   cmo 13858
            5.6.3  Miscellaneous theorems about integers   om2uz0i 13936
            5.6.4  Strong induction over upper sets of integers   uzsinds 13976
            5.6.5  Finitely supported functions over the nonnegative integers   fsuppmapnn0fiublem 13979
            5.6.6  The infinite sequence builder "seq" - extension   cseq 13990
            5.6.7  Integer powers   cexp 14050
            5.6.8  Ordered pair theorem for nonnegative integers   nn0le2msqi 14250
            5.6.9  Factorial function   cfa 14256
            5.6.10  The binomial coefficient operation   cbc 14285
            5.6.11  The ` # ` (set size) function   chash 14313
                  5.6.11.1  Proper unordered pairs and triples (sets of size 2 and 3)   hashprlei 14453
                  5.6.11.2  Functions with a domain containing at least two different elements   fundmge2nop0 14477
                  5.6.11.3  Finite induction on the size of the first component of a binary relation   hashdifsnp1 14481
      ​​*5.7  Words over a set
            5.7.1  Definitions and basic theorems   cword 14488
            5.7.2  Last symbol of a word   clsw 14536
            5.7.3  Concatenations of words   cconcat 14544
            5.7.4  Singleton words   cs1 14569
            5.7.5  Concatenations with singleton words   ccatws1cl 14590
            5.7.6  Subwords/substrings   csubstr 14614
            5.7.7  Prefixes of a word   cpfx 14644
            5.7.8  Subwords of subwords   swrdswrdlem 14678
            5.7.9  Subwords and concatenations   pfxcctswrd 14684
            5.7.10  Subwords of concatenations   swrdccatfn 14698
            5.7.11  Splicing words (substring replacement)   csplice 14723
            5.7.12  Reversing words   creverse 14732
            5.7.13  Repeated symbol words   creps 14742
            ​*5.7.14  Cyclical shifts of words   ccsh 14762
            5.7.15  Mapping words by a function   wrdco 14806
            5.7.16  Longer string literals   cs2 14816
      ​​*5.8  Reflexive and transitive closures of relations
            5.8.1  The reflexive and transitive properties of relations   coss12d 14943
            5.8.2  Basic properties of closures   cleq1lem 14953
            5.8.3  Definitions and basic properties of transitive closures   ctcl 14956
            5.8.4  Exponentiation of relations   crelexp 14990
            5.8.5  Reflexive-transitive closure as an indexed union   crtrcl 15026
            ​*5.8.6  Principle of transitive induction.   relexpindlem 15034
      ​5.9  Elementary real and complex functions
            5.9.1  The "shift" operation   cshi 15037
            5.9.2  Signum (sgn or sign) function   csgn 15057
            5.9.3  Real and imaginary parts; conjugate   ccj 15067
            5.9.4  Square root; absolute value   csqrt 15204
      ​5.10  Elementary limits and convergence
            5.10.1  Superior limit (lim sup)   clsp 15438
            5.10.2  Limits   cli 15452
            5.10.3  Finite and infinite sums   csu 15656
            5.10.4  The binomial theorem   binomlem 15799
            5.10.5  The inclusion/exclusion principle   incexclem 15806
            5.10.6  Infinite sums (cont.)   isumshft 15809
            5.10.7  Miscellaneous converging and diverging sequences   divrcnv 15822
            5.10.8  Arithmetic series   arisum 15830
            5.10.9  Geometric series   expcnv 15834
            5.10.10  Ratio test for infinite series convergence   cvgrat 15853
            5.10.11  Mertens' theorem   mertenslem1 15854
            5.10.12  Finite and infinite products   prodf 15857
                  5.10.12.1  Product sequences   prodf 15857
                  5.10.12.2  Non-trivial convergence   ntrivcvg 15867
                  5.10.12.3  Complex products   cprod 15873
                  5.10.12.4  Finite products   fprod 15909
                  5.10.12.5  Infinite products   iprodclim 15966
            5.10.13  Falling and Rising Factorial   cfallfac 15972
            5.10.14  Bernoulli polynomials and sums of k-th powers   cbp 16014
      ​5.11  Elementary trigonometry
            5.11.1  The exponential, sine, and cosine functions   ce 16029
                  5.11.1.1  The circle constant (tau = 2 pi)   ctau 16170
            5.11.2  _e is irrational   eirrlem 16172
      ​5.12  Cardinality of real and complex number subsets
            5.12.1  Countability of integers and rationals   xpnnen 16179
            5.12.2  The reals are uncountable   rpnnen2lem1 16182
​​*PART 6  ELEMENTARY NUMBER THEORY
      ​6.1  Elementary properties of divisibility
            6.1.1  Irrationality of square root of 2   sqrt2irrlem 16216
            6.1.2  Some Number sets are chains of proper subsets   nthruc 16220
            6.1.3  The divides relation   cdvds 16222
            ​*6.1.4  Even and odd numbers   evenelz 16304
            6.1.5  The division algorithm   divalglem0 16361
            6.1.6  Bit sequences   cbits 16385
            6.1.7  The greatest common divisor operator   cgcd 16460
            6.1.8  Bézout's identity   bezoutlem1 16506
            6.1.9  Algorithms   nn0seqcvgd 16532
            6.1.10  Euclid's Algorithm   eucalgval2 16543
            ​*6.1.11  The least common multiple   clcm 16550
            ​*6.1.12  Coprimality and Euclid's lemma   coprmgcdb 16611
            6.1.13  Cancellability of congruences   congr 16626
      ​6.2  Elementary prime number theory
            ​*6.2.1  Elementary properties   cprime 16633
            ​*6.2.2  Coprimality and Euclid's lemma (cont.)   coprm 16673
            6.2.3  Properties of the canonical representation of a rational   cnumer 16696
            6.2.4  Euler's theorem   codz 16723
            6.2.5  Arithmetic modulo a prime number   modprm1div 16757
            6.2.6  Pythagorean Triples   coprimeprodsq 16768
            6.2.7  The prime count function   cpc 16796
            6.2.8  Pocklington's theorem   prmpwdvds 16864
            6.2.9  Infinite primes theorem   unbenlem 16868
            6.2.10  Sum of prime reciprocals   prmreclem1 16876
            6.2.11  Fundamental theorem of arithmetic   1arithlem1 16883
            6.2.12  Lagrange's four-square theorem   cgz 16889
            6.2.13  Van der Waerden's theorem   cvdwa 16925
            6.2.14  Ramsey's theorem   cram 16959
            ​*6.2.15  Primorial function   cprmo 16991
            ​*6.2.16  Prime gaps   prmgaplem1 17009
            6.2.17  Decimal arithmetic (cont.)   dec2dvds 17023
            6.2.18  Cyclical shifts of words (cont.)   cshwsidrepsw 17054
            6.2.19  Specific prime numbers   prmlem0 17066
            6.2.20  Very large primes   1259lem1 17091
​PART 7  BASIC STRUCTURES
      ​7.1  Extensible structures
            ​*7.1.1  Basic definitions   cstr 17106
                  7.1.1.1  Extensible structures as structures with components   cstr 17106
                  7.1.1.2  Substitution of components   csts 17123
                  7.1.1.3  Slots   cslot 17141
                  ​*7.1.1.4  Structure component indices   cnx 17153
                  7.1.1.5  Base sets   cbs 17171
                  7.1.1.6  Base set restrictions   cress 17200
            7.1.2  Slot definitions   cplusg 17224
            7.1.3  Definition of the structure product   crest 17393
            7.1.4  Definition of the structure quotient   cordt 17472
      ​7.2  Moore spaces
            7.2.1  Moore closures   mrcflem 17577
            7.2.2  Independent sets in a Moore system   mrisval 17601
            7.2.3  Algebraic closure systems   isacs 17622
​PART 8  BASIC CATEGORY THEORY
      ​8.1  Categories
            8.1.1  Categories   ccat 17635
            8.1.2  Opposite category   coppc 17682
            8.1.3  Monomorphisms and epimorphisms   cmon 17702
            8.1.4  Sections, inverses, isomorphisms   csect 17718
            ​*8.1.5  Isomorphic objects   ccic 17769
            8.1.6  Subcategories   cssc 17781
            8.1.7  Functors   cfunc 17831
            8.1.8  Full & faithful functors   cful 17882
            8.1.9  Natural transformations and the functor category   cnat 17922
            8.1.10  Initial, terminal and zero objects of a category   cinito 17961
      ​8.2  Arrows (disjointified hom-sets)
            8.2.1  Identity and composition for arrows   cida 18033
      ​8.3  Examples of categories
            8.3.1  The category of sets   csetc 18055
            8.3.2  The category of categories   ccatc 18078
            ​*8.3.3  The category of extensible structures   fncnvimaeqv 18101
      ​8.4  Categorical constructions
            8.4.1  Product of categories   cxpc 18150
            8.4.2  Functor evaluation   cevlf 18192
            8.4.3  Hom functor   chof 18231
​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 18414
            9.5.2  Complete lattices   ccla 18481
            9.5.3  Distributive lattices   cdlat 18503
            9.5.4  Subset order structures   cipo 18510
      ​9.6  Posets, directed sets, and lattices as relations
            ​*9.6.1  Posets and lattices as relations   cps 18547
            9.6.2  Directed sets, nets   cdir 18577
​PART 10  BASIC ALGEBRAIC STRUCTURES
      ​10.1  Monoids
            ​*10.1.1  Magmas   cplusf 18588
            ​*10.1.2  Identity elements   mgmidmo 18611
            ​*10.1.3  Iterated sums in a magma   gsumvalx 18627
            10.1.4  Magma homomorphisms and submagmas   cmgmhm 18641
            ​*10.1.5  Semigroups   csgrp 18669
            ​*10.1.6  Definition and basic properties of monoids   cmnd 18685
            10.1.7  Monoid homomorphisms and submonoids   cmhm 18729
            ​*10.1.8  Iterated sums in a monoid   gsumvallem2 18777
            10.1.9  Free monoids   cfrmd 18790
                  ​*10.1.9.1  Monoid of endofunctions   cefmnd 18811
            10.1.10  Examples and counterexamples for magmas, semigroups and monoids   mgm2nsgrplem1 18861
      ​10.2  Groups
            10.2.1  Definition and basic properties   cgrp 18881
            ​*10.2.2  Group multiple operation   cmg 19014
            10.2.3  Subgroups and Quotient groups   csubg 19066
            ​*10.2.4  Cyclic monoids and groups   cycsubmel 19146
            10.2.5  Elementary theory of group homomorphisms   cghm 19158
            10.2.6  Isomorphisms of groups   cgim 19202
                  10.2.6.1  The first isomorphism theorem of groups   ghmquskerlem1 19225
            10.2.7  Group actions   cga 19231
            10.2.8  Centralizers and centers   ccntz 19257
            10.2.9  The opposite group   coppg 19287
            10.2.10  Symmetric groups   csymg 19312
                  ​*10.2.10.1  Definition and basic properties   csymg 19312
                  10.2.10.2  Cayley's theorem   cayleylem1 19358
                  10.2.10.3  Permutations fixing one element   symgfix2 19362
                  ​*10.2.10.4  Transpositions in the symmetric group   cpmtr 19387
                  10.2.10.5  The sign of a permutation   cpsgn 19435
            10.2.11  p-Groups and Sylow groups; Sylow's theorems   cod 19470
            10.2.12  Direct products   clsm 19580
                  10.2.12.1  Direct products (extension)   smndlsmidm 19602
            10.2.13  Free groups   cefg 19652
            10.2.14  Abelian groups   ccmn 19726
                  10.2.14.1  Definition and basic properties   ccmn 19726
                  10.2.14.2  Cyclic groups   ccyg 19823
                  10.2.14.3  Group sum operation   gsumval3a 19849
                  10.2.14.4  Group sums over (ranges of) integers   fsfnn0gsumfsffz 19929
                  10.2.14.5  Internal direct products   cdprd 19941
                  10.2.14.6  The Fundamental Theorem of Abelian Groups   ablfacrplem 20013
            10.2.15  Simple groups   csimpg 20038
                  10.2.15.1  Definition and basic properties   csimpg 20038
                  10.2.15.2  Classification of abelian simple groups   ablsimpnosubgd 20052
      ​10.3  Rings
            10.3.1  Multiplicative Group   cmgp 20065
            ​*10.3.2  Non-unital rings ("rngs")   crng 20083
            ​*10.3.3  Ring unity (multiplicative identity)   cur 20112
            10.3.4  Semirings   csrg 20117
                  ​*10.3.4.1  The binomial theorem for semirings   srgbinomlem1 20157
            10.3.5  Unital rings   crg 20164
            10.3.6  Opposite ring   coppr 20261
            10.3.7  Divisibility   cdsr 20282
            10.3.8  Ring primes   crpm 20360
            10.3.9  Homomorphisms of non-unital rings   crnghm 20362
            10.3.10  Ring homomorphisms   crh 20397
            10.3.11  Nonzero rings and zero rings   cnzr 20440
            10.3.12  Local rings   clring 20464
            10.3.13  Subrings   csubrng 20471
                  10.3.13.1  Subrings of non-unital rings   csubrng 20471
                  10.3.13.2  Subrings of unital rings   csubrg 20495
            10.3.14  Categories of rings   crngc 20538
                  ​*10.3.14.1  The category of non-unital rings   crngc 20538
                  ​*10.3.14.2  The category of (unital) rings   cringc 20567
                  10.3.14.3  Subcategories of the category of rings   srhmsubclem1 20599
      ​10.4  Division rings and fields
            10.4.1  Definition and basic properties   cdr 20613
            10.4.2  Sub-division rings   csdrg 20663
            10.4.3  Absolute value (abstract algebra)   cabv 20685
            10.4.4  Star rings   cstf 20712
      ​10.5  Left modules
            10.5.1  Definition and basic properties   clmod 20732
            10.5.2  Subspaces and spans in a left module   clss 20804
            10.5.3  Homomorphisms and isomorphisms of left modules   clmhm 20893
            10.5.4  Subspace sum; bases for a left module   clbs 20948
      ​10.6  Vector spaces
            10.6.1  Definition and basic properties   clvec 20976
      ​10.7  Subring algebras and ideals
            10.7.1  Subring algebras   csra 21045
            ​*10.7.2  Left ideals and spans   clidl 21091
            10.7.3  Two-sided ideals and quotient rings   c2idl 21132
                  ​*10.7.3.1  Condition for a non-unital ring to be unital   rngqiprng1elbas 21165
            10.7.4  Principal ideal rings. Divisibility in the integers   clpidl 21199
            10.7.5  Left regular elements. More kinds of rings   crlreg 21215
      ​10.8  The complex numbers as an algebraic extensible structure
            10.8.1  Definition and basic properties   cpsmet 21250
            ​*10.8.2  Ring of integers   czring 21359
                  ​*10.8.2.1  Example for a condition for a non-unital ring to be unital   pzriprnglem1 21394
            10.8.3  Algebraic constructions based on the complex numbers   czrh 21412
            10.8.4  Signs as subgroup of the complex numbers   cnmsgnsubg 21496
            10.8.5  Embedding of permutation signs into a ring   zrhpsgnmhm 21503
            10.8.6  The ordered field of real numbers   crefld 21523
      ​10.9  Generalized pre-Hilbert and Hilbert spaces
            10.9.1  Definition and basic properties   cphl 21543
            10.9.2  Orthocomplements and closed subspaces   cocv 21579
            10.9.3  Orthogonal projection and orthonormal bases   cpj 21621
​​*PART 11  BASIC LINEAR ALGEBRA
      ​11.1  Vectors and free modules
            *11.1.1  Direct sum of left modules   cdsmm 21652
            ​*11.1.2  Free modules   cfrlm 21667
            ​*11.1.3  Standard basis (unit vectors)   cuvc 21703
            ​*11.1.4  Independent sets and families   clindf 21725
            11.1.5  Characterization of free modules   lmimlbs 21757
      ​11.2  Associative algebras
            11.2.1  Definition and basic properties   casa 21771
      ​11.3  Abstract multivariate polynomials
            11.3.1  Definition and basic properties   cmps 21824
            11.3.2  Polynomial evaluation   ces 22003
            11.3.3  Additional definitions for (multivariate) polynomials   cslv 22041
            ​*11.3.4  Univariate polynomials   cps1 22081
            11.3.5  Univariate polynomial evaluation   ces1 22219
      ​​*11.4  Matrices
            *11.4.1  The matrix multiplication   cmmul 22272
            ​*11.4.2  Square matrices   cmat 22294
            ​*11.4.3  The matrix algebra   matmulr 22327
            ​*11.4.4  Matrices of dimension 0 and 1   mat0dimbas0 22355
            ​*11.4.5  The subalgebras of diagonal and scalar matrices   cdmat 22377
            ​*11.4.6  Multiplication of a matrix with a "column vector"   cmvmul 22429
            11.4.7  Replacement functions for a square matrix   cmarrep 22445
            11.4.8  Submatrices   csubma 22465
      ​11.5  The determinant
            11.5.1  Definition and basic properties   cmdat 22473
            11.5.2  Determinants of 2 x 2 -matrices   m2detleiblem1 22513
            11.5.3  The matrix adjugate/adjunct   cmadu 22521
            ​*11.5.4  Laplace expansion of determinants (special case)   symgmatr01lem 22542
            11.5.5  Inverse matrix   invrvald 22565
            ​*11.5.6  Cramer's rule   slesolvec 22568
      ​​*11.6  Polynomial matrices
            11.6.1  Basic properties   pmatring 22581
            ​*11.6.2  Constant polynomial matrices   ccpmat 22592
            ​*11.6.3  Collecting coefficients of polynomial matrices   cdecpmat 22651
            ​*11.6.4  Ring isomorphism between polynomial matrices and polynomials over matrices   cpm2mp 22681
      ​​*11.7  The characteristic polynomial
            *11.7.1  Definition and basic properties   cchpmat 22715
            ​*11.7.2  The characteristic factor function G   fvmptnn04if 22738
            ​*11.7.3  The Cayley-Hamilton theorem   cpmadurid 22756
​PART 12  BASIC TOPOLOGY
      ​12.1  Topology
            ​*12.1.1  Topological spaces   ctop 22782
                  12.1.1.1  Topologies   ctop 22782
                  12.1.1.2  Topologies on sets   ctopon 22799
                  12.1.1.3  Topological spaces   ctps 22821
            12.1.2  Topological bases   ctb 22835
            12.1.3  Examples of topologies   distop 22885
            12.1.4  Closure and interior   ccld 22907
            12.1.5  Neighborhoods   cnei 22988
            12.1.6  Limit points and perfect sets   clp 23025
            12.1.7  Subspace topologies   restrcl 23048
            12.1.8  Order topology   ordtbaslem 23079
            12.1.9  Limits and continuity in topological spaces   ccn 23115
            12.1.10  Separated spaces: T0, T1, T2 (Hausdorff) ...   ct0 23197
            12.1.11  Compactness   ccmp 23277
            12.1.12  Bolzano-Weierstrass theorem   bwth 23301
            12.1.13  Connectedness   cconn 23302
            12.1.14  First- and second-countability   c1stc 23328
            12.1.15  Local topological properties   clly 23355
            12.1.16  Refinements   cref 23393
            12.1.17  Compactly generated spaces   ckgen 23424
            12.1.18  Product topologies   ctx 23451
            12.1.19  Continuous function-builders   cnmptid 23552
            12.1.20  Quotient maps and quotient topology   ckq 23584
            12.1.21  Homeomorphisms   chmeo 23644
      ​12.2  Filters and filter bases
            12.2.1  Filter bases   elmptrab 23718
            12.2.2  Filters   cfil 23736
            12.2.3  Ultrafilters   cufil 23790
            12.2.4  Filter limits   cfm 23824
            12.2.5  Extension by continuity   ccnext 23950
            12.2.6  Topological groups   ctmd 23961
            12.2.7  Infinite group sum on topological groups   ctsu 24017
            12.2.8  Topological rings, fields, vector spaces   ctrg 24047
      ​12.3  Uniform Structures and Spaces
            12.3.1  Uniform structures   cust 24091
            12.3.2  The topology induced by an uniform structure   cutop 24122
            12.3.3  Uniform Spaces   cuss 24145
            12.3.4  Uniform continuity   cucn 24167
            12.3.5  Cauchy filters in uniform spaces   ccfilu 24178
            12.3.6  Complete uniform spaces   ccusp 24189
      ​12.4  Metric spaces
            12.4.1  Pseudometric spaces   ispsmet 24197
            12.4.2  Basic metric space properties   cxms 24210
            12.4.3  Metric space balls   blfvalps 24276
            12.4.4  Open sets of a metric space   mopnval 24331
            12.4.5  Continuity in metric spaces   metcnp3 24436
            12.4.6  The uniform structure generated by a metric   metuval 24445
            12.4.7  Examples of metric spaces   dscmet 24468
            ​*12.4.8  Normed algebraic structures   cnm 24472
            12.4.9  Normed space homomorphisms (bounded linear operators)   cnmo 24609
            12.4.10  Topology on the reals   qtopbaslem 24662
            12.4.11  Topological definitions using the reals   cii 24782
            12.4.12  Path homotopy   chtpy 24880
            12.4.13  The fundamental group   cpco 24914
      ​12.5  Metric subcomplex vector spaces
            12.5.1  Subcomplex modules   cclm 24976
            ​*12.5.2  Subcomplex vector spaces   ccvs 25037
            ​*12.5.3  Normed subcomplex vector spaces   isncvsngp 25064
            12.5.4  Subcomplex pre-Hilbert spaces   ccph 25081
            12.5.5  Convergence and completeness   ccfil 25167
            12.5.6  Baire's Category Theorem   bcthlem1 25239
            12.5.7  Banach spaces and subcomplex Hilbert spaces   ccms 25247
                  12.5.7.1  The complete ordered field of the real numbers   retopn 25294
            12.5.8  Euclidean spaces   crrx 25298
            12.5.9  Minimizing Vector Theorem   minveclem1 25339
            12.5.10  Projection Theorem   pjthlem1 25352
​PART 13  BASIC REAL AND COMPLEX ANALYSIS
      ​13.1  Continuity
            13.1.1  Intermediate value theorem   pmltpclem1 25364
      ​13.2  Integrals
            13.2.1  Lebesgue measure   covol 25378
            13.2.2  Lebesgue integration   cmbf 25530
                  13.2.2.1  Lesbesgue integral   cmbf 25530
                  13.2.2.2  Lesbesgue directed integral   cdit 25762
      ​13.3  Derivatives
            13.3.1  Real and complex differentiation   climc 25778
                  13.3.1.1  Derivatives of functions of one complex or real variable   climc 25778
                  13.3.1.2  Results on real differentiation   dvferm1lem 25903
​PART 14  BASIC REAL AND COMPLEX FUNCTIONS
      ​14.1  Polynomials
            14.1.1  Polynomial degrees   cmdg 25973
            14.1.2  The division algorithm for univariate polynomials   cmn1 26048
            14.1.3  Elementary properties of complex polynomials   cply 26105
            14.1.4  The division algorithm for polynomials   cquot 26212
            14.1.5  Algebraic numbers   caa 26236
            14.1.6  Liouville's approximation theorem   aalioulem1 26254
      ​14.2  Sequences and series
            14.2.1  Taylor polynomials and Taylor's theorem   ctayl 26274
            14.2.2  Uniform convergence   culm 26299
            14.2.3  Power series   pserval 26333
      ​14.3  Basic trigonometry
            14.3.1  The exponential, sine, and cosine functions (cont.)   efcn 26367
            14.3.2  Properties of pi = 3.14159...   pilem1 26375
            14.3.3  Mapping of the exponential function   efgh 26462
            14.3.4  The natural logarithm on complex numbers   clog 26475
            ​*14.3.5  Logarithms to an arbitrary base   clogb 26683
            14.3.6  Theorems of Pythagoras, isosceles triangles, and intersecting chords   angval 26720
            14.3.7  Solutions of quadratic, cubic, and quartic equations   quad2 26758
            14.3.8  Inverse trigonometric functions   casin 26781
            14.3.9  The Birthday Problem   log2ublem1 26865
            14.3.10  Areas in R^2   carea 26874
            14.3.11  More miscellaneous converging sequences   rlimcnp 26884
            14.3.12  Inequality of arithmetic and geometric means   cvxcl 26904
            14.3.13  Euler-Mascheroni constant   cem 26911
            14.3.14  Zeta function   czeta 26932
            14.3.15  Gamma function   clgam 26935
      ​14.4  Basic number theory
            14.4.1  Wilson's theorem   wilthlem1 26987
            14.4.2  The Fundamental Theorem of Algebra   ftalem1 26992
            14.4.3  The Basel problem (ζ(2) = π²/6)   basellem1 27000
            14.4.4  Number-theoretical functions   ccht 27010
            14.4.5  Perfect Number Theorem   mersenne 27147
            14.4.6  Characters of Z/nZ   cdchr 27152
            14.4.7  Bertrand's postulate   bcctr 27195
            ​*14.4.8  Quadratic residues and the Legendre symbol   clgs 27214
            ​*14.4.9  Gauss' Lemma   gausslemma2dlem0a 27276
            14.4.10  Quadratic reciprocity   lgseisenlem1 27295
            14.4.11  All primes 4n+1 are the sum of two squares   2sqlem1 27337
            14.4.12  Chebyshev's Weak Prime Number Theorem, Dirichlet's Theorem   chebbnd1lem1 27389
            14.4.13  The Prime Number Theorem   mudivsum 27450
            14.4.14  Ostrowski's theorem   abvcxp 27535
​PART 15  SURREAL NUMBERS
      ​​*15.1  Sign sequence representation and Alling's axioms
            15.1.1  Definitions and initial properties   csur 27560
            15.1.2  Ordering   sltsolem1 27595
            15.1.3  Birthday Function   bdayfo 27597
            15.1.4  Density   fvnobday 27598
            ​*15.1.5  Full-Eta Property   bdayimaon 27613
      ​15.2  Initial consequences of Alling's axioms
            15.2.1  Ordering Theorems   csle 27664
            15.2.2  Birthday Theorems   bdayfun 27692
      ​​*15.3  Conway cut representation
            15.3.1  Conway cuts   csslt 27700
            15.3.2  Zero and One   c0s 27742
            15.3.3  Cuts and Options   cmade 27756
            15.3.4  Cofinality and coinitiality   cofsslt 27825
      ​15.4  Induction and recursion
            15.4.1  Induction and recursion on one variable   cnorec 27841
            15.4.2  Induction and recursion on two variables   cnorec2 27852
      ​15.5  Surreal arithmetic
            15.5.1  Addition   cadds 27863
            15.5.2  Negation and Subtraction   cnegs 27919
            15.5.3  Multiplication   cmuls 27993
            15.5.4  Division   cdivs 28074
            15.5.5  Absolute value   cabss 28118
      ​15.6  Subsystems of surreals
            15.6.1  Ordinal numbers   cons 28131
            15.6.2  Surreal recursive sequences   cseqs 28143
            15.6.3  Natural numbers   cnn0s 28172
            15.6.4  Real numbers   creno 28208
​​*PART 16  ELEMENTARY GEOMETRY
      ​16.1  Definition and Tarski's Axioms of Geometry
            16.1.1  Justification for the congruence notation   tgjustf 28264
      ​16.2  Tarskian Geometry
            16.2.1  Congruence   tgcgrcomimp 28268
            16.2.2  Betweenness   tgbtwntriv2 28278
            16.2.3  Dimension   tglowdim1 28291
            16.2.4  Betweenness and Congruence   tgifscgr 28299
            16.2.5  Congruence of a series of points   ccgrg 28301
            16.2.6  Motions   cismt 28323
            16.2.7  Colinearity   tglng 28337
            16.2.8  Connectivity of betweenness   tgbtwnconn1lem1 28363
            16.2.9  Less-than relation in geometric congruences   cleg 28373
            16.2.10  Rays   chlg 28391
            16.2.11  Lines   btwnlng1 28410
            16.2.12  Point inversions   cmir 28443
            16.2.13  Right angles   crag 28484
            16.2.14  Half-planes   islnopp 28530
            16.2.15  Midpoints and Line Mirroring   cmid 28563
            16.2.16  Congruence of angles   ccgra 28598
            16.2.17  Angle Comparisons   cinag 28626
            16.2.18  Congruence Theorems   tgsas1 28645
            16.2.19  Equilateral triangles   ceqlg 28656
      ​16.3  Properties of geometries
            16.3.1  Isomorphisms between geometries   f1otrgds 28660
      ​16.4  Geometry in Hilbert spaces
            16.4.1  Geometry in the complex plane   cchhllem 28684
            16.4.2  Geometry in Euclidean spaces   cee 28686
                  16.4.2.1  Definition of the Euclidean space   cee 28686
                  16.4.2.2  Tarski's axioms for geometry for the Euclidean space   axdimuniq 28711
                  16.4.2.3  EE^n fulfills Tarski's Axioms   ceeng 28775
​​*PART 17  GRAPH THEORY
      ​*17.1  Vertices and edges
            17.1.1  The edge function extractor for extensible structures   cedgf 28786
            ​*17.1.2  Vertices and indexed edges   cvtx 28796
                  17.1.2.1  Definitions and basic properties   cvtx 28796
                  17.1.2.2  The vertices and edges of a graph represented as ordered pair   opvtxval 28803
                  17.1.2.3  The vertices and edges of a graph represented as extensible structure   funvtxdmge2val 28811
                  17.1.2.4  Representations of graphs without edges   snstrvtxval 28837
                  17.1.2.5  Degenerated cases of representations of graphs   vtxval0 28839
            17.1.3  Edges as range of the edge function   cedg 28847
      ​​*17.2  Undirected graphs
            17.2.1  Undirected hypergraphs   cuhgr 28856
            17.2.2  Undirected pseudographs and multigraphs   cupgr 28880
            ​*17.2.3  Loop-free graphs   umgrislfupgrlem 28922
            17.2.4  Edges as subsets of vertices of graphs   uhgredgiedgb 28926
            ​*17.2.5  Undirected simple graphs   cuspgr 28948
            17.2.6  Examples for graphs   usgr0e 29036
            17.2.7  Subgraphs   csubgr 29067
            17.2.8  Finite undirected simple graphs   cfusgr 29116
            17.2.9  Neighbors, complete graphs and universal vertices   cnbgr 29132
                  17.2.9.1  Neighbors   cnbgr 29132
                  17.2.9.2  Universal vertices   cuvtx 29185
                  17.2.9.3  Complete graphs   ccplgr 29209
            17.2.10  Vertex degree   cvtxdg 29266
            ​*17.2.11  Regular graphs   crgr 29356
      ​​*17.3  Walks, paths and cycles
            *17.3.1  Walks   cewlks 29396
            17.3.2  Walks for loop-free graphs   lfgrwlkprop 29488
            17.3.3  Trails   ctrls 29491
            17.3.4  Paths and simple paths   cpths 29513
            17.3.5  Closed walks   cclwlks 29571
            17.3.6  Circuits and cycles   ccrcts 29585
            ​*17.3.7  Walks as words   cwwlks 29623
            17.3.8  Walks/paths of length 2 (as length 3 strings)   2wlkdlem1 29723
            17.3.9  Walks in regular graphs   rusgrnumwwlkl1 29766
            ​*17.3.10  Closed walks as words   cclwwlk 29778
                  17.3.10.1  Closed walks as words   cclwwlk 29778
                  17.3.10.2  Closed walks of a fixed length as words   cclwwlkn 29821
                  17.3.10.3  Closed walks on a vertex of a fixed length as words   cclwwlknon 29884
            17.3.11  Examples for walks, trails and paths   0ewlk 29911
            17.3.12  Connected graphs   cconngr 29983
      ​17.4  Eulerian paths and the Konigsberg Bridge problem
            ​*17.4.1  Eulerian paths   ceupth 29994
            ​*17.4.2  The Königsberg Bridge problem   konigsbergvtx 30043
      ​17.5  The Friendship Theorem
            17.5.1  Friendship graphs - basics   cfrgr 30055
            17.5.2  The friendship theorem for small graphs   frgr1v 30068
            17.5.3  Theorems according to Mertzios and Unger   2pthfrgrrn 30079
            ​*17.5.4  Huneke's Proof of the Friendship Theorem   frgrncvvdeqlem1 30096
​PART 18  GUIDES AND MISCELLANEA
      ​18.1  Guides (conventions, explanations, and examples)
            ​*18.1.1  Conventions   conventions 30197
            18.1.2  Natural deduction   natded 30200
            ​*18.1.3  Natural deduction examples   ex-natded5.2 30201
            18.1.4  Definitional examples   ex-or 30218
            18.1.5  Other examples   aevdemo 30257
      ​18.2  Humor
            18.2.1  April Fool's theorem   avril1 30260
      ​18.3  (Future - to be reviewed and classified)
            18.3.1  Planar incidence geometry   cplig 30271
            ​*18.3.2  Aliases kept to prevent broken links   dummylink 30284
​​*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 30286
            19.1.2  Abelian groups   cablo 30341
      ​19.2  Complex vector spaces
            19.2.1  Definition and basic properties   cvc 30355
            19.2.2  Examples of complex vector spaces   cnaddabloOLD 30378
      ​19.3  Normed complex vector spaces
            19.3.1  Definition and basic properties   cnv 30381
            19.3.2  Examples of normed complex vector spaces   cnnv 30474
            19.3.3  Induced metric of a normed complex vector space   imsval 30482
            19.3.4  Inner product   cdip 30497
            19.3.5  Subspaces   css 30518
      ​19.4  Operators on complex vector spaces
            19.4.1  Definitions and basic properties   clno 30537
      ​19.5  Inner product (pre-Hilbert) spaces
            19.5.1  Definition and basic properties   ccphlo 30609
            19.5.2  Examples of pre-Hilbert spaces   cncph 30616
            19.5.3  Properties of pre-Hilbert spaces   isph 30619
      ​19.6  Complex Banach spaces
            19.6.1  Definition and basic properties   ccbn 30659
            19.6.2  Examples of complex Banach spaces   cnbn 30666
            19.6.3  Uniform Boundedness Theorem   ubthlem1 30667
            19.6.4  Minimizing Vector Theorem   minvecolem1 30671
      ​19.7  Complex Hilbert spaces
            19.7.1  Definition and basic properties   chlo 30682
            19.7.2  Standard axioms for a complex Hilbert space   hlex 30695
            19.7.3  Examples of complex Hilbert spaces   cnchl 30713
            19.7.4  Hellinger-Toeplitz Theorem   htthlem 30714
​​*PART 20  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      ​20.1  Axiomatization of complex pre-Hilbert spaces
            20.1.1  Basic Hilbert space definitions   chba 30716
            20.1.2  Preliminary ZFC lemmas   df-hnorm 30765
            ​*20.1.3  Derive the Hilbert space axioms from ZFC set theory   axhilex-zf 30778
            ​*20.1.4  Introduce the vector space axioms for a Hilbert space   ax-hilex 30796
            20.1.5  Vector operations   hvmulex 30808
            20.1.6  Inner product postulates for a Hilbert space   ax-hfi 30876
      ​20.2  Inner product and norms
            20.2.1  Inner product   his5 30883
            20.2.2  Norms   dfhnorm2 30919
            20.2.3  Relate Hilbert space to normed complex vector spaces   hilablo 30957
            20.2.4  Bunjakovaskij-Cauchy-Schwarz inequality   bcsiALT 30976
      ​20.3  Cauchy sequences and completeness axiom
            20.3.1  Cauchy sequences and limits   hcau 30981
            20.3.2  Derivation of the completeness axiom from ZF set theory   hilmet 30991
            20.3.3  Completeness postulate for a Hilbert space   ax-hcompl 30999
            20.3.4  Relate Hilbert space to ZFC pre-Hilbert and Hilbert spaces   hhcms 31000
      ​20.4  Subspaces and projections
            20.4.1  Subspaces   df-sh 31004
            20.4.2  Closed subspaces   df-ch 31018
            20.4.3  Orthocomplements   df-oc 31049
            20.4.4  Subspace sum, span, lattice join, lattice supremum   df-shs 31105
            20.4.5  Projection theorem   pjhthlem1 31188
            20.4.6  Projectors   df-pjh 31192
      ​20.5  Properties of Hilbert subspaces
            20.5.1  Orthomodular law   omlsilem 31199
            20.5.2  Projectors (cont.)   pjhtheu2 31213
            20.5.3  Hilbert lattice operations   sh0le 31237
            20.5.4  Span (cont.) and one-dimensional subspaces   spansn0 31338
            20.5.5  Commutes relation for Hilbert lattice elements   df-cm 31380
            20.5.6  Foulis-Holland theorem   fh1 31415
            20.5.7  Quantum Logic Explorer axioms   qlax1i 31424
            20.5.8  Orthogonal subspaces   chscllem1 31434
            20.5.9  Orthoarguesian laws 5OA and 3OA   5oalem1 31451
            20.5.10  Projectors (cont.)   pjorthi 31466
            20.5.11  Mayet's equation E_3   mayete3i 31525
      ​20.6  Operators on Hilbert spaces
            ​*20.6.1  Operator sum, difference, and scalar multiplication   df-hosum 31527
            20.6.2  Zero and identity operators   df-h0op 31545
            20.6.3  Operations on Hilbert space operators   hoaddcl 31555
            20.6.4  Linear, continuous, bounded, Hermitian, unitary operators and norms   df-nmop 31636
            20.6.5  Linear and continuous functionals and norms   df-nmfn 31642
            20.6.6  Adjoint   df-adjh 31646
            20.6.7  Dirac bra-ket notation   df-bra 31647
            20.6.8  Positive operators   df-leop 31649
            20.6.9  Eigenvectors, eigenvalues, spectrum   df-eigvec 31650
            20.6.10  Theorems about operators and functionals   nmopval 31653
            20.6.11  Riesz lemma   riesz3i 31859
            20.6.12  Adjoints (cont.)   cnlnadjlem1 31864
            20.6.13  Quantum computation error bound theorem   unierri 31901
            20.6.14  Dirac bra-ket notation (cont.)   branmfn 31902
            20.6.15  Positive operators (cont.)   leopg 31919
            20.6.16  Projectors as operators   pjhmopi 31943
      ​20.7  States on a Hilbert lattice and Godowski's equation
            20.7.1  States on a Hilbert lattice   df-st 32008
            20.7.2  Godowski's equation   golem1 32068
      ​20.8  Cover relation, atoms, exchange axiom, and modular symmetry
            20.8.1  Covers relation; modular pairs   df-cv 32076
            20.8.2  Atoms   df-at 32135
            20.8.3  Superposition principle   superpos 32151
            20.8.4  Atoms, exchange and covering properties, atomicity   chcv1 32152
            20.8.5  Irreducibility   chirredlem1 32187
            20.8.6  Atoms (cont.)   atcvat3i 32193
            20.8.7  Modular symmetry   mdsymlem1 32200
​PART 21  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      ​21.1  Mathboxes for user contributions
            21.1.1  Mathbox guidelines   mathbox 32239
      ​21.2  Mathbox for Stefan Allan
      ​21.3  Mathbox for Thierry Arnoux
            21.3.1  Propositional Calculus - misc additions   bian1d 32244
            21.3.2  Predicate Calculus   sbc2iedf 32251
                  21.3.2.1  Predicate Calculus - misc additions   sbc2iedf 32251
                  21.3.2.2  Restricted quantification - misc additions   ralcom4f 32253
                  21.3.2.3  Equality   eqtrb 32258
                  21.3.2.4  Double restricted existential uniqueness quantification   opsbc2ie 32260
                  21.3.2.5  Double restricted existential uniqueness quantification syntax   w2reu 32262
                  21.3.2.6  Substitution (without distinct variables) - misc additions   sbceqbidf 32271
                  21.3.2.7  Existential "at most one" - misc additions   mo5f 32273
                  21.3.2.8  Existential uniqueness - misc additions   reuxfrdf 32275
                  21.3.2.9  Restricted "at most one" - misc additions   rmoxfrd 32277
                  21.3.2.10  Restricted iota (description binder)   riotaeqbidva 32280
            21.3.3  General Set Theory   dmrab 32281
                  21.3.3.1  Class abstractions (a.k.a. class builders)   dmrab 32281
                  21.3.3.2  Image Sets   abrexdomjm 32288
                  21.3.3.3  Set relations and operations - misc additions   elunsn 32294
                  21.3.3.4  Unordered pairs   eqsnd 32310
                  21.3.3.5  Conditional operator - misc additions   ifeqeqx 32318
                  21.3.3.6  Set union   uniinn0 32326
                  21.3.3.7  Indexed union - misc additions   cbviunf 32331
                  21.3.3.8  Indexed intersection - misc additions   iinabrex 32344
                  21.3.3.9  Disjointness - misc additions   disjnf 32345
            21.3.4  Relations and Functions   xpdisjres 32373
                  21.3.4.1  Relations - misc additions   xpdisjres 32373
                  21.3.4.2  Functions - misc additions   ac6sf2 32393
                  21.3.4.3  Operations - misc additions   mpomptxf 32447
                  21.3.4.4  Support of a function   suppovss 32448
                  21.3.4.5  Explicit Functions with one or two points as a domain   cosnopne 32458
                  21.3.4.6  Isomorphisms - misc. additions   gtiso 32464
                  21.3.4.7  Disjointness (additional proof requiring functions)   disjdsct 32466
                  21.3.4.8  First and second members of an ordered pair - misc additions   df1stres 32467
                  21.3.4.9  Supremum - misc additions   supssd 32475
                  21.3.4.10  Finite Sets   imafi2 32477
                  21.3.4.11  Countable Sets   snct 32479
            21.3.5  Real and Complex Numbers   creq0 32501
                  21.3.5.1  Complex operations - misc. additions   creq0 32501
                  21.3.5.2  Ordering on reals - misc additions   lt2addrd 32505
                  21.3.5.3  Extended reals - misc additions   xrlelttric 32506
                  21.3.5.4  Extended nonnegative integers - misc additions   xnn0gt0 32523
                  21.3.5.5  Real number intervals - misc additions   joiniooico 32526
                  21.3.5.6  Finite intervals of integers - misc additions   uzssico 32536
                  21.3.5.7  Half-open integer ranges - misc additions   iundisjfi 32548
                  21.3.5.8  The ` # ` (set size) function - misc additions   hashunif 32559
                  21.3.5.9  The greatest common divisor operator - misc. additions   numdenneg 32562
                  21.3.5.10  Integers   nnindf 32564
                  21.3.5.11  Decimal numbers   dfdec100 32575
            ​*21.3.6  Decimal expansion   cdp2 32576
                  ​*21.3.6.1  Decimal point   cdp 32593
                  21.3.6.2  Division in the extended real number system   cxdiv 32622
            21.3.7  Words over a set - misc additions   wrdfd 32641
                  21.3.7.1  Splicing words (substring replacement)   splfv3 32661
                  21.3.7.2  Cyclic shift of words   1cshid 32662
            21.3.8  Extensible Structures   ressplusf 32666
                  21.3.8.1  Structure restriction operator   ressplusf 32666
                  21.3.8.2  The opposite group   oppgle 32669
                  21.3.8.3  Posets   ressprs 32672
                  21.3.8.4  Complete lattices   clatp0cl 32685
                  21.3.8.5  Order Theory   cmnt 32687
                  21.3.8.6  Extended reals Structure - misc additions   ax-xrssca 32713
                  21.3.8.7  The extended nonnegative real numbers commutative monoid   xrge0base 32723
            21.3.9  Algebra   abliso 32734
                  21.3.9.1  Monoids Homomorphisms   abliso 32734
                  21.3.9.2  Finitely supported group sums - misc additions   gsumsubg 32738
                  21.3.9.3  Centralizers and centers - misc additions   cntzun 32752
                  21.3.9.4  Totally ordered monoids and groups   comnd 32755
                  21.3.9.5  The symmetric group   symgfcoeu 32783
                  21.3.9.6  Transpositions   pmtridf1o 32793
                  21.3.9.7  Permutation Signs   psgnid 32796
                  21.3.9.8  Permutation cycles   ctocyc 32805
                  21.3.9.9  The Alternating Group   evpmval 32844
                  21.3.9.10  Signum in an ordered monoid   csgns 32857
                  21.3.9.11  The Archimedean property for generic ordered algebraic structures   cinftm 32862
                  21.3.9.12  Semiring left modules   cslmd 32885
                  21.3.9.13  Simple groups   prmsimpcyc 32913
                  21.3.9.14  Rings - misc additions   domnlcan 32914
                  21.3.9.15  Euclidean Domains   ceuf 32925
                  21.3.9.16  Division Rings   ringinveu 32931
                  21.3.9.17  Subfields   sdrgdvcl 32934
                  21.3.9.18  Field extensions generated by a set   cfldgen 32937
                  21.3.9.19  Totally ordered rings and fields   corng 32950
                  21.3.9.20  Ring homomorphisms - misc additions   rhmdvd 32973
                  21.3.9.21  Scalar restriction operation   cresv 32975
                  21.3.9.22  The commutative ring of gaussian integers   gzcrng 32995
                  21.3.9.23  The archimedean ordered field of real numbers   reofld 32996
                  21.3.9.24  The quotient map and quotient modules   qusker 33001
                  21.3.9.25  The ring of integers modulo ` N `   znfermltl 33018
                  21.3.9.26  Independent sets and families   islinds5 33019
                  ​*21.3.9.27  Subgroup sum / Sumset / Minkowski sum   elgrplsmsn 33039
                  21.3.9.28  The quotient map   qusmul 33054
                  21.3.9.29  Ideals   intlidl 33069
                  21.3.9.30  Prime Ideals   cprmidl 33086
                  21.3.9.31  Maximal Ideals   cmxidl 33108
                  21.3.9.32  The semiring of ideals of a ring   cidlsrg 33147
                  21.3.9.33  Unique factorization domains   cufd 33163
                  21.3.9.34  Univariate Polynomials   0ringmon1p 33168
                  21.3.9.35  Polynomial quotient and polynomial remainder   q1pdir 33205
                  21.3.9.36  The subring algebra   sra1r 33214
                  21.3.9.37  Division Ring Extensions   drgext0g 33221
                  21.3.9.38  Vector Spaces   lvecdimfi 33227
                  21.3.9.39  Vector Space Dimension   cldim 33228
            21.3.10  Field Extensions   cfldext 33262
                  21.3.10.1  Algebraic numbers   cirng 33293
                  21.3.10.2  Minimal polynomials   cminply 33302
            21.3.11  Matrices   csmat 33330
                  21.3.11.1  Submatrices   csmat 33330
                  21.3.11.2  Matrix literals   clmat 33348
                  21.3.11.3  Laplace expansion of determinants   mdetpmtr1 33360
            21.3.12  Topology   ist0cld 33370
                  21.3.12.1  Open maps   txomap 33371
                  21.3.12.2  Topology of the unit circle   qtopt1 33372
                  21.3.12.3  Refinements   reff 33376
                  21.3.12.4  Open cover refinement property   ccref 33379
                  21.3.12.5  Lindelöf spaces   cldlf 33389
                  21.3.12.6  Paracompact spaces   cpcmp 33392
                  ​*21.3.12.7  Spectrum of a ring   crspec 33399
                  21.3.12.8  Pseudometrics   cmetid 33423
                  21.3.12.9  Continuity - misc additions   hauseqcn 33435
                  21.3.12.10  Topology of the closed unit interval   elunitge0 33436
                  21.3.12.11  Topology of ` ( RR X. RR ) `   unicls 33440
                  21.3.12.12  Order topology - misc. additions   cnvordtrestixx 33450
                  21.3.12.13  Continuity in topological spaces - misc. additions   mndpluscn 33463
                  21.3.12.14  Topology of the extended nonnegative real numbers ordered monoid   xrge0hmph 33469
                  21.3.12.15  Limits - misc additions   lmlim 33484
                  21.3.12.16  Univariate polynomials   pl1cn 33492
            21.3.13  Uniform Stuctures and Spaces   chcmp 33493
                  21.3.13.1  Hausdorff uniform completion   chcmp 33493
            21.3.14  Topology and algebraic structures   zringnm 33495
                  21.3.14.1  The norm on the ring of the integer numbers   zringnm 33495
                  21.3.14.2  Topological ` ZZ ` -modules   zlm0 33497
                  21.3.14.3  Canonical embedding of the field of the rational numbers into a division ring   cqqh 33509
                  21.3.14.4  Canonical embedding of the real numbers into a complete ordered field   crrh 33530
                  21.3.14.5  Embedding from the extended real numbers into a complete lattice   cxrh 33553
                  21.3.14.6  Canonical embeddings into the ordered field of the real numbers   zrhre 33556
                  ​*21.3.14.7  Topological Manifolds   cmntop 33559
            21.3.15  Real and complex functions   nexple 33564
                  21.3.15.1  Integer powers - misc. additions   nexple 33564
                  21.3.15.2  Indicator Functions   cind 33565
                  21.3.15.3  Extended sum   cesum 33582
            21.3.16  Mixed Function/Constant operation   cofc 33650
            21.3.17  Abstract measure   csiga 33663
                  21.3.17.1  Sigma-Algebra   csiga 33663
                  21.3.17.2  Generated sigma-Algebra   csigagen 33693
                  ​*21.3.17.3  lambda and pi-Systems, Rings of Sets   ispisys 33707
                  21.3.17.4  The Borel algebra on the real numbers   cbrsiga 33736
                  21.3.17.5  Product Sigma-Algebra   csx 33743
                  21.3.17.6  Measures   cmeas 33750
                  21.3.17.7  The counting measure   cntmeas 33781
                  21.3.17.8  The Lebesgue measure - misc additions   voliune 33784
                  21.3.17.9  The Dirac delta measure   cdde 33787
                  21.3.17.10  The 'almost everywhere' relation   cae 33792
                  21.3.17.11  Measurable functions   cmbfm 33804
                  21.3.17.12  Borel Algebra on ` ( RR X. RR ) `   br2base 33825
                  ​*21.3.17.13  Caratheodory's extension theorem   coms 33847
            21.3.18  Integration   itgeq12dv 33882
                  21.3.18.1  Lebesgue integral - misc additions   itgeq12dv 33882
                  21.3.18.2  Bochner integral   citgm 33883
            21.3.19  Euler's partition theorem   oddpwdc 33910
            21.3.20  Sequences defined by strong recursion   csseq 33939
            21.3.21  Fibonacci Numbers   cfib 33952
            21.3.22  Probability   cprb 33963
                  21.3.22.1  Probability Theory   cprb 33963
                  21.3.22.2  Conditional Probabilities   ccprob 33987
                  21.3.22.3  Real-valued Random Variables   crrv 33996
                  21.3.22.4  Preimage set mapping operator   corvc 34011
                  21.3.22.5  Distribution Functions   orvcelval 34024
                  21.3.22.6  Cumulative Distribution Functions   orvclteel 34028
                  21.3.22.7  Probabilities - example   coinfliplem 34034
                  21.3.22.8  Bertrand's Ballot Problem   ballotlemoex 34041
            21.3.23  Signum (sgn or sign) function - misc. additions   sgncl 34094
                  21.3.23.1  Operations on words   ccatmulgnn0dir 34110
            21.3.24  Polynomials with real coefficients - misc additions   plymul02 34114
            21.3.25  Descartes's rule of signs   signspval 34120
                  21.3.25.1  Sign changes in a word over real numbers   signspval 34120
                  21.3.25.2  Counting sign changes in a word over real numbers   signslema 34130
            21.3.26  Number Theory   iblidicc 34160
                  21.3.26.1  Representations of a number as sums of integers   crepr 34176
                  21.3.26.2  Vinogradov Trigonometric Sums and the Circle Method   cvts 34203
                  21.3.26.3  The Ternary Goldbach Conjecture: Final Statement   ax-hgt749 34212
            21.3.27  Elementary Geometry   cstrkg2d 34232
                  ​*21.3.27.1  Two-dimensional geometry   cstrkg2d 34232
                  21.3.27.2  Outer Five Segment (not used, no need to move to main)   cafs 34237
            ​*21.3.28  LeftPad Project   clpad 34242
      ​​*21.4  Mathbox for Jonathan Ben-Naim
            21.4.1  First-order logic and set theory   bnj170 34265
            21.4.2  Well founded induction and recursion   bnj110 34425
            21.4.3  The existence of a minimal element in certain classes   bnj69 34577
            21.4.4  Well-founded induction   bnj1204 34579
            21.4.5  Well-founded recursion, part 1 of 3   bnj60 34629
            21.4.6  Well-founded recursion, part 2 of 3   bnj1500 34635
            21.4.7  Well-founded recursion, part 3 of 3   bnj1522 34639
      ​21.5  Mathbox for BTernaryTau
            21.5.1  ZF set theory   exdifsn 34640
                  21.5.1.1  Finitism   fineqvrep 34651
            21.5.2  Real and complex numbers   zltp1ne 34655
            21.5.3  Graph theory   lfuhgr 34663
                  21.5.3.1  Acyclic graphs   cacycgr 34688
      ​21.6  Mathbox for Mario Carneiro
            21.6.1  Predicate calculus with all distinct variables   ax-7d 34705
            21.6.2  Miscellaneous stuff   quartfull 34711
            21.6.3  Derangements and the Subfactorial   deranglem 34712
            21.6.4  The Erdős-Szekeres theorem   erdszelem1 34737
            21.6.5  The Kuratowski closure-complement theorem   kur14lem1 34752
            21.6.6  Retracts and sections   cretr 34763
            21.6.7  Path-connected and simply connected spaces   cpconn 34765
            21.6.8  Covering maps   ccvm 34801
            21.6.9  Normal numbers   snmlff 34875
            21.6.10  Godel-sets of formulas - part 1   cgoe 34879
            21.6.11  Godel-sets of formulas - part 2   cgon 34978
            21.6.12  Models of ZF   cgze 34992
            ​*21.6.13  Metamath formal systems   cmcn 35006
            21.6.14  Grammatical formal systems   cm0s 35131
            21.6.15  Models of formal systems   cmuv 35151
            21.6.16  Splitting fields   ccpms 35173
            21.6.17  p-adic number fields   czr 35187
      ​​*21.7  Mathbox for Filip Cernatescu
      ​21.8  Mathbox for Paul Chapman
            21.8.1  Real and complex numbers (cont.)   climuzcnv 35211
            21.8.2  Miscellaneous theorems   elfzm12 35215
      ​21.9  Mathbox for Adrian Ducourtial
            21.9.1  Propositional calculus   currybi 35224
            21.9.2  Clone theory   ccloneop 35225
      ​21.10  Mathbox for Scott Fenton
            21.10.1  ZFC Axioms in primitive form   axextprim 35231
            21.10.2  Untangled classes   untelirr 35238
            21.10.3  Extra propositional calculus theorems   3jaodd 35245
            21.10.4  Misc. Useful Theorems   nepss 35248
            21.10.5  Properties of real and complex numbers   sqdivzi 35258
            21.10.6  Infinite products   iprodefisumlem 35270
            21.10.7  Factorial limits   faclimlem1 35273
            21.10.8  Greatest common divisor and divisibility   gcd32 35279
            21.10.9  Properties of relationships   dftr6 35281
            21.10.10  Properties of functions and mappings   funpsstri 35297
            21.10.11  Set induction (or epsilon induction)   setinds 35310
            21.10.12  Ordinal numbers   elpotr 35313
            21.10.13  Defined equality axioms   axextdfeq 35329
            21.10.14  Hypothesis builders   hbntg 35337
            21.10.15  Well-founded zero, successor, and limits   cwsuc 35342
            21.10.16  Quantifier-free definitions   ctxp 35362
            21.10.17  Alternate ordered pairs   caltop 35488
            21.10.18  Geometry in the Euclidean space   cofs 35514
                  21.10.18.1  Congruence properties   cofs 35514
                  21.10.18.2  Betweenness properties   btwntriv2 35544
                  21.10.18.3  Segment Transportation   ctransport 35561
                  21.10.18.4  Properties relating betweenness and congruence   cifs 35567
                  21.10.18.5  Connectivity of betweenness   btwnconn1lem1 35619
                  21.10.18.6  Segment less than or equal to   csegle 35638
                  21.10.18.7  Outside-of relationship   coutsideof 35651
                  21.10.18.8  Lines and Rays   cline2 35666
            21.10.19  Forward difference   cfwddif 35690
            21.10.20  Rank theorems   rankung 35698
            21.10.21  Hereditarily Finite Sets   chf 35704
      ​21.11  Mathbox for Gino Giotto
            21.11.1  Study of ax-mulf usage.   mpomulnzcnf 35719
      ​21.12  Mathbox for Jeff Hankins
            21.12.1  Miscellany   a1i14 35720
            21.12.2  Basic topological facts   topbnd 35744
            21.12.3  Topology of the real numbers   ivthALT 35755
            21.12.4  Refinements   cfne 35756
            21.12.5  Neighborhood bases determine topologies   neibastop1 35779
            21.12.6  Lattice structure of topologies   topmtcl 35783
            21.12.7  Filter bases   fgmin 35790
            21.12.8  Directed sets, nets   tailfval 35792
      ​21.13  Mathbox for Anthony Hart
            21.13.1  Propositional Calculus   tb-ax1 35803
            21.13.2  Predicate Calculus   nalfal 35823
            21.13.3  Miscellaneous single axioms   meran1 35831
            21.13.4  Connective Symmetry   negsym1 35837
      ​21.14  Mathbox for Chen-Pang He
            21.14.1  Ordinal topology   ontopbas 35848
      ​21.15  Mathbox for Jeff Hoffman
            21.15.1  Inferences for finite induction on generic function values   fveleq 35871
            21.15.2  gdc.mm   nnssi2 35875
      ​21.16  Mathbox for Asger C. Ipsen
            21.16.1  Continuous nowhere differentiable functions   dnival 35882
      ​​*21.17  Mathbox for BJ
            *21.17.1  Propositional calculus   bj-mp2c 35951
                  *21.17.1.1  Derived rules of inference   bj-mp2c 35951
                  ​*21.17.1.2  A syntactic theorem   bj-0 35953
                  21.17.1.3  Minimal implicational calculus   bj-a1k 35955
                  ​*21.17.1.4  Positive calculus   bj-syl66ib 35966
                  21.17.1.5  Implication and negation   bj-con2com 35972
                  ​*21.17.1.6  Disjunction   bj-jaoi1 35983
                  ​*21.17.1.7  Logical equivalence   bj-dfbi4 35985
                  21.17.1.8  The conditional operator for propositions   bj-consensus 35990
                  ​*21.17.1.9  Propositional calculus: miscellaneous   bj-imbi12 35995
            ​*21.17.2  Modal logic   bj-axdd2 36005
            ​*21.17.3  Provability logic   cprvb 36010
            ​*21.17.4  First-order logic   bj-genr 36019
                  21.17.4.1  Adding ax-gen   bj-genr 36019
                  21.17.4.2  Adding ax-4   bj-2alim 36023
                  21.17.4.3  Adding ax-5   bj-ax12wlem 36056
                  21.17.4.4  Equality and substitution   bj-ssbeq 36065
                  21.17.4.5  Adding ax-6   bj-spimvwt 36081
                  21.17.4.6  Adding ax-7   bj-cbvexw 36088
                  21.17.4.7  Membership predicate, ax-8 and ax-9   bj-ax89 36090
                  21.17.4.8  Adding ax-11   bj-alcomexcom 36093
                  21.17.4.9  Adding ax-12   axc11n11 36095
                  21.17.4.10  Nonfreeness   wnnf 36136
                  21.17.4.11  Adding ax-13   bj-axc10 36196
                  ​*21.17.4.12  Removing dependencies on ax-13 (and ax-11)   bj-axc10v 36206
                  ​*21.17.4.13  Distinct var metavariables   bj-hbaeb2 36231
                  ​*21.17.4.14  Around ~ equsal   bj-equsal1t 36235
                  ​*21.17.4.15  Some Principia Mathematica proofs   stdpc5t 36240
                  21.17.4.16  Alternate definition of substitution   bj-sbsb 36250
                  21.17.4.17  Lemmas for substitution   bj-sbf3 36252
                  21.17.4.18  Existential uniqueness   bj-eu3f 36254
                  ​*21.17.4.19  First-order logic: miscellaneous   bj-sblem1 36255
            21.17.5  Set theory   eliminable1 36272
                  ​*21.17.5.1  Eliminability of class terms   eliminable1 36272
                  ​*21.17.5.2  Classes without the axiom of extensionality   bj-denoteslem 36284
                  21.17.5.3  Characterization among sets versus among classes   elelb 36311
                  ​*21.17.5.4  The nonfreeness quantifier for classes   bj-nfcsym 36313
                  ​*21.17.5.5  Lemmas for class substitution   bj-sbeqALT 36314
                  21.17.5.6  Removing some axiom requirements and disjoint variable conditions   bj-exlimvmpi 36325
                  ​*21.17.5.7  Class abstractions   bj-elabd2ALT 36339
                  21.17.5.8  Generalized class abstractions   bj-cgab 36347
                  ​*21.17.5.9  Restricted nonfreeness   wrnf 36355
                  ​*21.17.5.10  Russell's paradox   bj-ru0 36357
                  21.17.5.11  Curry's paradox in set theory   currysetlem 36360
                  ​*21.17.5.12  Some disjointness results   bj-n0i 36366
                  ​*21.17.5.13  Complements on direct products   bj-xpimasn 36370
                  ​*21.17.5.14  "Singletonization" and tagging   bj-snsetex 36378
                  ​*21.17.5.15  Tuples of classes   bj-cproj 36405
                  ​*21.17.5.16  Set theory: elementary operations relative to a universe   bj-rcleqf 36440
                  ​*21.17.5.17  Axioms for finite unions   bj-abex 36445
                  ​*21.17.5.18  Set theory: miscellaneous   eleq2w2ALT 36462
                  ​*21.17.5.19  Evaluation at a class   bj-evaleq 36487
                  21.17.5.20  Elementwise operations   celwise 36494
                  ​*21.17.5.21  Elementwise intersection (families of sets induced on a subset)   bj-rest00 36496
                  21.17.5.22  Moore collections (complements)   bj-raldifsn 36515
                  21.17.5.23  Maps-to notation for functions with three arguments   bj-0nelmpt 36531
                  ​*21.17.5.24  Currying   csethom 36537
                  ​*21.17.5.25  Setting components of extensible structures   cstrset 36549
            ​*21.17.6  Extended real and complex numbers, real and complex projective lines   bj-nfald 36552
                  21.17.6.1  Complements on class abstractions of ordered pairs and binary relations   bj-nfald 36552
                  ​*21.17.6.2  Identity relation (complements)   bj-opabssvv 36565
                  ​*21.17.6.3  Functionalized identity (diagonal in a Cartesian square)   cdiag2 36587
                  ​*21.17.6.4  Direct image and inverse image   cimdir 36593
                  ​*21.17.6.5  Extended numbers and projective lines as sets   cfractemp 36611
                  ​*21.17.6.6  Addition and opposite   caddcc 36652
                  ​*21.17.6.7  Order relation on the extended reals   cltxr 36656
                  ​*21.17.6.8  Argument, multiplication and inverse   carg 36658
                  21.17.6.9  The canonical bijection from the finite ordinals   ciomnn 36664
                  21.17.6.10  Divisibility   cnnbar 36675
            ​*21.17.7  Monoids   bj-smgrpssmgm 36683
                  ​*21.17.7.1  Finite sums in monoids   cfinsum 36698
            ​*21.17.8  Affine, Euclidean, and Cartesian geometry   bj-fvimacnv0 36701
                  *21.17.8.1  Real vector spaces   bj-fvimacnv0 36701
                  ​*21.17.8.2  Complex numbers (supplements)   bj-subcom 36723
                  ​*21.17.8.3  Barycentric coordinates   bj-bary1lem 36725
            21.17.9  Monoid of endomorphisms   cend 36728
      ​21.18  Mathbox for Jim Kingdon
                  21.18.0.1  Circle constant   taupilem3 36734
                  21.18.0.2  Number theory   dfgcd3 36739
                  21.18.0.3  Real numbers   irrdifflemf 36740
      ​21.19  Mathbox for ML
            21.19.1  Miscellaneous   csbrecsg 36743
            21.19.2  Cartesian exponentiation   cfinxp 36798
            21.19.3  Topology   iunctb2 36818
                  ​*21.19.3.1  Pi-base theorems   pibp16 36828
      ​21.20  Mathbox for Wolf Lammen
            21.20.1  1. Bootstrapping   wl-section-boot 36837
            21.20.2  Implication chains   wl-section-impchain 36861
            21.20.3  Theorems around the conditional operator   wl-ifp-ncond1 36879
            21.20.4  Alternative development of hadd, cadd   wl-df-3xor 36883
            21.20.5  An alternative axiom ~ ax-13   ax-wl-13v 36908
            21.20.6  Other stuff   wl-mps 36910
      ​21.21  Mathbox for Brendan Leahy
      ​21.22  Mathbox for Jeff Madsen
            21.22.1  Logic and set theory   unirep 37122
            21.22.2  Real and complex numbers; integers   filbcmb 37148
            21.22.3  Sequences and sums   sdclem2 37150
            21.22.4  Topology   subspopn 37160
            21.22.5  Metric spaces   metf1o 37163
            21.22.6  Continuous maps and homeomorphisms   constcncf 37170
            21.22.7  Boundedness   ctotbnd 37174
            21.22.8  Isometries   cismty 37206
            21.22.9  Heine-Borel Theorem   heibor1lem 37217
            21.22.10  Banach Fixed Point Theorem   bfplem1 37230
            21.22.11  Euclidean space   crrn 37233
            21.22.12  Intervals (continued)   ismrer1 37246
            21.22.13  Operation properties   cass 37250
            21.22.14  Groups and related structures   cmagm 37256
            21.22.15  Group homomorphism and isomorphism   cghomOLD 37291
            21.22.16  Rings   crngo 37302
            21.22.17  Division Rings   cdrng 37356
            21.22.18  Ring homomorphisms   crngohom 37368
            21.22.19  Commutative rings   ccm2 37397
            21.22.20  Ideals   cidl 37415
            21.22.21  Prime rings and integral domains   cprrng 37454
            21.22.22  Ideal generators   cigen 37467
      ​21.23  Mathbox for Giovanni Mascellani
            ​*21.23.1  Tools for automatic proof building   efald2 37486
            ​*21.23.2  Tseitin axioms   fald 37537
            ​*21.23.3  Equality deductions   iuneq2f 37564
            ​*21.23.4  Miscellanea   orcomdd 37575
      ​21.24  Mathbox for Peter Mazsa
            21.24.1  Notations   cxrn 37582
            21.24.2  Preparatory theorems   el2v1 37625
            21.24.3  Range Cartesian product   df-xrn 37780
            21.24.4  Cosets by ` R `   df-coss 37820
            21.24.5  Relations   df-rels 37894
            21.24.6  Subset relations   df-ssr 37907
            21.24.7  Reflexivity   df-refs 37919
            21.24.8  Converse reflexivity   df-cnvrefs 37934
            21.24.9  Symmetry   df-syms 37951
            21.24.10  Reflexivity and symmetry   symrefref2 37972
            21.24.11  Transitivity   df-trs 37981
            21.24.12  Equivalence relations   df-eqvrels 37993
            21.24.13  Redundancy   df-redunds 38032
            21.24.14  Domain quotients   df-dmqss 38047
            21.24.15  Equivalence relations on domain quotients   df-ers 38072
            21.24.16  Functions   df-funss 38089
            21.24.17  Disjoints vs. converse functions   df-disjss 38112
            21.24.18  Antisymmetry   df-antisymrel 38169
            21.24.19  Partitions: disjoints on domain quotients   df-parts 38174
            21.24.20  Partition-Equivalence Theorems   disjim 38190
      ​21.25  Mathbox for Rodolfo Medina
            21.25.1  Partitions   prtlem60 38262
      ​​*21.26  Mathbox for Norm Megill
            *21.26.1  Obsolete schemes ax-c4,c5,c7,c10,c11,c11n,c15,c9,c14,c16   ax-c5 38292
            ​*21.26.2  Rederive new axioms ax-4, ax-10, ax-6, ax-12, ax-13 from old   axc5 38302
            ​*21.26.3  Legacy theorems using obsolete axioms   ax5ALT 38316
            21.26.4  Experiments with weak deduction theorem   elimhyps 38370
            21.26.5  Miscellanea   cnaddcom 38381
            21.26.6  Atoms, hyperplanes, and covering in a left vector space (or module)   clsa 38383
            21.26.7  Functionals and kernels of a left vector space (or module)   clfn 38466
            21.26.8  Opposite rings and dual vector spaces   cld 38532
            21.26.9  Ortholattices and orthomodular lattices   cops 38581
            21.26.10  Atomic lattices with covering property   ccvr 38671
            21.26.11  Hilbert lattices   chlt 38759
            21.26.12  Projective geometries based on Hilbert lattices   clln 38901
            21.26.13  Construction of a vector space from a Hilbert lattice   cdlema1N 39201
            21.26.14  Construction of involution and inner product from a Hilbert lattice   clpoN 40890
      ​21.27  Mathbox for metakunt
            21.27.1  Commutative Semiring   ccsrg 41376
            21.27.2  General helpful statements   leexp1ad 41379
            21.27.3  Some gcd and lcm results   12gcd5e1 41411
            21.27.4  Least common multiple inequality theorem   3factsumint1 41429
            21.27.5  Logarithm inequalities   3exp7 41461
            21.27.6  Miscellaneous results for AKS formalisation   intlewftc 41469
            21.27.7  Sticks and stones   sticksstones1 41550
            21.27.8  Continuation AKS   aks6d1c6lem1 41574
            21.27.9  Permutation results   metakunt1 41577
            21.27.10  Unused lemmas scheduled for deletion   fac2xp3 41611
      ​21.28  Mathbox for Steven Nguyen
            21.28.1  Utility theorems   ioin9i8 41615
            21.28.2  Structures   nelsubginvcld 41656
            ​*21.28.3  Arithmetic theorems   c0exALT 41756
            21.28.4  Exponents and divisibility   oexpreposd 41803
            21.28.5  Real subtraction   cresub 41842
            ​*21.28.6  Projective spaces   cprjsp 41947
            21.28.7  Basic reductions for Fermat's Last Theorem   dffltz 41980
            ​*21.28.8  Exemplar theorems   iddii 42010
      ​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 42034
            21.31.2  Additional theory of functions   imaiinfv 42035
            21.31.3  Additional topology   elrfi 42036
            21.31.4  Characterization of closure operators. Kuratowski closure axioms   ismrcd1 42040
            21.31.5  Algebraic closure systems   cnacs 42044
            21.31.6  Miscellanea 1. Map utilities   constmap 42055
            21.31.7  Miscellanea for polynomials   mptfcl 42062
            21.31.8  Multivariate polynomials over the integers   cmzpcl 42063
            21.31.9  Miscellanea for Diophantine sets 1   coeq0i 42095
            21.31.10  Diophantine sets 1: definitions   cdioph 42097
            21.31.11  Diophantine sets 2 miscellanea   ellz1 42109
            21.31.12  Diophantine sets 2: union and intersection. Monotone Boolean algebra   diophin 42114
            21.31.13  Diophantine sets 3: construction   diophrex 42117
            21.31.14  Diophantine sets 4 miscellanea   2sbcrex 42126
            21.31.15  Diophantine sets 4: Quantification   rexrabdioph 42136
            21.31.16  Diophantine sets 5: Arithmetic sets   rabdiophlem1 42143
            21.31.17  Diophantine sets 6: reusability. renumbering of variables   eldioph4b 42153
            21.31.18  Pigeonhole Principle and cardinality helpers   fphpd 42158
            21.31.19  A non-closed set of reals is infinite   rencldnfilem 42162
            21.31.20  Lagrange's rational approximation theorem   irrapxlem1 42164
            21.31.21  Pell equations 1: A nontrivial solution always exists   pellexlem1 42171
            21.31.22  Pell equations 2: Algebraic number theory of the solution set   csquarenn 42178
            21.31.23  Pell equations 3: characterizing fundamental solution   infmrgelbi 42220
            ​*21.31.24  Logarithm laws generalized to an arbitrary base   reglogcl 42232
            21.31.25  Pell equations 4: the positive solution group is infinite cyclic   pellfund14 42240
            21.31.26  X and Y sequences 1: Definition and recurrence laws   crmx 42242
            21.31.27  Ordering and induction lemmas for the integers   monotuz 42284
            21.31.28  X and Y sequences 2: Order properties   rmxypos 42290
            21.31.29  Congruential equations   congtr 42308
            21.31.30  Alternating congruential equations   acongid 42318
            21.31.31  Additional theorems on integer divisibility   coprmdvdsb 42328
            21.31.32  X and Y sequences 3: Divisibility properties   jm2.18 42331
            21.31.33  X and Y sequences 4: Diophantine representability of Y   jm2.27a 42348
            21.31.34  X and Y sequences 5: Diophantine representability of X, ^, _C   rmxdiophlem 42358
            21.31.35  Uncategorized stuff not associated with a major project   setindtr 42367
            21.31.36  More equivalents of the Axiom of Choice   axac10 42376
            21.31.37  Finitely generated left modules   clfig 42413
            21.31.38  Noetherian left modules I   clnm 42421
            21.31.39  Addenda for structure powers   pwssplit4 42435
            21.31.40  Every set admits a group structure iff choice   unxpwdom3 42441
            21.31.41  Noetherian rings and left modules II   clnr 42455
            21.31.42  Hilbert's Basis Theorem   cldgis 42467
            21.31.43  Additional material on polynomials [DEPRECATED]   cmnc 42477
            21.31.44  Degree and minimal polynomial of algebraic numbers   cdgraa 42486
            21.31.45  Algebraic integers I   citgo 42503
            21.31.46  Endomorphism algebra   cmend 42521
            21.31.47  Cyclic groups and order   idomodle 42541
            21.31.48  Cyclotomic polynomials   ccytp 42547
            21.31.49  Miscellaneous topology   fgraphopab 42554
      ​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 42568
            21.34.2  Natural addition of Cantor normal forms   oawordex2 42678
            21.34.3  Surreal Contributions   abeqabi 42761
            21.34.4  Short Studies   nlimsuc 42794
                  21.34.4.1  Additional work on conditional logical operator   ifpan123g 42812
                  21.34.4.2  Sophisms   rp-fakeimass 42865
                  ​*21.34.4.3  Finite Sets   rp-isfinite5 42870
                  21.34.4.4  General Observations   intabssd 42872
                  21.34.4.5  Infinite Sets   pwelg 42913
                  ​*21.34.4.6  Finite intersection property   fipjust 42918
                  21.34.4.7  RP ADDTO: Subclasses and subsets   rababg 42927
                  21.34.4.8  RP ADDTO: The intersection of a class   elinintab 42928
                  21.34.4.9  RP ADDTO: Theorems requiring subset and intersection existence   elinintrab 42930
                  21.34.4.10  RP ADDTO: Relations   xpinintabd 42933
                  ​*21.34.4.11  RP ADDTO: Functions   elmapintab 42949
                  ​*21.34.4.12  RP ADDTO: Finite induction (for finite ordinals)   cnvcnvintabd 42953
                  21.34.4.13  RP ADDTO: First and second members of an ordered pair   elcnvlem 42954
                  21.34.4.14  RP ADDTO: The reflexive and transitive properties of relations   undmrnresiss 42957
                  21.34.4.15  RP ADDTO: Basic properties of closures   cleq2lem 42961
                  21.34.4.16  RP REPLACE: Definitions and basic properties of transitive closures   trcleq2lemRP 42983
                  ​*21.34.4.17  Additions for square root; absolute value   sqrtcvallem1 42984
            21.34.5  Additional statements on relations and subclasses   al3im 43000
                  21.34.5.1  Transitive relations (not to be confused with transitive classes).   trrelind 43018
                  21.34.5.2  Reflexive closures   crcl 43025
                  ​*21.34.5.3  Finite relationship composition.   relexp2 43030
                  21.34.5.4  Transitive closure of a relation   dftrcl3 43073
                  ​*21.34.5.5  Adapted from Frege   frege77d 43099
            ​*21.34.6  Propositions from _Begriffsschrift_   dfxor4 43119
                  *21.34.6.1  _Begriffsschrift_ Chapter I   dfxor4 43119
                  ​*21.34.6.2  _Begriffsschrift_ Notation hints   whe 43125
                  21.34.6.3  _Begriffsschrift_ Chapter II Implication   ax-frege1 43143
                  21.34.6.4  _Begriffsschrift_ Chapter II Implication and Negation   axfrege28 43182
                  ​*21.34.6.5  _Begriffsschrift_ Chapter II with logical equivalence   axfrege52a 43209
                  21.34.6.6  _Begriffsschrift_ Chapter II with equivalence of sets   axfrege52c 43240
                  ​*21.34.6.7  _Begriffsschrift_ Chapter II with equivalence of classes   frege53c 43267
                  ​*21.34.6.8  _Begriffsschrift_ Chapter III Properties hereditary in a sequence   dffrege69 43285
                  ​*21.34.6.9  _Begriffsschrift_ Chapter III Following in a sequence   dffrege76 43292
                  ​*21.34.6.10  _Begriffsschrift_ Chapter III Member of sequence   dffrege99 43315
                  ​*21.34.6.11  _Begriffsschrift_ Chapter III Single-valued procedures   dffrege115 43331
            ​*21.34.7  Exploring Topology via Seifert and Threlfall   enrelmap 43350
                  *21.34.7.1  Equinumerosity of sets of relations and maps   enrelmap 43350
                  ​*21.34.7.2  Generic Pseudoclosure Spaces, Pseudointerior Spaces, and Pseudoneighborhoods   or3or 43376
                  ​*21.34.7.3  Generic Neighborhood Spaces   gneispa 43483
            ​*21.34.8  Exploring Higher Homotopy via Kerodon   k0004lem1 43500
                  *21.34.8.1  Simplicial Sets   k0004lem1 43500
      ​21.35  Mathbox for Stanislas Polu
            21.35.1  IMO Problems   wwlemuld 43509
                  21.35.1.1  IMO 1972 B2   wwlemuld 43509
            ​*21.35.2  INT Inequalities Proof Generator   int-addcomd 43526
            ​*21.35.3  N-Digit Addition Proof Generator   unitadd 43548
            21.35.4  AM-GM (for k = 2,3,4)   gsumws3 43549
      ​21.36  Mathbox for Rohan Ridenour
            21.36.1  Misc   spALT 43554
            21.36.2  Monoid rings   cmnring 43566
            21.36.3  Shorter primitive equivalent of ax-groth   gru0eld 43589
                  21.36.3.1  Grothendieck universes are closed under collection   gru0eld 43589
                  21.36.3.2  Minimal universes   ismnu 43621
                  21.36.3.3  Primitive equivalent of ax-groth   expandan 43648
      ​21.37  Mathbox for Steve Rodriguez
            21.37.1  Miscellanea   nanorxor 43665
            21.37.2  Ratio test for infinite series convergence and divergence   dvgrat 43672
            21.37.3  Multiples   reldvds 43675
            21.37.4  Function operations   caofcan 43683
            21.37.5  Calculus   lhe4.4ex1a 43689
            21.37.6  The generalized binomial coefficient operation   cbcc 43696
            21.37.7  Binomial series   uzmptshftfval 43706
      ​21.38  Mathbox for Andrew Salmon
            21.38.1  Principia Mathematica * 10   pm10.12 43718
            21.38.2  Principia Mathematica * 11   2alanimi 43732
            21.38.3  Predicate Calculus   sbeqal1 43758
            21.38.4  Principia Mathematica * 13 and * 14   pm13.13a 43767
            21.38.5  Set Theory   elnev 43798
            21.38.6  Arithmetic   addcomgi 43816
            21.38.7  Geometry   cplusr 43817
      ​​*21.39  Mathbox for Alan Sare
            21.39.1  Auxiliary theorems for the Virtual Deduction tool   idiALT 43839
            21.39.2  Supplementary unification deductions   bi1imp 43843
            21.39.3  Conventional Metamath proofs, some derived from VD proofs   iidn3 43863
            21.39.4  What is Virtual Deduction?   wvd1 43931
            21.39.5  Virtual Deduction Theorems   df-vd1 43932
            21.39.6  Theorems proved using Virtual Deduction   trsspwALT 44180
            21.39.7  Theorems proved using Virtual Deduction with mmj2 assistance   simplbi2VD 44208
            21.39.8  Virtual Deduction transcriptions of textbook proofs   sb5ALTVD 44275
            21.39.9  Theorems proved using conjunction-form Virtual Deduction   elpwgdedVD 44279
            21.39.10  Theorems with a VD proof in conventional notation derived from a VD proof   suctrALT3 44286
            ​*21.39.11  Theorems with a proof in conventional notation derived from a VD proof   notnotrALT2 44289
      ​21.40  Mathbox for Glauco Siliprandi
            21.40.1  Miscellanea   evth2f 44300
            21.40.2  Functions   feq1dd 44463
            21.40.3  Ordering on real numbers - Real and complex numbers basic operations   sub2times 44577
            21.40.4  Real intervals   gtnelioc 44799
            21.40.5  Finite sums   fsummulc1f 44882
            21.40.6  Finite multiplication of numbers and finite multiplication of functions   fmul01 44891
            21.40.7  Limits   clim1fr1 44912
                  21.40.7.1  Inferior limit (lim inf)   clsi 45062
                  ​*21.40.7.2  Limits for sequences of extended real numbers   clsxlim 45129
            21.40.8  Trigonometry   coseq0 45175
            21.40.9  Continuous Functions   mulcncff 45181
            21.40.10  Derivatives   dvsinexp 45222
            21.40.11  Integrals   itgsin0pilem1 45261
            21.40.12  Stone Weierstrass theorem - real version   stoweidlem1 45312
            21.40.13  Wallis' product for π   wallispilem1 45376
            21.40.14  Stirling's approximation formula for ` n ` factorial   stirlinglem1 45385
            21.40.15  Dirichlet kernel   dirkerval 45402
            21.40.16  Fourier Series   fourierdlem1 45419
            21.40.17  e is transcendental   elaa2lem 45544
            21.40.18  n-dimensional Euclidean space   rrxtopn 45595
            21.40.19  Basic measure theory   csalg 45619
                  ​*21.40.19.1  σ-Algebras   csalg 45619
                  21.40.19.2  Sum of nonnegative extended reals   csumge0 45673
                  ​*21.40.19.3  Measures   cmea 45760
                  ​*21.40.19.4  Outer measures and Caratheodory's construction   come 45800
                  ​*21.40.19.5  Lebesgue measure on n-dimensional Real numbers   covoln 45847
                  ​*21.40.19.6  Measurable functions   csmblfn 46006
      ​21.41  Mathbox for Saveliy Skresanov
            21.41.1  Ceva's theorem   sigarval 46161
            21.41.2  Simple groups   simpcntrab 46181
      ​21.42  Mathbox for Ender Ting
            21.42.1  Increasing sequences and subsequences   et-ltneverrefl 46182
      ​21.43  Mathbox for Jarvin Udandy
      ​21.44  Mathbox for Adhemar
            ​*21.44.1  Minimal implicational calculus   adh-minim 46306
      ​21.45  Mathbox for Alexander van der Vekens
            21.45.1  General auxiliary theorems (1)   n0nsn2el 46330
                  21.45.1.1  Unordered and ordered pairs - extension for singletons   n0nsn2el 46330
                  21.45.1.2  Unordered and ordered pairs - extension for unordered pairs   elprneb 46334
                  21.45.1.3  Unordered and ordered pairs - extension for ordered pairs   oppr 46335
                  21.45.1.4  Relations - extension   eubrv 46340
                  21.45.1.5  Definite description binder (inverted iota) - extension   iota0def 46343
                  21.45.1.6  Functions - extension   fveqvfvv 46345
            21.45.2  Alternative for Russell's definition of a description binder   caiota 46386
            21.45.3  Double restricted existential uniqueness   r19.32 46401
                  21.45.3.1  Restricted quantification (extension)   r19.32 46401
                  21.45.3.2  Restricted uniqueness and "at most one" quantification   reuf1odnf 46410
                  21.45.3.3  Analogs to Existential uniqueness (double quantification)   2reu3 46413
                  21.45.3.4  Additional theorems for double restricted existential uniqueness   2reu8i 46416
            ​*21.45.4  Alternative definitions of function and operation values   wdfat 46419
                  21.45.4.1  Restricted quantification (extension)   ralbinrald 46425
                  21.45.4.2  The universal class (extension)   nvelim 46426
                  21.45.4.3  Introduce the Axiom of Power Sets (extension)   alneu 46427
                  21.45.4.4  Predicate "defined at"   dfateq12d 46429
                  21.45.4.5  Alternative definition of the value of a function   dfafv2 46435
                  21.45.4.6  Alternative definition of the value of an operation   aoveq123d 46481
            ​*21.45.5  Alternative definitions of function values (2)   cafv2 46511
            21.45.6  General auxiliary theorems (2)   an4com24 46571
                  21.45.6.1  Logical conjunction - extension   an4com24 46571
                  21.45.6.2  Abbreviated conjunction and disjunction of three wff's - extension   3an4ancom24 46572
                  21.45.6.3  Negated membership (alternative)   cnelbr 46574
                  21.45.6.4  The empty set - extension   ralralimp 46581
                  21.45.6.5  Indexed union and intersection - extension   otiunsndisjX 46582
                  21.45.6.6  Functions - extension   fvifeq 46583
                  21.45.6.7  Maps-to notation - extension   fvmptrab 46595
                  21.45.6.8  Subtraction - extension   cnambpcma 46597
                  21.45.6.9  Ordering on reals (cont.) - extension   leaddsuble 46600
                  21.45.6.10  Imaginary and complex number properties - extension   readdcnnred 46606
                  21.45.6.11  Nonnegative integers (as a subset of complex numbers) - extension   nn0resubcl 46611
                  21.45.6.12  Integers (as a subset of complex numbers) - extension   zgeltp1eq 46612
                  21.45.6.13  Decimal arithmetic - extension   1t10e1p1e11 46613
                  21.45.6.14  Upper sets of integers - extension   eluzge0nn0 46615
                  21.45.6.15  Infinity and the extended real number system (cont.) - extension   nltle2tri 46616
                  21.45.6.16  Finite intervals of integers - extension   ssfz12 46617
                  21.45.6.17  Half-open integer ranges - extension   fzopred 46625
                  21.45.6.18  The modulo (remainder) operation - extension   m1mod0mod1 46632
                  21.45.6.19  The infinite sequence builder "seq"   smonoord 46634
                  21.45.6.20  Finite and infinite sums - extension   fsummsndifre 46635
                  21.45.6.21  Extensible structures - extension   setsidel 46639
            ​*21.45.7  Preimages of function values   preimafvsnel 46642
            ​*21.45.8  Partitions of real intervals   ciccp 46676
            21.45.9  Shifting functions with an integer range domain   fargshiftfv 46702
            21.45.10  Words over a set (extension)   lswn0 46707
                  21.45.10.1  Last symbol of a word - extension   lswn0 46707
            21.45.11  Unordered pairs   wich 46708
                  21.45.11.1  Interchangeable setvar variables   wich 46708
                  21.45.11.2  Set of unordered pairs   sprid 46737
                  ​*21.45.11.3  Proper (unordered) pairs   prpair 46764
                  21.45.11.4  Set of proper unordered pairs   cprpr 46775
            21.45.12  Number theory (extension)   cfmtno 46790
                  ​*21.45.12.1  Fermat numbers   cfmtno 46790
                  ​*21.45.12.2  Mersenne primes   m2prm 46854
                  21.45.12.3  Proth's theorem   modexp2m1d 46875
                  21.45.12.4  Solutions of quadratic equations   quad1 46883
            ​*21.45.13  Even and odd numbers   ceven 46887
                  21.45.13.1  Definitions and basic properties   ceven 46887
                  21.45.13.2  Alternate definitions using the "divides" relation   dfeven2 46912
                  21.45.13.3  Alternate definitions using the "modulo" operation   dfeven3 46921
                  21.45.13.4  Alternate definitions using the "gcd" operation   iseven5 46927
                  21.45.13.5  Theorems of part 5 revised   zneoALTV 46932
                  21.45.13.6  Theorems of part 6 revised   odd2np1ALTV 46937
                  21.45.13.7  Theorems of AV's mathbox revised   0evenALTV 46951
                  21.45.13.8  Additional theorems   epoo 46966
                  21.45.13.9  Perfect Number Theorem (revised)   perfectALTVlem1 46984
            21.45.14  Number theory (extension 2)   cfppr 46987
                  ​*21.45.14.1  Fermat pseudoprimes   cfppr 46987
                  ​*21.45.14.2  Goldbach's conjectures   cgbe 47008
            21.45.15  Graph theory (extension)   cgrisom 47081
                  ​*21.45.15.1  Isomorphisms of graphs   cgrisom 47081
                  21.45.15.2  Loop-free graphs - extension   1hegrlfgr 47117
                  21.45.15.3  Walks - extension   cupwlks 47118
                  21.45.15.4  Edges of graphs expressed as sets of unordered pairs   upgredgssspr 47128
            21.45.16  Monoids (extension)   ovn0dmfun 47141
                  21.45.16.1  Auxiliary theorems   ovn0dmfun 47141
                  21.45.16.2  Magmas, Semigroups and Monoids (extension)   plusfreseq 47149
                  21.45.16.3  Examples and counterexamples for magmas, semigroups and monoids (extension)   opmpoismgm 47152
                  21.45.16.4  Group sum operation (extension 1)   gsumsplit2f 47165
            ​*21.45.17  Magmas and internal binary operations (alternate approach)   ccllaw 47168
                  *21.45.17.1  Laws for internal binary operations   ccllaw 47168
                  ​*21.45.17.2  Internal binary operations   cintop 47181
                  21.45.17.3  Alternative definitions for magmas and semigroups   cmgm2 47200
            21.45.18  Rings (extension)   lmod0rng 47214
                  21.45.18.1  Nonzero rings (extension)   lmod0rng 47214
                  21.45.18.2  Ideals as non-unital rings   lidldomn1 47216
                  21.45.18.3  The non-unital ring of even integers   0even 47222
                  21.45.18.4  A constructed not unital ring   cznrnglem 47244
                  ​*21.45.18.5  The category of non-unital rings (alternate definition)   crngcALTV 47248
                  ​*21.45.18.6  The category of (unital) rings (alternate definition)   cringcALTV 47272
            21.45.19  Basic algebraic structures (extension)   opeliun2xp 47319
                  21.45.19.1  Auxiliary theorems   opeliun2xp 47319
                  21.45.19.2  The binomial coefficient operation (extension)   bcpascm1 47338
                  21.45.19.3  The ` ZZ `-module ` ZZ X. ZZ `   zlmodzxzlmod 47341
                  21.45.19.4  Group sum operation (extension 2)   mgpsumunsn 47348
                  21.45.19.5  Symmetric groups (extension)   exple2lt6 47351
                  21.45.19.6  Divisibility (extension)   invginvrid 47354
                  21.45.19.7  The support of functions (extension)   rmsupp0 47355
                  21.45.19.8  Finitely supported functions (extension)   rmsuppfi 47360
                  21.45.19.9  Left modules (extension)   lmodvsmdi 47369
                  21.45.19.10  Associative algebras (extension)   assaascl0 47371
                  21.45.19.11  Univariate polynomials (extension)   ply1vr1smo 47373
                  21.45.19.12  Univariate polynomials (examples)   linply1 47384
            21.45.20  Linear algebra (extension)   cdmatalt 47387
                  ​*21.45.20.1  The subalgebras of diagonal and scalar matrices (extension)   cdmatalt 47387
                  ​*21.45.20.2  Linear combinations   clinc 47395
                  ​*21.45.20.3  Linear independence   clininds 47431
                  21.45.20.4  Simple left modules and the ` ZZ `-module   lmod1lem1 47478
                  21.45.20.5  Differences between (left) modules and (left) vector spaces   lvecpsslmod 47498
            21.45.21  Complexity theory   suppdm 47501
                  21.45.21.1  Auxiliary theorems   suppdm 47501
                  21.45.21.2  The modulo (remainder) operation (extension)   fldivmod 47514
                  21.45.21.3  Even and odd integers   nn0onn0ex 47519
                  21.45.21.4  The natural logarithm on complex numbers (extension)   logcxp0 47531
                  21.45.21.5  Division of functions   cfdiv 47533
                  21.45.21.6  Upper bounds   cbigo 47543
                  21.45.21.7  Logarithm to an arbitrary base (extension)   rege1logbrege0 47554
                  ​*21.45.21.8  The binary logarithm   fldivexpfllog2 47561
                  21.45.21.9  Binary length   cblen 47565
                  ​*21.45.21.10  Digits   cdig 47591
                  21.45.21.11  Nonnegative integer as sum of its shifted digits   dignn0flhalflem1 47611
                  21.45.21.12  Algorithms for the multiplication of nonnegative integers   nn0mulfsum 47620
                  ​*21.45.21.13  N-ary functions   cnaryf 47622
                  ​*21.45.21.14  The Ackermann function   citco 47653
            21.45.22  Elementary geometry (extension)   fv1prop 47695
                  21.45.22.1  Auxiliary theorems   fv1prop 47695
                  21.45.22.2  Real euclidean space of dimension 2   rrx2pxel 47707
                  21.45.22.3  Spheres and lines in real Euclidean spaces   cline 47723
      ​21.46  Mathbox for Zhi Wang
            21.46.1  Propositional calculus   pm4.71da 47785
            21.46.2  Predicate calculus with equality   dtrucor3 47794
                  21.46.2.1  Axiom scheme ax-5 (Distinctness)   dtrucor3 47794
            21.46.3  ZF Set Theory - start with the Axiom of Extensionality   ralbidb 47795
                  21.46.3.1  Restricted quantification   ralbidb 47795
                  21.46.3.2  The empty set   ssdisjd 47801
                  21.46.3.3  Unordered and ordered pairs   vsn 47805
                  21.46.3.4  The union of a class   unilbss 47811
            21.46.4  ZF Set Theory - add the Axiom of Replacement   inpw 47812
                  21.46.4.1  Theorems requiring subset and intersection existence   inpw 47812
            21.46.5  ZF Set Theory - add the Axiom of Power Sets   mof0 47813
                  21.46.5.1  Functions   mof0 47813
                  21.46.5.2  Operations   fvconstr 47831
            21.46.6  ZF Set Theory - add the Axiom of Union   fvconst0ci 47834
                  21.46.6.1  Equinumerosity   fvconst0ci 47834
            21.46.7  Order sets   iccin 47838
                  21.46.7.1  Real number intervals   iccin 47838
            21.46.8  Moore spaces   mreuniss 47841
            ​*21.46.9  Topology   clduni 47842
                  21.46.9.1  Closure and interior   clduni 47842
                  21.46.9.2  Neighborhoods   neircl 47846
                  21.46.9.3  Subspace topologies   restcls2lem 47854
                  21.46.9.4  Limits and continuity in topological spaces   cnneiima 47858
                  21.46.9.5  Topological definitions using the reals   iooii 47859
                  21.46.9.6  Separated sets   sepnsepolem1 47863
                  21.46.9.7  Separated spaces: T0, T1, T2 (Hausdorff) ...   isnrm4 47872
            21.46.10  Preordered sets and directed sets using extensible structures   isprsd 47897
            21.46.11  Posets and lattices using extensible structures   lubeldm2 47898
                  21.46.11.1  Posets   lubeldm2 47898
                  21.46.11.2  Lattices   toslat 47916
                  21.46.11.3  Subset order structures   intubeu 47918
            21.46.12  Categories   catprslem 47939
                  21.46.12.1  Categories   catprslem 47939
                  21.46.12.2  Monomorphisms and epimorphisms   idmon 47945
                  21.46.12.3  Functors   funcf2lem 47947
            21.46.13  Examples of categories   cthinc 47948
                  21.46.13.1  Thin categories   cthinc 47948
                  21.46.13.2  Preordered sets as thin categories   cprstc 47991
                  21.46.13.3  Monoids as categories   cmndtc 48012
      ​21.47  Mathbox for Emmett Weisz
            ​*21.47.1  Miscellaneous Theorems   nfintd 48027
            21.47.2  Set Recursion   csetrecs 48037
                  ​*21.47.2.1  Basic Properties of Set Recursion   csetrecs 48037
                  21.47.2.2  Examples and properties of set recursion   elsetrecslem 48053
            ​*21.47.3  Construction of Games and Surreal Numbers   cpg 48063
      ​​*21.48  Mathbox for David A. Wheeler
            21.48.1  Natural deduction   sbidd 48072
            ​*21.48.2  Greater than, greater than or equal to.   cge-real 48074
            ​*21.48.3  Hyperbolic trigonometric functions   csinh 48084
            ​*21.48.4  Reciprocal trigonometric functions (sec, csc, cot)   csec 48095
            ​*21.48.5  Identities for "if"   ifnmfalse 48117
            ​*21.48.6  Logarithms generalized to arbitrary base using ` logb `   logb2aval 48118
            ​*21.48.7  Logarithm laws generalized to an arbitrary base - log_   clog- 48119
            ​*21.48.8  Formally define notions such as reflexivity   wreflexive 48121
            ​*21.48.9  Algebra helpers   comraddi 48125
            ​*21.48.10  Algebra helper examples   i2linesi 48134
            ​*21.48.11  Formal methods "surprises"   alimp-surprise 48136
            ​*21.48.12  Allsome quantifier   walsi 48142
            ​*21.48.13  Miscellaneous   5m4e1 48153
            21.48.14  Theorems about algebraic numbers   aacllem 48157
      ​21.49  Mathbox for Kunhao Zheng
            21.49.1  Weighted AM-GM inequality   amgmwlem 48158