TOC of Theorem List - Metamath Proof Explorer

Table of Contents Summary

PART 1  CLASSICAL FIRST-ORDER LOGIC WITH EQUALITY
      1.1  Pre-logic
      1.2  Propositional calculus
      1.3  Other axiomatizations related to classical propositional calculus
      1.4  Predicate calculus with equality: Tarski's system S2 (1 rule, 6 schemes)
      1.5  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes)
      1.6  Uniqueness and unique existence
      1.7  Other axiomatizations related to classical predicate calculus
PART 2  ZF (ZERMELO-FRAENKEL) SET THEORY
      2.1  ZF Set Theory - start with the Axiom of Extensionality
      2.2  ZF Set Theory - add the Axiom of Replacement
      2.3  ZF Set Theory - add the Axiom of Power Sets
      2.4  ZF Set Theory - add the Axiom of Union
      2.5  ZF Set Theory - add the Axiom of Regularity
      2.6  ZF Set Theory - add the Axiom of Infinity
PART 3  ZFC (ZERMELO-FRAENKEL WITH CHOICE) SET THEORY
      3.1  ZFC Set Theory - add Countable Choice and Dependent Choice
      3.2  ZFC Set Theory - add the Axiom of Choice
      3.3  ZFC Axioms with no distinct variable requirements
      3.4  The Generalized Continuum Hypothesis
PART 4  TG (TARSKI-GROTHENDIECK) SET THEORY
      4.1  Inaccessibles
      4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
PART 5  REAL AND COMPLEX NUMBERS
      5.1  Construction and axiomatization of real and complex numbers
      5.2  Derive the basic properties from the field axioms
      5.3  Real and complex numbers - basic operations
      5.4  Integer sets
      5.5  Order sets
      5.6  Elementary integer functions
      5.7  Words over a set
      5.8  Reflexive and transitive closures of relations
      5.9  Elementary real and complex functions
      5.10  Elementary limits and convergence
      5.11  Elementary trigonometry
      5.12  Cardinality of real and complex number subsets
PART 6  ELEMENTARY NUMBER THEORY
      6.1  Elementary properties of divisibility
      6.2  Elementary prime number theory
PART 7  BASIC STRUCTURES
      7.1  Extensible structures
      7.2  Moore spaces
PART 8  BASIC CATEGORY THEORY
      8.1  Categories
      8.2  Arrows (disjointified hom-sets)
      8.3  Examples of categories
      8.4  Categorical constructions
PART 9  BASIC ORDER THEORY
      9.1  Dual of an order structure
      9.2  Preordered sets and directed sets
      9.3  Partially ordered sets (posets)
      9.4  Totally ordered sets (tosets)
      9.5  Lattices
      9.6  Posets, directed sets, and lattices as relations
PART 10  BASIC ALGEBRAIC STRUCTURES
      10.1  Monoids
      10.2  Groups
      10.3  Rings
      10.4  Division rings and fields
      10.5  Left modules
      10.6  Vector spaces
      10.7  Subring algebras and ideals
      10.8  The complex numbers as an algebraic extensible structure
      10.9  Generalized pre-Hilbert and Hilbert spaces
PART 11  BASIC LINEAR ALGEBRA
      11.1  Vectors and free modules
      11.2  Associative algebras
      11.3  Abstract multivariate polynomials
      11.4  Matrices
      11.5  The determinant
      11.6  Polynomial matrices
      11.7  The characteristic polynomial
PART 12  BASIC TOPOLOGY
      12.1  Topology
      12.2  Filters and filter bases
      12.3  Uniform Structures and Spaces
      12.4  Metric spaces
      12.5  Metric subcomplex vector spaces
PART 13  BASIC REAL AND COMPLEX ANALYSIS
      13.1  Continuity
      13.2  Integrals
      13.3  Derivatives
PART 14  BASIC REAL AND COMPLEX FUNCTIONS
      14.1  Polynomials
      14.2  Sequences and series
      14.3  Basic trigonometry
      14.4  Basic number theory
PART 15  SURREAL NUMBERS
      15.1  Sign sequence representation and Alling's axioms
      15.2  Initial consequences of Alling's axioms
      15.3  Conway cut representation
      15.4  Induction and recursion
      15.5  Surreal arithmetic
      15.6  Subsystems of surreals
PART 16  ELEMENTARY GEOMETRY
      16.1  Definition and Tarski's Axioms of Geometry
      16.2  Tarskian Geometry
      16.3  Properties of geometries
      16.4  Geometry in Hilbert spaces
PART 17  GRAPH THEORY
      17.1  Vertices and edges
      17.2  Undirected graphs
      17.3  Walks, paths and cycles
      17.4  Eulerian paths and the Konigsberg Bridge problem
      17.5  The Friendship Theorem
PART 18  GUIDES AND MISCELLANEA
      18.1  Guides (conventions, explanations, and examples)
      18.2  Humor
      18.3  (Future - to be reviewed and classified)
PART 19  COMPLEX TOPOLOGICAL VECTOR SPACES (DEPRECATED)
      19.1  Additional material on group theory (deprecated)
      19.2  Complex vector spaces
      19.3  Normed complex vector spaces
      19.4  Operators on complex vector spaces
      19.5  Inner product (pre-Hilbert) spaces
      19.6  Complex Banach spaces
      19.7  Complex Hilbert spaces
PART 20  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      20.1  Axiomatization of complex pre-Hilbert spaces
      20.2  Inner product and norms
      20.3  Cauchy sequences and completeness axiom
      20.4  Subspaces and projections
      20.5  Properties of Hilbert subspaces
      20.6  Operators on Hilbert spaces
      20.7  States on a Hilbert lattice and Godowski's equation
      20.8  Cover relation, atoms, exchange axiom, and modular symmetry
PART 21  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      ​21.1  Mathboxes for user contributions
      21.2  Mathbox for Stefan Allan
      21.3  Mathbox for Thierry Arnoux
      21.4  Mathbox for Jonathan Ben-Naim
      21.5  Mathbox for BTernaryTau
      21.6  Mathbox for Mario Carneiro
      21.7  Mathbox for Filip Cernatescu
      21.8  Mathbox for Paul Chapman
      21.9  Mathbox for Hongxiu Chen
      21.10  Mathbox for Adrian Ducourtial
      21.11  Mathbox for Scott Fenton
      21.12  Mathbox for Gino Giotto
      21.13  Mathbox for Jeff Hankins
      21.14  Mathbox for Anthony Hart
      21.15  Mathbox for Chen-Pang He
      21.16  Mathbox for Jeff Hoffman
      21.17  Mathbox for Asger C. Ipsen
      21.18  Mathbox for BJ
      21.19  Mathbox for Jim Kingdon
      21.20  Mathbox for ML
      21.21  Mathbox for Wolf Lammen
      21.22  Mathbox for Brendan Leahy
      21.23  Mathbox for Jeff Madsen
      21.24  Mathbox for Giovanni Mascellani
      21.25  Mathbox for Peter Mazsa
      21.26  Mathbox for Rodolfo Medina
      21.27  Mathbox for Norm Megill
      21.28  Mathbox for metakunt
      21.29  Mathbox for Steven Nguyen
      21.30  Mathbox for Igor Ieskov
      21.31  Mathbox for OpenAI
      21.32  Mathbox for Stefan O'Rear
      21.33  Mathbox for Noam Pasman
      21.34  Mathbox for Jon Pennant
      21.35  Mathbox for Richard Penner
      21.36  Mathbox for Stanislas Polu
      21.37  Mathbox for Rohan Ridenour
      21.38  Mathbox for Steve Rodriguez
      21.39  Mathbox for Andrew Salmon
      21.40  Mathbox for Alan Sare
      21.41  Mathbox for Glauco Siliprandi
      21.42  Mathbox for Saveliy Skresanov
      21.43  Mathbox for Ender Ting
      21.44  Mathbox for Jarvin Udandy
      21.45  Mathbox for Adhemar
      21.46  Mathbox for Alexander van der Vekens
      21.47  Mathbox for Zhi Wang
      21.48  Mathbox for Emmett Weisz
      21.49  Mathbox for David A. Wheeler
      21.50  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 394
            ​*1.2.7  Logical disjunction   wo 845
            ​*1.2.8  Mixed connectives   jaao 952
            ​*1.2.9  The conditional operator for propositions   wif 1060
            ​*1.2.10  The weak deduction theorem for propositional calculus   elimh 1080
            1.2.11  Abbreviated conjunction and disjunction of three wff's   w3o 1083
            1.2.12  Logical "nand" (Sheffer stroke)   wnan 1485
            1.2.13  Logical "xor"   wxo 1505
            1.2.14  Logical "nor"   wnor 1522
            1.2.15  True and false constants   wal 1532
                  ​*1.2.15.1  Universal quantifier for use by df-tru   wal 1532
                  ​*1.2.15.2  Equality predicate for use by df-tru   cv 1533
                  1.2.15.3  The true constant   wtru 1535
                  1.2.15.4  The false constant   wfal 1546
            ​*1.2.16  Truth tables   truimtru 1557
                  1.2.16.1  Implication   truimtru 1557
                  1.2.16.2  Negation   nottru 1561
                  1.2.16.3  Equivalence   trubitru 1563
                  1.2.16.4  Conjunction   truantru 1567
                  1.2.16.5  Disjunction   truortru 1571
                  1.2.16.6  Alternative denial   trunantru 1575
                  1.2.16.7  Exclusive disjunction   truxortru 1579
                  1.2.16.8  Joint denial   trunortru 1583
            ​*1.2.17  Half adder and full adder in propositional calculus   whad 1587
                  1.2.17.1  Full adder: sum   whad 1587
                  1.2.17.2  Full adder: carry   wcad 1600
      ​1.3  Other axiomatizations related to classical propositional calculus
            ​*1.3.1  Minimal implicational calculus   minimp 1616
            ​*1.3.2  Implicational Calculus   impsingle 1622
            1.3.3  Derive the Lukasiewicz axioms from Meredith's sole axiom   meredith 1636
            1.3.4  Derive the standard axioms from the Lukasiewicz axioms   luklem1 1653
            ​*1.3.5  Derive Nicod's axiom from the standard axioms   nic-dfim 1664
            1.3.6  Derive the Lukasiewicz axioms from Nicod's axiom   nic-imp 1670
            1.3.7  Derive Nicod's Axiom from Lukasiewicz's First Sheffer Stroke Axiom   lukshef-ax1 1689
            1.3.8  Derive the Lukasiewicz Axioms from the Tarski-Bernays-Wajsberg Axioms   tbw-bijust 1693
            1.3.9  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's First CO Axiom   merco1 1708
            1.3.10  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's Second CO Axiom   merco2 1731
            1.3.11  Derive the Lukasiewicz axioms from the Russell-Bernays Axioms   rb-bijust 1744
            ​*1.3.12  Stoic logic non-modal portion (Chrysippus of Soli)   mptnan 1763
      ​​*1.4  Predicate calculus with equality: Tarski's system S2 (1 rule, 6 schemes)
            *1.4.1  Universal quantifier (continued); define "exists" and "not free"   wex 1774
                  1.4.1.1  Existential quantifier   wex 1774
                  1.4.1.2  Nonfreeness predicate   wnf 1778
            1.4.2  Rule scheme ax-gen (Generalization)   ax-gen 1790
            1.4.3  Axiom scheme ax-4 (Quantified Implication)   ax-4 1804
                  ​*1.4.3.1  The empty domain of discourse   empty 1902
            1.4.4  Axiom scheme ax-5 (Distinctness) - first use of $d   ax-5 1906
            ​*1.4.5  Equality predicate (continued)   weq 1959
            1.4.6  Axiom scheme ax-6 (Existence)   ax-6 1964
            1.4.7  Axiom scheme ax-7 (Equality)   ax-7 2004
            1.4.8  Define proper substitution   sbjust 2059
            1.4.9  Membership predicate   wcel 2099
            1.4.10  Axiom scheme ax-8 (Left Equality for Binary Predicate)   ax-8 2101
            1.4.11  Axiom scheme ax-9 (Right Equality for Binary Predicate)   ax-9 2109
            ​*1.4.12  Logical redundancy of ax-10 , ax-11 , ax-12 , ax-13   ax6dgen 2117
      ​​*1.5  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes)
            1.5.1  Axiom scheme ax-10 (Quantified Negation)   ax-10 2130
            1.5.2  Axiom scheme ax-11 (Quantifier Commutation)   ax-11 2147
            1.5.3  Axiom scheme ax-12 (Substitution)   ax-12 2167
            1.5.4  Axiom scheme ax-13 (Quantified Equality)   ax-13 2366
      ​1.6  Uniqueness and unique existence
            1.6.1  Uniqueness: the at-most-one quantifier   wmo 2527
            1.6.2  Unique existence: the unique existential quantifier   weu 2557
      ​1.7  Other axiomatizations related to classical predicate calculus
            ​*1.7.1  Aristotelian logic: Assertic syllogisms   barbara 2652
            ​*1.7.2  Intuitionistic logic   axia1 2682
​​*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 2697
            2.1.2  Classes   cab 2703
                  2.1.2.1  Class abstractions   cab 2703
                  ​*2.1.2.2  Class equality   df-cleq 2718
                  2.1.2.3  Class membership   df-clel 2803
                  2.1.2.4  Elementary properties of class abstractions   eqabdv 2860
            2.1.3  Class form not-free predicate   wnfc 2876
            2.1.4  Negated equality and membership   wne 2930
                  2.1.4.1  Negated equality   wne 2930
                  2.1.4.2  Negated membership   wnel 3036
            2.1.5  Restricted quantification   wral 3051
                  2.1.5.1  Restricted universal and existential quantification   wral 3051
                  2.1.5.2  Restricted existential uniqueness and at-most-one quantifier   wreu 3362
                  2.1.5.3  Restricted class abstraction   crab 3419
            2.1.6  The universal class   cvv 3462
            ​*2.1.7  Conditional equality (experimental)   wcdeq 3756
            2.1.8  Russell's Paradox   rru 3772
            2.1.9  Proper substitution of classes for sets   wsbc 3775
            2.1.10  Proper substitution of classes for sets into classes   csb 3891
            2.1.11  Define basic set operations and relations   cdif 3943
            2.1.12  Subclasses and subsets   df-ss 3963
            2.1.13  The difference, union, and intersection of two classes   dfdif3 4110
                  2.1.13.1  The difference of two classes   dfdif3 4110
                  2.1.13.2  The union of two classes   elun 4145
                  2.1.13.3  The intersection of two classes   elini 4191
                  2.1.13.4  The symmetric difference of two classes   csymdif 4240
                  2.1.13.5  Combinations of difference, union, and intersection of two classes   unabs 4253
                  2.1.13.6  Class abstractions with difference, union, and intersection of two classes   unabw 4296
                  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 4523
            ​*2.1.16  The weak deduction theorem for set theory   dedth 4581
            2.1.17  Power classes   cpw 4597
            2.1.18  Unordered and ordered pairs   snjust 4622
            2.1.19  The union of a class   cuni 4905
            2.1.20  The intersection of a class   cint 4946
            2.1.21  Indexed union and intersection   ciun 4993
            2.1.22  Disjointness   wdisj 5110
            2.1.23  Binary relations   wbr 5145
            2.1.24  Ordered-pair class abstractions (class builders)   copab 5207
            2.1.25  Functions in maps-to notation   cmpt 5228
            2.1.26  Transitive classes   wtr 5262
      ​2.2  ZF Set Theory - add the Axiom of Replacement
            2.2.1  Introduce the Axiom of Replacement   ax-rep 5282
            2.2.2  Derive the Axiom of Separation   axsepgfromrep 5294
            2.2.3  Derive the Null Set Axiom   axnulALT 5301
            2.2.4  Theorems requiring subset and intersection existence   nalset 5310
            2.2.5  Theorems requiring empty set existence   class2set 5351
      ​2.3  ZF Set Theory - add the Axiom of Power Sets
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 5361
            2.3.2  Derive the Axiom of Pairing   axprlem1 5419
            2.3.3  Ordered pair theorem   opnz 5471
            2.3.4  Ordered-pair class abstractions (cont.)   opabidw 5522
            2.3.5  Power class of union and intersection   pwin 5568
            2.3.6  The identity relation   cid 5571
            2.3.7  The membership relation (or epsilon relation)   cep 5577
            ​*2.3.8  Partial and total orderings   wpo 5584
            2.3.9  Founded and well-ordering relations   wfr 5626
            2.3.10  Relations   cxp 5672
            2.3.11  The Predecessor Class   cpred 6303
            2.3.12  Well-founded induction (variant)   frpomin 6345
            2.3.13  Well-ordered induction   tz6.26 6352
            2.3.14  Ordinals   word 6367
            2.3.15  Definite description binder (inverted iota)   cio 6496
            2.3.16  Functions   wfun 6540
            2.3.17  Cantor's Theorem   canth 7369
            2.3.18  Restricted iota (description binder)   crio 7371
            2.3.19  Operations   co 7416
                  2.3.19.1  Variable-to-class conversion for operations   caovclg 7610
            2.3.20  Maps-to notation   mpondm0 7658
            2.3.21  Function operation   cof 7680
            2.3.22  Proper subset relation   crpss 7725
      ​2.4  ZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 7738
            2.4.2  Ordinals (continued)   epweon 7775
            2.4.3  Transfinite induction   tfi 7855
            2.4.4  The natural numbers (i.e., finite ordinals)   com 7868
            2.4.5  Peano's postulates   peano1 7892
            2.4.6  Finite induction (for finite ordinals)   find 7900
            2.4.7  Relations and functions (cont.)   dmexg 7906
            2.4.8  First and second members of an ordered pair   c1st 7993
            2.4.9  Induction on Cartesian products   frpoins3xpg 8146
            2.4.10  Ordering on Cartesian products   xpord2lem 8148
            2.4.11  Ordering Ordinal Sequences   orderseqlem 8163
            ​*2.4.12  The support of functions   csupp 8166
            ​*2.4.13  Special maps-to operations   opeliunxp2f 8217
            2.4.14  Function transposition   ctpos 8232
            2.4.15  Curry and uncurry   ccur 8272
            2.4.16  Undefined values   cund 8279
            2.4.17  Well-founded recursion   cfrecs 8287
            2.4.18  Well-ordered recursion   cwrecs 8318
            2.4.19  Functions on ordinals; strictly monotone ordinal functions   iunon 8361
            2.4.20  "Strong" transfinite recursion   crecs 8392
            2.4.21  Recursive definition generator   crdg 8431
            2.4.22  Finite recursion   frfnom 8457
            2.4.23  Ordinal arithmetic   c1o 8481
            2.4.24  Natural number arithmetic   nna0 8626
            2.4.25  Natural addition   cnadd 8687
            2.4.26  Equivalence relations and classes   wer 8723
            2.4.27  The mapping operation   cmap 8847
            2.4.28  Infinite Cartesian products   cixp 8918
            2.4.29  Equinumerosity   cen 8963
            2.4.30  Schroeder-Bernstein Theorem   sbthlem1 9113
            2.4.31  Equinumerosity (cont.)   xpf1o 9169
            2.4.32  Finite sets   dif1enlem 9186
            2.4.33  Pigeonhole Principle   phplem1 9234
            2.4.34  Finite sets (cont.)   onomeneq 9255
            2.4.35  Finitely supported functions   cfsupp 9398
            2.4.36  Finite intersections   cfi 9446
            2.4.37  Hall's marriage theorem   marypha1lem 9469
            2.4.38  Supremum and infimum   csup 9476
            2.4.39  Ordinal isomorphism, Hartogs's theorem   coi 9545
            2.4.40  Hartogs function   char 9592
            2.4.41  Weak dominance   cwdom 9600
      ​2.5  ZF Set Theory - add the Axiom of Regularity
            2.5.1  Introduce the Axiom of Regularity   ax-reg 9628
            2.5.2  Axiom of Infinity equivalents   inf0 9657
      ​2.6  ZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-inf 9674
            2.6.2  Existence of omega (the set of natural numbers)   omex 9679
            2.6.3  Cantor normal form   ccnf 9697
            2.6.4  Transitive closure of a relation   cttrcl 9743
            2.6.5  Transitive closure   trcl 9764
            2.6.6  Well-Founded Induction   frmin 9785
            2.6.7  Well-Founded Recursion   frr3g 9792
            2.6.8  Rank   cr1 9798
            2.6.9  Scott's trick; collection principle; Hilbert's epsilon   scottex 9921
            2.6.10  Disjoint union   cdju 9934
            2.6.11  Cardinal numbers   ccrd 9971
            2.6.12  Axiom of Choice equivalents   wac 10151
            ​*2.6.13  Cardinal number arithmetic   undjudom 10203
            2.6.14  The Ackermann bijection   ackbij2lem1 10253
            2.6.15  Cofinality (without Axiom of Choice)   cflem 10280
            2.6.16  Eight inequivalent definitions of finite set   sornom 10311
            2.6.17  Hereditarily size-limited sets without Choice   itunifval 10450
​​*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 10469
            3.1.2  Introduce the Axiom of Dependent Choice   ax-dc 10480
      ​3.2  ZFC Set Theory - add the Axiom of Choice
            3.2.1  Introduce the Axiom of Choice   ax-ac 10493
            3.2.2  AC equivalents: well-ordering, Zorn's lemma   numthcor 10528
            3.2.3  Cardinal number theorems using Axiom of Choice   cardval 10580
            3.2.4  Cardinal number arithmetic using Axiom of Choice   iunctb 10608
            3.2.5  Cofinality using the Axiom of Choice   alephreg 10616
      ​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 10654
            3.4.2  Derivation of the Axiom of Choice   gchaclem 10712
​​*PART 4  TG (TARSKI-GROTHENDIECK) SET THEORY
      ​4.1  Inaccessibles
            4.1.1  Weakly and strongly inaccessible cardinals   cwina 10716
            4.1.2  Weak universes   cwun 10734
            4.1.3  Tarski classes   ctsk 10782
            4.1.4  Grothendieck universes   cgru 10824
      ​4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
            4.2.1  Introduce the Tarski-Grothendieck Axiom   ax-groth 10857
            4.2.2  Derive the Power Set, Infinity and Choice Axioms   grothpw 10860
            4.2.3  Tarski map function   ctskm 10871
​​*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 10878
            5.1.2  Final derivation of real and complex number postulates   axaddf 11179
            5.1.3  Real and complex number postulates restated as axioms   ax-cnex 11205
      ​5.2  Derive the basic properties from the field axioms
            5.2.1  Some deductions from the field axioms for complex numbers   cnex 11230
            5.2.2  Infinity and the extended real number system   cpnf 11286
            5.2.3  Restate the ordering postulates with extended real "less than"   axlttri 11326
            5.2.4  Ordering on reals   lttr 11331
            5.2.5  Initial properties of the complex numbers   mul12 11420
      ​5.3  Real and complex numbers - basic operations
            5.3.1  Addition   add12 11472
            5.3.2  Subtraction   cmin 11485
            5.3.3  Multiplication   kcnktkm1cn 11686
            5.3.4  Ordering on reals (cont.)   gt0ne0 11720
            5.3.5  Reciprocals   ixi 11884
            5.3.6  Division   cdiv 11912
            5.3.7  Ordering on reals (cont.)   elimgt0 12097
            5.3.8  Completeness Axiom and Suprema   fimaxre 12204
            5.3.9  Imaginary and complex number properties   inelr 12248
            5.3.10  Function operation analogue theorems   ofsubeq0 12255
      ​5.4  Integer sets
            5.4.1  Positive integers (as a subset of complex numbers)   cn 12258
            5.4.2  Principle of mathematical induction   nnind 12276
            ​*5.4.3  Decimal representation of numbers   c2 12313
            ​*5.4.4  Some properties of specific numbers   neg1cn 12372
            5.4.5  Simple number properties   halfcl 12483
            5.4.6  The Archimedean property   nnunb 12514
            5.4.7  Nonnegative integers (as a subset of complex numbers)   cn0 12518
            ​*5.4.8  Extended nonnegative integers   cxnn0 12590
            5.4.9  Integers (as a subset of complex numbers)   cz 12604
            5.4.10  Decimal arithmetic   cdc 12723
            5.4.11  Upper sets of integers   cuz 12868
            5.4.12  Well-ordering principle for bounded-below sets of integers   uzwo3 12973
            5.4.13  Rational numbers (as a subset of complex numbers)   cq 12978
            5.4.14  Existence of the set of complex numbers   rpnnen1lem2 13007
      ​5.5  Order sets
            5.5.1  Positive reals (as a subset of complex numbers)   crp 13022
            5.5.2  Infinity and the extended real number system (cont.)   cxne 13137
            5.5.3  Supremum and infimum on the extended reals   xrsupexmnf 13332
            5.5.4  Real number intervals   cioo 13372
            5.5.5  Finite intervals of integers   cfz 13532
            ​*5.5.6  Finite intervals of nonnegative integers   elfz2nn0 13640
            5.5.7  Half-open integer ranges   cfzo 13675
      ​5.6  Elementary integer functions
            5.6.1  The floor and ceiling functions   cfl 13804
            5.6.2  The modulo (remainder) operation   cmo 13883
            5.6.3  Miscellaneous theorems about integers   om2uz0i 13961
            5.6.4  Strong induction over upper sets of integers   uzsinds 14001
            5.6.5  Finitely supported functions over the nonnegative integers   fsuppmapnn0fiublem 14004
            5.6.6  The infinite sequence builder "seq" - extension   cseq 14015
            5.6.7  Integer powers   cexp 14075
            5.6.8  Ordered pair theorem for nonnegative integers   nn0le2msqi 14279
            5.6.9  Factorial function   cfa 14285
            5.6.10  The binomial coefficient operation   cbc 14314
            5.6.11  The ` # ` (set size) function   chash 14342
                  5.6.11.1  Proper unordered pairs and triples (sets of size 2 and 3)   hashprlei 14482
                  5.6.11.2  Functions with a domain containing at least two different elements   fundmge2nop0 14506
                  5.6.11.3  Finite induction on the size of the first component of a binary relation   hashdifsnp1 14510
      ​​*5.7  Words over a set
            5.7.1  Definitions and basic theorems   cword 14517
            5.7.2  Last symbol of a word   clsw 14565
            5.7.3  Concatenations of words   cconcat 14573
            5.7.4  Singleton words   cs1 14598
            5.7.5  Concatenations with singleton words   ccatws1cl 14619
            5.7.6  Subwords/substrings   csubstr 14643
            5.7.7  Prefixes of a word   cpfx 14673
            5.7.8  Subwords of subwords   swrdswrdlem 14707
            5.7.9  Subwords and concatenations   pfxcctswrd 14713
            5.7.10  Subwords of concatenations   swrdccatfn 14727
            5.7.11  Splicing words (substring replacement)   csplice 14752
            5.7.12  Reversing words   creverse 14761
            5.7.13  Repeated symbol words   creps 14771
            ​*5.7.14  Cyclical shifts of words   ccsh 14791
            5.7.15  Mapping words by a function   wrdco 14835
            5.7.16  Longer string literals   cs2 14845
      ​​*5.8  Reflexive and transitive closures of relations
            5.8.1  The reflexive and transitive properties of relations   coss12d 14972
            5.8.2  Basic properties of closures   cleq1lem 14982
            5.8.3  Definitions and basic properties of transitive closures   ctcl 14985
            5.8.4  Exponentiation of relations   crelexp 15019
            5.8.5  Reflexive-transitive closure as an indexed union   crtrcl 15055
            ​*5.8.6  Principle of transitive induction.   relexpindlem 15063
      ​5.9  Elementary real and complex functions
            5.9.1  The "shift" operation   cshi 15066
            5.9.2  Signum (sgn or sign) function   csgn 15086
            5.9.3  Real and imaginary parts; conjugate   ccj 15096
            5.9.4  Square root; absolute value   csqrt 15233
      ​5.10  Elementary limits and convergence
            5.10.1  Superior limit (lim sup)   clsp 15467
            5.10.2  Limits   cli 15481
            5.10.3  Finite and infinite sums   csu 15685
            5.10.4  The binomial theorem   binomlem 15828
            5.10.5  The inclusion/exclusion principle   incexclem 15835
            5.10.6  Infinite sums (cont.)   isumshft 15838
            5.10.7  Miscellaneous converging and diverging sequences   divrcnv 15851
            5.10.8  Arithmetic series   arisum 15859
            5.10.9  Geometric series   expcnv 15863
            5.10.10  Ratio test for infinite series convergence   cvgrat 15882
            5.10.11  Mertens' theorem   mertenslem1 15883
            5.10.12  Finite and infinite products   prodf 15886
                  5.10.12.1  Product sequences   prodf 15886
                  5.10.12.2  Non-trivial convergence   ntrivcvg 15896
                  5.10.12.3  Complex products   cprod 15902
                  5.10.12.4  Finite products   fprod 15938
                  5.10.12.5  Infinite products   iprodclim 15995
            5.10.13  Falling and Rising Factorial   cfallfac 16001
            5.10.14  Bernoulli polynomials and sums of k-th powers   cbp 16043
      ​5.11  Elementary trigonometry
            5.11.1  The exponential, sine, and cosine functions   ce 16058
                  5.11.1.1  The circle constant (tau = 2 pi)   ctau 16199
            5.11.2  _e is irrational   eirrlem 16201
      ​5.12  Cardinality of real and complex number subsets
            5.12.1  Countability of integers and rationals   xpnnen 16208
            5.12.2  The reals are uncountable   rpnnen2lem1 16211
​​*PART 6  ELEMENTARY NUMBER THEORY
      ​6.1  Elementary properties of divisibility
            6.1.1  Irrationality of square root of 2   sqrt2irrlem 16245
            6.1.2  Some Number sets are chains of proper subsets   nthruc 16249
            6.1.3  The divides relation   cdvds 16251
            ​*6.1.4  Even and odd numbers   evenelz 16333
            6.1.5  The division algorithm   divalglem0 16390
            6.1.6  Bit sequences   cbits 16414
            6.1.7  The greatest common divisor operator   cgcd 16489
            6.1.8  Bézout's identity   bezoutlem1 16535
            6.1.9  Algorithms   nn0seqcvgd 16566
            6.1.10  Euclid's Algorithm   eucalgval2 16577
            ​*6.1.11  The least common multiple   clcm 16584
            ​*6.1.12  Coprimality and Euclid's lemma   coprmgcdb 16645
            6.1.13  Cancellability of congruences   congr 16660
      ​6.2  Elementary prime number theory
            ​*6.2.1  Elementary properties   cprime 16667
            ​*6.2.2  Coprimality and Euclid's lemma (cont.)   coprm 16707
            6.2.3  Properties of the canonical representation of a rational   cnumer 16730
            6.2.4  Euler's theorem   codz 16760
            6.2.5  Arithmetic modulo a prime number   modprm1div 16794
            6.2.6  Pythagorean Triples   coprimeprodsq 16805
            6.2.7  The prime count function   cpc 16833
            6.2.8  Pocklington's theorem   prmpwdvds 16901
            6.2.9  Infinite primes theorem   unbenlem 16905
            6.2.10  Sum of prime reciprocals   prmreclem1 16913
            6.2.11  Fundamental theorem of arithmetic   1arithlem1 16920
            6.2.12  Lagrange's four-square theorem   cgz 16926
            6.2.13  Van der Waerden's theorem   cvdwa 16962
            6.2.14  Ramsey's theorem   cram 16996
            ​*6.2.15  Primorial function   cprmo 17028
            ​*6.2.16  Prime gaps   prmgaplem1 17046
            6.2.17  Decimal arithmetic (cont.)   dec2dvds 17060
            6.2.18  Cyclical shifts of words (cont.)   cshwsidrepsw 17091
            6.2.19  Specific prime numbers   prmlem0 17103
            6.2.20  Very large primes   1259lem1 17128
​PART 7  BASIC STRUCTURES
      ​7.1  Extensible structures
            ​*7.1.1  Basic definitions   cstr 17143
                  7.1.1.1  Extensible structures as structures with components   cstr 17143
                  7.1.1.2  Substitution of components   csts 17160
                  7.1.1.3  Slots   cslot 17178
                  ​*7.1.1.4  Structure component indices   cnx 17190
                  7.1.1.5  Base sets   cbs 17208
                  7.1.1.6  Base set restrictions   cress 17237
            7.1.2  Slot definitions   cplusg 17261
            7.1.3  Definition of the structure product   crest 17430
            7.1.4  Definition of the structure quotient   cordt 17509
      ​7.2  Moore spaces
            7.2.1  Moore closures   mrcflem 17614
            7.2.2  Independent sets in a Moore system   mrisval 17638
            7.2.3  Algebraic closure systems   isacs 17659
​PART 8  BASIC CATEGORY THEORY
      ​8.1  Categories
            8.1.1  Categories   ccat 17672
            8.1.2  Opposite category   coppc 17719
            8.1.3  Monomorphisms and epimorphisms   cmon 17739
            8.1.4  Sections, inverses, isomorphisms   csect 17755
            ​*8.1.5  Isomorphic objects   ccic 17806
            8.1.6  Subcategories   cssc 17818
            8.1.7  Functors   cfunc 17868
            8.1.8  Full & faithful functors   cful 17919
            8.1.9  Natural transformations and the functor category   cnat 17959
            8.1.10  Initial, terminal and zero objects of a category   cinito 17998
      ​8.2  Arrows (disjointified hom-sets)
            8.2.1  Identity and composition for arrows   cida 18070
      ​8.3  Examples of categories
            8.3.1  The category of sets   csetc 18092
            8.3.2  The category of categories   ccatc 18115
            ​*8.3.3  The category of extensible structures   fncnvimaeqv 18138
      ​8.4  Categorical constructions
            8.4.1  Product of categories   cxpc 18187
            8.4.2  Functor evaluation   cevlf 18229
            8.4.3  Hom functor   chof 18268
​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 18451
            9.5.2  Complete lattices   ccla 18518
            9.5.3  Distributive lattices   cdlat 18540
            9.5.4  Subset order structures   cipo 18547
      ​9.6  Posets, directed sets, and lattices as relations
            ​*9.6.1  Posets and lattices as relations   cps 18584
            9.6.2  Directed sets, nets   cdir 18614
​PART 10  BASIC ALGEBRAIC STRUCTURES
      ​10.1  Monoids
            ​*10.1.1  Magmas   cplusf 18625
            ​*10.1.2  Identity elements   mgmidmo 18648
            ​*10.1.3  Iterated sums in a magma   gsumvalx 18664
            10.1.4  Magma homomorphisms and submagmas   cmgmhm 18678
            ​*10.1.5  Semigroups   csgrp 18706
            ​*10.1.6  Definition and basic properties of monoids   cmnd 18722
            10.1.7  Monoid homomorphisms and submonoids   cmhm 18766
            ​*10.1.8  Iterated sums in a monoid   gsumvallem2 18819
            10.1.9  Free monoids   cfrmd 18832
                  ​*10.1.9.1  Monoid of endofunctions   cefmnd 18853
            10.1.10  Examples and counterexamples for magmas, semigroups and monoids   mgm2nsgrplem1 18903
      ​10.2  Groups
            10.2.1  Definition and basic properties   cgrp 18923
            ​*10.2.2  Group multiple operation   cmg 19057
            10.2.3  Subgroups and Quotient groups   csubg 19110
            ​*10.2.4  Cyclic monoids and groups   cycsubmel 19190
            10.2.5  Elementary theory of group homomorphisms   cghm 19202
            10.2.6  Isomorphisms of groups   cgim 19247
                  10.2.6.1  The first isomorphism theorem of groups   ghmqusnsglem1 19270
            10.2.7  Group actions   cga 19279
            10.2.8  Centralizers and centers   ccntz 19305
            10.2.9  The opposite group   coppg 19335
            10.2.10  Symmetric groups   csymg 19360
                  ​*10.2.10.1  Definition and basic properties   csymg 19360
                  10.2.10.2  Cayley's theorem   cayleylem1 19406
                  10.2.10.3  Permutations fixing one element   symgfix2 19410
                  ​*10.2.10.4  Transpositions in the symmetric group   cpmtr 19435
                  10.2.10.5  The sign of a permutation   cpsgn 19483
            10.2.11  p-Groups and Sylow groups; Sylow's theorems   cod 19518
            10.2.12  Direct products   clsm 19628
                  10.2.12.1  Direct products (extension)   smndlsmidm 19650
            10.2.13  Free groups   cefg 19700
            10.2.14  Abelian groups   ccmn 19774
                  10.2.14.1  Definition and basic properties   ccmn 19774
                  10.2.14.2  Cyclic groups   ccyg 19871
                  10.2.14.3  Group sum operation   gsumval3a 19897
                  10.2.14.4  Group sums over (ranges of) integers   fsfnn0gsumfsffz 19977
                  10.2.14.5  Internal direct products   cdprd 19989
                  10.2.14.6  The Fundamental Theorem of Abelian Groups   ablfacrplem 20061
            10.2.15  Simple groups   csimpg 20086
                  10.2.15.1  Definition and basic properties   csimpg 20086
                  10.2.15.2  Classification of abelian simple groups   ablsimpnosubgd 20100
      ​10.3  Rings
            10.3.1  Multiplicative Group   cmgp 20113
            ​*10.3.2  Non-unital rings ("rngs")   crng 20131
            ​*10.3.3  Ring unity (multiplicative identity)   cur 20160
            10.3.4  Semirings   csrg 20165
                  ​*10.3.4.1  The binomial theorem for semirings   srgbinomlem1 20205
            10.3.5  Unital rings   crg 20212
            10.3.6  Opposite ring   coppr 20311
            10.3.7  Divisibility   cdsr 20332
            10.3.8  Ring primes   crpm 20410
            10.3.9  Homomorphisms of non-unital rings   crnghm 20412
            10.3.10  Ring homomorphisms   crh 20447
            10.3.11  Nonzero rings and zero rings   cnzr 20490
            10.3.12  Local rings   clring 20516
            10.3.13  Subrings   csubrng 20523
                  10.3.13.1  Subrings of non-unital rings   csubrng 20523
                  10.3.13.2  Subrings of unital rings   csubrg 20547
            10.3.14  Categories of rings   crngc 20590
                  ​*10.3.14.1  The category of non-unital rings   crngc 20590
                  ​*10.3.14.2  The category of (unital) rings   cringc 20619
                  10.3.14.3  Subcategories of the category of rings   srhmsubclem1 20651
            10.3.15  Left regular elements and domains   crlreg 20665
      ​10.4  Division rings and fields
            10.4.1  Definition and basic properties   cdr 20703
            10.4.2  Sub-division rings   csdrg 20761
            10.4.3  Absolute value (abstract algebra)   cabv 20783
            10.4.4  Star rings   cstf 20812
      ​10.5  Left modules
            10.5.1  Definition and basic properties   clmod 20832
            10.5.2  Subspaces and spans in a left module   clss 20904
            10.5.3  Homomorphisms and isomorphisms of left modules   clmhm 20993
            10.5.4  Subspace sum; bases for a left module   clbs 21048
      ​10.6  Vector spaces
            10.6.1  Definition and basic properties   clvec 21076
      ​10.7  Subring algebras and ideals
            10.7.1  Subring algebras   csra 21145
            ​*10.7.2  Left ideals and spans   clidl 21191
            10.7.3  Two-sided ideals and quotient rings   c2idl 21234
                  ​*10.7.3.1  Condition for a non-unital ring to be unital   rngqiprng1elbas 21271
            10.7.4  Principal ideal rings. Divisibility in the integers   clpidl 21305
            10.7.5  Principal ideal domains   cpid 21321
      ​10.8  The complex numbers as an algebraic extensible structure
            10.8.1  Definition and basic properties   cpsmet 21323
            ​*10.8.2  Ring of integers   czring 21432
                  ​*10.8.2.1  Example for a condition for a non-unital ring to be unital   pzriprnglem1 21467
            10.8.3  Algebraic constructions based on the complex numbers   czrh 21485
            10.8.4  Signs as subgroup of the complex numbers   cnmsgnsubg 21569
            10.8.5  Embedding of permutation signs into a ring   zrhpsgnmhm 21576
            10.8.6  The ordered field of real numbers   crefld 21596
      ​10.9  Generalized pre-Hilbert and Hilbert spaces
            10.9.1  Definition and basic properties   cphl 21616
            10.9.2  Orthocomplements and closed subspaces   cocv 21652
            10.9.3  Orthogonal projection and orthonormal bases   cpj 21694
​​*PART 11  BASIC LINEAR ALGEBRA
      ​11.1  Vectors and free modules
            *11.1.1  Direct sum of left modules   cdsmm 21725
            ​*11.1.2  Free modules   cfrlm 21740
            ​*11.1.3  Standard basis (unit vectors)   cuvc 21776
            ​*11.1.4  Independent sets and families   clindf 21798
            11.1.5  Characterization of free modules   lmimlbs 21830
      ​11.2  Associative algebras
            11.2.1  Definition and basic properties   casa 21844
      ​11.3  Abstract multivariate polynomials
            11.3.1  Definition and basic properties   cmps 21897
            11.3.2  Polynomial evaluation   ces 22081
            11.3.3  Additional definitions for (multivariate) polynomials   cslv 22119
            ​*11.3.4  Univariate polynomials   cps1 22160
            11.3.5  Univariate polynomial evaluation   ces1 22301
                  11.3.5.1  Specialization of polynomial evaluation as a ring homomorphism   evls1scafv 22354
      ​​*11.4  Matrices
            *11.4.1  The matrix multiplication   cmmul 22378
            ​*11.4.2  Square matrices   cmat 22395
            ​*11.4.3  The matrix algebra   matmulr 22428
            ​*11.4.4  Matrices of dimension 0 and 1   mat0dimbas0 22456
            ​*11.4.5  The subalgebras of diagonal and scalar matrices   cdmat 22478
            ​*11.4.6  Multiplication of a matrix with a "column vector"   cmvmul 22530
            11.4.7  Replacement functions for a square matrix   cmarrep 22546
            11.4.8  Submatrices   csubma 22566
      ​11.5  The determinant
            11.5.1  Definition and basic properties   cmdat 22574
            11.5.2  Determinants of 2 x 2 -matrices   m2detleiblem1 22614
            11.5.3  The matrix adjugate/adjunct   cmadu 22622
            ​*11.5.4  Laplace expansion of determinants (special case)   symgmatr01lem 22643
            11.5.5  Inverse matrix   invrvald 22666
            ​*11.5.6  Cramer's rule   slesolvec 22669
      ​​*11.6  Polynomial matrices
            11.6.1  Basic properties   pmatring 22682
            ​*11.6.2  Constant polynomial matrices   ccpmat 22693
            ​*11.6.3  Collecting coefficients of polynomial matrices   cdecpmat 22752
            ​*11.6.4  Ring isomorphism between polynomial matrices and polynomials over matrices   cpm2mp 22782
      ​​*11.7  The characteristic polynomial
            *11.7.1  Definition and basic properties   cchpmat 22816
            ​*11.7.2  The characteristic factor function G   fvmptnn04if 22839
            ​*11.7.3  The Cayley-Hamilton theorem   cpmadurid 22857
​PART 12  BASIC TOPOLOGY
      ​12.1  Topology
            ​*12.1.1  Topological spaces   ctop 22883
                  12.1.1.1  Topologies   ctop 22883
                  12.1.1.2  Topologies on sets   ctopon 22900
                  12.1.1.3  Topological spaces   ctps 22922
            12.1.2  Topological bases   ctb 22936
            12.1.3  Examples of topologies   distop 22986
            12.1.4  Closure and interior   ccld 23008
            12.1.5  Neighborhoods   cnei 23089
            12.1.6  Limit points and perfect sets   clp 23126
            12.1.7  Subspace topologies   restrcl 23149
            12.1.8  Order topology   ordtbaslem 23180
            12.1.9  Limits and continuity in topological spaces   ccn 23216
            12.1.10  Separated spaces: T0, T1, T2 (Hausdorff) ...   ct0 23298
            12.1.11  Compactness   ccmp 23378
            12.1.12  Bolzano-Weierstrass theorem   bwth 23402
            12.1.13  Connectedness   cconn 23403
            12.1.14  First- and second-countability   c1stc 23429
            12.1.15  Local topological properties   clly 23456
            12.1.16  Refinements   cref 23494
            12.1.17  Compactly generated spaces   ckgen 23525
            12.1.18  Product topologies   ctx 23552
            12.1.19  Continuous function-builders   cnmptid 23653
            12.1.20  Quotient maps and quotient topology   ckq 23685
            12.1.21  Homeomorphisms   chmeo 23745
      ​12.2  Filters and filter bases
            12.2.1  Filter bases   elmptrab 23819
            12.2.2  Filters   cfil 23837
            12.2.3  Ultrafilters   cufil 23891
            12.2.4  Filter limits   cfm 23925
            12.2.5  Extension by continuity   ccnext 24051
            12.2.6  Topological groups   ctmd 24062
            12.2.7  Infinite group sum on topological groups   ctsu 24118
            12.2.8  Topological rings, fields, vector spaces   ctrg 24148
      ​12.3  Uniform Structures and Spaces
            12.3.1  Uniform structures   cust 24192
            12.3.2  The topology induced by an uniform structure   cutop 24223
            12.3.3  Uniform Spaces   cuss 24246
            12.3.4  Uniform continuity   cucn 24268
            12.3.5  Cauchy filters in uniform spaces   ccfilu 24279
            12.3.6  Complete uniform spaces   ccusp 24290
      ​12.4  Metric spaces
            12.4.1  Pseudometric spaces   ispsmet 24298
            12.4.2  Basic metric space properties   cxms 24311
            12.4.3  Metric space balls   blfvalps 24377
            12.4.4  Open sets of a metric space   mopnval 24432
            12.4.5  Continuity in metric spaces   metcnp3 24537
            12.4.6  The uniform structure generated by a metric   metuval 24546
            12.4.7  Examples of metric spaces   dscmet 24569
            ​*12.4.8  Normed algebraic structures   cnm 24573
            12.4.9  Normed space homomorphisms (bounded linear operators)   cnmo 24710
            12.4.10  Topology on the reals   qtopbaslem 24763
            12.4.11  Topological definitions using the reals   cii 24883
            12.4.12  Path homotopy   chtpy 24981
            12.4.13  The fundamental group   cpco 25015
      ​12.5  Metric subcomplex vector spaces
            12.5.1  Subcomplex modules   cclm 25077
            ​*12.5.2  Subcomplex vector spaces   ccvs 25138
            ​*12.5.3  Normed subcomplex vector spaces   isncvsngp 25165
            12.5.4  Subcomplex pre-Hilbert spaces   ccph 25182
            12.5.5  Convergence and completeness   ccfil 25268
            12.5.6  Baire's Category Theorem   bcthlem1 25340
            12.5.7  Banach spaces and subcomplex Hilbert spaces   ccms 25348
                  12.5.7.1  The complete ordered field of the real numbers   retopn 25395
            12.5.8  Euclidean spaces   crrx 25399
            12.5.9  Minimizing Vector Theorem   minveclem1 25440
            12.5.10  Projection Theorem   pjthlem1 25453
​PART 13  BASIC REAL AND COMPLEX ANALYSIS
      ​13.1  Continuity
            13.1.1  Intermediate value theorem   pmltpclem1 25465
      ​13.2  Integrals
            13.2.1  Lebesgue measure   covol 25479
            13.2.2  Lebesgue integration   cmbf 25631
                  13.2.2.1  Lesbesgue integral   cmbf 25631
                  13.2.2.2  Lesbesgue directed integral   cdit 25863
      ​13.3  Derivatives
            13.3.1  Real and complex differentiation   climc 25879
                  13.3.1.1  Derivatives of functions of one complex or real variable   climc 25879
                  13.3.1.2  Results on real differentiation   dvferm1lem 26004
​PART 14  BASIC REAL AND COMPLEX FUNCTIONS
      ​14.1  Polynomials
            14.1.1  Polynomial degrees   cmdg 26074
            14.1.2  The division algorithm for univariate polynomials   cmn1 26150
            14.1.3  Elementary properties of complex polynomials   cply 26208
            14.1.4  The division algorithm for polynomials   cquot 26315
            14.1.5  Algebraic numbers   caa 26339
            14.1.6  Liouville's approximation theorem   aalioulem1 26357
      ​14.2  Sequences and series
            14.2.1  Taylor polynomials and Taylor's theorem   ctayl 26377
            14.2.2  Uniform convergence   culm 26402
            14.2.3  Power series   pserval 26436
      ​14.3  Basic trigonometry
            14.3.1  The exponential, sine, and cosine functions (cont.)   efcn 26470
            14.3.2  Properties of pi = 3.14159...   pilem1 26478
            14.3.3  Mapping of the exponential function   efgh 26565
            14.3.4  The natural logarithm on complex numbers   clog 26578
            ​*14.3.5  Logarithms to an arbitrary base   clogb 26789
            14.3.6  Theorems of Pythagoras, isosceles triangles, and intersecting chords   angval 26826
            14.3.7  Solutions of quadratic, cubic, and quartic equations   quad2 26864
            14.3.8  Inverse trigonometric functions   casin 26887
            14.3.9  The Birthday Problem   log2ublem1 26971
            14.3.10  Areas in R^2   carea 26980
            14.3.11  More miscellaneous converging sequences   rlimcnp 26990
            14.3.12  Inequality of arithmetic and geometric means   cvxcl 27010
            14.3.13  Euler-Mascheroni constant   cem 27017
            14.3.14  Zeta function   czeta 27038
            14.3.15  Gamma function   clgam 27041
      ​14.4  Basic number theory
            14.4.1  Wilson's theorem   wilthlem1 27093
            14.4.2  The Fundamental Theorem of Algebra   ftalem1 27098
            14.4.3  The Basel problem (ζ(2) = π²/6)   basellem1 27106
            14.4.4  Number-theoretical functions   ccht 27116
            14.4.5  Perfect Number Theorem   mersenne 27253
            14.4.6  Characters of Z/nZ   cdchr 27258
            14.4.7  Bertrand's postulate   bcctr 27301
            ​*14.4.8  Quadratic residues and the Legendre symbol   clgs 27320
            ​*14.4.9  Gauss' Lemma   gausslemma2dlem0a 27382
            14.4.10  Quadratic reciprocity   lgseisenlem1 27401
            14.4.11  All primes 4n+1 are the sum of two squares   2sqlem1 27443
            14.4.12  Chebyshev's Weak Prime Number Theorem, Dirichlet's Theorem   chebbnd1lem1 27495
            14.4.13  The Prime Number Theorem   mudivsum 27556
            14.4.14  Ostrowski's theorem   abvcxp 27641
​PART 15  SURREAL NUMBERS
      ​​*15.1  Sign sequence representation and Alling's axioms
            15.1.1  Definitions and initial properties   csur 27666
            15.1.2  Ordering   sltsolem1 27702
            15.1.3  Birthday Function   bdayfo 27704
            15.1.4  Density   fvnobday 27705
            ​*15.1.5  Full-Eta Property   bdayimaon 27720
      ​15.2  Initial consequences of Alling's axioms
            15.2.1  Ordering Theorems   csle 27771
            15.2.2  Birthday Theorems   bdayfun 27799
      ​​*15.3  Conway cut representation
            15.3.1  Conway cuts   csslt 27807
            15.3.2  Zero and One   c0s 27849
            15.3.3  Cuts and Options   cmade 27863
            15.3.4  Cofinality and coinitiality   cofsslt 27932
      ​15.4  Induction and recursion
            15.4.1  Induction and recursion on one variable   cnorec 27948
            15.4.2  Induction and recursion on two variables   cnorec2 27959
      ​15.5  Surreal arithmetic
            15.5.1  Addition   cadds 27970
            15.5.2  Negation and Subtraction   cnegs 28026
            15.5.3  Multiplication   cmuls 28104
            15.5.4  Division   cdivs 28185
            15.5.5  Absolute value   cabss 28229
      ​15.6  Subsystems of surreals
            15.6.1  Ordinal numbers   cons 28242
            15.6.2  Surreal recursive sequences   cseqs 28254
            15.6.3  Natural numbers   cnn0s 28283
            15.6.4  Integers   czs 28325
            15.6.5  Real numbers   creno 28341
​​*PART 16  ELEMENTARY GEOMETRY
      ​16.1  Definition and Tarski's Axioms of Geometry
            16.1.1  Justification for the congruence notation   tgjustf 28397
      ​16.2  Tarskian Geometry
            16.2.1  Congruence   tgcgrcomimp 28401
            16.2.2  Betweenness   tgbtwntriv2 28411
            16.2.3  Dimension   tglowdim1 28424
            16.2.4  Betweenness and Congruence   tgifscgr 28432
            16.2.5  Congruence of a series of points   ccgrg 28434
            16.2.6  Motions   cismt 28456
            16.2.7  Colinearity   tglng 28470
            16.2.8  Connectivity of betweenness   tgbtwnconn1lem1 28496
            16.2.9  Less-than relation in geometric congruences   cleg 28506
            16.2.10  Rays   chlg 28524
            16.2.11  Lines   btwnlng1 28543
            16.2.12  Point inversions   cmir 28576
            16.2.13  Right angles   crag 28617
            16.2.14  Half-planes   islnopp 28663
            16.2.15  Midpoints and Line Mirroring   cmid 28696
            16.2.16  Congruence of angles   ccgra 28731
            16.2.17  Angle Comparisons   cinag 28759
            16.2.18  Congruence Theorems   tgsas1 28778
            16.2.19  Equilateral triangles   ceqlg 28789
      ​16.3  Properties of geometries
            16.3.1  Isomorphisms between geometries   f1otrgds 28793
      ​16.4  Geometry in Hilbert spaces
            16.4.1  Geometry in the complex plane   cchhllem 28817
            16.4.2  Geometry in Euclidean spaces   cee 28819
                  16.4.2.1  Definition of the Euclidean space   cee 28819
                  16.4.2.2  Tarski's axioms for geometry for the Euclidean space   axdimuniq 28844
                  16.4.2.3  EE^n fulfills Tarski's Axioms   ceeng 28908
​​*PART 17  GRAPH THEORY
      ​*17.1  Vertices and edges
            17.1.1  The edge function extractor for extensible structures   cedgf 28919
            ​*17.1.2  Vertices and indexed edges   cvtx 28929
                  17.1.2.1  Definitions and basic properties   cvtx 28929
                  17.1.2.2  The vertices and edges of a graph represented as ordered pair   opvtxval 28936
                  17.1.2.3  The vertices and edges of a graph represented as extensible structure   funvtxdmge2val 28944
                  17.1.2.4  Representations of graphs without edges   snstrvtxval 28970
                  17.1.2.5  Degenerated cases of representations of graphs   vtxval0 28972
            17.1.3  Edges as range of the edge function   cedg 28980
      ​​*17.2  Undirected graphs
            17.2.1  Undirected hypergraphs   cuhgr 28989
            17.2.2  Undirected pseudographs and multigraphs   cupgr 29013
            ​*17.2.3  Loop-free graphs   umgrislfupgrlem 29055
            17.2.4  Edges as subsets of vertices of graphs   uhgredgiedgb 29059
            ​*17.2.5  Undirected simple graphs   cuspgr 29081
            17.2.6  Examples for graphs   usgr0e 29169
            17.2.7  Subgraphs   csubgr 29200
            17.2.8  Finite undirected simple graphs   cfusgr 29249
            17.2.9  Neighbors, complete graphs and universal vertices   cnbgr 29265
                  17.2.9.1  Neighbors   cnbgr 29265
                  17.2.9.2  Universal vertices   cuvtx 29318
                  17.2.9.3  Complete graphs   ccplgr 29342
            17.2.10  Vertex degree   cvtxdg 29399
            ​*17.2.11  Regular graphs   crgr 29489
      ​​*17.3  Walks, paths and cycles
            *17.3.1  Walks   cewlks 29529
            17.3.2  Walks for loop-free graphs   lfgrwlkprop 29621
            17.3.3  Trails   ctrls 29624
            17.3.4  Paths and simple paths   cpths 29646
            17.3.5  Closed walks   cclwlks 29704
            17.3.6  Circuits and cycles   ccrcts 29718
            ​*17.3.7  Walks as words   cwwlks 29756
            17.3.8  Walks/paths of length 2 (as length 3 strings)   2wlkdlem1 29856
            17.3.9  Walks in regular graphs   rusgrnumwwlkl1 29899
            ​*17.3.10  Closed walks as words   cclwwlk 29911
                  17.3.10.1  Closed walks as words   cclwwlk 29911
                  17.3.10.2  Closed walks of a fixed length as words   cclwwlkn 29954
                  17.3.10.3  Closed walks on a vertex of a fixed length as words   cclwwlknon 30017
            17.3.11  Examples for walks, trails and paths   0ewlk 30044
            17.3.12  Connected graphs   cconngr 30116
      ​17.4  Eulerian paths and the Konigsberg Bridge problem
            ​*17.4.1  Eulerian paths   ceupth 30127
            ​*17.4.2  The Königsberg Bridge problem   konigsbergvtx 30176
      ​17.5  The Friendship Theorem
            17.5.1  Friendship graphs - basics   cfrgr 30188
            17.5.2  The friendship theorem for small graphs   frgr1v 30201
            17.5.3  Theorems according to Mertzios and Unger   2pthfrgrrn 30212
            ​*17.5.4  Huneke's Proof of the Friendship Theorem   frgrncvvdeqlem1 30229
​PART 18  GUIDES AND MISCELLANEA
      ​18.1  Guides (conventions, explanations, and examples)
            ​*18.1.1  Conventions   conventions 30330
            18.1.2  Natural deduction   natded 30333
            ​*18.1.3  Natural deduction examples   ex-natded5.2 30334
            18.1.4  Definitional examples   ex-or 30351
            18.1.5  Other examples   aevdemo 30390
      ​18.2  Humor
            18.2.1  April Fool's theorem   avril1 30393
      ​18.3  (Future - to be reviewed and classified)
            18.3.1  Planar incidence geometry   cplig 30404
            ​*18.3.2  Aliases kept to prevent broken links   dummylink 30417
​​*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 30419
            19.1.2  Abelian groups   cablo 30474
      ​19.2  Complex vector spaces
            19.2.1  Definition and basic properties   cvc 30488
            19.2.2  Examples of complex vector spaces   cnaddabloOLD 30511
      ​19.3  Normed complex vector spaces
            19.3.1  Definition and basic properties   cnv 30514
            19.3.2  Examples of normed complex vector spaces   cnnv 30607
            19.3.3  Induced metric of a normed complex vector space   imsval 30615
            19.3.4  Inner product   cdip 30630
            19.3.5  Subspaces   css 30651
      ​19.4  Operators on complex vector spaces
            19.4.1  Definitions and basic properties   clno 30670
      ​19.5  Inner product (pre-Hilbert) spaces
            19.5.1  Definition and basic properties   ccphlo 30742
            19.5.2  Examples of pre-Hilbert spaces   cncph 30749
            19.5.3  Properties of pre-Hilbert spaces   isph 30752
      ​19.6  Complex Banach spaces
            19.6.1  Definition and basic properties   ccbn 30792
            19.6.2  Examples of complex Banach spaces   cnbn 30799
            19.6.3  Uniform Boundedness Theorem   ubthlem1 30800
            19.6.4  Minimizing Vector Theorem   minvecolem1 30804
      ​19.7  Complex Hilbert spaces
            19.7.1  Definition and basic properties   chlo 30815
            19.7.2  Standard axioms for a complex Hilbert space   hlex 30828
            19.7.3  Examples of complex Hilbert spaces   cnchl 30846
            19.7.4  Hellinger-Toeplitz Theorem   htthlem 30847
​​*PART 20  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      ​20.1  Axiomatization of complex pre-Hilbert spaces
            20.1.1  Basic Hilbert space definitions   chba 30849
            20.1.2  Preliminary ZFC lemmas   df-hnorm 30898
            ​*20.1.3  Derive the Hilbert space axioms from ZFC set theory   axhilex-zf 30911
            ​*20.1.4  Introduce the vector space axioms for a Hilbert space   ax-hilex 30929
            20.1.5  Vector operations   hvmulex 30941
            20.1.6  Inner product postulates for a Hilbert space   ax-hfi 31009
      ​20.2  Inner product and norms
            20.2.1  Inner product   his5 31016
            20.2.2  Norms   dfhnorm2 31052
            20.2.3  Relate Hilbert space to normed complex vector spaces   hilablo 31090
            20.2.4  Bunjakovaskij-Cauchy-Schwarz inequality   bcsiALT 31109
      ​20.3  Cauchy sequences and completeness axiom
            20.3.1  Cauchy sequences and limits   hcau 31114
            20.3.2  Derivation of the completeness axiom from ZF set theory   hilmet 31124
            20.3.3  Completeness postulate for a Hilbert space   ax-hcompl 31132
            20.3.4  Relate Hilbert space to ZFC pre-Hilbert and Hilbert spaces   hhcms 31133
      ​20.4  Subspaces and projections
            20.4.1  Subspaces   df-sh 31137
            20.4.2  Closed subspaces   df-ch 31151
            20.4.3  Orthocomplements   df-oc 31182
            20.4.4  Subspace sum, span, lattice join, lattice supremum   df-shs 31238
            20.4.5  Projection theorem   pjhthlem1 31321
            20.4.6  Projectors   df-pjh 31325
      ​20.5  Properties of Hilbert subspaces
            20.5.1  Orthomodular law   omlsilem 31332
            20.5.2  Projectors (cont.)   pjhtheu2 31346
            20.5.3  Hilbert lattice operations   sh0le 31370
            20.5.4  Span (cont.) and one-dimensional subspaces   spansn0 31471
            20.5.5  Commutes relation for Hilbert lattice elements   df-cm 31513
            20.5.6  Foulis-Holland theorem   fh1 31548
            20.5.7  Quantum Logic Explorer axioms   qlax1i 31557
            20.5.8  Orthogonal subspaces   chscllem1 31567
            20.5.9  Orthoarguesian laws 5OA and 3OA   5oalem1 31584
            20.5.10  Projectors (cont.)   pjorthi 31599
            20.5.11  Mayet's equation E_3   mayete3i 31658
      ​20.6  Operators on Hilbert spaces
            ​*20.6.1  Operator sum, difference, and scalar multiplication   df-hosum 31660
            20.6.2  Zero and identity operators   df-h0op 31678
            20.6.3  Operations on Hilbert space operators   hoaddcl 31688
            20.6.4  Linear, continuous, bounded, Hermitian, unitary operators and norms   df-nmop 31769
            20.6.5  Linear and continuous functionals and norms   df-nmfn 31775
            20.6.6  Adjoint   df-adjh 31779
            20.6.7  Dirac bra-ket notation   df-bra 31780
            20.6.8  Positive operators   df-leop 31782
            20.6.9  Eigenvectors, eigenvalues, spectrum   df-eigvec 31783
            20.6.10  Theorems about operators and functionals   nmopval 31786
            20.6.11  Riesz lemma   riesz3i 31992
            20.6.12  Adjoints (cont.)   cnlnadjlem1 31997
            20.6.13  Quantum computation error bound theorem   unierri 32034
            20.6.14  Dirac bra-ket notation (cont.)   branmfn 32035
            20.6.15  Positive operators (cont.)   leopg 32052
            20.6.16  Projectors as operators   pjhmopi 32076
      ​20.7  States on a Hilbert lattice and Godowski's equation
            20.7.1  States on a Hilbert lattice   df-st 32141
            20.7.2  Godowski's equation   golem1 32201
      ​20.8  Cover relation, atoms, exchange axiom, and modular symmetry
            20.8.1  Covers relation; modular pairs   df-cv 32209
            20.8.2  Atoms   df-at 32268
            20.8.3  Superposition principle   superpos 32284
            20.8.4  Atoms, exchange and covering properties, atomicity   chcv1 32285
            20.8.5  Irreducibility   chirredlem1 32320
            20.8.6  Atoms (cont.)   atcvat3i 32326
            20.8.7  Modular symmetry   mdsymlem1 32333
​PART 21  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      ​21.1  Mathboxes for user contributions
            21.1.1  Mathbox guidelines   mathbox 32372
      ​21.2  Mathbox for Stefan Allan
      ​21.3  Mathbox for Thierry Arnoux
            21.3.1  Propositional Calculus - misc additions   an42ds 32377
            21.3.2  Predicate Calculus   sbc2iedf 32392
                  21.3.2.1  Predicate Calculus - misc additions   sbc2iedf 32392
                  21.3.2.2  Restricted quantification - misc additions   ralcom4f 32394
                  21.3.2.3  Equality   eqtrb 32399
                  21.3.2.4  Double restricted existential uniqueness quantification   opsbc2ie 32401
                  21.3.2.5  Double restricted existential uniqueness quantification syntax   w2reu 32403
                  21.3.2.6  Substitution (without distinct variables) - misc additions   sbceqbidf 32412
                  21.3.2.7  Existential "at most one" - misc additions   mo5f 32414
                  21.3.2.8  Existential uniqueness - misc additions   reuxfrdf 32416
                  21.3.2.9  Restricted "at most one" - misc additions   rmoxfrd 32418
                  21.3.2.10  Restricted iota (description binder)   riotaeqbidva 32421
            21.3.3  General Set Theory   dmrab 32422
                  21.3.3.1  Class abstractions (a.k.a. class builders)   dmrab 32422
                  21.3.3.2  Image Sets   abrexdomjm 32433
                  21.3.3.3  Set relations and operations - misc additions   elunsn 32439
                  21.3.3.4  Unordered pairs   elpreq 32456
                  21.3.3.5  Conditional operator - misc additions   ifeqeqx 32463
                  21.3.3.6  Set union   uniinn0 32471
                  21.3.3.7  Indexed union - misc additions   cbviunf 32476
                  21.3.3.8  Indexed intersection - misc additions   iinabrex 32489
                  21.3.3.9  Disjointness - misc additions   disjnf 32490
            21.3.4  Relations and Functions   xpdisjres 32518
                  21.3.4.1  Relations - misc additions   xpdisjres 32518
                  21.3.4.2  Functions - misc additions   feq2dd 32540
                  21.3.4.3  Operations - misc additions   mpomptxf 32595
                  21.3.4.4  Support of a function   suppovss 32597
                  21.3.4.5  Explicit Functions with one or two points as a domain   cosnopne 32606
                  21.3.4.6  Isomorphisms - misc. additions   gtiso 32612
                  21.3.4.7  Disjointness (additional proof requiring functions)   disjdsct 32614
                  21.3.4.8  First and second members of an ordered pair - misc additions   df1stres 32615
                  21.3.4.9  Supremum - misc additions   supssd 32623
                  21.3.4.10  Finite Sets   imafi2 32625
                  21.3.4.11  Countable Sets   snct 32627
            21.3.5  Real and Complex Numbers   creq0 32649
                  21.3.5.1  Complex operations - misc. additions   creq0 32649
                  21.3.5.2  Ordering on reals - misc additions   lt2addrd 32658
                  21.3.5.3  Extended reals - misc additions   xrlelttric 32659
                  21.3.5.4  Extended nonnegative integers - misc additions   xnn0gt0 32676
                  21.3.5.5  Real number intervals - misc additions   joiniooico 32679
                  21.3.5.6  Finite intervals of integers - misc additions   uzssico 32689
                  21.3.5.7  Half-open integer ranges - misc additions   iundisjfi 32701
                  21.3.5.8  The ` # ` (set size) function - misc additions   hashunif 32713
                  21.3.5.9  The greatest common divisor operator - misc. additions   znumd 32716
                  21.3.5.10  Integers   nn0split01 32721
                  21.3.5.11  Decimal numbers   dfdec100 32734
            ​*21.3.6  Decimal expansion   cdp2 32735
                  ​*21.3.6.1  Decimal point   cdp 32752
                  21.3.6.2  Division in the extended real number system   cxdiv 32781
            21.3.7  Words over a set - misc additions   wrdfd 32800
                  21.3.7.1  Splicing words (substring replacement)   splfv3 32825
                  21.3.7.2  Cyclic shift of words   1cshid 32826
            21.3.8  Extensible Structures   ressplusf 32830
                  21.3.8.1  Structure restriction operator   ressplusf 32830
                  21.3.8.2  The opposite group   oppgle 32833
                  21.3.8.3  Posets   ressprs 32836
                  21.3.8.4  Complete lattices   clatp0cl 32849
                  21.3.8.5  Order Theory   cmnt 32851
                  21.3.8.6  Chains   cchn 32877
                  21.3.8.7  Extended reals Structure - misc additions   ax-xrssca 32887
                  21.3.8.8  The extended nonnegative real numbers commutative monoid   xrge0base 32897
            21.3.9  Algebra   cmn4d 32908
                  21.3.9.1  Monoids   cmn4d 32908
                  21.3.9.2  Monoids Homomorphisms   abliso 32912
                  21.3.9.3  Groups - misc additions   subgcld 32916
                  21.3.9.4  Finitely supported group sums - misc additions   gsumsubg 32918
                  21.3.9.5  Centralizers and centers - misc additions   cntzun 32933
                  21.3.9.6  Totally ordered monoids and groups   comnd 32936
                  21.3.9.7  The symmetric group   symgfcoeu 32964
                  21.3.9.8  Transpositions   pmtridf1o 32976
                  21.3.9.9  Permutation Signs   psgnid 32979
                  21.3.9.10  Permutation cycles   ctocyc 32988
                  21.3.9.11  The Alternating Group   evpmval 33027
                  21.3.9.12  Signum in an ordered monoid   csgns 33040
                  21.3.9.13  The Archimedean property for generic ordered algebraic structures   cinftm 33045
                  21.3.9.14  Semiring left modules   cslmd 33068
                  21.3.9.15  Simple groups   prmsimpcyc 33096
                  21.3.9.16  Rings - misc additions   cringmul32d 33097
                  21.3.9.17  The zero ring   irrednzr 33110
                  21.3.9.18  Localization of rings   cerl 33113
                  21.3.9.19  Integral Domains   domnmuln0rd 33134
                  21.3.9.20  Euclidean Domains   ceuf 33145
                  21.3.9.21  Division Rings   ringinveu 33151
                  21.3.9.22  Subfields   sdrgdvcl 33154
                  21.3.9.23  Field of fractions   cfrac 33157
                  21.3.9.24  Field extensions generated by a set   cfldgen 33165
                  21.3.9.25  Totally ordered rings and fields   corng 33178
                  21.3.9.26  Ring homomorphisms - misc additions   rhmdvd 33201
                  21.3.9.27  Scalar restriction operation   cresv 33203
                  21.3.9.28  The commutative ring of gaussian integers   gzcrng 33223
                  21.3.9.29  The archimedean ordered field of real numbers   cnfldfld 33224
                  21.3.9.30  The quotient map and quotient modules   qusker 33230
                  21.3.9.31  The ring of integers modulo ` N `   znfermltl 33247
                  21.3.9.32  Independent sets and families   islinds5 33248
                  21.3.9.33  Ring associates, ring units   dvdsruassoi 33265
                  ​*21.3.9.34  Subgroup sum / Sumset / Minkowski sum   elgrplsmsn 33271
                  21.3.9.35  The quotient map   quslsm 33286
                  21.3.9.36  Ideals   lidlmcld 33300
                  21.3.9.37  Prime Ideals   cprmidl 33316
                  21.3.9.38  Maximal Ideals   cmxidl 33340
                  21.3.9.39  The semiring of ideals of a ring   cidlsrg 33381
                  21.3.9.40  Prime Elements   rprmval 33397
                  21.3.9.41  Unique factorization domains   cufd 33419
                  21.3.9.42  The ring of integers   zringidom 33432
                  21.3.9.43  Univariate Polynomials   0ringmon1p 33436
                  21.3.9.44  Polynomial quotient and polynomial remainder   q1pdir 33476
                  21.3.9.45  The subring algebra   sra1r 33485
                  21.3.9.46  Division Ring Extensions   drgext0g 33492
                  21.3.9.47  Vector Spaces   lvecdimfi 33498
                  21.3.9.48  Vector Space Dimension   cldim 33499
            21.3.10  Field Extensions   cfldext 33533
                  21.3.10.1  Algebraic numbers   cirng 33565
                  21.3.10.2  Minimal polynomials   cminply 33574
                  21.3.10.3  Quadratic Field Extensions   rtelextdg2lem 33599
                  21.3.10.4  Towers of quadratic extentions   fldext2chn 33601
            ​*21.3.11  Constructible Numbers   cconstr 33602
                  21.3.11.1  Impossible constructions   2sqr3minply 33620
            21.3.12  Matrices   csmat 33621
                  21.3.12.1  Submatrices   csmat 33621
                  21.3.12.2  Matrix literals   clmat 33639
                  21.3.12.3  Laplace expansion of determinants   mdetpmtr1 33651
            21.3.13  Topology   ist0cld 33661
                  21.3.13.1  Open maps   txomap 33662
                  21.3.13.2  Topology of the unit circle   qtopt1 33663
                  21.3.13.3  Refinements   reff 33667
                  21.3.13.4  Open cover refinement property   ccref 33670
                  21.3.13.5  Lindelöf spaces   cldlf 33680
                  21.3.13.6  Paracompact spaces   cpcmp 33683
                  ​*21.3.13.7  Spectrum of a ring   crspec 33690
                  21.3.13.8  Pseudometrics   cmetid 33714
                  21.3.13.9  Continuity - misc additions   hauseqcn 33726
                  21.3.13.10  Topology of the closed unit interval   elunitge0 33727
                  21.3.13.11  Topology of ` ( RR X. RR ) `   unicls 33731
                  21.3.13.12  Order topology - misc. additions   cnvordtrestixx 33741
                  21.3.13.13  Continuity in topological spaces - misc. additions   mndpluscn 33754
                  21.3.13.14  Topology of the extended nonnegative real numbers ordered monoid   xrge0hmph 33760
                  21.3.13.15  Limits - misc additions   lmlim 33775
                  21.3.13.16  Univariate polynomials   pl1cn 33783
            21.3.14  Uniform Stuctures and Spaces   chcmp 33784
                  21.3.14.1  Hausdorff uniform completion   chcmp 33784
            21.3.15  Topology and algebraic structures   zringnm 33786
                  21.3.15.1  The norm on the ring of the integer numbers   zringnm 33786
                  21.3.15.2  Topological ` ZZ ` -modules   zlm0 33788
                  21.3.15.3  Canonical embedding of the field of the rational numbers into a division ring   cqqh 33800
                  21.3.15.4  Canonical embedding of the real numbers into a complete ordered field   crrh 33821
                  21.3.15.5  Embedding from the extended real numbers into a complete lattice   cxrh 33844
                  21.3.15.6  Canonical embeddings into the ordered field of the real numbers   zrhre 33847
                  ​*21.3.15.7  Topological Manifolds   cmntop 33850
            21.3.16  Real and complex functions   nexple 33855
                  21.3.16.1  Integer powers - misc. additions   nexple 33855
                  21.3.16.2  Indicator Functions   cind 33856
                  21.3.16.3  Extended sum   cesum 33873
            21.3.17  Mixed Function/Constant operation   cofc 33941
            21.3.18  Abstract measure   csiga 33954
                  21.3.18.1  Sigma-Algebra   csiga 33954
                  21.3.18.2  Generated sigma-Algebra   csigagen 33984
                  ​*21.3.18.3  lambda and pi-Systems, Rings of Sets   ispisys 33998
                  21.3.18.4  The Borel algebra on the real numbers   cbrsiga 34027
                  21.3.18.5  Product Sigma-Algebra   csx 34034
                  21.3.18.6  Measures   cmeas 34041
                  21.3.18.7  The counting measure   cntmeas 34072
                  21.3.18.8  The Lebesgue measure - misc additions   voliune 34075
                  21.3.18.9  The Dirac delta measure   cdde 34078
                  21.3.18.10  The 'almost everywhere' relation   cae 34083
                  21.3.18.11  Measurable functions   cmbfm 34095
                  21.3.18.12  Borel Algebra on ` ( RR X. RR ) `   br2base 34116
                  ​*21.3.18.13  Caratheodory's extension theorem   coms 34138
            21.3.19  Integration   itgeq12dv 34173
                  21.3.19.1  Lebesgue integral - misc additions   itgeq12dv 34173
                  21.3.19.2  Bochner integral   citgm 34174
            21.3.20  Euler's partition theorem   oddpwdc 34201
            21.3.21  Sequences defined by strong recursion   csseq 34230
            21.3.22  Fibonacci Numbers   cfib 34243
            21.3.23  Probability   cprb 34254
                  21.3.23.1  Probability Theory   cprb 34254
                  21.3.23.2  Conditional Probabilities   ccprob 34278
                  21.3.23.3  Real-valued Random Variables   crrv 34287
                  21.3.23.4  Preimage set mapping operator   corvc 34302
                  21.3.23.5  Distribution Functions   orvcelval 34315
                  21.3.23.6  Cumulative Distribution Functions   orvclteel 34319
                  21.3.23.7  Probabilities - example   coinfliplem 34325
                  21.3.23.8  Bertrand's Ballot Problem   ballotlemoex 34332
            21.3.24  Signum (sgn or sign) function - misc. additions   sgncl 34385
                  21.3.24.1  Operations on words   ccatmulgnn0dir 34401
            21.3.25  Polynomials with real coefficients - misc additions   plymul02 34405
            21.3.26  Descartes's rule of signs   signspval 34411
                  21.3.26.1  Sign changes in a word over real numbers   signspval 34411
                  21.3.26.2  Counting sign changes in a word over real numbers   signslema 34421
            21.3.27  Number Theory   iblidicc 34451
                  21.3.27.1  Representations of a number as sums of integers   crepr 34467
                  21.3.27.2  Vinogradov Trigonometric Sums and the Circle Method   cvts 34494
                  21.3.27.3  The Ternary Goldbach Conjecture: Final Statement   ax-hgt749 34503
            21.3.28  Elementary Geometry   cstrkg2d 34523
                  ​*21.3.28.1  Two-dimensional geometry   cstrkg2d 34523
                  21.3.28.2  Outer Five Segment (not used, no need to move to main)   cafs 34528
            ​*21.3.29  LeftPad Project   clpad 34533
      ​​*21.4  Mathbox for Jonathan Ben-Naim
            21.4.1  First-order logic and set theory   bnj170 34556
            21.4.2  Well founded induction and recursion   bnj110 34716
            21.4.3  The existence of a minimal element in certain classes   bnj69 34868
            21.4.4  Well-founded induction   bnj1204 34870
            21.4.5  Well-founded recursion, part 1 of 3   bnj60 34920
            21.4.6  Well-founded recursion, part 2 of 3   bnj1500 34926
            21.4.7  Well-founded recursion, part 3 of 3   bnj1522 34930
      ​21.5  Mathbox for BTernaryTau
            21.5.1  ZF set theory   exdifsn 34931
                  21.5.1.1  Finitism   prcinf 34943
                  21.5.1.2  Global choice   gblacfnacd 34948
            21.5.2  Real and complex numbers   zltp1ne 34950
            21.5.3  Graph theory   lfuhgr 34958
                  21.5.3.1  Acyclic graphs   cacycgr 34983
      ​21.6  Mathbox for Mario Carneiro
            21.6.1  Predicate calculus with all distinct variables   ax-7d 35000
            21.6.2  Miscellaneous stuff   quartfull 35006
            21.6.3  Derangements and the Subfactorial   deranglem 35007
            21.6.4  The Erdős-Szekeres theorem   erdszelem1 35032
            21.6.5  The Kuratowski closure-complement theorem   kur14lem1 35047
            21.6.6  Retracts and sections   cretr 35058
            21.6.7  Path-connected and simply connected spaces   cpconn 35060
            21.6.8  Covering maps   ccvm 35096
            21.6.9  Normal numbers   snmlff 35170
            21.6.10  Godel-sets of formulas - part 1   cgoe 35174
            21.6.11  Godel-sets of formulas - part 2   cgon 35273
            21.6.12  Models of ZF   cgze 35287
            ​*21.6.13  Metamath formal systems   cmcn 35301
            21.6.14  Grammatical formal systems   cm0s 35426
            21.6.15  Models of formal systems   cmuv 35446
            21.6.16  Splitting fields   ccpms 35468
            21.6.17  p-adic number fields   czr 35488
      ​​*21.7  Mathbox for Filip Cernatescu
      ​21.8  Mathbox for Paul Chapman
            21.8.1  Real and complex numbers (cont.)   climuzcnv 35512
            21.8.2  Miscellaneous theorems   elfzm12 35516
      ​21.9  Mathbox for Hongxiu Chen
      ​21.10  Mathbox for Adrian Ducourtial
            21.10.1  Propositional calculus   currybi 35529
            21.10.2  Clone theory   ccloneop 35530
      ​21.11  Mathbox for Scott Fenton
            21.11.1  ZFC Axioms in primitive form   axextprim 35536
            21.11.2  Untangled classes   untelirr 35543
            21.11.3  Extra propositional calculus theorems   3jaodd 35550
            21.11.4  Misc. Useful Theorems   nepss 35553
            21.11.5  Properties of real and complex numbers   sqdivzi 35563
            21.11.6  Infinite products   iprodefisumlem 35575
            21.11.7  Factorial limits   faclimlem1 35578
            21.11.8  Greatest common divisor and divisibility   gcd32 35584
            21.11.9  Properties of relationships   dftr6 35586
            21.11.10  Properties of functions and mappings   funpsstri 35602
            21.11.11  Set induction (or epsilon induction)   setinds 35615
            21.11.12  Ordinal numbers   elpotr 35618
            21.11.13  Defined equality axioms   axextdfeq 35634
            21.11.14  Hypothesis builders   hbntg 35642
            21.11.15  Well-founded zero, successor, and limits   cwsuc 35647
            21.11.16  Quantifier-free definitions   ctxp 35667
            21.11.17  Alternate ordered pairs   caltop 35793
            21.11.18  Geometry in the Euclidean space   cofs 35819
                  21.11.18.1  Congruence properties   cofs 35819
                  21.11.18.2  Betweenness properties   btwntriv2 35849
                  21.11.18.3  Segment Transportation   ctransport 35866
                  21.11.18.4  Properties relating betweenness and congruence   cifs 35872
                  21.11.18.5  Connectivity of betweenness   btwnconn1lem1 35924
                  21.11.18.6  Segment less than or equal to   csegle 35943
                  21.11.18.7  Outside-of relationship   coutsideof 35956
                  21.11.18.8  Lines and Rays   cline2 35971
            21.11.19  Forward difference   cfwddif 35995
            21.11.20  Rank theorems   rankung 36003
            21.11.21  Hereditarily Finite Sets   chf 36009
      ​21.12  Mathbox for Gino Giotto
            21.12.1  Study of ax-mulf usage.   mpomulnzcnf 36024
      ​21.13  Mathbox for Jeff Hankins
            21.13.1  Miscellany   a1i14 36025
            21.13.2  Basic topological facts   topbnd 36049
            21.13.3  Topology of the real numbers   ivthALT 36060
            21.13.4  Refinements   cfne 36061
            21.13.5  Neighborhood bases determine topologies   neibastop1 36084
            21.13.6  Lattice structure of topologies   topmtcl 36088
            21.13.7  Filter bases   fgmin 36095
            21.13.8  Directed sets, nets   tailfval 36097
      ​21.14  Mathbox for Anthony Hart
            21.14.1  Propositional Calculus   tb-ax1 36108
            21.14.2  Predicate Calculus   nalfal 36128
            21.14.3  Miscellaneous single axioms   meran1 36136
            21.14.4  Connective Symmetry   negsym1 36142
      ​21.15  Mathbox for Chen-Pang He
            21.15.1  Ordinal topology   ontopbas 36153
      ​21.16  Mathbox for Jeff Hoffman
            21.16.1  Inferences for finite induction on generic function values   fveleq 36176
            21.16.2  gdc.mm   nnssi2 36180
      ​21.17  Mathbox for Asger C. Ipsen
            21.17.1  Continuous nowhere differentiable functions   dnival 36187
      ​​*21.18  Mathbox for BJ
            *21.18.1  Propositional calculus   bj-mp2c 36256
                  *21.18.1.1  Derived rules of inference   bj-mp2c 36256
                  ​*21.18.1.2  A syntactic theorem   bj-0 36258
                  21.18.1.3  Minimal implicational calculus   bj-a1k 36260
                  ​*21.18.1.4  Positive calculus   bj-syl66ib 36271
                  21.18.1.5  Implication and negation   bj-con2com 36277
                  ​*21.18.1.6  Disjunction   bj-jaoi1 36288
                  ​*21.18.1.7  Logical equivalence   bj-dfbi4 36290
                  21.18.1.8  The conditional operator for propositions   bj-consensus 36295
                  ​*21.18.1.9  Propositional calculus: miscellaneous   bj-imbi12 36300
            ​*21.18.2  Modal logic   bj-axdd2 36310
            ​*21.18.3  Provability logic   cprvb 36315
            ​*21.18.4  First-order logic   bj-genr 36324
                  21.18.4.1  Adding ax-gen   bj-genr 36324
                  21.18.4.2  Adding ax-4   bj-2alim 36328
                  21.18.4.3  Adding ax-5   bj-ax12wlem 36361
                  21.18.4.4  Equality and substitution   bj-ssbeq 36370
                  21.18.4.5  Adding ax-6   bj-spimvwt 36386
                  21.18.4.6  Adding ax-7   bj-cbvexw 36393
                  21.18.4.7  Membership predicate, ax-8 and ax-9   bj-ax89 36395
                  21.18.4.8  Adding ax-11   bj-alcomexcom 36398
                  21.18.4.9  Adding ax-12   axc11n11 36400
                  21.18.4.10  Nonfreeness   wnnf 36441
                  21.18.4.11  Adding ax-13   bj-axc10 36501
                  ​*21.18.4.12  Removing dependencies on ax-13 (and ax-11)   bj-axc10v 36511
                  ​*21.18.4.13  Distinct var metavariables   bj-hbaeb2 36536
                  ​*21.18.4.14  Around ~ equsal   bj-equsal1t 36540
                  ​*21.18.4.15  Some Principia Mathematica proofs   stdpc5t 36545
                  21.18.4.16  Alternate definition of substitution   bj-sbsb 36555
                  21.18.4.17  Lemmas for substitution   bj-sbf3 36557
                  21.18.4.18  Existential uniqueness   bj-eu3f 36559
                  ​*21.18.4.19  First-order logic: miscellaneous   bj-sblem1 36560
            21.18.5  Set theory   eliminable1 36577
                  ​*21.18.5.1  Eliminability of class terms   eliminable1 36577
                  ​*21.18.5.2  Classes without the axiom of extensionality   bj-denoteslem 36589
                  21.18.5.3  Characterization among sets versus among classes   elelb 36616
                  ​*21.18.5.4  The nonfreeness quantifier for classes   bj-nfcsym 36618
                  ​*21.18.5.5  Lemmas for class substitution   bj-sbeqALT 36619
                  21.18.5.6  Removing some axiom requirements and disjoint variable conditions   bj-exlimvmpi 36630
                  ​*21.18.5.7  Class abstractions   bj-elabd2ALT 36644
                  21.18.5.8  Generalized class abstractions   bj-cgab 36652
                  ​*21.18.5.9  Restricted nonfreeness   wrnf 36660
                  ​*21.18.5.10  Russell's paradox   bj-ru0 36662
                  21.18.5.11  Curry's paradox in set theory   currysetlem 36665
                  ​*21.18.5.12  Some disjointness results   bj-n0i 36671
                  ​*21.18.5.13  Complements on direct products   bj-xpimasn 36675
                  ​*21.18.5.14  "Singletonization" and tagging   bj-snsetex 36683
                  ​*21.18.5.15  Tuples of classes   bj-cproj 36710
                  ​*21.18.5.16  Set theory: elementary operations relative to a universe   bj-rcleqf 36745
                  ​*21.18.5.17  Axioms for finite unions   bj-abex 36750
                  ​*21.18.5.18  Set theory: miscellaneous   eleq2w2ALT 36767
                  ​*21.18.5.19  Evaluation at a class   bj-evaleq 36792
                  21.18.5.20  Elementwise operations   celwise 36799
                  ​*21.18.5.21  Elementwise intersection (families of sets induced on a subset)   bj-rest00 36801
                  21.18.5.22  Moore collections (complements)   bj-raldifsn 36820
                  21.18.5.23  Maps-to notation for functions with three arguments   bj-0nelmpt 36836
                  ​*21.18.5.24  Currying   csethom 36842
                  ​*21.18.5.25  Setting components of extensible structures   cstrset 36854
            ​*21.18.6  Extended real and complex numbers, real and complex projective lines   bj-nfald 36857
                  21.18.6.1  Complements on class abstractions of ordered pairs and binary relations   bj-nfald 36857
                  ​*21.18.6.2  Identity relation (complements)   bj-opabssvv 36870
                  ​*21.18.6.3  Functionalized identity (diagonal in a Cartesian square)   cdiag2 36892
                  ​*21.18.6.4  Direct image and inverse image   cimdir 36898
                  ​*21.18.6.5  Extended numbers and projective lines as sets   cfractemp 36916
                  ​*21.18.6.6  Addition and opposite   caddcc 36957
                  ​*21.18.6.7  Order relation on the extended reals   cltxr 36961
                  ​*21.18.6.8  Argument, multiplication and inverse   carg 36963
                  21.18.6.9  The canonical bijection from the finite ordinals   ciomnn 36969
                  21.18.6.10  Divisibility   cnnbar 36980
            ​*21.18.7  Monoids   bj-smgrpssmgm 36988
                  ​*21.18.7.1  Finite sums in monoids   cfinsum 37003
            ​*21.18.8  Affine, Euclidean, and Cartesian geometry   bj-fvimacnv0 37006
                  *21.18.8.1  Real vector spaces   bj-fvimacnv0 37006
                  ​*21.18.8.2  Complex numbers (supplements)   bj-subcom 37028
                  ​*21.18.8.3  Barycentric coordinates   bj-bary1lem 37030
            21.18.9  Monoid of endomorphisms   cend 37033
      ​21.19  Mathbox for Jim Kingdon
                  21.19.0.1  Circle constant   taupilem3 37039
                  21.19.0.2  Number theory   dfgcd3 37044
                  21.19.0.3  Real numbers   irrdifflemf 37045
      ​21.20  Mathbox for ML
            21.20.1  Miscellaneous   csbrecsg 37048
            21.20.2  Cartesian exponentiation   cfinxp 37103
            21.20.3  Topology   iunctb2 37123
                  ​*21.20.3.1  Pi-base theorems   pibp16 37133
      ​21.21  Mathbox for Wolf Lammen
            21.21.1  1. Bootstrapping   wl-section-boot 37142
            21.21.2  Implication chains   wl-section-impchain 37166
            21.21.3  Theorems around the conditional operator   wl-ifp-ncond1 37184
            21.21.4  Alternative development of hadd, cadd   wl-df-3xor 37188
            21.21.5  An alternative axiom ~ ax-13   ax-wl-13v 37213
            21.21.6  Other stuff   wl-mps 37215
      ​21.22  Mathbox for Brendan Leahy
      ​21.23  Mathbox for Jeff Madsen
            21.23.1  Logic and set theory   unirep 37428
            21.23.2  Real and complex numbers; integers   filbcmb 37454
            21.23.3  Sequences and sums   sdclem2 37456
            21.23.4  Topology   subspopn 37466
            21.23.5  Metric spaces   metf1o 37469
            21.23.6  Continuous maps and homeomorphisms   constcncf 37476
            21.23.7  Boundedness   ctotbnd 37480
            21.23.8  Isometries   cismty 37512
            21.23.9  Heine-Borel Theorem   heibor1lem 37523
            21.23.10  Banach Fixed Point Theorem   bfplem1 37536
            21.23.11  Euclidean space   crrn 37539
            21.23.12  Intervals (continued)   ismrer1 37552
            21.23.13  Operation properties   cass 37556
            21.23.14  Groups and related structures   cmagm 37562
            21.23.15  Group homomorphism and isomorphism   cghomOLD 37597
            21.23.16  Rings   crngo 37608
            21.23.17  Division Rings   cdrng 37662
            21.23.18  Ring homomorphisms   crngohom 37674
            21.23.19  Commutative rings   ccm2 37703
            21.23.20  Ideals   cidl 37721
            21.23.21  Prime rings and integral domains   cprrng 37760
            21.23.22  Ideal generators   cigen 37773
      ​21.24  Mathbox for Giovanni Mascellani
            ​*21.24.1  Tools for automatic proof building   efald2 37792
            ​*21.24.2  Tseitin axioms   fald 37843
            ​*21.24.3  Equality deductions   iuneq2f 37870
            ​*21.24.4  Miscellanea   orcomdd 37881
      ​21.25  Mathbox for Peter Mazsa
            21.25.1  Notations   cxrn 37888
            21.25.2  Preparatory theorems   el2v1 37931
            21.25.3  Range Cartesian product   df-xrn 38082
            21.25.4  Cosets by ` R `   df-coss 38122
            21.25.5  Relations   df-rels 38196
            21.25.6  Subset relations   df-ssr 38209
            21.25.7  Reflexivity   df-refs 38221
            21.25.8  Converse reflexivity   df-cnvrefs 38236
            21.25.9  Symmetry   df-syms 38253
            21.25.10  Reflexivity and symmetry   symrefref2 38274
            21.25.11  Transitivity   df-trs 38283
            21.25.12  Equivalence relations   df-eqvrels 38295
            21.25.13  Redundancy   df-redunds 38334
            21.25.14  Domain quotients   df-dmqss 38349
            21.25.15  Equivalence relations on domain quotients   df-ers 38374
            21.25.16  Functions   df-funss 38391
            21.25.17  Disjoints vs. converse functions   df-disjss 38414
            21.25.18  Antisymmetry   df-antisymrel 38471
            21.25.19  Partitions: disjoints on domain quotients   df-parts 38476
            21.25.20  Partition-Equivalence Theorems   disjim 38492
      ​21.26  Mathbox for Rodolfo Medina
            21.26.1  Partitions   prtlem60 38564
      ​​*21.27  Mathbox for Norm Megill
            *21.27.1  Obsolete schemes ax-c4,c5,c7,c10,c11,c11n,c15,c9,c14,c16   ax-c5 38594
            ​*21.27.2  Rederive new axioms ax-4, ax-10, ax-6, ax-12, ax-13 from old   axc5 38604
            ​*21.27.3  Legacy theorems using obsolete axioms   ax5ALT 38618
            21.27.4  Experiments with weak deduction theorem   elimhyps 38672
            21.27.5  Miscellanea   cnaddcom 38683
            21.27.6  Atoms, hyperplanes, and covering in a left vector space (or module)   clsa 38685
            21.27.7  Functionals and kernels of a left vector space (or module)   clfn 38768
            21.27.8  Opposite rings and dual vector spaces   cld 38834
            21.27.9  Ortholattices and orthomodular lattices   cops 38883
            21.27.10  Atomic lattices with covering property   ccvr 38973
            21.27.11  Hilbert lattices   chlt 39061
            21.27.12  Projective geometries based on Hilbert lattices   clln 39203
            21.27.13  Construction of a vector space from a Hilbert lattice   cdlema1N 39503
            21.27.14  Construction of involution and inner product from a Hilbert lattice   clpoN 41192
      ​21.28  Mathbox for metakunt
            21.28.1  Commutative Semiring   ccsrg 41678
            21.28.2  General helpful statements   rhmzrhval 41681
            21.28.3  Some gcd and lcm results   12gcd5e1 41715
            21.28.4  Least common multiple inequality theorem   3factsumint1 41733
            21.28.5  Logarithm inequalities   3exp7 41765
            21.28.6  Miscellaneous results for AKS formalisation   intlewftc 41773
            21.28.7  Sticks and stones   sticksstones1 41858
            21.28.8  Continuation AKS   aks6d1c6lem1 41882
            21.28.9  Permutation results   metakunt1 41913
            21.28.10  Unused lemmas scheduled for deletion   fac2xp3 41947
      ​21.29  Mathbox for Steven Nguyen
            21.29.1  Utility theorems   intnanrt 41951
            ​*21.29.2  Arithmetic theorems   c0exALT 41998
            21.29.3  Exponents and divisibility   oexpreposd 42048
            21.29.4  Real subtraction   cresub 42076
            21.29.5  Structures   nelsubginvcld 42186
            ​*21.29.6  Projective spaces   cprjsp 42291
            21.29.7  Basic reductions for Fermat's Last Theorem   dffltz 42324
            ​*21.29.8  Exemplar theorems   iddii 42354
                  ​*21.29.8.1  Standard replacements of ax-10 , ax-11 , ax-12   nfa1w 42368
      ​21.30  Mathbox for Igor Ieskov
      ​21.31  Mathbox for OpenAI
      ​21.32  Mathbox for Stefan O'Rear
            21.32.1  Additional elementary logic and set theory   moxfr 42386
            21.32.2  Additional theory of functions   imaiinfv 42387
            21.32.3  Additional topology   elrfi 42388
            21.32.4  Characterization of closure operators. Kuratowski closure axioms   ismrcd1 42392
            21.32.5  Algebraic closure systems   cnacs 42396
            21.32.6  Miscellanea 1. Map utilities   constmap 42407
            21.32.7  Miscellanea for polynomials   mptfcl 42414
            21.32.8  Multivariate polynomials over the integers   cmzpcl 42415
            21.32.9  Miscellanea for Diophantine sets 1   coeq0i 42447
            21.32.10  Diophantine sets 1: definitions   cdioph 42449
            21.32.11  Diophantine sets 2 miscellanea   ellz1 42461
            21.32.12  Diophantine sets 2: union and intersection. Monotone Boolean algebra   diophin 42466
            21.32.13  Diophantine sets 3: construction   diophrex 42469
            21.32.14  Diophantine sets 4 miscellanea   2sbcrex 42478
            21.32.15  Diophantine sets 4: Quantification   rexrabdioph 42488
            21.32.16  Diophantine sets 5: Arithmetic sets   rabdiophlem1 42495
            21.32.17  Diophantine sets 6: reusability. renumbering of variables   eldioph4b 42505
            21.32.18  Pigeonhole Principle and cardinality helpers   fphpd 42510
            21.32.19  A non-closed set of reals is infinite   rencldnfilem 42514
            21.32.20  Lagrange's rational approximation theorem   irrapxlem1 42516
            21.32.21  Pell equations 1: A nontrivial solution always exists   pellexlem1 42523
            21.32.22  Pell equations 2: Algebraic number theory of the solution set   csquarenn 42530
            21.32.23  Pell equations 3: characterizing fundamental solution   infmrgelbi 42572
            ​*21.32.24  Logarithm laws generalized to an arbitrary base   reglogcl 42584
            21.32.25  Pell equations 4: the positive solution group is infinite cyclic   pellfund14 42592
            21.32.26  X and Y sequences 1: Definition and recurrence laws   crmx 42594
            21.32.27  Ordering and induction lemmas for the integers   monotuz 42636
            21.32.28  X and Y sequences 2: Order properties   rmxypos 42642
            21.32.29  Congruential equations   congtr 42660
            21.32.30  Alternating congruential equations   acongid 42670
            21.32.31  Additional theorems on integer divisibility   coprmdvdsb 42680
            21.32.32  X and Y sequences 3: Divisibility properties   jm2.18 42683
            21.32.33  X and Y sequences 4: Diophantine representability of Y   jm2.27a 42700
            21.32.34  X and Y sequences 5: Diophantine representability of X, ^, _C   rmxdiophlem 42710
            21.32.35  Uncategorized stuff not associated with a major project   setindtr 42719
            21.32.36  More equivalents of the Axiom of Choice   axac10 42728
            21.32.37  Finitely generated left modules   clfig 42765
            21.32.38  Noetherian left modules I   clnm 42773
            21.32.39  Addenda for structure powers   pwssplit4 42787
            21.32.40  Every set admits a group structure iff choice   unxpwdom3 42793
            21.32.41  Noetherian rings and left modules II   clnr 42807
            21.32.42  Hilbert's Basis Theorem   cldgis 42819
            21.32.43  Additional material on polynomials [DEPRECATED]   cmnc 42829
            21.32.44  Degree and minimal polynomial of algebraic numbers   cdgraa 42838
            21.32.45  Algebraic integers I   citgo 42855
            21.32.46  Endomorphism algebra   cmend 42873
            21.32.47  Cyclic groups and order   idomodle 42893
            21.32.48  Cyclotomic polynomials   ccytp 42899
            21.32.49  Miscellaneous topology   fgraphopab 42905
      ​21.33  Mathbox for Noam Pasman
      ​21.34  Mathbox for Jon Pennant
      ​21.35  Mathbox for Richard Penner
            21.35.1  Set Theory and Ordinal Numbers   uniel 42919
            21.35.2  Natural addition of Cantor normal forms   oawordex2 43029
            21.35.3  Surreal Contributions   abeqabi 43112
            21.35.4  Short Studies   nlimsuc 43145
                  21.35.4.1  Additional work on conditional logical operator   ifpan123g 43163
                  21.35.4.2  Sophisms   rp-fakeimass 43216
                  ​*21.35.4.3  Finite Sets   rp-isfinite5 43221
                  21.35.4.4  General Observations   intabssd 43223
                  21.35.4.5  Infinite Sets   pwelg 43264
                  ​*21.35.4.6  Finite intersection property   fipjust 43269
                  21.35.4.7  RP ADDTO: Subclasses and subsets   rababg 43278
                  21.35.4.8  RP ADDTO: The intersection of a class   elinintab 43279
                  21.35.4.9  RP ADDTO: Theorems requiring subset and intersection existence   elinintrab 43281
                  21.35.4.10  RP ADDTO: Relations   xpinintabd 43284
                  ​*21.35.4.11  RP ADDTO: Functions   elmapintab 43300
                  ​*21.35.4.12  RP ADDTO: Finite induction (for finite ordinals)   cnvcnvintabd 43304
                  21.35.4.13  RP ADDTO: First and second members of an ordered pair   elcnvlem 43305
                  21.35.4.14  RP ADDTO: The reflexive and transitive properties of relations   undmrnresiss 43308
                  21.35.4.15  RP ADDTO: Basic properties of closures   cleq2lem 43312
                  21.35.4.16  RP REPLACE: Definitions and basic properties of transitive closures   trcleq2lemRP 43334
                  ​*21.35.4.17  Additions for square root; absolute value   sqrtcvallem1 43335
            21.35.5  Additional statements on relations and subclasses   al3im 43351
                  21.35.5.1  Transitive relations (not to be confused with transitive classes).   trrelind 43369
                  21.35.5.2  Reflexive closures   crcl 43376
                  ​*21.35.5.3  Finite relationship composition.   relexp2 43381
                  21.35.5.4  Transitive closure of a relation   dftrcl3 43424
                  ​*21.35.5.5  Adapted from Frege   frege77d 43450
            ​*21.35.6  Propositions from _Begriffsschrift_   dfxor4 43470
                  *21.35.6.1  _Begriffsschrift_ Chapter I   dfxor4 43470
                  ​*21.35.6.2  _Begriffsschrift_ Notation hints   whe 43476
                  21.35.6.3  _Begriffsschrift_ Chapter II Implication   ax-frege1 43494
                  21.35.6.4  _Begriffsschrift_ Chapter II Implication and Negation   axfrege28 43533
                  ​*21.35.6.5  _Begriffsschrift_ Chapter II with logical equivalence   axfrege52a 43560
                  21.35.6.6  _Begriffsschrift_ Chapter II with equivalence of sets   axfrege52c 43591
                  ​*21.35.6.7  _Begriffsschrift_ Chapter II with equivalence of classes   frege53c 43618
                  ​*21.35.6.8  _Begriffsschrift_ Chapter III Properties hereditary in a sequence   dffrege69 43636
                  ​*21.35.6.9  _Begriffsschrift_ Chapter III Following in a sequence   dffrege76 43643
                  ​*21.35.6.10  _Begriffsschrift_ Chapter III Member of sequence   dffrege99 43666
                  ​*21.35.6.11  _Begriffsschrift_ Chapter III Single-valued procedures   dffrege115 43682
            ​*21.35.7  Exploring Topology via Seifert and Threlfall   enrelmap 43701
                  *21.35.7.1  Equinumerosity of sets of relations and maps   enrelmap 43701
                  ​*21.35.7.2  Generic Pseudoclosure Spaces, Pseudointerior Spaces, and Pseudoneighborhoods   or3or 43727
                  ​*21.35.7.3  Generic Neighborhood Spaces   gneispa 43834
            ​*21.35.8  Exploring Higher Homotopy via Kerodon   k0004lem1 43851
                  *21.35.8.1  Simplicial Sets   k0004lem1 43851
      ​21.36  Mathbox for Stanislas Polu
            21.36.1  IMO Problems   wwlemuld 43860
                  21.36.1.1  IMO 1972 B2   wwlemuld 43860
            ​*21.36.2  INT Inequalities Proof Generator   int-addcomd 43877
            ​*21.36.3  N-Digit Addition Proof Generator   unitadd 43899
            21.36.4  AM-GM (for k = 2,3,4)   gsumws3 43900
      ​21.37  Mathbox for Rohan Ridenour
            21.37.1  Misc   spALT 43905
            21.37.2  Monoid rings   cmnring 43917
            21.37.3  Shorter primitive equivalent of ax-groth   gru0eld 43940
                  21.37.3.1  Grothendieck universes are closed under collection   gru0eld 43940
                  21.37.3.2  Minimal universes   ismnu 43972
                  21.37.3.3  Primitive equivalent of ax-groth   expandan 43999
      ​21.38  Mathbox for Steve Rodriguez
            21.38.1  Miscellanea   nanorxor 44016
            21.38.2  Ratio test for infinite series convergence and divergence   dvgrat 44023
            21.38.3  Multiples   reldvds 44026
            21.38.4  Function operations   caofcan 44034
            21.38.5  Calculus   lhe4.4ex1a 44040
            21.38.6  The generalized binomial coefficient operation   cbcc 44047
            21.38.7  Binomial series   uzmptshftfval 44057
      ​21.39  Mathbox for Andrew Salmon
            21.39.1  Principia Mathematica * 10   pm10.12 44069
            21.39.2  Principia Mathematica * 11   2alanimi 44083
            21.39.3  Predicate Calculus   sbeqal1 44109
            21.39.4  Principia Mathematica * 13 and * 14   pm13.13a 44118
            21.39.5  Set Theory   elnev 44149
            21.39.6  Arithmetic   addcomgi 44167
            21.39.7  Geometry   cplusr 44168
      ​​*21.40  Mathbox for Alan Sare
            21.40.1  Auxiliary theorems for the Virtual Deduction tool   idiALT 44190
            21.40.2  Supplementary unification deductions   bi1imp 44194
            21.40.3  Conventional Metamath proofs, some derived from VD proofs   iidn3 44214
            21.40.4  What is Virtual Deduction?   wvd1 44282
            21.40.5  Virtual Deduction Theorems   df-vd1 44283
            21.40.6  Theorems proved using Virtual Deduction   trsspwALT 44531
            21.40.7  Theorems proved using Virtual Deduction with mmj2 assistance   simplbi2VD 44559
            21.40.8  Virtual Deduction transcriptions of textbook proofs   sb5ALTVD 44626
            21.40.9  Theorems proved using conjunction-form Virtual Deduction   elpwgdedVD 44630
            21.40.10  Theorems with a VD proof in conventional notation derived from a VD proof   suctrALT3 44637
            ​*21.40.11  Theorems with a proof in conventional notation derived from a VD proof   notnotrALT2 44640
      ​21.41  Mathbox for Glauco Siliprandi
            21.41.1  Miscellanea   evth2f 44651
            21.41.2  Functions   feq1dd 44810
            21.41.3  Ordering on real numbers - Real and complex numbers basic operations   sub2times 44923
            21.41.4  Real intervals   gtnelioc 45145
            21.41.5  Finite sums   fsummulc1f 45228
            21.41.6  Finite multiplication of numbers and finite multiplication of functions   fmul01 45237
            21.41.7  Limits   clim1fr1 45258
                  21.41.7.1  Inferior limit (lim inf)   clsi 45408
                  ​*21.41.7.2  Limits for sequences of extended real numbers   clsxlim 45475
            21.41.8  Trigonometry   coseq0 45521
            21.41.9  Continuous Functions   mulcncff 45527
            21.41.10  Derivatives   dvsinexp 45568
            21.41.11  Integrals   itgsin0pilem1 45607
            21.41.12  Stone Weierstrass theorem - real version   stoweidlem1 45658
            21.41.13  Wallis' product for π   wallispilem1 45722
            21.41.14  Stirling's approximation formula for ` n ` factorial   stirlinglem1 45731
            21.41.15  Dirichlet kernel   dirkerval 45748
            21.41.16  Fourier Series   fourierdlem1 45765
            21.41.17  e is transcendental   elaa2lem 45890
            21.41.18  n-dimensional Euclidean space   rrxtopn 45941
            21.41.19  Basic measure theory   csalg 45965
                  ​*21.41.19.1  σ-Algebras   csalg 45965
                  21.41.19.2  Sum of nonnegative extended reals   csumge0 46019
                  ​*21.41.19.3  Measures   cmea 46106
                  ​*21.41.19.4  Outer measures and Caratheodory's construction   come 46146
                  ​*21.41.19.5  Lebesgue measure on n-dimensional Real numbers   covoln 46193
                  ​*21.41.19.6  Measurable functions   csmblfn 46352
      ​21.42  Mathbox for Saveliy Skresanov
            21.42.1  Ceva's theorem   sigarval 46507
            21.42.2  Simple groups   simpcntrab 46527
      ​21.43  Mathbox for Ender Ting
            21.43.1  Increasing sequences and subsequences   et-ltneverrefl 46528
      ​21.44  Mathbox for Jarvin Udandy
      ​21.45  Mathbox for Adhemar
            ​*21.45.1  Minimal implicational calculus   adh-minim 46652
      ​21.46  Mathbox for Alexander van der Vekens
            21.46.1  General auxiliary theorems (1)   n0nsn2el 46676
                  21.46.1.1  Unordered and ordered pairs - extension for singletons   n0nsn2el 46676
                  21.46.1.2  Unordered and ordered pairs - extension for unordered pairs   elprneb 46680
                  21.46.1.3  Unordered and ordered pairs - extension for ordered pairs   oppr 46681
                  21.46.1.4  Relations - extension   eubrv 46686
                  21.46.1.5  Definite description binder (inverted iota) - extension   iota0def 46689
                  21.46.1.6  Functions - extension   fveqvfvv 46691
            21.46.2  Alternative for Russell's definition of a description binder   caiota 46732
            21.46.3  Double restricted existential uniqueness   r19.32 46747
                  21.46.3.1  Restricted quantification (extension)   r19.32 46747
                  21.46.3.2  Restricted uniqueness and "at most one" quantification   reuf1odnf 46756
                  21.46.3.3  Analogs to Existential uniqueness (double quantification)   2reu3 46759
                  21.46.3.4  Additional theorems for double restricted existential uniqueness   2reu8i 46762
            ​*21.46.4  Alternative definitions of function and operation values   wdfat 46765
                  21.46.4.1  Restricted quantification (extension)   ralbinrald 46771
                  21.46.4.2  The universal class (extension)   nvelim 46772
                  21.46.4.3  Introduce the Axiom of Power Sets (extension)   alneu 46773
                  21.46.4.4  Predicate "defined at"   dfateq12d 46775
                  21.46.4.5  Alternative definition of the value of a function   dfafv2 46781
                  21.46.4.6  Alternative definition of the value of an operation   aoveq123d 46827
            ​*21.46.5  Alternative definitions of function values (2)   cafv2 46857
            21.46.6  General auxiliary theorems (2)   an4com24 46917
                  21.46.6.1  Logical conjunction - extension   an4com24 46917
                  21.46.6.2  Abbreviated conjunction and disjunction of three wff's - extension   3an4ancom24 46918
                  21.46.6.3  Negated membership (alternative)   cnelbr 46920
                  21.46.6.4  The empty set - extension   ralralimp 46927
                  21.46.6.5  Indexed union and intersection - extension   otiunsndisjX 46928
                  21.46.6.6  Functions - extension   fvifeq 46929
                  21.46.6.7  Maps-to notation - extension   fvmptrab 46941
                  21.46.6.8  Subtraction - extension   cnambpcma 46943
                  21.46.6.9  Ordering on reals (cont.) - extension   leaddsuble 46946
                  21.46.6.10  Imaginary and complex number properties - extension   readdcnnred 46952
                  21.46.6.11  Nonnegative integers (as a subset of complex numbers) - extension   nn0resubcl 46957
                  21.46.6.12  Integers (as a subset of complex numbers) - extension   zgeltp1eq 46958
                  21.46.6.13  Decimal arithmetic - extension   1t10e1p1e11 46959
                  21.46.6.14  Upper sets of integers - extension   eluzge0nn0 46961
                  21.46.6.15  Infinity and the extended real number system (cont.) - extension   nltle2tri 46962
                  21.46.6.16  Finite intervals of integers - extension   ssfz12 46963
                  21.46.6.17  Half-open integer ranges - extension   fzopred 46971
                  21.46.6.18  The modulo (remainder) operation - extension   m1mod0mod1 46977
                  21.46.6.19  The infinite sequence builder "seq"   smonoord 46979
                  21.46.6.20  Finite and infinite sums - extension   fsummsndifre 46980
                  21.46.6.21  Extensible structures - extension   setsidel 46984
            ​*21.46.7  Preimages of function values   preimafvsnel 46987
            ​*21.46.8  Partitions of real intervals   ciccp 47021
            21.46.9  Shifting functions with an integer range domain   fargshiftfv 47047
            21.46.10  Words over a set (extension)   lswn0 47052
                  21.46.10.1  Last symbol of a word - extension   lswn0 47052
            21.46.11  Unordered pairs   wich 47053
                  21.46.11.1  Interchangeable setvar variables   wich 47053
                  21.46.11.2  Set of unordered pairs   sprid 47082
                  ​*21.46.11.3  Proper (unordered) pairs   prpair 47109
                  21.46.11.4  Set of proper unordered pairs   cprpr 47120
            21.46.12  Number theory (extension)   cfmtno 47135
                  ​*21.46.12.1  Fermat numbers   cfmtno 47135
                  ​*21.46.12.2  Mersenne primes   m2prm 47199
                  21.46.12.3  Proth's theorem   modexp2m1d 47220
                  21.46.12.4  Solutions of quadratic equations   quad1 47228
            ​*21.46.13  Even and odd numbers   ceven 47232
                  21.46.13.1  Definitions and basic properties   ceven 47232
                  21.46.13.2  Alternate definitions using the "divides" relation   dfeven2 47257
                  21.46.13.3  Alternate definitions using the "modulo" operation   dfeven3 47266
                  21.46.13.4  Alternate definitions using the "gcd" operation   iseven5 47272
                  21.46.13.5  Theorems of part 5 revised   zneoALTV 47277
                  21.46.13.6  Theorems of part 6 revised   odd2np1ALTV 47282
                  21.46.13.7  Theorems of AV's mathbox revised   0evenALTV 47296
                  21.46.13.8  Additional theorems   epoo 47311
                  21.46.13.9  Perfect Number Theorem (revised)   perfectALTVlem1 47329
            21.46.14  Number theory (extension 2)   cfppr 47332
                  ​*21.46.14.1  Fermat pseudoprimes   cfppr 47332
                  ​*21.46.14.2  Goldbach's conjectures   cgbe 47353
            21.46.15  Graph theory (extension)   cclnbgr 47426
                  21.46.15.1  Closed neighborhood of a vertex   cclnbgr 47426
                  ​*21.46.15.2  Semiclosed and semiopen neighborhoods (experimental)   dfsclnbgr2 47449
                  21.46.15.3  Induced subgraphs   cisubgr 47463
                  ​*21.46.15.4  Isomorphisms of graphs   cgrisom 47475
                  ​*21.46.15.5  Local isomorphisms of graphs   cgrlim 47518
                  21.46.15.6  Loop-free graphs - extension   1hegrlfgr 47545
                  21.46.15.7  Walks - extension   cupwlks 47546
                  21.46.15.8  Edges of graphs expressed as sets of unordered pairs   upgredgssspr 47556
            21.46.16  Monoids (extension)   ovn0dmfun 47569
                  21.46.16.1  Auxiliary theorems   ovn0dmfun 47569
                  21.46.16.2  Magmas, Semigroups and Monoids (extension)   plusfreseq 47577
                  21.46.16.3  Examples and counterexamples for magmas, semigroups and monoids (extension)   opmpoismgm 47580
                  21.46.16.4  Group sum operation (extension 1)   gsumsplit2f 47593
            ​*21.46.17  Magmas and internal binary operations (alternate approach)   ccllaw 47596
                  *21.46.17.1  Laws for internal binary operations   ccllaw 47596
                  ​*21.46.17.2  Internal binary operations   cintop 47609
                  21.46.17.3  Alternative definitions for magmas and semigroups   cmgm2 47628
            21.46.18  Rings (extension)   lmod0rng 47642
                  21.46.18.1  Nonzero rings (extension)   lmod0rng 47642
                  21.46.18.2  Ideals as non-unital rings   lidldomn1 47644
                  21.46.18.3  The non-unital ring of even integers   0even 47650
                  21.46.18.4  A constructed not unital ring   cznrnglem 47672
                  ​*21.46.18.5  The category of non-unital rings (alternate definition)   crngcALTV 47676
                  ​*21.46.18.6  The category of (unital) rings (alternate definition)   cringcALTV 47700
            21.46.19  Basic algebraic structures (extension)   opeliun2xp 47747
                  21.46.19.1  Auxiliary theorems   opeliun2xp 47747
                  21.46.19.2  The binomial coefficient operation (extension)   bcpascm1 47766
                  21.46.19.3  The ` ZZ `-module ` ZZ X. ZZ `   zlmodzxzlmod 47769
                  21.46.19.4  Group sum operation (extension 2)   mgpsumunsn 47776
                  21.46.19.5  Symmetric groups (extension)   exple2lt6 47779
                  21.46.19.6  Divisibility (extension)   invginvrid 47782
                  21.46.19.7  The support of functions (extension)   rmsupp0 47783
                  21.46.19.8  Finitely supported functions (extension)   rmsuppfi 47788
                  21.46.19.9  Left modules (extension)   lmodvsmdi 47797
                  21.46.19.10  Associative algebras (extension)   assaascl0 47799
                  21.46.19.11  Univariate polynomials (extension)   ply1vr1smo 47801
                  21.46.19.12  Univariate polynomials (examples)   linply1 47812
            21.46.20  Linear algebra (extension)   cdmatalt 47815
                  ​*21.46.20.1  The subalgebras of diagonal and scalar matrices (extension)   cdmatalt 47815
                  ​*21.46.20.2  Linear combinations   clinc 47823
                  ​*21.46.20.3  Linear independence   clininds 47859
                  21.46.20.4  Simple left modules and the ` ZZ `-module   lmod1lem1 47906
                  21.46.20.5  Differences between (left) modules and (left) vector spaces   lvecpsslmod 47926
            21.46.21  Complexity theory   suppdm 47929
                  21.46.21.1  Auxiliary theorems   suppdm 47929
                  21.46.21.2  The modulo (remainder) operation (extension)   fldivmod 47942
                  21.46.21.3  Even and odd integers   nn0onn0ex 47947
                  21.46.21.4  The natural logarithm on complex numbers (extension)   logcxp0 47959
                  21.46.21.5  Division of functions   cfdiv 47961
                  21.46.21.6  Upper bounds   cbigo 47971
                  21.46.21.7  Logarithm to an arbitrary base (extension)   rege1logbrege0 47982
                  ​*21.46.21.8  The binary logarithm   fldivexpfllog2 47989
                  21.46.21.9  Binary length   cblen 47993
                  ​*21.46.21.10  Digits   cdig 48019
                  21.46.21.11  Nonnegative integer as sum of its shifted digits   dignn0flhalflem1 48039
                  21.46.21.12  Algorithms for the multiplication of nonnegative integers   nn0mulfsum 48048
                  ​*21.46.21.13  N-ary functions   cnaryf 48050
                  ​*21.46.21.14  The Ackermann function   citco 48081
            21.46.22  Elementary geometry (extension)   fv1prop 48123
                  21.46.22.1  Auxiliary theorems   fv1prop 48123
                  21.46.22.2  Real euclidean space of dimension 2   rrx2pxel 48135
                  21.46.22.3  Spheres and lines in real Euclidean spaces   cline 48151
      ​21.47  Mathbox for Zhi Wang
            21.47.1  Propositional calculus   pm4.71da 48213
            21.47.2  Predicate calculus with equality   dtrucor3 48222
                  21.47.2.1  Axiom scheme ax-5 (Distinctness)   dtrucor3 48222
            21.47.3  ZF Set Theory - start with the Axiom of Extensionality   ralbidb 48223
                  21.47.3.1  Restricted quantification   ralbidb 48223
                  21.47.3.2  The empty set   ssdisjd 48229
                  21.47.3.3  Unordered and ordered pairs   vsn 48233
                  21.47.3.4  The union of a class   unilbss 48239
            21.47.4  ZF Set Theory - add the Axiom of Replacement   inpw 48240
                  21.47.4.1  Theorems requiring subset and intersection existence   inpw 48240
            21.47.5  ZF Set Theory - add the Axiom of Power Sets   mof0 48241
                  21.47.5.1  Functions   mof0 48241
                  21.47.5.2  Operations   fvconstr 48259
            21.47.6  ZF Set Theory - add the Axiom of Union   fvconst0ci 48262
                  21.47.6.1  Equinumerosity   fvconst0ci 48262
            21.47.7  Order sets   iccin 48266
                  21.47.7.1  Real number intervals   iccin 48266
            21.47.8  Moore spaces   mreuniss 48269
            ​*21.47.9  Topology   clduni 48270
                  21.47.9.1  Closure and interior   clduni 48270
                  21.47.9.2  Neighborhoods   neircl 48274
                  21.47.9.3  Subspace topologies   restcls2lem 48282
                  21.47.9.4  Limits and continuity in topological spaces   cnneiima 48286
                  21.47.9.5  Topological definitions using the reals   iooii 48287
                  21.47.9.6  Separated sets   sepnsepolem1 48291
                  21.47.9.7  Separated spaces: T0, T1, T2 (Hausdorff) ...   isnrm4 48300
            21.47.10  Preordered sets and directed sets using extensible structures   isprsd 48325
            21.47.11  Posets and lattices using extensible structures   lubeldm2 48326
                  21.47.11.1  Posets   lubeldm2 48326
                  21.47.11.2  Lattices   toslat 48344
                  21.47.11.3  Subset order structures   intubeu 48346
            21.47.12  Categories   catprslem 48367
                  21.47.12.1  Categories   catprslem 48367
                  21.47.12.2  Monomorphisms and epimorphisms   idmon 48373
                  21.47.12.3  Functors   funcf2lem 48375
            21.47.13  Examples of categories   cthinc 48376
                  21.47.13.1  Thin categories   cthinc 48376
                  21.47.13.2  Preordered sets as thin categories   cprstc 48419
                  21.47.13.3  Monoids as categories   cmndtc 48440
      ​21.48  Mathbox for Emmett Weisz
            ​*21.48.1  Miscellaneous Theorems   nfintd 48455
            21.48.2  Set Recursion   csetrecs 48465
                  ​*21.48.2.1  Basic Properties of Set Recursion   csetrecs 48465
                  21.48.2.2  Examples and properties of set recursion   elsetrecslem 48481
            ​*21.48.3  Construction of Games and Surreal Numbers   cpg 48491
      ​​*21.49  Mathbox for David A. Wheeler
            21.49.1  Natural deduction   sbidd 48500
            ​*21.49.2  Greater than, greater than or equal to.   cge-real 48502
            ​*21.49.3  Hyperbolic trigonometric functions   csinh 48512
            ​*21.49.4  Reciprocal trigonometric functions (sec, csc, cot)   csec 48523
            ​*21.49.5  Identities for "if"   ifnmfalse 48545
            ​*21.49.6  Logarithms generalized to arbitrary base using ` logb `   logb2aval 48546
            ​*21.49.7  Logarithm laws generalized to an arbitrary base - log_   clog- 48547
            ​*21.49.8  Formally define notions such as reflexivity   wreflexive 48549
            ​*21.49.9  Algebra helpers   comraddi 48553
            ​*21.49.10  Algebra helper examples   i2linesi 48562
            ​*21.49.11  Formal methods "surprises"   alimp-surprise 48564
            ​*21.49.12  Allsome quantifier   walsi 48570
            ​*21.49.13  Miscellaneous   5m4e1 48581
            21.49.14  Theorems about algebraic numbers   aacllem 48585
      ​21.50  Mathbox for Kunhao Zheng
            21.50.1  Weighted AM-GM inequality   amgmwlem 48586