TOC of Theorem List - Metamath Proof Explorer

Table of Contents Summary

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

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

​​*PART 1  CLASSICAL FIRST-ORDER LOGIC WITH EQUALITY
      ​*1.1  Pre-logic
            *1.1.1  Inferences for assisting proof development   idi 1
      ​​*1.2  Propositional calculus
            1.2.1  Recursively define primitive wffs for propositional calculus   wn 3
            ​*1.2.2  The axioms of propositional calculus   ax-mp 5
            ​*1.2.3  Logical implication   mp2 9
            ​*1.2.4  Logical negation   con4 113
            ​*1.2.5  Logical equivalence   wb 205
            ​*1.2.6  Logical conjunction   wa 396
            ​*1.2.7  Logical disjunction   wo 845
            ​*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 1083
            1.2.11  Abbreviated conjunction and disjunction of three wff's   w3o 1086
            1.2.12  Logical "nand" (Sheffer stroke)   wnan 1489
            1.2.13  Logical "xor"   wxo 1509
            1.2.14  Logical "nor"   wnor 1528
            1.2.15  True and false constants   wal 1539
                  ​*1.2.15.1  Universal quantifier for use by df-tru   wal 1539
                  ​*1.2.15.2  Equality predicate for use by df-tru   cv 1540
                  1.2.15.3  The true constant   wtru 1542
                  1.2.15.4  The false constant   wfal 1553
            ​*1.2.16  Truth tables   truimtru 1564
                  1.2.16.1  Implication   truimtru 1564
                  1.2.16.2  Negation   nottru 1568
                  1.2.16.3  Equivalence   trubitru 1570
                  1.2.16.4  Conjunction   truantru 1574
                  1.2.16.5  Disjunction   truortru 1578
                  1.2.16.6  Alternative denial   trunantru 1582
                  1.2.16.7  Exclusive disjunction   truxortru 1586
                  1.2.16.8  Joint denial   trunortru 1590
            ​*1.2.17  Half adder and full adder in propositional calculus   whad 1594
                  1.2.17.1  Full adder: sum   whad 1594
                  1.2.17.2  Full adder: carry   wcad 1607
      ​1.3  Other axiomatizations related to classical propositional calculus
            ​*1.3.1  Minimal implicational calculus   minimp 1623
            ​*1.3.2  Implicational Calculus   impsingle 1629
            1.3.3  Derive the Lukasiewicz axioms from Meredith's sole axiom   meredith 1643
            1.3.4  Derive the standard axioms from the Lukasiewicz axioms   luklem1 1660
            ​*1.3.5  Derive Nicod's axiom from the standard axioms   nic-dfim 1671
            1.3.6  Derive the Lukasiewicz axioms from Nicod's axiom   nic-imp 1677
            1.3.7  Derive Nicod's Axiom from Lukasiewicz's First Sheffer Stroke Axiom   lukshef-ax1 1696
            1.3.8  Derive the Lukasiewicz Axioms from the Tarski-Bernays-Wajsberg Axioms   tbw-bijust 1700
            1.3.9  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's First CO Axiom   merco1 1715
            1.3.10  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's Second CO Axiom   merco2 1738
            1.3.11  Derive the Lukasiewicz axioms from the Russell-Bernays Axioms   rb-bijust 1751
            ​*1.3.12  Stoic logic non-modal portion (Chrysippus of Soli)   mptnan 1770
      ​​*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 1781
                  1.4.1.1  Existential quantifier   wex 1781
                  1.4.1.2  Nonfreeness predicate   wnf 1785
            1.4.2  Rule scheme ax-gen (Generalization)   ax-gen 1797
            1.4.3  Axiom scheme ax-4 (Quantified Implication)   ax-4 1811
                  ​*1.4.3.1  The empty domain of discourse   empty 1909
            1.4.4  Axiom scheme ax-5 (Distinctness) - first use of $d   ax-5 1913
            ​*1.4.5  Equality predicate (continued)   weq 1966
            1.4.6  Axiom scheme ax-6 (Existence)   ax-6 1971
            1.4.7  Axiom scheme ax-7 (Equality)   ax-7 2011
            1.4.8  Define proper substitution   sbjust 2066
            1.4.9  Membership predicate   wcel 2106
            1.4.10  Axiom scheme ax-8 (Left Equality for Binary Predicate)   ax-8 2108
            1.4.11  Axiom scheme ax-9 (Right Equality for Binary Predicate)   ax-9 2116
            ​*1.4.12  Logical redundancy of ax-10 , ax-11 , ax-12 , ax-13   ax6dgen 2124
      ​​*1.5  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes)
            1.5.1  Axiom scheme ax-10 (Quantified Negation)   ax-10 2137
            1.5.2  Axiom scheme ax-11 (Quantifier Commutation)   ax-11 2154
            1.5.3  Axiom scheme ax-12 (Substitution)   ax-12 2171
            1.5.4  Axiom scheme ax-13 (Quantified Equality)   ax-13 2371
      ​1.6  Uniqueness and unique existence
            1.6.1  Uniqueness: the at-most-one quantifier   wmo 2532
            1.6.2  Unique existence: the unique existential quantifier   weu 2562
      ​1.7  Other axiomatizations related to classical predicate calculus
            ​*1.7.1  Aristotelian logic: Assertic syllogisms   barbara 2658
            ​*1.7.2  Intuitionistic logic   axia1 2688
​​*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 2703
            2.1.2  Classes   cab 2709
                  2.1.2.1  Class abstractions   cab 2709
                  ​*2.1.2.2  Class equality   df-cleq 2724
                  2.1.2.3  Class membership   df-clel 2810
                  2.1.2.4  Elementary properties of class abstractions   eqabdv 2867
            2.1.3  Class form not-free predicate   wnfc 2883
            2.1.4  Negated equality and membership   wne 2940
                  2.1.4.1  Negated equality   wne 2940
                  2.1.4.2  Negated membership   wnel 3046
            2.1.5  Restricted quantification   wral 3061
                  2.1.5.1  Restricted universal and existential quantification   wral 3061
                  2.1.5.2  Restricted existential uniqueness and at-most-one quantifier   wreu 3374
                  2.1.5.3  Restricted class abstraction   crab 3432
            2.1.6  The universal class   cvv 3474
            ​*2.1.7  Conditional equality (experimental)   wcdeq 3759
            2.1.8  Russell's Paradox   rru 3775
            2.1.9  Proper substitution of classes for sets   wsbc 3777
            2.1.10  Proper substitution of classes for sets into classes   csb 3893
            2.1.11  Define basic set operations and relations   cdif 3945
            2.1.12  Subclasses and subsets   df-ss 3965
            2.1.13  The difference, union, and intersection of two classes   dfdif3 4114
                  2.1.13.1  The difference of two classes   dfdif3 4114
                  2.1.13.2  The union of two classes   elun 4148
                  2.1.13.3  The intersection of two classes   elini 4193
                  2.1.13.4  The symmetric difference of two classes   csymdif 4241
                  2.1.13.5  Combinations of difference, union, and intersection of two classes   unabs 4254
                  2.1.13.6  Class abstractions with difference, union, and intersection of two classes   unabw 4297
                  2.1.13.7  Restricted uniqueness with difference, union, and intersection   reuun2 4314
            2.1.14  The empty set   c0 4322
            ​*2.1.15  The conditional operator for classes   cif 4528
            ​*2.1.16  The weak deduction theorem for set theory   dedth 4586
            2.1.17  Power classes   cpw 4602
            2.1.18  Unordered and ordered pairs   snjust 4627
            2.1.19  The union of a class   cuni 4908
            2.1.20  The intersection of a class   cint 4950
            2.1.21  Indexed union and intersection   ciun 4997
            2.1.22  Disjointness   wdisj 5113
            2.1.23  Binary relations   wbr 5148
            2.1.24  Ordered-pair class abstractions (class builders)   copab 5210
            2.1.25  Functions in maps-to notation   cmpt 5231
            2.1.26  Transitive classes   wtr 5265
      ​2.2  ZF Set Theory - add the Axiom of Replacement
            2.2.1  Introduce the Axiom of Replacement   ax-rep 5285
            2.2.2  Derive the Axiom of Separation   axsepgfromrep 5297
            2.2.3  Derive the Null Set Axiom   axnulALT 5304
            2.2.4  Theorems requiring subset and intersection existence   nalset 5313
            2.2.5  Theorems requiring empty set existence   class2set 5353
      ​2.3  ZF Set Theory - add the Axiom of Power Sets
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 5363
            2.3.2  Derive the Axiom of Pairing   axprlem1 5421
            2.3.3  Ordered pair theorem   opnz 5473
            2.3.4  Ordered-pair class abstractions (cont.)   opabidw 5524
            2.3.5  Power class of union and intersection   pwin 5570
            2.3.6  The identity relation   cid 5573
            2.3.7  The membership relation (or epsilon relation)   cep 5579
            ​*2.3.8  Partial and total orderings   wpo 5586
            2.3.9  Founded and well-ordering relations   wfr 5628
            2.3.10  Relations   cxp 5674
            2.3.11  The Predecessor Class   cpred 6299
            2.3.12  Well-founded induction (variant)   frpomin 6341
            2.3.13  Well-ordered induction   tz6.26 6348
            2.3.14  Ordinals   word 6363
            2.3.15  Definite description binder (inverted iota)   cio 6493
            2.3.16  Functions   wfun 6537
            2.3.17  Cantor's Theorem   canth 7364
            2.3.18  Restricted iota (description binder)   crio 7366
            2.3.19  Operations   co 7411
                  2.3.19.1  Variable-to-class conversion for operations   caovclg 7601
            2.3.20  Maps-to notation   mpondm0 7649
            2.3.21  Function operation   cof 7670
            2.3.22  Proper subset relation   crpss 7714
      ​2.4  ZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 7727
            2.4.2  Ordinals (continued)   epweon 7764
            2.4.3  Transfinite induction   tfi 7844
            2.4.4  The natural numbers (i.e., finite ordinals)   com 7857
            2.4.5  Peano's postulates   peano1 7881
            2.4.6  Finite induction (for finite ordinals)   find 7889
            2.4.7  Relations and functions (cont.)   dmexg 7896
            2.4.8  First and second members of an ordered pair   c1st 7975
            2.4.9  Induction on Cartesian products   frpoins3xpg 8128
            2.4.10  Ordering on Cartesian products   xpord2lem 8130
            2.4.11  Ordering Ordinal Sequences   orderseqlem 8145
            ​*2.4.12  The support of functions   csupp 8148
            ​*2.4.13  Special maps-to operations   opeliunxp2f 8197
            2.4.14  Function transposition   ctpos 8212
            2.4.15  Curry and uncurry   ccur 8252
            2.4.16  Undefined values   cund 8259
            2.4.17  Well-founded recursion   cfrecs 8267
            2.4.18  Well-ordered recursion   cwrecs 8298
            2.4.19  Functions on ordinals; strictly monotone ordinal functions   iunon 8341
            2.4.20  "Strong" transfinite recursion   crecs 8372
            2.4.21  Recursive definition generator   crdg 8411
            2.4.22  Finite recursion   frfnom 8437
            2.4.23  Ordinal arithmetic   c1o 8461
            2.4.24  Natural number arithmetic   nna0 8606
            2.4.25  Natural addition   cnadd 8666
            2.4.26  Equivalence relations and classes   wer 8702
            2.4.27  The mapping operation   cmap 8822
            2.4.28  Infinite Cartesian products   cixp 8893
            2.4.29  Equinumerosity   cen 8938
            2.4.30  Schroeder-Bernstein Theorem   sbthlem1 9085
            2.4.31  Equinumerosity (cont.)   xpf1o 9141
            2.4.32  Finite sets   dif1enlem 9158
            2.4.33  Pigeonhole Principle   phplem1 9209
            2.4.34  Finite sets (cont.)   onomeneq 9230
            2.4.35  Finitely supported functions   cfsupp 9363
            2.4.36  Finite intersections   cfi 9407
            2.4.37  Hall's marriage theorem   marypha1lem 9430
            2.4.38  Supremum and infimum   csup 9437
            2.4.39  Ordinal isomorphism, Hartogs's theorem   coi 9506
            2.4.40  Hartogs function   char 9553
            2.4.41  Weak dominance   cwdom 9561
      ​2.5  ZF Set Theory - add the Axiom of Regularity
            2.5.1  Introduce the Axiom of Regularity   ax-reg 9589
            2.5.2  Axiom of Infinity equivalents   inf0 9618
      ​2.6  ZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-inf 9635
            2.6.2  Existence of omega (the set of natural numbers)   omex 9640
            2.6.3  Cantor normal form   ccnf 9658
            2.6.4  Transitive closure of a relation   cttrcl 9704
            2.6.5  Transitive closure   trcl 9725
            2.6.6  Well-Founded Induction   frmin 9746
            2.6.7  Well-Founded Recursion   frr3g 9753
            2.6.8  Rank   cr1 9759
            2.6.9  Scott's trick; collection principle; Hilbert's epsilon   scottex 9882
            2.6.10  Disjoint union   cdju 9895
            2.6.11  Cardinal numbers   ccrd 9932
            2.6.12  Axiom of Choice equivalents   wac 10112
            ​*2.6.13  Cardinal number arithmetic   undjudom 10164
            2.6.14  The Ackermann bijection   ackbij2lem1 10216
            2.6.15  Cofinality (without Axiom of Choice)   cflem 10243
            2.6.16  Eight inequivalent definitions of finite set   sornom 10274
            2.6.17  Hereditarily size-limited sets without Choice   itunifval 10413
​​*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 10432
            3.1.2  Introduce the Axiom of Dependent Choice   ax-dc 10443
      ​3.2  ZFC Set Theory - add the Axiom of Choice
            3.2.1  Introduce the Axiom of Choice   ax-ac 10456
            3.2.2  AC equivalents: well-ordering, Zorn's lemma   numthcor 10491
            3.2.3  Cardinal number theorems using Axiom of Choice   cardval 10543
            3.2.4  Cardinal number arithmetic using Axiom of Choice   iunctb 10571
            3.2.5  Cofinality using the Axiom of Choice   alephreg 10579
      ​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 10617
            3.4.2  Derivation of the Axiom of Choice   gchaclem 10675
​​*PART 4  TG (TARSKI-GROTHENDIECK) SET THEORY
      ​4.1  Inaccessibles
            4.1.1  Weakly and strongly inaccessible cardinals   cwina 10679
            4.1.2  Weak universes   cwun 10697
            4.1.3  Tarski classes   ctsk 10745
            4.1.4  Grothendieck universes   cgru 10787
      ​4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
            4.2.1  Introduce the Tarski-Grothendieck Axiom   ax-groth 10820
            4.2.2  Derive the Power Set, Infinity and Choice Axioms   grothpw 10823
            4.2.3  Tarski map function   ctskm 10834
​​*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 10841
            5.1.2  Final derivation of real and complex number postulates   axaddf 11142
            5.1.3  Real and complex number postulates restated as axioms   ax-cnex 11168
      ​5.2  Derive the basic properties from the field axioms
            5.2.1  Some deductions from the field axioms for complex numbers   cnex 11193
            5.2.2  Infinity and the extended real number system   cpnf 11247
            5.2.3  Restate the ordering postulates with extended real "less than"   axlttri 11287
            5.2.4  Ordering on reals   lttr 11292
            5.2.5  Initial properties of the complex numbers   mul12 11381
      ​5.3  Real and complex numbers - basic operations
            5.3.1  Addition   add12 11433
            5.3.2  Subtraction   cmin 11446
            5.3.3  Multiplication   kcnktkm1cn 11647
            5.3.4  Ordering on reals (cont.)   gt0ne0 11681
            5.3.5  Reciprocals   ixi 11845
            5.3.6  Division   cdiv 11873
            5.3.7  Ordering on reals (cont.)   elimgt0 12054
            5.3.8  Completeness Axiom and Suprema   fimaxre 12160
            5.3.9  Imaginary and complex number properties   inelr 12204
            5.3.10  Function operation analogue theorems   ofsubeq0 12211
      ​5.4  Integer sets
            5.4.1  Positive integers (as a subset of complex numbers)   cn 12214
            5.4.2  Principle of mathematical induction   nnind 12232
            ​*5.4.3  Decimal representation of numbers   c2 12269
            ​*5.4.4  Some properties of specific numbers   neg1cn 12328
            5.4.5  Simple number properties   halfcl 12439
            5.4.6  The Archimedean property   nnunb 12470
            5.4.7  Nonnegative integers (as a subset of complex numbers)   cn0 12474
            ​*5.4.8  Extended nonnegative integers   cxnn0 12546
            5.4.9  Integers (as a subset of complex numbers)   cz 12560
            5.4.10  Decimal arithmetic   cdc 12679
            5.4.11  Upper sets of integers   cuz 12824
            5.4.12  Well-ordering principle for bounded-below sets of integers   uzwo3 12929
            5.4.13  Rational numbers (as a subset of complex numbers)   cq 12934
            5.4.14  Existence of the set of complex numbers   rpnnen1lem2 12963
      ​5.5  Order sets
            5.5.1  Positive reals (as a subset of complex numbers)   crp 12976
            5.5.2  Infinity and the extended real number system (cont.)   cxne 13091
            5.5.3  Supremum and infimum on the extended reals   xrsupexmnf 13286
            5.5.4  Real number intervals   cioo 13326
            5.5.5  Finite intervals of integers   cfz 13486
            ​*5.5.6  Finite intervals of nonnegative integers   elfz2nn0 13594
            5.5.7  Half-open integer ranges   cfzo 13629
      ​5.6  Elementary integer functions
            5.6.1  The floor and ceiling functions   cfl 13757
            5.6.2  The modulo (remainder) operation   cmo 13836
            5.6.3  Miscellaneous theorems about integers   om2uz0i 13914
            5.6.4  Strong induction over upper sets of integers   uzsinds 13954
            5.6.5  Finitely supported functions over the nonnegative integers   fsuppmapnn0fiublem 13957
            5.6.6  The infinite sequence builder "seq" - extension   cseq 13968
            5.6.7  Integer powers   cexp 14029
            5.6.8  Ordered pair theorem for nonnegative integers   nn0le2msqi 14229
            5.6.9  Factorial function   cfa 14235
            5.6.10  The binomial coefficient operation   cbc 14264
            5.6.11  The ` # ` (set size) function   chash 14292
                  5.6.11.1  Proper unordered pairs and triples (sets of size 2 and 3)   hashprlei 14431
                  5.6.11.2  Functions with a domain containing at least two different elements   fundmge2nop0 14455
                  5.6.11.3  Finite induction on the size of the first component of a binary relation   hashdifsnp1 14459
      ​​*5.7  Words over a set
            5.7.1  Definitions and basic theorems   cword 14466
            5.7.2  Last symbol of a word   clsw 14514
            5.7.3  Concatenations of words   cconcat 14522
            5.7.4  Singleton words   cs1 14547
            5.7.5  Concatenations with singleton words   ccatws1cl 14568
            5.7.6  Subwords/substrings   csubstr 14592
            5.7.7  Prefixes of a word   cpfx 14622
            5.7.8  Subwords of subwords   swrdswrdlem 14656
            5.7.9  Subwords and concatenations   pfxcctswrd 14662
            5.7.10  Subwords of concatenations   swrdccatfn 14676
            5.7.11  Splicing words (substring replacement)   csplice 14701
            5.7.12  Reversing words   creverse 14710
            5.7.13  Repeated symbol words   creps 14720
            ​*5.7.14  Cyclical shifts of words   ccsh 14740
            5.7.15  Mapping words by a function   wrdco 14784
            5.7.16  Longer string literals   cs2 14794
      ​​*5.8  Reflexive and transitive closures of relations
            5.8.1  The reflexive and transitive properties of relations   coss12d 14921
            5.8.2  Basic properties of closures   cleq1lem 14931
            5.8.3  Definitions and basic properties of transitive closures   ctcl 14934
            5.8.4  Exponentiation of relations   crelexp 14968
            5.8.5  Reflexive-transitive closure as an indexed union   crtrcl 15004
            ​*5.8.6  Principle of transitive induction.   relexpindlem 15012
      ​5.9  Elementary real and complex functions
            5.9.1  The "shift" operation   cshi 15015
            5.9.2  Signum (sgn or sign) function   csgn 15035
            5.9.3  Real and imaginary parts; conjugate   ccj 15045
            5.9.4  Square root; absolute value   csqrt 15182
      ​5.10  Elementary limits and convergence
            5.10.1  Superior limit (lim sup)   clsp 15416
            5.10.2  Limits   cli 15430
            5.10.3  Finite and infinite sums   csu 15634
            5.10.4  The binomial theorem   binomlem 15777
            5.10.5  The inclusion/exclusion principle   incexclem 15784
            5.10.6  Infinite sums (cont.)   isumshft 15787
            5.10.7  Miscellaneous converging and diverging sequences   divrcnv 15800
            5.10.8  Arithmetic series   arisum 15808
            5.10.9  Geometric series   expcnv 15812
            5.10.10  Ratio test for infinite series convergence   cvgrat 15831
            5.10.11  Mertens' theorem   mertenslem1 15832
            5.10.12  Finite and infinite products   prodf 15835
                  5.10.12.1  Product sequences   prodf 15835
                  5.10.12.2  Non-trivial convergence   ntrivcvg 15845
                  5.10.12.3  Complex products   cprod 15851
                  5.10.12.4  Finite products   fprod 15887
                  5.10.12.5  Infinite products   iprodclim 15944
            5.10.13  Falling and Rising Factorial   cfallfac 15950
            5.10.14  Bernoulli polynomials and sums of k-th powers   cbp 15992
      ​5.11  Elementary trigonometry
            5.11.1  The exponential, sine, and cosine functions   ce 16007
                  5.11.1.1  The circle constant (tau = 2 pi)   ctau 16147
            5.11.2  _e is irrational   eirrlem 16149
      ​5.12  Cardinality of real and complex number subsets
            5.12.1  Countability of integers and rationals   xpnnen 16156
            5.12.2  The reals are uncountable   rpnnen2lem1 16159
​​*PART 6  ELEMENTARY NUMBER THEORY
      ​6.1  Elementary properties of divisibility
            6.1.1  Irrationality of square root of 2   sqrt2irrlem 16193
            6.1.2  Some Number sets are chains of proper subsets   nthruc 16197
            6.1.3  The divides relation   cdvds 16199
            ​*6.1.4  Even and odd numbers   evenelz 16281
            6.1.5  The division algorithm   divalglem0 16338
            6.1.6  Bit sequences   cbits 16362
            6.1.7  The greatest common divisor operator   cgcd 16437
            6.1.8  Bézout's identity   bezoutlem1 16483
            6.1.9  Algorithms   nn0seqcvgd 16509
            6.1.10  Euclid's Algorithm   eucalgval2 16520
            ​*6.1.11  The least common multiple   clcm 16527
            ​*6.1.12  Coprimality and Euclid's lemma   coprmgcdb 16588
            6.1.13  Cancellability of congruences   congr 16603
      ​6.2  Elementary prime number theory
            ​*6.2.1  Elementary properties   cprime 16610
            ​*6.2.2  Coprimality and Euclid's lemma (cont.)   coprm 16650
            6.2.3  Properties of the canonical representation of a rational   cnumer 16671
            6.2.4  Euler's theorem   codz 16698
            6.2.5  Arithmetic modulo a prime number   modprm1div 16732
            6.2.6  Pythagorean Triples   coprimeprodsq 16743
            6.2.7  The prime count function   cpc 16771
            6.2.8  Pocklington's theorem   prmpwdvds 16839
            6.2.9  Infinite primes theorem   unbenlem 16843
            6.2.10  Sum of prime reciprocals   prmreclem1 16851
            6.2.11  Fundamental theorem of arithmetic   1arithlem1 16858
            6.2.12  Lagrange's four-square theorem   cgz 16864
            6.2.13  Van der Waerden's theorem   cvdwa 16900
            6.2.14  Ramsey's theorem   cram 16934
            ​*6.2.15  Primorial function   cprmo 16966
            ​*6.2.16  Prime gaps   prmgaplem1 16984
            6.2.17  Decimal arithmetic (cont.)   dec2dvds 16998
            6.2.18  Cyclical shifts of words (cont.)   cshwsidrepsw 17029
            6.2.19  Specific prime numbers   prmlem0 17041
            6.2.20  Very large primes   1259lem1 17066
​PART 7  BASIC STRUCTURES
      ​7.1  Extensible structures
            ​*7.1.1  Basic definitions   cstr 17081
                  7.1.1.1  Extensible structures as structures with components   cstr 17081
                  7.1.1.2  Substitution of components   csts 17098
                  7.1.1.3  Slots   cslot 17116
                  ​*7.1.1.4  Structure component indices   cnx 17128
                  7.1.1.5  Base sets   cbs 17146
                  7.1.1.6  Base set restrictions   cress 17175
            7.1.2  Slot definitions   cplusg 17199
            7.1.3  Definition of the structure product   crest 17368
            7.1.4  Definition of the structure quotient   cordt 17447
      ​7.2  Moore spaces
            7.2.1  Moore closures   mrcflem 17552
            7.2.2  Independent sets in a Moore system   mrisval 17576
            7.2.3  Algebraic closure systems   isacs 17597
​PART 8  BASIC CATEGORY THEORY
      ​8.1  Categories
            8.1.1  Categories   ccat 17610
            8.1.2  Opposite category   coppc 17657
            8.1.3  Monomorphisms and epimorphisms   cmon 17677
            8.1.4  Sections, inverses, isomorphisms   csect 17693
            ​*8.1.5  Isomorphic objects   ccic 17744
            8.1.6  Subcategories   cssc 17756
            8.1.7  Functors   cfunc 17806
            8.1.8  Full & faithful functors   cful 17855
            8.1.9  Natural transformations and the functor category   cnat 17894
            8.1.10  Initial, terminal and zero objects of a category   cinito 17933
      ​8.2  Arrows (disjointified hom-sets)
            8.2.1  Identity and composition for arrows   cida 18005
      ​8.3  Examples of categories
            8.3.1  The category of sets   csetc 18027
            8.3.2  The category of categories   ccatc 18050
            ​*8.3.3  The category of extensible structures   fncnvimaeqv 18073
      ​8.4  Categorical constructions
            8.4.1  Product of categories   cxpc 18122
            8.4.2  Functor evaluation   cevlf 18164
            8.4.3  Hom functor   chof 18203
​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 18386
            9.5.2  Complete lattices   ccla 18453
            9.5.3  Distributive lattices   cdlat 18475
            9.5.4  Subset order structures   cipo 18482
      ​9.6  Posets, directed sets, and lattices as relations
            ​*9.6.1  Posets and lattices as relations   cps 18519
            9.6.2  Directed sets, nets   cdir 18549
​PART 10  BASIC ALGEBRAIC STRUCTURES
      ​10.1  Monoids
            ​*10.1.1  Magmas   cplusf 18560
            ​*10.1.2  Identity elements   mgmidmo 18581
            ​*10.1.3  Iterated sums in a magma   gsumvalx 18597
            ​*10.1.4  Semigroups   csgrp 18611
            ​*10.1.5  Definition and basic properties of monoids   cmnd 18627
            10.1.6  Monoid homomorphisms and submonoids   cmhm 18671
            ​*10.1.7  Iterated sums in a monoid   gsumvallem2 18717
            10.1.8  Free monoids   cfrmd 18730
                  ​*10.1.8.1  Monoid of endofunctions   cefmnd 18751
            10.1.9  Examples and counterexamples for magmas, semigroups and monoids   mgm2nsgrplem1 18801
      ​10.2  Groups
            10.2.1  Definition and basic properties   cgrp 18821
            ​*10.2.2  Group multiple operation   cmg 18952
            10.2.3  Subgroups and Quotient groups   csubg 19002
            ​*10.2.4  Cyclic monoids and groups   cycsubmel 19079
            10.2.5  Elementary theory of group homomorphisms   cghm 19091
            10.2.6  Isomorphisms of groups   cgim 19133
            10.2.7  Group actions   cga 19155
            10.2.8  Centralizers and centers   ccntz 19181
            10.2.9  The opposite group   coppg 19211
            10.2.10  Symmetric groups   csymg 19236
                  ​*10.2.10.1  Definition and basic properties   csymg 19236
                  10.2.10.2  Cayley's theorem   cayleylem1 19282
                  10.2.10.3  Permutations fixing one element   symgfix2 19286
                  ​*10.2.10.4  Transpositions in the symmetric group   cpmtr 19311
                  10.2.10.5  The sign of a permutation   cpsgn 19359
            10.2.11  p-Groups and Sylow groups; Sylow's theorems   cod 19394
            10.2.12  Direct products   clsm 19504
                  10.2.12.1  Direct products (extension)   smndlsmidm 19526
            10.2.13  Free groups   cefg 19576
            10.2.14  Abelian groups   ccmn 19650
                  10.2.14.1  Definition and basic properties   ccmn 19650
                  10.2.14.2  Cyclic groups   ccyg 19747
                  10.2.14.3  Group sum operation   gsumval3a 19773
                  10.2.14.4  Group sums over (ranges of) integers   fsfnn0gsumfsffz 19853
                  10.2.14.5  Internal direct products   cdprd 19865
                  10.2.14.6  The Fundamental Theorem of Abelian Groups   ablfacrplem 19937
            10.2.15  Simple groups   csimpg 19962
                  10.2.15.1  Definition and basic properties   csimpg 19962
                  10.2.15.2  Classification of abelian simple groups   ablsimpnosubgd 19976
      ​10.3  Rings
            10.3.1  Multiplicative Group   cmgp 19989
            ​*10.3.2  Ring unity (multiplicative identity)   cur 20006
            10.3.3  Semirings   csrg 20011
                  ​*10.3.3.1  The binomial theorem for semirings   srgbinomlem1 20051
            10.3.4  Definition and basic properties of unital rings   crg 20058
            10.3.5  Opposite ring   coppr 20153
            10.3.6  Divisibility   cdsr 20172
            10.3.7  Ring primes   crpm 20250
            10.3.8  Ring homomorphisms   crh 20252
            10.3.9  Nonzero rings and zero rings   cnzr 20295
            10.3.10  Local rings   clring 20312
            10.3.11  Subrings of a ring   csubrg 20319
      ​10.4  Division rings and fields
            10.4.1  Definition and basic properties   cdr 20361
            10.4.2  Sub-division rings   csdrg 20406
            10.4.3  Absolute value (abstract algebra)   cabv 20428
            10.4.4  Star rings   cstf 20455
      ​10.5  Left modules
            10.5.1  Definition and basic properties   clmod 20475
            10.5.2  Subspaces and spans in a left module   clss 20547
            10.5.3  Homomorphisms and isomorphisms of left modules   clmhm 20635
            10.5.4  Subspace sum; bases for a left module   clbs 20690
      ​10.6  Vector spaces
            10.6.1  Definition and basic properties   clvec 20718
      ​10.7  Ideals
            10.7.1  The subring algebra; ideals   csra 20787
            10.7.2  Two-sided ideals and quotient rings   c2idl 20862
            10.7.3  Principal ideal rings. Divisibility in the integers   clpidl 20885
            10.7.4  Left regular elements. More kinds of rings   crlreg 20901
      ​10.8  The complex numbers as an algebraic extensible structure
            10.8.1  Definition and basic properties   cpsmet 20934
            ​*10.8.2  Ring of integers   czring 21023
            10.8.3  Algebraic constructions based on the complex numbers   czrh 21055
            10.8.4  Signs as subgroup of the complex numbers   cnmsgnsubg 21136
            10.8.5  Embedding of permutation signs into a ring   zrhpsgnmhm 21143
            10.8.6  The ordered field of real numbers   crefld 21163
      ​10.9  Generalized pre-Hilbert and Hilbert spaces
            10.9.1  Definition and basic properties   cphl 21183
            10.9.2  Orthocomplements and closed subspaces   cocv 21219
            10.9.3  Orthogonal projection and orthonormal bases   cpj 21261
​​*PART 11  BASIC LINEAR ALGEBRA
      ​11.1  Vectors and free modules
            *11.1.1  Direct sum of left modules   cdsmm 21292
            ​*11.1.2  Free modules   cfrlm 21307
            ​*11.1.3  Standard basis (unit vectors)   cuvc 21343
            ​*11.1.4  Independent sets and families   clindf 21365
            11.1.5  Characterization of free modules   lmimlbs 21397
      ​11.2  Associative algebras
            11.2.1  Definition and basic properties   casa 21411
      ​11.3  Abstract multivariate polynomials
            11.3.1  Definition and basic properties   cmps 21463
            11.3.2  Polynomial evaluation   ces 21639
            11.3.3  Additional definitions for (multivariate) polynomials   cslv 21677
            ​*11.3.4  Univariate polynomials   cps1 21705
            11.3.5  Univariate polynomial evaluation   ces1 21839
      ​​*11.4  Matrices
            *11.4.1  The matrix multiplication   cmmul 21892
            ​*11.4.2  Square matrices   cmat 21914
            ​*11.4.3  The matrix algebra   matmulr 21947
            ​*11.4.4  Matrices of dimension 0 and 1   mat0dimbas0 21975
            ​*11.4.5  The subalgebras of diagonal and scalar matrices   cdmat 21997
            ​*11.4.6  Multiplication of a matrix with a "column vector"   cmvmul 22049
            11.4.7  Replacement functions for a square matrix   cmarrep 22065
            11.4.8  Submatrices   csubma 22085
      ​11.5  The determinant
            11.5.1  Definition and basic properties   cmdat 22093
            11.5.2  Determinants of 2 x 2 -matrices   m2detleiblem1 22133
            11.5.3  The matrix adjugate/adjunct   cmadu 22141
            ​*11.5.4  Laplace expansion of determinants (special case)   symgmatr01lem 22162
            11.5.5  Inverse matrix   invrvald 22185
            ​*11.5.6  Cramer's rule   slesolvec 22188
      ​​*11.6  Polynomial matrices
            11.6.1  Basic properties   pmatring 22201
            ​*11.6.2  Constant polynomial matrices   ccpmat 22212
            ​*11.6.3  Collecting coefficients of polynomial matrices   cdecpmat 22271
            ​*11.6.4  Ring isomorphism between polynomial matrices and polynomials over matrices   cpm2mp 22301
      ​​*11.7  The characteristic polynomial
            *11.7.1  Definition and basic properties   cchpmat 22335
            ​*11.7.2  The characteristic factor function G   fvmptnn04if 22358
            ​*11.7.3  The Cayley-Hamilton theorem   cpmadurid 22376
​PART 12  BASIC TOPOLOGY
      ​12.1  Topology
            ​*12.1.1  Topological spaces   ctop 22402
                  12.1.1.1  Topologies   ctop 22402
                  12.1.1.2  Topologies on sets   ctopon 22419
                  12.1.1.3  Topological spaces   ctps 22441
            12.1.2  Topological bases   ctb 22455
            12.1.3  Examples of topologies   distop 22505
            12.1.4  Closure and interior   ccld 22527
            12.1.5  Neighborhoods   cnei 22608
            12.1.6  Limit points and perfect sets   clp 22645
            12.1.7  Subspace topologies   restrcl 22668
            12.1.8  Order topology   ordtbaslem 22699
            12.1.9  Limits and continuity in topological spaces   ccn 22735
            12.1.10  Separated spaces: T0, T1, T2 (Hausdorff) ...   ct0 22817
            12.1.11  Compactness   ccmp 22897
            12.1.12  Bolzano-Weierstrass theorem   bwth 22921
            12.1.13  Connectedness   cconn 22922
            12.1.14  First- and second-countability   c1stc 22948
            12.1.15  Local topological properties   clly 22975
            12.1.16  Refinements   cref 23013
            12.1.17  Compactly generated spaces   ckgen 23044
            12.1.18  Product topologies   ctx 23071
            12.1.19  Continuous function-builders   cnmptid 23172
            12.1.20  Quotient maps and quotient topology   ckq 23204
            12.1.21  Homeomorphisms   chmeo 23264
      ​12.2  Filters and filter bases
            12.2.1  Filter bases   elmptrab 23338
            12.2.2  Filters   cfil 23356
            12.2.3  Ultrafilters   cufil 23410
            12.2.4  Filter limits   cfm 23444
            12.2.5  Extension by continuity   ccnext 23570
            12.2.6  Topological groups   ctmd 23581
            12.2.7  Infinite group sum on topological groups   ctsu 23637
            12.2.8  Topological rings, fields, vector spaces   ctrg 23667
      ​12.3  Uniform Structures and Spaces
            12.3.1  Uniform structures   cust 23711
            12.3.2  The topology induced by an uniform structure   cutop 23742
            12.3.3  Uniform Spaces   cuss 23765
            12.3.4  Uniform continuity   cucn 23787
            12.3.5  Cauchy filters in uniform spaces   ccfilu 23798
            12.3.6  Complete uniform spaces   ccusp 23809
      ​12.4  Metric spaces
            12.4.1  Pseudometric spaces   ispsmet 23817
            12.4.2  Basic metric space properties   cxms 23830
            12.4.3  Metric space balls   blfvalps 23896
            12.4.4  Open sets of a metric space   mopnval 23951
            12.4.5  Continuity in metric spaces   metcnp3 24056
            12.4.6  The uniform structure generated by a metric   metuval 24065
            12.4.7  Examples of metric spaces   dscmet 24088
            ​*12.4.8  Normed algebraic structures   cnm 24092
            12.4.9  Normed space homomorphisms (bounded linear operators)   cnmo 24229
            12.4.10  Topology on the reals   qtopbaslem 24282
            12.4.11  Topological definitions using the reals   cii 24398
            12.4.12  Path homotopy   chtpy 24490
            12.4.13  The fundamental group   cpco 24523
      ​12.5  Metric subcomplex vector spaces
            12.5.1  Subcomplex modules   cclm 24585
            ​*12.5.2  Subcomplex vector spaces   ccvs 24646
            ​*12.5.3  Normed subcomplex vector spaces   isncvsngp 24673
            12.5.4  Subcomplex pre-Hilbert spaces   ccph 24690
            12.5.5  Convergence and completeness   ccfil 24776
            12.5.6  Baire's Category Theorem   bcthlem1 24848
            12.5.7  Banach spaces and subcomplex Hilbert spaces   ccms 24856
                  12.5.7.1  The complete ordered field of the real numbers   retopn 24903
            12.5.8  Euclidean spaces   crrx 24907
            12.5.9  Minimizing Vector Theorem   minveclem1 24948
            12.5.10  Projection Theorem   pjthlem1 24961
​PART 13  BASIC REAL AND COMPLEX ANALYSIS
      ​13.1  Continuity
            13.1.1  Intermediate value theorem   pmltpclem1 24972
      ​13.2  Integrals
            13.2.1  Lebesgue measure   covol 24986
            13.2.2  Lebesgue integration   cmbf 25138
                  13.2.2.1  Lesbesgue integral   cmbf 25138
                  13.2.2.2  Lesbesgue directed integral   cdit 25370
      ​13.3  Derivatives
            13.3.1  Real and complex differentiation   climc 25386
                  13.3.1.1  Derivatives of functions of one complex or real variable   climc 25386
                  13.3.1.2  Results on real differentiation   dvferm1lem 25508
​PART 14  BASIC REAL AND COMPLEX FUNCTIONS
      ​14.1  Polynomials
            14.1.1  Polynomial degrees   cmdg 25575
            14.1.2  The division algorithm for univariate polynomials   cmn1 25650
            14.1.3  Elementary properties of complex polynomials   cply 25705
            14.1.4  The division algorithm for polynomials   cquot 25810
            14.1.5  Algebraic numbers   caa 25834
            14.1.6  Liouville's approximation theorem   aalioulem1 25852
      ​14.2  Sequences and series
            14.2.1  Taylor polynomials and Taylor's theorem   ctayl 25872
            14.2.2  Uniform convergence   culm 25895
            14.2.3  Power series   pserval 25929
      ​14.3  Basic trigonometry
            14.3.1  The exponential, sine, and cosine functions (cont.)   efcn 25962
            14.3.2  Properties of pi = 3.14159...   pilem1 25970
            14.3.3  Mapping of the exponential function   efgh 26057
            14.3.4  The natural logarithm on complex numbers   clog 26070
            ​*14.3.5  Logarithms to an arbitrary base   clogb 26276
            14.3.6  Theorems of Pythagoras, isosceles triangles, and intersecting chords   angval 26313
            14.3.7  Solutions of quadratic, cubic, and quartic equations   quad2 26351
            14.3.8  Inverse trigonometric functions   casin 26374
            14.3.9  The Birthday Problem   log2ublem1 26458
            14.3.10  Areas in R^2   carea 26467
            14.3.11  More miscellaneous converging sequences   rlimcnp 26477
            14.3.12  Inequality of arithmetic and geometric means   cvxcl 26496
            14.3.13  Euler-Mascheroni constant   cem 26503
            14.3.14  Zeta function   czeta 26524
            14.3.15  Gamma function   clgam 26527
      ​14.4  Basic number theory
            14.4.1  Wilson's theorem   wilthlem1 26579
            14.4.2  The Fundamental Theorem of Algebra   ftalem1 26584
            14.4.3  The Basel problem (ζ(2) = π²/6)   basellem1 26592
            14.4.4  Number-theoretical functions   ccht 26602
            14.4.5  Perfect Number Theorem   mersenne 26737
            14.4.6  Characters of Z/nZ   cdchr 26742
            14.4.7  Bertrand's postulate   bcctr 26785
            ​*14.4.8  Quadratic residues and the Legendre symbol   clgs 26804
            ​*14.4.9  Gauss' Lemma   gausslemma2dlem0a 26866
            14.4.10  Quadratic reciprocity   lgseisenlem1 26885
            14.4.11  All primes 4n+1 are the sum of two squares   2sqlem1 26927
            14.4.12  Chebyshev's Weak Prime Number Theorem, Dirichlet's Theorem   chebbnd1lem1 26979
            14.4.13  The Prime Number Theorem   mudivsum 27040
            14.4.14  Ostrowski's theorem   abvcxp 27125
​PART 15  SURREAL NUMBERS
      ​​*15.1  Sign sequence representation and Alling's axioms
            15.1.1  Definitions and initial properties   csur 27150
            15.1.2  Ordering   sltsolem1 27185
            15.1.3  Birthday Function   bdayfo 27187
            15.1.4  Density   fvnobday 27188
            ​*15.1.5  Full-Eta Property   bdayimaon 27203
      ​15.2  Initial consequences of Alling's axioms
            15.2.1  Ordering Theorems   csle 27254
            15.2.2  Birthday Theorems   bdayfun 27281
      ​​*15.3  Conway cut representation
            15.3.1  Conway cuts   csslt 27289
            15.3.2  Zero and One   c0s 27331
            15.3.3  Cuts and Options   cmade 27345
            15.3.4  Cofinality and coinitiality   cofsslt 27414
      ​15.4  Induction and recursion
            15.4.1  Induction and recursion on one variable   cnorec 27430
            15.4.2  Induction and recursion on two variables   cnorec2 27441
      ​15.5  Surreal arithmetic
            15.5.1  Addition   cadds 27452
            15.5.2  Negation and Subtraction   cnegs 27504
            15.5.3  Multiplication   cmuls 27572
            15.5.4  Division   cdivs 27645
      ​15.6  Subsystems of surreals
            15.6.1  Ordinal numbers   cons 27688
            15.6.2  Natural numbers   cnn0s 27700
​​*PART 16  ELEMENTARY GEOMETRY
      ​16.1  Definition and Tarski's Axioms of Geometry
            16.1.1  Justification for the congruence notation   tgjustf 27762
      ​16.2  Tarskian Geometry
            16.2.1  Congruence   tgcgrcomimp 27766
            16.2.2  Betweenness   tgbtwntriv2 27776
            16.2.3  Dimension   tglowdim1 27789
            16.2.4  Betweenness and Congruence   tgifscgr 27797
            16.2.5  Congruence of a series of points   ccgrg 27799
            16.2.6  Motions   cismt 27821
            16.2.7  Colinearity   tglng 27835
            16.2.8  Connectivity of betweenness   tgbtwnconn1lem1 27861
            16.2.9  Less-than relation in geometric congruences   cleg 27871
            16.2.10  Rays   chlg 27889
            16.2.11  Lines   btwnlng1 27908
            16.2.12  Point inversions   cmir 27941
            16.2.13  Right angles   crag 27982
            16.2.14  Half-planes   islnopp 28028
            16.2.15  Midpoints and Line Mirroring   cmid 28061
            16.2.16  Congruence of angles   ccgra 28096
            16.2.17  Angle Comparisons   cinag 28124
            16.2.18  Congruence Theorems   tgsas1 28143
            16.2.19  Equilateral triangles   ceqlg 28154
      ​16.3  Properties of geometries
            16.3.1  Isomorphisms between geometries   f1otrgds 28158
      ​16.4  Geometry in Hilbert spaces
            16.4.1  Geometry in the complex plane   cchhllem 28182
            16.4.2  Geometry in Euclidean spaces   cee 28184
                  16.4.2.1  Definition of the Euclidean space   cee 28184
                  16.4.2.2  Tarski's axioms for geometry for the Euclidean space   axdimuniq 28209
                  16.4.2.3  EE^n fulfills Tarski's Axioms   ceeng 28273
​​*PART 17  GRAPH THEORY
      ​*17.1  Vertices and edges
            17.1.1  The edge function extractor for extensible structures   cedgf 28284
            ​*17.1.2  Vertices and indexed edges   cvtx 28294
                  17.1.2.1  Definitions and basic properties   cvtx 28294
                  17.1.2.2  The vertices and edges of a graph represented as ordered pair   opvtxval 28301
                  17.1.2.3  The vertices and edges of a graph represented as extensible structure   funvtxdmge2val 28309
                  17.1.2.4  Representations of graphs without edges   snstrvtxval 28335
                  17.1.2.5  Degenerated cases of representations of graphs   vtxval0 28337
            17.1.3  Edges as range of the edge function   cedg 28345
      ​​*17.2  Undirected graphs
            17.2.1  Undirected hypergraphs   cuhgr 28354
            17.2.2  Undirected pseudographs and multigraphs   cupgr 28378
            ​*17.2.3  Loop-free graphs   umgrislfupgrlem 28420
            17.2.4  Edges as subsets of vertices of graphs   uhgredgiedgb 28424
            ​*17.2.5  Undirected simple graphs   cuspgr 28446
            17.2.6  Examples for graphs   usgr0e 28531
            17.2.7  Subgraphs   csubgr 28562
            17.2.8  Finite undirected simple graphs   cfusgr 28611
            17.2.9  Neighbors, complete graphs and universal vertices   cnbgr 28627
                  17.2.9.1  Neighbors   cnbgr 28627
                  17.2.9.2  Universal vertices   cuvtx 28680
                  17.2.9.3  Complete graphs   ccplgr 28704
            17.2.10  Vertex degree   cvtxdg 28760
            ​*17.2.11  Regular graphs   crgr 28850
      ​​*17.3  Walks, paths and cycles
            *17.3.1  Walks   cewlks 28890
            17.3.2  Walks for loop-free graphs   lfgrwlkprop 28982
            17.3.3  Trails   ctrls 28985
            17.3.4  Paths and simple paths   cpths 29007
            17.3.5  Closed walks   cclwlks 29065
            17.3.6  Circuits and cycles   ccrcts 29079
            ​*17.3.7  Walks as words   cwwlks 29117
            17.3.8  Walks/paths of length 2 (as length 3 strings)   2wlkdlem1 29217
            17.3.9  Walks in regular graphs   rusgrnumwwlkl1 29260
            ​*17.3.10  Closed walks as words   cclwwlk 29272
                  17.3.10.1  Closed walks as words   cclwwlk 29272
                  17.3.10.2  Closed walks of a fixed length as words   cclwwlkn 29315
                  17.3.10.3  Closed walks on a vertex of a fixed length as words   cclwwlknon 29378
            17.3.11  Examples for walks, trails and paths   0ewlk 29405
            17.3.12  Connected graphs   cconngr 29477
      ​17.4  Eulerian paths and the Konigsberg Bridge problem
            ​*17.4.1  Eulerian paths   ceupth 29488
            ​*17.4.2  The Königsberg Bridge problem   konigsbergvtx 29537
      ​17.5  The Friendship Theorem
            17.5.1  Friendship graphs - basics   cfrgr 29549
            17.5.2  The friendship theorem for small graphs   frgr1v 29562
            17.5.3  Theorems according to Mertzios and Unger   2pthfrgrrn 29573
            ​*17.5.4  Huneke's Proof of the Friendship Theorem   frgrncvvdeqlem1 29590
​PART 18  GUIDES AND MISCELLANEA
      ​18.1  Guides (conventions, explanations, and examples)
            ​*18.1.1  Conventions   conventions 29691
            18.1.2  Natural deduction   natded 29694
            ​*18.1.3  Natural deduction examples   ex-natded5.2 29695
            18.1.4  Definitional examples   ex-or 29712
            18.1.5  Other examples   aevdemo 29751
      ​18.2  Humor
            18.2.1  April Fool's theorem   avril1 29754
      ​18.3  (Future - to be reviewed and classified)
            18.3.1  Planar incidence geometry   cplig 29765
            ​*18.3.2  Aliases kept to prevent broken links   dummylink 29778
​​*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 29780
            19.1.2  Abelian groups   cablo 29835
      ​19.2  Complex vector spaces
            19.2.1  Definition and basic properties   cvc 29849
            19.2.2  Examples of complex vector spaces   cnaddabloOLD 29872
      ​19.3  Normed complex vector spaces
            19.3.1  Definition and basic properties   cnv 29875
            19.3.2  Examples of normed complex vector spaces   cnnv 29968
            19.3.3  Induced metric of a normed complex vector space   imsval 29976
            19.3.4  Inner product   cdip 29991
            19.3.5  Subspaces   css 30012
      ​19.4  Operators on complex vector spaces
            19.4.1  Definitions and basic properties   clno 30031
      ​19.5  Inner product (pre-Hilbert) spaces
            19.5.1  Definition and basic properties   ccphlo 30103
            19.5.2  Examples of pre-Hilbert spaces   cncph 30110
            19.5.3  Properties of pre-Hilbert spaces   isph 30113
      ​19.6  Complex Banach spaces
            19.6.1  Definition and basic properties   ccbn 30153
            19.6.2  Examples of complex Banach spaces   cnbn 30160
            19.6.3  Uniform Boundedness Theorem   ubthlem1 30161
            19.6.4  Minimizing Vector Theorem   minvecolem1 30165
      ​19.7  Complex Hilbert spaces
            19.7.1  Definition and basic properties   chlo 30176
            19.7.2  Standard axioms for a complex Hilbert space   hlex 30189
            19.7.3  Examples of complex Hilbert spaces   cnchl 30207
            19.7.4  Hellinger-Toeplitz Theorem   htthlem 30208
​​*PART 20  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      ​20.1  Axiomatization of complex pre-Hilbert spaces
            20.1.1  Basic Hilbert space definitions   chba 30210
            20.1.2  Preliminary ZFC lemmas   df-hnorm 30259
            ​*20.1.3  Derive the Hilbert space axioms from ZFC set theory   axhilex-zf 30272
            ​*20.1.4  Introduce the vector space axioms for a Hilbert space   ax-hilex 30290
            20.1.5  Vector operations   hvmulex 30302
            20.1.6  Inner product postulates for a Hilbert space   ax-hfi 30370
      ​20.2  Inner product and norms
            20.2.1  Inner product   his5 30377
            20.2.2  Norms   dfhnorm2 30413
            20.2.3  Relate Hilbert space to normed complex vector spaces   hilablo 30451
            20.2.4  Bunjakovaskij-Cauchy-Schwarz inequality   bcsiALT 30470
      ​20.3  Cauchy sequences and completeness axiom
            20.3.1  Cauchy sequences and limits   hcau 30475
            20.3.2  Derivation of the completeness axiom from ZF set theory   hilmet 30485
            20.3.3  Completeness postulate for a Hilbert space   ax-hcompl 30493
            20.3.4  Relate Hilbert space to ZFC pre-Hilbert and Hilbert spaces   hhcms 30494
      ​20.4  Subspaces and projections
            20.4.1  Subspaces   df-sh 30498
            20.4.2  Closed subspaces   df-ch 30512
            20.4.3  Orthocomplements   df-oc 30543
            20.4.4  Subspace sum, span, lattice join, lattice supremum   df-shs 30599
            20.4.5  Projection theorem   pjhthlem1 30682
            20.4.6  Projectors   df-pjh 30686
      ​20.5  Properties of Hilbert subspaces
            20.5.1  Orthomodular law   omlsilem 30693
            20.5.2  Projectors (cont.)   pjhtheu2 30707
            20.5.3  Hilbert lattice operations   sh0le 30731
            20.5.4  Span (cont.) and one-dimensional subspaces   spansn0 30832
            20.5.5  Commutes relation for Hilbert lattice elements   df-cm 30874
            20.5.6  Foulis-Holland theorem   fh1 30909
            20.5.7  Quantum Logic Explorer axioms   qlax1i 30918
            20.5.8  Orthogonal subspaces   chscllem1 30928
            20.5.9  Orthoarguesian laws 5OA and 3OA   5oalem1 30945
            20.5.10  Projectors (cont.)   pjorthi 30960
            20.5.11  Mayet's equation E_3   mayete3i 31019
      ​20.6  Operators on Hilbert spaces
            ​*20.6.1  Operator sum, difference, and scalar multiplication   df-hosum 31021
            20.6.2  Zero and identity operators   df-h0op 31039
            20.6.3  Operations on Hilbert space operators   hoaddcl 31049
            20.6.4  Linear, continuous, bounded, Hermitian, unitary operators and norms   df-nmop 31130
            20.6.5  Linear and continuous functionals and norms   df-nmfn 31136
            20.6.6  Adjoint   df-adjh 31140
            20.6.7  Dirac bra-ket notation   df-bra 31141
            20.6.8  Positive operators   df-leop 31143
            20.6.9  Eigenvectors, eigenvalues, spectrum   df-eigvec 31144
            20.6.10  Theorems about operators and functionals   nmopval 31147
            20.6.11  Riesz lemma   riesz3i 31353
            20.6.12  Adjoints (cont.)   cnlnadjlem1 31358
            20.6.13  Quantum computation error bound theorem   unierri 31395
            20.6.14  Dirac bra-ket notation (cont.)   branmfn 31396
            20.6.15  Positive operators (cont.)   leopg 31413
            20.6.16  Projectors as operators   pjhmopi 31437
      ​20.7  States on a Hilbert lattice and Godowski's equation
            20.7.1  States on a Hilbert lattice   df-st 31502
            20.7.2  Godowski's equation   golem1 31562
      ​20.8  Cover relation, atoms, exchange axiom, and modular symmetry
            20.8.1  Covers relation; modular pairs   df-cv 31570
            20.8.2  Atoms   df-at 31629
            20.8.3  Superposition principle   superpos 31645
            20.8.4  Atoms, exchange and covering properties, atomicity   chcv1 31646
            20.8.5  Irreducibility   chirredlem1 31681
            20.8.6  Atoms (cont.)   atcvat3i 31687
            20.8.7  Modular symmetry   mdsymlem1 31694
​PART 21  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      ​21.1  Mathboxes for user contributions
            21.1.1  Mathbox guidelines   mathbox 31733
      ​21.2  Mathbox for Stefan Allan
      ​21.3  Mathbox for Thierry Arnoux
            21.3.1  Propositional Calculus - misc additions   bian1d 31738
            21.3.2  Predicate Calculus   sbc2iedf 31745
                  21.3.2.1  Predicate Calculus - misc additions   sbc2iedf 31745
                  21.3.2.2  Restricted quantification - misc additions   ralcom4f 31747
                  21.3.2.3  Equality   eqtrb 31752
                  21.3.2.4  Double restricted existential uniqueness quantification   opsbc2ie 31754
                  21.3.2.5  Double restricted existential uniqueness quantification syntax   w2reu 31756
                  21.3.2.6  Substitution (without distinct variables) - misc additions   sbceqbidf 31765
                  21.3.2.7  Existential "at most one" - misc additions   mo5f 31767
                  21.3.2.8  Existential uniqueness - misc additions   reuxfrdf 31769
                  21.3.2.9  Restricted "at most one" - misc additions   rmoxfrd 31771
                  21.3.2.10  Restricted iota (description binder)   riotaeqbidva 31774
            21.3.3  General Set Theory   dmrab 31775
                  21.3.3.1  Class abstractions (a.k.a. class builders)   dmrab 31775
                  21.3.3.2  Image Sets   abrexdomjm 31782
                  21.3.3.3  Set relations and operations - misc additions   elunsn 31788
                  21.3.3.4  Unordered pairs   eqsnd 31804
                  21.3.3.5  Conditional operator - misc additions   ifeqeqx 31812
                  21.3.3.6  Set union   uniinn0 31820
                  21.3.3.7  Indexed union - misc additions   cbviunf 31825
                  21.3.3.8  Indexed intersection - misc additions   iinabrex 31838
                  21.3.3.9  Disjointness - misc additions   disjnf 31839
            21.3.4  Relations and Functions   xpdisjres 31867
                  21.3.4.1  Relations - misc additions   xpdisjres 31867
                  21.3.4.2  Functions - misc additions   ac6sf2 31887
                  21.3.4.3  Operations - misc additions   mpomptxf 31943
                  21.3.4.4  Support of a function   suppovss 31944
                  21.3.4.5  Explicit Functions with one or two points as a domain   cosnopne 31954
                  21.3.4.6  Isomorphisms - misc. additions   gtiso 31960
                  21.3.4.7  Disjointness (additional proof requiring functions)   disjdsct 31962
                  21.3.4.8  First and second members of an ordered pair - misc additions   df1stres 31963
                  21.3.4.9  Equivalence relations and classes   ecref 31971
                  21.3.4.10  Supremum - misc additions   supssd 31972
                  21.3.4.11  Finite Sets   imafi2 31974
                  21.3.4.12  Countable Sets   snct 31976
            21.3.5  Real and Complex Numbers   creq0 31998
                  21.3.5.1  Complex operations - misc. additions   creq0 31998
                  21.3.5.2  Ordering on reals - misc additions   lt2addrd 32002
                  21.3.5.3  Extended reals - misc additions   xrlelttric 32003
                  21.3.5.4  Extended nonnegative integers - misc additions   xnn0gt0 32020
                  21.3.5.5  Real number intervals - misc additions   joiniooico 32023
                  21.3.5.6  Finite intervals of integers - misc additions   uzssico 32033
                  21.3.5.7  Half-open integer ranges - misc additions   iundisjfi 32045
                  21.3.5.8  The ` # ` (set size) function - misc additions   hashunif 32056
                  21.3.5.9  The greatest common divisor operator - misc. additions   dvdszzq 32059
                  21.3.5.10  Integers   nnindf 32063
                  21.3.5.11  Decimal numbers   dfdec100 32074
            ​*21.3.6  Decimal expansion   cdp2 32075
                  ​*21.3.6.1  Decimal point   cdp 32092
                  21.3.6.2  Division in the extended real number system   cxdiv 32121
            21.3.7  Words over a set - misc additions   wrdfd 32140
                  21.3.7.1  Splicing words (substring replacement)   splfv3 32160
                  21.3.7.2  Cyclic shift of words   1cshid 32161
            21.3.8  Extensible Structures   ressplusf 32165
                  21.3.8.1  Structure restriction operator   ressplusf 32165
                  21.3.8.2  The opposite group   oppgle 32168
                  21.3.8.3  Posets   ressprs 32171
                  21.3.8.4  Complete lattices   clatp0cl 32184
                  21.3.8.5  Order Theory   cmnt 32186
                  21.3.8.6  Extended reals Structure - misc additions   ax-xrssca 32212
                  21.3.8.7  The extended nonnegative real numbers commutative monoid   xrge0base 32224
            21.3.9  Algebra   abliso 32235
                  21.3.9.1  Monoids Homomorphisms   abliso 32235
                  21.3.9.2  Finitely supported group sums - misc additions   gsumsubg 32239
                  21.3.9.3  Centralizers and centers - misc additions   cntzun 32253
                  21.3.9.4  Totally ordered monoids and groups   comnd 32256
                  21.3.9.5  The symmetric group   symgfcoeu 32284
                  21.3.9.6  Transpositions   pmtridf1o 32294
                  21.3.9.7  Permutation Signs   psgnid 32297
                  21.3.9.8  Permutation cycles   ctocyc 32306
                  21.3.9.9  The Alternating Group   evpmval 32345
                  21.3.9.10  Signum in an ordered monoid   csgns 32358
                  21.3.9.11  The Archimedean property for generic ordered algebraic structures   cinftm 32363
                  21.3.9.12  Semiring left modules   cslmd 32386
                  21.3.9.13  Simple groups   prmsimpcyc 32414
                  21.3.9.14  Rings - misc additions   idomdomd 32415
                  21.3.9.15  Euclidean Domains   ceuf 32429
                  21.3.9.16  Division Rings   ringinveu 32435
                  21.3.9.17  Subfields   sdrgdvcl 32438
                  21.3.9.18  Field extensions generated by a set   cfldgen 32441
                  21.3.9.19  Totally ordered rings and fields   corng 32454
                  21.3.9.20  Ring homomorphisms - misc additions   rhmdvd 32477
                  21.3.9.21  Scalar restriction operation   cresv 32479
                  21.3.9.22  The commutative ring of gaussian integers   gzcrng 32499
                  21.3.9.23  The archimedean ordered field of real numbers   reofld 32500
                  21.3.9.24  The quotient map and quotient modules   qusker 32505
                  21.3.9.25  The ring of integers modulo ` N `   fermltlchr 32523
                  21.3.9.26  Independent sets and families   islinds5 32525
                  ​*21.3.9.27  Subgroup sum / Sumset / Minkowski sum   elgrplsmsn 32545
                  21.3.9.28  The quotient map   qusmul 32560
                  21.3.9.29  Ideals   intlidl 32581
                  21.3.9.30  Prime Ideals   cprmidl 32598
                  21.3.9.31  Maximal Ideals   cmxidl 32620
                  21.3.9.32  The semiring of ideals of a ring   cidlsrg 32659
                  21.3.9.33  Unique factorization domains   cufd 32675
                  21.3.9.34  Associative algebras   asclmulg 32680
                  21.3.9.35  Univariate Polynomials   0ringmon1p 32681
                  21.3.9.36  Polynomial quotient and polynomial remainder   q1pdir 32719
                  21.3.9.37  The subring algebra   sra1r 32728
                  21.3.9.38  Division Ring Extensions   drgext0g 32735
                  21.3.9.39  Vector Spaces   lvecdimfi 32741
                  21.3.9.40  Vector Space Dimension   cldim 32742
            21.3.10  Field Extensions   cfldext 32776
                  21.3.10.1  Algebraic numbers   cirng 32807
                  21.3.10.2  Minimal polynomials   cminply 32816
            21.3.11  Matrices   csmat 32842
                  21.3.11.1  Submatrices   csmat 32842
                  21.3.11.2  Matrix literals   clmat 32860
                  21.3.11.3  Laplace expansion of determinants   mdetpmtr1 32872
            21.3.12  Topology   ist0cld 32882
                  21.3.12.1  Open maps   txomap 32883
                  21.3.12.2  Topology of the unit circle   qtopt1 32884
                  21.3.12.3  Refinements   reff 32888
                  21.3.12.4  Open cover refinement property   ccref 32891
                  21.3.12.5  Lindelöf spaces   cldlf 32901
                  21.3.12.6  Paracompact spaces   cpcmp 32904
                  ​*21.3.12.7  Spectrum of a ring   crspec 32911
                  21.3.12.8  Pseudometrics   cmetid 32935
                  21.3.12.9  Continuity - misc additions   hauseqcn 32947
                  21.3.12.10  Topology of the closed unit interval   elunitge0 32948
                  21.3.12.11  Topology of ` ( RR X. RR ) `   unicls 32952
                  21.3.12.12  Order topology - misc. additions   cnvordtrestixx 32962
                  21.3.12.13  Continuity in topological spaces - misc. additions   mndpluscn 32975
                  21.3.12.14  Topology of the extended nonnegative real numbers ordered monoid   xrge0hmph 32981
                  21.3.12.15  Limits - misc additions   lmlim 32996
                  21.3.12.16  Univariate polynomials   pl1cn 33004
            21.3.13  Uniform Stuctures and Spaces   chcmp 33005
                  21.3.13.1  Hausdorff uniform completion   chcmp 33005
            21.3.14  Topology and algebraic structures   zringnm 33007
                  21.3.14.1  The norm on the ring of the integer numbers   zringnm 33007
                  21.3.14.2  Topological ` ZZ ` -modules   zlm0 33009
                  21.3.14.3  Canonical embedding of the field of the rational numbers into a division ring   cqqh 33021
                  21.3.14.4  Canonical embedding of the real numbers into a complete ordered field   crrh 33042
                  21.3.14.5  Embedding from the extended real numbers into a complete lattice   cxrh 33065
                  21.3.14.6  Canonical embeddings into the ordered field of the real numbers   zrhre 33068
                  ​*21.3.14.7  Topological Manifolds   cmntop 33071
            21.3.15  Real and complex functions   nexple 33076
                  21.3.15.1  Integer powers - misc. additions   nexple 33076
                  21.3.15.2  Indicator Functions   cind 33077
                  21.3.15.3  Extended sum   cesum 33094
            21.3.16  Mixed Function/Constant operation   cofc 33162
            21.3.17  Abstract measure   csiga 33175
                  21.3.17.1  Sigma-Algebra   csiga 33175
                  21.3.17.2  Generated sigma-Algebra   csigagen 33205
                  ​*21.3.17.3  lambda and pi-Systems, Rings of Sets   ispisys 33219
                  21.3.17.4  The Borel algebra on the real numbers   cbrsiga 33248
                  21.3.17.5  Product Sigma-Algebra   csx 33255
                  21.3.17.6  Measures   cmeas 33262
                  21.3.17.7  The counting measure   cntmeas 33293
                  21.3.17.8  The Lebesgue measure - misc additions   voliune 33296
                  21.3.17.9  The Dirac delta measure   cdde 33299
                  21.3.17.10  The 'almost everywhere' relation   cae 33304
                  21.3.17.11  Measurable functions   cmbfm 33316
                  21.3.17.12  Borel Algebra on ` ( RR X. RR ) `   br2base 33337
                  ​*21.3.17.13  Caratheodory's extension theorem   coms 33359
            21.3.18  Integration   itgeq12dv 33394
                  21.3.18.1  Lebesgue integral - misc additions   itgeq12dv 33394
                  21.3.18.2  Bochner integral   citgm 33395
            21.3.19  Euler's partition theorem   oddpwdc 33422
            21.3.20  Sequences defined by strong recursion   csseq 33451
            21.3.21  Fibonacci Numbers   cfib 33464
            21.3.22  Probability   cprb 33475
                  21.3.22.1  Probability Theory   cprb 33475
                  21.3.22.2  Conditional Probabilities   ccprob 33499
                  21.3.22.3  Real-valued Random Variables   crrv 33508
                  21.3.22.4  Preimage set mapping operator   corvc 33523
                  21.3.22.5  Distribution Functions   orvcelval 33536
                  21.3.22.6  Cumulative Distribution Functions   orvclteel 33540
                  21.3.22.7  Probabilities - example   coinfliplem 33546
                  21.3.22.8  Bertrand's Ballot Problem   ballotlemoex 33553
            21.3.23  Signum (sgn or sign) function - misc. additions   sgncl 33606
                  21.3.23.1  Operations on words   ccatmulgnn0dir 33622
            21.3.24  Polynomials with real coefficients - misc additions   plymul02 33626
            21.3.25  Descartes's rule of signs   signspval 33632
                  21.3.25.1  Sign changes in a word over real numbers   signspval 33632
                  21.3.25.2  Counting sign changes in a word over real numbers   signslema 33642
            21.3.26  Number Theory   efcld 33672
                  21.3.26.1  Representations of a number as sums of integers   crepr 33689
                  21.3.26.2  Vinogradov Trigonometric Sums and the Circle Method   cvts 33716
                  21.3.26.3  The Ternary Goldbach Conjecture: Final Statement   ax-hgt749 33725
            21.3.27  Elementary Geometry   cstrkg2d 33745
                  ​*21.3.27.1  Two-dimensional geometry   cstrkg2d 33745
                  21.3.27.2  Outer Five Segment (not used, no need to move to main)   cafs 33750
            ​*21.3.28  LeftPad Project   clpad 33755
      ​​*21.4  Mathbox for Jonathan Ben-Naim
            21.4.1  First-order logic and set theory   bnj170 33778
            21.4.2  Well founded induction and recursion   bnj110 33938
            21.4.3  The existence of a minimal element in certain classes   bnj69 34090
            21.4.4  Well-founded induction   bnj1204 34092
            21.4.5  Well-founded recursion, part 1 of 3   bnj60 34142
            21.4.6  Well-founded recursion, part 2 of 3   bnj1500 34148
            21.4.7  Well-founded recursion, part 3 of 3   bnj1522 34152
      ​21.5  Mathbox for BTernaryTau
            21.5.1  ZF set theory   exdifsn 34153
                  21.5.1.1  Finitism   fineqvrep 34164
            21.5.2  Real and complex numbers   zltp1ne 34168
            21.5.3  Graph theory   lfuhgr 34177
                  21.5.3.1  Acyclic graphs   cacycgr 34202
      ​21.6  Mathbox for Mario Carneiro
            21.6.1  Predicate calculus with all distinct variables   ax-7d 34219
            21.6.2  Miscellaneous stuff   quartfull 34225
            21.6.3  Derangements and the Subfactorial   deranglem 34226
            21.6.4  The Erdős-Szekeres theorem   erdszelem1 34251
            21.6.5  The Kuratowski closure-complement theorem   kur14lem1 34266
            21.6.6  Retracts and sections   cretr 34277
            21.6.7  Path-connected and simply connected spaces   cpconn 34279
            21.6.8  Covering maps   ccvm 34315
            21.6.9  Normal numbers   snmlff 34389
            21.6.10  Godel-sets of formulas - part 1   cgoe 34393
            21.6.11  Godel-sets of formulas - part 2   cgon 34492
            21.6.12  Models of ZF   cgze 34506
            ​*21.6.13  Metamath formal systems   cmcn 34520
            21.6.14  Grammatical formal systems   cm0s 34645
            21.6.15  Models of formal systems   cmuv 34665
            21.6.16  Splitting fields   ccpms 34687
            21.6.17  p-adic number fields   czr 34701
      ​​*21.7  Mathbox for Filip Cernatescu
      ​21.8  Mathbox for Paul Chapman
            21.8.1  Real and complex numbers (cont.)   climuzcnv 34725
            21.8.2  Miscellaneous theorems   elfzm12 34729
      ​21.9  Mathbox for Adrian Ducourtial
      ​21.10  Mathbox for Scott Fenton
            21.10.1  ZFC Axioms in primitive form   axextprim 34739
            21.10.2  Untangled classes   untelirr 34746
            21.10.3  Extra propositional calculus theorems   3jaodd 34753
            21.10.4  Misc. Useful Theorems   nepss 34756
            21.10.5  Properties of real and complex numbers   sqdivzi 34766
            21.10.6  Infinite products   iprodefisumlem 34779
            21.10.7  Factorial limits   faclimlem1 34782
            21.10.8  Greatest common divisor and divisibility   gcd32 34788
            21.10.9  Properties of relationships   dftr6 34790
            21.10.10  Properties of functions and mappings   funpsstri 34806
            21.10.11  Set induction (or epsilon induction)   setinds 34819
            21.10.12  Ordinal numbers   elpotr 34822
            21.10.13  Defined equality axioms   axextdfeq 34838
            21.10.14  Hypothesis builders   hbntg 34846
            21.10.15  Well-founded zero, successor, and limits   cwsuc 34851
            21.10.16  Quantifier-free definitions   ctxp 34871
            21.10.17  Alternate ordered pairs   caltop 34997
            21.10.18  Geometry in the Euclidean space   cofs 35023
                  21.10.18.1  Congruence properties   cofs 35023
                  21.10.18.2  Betweenness properties   btwntriv2 35053
                  21.10.18.3  Segment Transportation   ctransport 35070
                  21.10.18.4  Properties relating betweenness and congruence   cifs 35076
                  21.10.18.5  Connectivity of betweenness   btwnconn1lem1 35128
                  21.10.18.6  Segment less than or equal to   csegle 35147
                  21.10.18.7  Outside-of relationship   coutsideof 35160
                  21.10.18.8  Lines and Rays   cline2 35175
            21.10.19  Forward difference   cfwddif 35199
            21.10.20  Rank theorems   rankung 35207
            21.10.21  Hereditarily Finite Sets   chf 35213
      ​21.11  Mathbox for Gino Giotto
            21.11.1  Study of ax-mulf usage.   mpomulf 35228
                  21.11.1.1  Miscellaneous   gg-cnfldex 35249
                  21.11.1.2  Theorems avoiding ax-addf   mpoaddf 35254
                  21.11.1.3  Revision of df-cnfld   gg-dfcnfld 35256
                  21.11.1.4  Replace cnfldmul with mpocnfldmul   gg-cncrng 35269
      ​21.12  Mathbox for Jeff Hankins
            21.12.1  Miscellany   a1i14 35271
            21.12.2  Basic topological facts   topbnd 35295
            21.12.3  Topology of the real numbers   ivthALT 35306
            21.12.4  Refinements   cfne 35307
            21.12.5  Neighborhood bases determine topologies   neibastop1 35330
            21.12.6  Lattice structure of topologies   topmtcl 35334
            21.12.7  Filter bases   fgmin 35341
            21.12.8  Directed sets, nets   tailfval 35343
      ​21.13  Mathbox for Anthony Hart
            21.13.1  Propositional Calculus   tb-ax1 35354
            21.13.2  Predicate Calculus   nalfal 35374
            21.13.3  Miscellaneous single axioms   meran1 35382
            21.13.4  Connective Symmetry   negsym1 35388
      ​21.14  Mathbox for Chen-Pang He
            21.14.1  Ordinal topology   ontopbas 35399
      ​21.15  Mathbox for Jeff Hoffman
            21.15.1  Inferences for finite induction on generic function values   fveleq 35422
            21.15.2  gdc.mm   nnssi2 35426
      ​21.16  Mathbox for Asger C. Ipsen
            21.16.1  Continuous nowhere differentiable functions   dnival 35433
      ​​*21.17  Mathbox for BJ
            *21.17.1  Propositional calculus   bj-mp2c 35502
                  *21.17.1.1  Derived rules of inference   bj-mp2c 35502
                  ​*21.17.1.2  A syntactic theorem   bj-0 35504
                  21.17.1.3  Minimal implicational calculus   bj-a1k 35506
                  ​*21.17.1.4  Positive calculus   bj-syl66ib 35517
                  21.17.1.5  Implication and negation   bj-con2com 35523
                  ​*21.17.1.6  Disjunction   bj-jaoi1 35534
                  ​*21.17.1.7  Logical equivalence   bj-dfbi4 35536
                  21.17.1.8  The conditional operator for propositions   bj-consensus 35541
                  ​*21.17.1.9  Propositional calculus: miscellaneous   bj-imbi12 35546
            ​*21.17.2  Modal logic   bj-axdd2 35556
            ​*21.17.3  Provability logic   cprvb 35561
            ​*21.17.4  First-order logic   bj-genr 35570
                  21.17.4.1  Adding ax-gen   bj-genr 35570
                  21.17.4.2  Adding ax-4   bj-2alim 35574
                  21.17.4.3  Adding ax-5   bj-ax12wlem 35607
                  21.17.4.4  Equality and substitution   bj-ssbeq 35616
                  21.17.4.5  Adding ax-6   bj-spimvwt 35632
                  21.17.4.6  Adding ax-7   bj-cbvexw 35639
                  21.17.4.7  Membership predicate, ax-8 and ax-9   bj-ax89 35641
                  21.17.4.8  Adding ax-11   bj-alcomexcom 35644
                  21.17.4.9  Adding ax-12   axc11n11 35646
                  21.17.4.10  Nonfreeness   wnnf 35687
                  21.17.4.11  Adding ax-13   bj-axc10 35747
                  ​*21.17.4.12  Removing dependencies on ax-13 (and ax-11)   bj-axc10v 35757
                  ​*21.17.4.13  Distinct var metavariables   bj-hbaeb2 35782
                  ​*21.17.4.14  Around ~ equsal   bj-equsal1t 35786
                  ​*21.17.4.15  Some Principia Mathematica proofs   stdpc5t 35791
                  21.17.4.16  Alternate definition of substitution   bj-sbsb 35801
                  21.17.4.17  Lemmas for substitution   bj-sbf3 35803
                  21.17.4.18  Existential uniqueness   bj-eu3f 35806
                  ​*21.17.4.19  First-order logic: miscellaneous   bj-sblem1 35807
            21.17.5  Set theory   eliminable1 35824
                  ​*21.17.5.1  Eliminability of class terms   eliminable1 35824
                  ​*21.17.5.2  Classes without the axiom of extensionality   bj-denoteslem 35836
                  21.17.5.3  Characterization among sets versus among classes   elelb 35863
                  ​*21.17.5.4  The nonfreeness quantifier for classes   bj-nfcsym 35865
                  ​*21.17.5.5  Lemmas for class substitution   bj-sbeqALT 35866
                  21.17.5.6  Removing some axiom requirements and disjoint variable conditions   bj-exlimvmpi 35877
                  ​*21.17.5.7  Class abstractions   bj-elabd2ALT 35891
                  21.17.5.8  Generalized class abstractions   bj-cgab 35899
                  ​*21.17.5.9  Restricted nonfreeness   wrnf 35907
                  ​*21.17.5.10  Russell's paradox   bj-ru0 35909
                  21.17.5.11  Curry's paradox in set theory   currysetlem 35912
                  ​*21.17.5.12  Some disjointness results   bj-n0i 35918
                  ​*21.17.5.13  Complements on direct products   bj-xpimasn 35922
                  ​*21.17.5.14  "Singletonization" and tagging   bj-snsetex 35930
                  ​*21.17.5.15  Tuples of classes   bj-cproj 35957
                  ​*21.17.5.16  Set theory: elementary operations relative to a universe   bj-rcleqf 35992
                  ​*21.17.5.17  Axioms for finite unions   bj-abex 35997
                  ​*21.17.5.18  Set theory: miscellaneous   eleq2w2ALT 36014
                  ​*21.17.5.19  Evaluation at a class   bj-evaleq 36039
                  21.17.5.20  Elementwise operations   celwise 36046
                  ​*21.17.5.21  Elementwise intersection (families of sets induced on a subset)   bj-rest00 36048
                  21.17.5.22  Moore collections (complements)   bj-raldifsn 36067
                  21.17.5.23  Maps-to notation for functions with three arguments   bj-0nelmpt 36083
                  ​*21.17.5.24  Currying   csethom 36089
                  ​*21.17.5.25  Setting components of extensible structures   cstrset 36101
            ​*21.17.6  Extended real and complex numbers, real and complex projective lines   bj-nfald 36104
                  21.17.6.1  Complements on class abstractions of ordered pairs and binary relations   bj-nfald 36104
                  ​*21.17.6.2  Identity relation (complements)   bj-opabssvv 36117
                  ​*21.17.6.3  Functionalized identity (diagonal in a Cartesian square)   cdiag2 36139
                  ​*21.17.6.4  Direct image and inverse image   cimdir 36145
                  ​*21.17.6.5  Extended numbers and projective lines as sets   cfractemp 36163
                  ​*21.17.6.6  Addition and opposite   caddcc 36204
                  ​*21.17.6.7  Order relation on the extended reals   cltxr 36208
                  ​*21.17.6.8  Argument, multiplication and inverse   carg 36210
                  21.17.6.9  The canonical bijection from the finite ordinals   ciomnn 36216
                  21.17.6.10  Divisibility   cnnbar 36227
            ​*21.17.7  Monoids   bj-smgrpssmgm 36235
                  ​*21.17.7.1  Finite sums in monoids   cfinsum 36250
            ​*21.17.8  Affine, Euclidean, and Cartesian geometry   bj-fvimacnv0 36253
                  *21.17.8.1  Real vector spaces   bj-fvimacnv0 36253
                  ​*21.17.8.2  Complex numbers (supplements)   bj-subcom 36275
                  ​*21.17.8.3  Barycentric coordinates   bj-bary1lem 36277
            21.17.9  Monoid of endomorphisms   cend 36280
      ​21.18  Mathbox for Jim Kingdon
                  21.18.0.1  Circle constant   taupilem3 36286
                  21.18.0.2  Number theory   dfgcd3 36291
                  21.18.0.3  Real numbers   irrdifflemf 36292
      ​21.19  Mathbox for ML
            21.19.1  Miscellaneous   csbrecsg 36295
            21.19.2  Cartesian exponentiation   cfinxp 36350
            21.19.3  Topology   iunctb2 36370
                  ​*21.19.3.1  Pi-base theorems   pibp16 36380
      ​21.20  Mathbox for Wolf Lammen
            21.20.1  1. Bootstrapping   wl-section-boot 36389
            21.20.2  Implication chains   wl-section-impchain 36413
            21.20.3  Theorems around the conditional operator   wl-ifp-ncond1 36431
            21.20.4  Alternative development of hadd, cadd   wl-df-3xor 36435
            21.20.5  An alternative axiom ~ ax-13   ax-wl-13v 36460
            21.20.6  Other stuff   wl-mps 36462
      ​21.21  Mathbox for Brendan Leahy
      ​21.22  Mathbox for Jeff Madsen
            21.22.1  Logic and set theory   unirep 36668
            21.22.2  Real and complex numbers; integers   filbcmb 36694
            21.22.3  Sequences and sums   sdclem2 36696
            21.22.4  Topology   subspopn 36706
            21.22.5  Metric spaces   metf1o 36709
            21.22.6  Continuous maps and homeomorphisms   constcncf 36716
            21.22.7  Boundedness   ctotbnd 36720
            21.22.8  Isometries   cismty 36752
            21.22.9  Heine-Borel Theorem   heibor1lem 36763
            21.22.10  Banach Fixed Point Theorem   bfplem1 36776
            21.22.11  Euclidean space   crrn 36779
            21.22.12  Intervals (continued)   ismrer1 36792
            21.22.13  Operation properties   cass 36796
            21.22.14  Groups and related structures   cmagm 36802
            21.22.15  Group homomorphism and isomorphism   cghomOLD 36837
            21.22.16  Rings   crngo 36848
            21.22.17  Division Rings   cdrng 36902
            21.22.18  Ring homomorphisms   crnghom 36914
            21.22.19  Commutative rings   ccm2 36943
            21.22.20  Ideals   cidl 36961
            21.22.21  Prime rings and integral domains   cprrng 37000
            21.22.22  Ideal generators   cigen 37013
      ​21.23  Mathbox for Giovanni Mascellani
            ​*21.23.1  Tools for automatic proof building   efald2 37032
            ​*21.23.2  Tseitin axioms   fald 37083
            ​*21.23.3  Equality deductions   iuneq2f 37110
            ​*21.23.4  Miscellanea   orcomdd 37121
      ​21.24  Mathbox for Peter Mazsa
            21.24.1  Notations   cxrn 37128
            21.24.2  Preparatory theorems   el2v1 37171
            21.24.3  Range Cartesian product   df-xrn 37327
            21.24.4  Cosets by ` R `   df-coss 37367
            21.24.5  Relations   df-rels 37441
            21.24.6  Subset relations   df-ssr 37454
            21.24.7  Reflexivity   df-refs 37466
            21.24.8  Converse reflexivity   df-cnvrefs 37481
            21.24.9  Symmetry   df-syms 37498
            21.24.10  Reflexivity and symmetry   symrefref2 37519
            21.24.11  Transitivity   df-trs 37528
            21.24.12  Equivalence relations   df-eqvrels 37540
            21.24.13  Redundancy   df-redunds 37579
            21.24.14  Domain quotients   df-dmqss 37594
            21.24.15  Equivalence relations on domain quotients   df-ers 37619
            21.24.16  Functions   df-funss 37636
            21.24.17  Disjoints vs. converse functions   df-disjss 37659
            21.24.18  Antisymmetry   df-antisymrel 37716
            21.24.19  Partitions: disjoints on domain quotients   df-parts 37721
            21.24.20  Partition-Equivalence Theorems   disjim 37737
      ​21.25  Mathbox for Rodolfo Medina
            21.25.1  Partitions   prtlem60 37809
      ​​*21.26  Mathbox for Norm Megill
            *21.26.1  Obsolete schemes ax-c4,c5,c7,c10,c11,c11n,c15,c9,c14,c16   ax-c5 37839
            ​*21.26.2  Rederive new axioms ax-4, ax-10, ax-6, ax-12, ax-13 from old   axc5 37849
            ​*21.26.3  Legacy theorems using obsolete axioms   ax5ALT 37863
            21.26.4  Experiments with weak deduction theorem   elimhyps 37917
            21.26.5  Miscellanea   cnaddcom 37928
            21.26.6  Atoms, hyperplanes, and covering in a left vector space (or module)   clsa 37930
            21.26.7  Functionals and kernels of a left vector space (or module)   clfn 38013
            21.26.8  Opposite rings and dual vector spaces   cld 38079
            21.26.9  Ortholattices and orthomodular lattices   cops 38128
            21.26.10  Atomic lattices with covering property   ccvr 38218
            21.26.11  Hilbert lattices   chlt 38306
            21.26.12  Projective geometries based on Hilbert lattices   clln 38448
            21.26.13  Construction of a vector space from a Hilbert lattice   cdlema1N 38748
            21.26.14  Construction of involution and inner product from a Hilbert lattice   clpoN 40437
      ​21.27  Mathbox for metakunt
            21.27.1  General helpful statements   leexp1ad 40923
            21.27.2  Some gcd and lcm results   12gcd5e1 40954
            21.27.3  Least common multiple inequality theorem   3factsumint1 40972
            21.27.4  Logarithm inequalities   3exp7 41004
            21.27.5  Miscellaneous results for AKS formalisation   intlewftc 41012
            21.27.6  Sticks and stones   sticksstones1 41048
            21.27.7  Permutation results   metakunt1 41071
            21.27.8  Unused lemmas scheduled for deletion   andiff 41105
      ​21.28  Mathbox for Steven Nguyen
            21.28.1  Utility theorems   ioin9i8 41110
            21.28.2  Structures   nelsubginvcld 41156
            ​*21.28.3  Arithmetic theorems   c0exALT 41255
            21.28.4  Exponents and divisibility   oexpreposd 41294
            21.28.5  Real subtraction   cresub 41320
            ​*21.28.6  Projective spaces   cprjsp 41425
            21.28.7  Basic reductions for Fermat's Last Theorem   dffltz 41458
            ​*21.28.8  Exemplar theorems   iddii 41488
      ​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 41512
            21.31.2  Additional theory of functions   imaiinfv 41513
            21.31.3  Additional topology   elrfi 41514
            21.31.4  Characterization of closure operators. Kuratowski closure axioms   ismrcd1 41518
            21.31.5  Algebraic closure systems   cnacs 41522
            21.31.6  Miscellanea 1. Map utilities   constmap 41533
            21.31.7  Miscellanea for polynomials   mptfcl 41540
            21.31.8  Multivariate polynomials over the integers   cmzpcl 41541
            21.31.9  Miscellanea for Diophantine sets 1   coeq0i 41573
            21.31.10  Diophantine sets 1: definitions   cdioph 41575
            21.31.11  Diophantine sets 2 miscellanea   ellz1 41587
            21.31.12  Diophantine sets 2: union and intersection. Monotone Boolean algebra   diophin 41592
            21.31.13  Diophantine sets 3: construction   diophrex 41595
            21.31.14  Diophantine sets 4 miscellanea   2sbcrex 41604
            21.31.15  Diophantine sets 4: Quantification   rexrabdioph 41614
            21.31.16  Diophantine sets 5: Arithmetic sets   rabdiophlem1 41621
            21.31.17  Diophantine sets 6: reusability. renumbering of variables   eldioph4b 41631
            21.31.18  Pigeonhole Principle and cardinality helpers   fphpd 41636
            21.31.19  A non-closed set of reals is infinite   rencldnfilem 41640
            21.31.20  Lagrange's rational approximation theorem   irrapxlem1 41642
            21.31.21  Pell equations 1: A nontrivial solution always exists   pellexlem1 41649
            21.31.22  Pell equations 2: Algebraic number theory of the solution set   csquarenn 41656
            21.31.23  Pell equations 3: characterizing fundamental solution   infmrgelbi 41698
            ​*21.31.24  Logarithm laws generalized to an arbitrary base   reglogcl 41710
            21.31.25  Pell equations 4: the positive solution group is infinite cyclic   pellfund14 41718
            21.31.26  X and Y sequences 1: Definition and recurrence laws   crmx 41720
            21.31.27  Ordering and induction lemmas for the integers   monotuz 41762
            21.31.28  X and Y sequences 2: Order properties   rmxypos 41768
            21.31.29  Congruential equations   congtr 41786
            21.31.30  Alternating congruential equations   acongid 41796
            21.31.31  Additional theorems on integer divisibility   coprmdvdsb 41806
            21.31.32  X and Y sequences 3: Divisibility properties   jm2.18 41809
            21.31.33  X and Y sequences 4: Diophantine representability of Y   jm2.27a 41826
            21.31.34  X and Y sequences 5: Diophantine representability of X, ^, _C   rmxdiophlem 41836
            21.31.35  Uncategorized stuff not associated with a major project   setindtr 41845
            21.31.36  More equivalents of the Axiom of Choice   axac10 41854
            21.31.37  Finitely generated left modules   clfig 41891
            21.31.38  Noetherian left modules I   clnm 41899
            21.31.39  Addenda for structure powers   pwssplit4 41913
            21.31.40  Every set admits a group structure iff choice   unxpwdom3 41919
            21.31.41  Noetherian rings and left modules II   clnr 41933
            21.31.42  Hilbert's Basis Theorem   cldgis 41945
            21.31.43  Additional material on polynomials [DEPRECATED]   cmnc 41955
            21.31.44  Degree and minimal polynomial of algebraic numbers   cdgraa 41964
            21.31.45  Algebraic integers I   citgo 41981
            21.31.46  Endomorphism algebra   cmend 41999
            21.31.47  Cyclic groups and order   idomrootle 42019
            21.31.48  Cyclotomic polynomials   ccytp 42026
            21.31.49  Miscellaneous topology   fgraphopab 42034
      ​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 42048
            21.34.2  Natural addition of Cantor normal forms   oawordex2 42158
            21.34.3  Surreal Contributions   abeqabi 42241
            21.34.4  Short Studies   nlimsuc 42274
                  21.34.4.1  Additional work on conditional logical operator   ifpan123g 42292
                  21.34.4.2  Sophisms   rp-fakeimass 42345
                  ​*21.34.4.3  Finite Sets   rp-isfinite5 42350
                  21.34.4.4  General Observations   intabssd 42352
                  21.34.4.5  Infinite Sets   pwelg 42393
                  ​*21.34.4.6  Finite intersection property   fipjust 42398
                  21.34.4.7  RP ADDTO: Subclasses and subsets   rababg 42407
                  21.34.4.8  RP ADDTO: The intersection of a class   elinintab 42408
                  21.34.4.9  RP ADDTO: Theorems requiring subset and intersection existence   elinintrab 42410
                  21.34.4.10  RP ADDTO: Relations   xpinintabd 42413
                  ​*21.34.4.11  RP ADDTO: Functions   elmapintab 42429
                  ​*21.34.4.12  RP ADDTO: Finite induction (for finite ordinals)   cnvcnvintabd 42433
                  21.34.4.13  RP ADDTO: First and second members of an ordered pair   elcnvlem 42434
                  21.34.4.14  RP ADDTO: The reflexive and transitive properties of relations   undmrnresiss 42437
                  21.34.4.15  RP ADDTO: Basic properties of closures   cleq2lem 42441
                  21.34.4.16  RP REPLACE: Definitions and basic properties of transitive closures   trcleq2lemRP 42463
                  ​*21.34.4.17  Additions for square root; absolute value   sqrtcvallem1 42464
            21.34.5  Additional statements on relations and subclasses   al3im 42480
                  21.34.5.1  Transitive relations (not to be confused with transitive classes).   trrelind 42498
                  21.34.5.2  Reflexive closures   crcl 42505
                  ​*21.34.5.3  Finite relationship composition.   relexp2 42510
                  21.34.5.4  Transitive closure of a relation   dftrcl3 42553
                  ​*21.34.5.5  Adapted from Frege   frege77d 42579
            ​*21.34.6  Propositions from _Begriffsschrift_   dfxor4 42599
                  *21.34.6.1  _Begriffsschrift_ Chapter I   dfxor4 42599
                  ​*21.34.6.2  _Begriffsschrift_ Notation hints   whe 42605
                  21.34.6.3  _Begriffsschrift_ Chapter II Implication   ax-frege1 42623
                  21.34.6.4  _Begriffsschrift_ Chapter II Implication and Negation   axfrege28 42662
                  ​*21.34.6.5  _Begriffsschrift_ Chapter II with logical equivalence   axfrege52a 42689
                  21.34.6.6  _Begriffsschrift_ Chapter II with equivalence of sets   axfrege52c 42720
                  ​*21.34.6.7  _Begriffsschrift_ Chapter II with equivalence of classes   frege53c 42747
                  ​*21.34.6.8  _Begriffsschrift_ Chapter III Properties hereditary in a sequence   dffrege69 42765
                  ​*21.34.6.9  _Begriffsschrift_ Chapter III Following in a sequence   dffrege76 42772
                  ​*21.34.6.10  _Begriffsschrift_ Chapter III Member of sequence   dffrege99 42795
                  ​*21.34.6.11  _Begriffsschrift_ Chapter III Single-valued procedures   dffrege115 42811
            ​*21.34.7  Exploring Topology via Seifert and Threlfall   enrelmap 42830
                  *21.34.7.1  Equinumerosity of sets of relations and maps   enrelmap 42830
                  ​*21.34.7.2  Generic Pseudoclosure Spaces, Pseudointerior Spaces, and Pseudoneighborhoods   or3or 42856
                  ​*21.34.7.3  Generic Neighborhood Spaces   gneispa 42963
            ​*21.34.8  Exploring Higher Homotopy via Kerodon   k0004lem1 42980
                  *21.34.8.1  Simplicial Sets   k0004lem1 42980
      ​21.35  Mathbox for Stanislas Polu
            21.35.1  IMO Problems   wwlemuld 42989
                  21.35.1.1  IMO 1972 B2   wwlemuld 42989
            ​*21.35.2  INT Inequalities Proof Generator   int-addcomd 43007
            ​*21.35.3  N-Digit Addition Proof Generator   unitadd 43029
            21.35.4  AM-GM (for k = 2,3,4)   gsumws3 43030
      ​21.36  Mathbox for Rohan Ridenour
            21.36.1  Misc   spALT 43035
            21.36.2  Monoid rings   cmnring 43047
            21.36.3  Shorter primitive equivalent of ax-groth   gru0eld 43070
                  21.36.3.1  Grothendieck universes are closed under collection   gru0eld 43070
                  21.36.3.2  Minimal universes   ismnu 43102
                  21.36.3.3  Primitive equivalent of ax-groth   expandan 43129
      ​21.37  Mathbox for Steve Rodriguez
            21.37.1  Miscellanea   nanorxor 43146
            21.37.2  Ratio test for infinite series convergence and divergence   dvgrat 43153
            21.37.3  Multiples   reldvds 43156
            21.37.4  Function operations   caofcan 43164
            21.37.5  Calculus   lhe4.4ex1a 43170
            21.37.6  The generalized binomial coefficient operation   cbcc 43177
            21.37.7  Binomial series   uzmptshftfval 43187
      ​21.38  Mathbox for Andrew Salmon
            21.38.1  Principia Mathematica * 10   pm10.12 43199
            21.38.2  Principia Mathematica * 11   2alanimi 43213
            21.38.3  Predicate Calculus   sbeqal1 43239
            21.38.4  Principia Mathematica * 13 and * 14   pm13.13a 43248
            21.38.5  Set Theory   elnev 43279
            21.38.6  Arithmetic   addcomgi 43297
            21.38.7  Geometry   cplusr 43298
      ​​*21.39  Mathbox for Alan Sare
            21.39.1  Auxiliary theorems for the Virtual Deduction tool   idiALT 43320
            21.39.2  Supplementary unification deductions   bi1imp 43324
            21.39.3  Conventional Metamath proofs, some derived from VD proofs   iidn3 43344
            21.39.4  What is Virtual Deduction?   wvd1 43412
            21.39.5  Virtual Deduction Theorems   df-vd1 43413
            21.39.6  Theorems proved using Virtual Deduction   trsspwALT 43661
            21.39.7  Theorems proved using Virtual Deduction with mmj2 assistance   simplbi2VD 43689
            21.39.8  Virtual Deduction transcriptions of textbook proofs   sb5ALTVD 43756
            21.39.9  Theorems proved using conjunction-form Virtual Deduction   elpwgdedVD 43760
            21.39.10  Theorems with a VD proof in conventional notation derived from a VD proof   suctrALT3 43767
            ​*21.39.11  Theorems with a proof in conventional notation derived from a VD proof   notnotrALT2 43770
      ​21.40  Mathbox for Glauco Siliprandi
            21.40.1  Miscellanea   evth2f 43781
            21.40.2  Functions   feq1dd 43945
            21.40.3  Ordering on real numbers - Real and complex numbers basic operations   sub2times 44061
            21.40.4  Real intervals   gtnelioc 44283
            21.40.5  Finite sums   fsummulc1f 44366
            21.40.6  Finite multiplication of numbers and finite multiplication of functions   fmul01 44375
            21.40.7  Limits   clim1fr1 44396
                  21.40.7.1  Inferior limit (lim inf)   clsi 44546
                  ​*21.40.7.2  Limits for sequences of extended real numbers   clsxlim 44613
            21.40.8  Trigonometry   coseq0 44659
            21.40.9  Continuous Functions   mulcncff 44665
            21.40.10  Derivatives   dvsinexp 44706
            21.40.11  Integrals   itgsin0pilem1 44745
            21.40.12  Stone Weierstrass theorem - real version   stoweidlem1 44796
            21.40.13  Wallis' product for π   wallispilem1 44860
            21.40.14  Stirling's approximation formula for ` n ` factorial   stirlinglem1 44869
            21.40.15  Dirichlet kernel   dirkerval 44886
            21.40.16  Fourier Series   fourierdlem1 44903
            21.40.17  e is transcendental   elaa2lem 45028
            21.40.18  n-dimensional Euclidean space   rrxtopn 45079
            21.40.19  Basic measure theory   csalg 45103
                  ​*21.40.19.1  σ-Algebras   csalg 45103
                  21.40.19.2  Sum of nonnegative extended reals   csumge0 45157
                  ​*21.40.19.3  Measures   cmea 45244
                  ​*21.40.19.4  Outer measures and Caratheodory's construction   come 45284
                  ​*21.40.19.5  Lebesgue measure on n-dimensional Real numbers   covoln 45331
                  ​*21.40.19.6  Measurable functions   csmblfn 45490
      ​21.41  Mathbox for Saveliy Skresanov
            21.41.1  Ceva's theorem   sigarval 45645
            21.41.2  Simple groups   simpcntrab 45665
      ​21.42  Mathbox for Ender Ting
            21.42.1  Increasing sequences and subsequences   et-ltneverrefl 45666
      ​21.43  Mathbox for Jarvin Udandy
      ​21.44  Mathbox for Adhemar
            ​*21.44.1  Minimal implicational calculus   adh-minim 45790
      ​21.45  Mathbox for Alexander van der Vekens
            21.45.1  General auxiliary theorems (1)   n0nsn2el 45814
                  21.45.1.1  Unordered and ordered pairs - extension for singletons   n0nsn2el 45814
                  21.45.1.2  Unordered and ordered pairs - extension for unordered pairs   elprneb 45818
                  21.45.1.3  Unordered and ordered pairs - extension for ordered pairs   oppr 45819
                  21.45.1.4  Relations - extension   eubrv 45824
                  21.45.1.5  Definite description binder (inverted iota) - extension   iota0def 45827
                  21.45.1.6  Functions - extension   fveqvfvv 45829
            21.45.2  Alternative for Russell's definition of a description binder   caiota 45870
            21.45.3  Double restricted existential uniqueness   r19.32 45885
                  21.45.3.1  Restricted quantification (extension)   r19.32 45885
                  21.45.3.2  Restricted uniqueness and "at most one" quantification   reuf1odnf 45894
                  21.45.3.3  Analogs to Existential uniqueness (double quantification)   2reu3 45897
                  21.45.3.4  Additional theorems for double restricted existential uniqueness   2reu8i 45900
            ​*21.45.4  Alternative definitions of function and operation values   wdfat 45903
                  21.45.4.1  Restricted quantification (extension)   ralbinrald 45909
                  21.45.4.2  The universal class (extension)   nvelim 45910
                  21.45.4.3  Introduce the Axiom of Power Sets (extension)   alneu 45911
                  21.45.4.4  Predicate "defined at"   dfateq12d 45913
                  21.45.4.5  Alternative definition of the value of a function   dfafv2 45919
                  21.45.4.6  Alternative definition of the value of an operation   aoveq123d 45965
            ​*21.45.5  Alternative definitions of function values (2)   cafv2 45995
            21.45.6  General auxiliary theorems (2)   an4com24 46055
                  21.45.6.1  Logical conjunction - extension   an4com24 46055
                  21.45.6.2  Abbreviated conjunction and disjunction of three wff's - extension   3an4ancom24 46056
                  21.45.6.3  Negated membership (alternative)   cnelbr 46058
                  21.45.6.4  The empty set - extension   ralralimp 46065
                  21.45.6.5  Indexed union and intersection - extension   otiunsndisjX 46066
                  21.45.6.6  Functions - extension   fvifeq 46067
                  21.45.6.7  Maps-to notation - extension   fvmptrab 46079
                  21.45.6.8  Subtraction - extension   cnambpcma 46081
                  21.45.6.9  Ordering on reals (cont.) - extension   leaddsuble 46084
                  21.45.6.10  Imaginary and complex number properties - extension   readdcnnred 46090
                  21.45.6.11  Nonnegative integers (as a subset of complex numbers) - extension   nn0resubcl 46095
                  21.45.6.12  Integers (as a subset of complex numbers) - extension   zgeltp1eq 46096
                  21.45.6.13  Decimal arithmetic - extension   1t10e1p1e11 46097
                  21.45.6.14  Upper sets of integers - extension   eluzge0nn0 46099
                  21.45.6.15  Infinity and the extended real number system (cont.) - extension   nltle2tri 46100
                  21.45.6.16  Finite intervals of integers - extension   ssfz12 46101
                  21.45.6.17  Half-open integer ranges - extension   fzopred 46109
                  21.45.6.18  The modulo (remainder) operation - extension   m1mod0mod1 46116
                  21.45.6.19  The infinite sequence builder "seq"   smonoord 46118
                  21.45.6.20  Finite and infinite sums - extension   fsummsndifre 46119
                  21.45.6.21  Extensible structures - extension   setsidel 46123
            ​*21.45.7  Preimages of function values   preimafvsnel 46126
            ​*21.45.8  Partitions of real intervals   ciccp 46160
            21.45.9  Shifting functions with an integer range domain   fargshiftfv 46186
            21.45.10  Words over a set (extension)   lswn0 46191
                  21.45.10.1  Last symbol of a word - extension   lswn0 46191
            21.45.11  Unordered pairs   wich 46192
                  21.45.11.1  Interchangeable setvar variables   wich 46192
                  21.45.11.2  Set of unordered pairs   sprid 46221
                  ​*21.45.11.3  Proper (unordered) pairs   prpair 46248
                  21.45.11.4  Set of proper unordered pairs   cprpr 46259
            21.45.12  Number theory (extension)   cfmtno 46274
                  ​*21.45.12.1  Fermat numbers   cfmtno 46274
                  ​*21.45.12.2  Mersenne primes   m2prm 46338
                  21.45.12.3  Proth's theorem   modexp2m1d 46359
                  21.45.12.4  Solutions of quadratic equations   quad1 46367
            ​*21.45.13  Even and odd numbers   ceven 46371
                  21.45.13.1  Definitions and basic properties   ceven 46371
                  21.45.13.2  Alternate definitions using the "divides" relation   dfeven2 46396
                  21.45.13.3  Alternate definitions using the "modulo" operation   dfeven3 46405
                  21.45.13.4  Alternate definitions using the "gcd" operation   iseven5 46411
                  21.45.13.5  Theorems of part 5 revised   zneoALTV 46416
                  21.45.13.6  Theorems of part 6 revised   odd2np1ALTV 46421
                  21.45.13.7  Theorems of AV's mathbox revised   0evenALTV 46435
                  21.45.13.8  Additional theorems   epoo 46450
                  21.45.13.9  Perfect Number Theorem (revised)   perfectALTVlem1 46468
            21.45.14  Number theory (extension 2)   cfppr 46471
                  ​*21.45.14.1  Fermat pseudoprimes   cfppr 46471
                  ​*21.45.14.2  Goldbach's conjectures   cgbe 46492
            21.45.15  Graph theory (extension)   cgrisom 46565
                  ​*21.45.15.1  Isomorphic graphs   cgrisom 46565
                  21.45.15.2  Loop-free graphs - extension   1hegrlfgr 46589
                  21.45.15.3  Walks - extension   cupwlks 46590
                  21.45.15.4  Edges of graphs expressed as sets of unordered pairs   upgredgssspr 46600
            21.45.16  Monoids (extension)   ovn0dmfun 46613
                  21.45.16.1  Auxiliary theorems   ovn0dmfun 46613
                  21.45.16.2  Magmas, Semigroups and Monoids (extension)   plusfreseq 46621
                  21.45.16.3  Magma homomorphisms and submagmas   cmgmhm 46626
                  21.45.16.4  Examples and counterexamples for magmas, semigroups and monoids (extension)   opmpoismgm 46656
                  21.45.16.5  Group sum operation (extension 1)   gsumsplit2f 46669
            ​*21.45.17  Magmas and internal binary operations (alternate approach)   ccllaw 46672
                  *21.45.17.1  Laws for internal binary operations   ccllaw 46672
                  ​*21.45.17.2  Internal binary operations   cintop 46685
                  21.45.17.3  Alternative definitions for magmas and semigroups   cmgm2 46704
            21.45.18  Categories (extension)   idfusubc0 46718
                  21.45.18.1  Subcategories (extension)   idfusubc0 46718
            21.45.19  Rings (extension)   lmod0rng 46721
                  21.45.19.1  Nonzero rings (extension)   lmod0rng 46721
                  ​*21.45.19.2  Non-unital rings ("rngs")   crng 46727
                  21.45.19.3  Rng homomorphisms   crngh 46762
                  21.45.19.4  Ring homomorphisms (extension)   rhmfn 46800
                  21.45.19.5  Subrings of non-unital rings   csubrng 46803
                  ​*21.45.19.6  Ideals of non-unital rings   rnglidlmcl 46827
                  ​*21.45.19.7  Condition for a non-unital ring to be unital   rngqiprng1elbas 46850
                  21.45.19.8  Ideals as non-unital rings   lidldomn1 46902
                  21.45.19.9  The non-unital ring of even integers   0even 46908
                  21.45.19.10  A constructed not unital ring   cznrnglem 46930
                  ​*21.45.19.11  The category of non-unital rings   crngc 46934
                  ​*21.45.19.12  The category of (unital) rings   cringc 46980
                  21.45.19.13  Subcategories of the category of rings   srhmsubclem1 47050
            21.45.20  Basic algebraic structures (extension)   opeliun2xp 47087
                  21.45.20.1  Auxiliary theorems   opeliun2xp 47087
                  21.45.20.2  The binomial coefficient operation (extension)   bcpascm1 47106
                  21.45.20.3  The ` ZZ `-module ` ZZ X. ZZ `   zlmodzxzlmod 47109
                  21.45.20.4  Group sum operation (extension 2)   mgpsumunsn 47116
                  21.45.20.5  Symmetric groups (extension)   exple2lt6 47119
                  21.45.20.6  Divisibility (extension)   invginvrid 47122
                  21.45.20.7  The support of functions (extension)   rmsupp0 47123
                  21.45.20.8  Finitely supported functions (extension)   rmsuppfi 47128
                  21.45.20.9  Left modules (extension)   lmodvsmdi 47137
                  21.45.20.10  Associative algebras (extension)   assaascl0 47139
                  21.45.20.11  Univariate polynomials (extension)   ply1vr1smo 47141
                  21.45.20.12  Univariate polynomials (examples)   linply1 47152
            21.45.21  Linear algebra (extension)   cdmatalt 47155
                  ​*21.45.21.1  The subalgebras of diagonal and scalar matrices (extension)   cdmatalt 47155
                  ​*21.45.21.2  Linear combinations   clinc 47163
                  ​*21.45.21.3  Linear independence   clininds 47199
                  21.45.21.4  Simple left modules and the ` ZZ `-module   lmod1lem1 47246
                  21.45.21.5  Differences between (left) modules and (left) vector spaces   lvecpsslmod 47266
            21.45.22  Complexity theory   suppdm 47269
                  21.45.22.1  Auxiliary theorems   suppdm 47269
                  21.45.22.2  The modulo (remainder) operation (extension)   fldivmod 47282
                  21.45.22.3  Even and odd integers   nn0onn0ex 47287
                  21.45.22.4  The natural logarithm on complex numbers (extension)   logcxp0 47299
                  21.45.22.5  Division of functions   cfdiv 47301
                  21.45.22.6  Upper bounds   cbigo 47311
                  21.45.22.7  Logarithm to an arbitrary base (extension)   rege1logbrege0 47322
                  ​*21.45.22.8  The binary logarithm   fldivexpfllog2 47329
                  21.45.22.9  Binary length   cblen 47333
                  ​*21.45.22.10  Digits   cdig 47359
                  21.45.22.11  Nonnegative integer as sum of its shifted digits   dignn0flhalflem1 47379
                  21.45.22.12  Algorithms for the multiplication of nonnegative integers   nn0mulfsum 47388
                  ​*21.45.22.13  N-ary functions   cnaryf 47390
                  ​*21.45.22.14  The Ackermann function   citco 47421
            21.45.23  Elementary geometry (extension)   fv1prop 47463
                  21.45.23.1  Auxiliary theorems   fv1prop 47463
                  21.45.23.2  Real euclidean space of dimension 2   rrx2pxel 47475
                  21.45.23.3  Spheres and lines in real Euclidean spaces   cline 47491
      ​21.46  Mathbox for Zhi Wang
            21.46.1  Propositional calculus   pm4.71da 47553
            21.46.2  Predicate calculus with equality   dtrucor3 47562
                  21.46.2.1  Axiom scheme ax-5 (Distinctness)   dtrucor3 47562
            21.46.3  ZF Set Theory - start with the Axiom of Extensionality   ralbidb 47563
                  21.46.3.1  Restricted quantification   ralbidb 47563
                  21.46.3.2  The empty set   ssdisjd 47570
                  21.46.3.3  Unordered and ordered pairs   vsn 47574
                  21.46.3.4  The union of a class   unilbss 47580
            21.46.4  ZF Set Theory - add the Axiom of Replacement   inpw 47581
                  21.46.4.1  Theorems requiring subset and intersection existence   inpw 47581
            21.46.5  ZF Set Theory - add the Axiom of Power Sets   mof0 47582
                  21.46.5.1  Functions   mof0 47582
                  21.46.5.2  Operations   fvconstr 47600
            21.46.6  ZF Set Theory - add the Axiom of Union   fvconst0ci 47603
                  21.46.6.1  Equinumerosity   fvconst0ci 47603
            21.46.7  Order sets   iccin 47607
                  21.46.7.1  Real number intervals   iccin 47607
            21.46.8  Moore spaces   mreuniss 47610
            ​*21.46.9  Topology   clduni 47611
                  21.46.9.1  Closure and interior   clduni 47611
                  21.46.9.2  Neighborhoods   neircl 47615
                  21.46.9.3  Subspace topologies   restcls2lem 47623
                  21.46.9.4  Limits and continuity in topological spaces   cnneiima 47627
                  21.46.9.5  Topological definitions using the reals   iooii 47628
                  21.46.9.6  Separated sets   sepnsepolem1 47632
                  21.46.9.7  Separated spaces: T0, T1, T2 (Hausdorff) ...   isnrm4 47641
            21.46.10  Preordered sets and directed sets using extensible structures   isprsd 47666
            21.46.11  Posets and lattices using extensible structures   lubeldm2 47667
                  21.46.11.1  Posets   lubeldm2 47667
                  21.46.11.2  Lattices   toslat 47685
                  21.46.11.3  Subset order structures   intubeu 47687
            21.46.12  Categories   catprslem 47708
                  21.46.12.1  Categories   catprslem 47708
                  21.46.12.2  Monomorphisms and epimorphisms   idmon 47714
                  21.46.12.3  Functors   funcf2lem 47716
            21.46.13  Examples of categories   cthinc 47717
                  21.46.13.1  Thin categories   cthinc 47717
                  21.46.13.2  Preordered sets as thin categories   cprstc 47760
                  21.46.13.3  Monoids as categories   cmndtc 47781
      ​21.47  Mathbox for Emmett Weisz
            ​*21.47.1  Miscellaneous Theorems   nfintd 47796
            21.47.2  Set Recursion   csetrecs 47806
                  ​*21.47.2.1  Basic Properties of Set Recursion   csetrecs 47806
                  21.47.2.2  Examples and properties of set recursion   elsetrecslem 47822
            ​*21.47.3  Construction of Games and Surreal Numbers   cpg 47832
      ​​*21.48  Mathbox for David A. Wheeler
            21.48.1  Natural deduction   sbidd 47841
            ​*21.48.2  Greater than, greater than or equal to.   cge-real 47843
            ​*21.48.3  Hyperbolic trigonometric functions   csinh 47853
            ​*21.48.4  Reciprocal trigonometric functions (sec, csc, cot)   csec 47864
            ​*21.48.5  Identities for "if"   ifnmfalse 47886
            ​*21.48.6  Logarithms generalized to arbitrary base using ` logb `   logb2aval 47887
            ​*21.48.7  Logarithm laws generalized to an arbitrary base - log_   clog- 47888
            ​*21.48.8  Formally define notions such as reflexivity   wreflexive 47890
            ​*21.48.9  Algebra helpers   comraddi 47894
            ​*21.48.10  Algebra helper examples   i2linesi 47903
            ​*21.48.11  Formal methods "surprises"   alimp-surprise 47905
            ​*21.48.12  Allsome quantifier   walsi 47911
            ​*21.48.13  Miscellaneous   5m4e1 47922
            21.48.14  Theorems about algebraic numbers   aacllem 47926
      ​21.49  Mathbox for Kunhao Zheng
            21.49.1  Weighted AM-GM inequality   amgmwlem 47927