TOC of Theorem List - Metamath Proof Explorer

Table of Contents Summary

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

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

​​*PART 1  CLASSICAL FIRST-ORDER LOGIC WITH EQUALITY
      ​*1.1  Pre-logic
            *1.1.1  Inferences for assisting proof development   idi 1
      ​​*1.2  Propositional calculus
            1.2.1  Recursively define primitive wffs for propositional calculus   wn 3
            ​*1.2.2  The axioms of propositional calculus   ax-mp 5
            ​*1.2.3  Logical implication   mp2 9
            ​*1.2.4  Logical negation   con4 114
            ​*1.2.5  Logical equivalence   wb 209
            ​*1.2.6  Logical conjunction   wa 401
            ​*1.2.7  Logical disjunction   wo 861
            ​*1.2.8  Mixed connectives   jaao 969
            ​*1.2.9  The conditional operator for propositions   wif 1078
            ​*1.2.10  The weak deduction theorem for propositional calculus   elimh 1099
            1.2.11  Abbreviated conjunction and disjunction of three wff's   w3o 1102
            1.2.12  Logical "nand" (Sheffer stroke)   wnan 1521
            1.2.13  Logical "xor"   wxo 1541
            1.2.14  Logical "nor"   wnor 1558
            1.2.15  True and false constants   wal 1568
                  ​*1.2.15.1  Universal quantifier for use by df-tru   wal 1568
                  ​*1.2.15.2  Equality predicate for use by df-tru   cv 1569
                  1.2.15.3  The true constant   wtru 1571
                  1.2.15.4  The false constant   wfal 1582
            ​*1.2.16  Truth tables   truimtru 1593
                  1.2.16.1  Implication   truimtru 1593
                  1.2.16.2  Negation   nottru 1597
                  1.2.16.3  Equivalence   trubitru 1599
                  1.2.16.4  Conjunction   truantru 1603
                  1.2.16.5  Disjunction   truortru 1607
                  1.2.16.6  Alternative denial   trunantru 1611
                  1.2.16.7  Exclusive disjunction   truxortru 1615
                  1.2.16.8  Joint denial   trunortru 1619
            ​*1.2.17  Half adder and full adder in propositional calculus   whad 1623
                  1.2.17.1  Full adder: sum   whad 1623
                  1.2.17.2  Full adder: carry   wcad 1639
      ​1.3  Other axiomatizations related to classical propositional calculus
            ​*1.3.1  Minimal implicational calculus   minimp 1654
            ​*1.3.2  Implicational Calculus   impsingle 1660
            1.3.3  Derive the Lukasiewicz axioms from Meredith's sole axiom   meredith 1674
            1.3.4  Derive the standard axioms from the Lukasiewicz axioms   luklem1 1691
            ​*1.3.5  Derive Nicod's axiom from the standard axioms   nic-dfim 1702
            1.3.6  Derive the Lukasiewicz axioms from Nicod's axiom   nic-imp 1708
            1.3.7  Derive Nicod's Axiom from Lukasiewicz's First Sheffer Stroke Axiom   lukshef-ax1 1727
            1.3.8  Derive the Lukasiewicz Axioms from the Tarski-Bernays-Wajsberg Axioms   tbw-bijust 1731
            1.3.9  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's First CO Axiom   merco1 1746
            1.3.10  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's Second CO Axiom   merco2 1769
            1.3.11  Derive the Lukasiewicz axioms from the Russell-Bernays Axioms   rb-bijust 1782
            ​*1.3.12  Stoic logic non-modal portion (Chrysippus of Soli)   mptnan 1801
      ​​*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 1812
                  1.4.1.1  Existential quantifier   wex 1812
                  1.4.1.2  Nonfreeness predicate   wnf 1816
            1.4.2  Rule scheme ax-gen (Generalization)   ax-gen 1828
            1.4.3  Axiom scheme ax-4 (Quantified Implication)   ax-4 1842
                  ​*1.4.3.1  The empty domain of discourse   empty 1939
            1.4.4  Axiom scheme ax-5 (Distinctness) - first use of $d   ax-5 1943
            ​*1.4.5  Equality predicate (continued)   weq 1995
            1.4.6  Axiom scheme ax-6 (Existence)   ax-6 2000
            1.4.7  Axiom scheme ax-7 (Equality)   ax-7 2041
            1.4.8  Define proper substitution   sbjust 2098
            1.4.9  Membership predicate   wcel 2145
            1.4.10  Axiom scheme ax-8 (Left Equality for Binary Predicate)   ax-8 2147
            1.4.11  Axiom scheme ax-9 (Right Equality for Binary Predicate)   ax-9 2155
            ​*1.4.12  Logical redundancy of ax-10 , ax-11 , ax-12 , ax-13   ax6dgen 2165
      ​​*1.5  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes)
            1.5.1  Axiom scheme ax-10 (Quantified Negation)   ax-10 2178
            1.5.2  Axiom scheme ax-11 (Quantifier Commutation)   ax-11 2194
            1.5.3  Axiom scheme ax-12 (Substitution)   ax-12 2213
            1.5.4  Axiom scheme ax-13 (Quantified Equality)   ax-13 2402
      ​1.6  Uniqueness and unique existence
            1.6.1  Uniqueness: the at-most-one quantifier   wmo 2563
            1.6.2  Unique existence: the unique existential quantifier   weu 2594
      ​1.7  Other axiomatizations related to classical predicate calculus
            ​*1.7.1  Aristotelian logic: Assertic syllogisms   barbara 2688
            ​*1.7.2  Intuitionistic logic   axia1 2718
​​*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 2733
            2.1.2  Classes   cab 2739
                  2.1.2.1  Class abstractions   cab 2739
                  ​*2.1.2.2  Class equality   df-cleq 2753
                  2.1.2.3  Class membership   df-clel 2836
                  2.1.2.4  Elementary properties of class abstractions   eqabdv 2894
            2.1.3  Class form not-free predicate   wnfc 2908
            2.1.4  Negated equality and membership   wne 2956
                  2.1.4.1  Negated equality   wne 2956
                  2.1.4.2  Negated membership   wnel 3062
            2.1.5  Restricted quantification   wral 3077
                  2.1.5.1  Restricted universal and existential quantification   wral 3077
                  2.1.5.2  Restricted existential uniqueness and at-most-one quantifier   wreu 3364
                  2.1.5.3  Restricted class abstraction   crab 3413
            2.1.6  The universal class   cvv 3451
            ​*2.1.7  Conditional equality (experimental)   wcdeq 3721
            2.1.8  Russell's Paradox   rru 3737
            2.1.9  Proper substitution of classes for sets   wsbc 3739
            2.1.10  Proper substitution of classes for sets into classes   csb 3847
            2.1.11  Define basic set operations and relations   cdif 3896
            2.1.12  Subclasses and subsets   df-ss 3916
            2.1.13  The difference, union, and intersection of two classes   dfdif3 4066
                  2.1.13.1  The difference of two classes   dfdif3 4066
                  2.1.13.2  The union of two classes   elun 4100
                  2.1.13.3  The intersection of two classes   elini 4145
                  2.1.13.4  The symmetric difference of two classes   csymdif 4198
                  2.1.13.5  Combinations of difference, union, and intersection of two classes   unabs 4211
                  2.1.13.6  Class abstractions with difference, union, and intersection of two classes   unabw 4253
                  2.1.13.7  Restricted uniqueness with difference, union, and intersection   reuun2 4271
            2.1.14  The empty set   c0 4279
            ​*2.1.15  The conditional operator for classes   cif 4482
            ​*2.1.16  The weak deduction theorem for set theory   dedth 4541
            2.1.17  Power classes   cpw 4557
            2.1.18  Unordered and ordered pairs   snjust 4583
            2.1.19  The union of a class   cuni 4867
            2.1.20  The intersection of a class   cint 4907
            2.1.21  Indexed union and intersection   ciun 4951
            2.1.22  Disjointness   wdisj 5070
            2.1.23  Binary relations   wbr 5103
            2.1.24  Ordered-pair class abstractions (class builders)   copab 5167
            2.1.25  Functions in maps-to notation   cmpt 5186
            2.1.26  Transitive classes   wtr 5212
      ​2.2  ZF Set Theory - add the Axiom of Replacement
            2.2.1  Introduce the Axiom of Replacement   ax-rep 5232
            2.2.2  Derive the Axiom of Separation   axsepgfromrep 5247
            2.2.3  Derive the Null Set Axiom   axnulALT 5258
            2.2.4  Theorems requiring subset and intersection existence   exnelv 5267
            2.2.5  Theorems requiring empty set existence   class2set 5316
      ​2.3  ZF Set Theory - add the Axiom of Power Sets
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 5327
            2.3.2  Derive the Axiom of Pairing   axprlem1 5385
            2.3.3  Ordered pair theorem   opnz 5442
            2.3.4  Ordered-pair class abstractions (cont.)   opabidw 5498
            2.3.5  Power class of union and intersection   pwin 5542
            2.3.6  The identity relation   cid 5545
            2.3.7  The membership relation (or epsilon relation)   cep 5550
            ​*2.3.8  Partial and total orderings   wpo 5557
            2.3.9  Founded and well-ordering relations   wfr 5601
            2.3.10  Relations   cxp 5649
            2.3.11  The Predecessor Class   cpred 6303
            2.3.12  Well-founded induction (variant)   frpomin 6343
            2.3.13  Well-ordered induction   tz6.26 6350
            2.3.14  Ordinals   word 6361
            2.3.15  Definite description binder (inverted iota)   cio 6492
            2.3.16  Functions   wfun 6532
            2.3.17  Cantor's Theorem   canth 7374
            2.3.18  Restricted iota (description binder)   crio 7376
            2.3.19  Operations   co 7420
                  2.3.19.1  Variable-to-class conversion for operations   caovclg 7613
            2.3.20  Operations in maps-to notation   mpondm0 7661
            2.3.21  Functions with three arguments in maps-to notation   cmpt3 7683
            2.3.22  Function operation   cof 7691
            2.3.23  Proper subset relation   crpss 7738
      ​2.4  ZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 7751
            2.4.2  Ordinals (continued)   epweon 7789
            2.4.3  Transfinite induction   tfi 7864
            2.4.4  The natural numbers (i.e., finite ordinals)   com 7877
            2.4.5  Peano's postulates   peano1 7900
            2.4.6  Finite induction (for finite ordinals)   find 7907
            2.4.7  Relations and functions (cont.)   dmexg 7913
            2.4.8  First and second members of an ordered pair   c1st 7999
            2.4.9  Induction on Cartesian products   frpoins3xpg 8157
            2.4.10  Ordering on Cartesian products   xpord2lem 8159
            2.4.11  Ordering Ordinal Sequences   orderseqlem 8174
            ​*2.4.12  The support of functions   csupp 8177
            ​*2.4.13  Special maps-to operations   opeliunxp2f 8227
            2.4.14  Function transposition   ctpos 8242
            2.4.15  Curry and uncurry   ccur 8282
            2.4.16  Undefined values   cund 8289
            2.4.17  Well-founded recursion   cfrecs 8298
            2.4.18  Well-ordered recursion   cwrecs 8329
            2.4.19  Functions on ordinals; strictly monotone ordinal functions   iunon 8347
            2.4.20  "Strong" transfinite recursion   crecs 8378
            2.4.21  Recursive definition generator   crdg 8417
            2.4.22  Finite recursion   frfnom 8443
            2.4.23  Ordinal arithmetic   c1o 8469
            2.4.24  Natural number arithmetic   nna0 8613
            2.4.25  Natural addition   cnadd 8674
            2.4.26  Equivalence relations and classes   wer 8714
            2.4.27  The mapping operation   cmap 8847
            2.4.28  Infinite Cartesian products   cixp 8925
            2.4.29  Equinumerosity   cen 8970
            2.4.30  Schroeder-Bernstein Theorem   sbthlem1 9106
            2.4.31  Equinumerosity (cont.)   xpf1o 9158
            2.4.32  Finite sets   dif1enlem 9175
            2.4.33  Pigeonhole Principle   phplem1 9219
            2.4.34  Finite sets (cont.)   onomeneq 9229
            2.4.35  Finitely supported functions   cfsupp 9353
            2.4.36  Finite intersections   cfi 9402
            2.4.37  Hall's marriage theorem   marypha1lem 9425
            2.4.38  Supremum and infimum   csup 9432
            2.4.39  Ordinal isomorphism, Hartogs's theorem   coi 9503
            2.4.40  Hartogs function   char 9550
            2.4.41  Weak dominance   cwdom 9558
      ​2.5  ZF Set Theory - add the Axiom of Regularity
            2.5.1  Introduce the Axiom of Regularity   ax-reg 9586
            2.5.2  Axiom of Infinity equivalents   inf0 9622
      ​2.6  ZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-inf 9639
            2.6.2  Existence of omega (the set of natural numbers)   omex 9644
            2.6.3  Cantor normal form   ccnf 9662
            2.6.4  Transitive closure of a relation   cttrcl 9708
            2.6.5  Transitive closure   trcl 9729
            2.6.6  Set induction (or epsilon induction)   setind 9748
            2.6.7  Well-Founded Induction   frmin 9753
            2.6.8  Well-Founded Recursion   frr3g 9760
            2.6.9  Rank   cr1 9766
            2.6.10  Hereditarily finite sets   chf 9904
            2.6.11  Scott's trick; collection principle; Hilbert's epsilon   cscott 9928
            2.6.12  Set Recursion   csetrecs 9964
                  ​*2.6.12.1  Basic Properties of Set Recursion   csetrecs 9964
            2.6.13  Disjoint union   cdju 9979
            2.6.14  Cardinal numbers   ccrd 10016
            2.6.15  Axiom of Choice equivalents   wac 10194
            ​*2.6.16  Cardinal number arithmetic   undjudom 10246
            2.6.17  The Ackermann bijection   ackbij2lem1 10296
            2.6.18  Cofinality (without Axiom of Choice)   cflem 10323
            2.6.19  Eight inequivalent definitions of finite set   sornom 10355
            2.6.20  Hereditarily size-limited sets without Choice   itunifval 10494
​​*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 10513
            3.1.2  Introduce the Axiom of Dependent Choice   ax-dc 10524
      ​3.2  ZFC Set Theory - add the Axiom of Choice
            3.2.1  Introduce the Axiom of Choice   ax-ac 10537
            3.2.2  AC equivalents: well-ordering, Zorn's lemma   numthcor 10572
            3.2.3  Cardinal number theorems using Axiom of Choice   cardval 10630
            3.2.4  Cardinal number arithmetic using Axiom of Choice   iunctb 10659
            3.2.5  Cofinality using the Axiom of Choice   alephreg 10667
      ​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 10705
            3.4.2  Derivation of the Axiom of Choice   gchaclem 10763
​​*PART 4  TG (TARSKI-GROTHENDIECK) SET THEORY
      ​4.1  Inaccessibles
            4.1.1  Weakly and strongly inaccessible cardinals   cwina 10767
            4.1.2  Weak universes   cwun 10785
            4.1.3  Tarski classes   ctsk 10833
            4.1.4  Grothendieck universes   cgru 10875
      ​4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
            4.2.1  Introduce the Tarski-Grothendieck Axiom   ax-groth 10908
            4.2.2  Derive the Power Set, Infinity and Choice Axioms   grothpw 10911
            4.2.3  Tarski map function   ctskm 10922
​​*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 10929
            5.1.2  Final derivation of real and complex number postulates   axaddf 11230
            5.1.3  Real and complex number postulates restated as axioms   ax-cnex 11256
      ​5.2  Derive the basic properties from the field axioms
            5.2.1  Some deductions from the field axioms for complex numbers   cnex 11281
            5.2.2  Infinity and the extended real number system   cpnf 11340
            5.2.3  Restate the ordering postulates with extended real "less than"   axlttri 11381
            5.2.4  Ordering on reals   lttr 11386
            5.2.5  Initial properties of the complex numbers   mul12 11475
      ​5.3  Real and complex numbers - basic operations
            5.3.1  Addition   add12 11528
            5.3.2  Subtraction   cmin 11541
            5.3.3  Multiplication   kcnktkm1cn 11747
            5.3.4  Ordering on reals (cont.)   gt0ne0 11781
            5.3.5  Reciprocals   ixi 11945
            5.3.6  Division   cdiv 11973
            5.3.7  Ordering on reals (cont.)   elimgt0 12155
            5.3.8  Completeness Axiom and Suprema   fimaxre 12261
            5.3.9  Imaginary and complex number properties   neg1cn 12305
            5.3.10  Function operation analogue theorems   ofsubeq0 12317
            ​*5.3.11  Indicator Functions   cind 12320
      ​5.4  Integer sets
            5.4.1  Positive integers (as a subset of complex numbers)   cn 12335
            5.4.2  Principle of mathematical induction   nnind 12353
            ​*5.4.3  Decimal representation of numbers   c2 12397
            ​*5.4.4  Some properties of specific numbers   1pneg1e0 12460
            5.4.5  Simple number properties   halfcl 12572
            5.4.6  The Archimedean property   nnunb 12602
            5.4.7  Nonnegative integers (as a subset of complex numbers)   cn0 12606
            ​*5.4.8  Extended nonnegative integers   cxnn0 12679
            5.4.9  Integers (as a subset of complex numbers)   cz 12693
            5.4.10  Decimal arithmetic   cdc 12814
            5.4.11  Upper sets of integers   cuz 12965
            5.4.12  Well-ordering principle for bounded-below sets of integers   uzwo3 13070
            5.4.13  Rational numbers (as a subset of complex numbers)   cq 13075
            5.4.14  Existence of the set of complex numbers   rpnnen1lem2 13105
      ​5.5  Order sets
            5.5.1  Positive reals (as a subset of complex numbers)   crp 13120
            5.5.2  Infinity and the extended real number system (cont.)   cxne 13238
            5.5.3  Supremum and infimum on the extended reals   xrsupexmnf 13435
            5.5.4  Real number intervals   cioo 13476
            5.5.5  Finite intervals of integers   cfz 13639
            ​*5.5.6  Finite intervals of nonnegative integers   elfz2nn0 13752
            5.5.7  Half-open integer ranges   cfzo 13788
      ​5.6  Elementary integer functions
            5.6.1  The floor and ceiling functions   cfl 13930
            5.6.2  The modulo (remainder) operation   cmo 14009
            5.6.3  Miscellaneous theorems about integers   om2uz0i 14090
            5.6.4  Strong induction over upper sets of integers   uzsinds 14130
            5.6.5  Finitely supported functions over the nonnegative integers   fsuppmapnn0fiublem 14133
            5.6.6  The infinite sequence builder "seq" - extension   cseq 14144
            5.6.7  Integer powers   cexp 14204
            5.6.8  Ordered pair theorem for nonnegative integers   nn0le2msqi 14411
            5.6.9  Factorial function   cfa 14417
            5.6.10  The binomial coefficient operation   cbc 14446
            5.6.11  The ` # ` (set size) function   chash 14474
                  5.6.11.1  Proper unordered pairs and triples (sets of size 2 and 3)   hashprlei 14613
                  5.6.11.2  Functions with a domain containing at least two different elements   fundmge2nop0 14647
                  5.6.11.3  Finite induction on the size of the first component of a binary relation   hashdifsnp1 14651
      ​​*5.7  Words over a set
            5.7.1  Definitions and basic theorems   cword 14658
            5.7.2  Last symbol of a word   clsw 14707
            5.7.3  Concatenations of words   cconcat 14715
            5.7.4  Singleton words   cs1 14742
            5.7.5  Concatenations with singleton words   ccatws1cl 14764
            5.7.6  Subwords/substrings   csubstr 14788
            5.7.7  Prefixes of a word   cpfx 14820
            5.7.8  Subwords of subwords   swrdswrdlem 14853
            5.7.9  Subwords and concatenations   pfxcctswrd 14859
            5.7.10  Subwords of concatenations   swrdccatfn 14873
            5.7.11  Splicing words (substring replacement)   csplice 14898
            5.7.12  Reversing words   creverse 14907
            5.7.13  Repeated symbol words   creps 14919
            ​*5.7.14  Cyclical shifts of words   ccsh 14939
            5.7.15  Mapping words by a function   wrdco 14982
            5.7.16  Longer string literals   cs2 14992
      ​​*5.8  Reflexive and transitive closures of relations
            5.8.1  The reflexive and transitive properties of relations   coss12d 15125
            5.8.2  Basic properties of closures   cleq1lem 15135
            5.8.3  Definitions and basic properties of transitive closures   ctcl 15138
            5.8.4  Exponentiation of relations   crelexp 15172
            5.8.5  Reflexive-transitive closure as an indexed union   crtrcl 15208
            ​*5.8.6  Principle of transitive induction   relexpindlem 15216
      ​5.9  Elementary real and complex functions
            5.9.1  The "shift" operation   cshi 15219
            5.9.2  Signum (sgn or sign) function   csgn 15239
            5.9.3  Real and imaginary parts; conjugate   ccj 15263
            5.9.4  Square root; absolute value   csqrt 15400
      ​5.10  Elementary limits and convergence
            5.10.1  Superior limit (lim sup)   clsp 15637
            5.10.2  Limits   cli 15651
            5.10.3  Finite and infinite sums   csu 15853
            5.10.4  The binomial theorem   binomlem 15998
            5.10.5  The inclusion/exclusion principle   incexclem 16005
            5.10.6  Infinite sums (cont.)   isumshft 16008
            5.10.7  Miscellaneous converging and diverging sequences   divrcnv 16021
            5.10.8  Arithmetic series   arisum 16029
            5.10.9  Geometric series   expcnv 16033
            5.10.10  Ratio test for infinite series convergence   cvgrat 16052
            5.10.11  Mertens' theorem   mertenslem1 16053
            5.10.12  Finite and infinite products   prodf 16056
                  5.10.12.1  Product sequences   prodf 16056
                  5.10.12.2  Non-trivial convergence   ntrivcvg 16066
                  5.10.12.3  Complex products   cprod 16072
                  5.10.12.4  Finite products   fprod 16108
                  5.10.12.5  Infinite products   iprodclim 16165
            5.10.13  Falling and Rising Factorial   cfallfac 16171
            5.10.14  Bernoulli polynomials and sums of k-th powers   cbp 16212
      ​5.11  Elementary trigonometry
            5.11.1  The exponential, sine, and cosine functions   ce 16227
                  5.11.1.1  The circle constant (tau = 2 pi)   ctau 16370
            5.11.2  _e is irrational   eirrlem 16372
      ​5.12  Cardinality of real and complex number subsets
            5.12.1  Countability of integers and rationals   xpnnen 16379
            5.12.2  The reals are uncountable   rpnnen2lem1 16382
​​*PART 6  ELEMENTARY NUMBER THEORY
      ​6.1  Elementary properties of divisibility
            6.1.1  Irrationality of square root of 2   sqrt2irrlem 16416
            6.1.2  Some Number sets are chains of proper subsets   nthruc 16420
            6.1.3  The divides relation   cdvds 16422
            ​*6.1.4  Even and odd numbers   evenelz 16506
            6.1.5  The division algorithm   divalglem0 16563
            6.1.6  Bit sequences   cbits 16589
            6.1.7  The greatest common divisor operator   cgcd 16664
            6.1.8  Bézout's identity   bezoutlem1 16712
            6.1.9  Algorithms   nn0seqcvgd 16745
            6.1.10  Euclid's Algorithm   eucalgval2 16756
            ​*6.1.11  The least common multiple   clcm 16763
            ​*6.1.12  Coprimality and Euclid's lemma   coprmgcdb 16824
            6.1.13  Cancellability of congruences   congr 16839
      ​6.2  Elementary prime number theory
            ​*6.2.1  Elementary properties   cprime 16846
            ​*6.2.2  Coprimality and Euclid's lemma (cont.)   coprm 16887
            6.2.3  Properties of the canonical representation of a rational   cnumer 16909
            6.2.4  Euler's theorem   codz 16940
            6.2.5  Arithmetic modulo a prime number   modprm1div 16975
            6.2.6  Pythagorean Triples   coprimeprodsq 16986
            6.2.7  The prime count function   cpc 17014
            6.2.8  Pocklington's theorem   prmpwdvds 17082
            6.2.9  Infinite primes theorem   unbenlem 17086
            6.2.10  Sum of prime reciprocals   prmreclem1 17094
            6.2.11  Fundamental theorem of arithmetic   1arithlem1 17101
            6.2.12  Lagrange's four-square theorem   cgz 17107
            6.2.13  Van der Waerden's theorem   cvdwa 17143
            6.2.14  Ramsey's theorem   cram 17177
            ​*6.2.15  Primorial function   cprmo 17209
            ​*6.2.16  Prime gaps   prmgaplem1 17227
            6.2.17  Decimal arithmetic (cont.)   dec2dvds 17241
            6.2.18  Cyclical shifts of words (cont.)   cshwsidrepsw 17271
            6.2.19  Specific prime numbers   prmlem0 17283
            6.2.20  Very large primes   1259lem1 17309
​PART 7  BASIC STRUCTURES
      ​7.1  Extensible structures
            ​*7.1.1  Basic definitions   cstr 17324
                  7.1.1.1  Extensible structures as structures with components   cstr 17324
                  7.1.1.2  Substitution of components   csts 17341
                  7.1.1.3  Slots   cslot 17359
                  ​*7.1.1.4  Structure component indices   cnx 17371
                  7.1.1.5  Base sets   cbs 17387
                  7.1.1.6  Base set restrictions   cress 17408
            7.1.2  Slot definitions   cplusg 17428
            7.1.3  Definition of the structure product   crest 17591
            7.1.4  Definition of the structure quotient   cordt 17671
      ​7.2  Moore spaces
            7.2.1  Moore closures   mrcflem 17780
            7.2.2  Independent sets in a Moore system   mrisval 17804
            7.2.3  Algebraic closure systems   isacs 17825
​PART 8  BASIC CATEGORY THEORY
      ​8.1  Categories
            8.1.1  Categories   ccat 17838
            8.1.2  Opposite category   coppc 17885
            8.1.3  Monomorphisms and epimorphisms   cmon 17903
            8.1.4  Sections, inverses, isomorphisms   csect 17919
            ​*8.1.5  Isomorphic objects   ccic 17970
            8.1.6  Subcategories   cssc 17982
            8.1.7  Functors   cfunc 18029
            8.1.8  Full & faithful functors   cful 18079
            8.1.9  Natural transformations and the functor category   cnat 18119
            8.1.10  Initial, terminal and zero objects of a category   cinito 18156
      ​8.2  Arrows (disjointified hom-sets)
            8.2.1  Identity and composition for arrows   cida 18228
      ​8.3  Examples of categories
            8.3.1  The category of sets   csetc 18250
            8.3.2  The category of categories   ccatc 18273
            ​*8.3.3  The category of extensible structures   fncnvimaeqv 18294
      ​8.4  Categorical constructions
            8.4.1  Product of categories   cxpc 18342
            8.4.2  Functor evaluation   cevlf 18383
            8.4.3  Hom functor   chof 18422
​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 18605
            9.5.2  Complete lattices   ccla 18672
            9.5.3  Distributive lattices   cdlat 18694
            9.5.4  Subset order structures   cipo 18701
      ​9.6  Posets, directed sets, and lattices as relations
            ​*9.6.1  Posets and lattices as relations   cps 18738
            9.6.2  Directed sets, nets   cdir 18768
      ​9.7  Chains
​PART 10  BASIC ALGEBRAIC STRUCTURES
      ​10.1  Monoids
            ​*10.1.1  Magmas   cplusf 18813
            ​*10.1.2  Identity elements   mgmidmo 18838
            ​*10.1.3  Iterated sums in a magma   gsumvalx 18865
            10.1.4  Magma homomorphisms and submagmas   cmgmhm 18879
            ​*10.1.5  Semigroups   csgrp 18907
            ​*10.1.6  Definition and basic properties of monoids   cmnd 18923
            10.1.7  Monoid homomorphisms and submonoids   cmhm 18976
            ​*10.1.8  Iterated sums in a monoid   gsumvallem2 19030
            10.1.9  Free monoids   cfrmd 19043
                  ​*10.1.9.1  Monoid of endofunctions   cefmnd 19064
            10.1.10  Examples and counterexamples for magmas, semigroups and monoids   mgm2nsgrplem1 19117
      ​10.2  Groups
            10.2.1  Definition and basic properties   cgrp 19144
            ​*10.2.2  Group multiple operation   cmg 19277
            10.2.3  Subgroups and Quotient groups   csubg 19330
            ​*10.2.4  Cyclic monoids and groups   cycsubmel 19415
            10.2.5  Elementary theory of group homomorphisms   cghm 19427
            10.2.6  Isomorphisms of groups   cgim 19471
                  10.2.6.1  The first isomorphism theorem of groups   ghmqusnsglem1 19494
            10.2.7  Group actions   cga 19503
            10.2.8  Centralizers and centers   ccntz 19529
            10.2.9  The opposite group   coppg 19559
            10.2.10  Symmetric groups   csymg 19583
                  ​*10.2.10.1  Definition and basic properties   csymg 19583
                  10.2.10.2  Cayley's theorem   cayleylem1 19626
                  10.2.10.3  Permutations fixing one element   symgfix2 19630
                  ​*10.2.10.4  Transpositions in the symmetric group   cpmtr 19655
                  10.2.10.5  The sign of a permutation   cpsgn 19703
            10.2.11  p-Groups and Sylow groups; Sylow's theorems   cod 19738
            10.2.12  Direct products   clsm 19848
                  10.2.12.1  Direct products (extension)   smndlsmidm 19870
            10.2.13  Free groups   cefg 19920
            10.2.14  Abelian groups   ccmn 19994
                  10.2.14.1  Definition and basic properties   ccmn 19994
                  10.2.14.2  Cyclic groups   ccyg 20091
                  10.2.14.3  Group sum operation   gsumval3a 20117
                  10.2.14.4  Group sums over (ranges of) integers   fsfnn0gsumfsffz 20197
                  10.2.14.5  Internal direct products   cdprd 20209
                  10.2.14.6  The Fundamental Theorem of Abelian Groups   ablfacrplem 20281
            10.2.15  Simple groups   csimpg 20306
                  10.2.15.1  Definition and basic properties   csimpg 20306
                  10.2.15.2  Classification of abelian simple groups   ablsimpnosubgd 20320
            10.2.16  Totally ordered monoids and groups   comnd 20333
      ​10.3  Rings
            10.3.1  Multiplicative Group   cmgp 20360
            ​*10.3.2  Non-unital rings ("rngs")   crng 20374
            ​*10.3.3  Ring unity (multiplicative identity)   cur 20407
            10.3.4  Semirings   csrg 20412
                  ​*10.3.4.1  The binomial theorem for semirings   srgbinomlem1 20452
            10.3.5  Unital rings   crg 20459
            10.3.6  Opposite ring   coppr 20566
            10.3.7  Divisibility   cdsr 20584
            10.3.8  Ring primes   crpm 20662
            10.3.9  Homomorphisms of non-unital rings   crnghm 20664
            10.3.10  Ring homomorphisms   crh 20699
            10.3.11  Nonzero rings and zero rings   cnzr 20762
            10.3.12  Local rings   clring 20790
            10.3.13  Subrings   csubrng 20797
                  10.3.13.1  Subrings of non-unital rings   csubrng 20797
                  10.3.13.2  Subrings of unital rings   csubrg 20821
                  10.3.13.3  Subrings generated by a subset   crgspn 20862
            10.3.14  Categories of rings   crngc 20868
                  ​*10.3.14.1  The category of non-unital rings   crngc 20868
                  ​*10.3.14.2  The category of (unital) rings   cringc 20897
                  10.3.14.3  Subcategories of the category of rings   srhmsubclem1 20929
            10.3.15  Left regular elements and domains   crlreg 20943
      ​10.4  Division rings and fields
            10.4.1  Definition and basic properties   cdr 20980
            10.4.2  Sub-division rings   csdrg 21043
            10.4.3  Absolute value (abstract algebra)   cabv 21065
            10.4.4  Star rings   cstf 21094
            10.4.5  Totally ordered rings and fields   corng 21114
      ​10.5  Left modules
            10.5.1  Definition and basic properties   clmod 21135
            10.5.2  Subspaces and spans in a left module   clss 21206
            10.5.3  Homomorphisms and isomorphisms of left modules   clmhm 21294
            10.5.4  Subspace sum; bases for a left module   clbs 21349
      ​10.6  Vector spaces
            10.6.1  Definition and basic properties   clvec 21377
      ​10.7  Subring algebras and ideals
            10.7.1  Subring algebras   csra 21446
            ​*10.7.2  Left ideals and spans   clidl 21484
            10.7.3  Two-sided ideals and quotient rings   c2idl 21542
                  ​*10.7.3.1  Condition for a non-unital ring to be unital   rngqiprng1elbas 21582
                  10.7.3.2  Prime Ideals   cprmidl 21616
            10.7.4  Principal ideal rings. Divisibility in the integers   clpidl 21644
            10.7.5  Principal ideal domains   cpid 21660
      ​10.8  The complex numbers as an algebraic extensible structure
            10.8.1  Definition and basic properties   cpsmet 21662
            ​*10.8.2  Ring of integers   czring 21752
                  ​*10.8.2.1  Example for a condition for a non-unital ring to be unital   pzriprnglem1 21787
            10.8.3  Algebraic constructions based on the complex numbers   czrh 21805
            10.8.4  Signs as subgroup of the complex numbers   cnmsgnsubg 21883
            10.8.5  Embedding of permutation signs into a ring   zrhpsgnmhm 21890
            10.8.6  The ordered field of real numbers   crefld 21910
      ​10.9  Generalized pre-Hilbert and Hilbert spaces
            10.9.1  Definition and basic properties   cphl 21930
            10.9.2  Orthocomplements and closed subspaces   cocv 21966
            10.9.3  Orthogonal projection and orthonormal bases   cpj 22006
​​*PART 11  BASIC LINEAR ALGEBRA
      ​11.1  Vectors and free modules
            *11.1.1  Direct sum of left modules   cdsmm 22037
            ​*11.1.2  Free modules   cfrlm 22052
            ​*11.1.3  Standard basis (unit vectors)   cuvc 22088
            ​*11.1.4  Independent sets and families   clindf 22110
            11.1.5  Characterization of free modules   lmimlbs 22142
      ​11.2  Associative algebras
            11.2.1  Definition and basic properties   casa 22158
      ​11.3  Abstract multivariate polynomials
            11.3.1  Definition and basic properties   cmps 22212
            11.3.2  Polynomial evaluation   ces 22381
            11.3.3  The "variable selection" function   cslv 22425
            11.3.4  Additional definitions for (multivariate) polynomials   cmhp 22454
            ​*11.3.5  Univariate polynomials   cps1 22493
            11.3.6  Univariate polynomial evaluation   ces1 22631
                  11.3.6.1  Specialization of polynomial evaluation as a ring homomorphism   evls1scafv 22684
      ​​*11.4  Matrices
            *11.4.1  The matrix multiplication   cmmul 22705
            ​*11.4.2  Square matrices   cmat 22722
            ​*11.4.3  The matrix algebra   matmulr 22753
            ​*11.4.4  Matrices of dimension 0 and 1   mat0dimbas0 22781
            ​*11.4.5  The subalgebras of diagonal and scalar matrices   cdmat 22803
            ​*11.4.6  Multiplication of a matrix with a "column vector"   cmvmul 22855
            11.4.7  Replacement functions for a square matrix   cmarrep 22871
            11.4.8  Submatrices   csubma 22891
      ​11.5  The determinant
            11.5.1  Definition and basic properties   cmdat 22899
            11.5.2  Determinants of 2 x 2 -matrices   m2detleiblem1 22939
            11.5.3  The matrix adjugate/adjunct   cmadu 22947
            ​*11.5.4  Laplace expansion of determinants (special case)   symgmatr01lem 22968
            11.5.5  Inverse matrix   invrvald 22991
            ​*11.5.6  Cramer's rule   slesolvec 22997
      ​​*11.6  Polynomial matrices
            11.6.1  Basic properties   pmatring 23010
            ​*11.6.2  Constant polynomial matrices   ccpmat 23021
            ​*11.6.3  Collecting coefficients of polynomial matrices   cdecpmat 23080
            ​*11.6.4  Ring isomorphism between polynomial matrices and polynomials over matrices   cpm2mp 23110
      ​​*11.7  The characteristic polynomial
            *11.7.1  Definition and basic properties   cchpmat 23144
            ​*11.7.2  The characteristic factor function G   fvmptnn04if 23167
            ​*11.7.3  The Cayley-Hamilton theorem   cpmadurid 23185
​PART 12  BASIC TOPOLOGY
      ​12.1  Topology
            ​*12.1.1  Topological spaces   ctop 23211
                  12.1.1.1  Topologies   ctop 23211
                  12.1.1.2  Topologies on sets   ctopon 23228
                  12.1.1.3  Topological spaces   ctps 23250
            12.1.2  Topological bases   ctb 23263
            12.1.3  Examples of topologies   distop 23313
            12.1.4  Closure and interior   ccld 23334
            12.1.5  Neighborhoods   cnei 23415
            12.1.6  Limit points and perfect sets   clp 23452
            12.1.7  Subspace topologies   restrcl 23475
            12.1.8  Order topology   ordtbaslem 23506
            12.1.9  Limits and continuity in topological spaces   ccn 23542
            12.1.10  Separated spaces: T0, T1, T2 (Hausdorff) ...   ct0 23624
            12.1.11  Compactness   ccmp 23704
            12.1.12  Bolzano-Weierstrass theorem   bwth 23728
            12.1.13  Connectedness   cconn 23729
            12.1.14  First- and second-countability   c1stc 23755
            12.1.15  Local topological properties   clly 23783
            12.1.16  Refinements   cref 23821
            12.1.17  Compactly generated spaces   ckgen 23852
            12.1.18  Product topologies   ctx 23879
            12.1.19  Continuous function-builders   cnmptid 23980
            12.1.20  Quotient maps and quotient topology   ckq 24012
            12.1.21  Homeomorphisms   chmeo 24072
      ​12.2  Filters and filter bases
            12.2.1  Filter bases   elmptrab 24146
            12.2.2  Filters   cfil 24164
            12.2.3  Ultrafilters   cufil 24218
            12.2.4  Filter limits   cfm 24252
            12.2.5  Extension by continuity   ccnext 24378
            12.2.6  Topological groups   ctmd 24389
            12.2.7  Infinite group sum on topological groups   ctsu 24445
            12.2.8  Topological rings, fields, vector spaces   ctrg 24475
      ​12.3  Uniform Structures and Spaces
            12.3.1  Uniform structures   cust 24519
            12.3.2  The topology induced by an uniform structure   cutop 24549
            12.3.3  Uniform Spaces   cuss 24572
            12.3.4  Uniform continuity   cucn 24593
            12.3.5  Cauchy filters in uniform spaces   ccfilu 24604
            12.3.6  Complete uniform spaces   ccusp 24615
      ​12.4  Metric spaces
            12.4.1  Pseudometric spaces   ispsmet 24623
            12.4.2  Basic metric space properties   cxms 24636
            12.4.3  Metric space balls   blfvalps 24702
            12.4.4  Open sets of a metric space   mopnval 24757
            12.4.5  Continuity in metric spaces   metcnp3 24859
            12.4.6  The uniform structure generated by a metric   metuval 24868
            12.4.7  Examples of metric spaces   dscmet 24891
            ​*12.4.8  Normed algebraic structures   cnm 24895
            12.4.9  Normed space homomorphisms (bounded linear operators)   cnmo 25024
            12.4.10  Topology on the reals   qtopbaslem 25077
            12.4.11  Topological definitions using the reals   cii 25196
            12.4.12  Path homotopy   chtpy 25288
            12.4.13  The fundamental group   cpco 25321
      ​12.5  Metric subcomplex vector spaces
            12.5.1  Subcomplex modules   cclm 25383
            ​*12.5.2  Subcomplex vector spaces   ccvs 25444
            ​*12.5.3  Normed subcomplex vector spaces   isncvsngp 25470
            12.5.4  Subcomplex pre-Hilbert spaces   ccph 25487
            12.5.5  Convergence and completeness   ccfil 25573
            12.5.6  Baire's Category Theorem   bcthlem1 25645
            12.5.7  Banach spaces and subcomplex Hilbert spaces   ccms 25653
                  12.5.7.1  The complete ordered field of the real numbers   retopn 25700
            12.5.8  Euclidean spaces   crrx 25704
            12.5.9  Minimizing Vector Theorem   minveclem1 25745
            12.5.10  Projection Theorem   pjthlem1 25758
​PART 13  BASIC REAL AND COMPLEX ANALYSIS
      ​13.1  Continuity
            13.1.1  Intermediate value theorem   pmltpclem1 25769
      ​13.2  Integrals
            13.2.1  Lebesgue measure   covol 25783
            13.2.2  Lebesgue integration   cmbf 25935
                  13.2.2.1  Lesbesgue integral   cmbf 25935
                  13.2.2.2  Lesbesgue directed integral   cdit 26166
      ​13.3  Derivatives
            13.3.1  Real and complex differentiation   climc 26182
                  13.3.1.1  Derivatives of functions of one complex or real variable   climc 26182
                  13.3.1.2  Results on real differentiation   dvferm1lem 26304
​PART 14  BASIC REAL AND COMPLEX FUNCTIONS
      ​14.1  Polynomials
            14.1.1  Polynomial degrees   cmdg 26371
            14.1.2  The division algorithm for univariate polynomials   cmn1 26444
            14.1.3  Elementary properties of complex polynomials   cply 26502
            14.1.4  The division algorithm for polynomials   cquot 26611
            14.1.5  Algebraic numbers   caa 26637
            14.1.6  Liouville's approximation theorem   aalioulem1 26659
      ​14.2  Sequences and series
            14.2.1  Taylor polynomials and Taylor's theorem   ctayl 26680
            14.2.2  Uniform convergence   culm 26703
            14.2.3  Power series   pserval 26737
      ​14.3  Basic trigonometry
            14.3.1  The exponential, sine, and cosine functions (cont.)   efcn 26770
            14.3.2  Properties of pi = 3.14159...   pilem1 26778
            14.3.3  Mapping of the exponential function   efgh 26869
            14.3.4  The natural logarithm on complex numbers   clog 26882
            ​*14.3.5  Logarithms to an arbitrary base   clogb 27092
            14.3.6  Theorems of Pythagoras, isosceles triangles, and intersecting chords   angval 27129
            14.3.7  Solutions of quadratic, cubic, and quartic equations   quad2 27167
            14.3.8  Inverse trigonometric functions   casin 27190
            14.3.9  The Birthday Problem   log2ublem1 27274
            14.3.10  Areas in R^2   carea 27283
            14.3.11  More miscellaneous converging sequences   rlimcnp 27293
            14.3.12  Inequality of arithmetic and geometric means   cvxcl 27312
            14.3.13  Euler-Mascheroni constant   cem 27319
            14.3.14  Zeta function   czeta 27340
            14.3.15  Gamma function   clgam 27343
      ​14.4  Basic number theory
            14.4.1  Wilson's theorem   wilthlem1 27395
            14.4.2  The Fundamental Theorem of Algebra   ftalem1 27400
            14.4.3  The Basel problem (ζ(2) = π²/6)   basellem1 27408
            14.4.4  Number-theoretical functions   ccht 27418
            14.4.5  Perfect Number Theorem   mersenne 27554
            14.4.6  Characters of Z/nZ   cdchr 27559
            14.4.7  Bertrand's postulate   bcctr 27602
            ​*14.4.8  Quadratic residues and the Legendre symbol   clgs 27621
            ​*14.4.9  Gauss' Lemma   gausslemma2dlem0a 27683
            14.4.10  Quadratic reciprocity   lgseisenlem1 27702
            14.4.11  All primes 4n+1 are the sum of two squares   2sqlem1 27744
            14.4.12  Chebyshev's Weak Prime Number Theorem, Dirichlet's Theorem   chebbnd1lem1 27796
            14.4.13  The Prime Number Theorem   mudivsum 27857
            14.4.14  Ostrowski's theorem   abvcxp 27942
            14.4.15  Fermat's Last Theorem   infdesc 27967
                  14.4.15.1  Basic reductions for Fermat's Last Theorem   infdesc 27967
                  ​*14.4.15.2  Proof of Fermat's Last Theorem (work in progress)   fltoprmlem1 27993
​PART 15  SURREAL NUMBERS
      ​​*15.1  Sign sequence representation and Alling's axioms
            15.1.1  Definitions and initial properties   csur 27997
            15.1.2  Ordering   ltssolem1 28032
            15.1.3  Birthday Function   bdayfo 28034
            15.1.4  Density   fvnobday 28035
            ​*15.1.5  Full-Eta Property   bdayimaon 28050
      ​15.2  Initial consequences of Alling's axioms
            15.2.1  Ordering Theorems   cles 28101
            15.2.2  Birthday Theorems   bdayfun 28133
      ​​*15.3  Conway cut representation
            15.3.1  Conway cuts   cslts 28143
            15.3.2  Zero and One   c0s 28191
            15.3.3  Cuts and Options   cmade 28208
            15.3.4  Cofinality and coinitiality   cofslts 28304
      ​15.4  Induction and recursion
            15.4.1  Induction and recursion on one variable   cnorec 28323
            15.4.2  Induction and recursion on two variables   cnorec2 28334
      ​15.5  Surreal arithmetic
            15.5.1  Addition   cadds 28345
            15.5.2  Negation and Subtraction   cnegs 28405
            15.5.3  Multiplication   cmuls 28492
            15.5.4  Division   cdivs 28573
            15.5.5  Absolute value   cabss 28623
      ​15.6  Subsystems of surreals
            15.6.1  Ordinal numbers   cons 28637
            15.6.2  Surreal recursive sequences   cseqs 28669
            15.6.3  Natural numbers   cn0s 28698
            15.6.4  Integers   czs 28764
            15.6.5  Dyadic fractions   c2s 28796
            15.6.6  Real numbers   creno 28875
​​*PART 16  ELEMENTARY GEOMETRY
      ​16.1  Definition and Tarski's Axioms of Geometry
            16.1.1  Justification for the congruence notation   tgjustf 28935
      ​16.2  Tarskian Geometry
            16.2.1  Congruence   tgcgrcomimp 28939
            16.2.2  Betweenness   tgbtwntriv2 28950
            16.2.3  Dimension   tglowdim1 28963
            16.2.4  Betweenness and Congruence   tgifscgr 28971
            16.2.5  Congruence of a series of points   ccgrg 28973
            16.2.6  Motions   cismt 28995
            16.2.7  Colinearity   tglng 29009
            16.2.8  Connectivity of betweenness   tgbtwnconn1lem1 29035
            16.2.9  Less-than relation in geometric congruences   cleg 29045
            16.2.10  Rays   chlg 29063
            16.2.11  Lines   btwnlng1 29087
            16.2.12  Point inversions   cmir 29124
            16.2.13  Right angles   crag 29168
            16.2.14  Half-planes   islnopp 29215
            16.2.15  Planes   cplng 29251
            16.2.16  Midpoints and Line Mirroring   cmid 29277
            16.2.17  Congruence of angles   ccgra 29314
            16.2.18  Angle Comparisons   cinag 29354
            16.2.19  Angle Addition   cangmgm 29373
            16.2.20  Congruence Theorems   tgsas1 29399
            16.2.21  Equilateral triangles   ceqlg 29410
            16.2.22  Parallel lines   cprlng 29414
      ​16.3  Properties of geometries
            16.3.1  Isomorphisms between geometries   f1otrgds 29446
      ​16.4  Geometry in Hilbert spaces
            16.4.1  Geometry in the complex plane   cchhllem 29464
            16.4.2  Geometry in Euclidean spaces   cee 29465
                  16.4.2.1  Definition of the Euclidean space   cee 29465
                  16.4.2.2  Tarski's axioms for geometry for the Euclidean space   axdimuniq 29491
                  16.4.2.3  EE^n fulfills Tarski's Axioms   ceeng 29555
​​*PART 17  GRAPH THEORY
      ​*17.1  Vertices and edges
            17.1.1  The edge function extractor for extensible structures   cedgf 29566
            ​*17.1.2  Vertices and indexed edges   cvtx 29574
                  17.1.2.1  Definitions and basic properties   cvtx 29574
                  17.1.2.2  The vertices and edges of a graph represented as ordered pair   opvtxval 29581
                  17.1.2.3  The vertices and edges of a graph represented as extensible structure   funvtxdmge2val 29589
                  17.1.2.4  Representations of graphs without edges   snstrvtxval 29615
                  17.1.2.5  Degenerated cases of representations of graphs   vtxval0 29617
            17.1.3  Edges as range of the edge function   cedg 29625
      ​​*17.2  Undirected graphs
            17.2.1  Undirected hypergraphs   cuhgr 29634
            17.2.2  Undirected pseudographs and multigraphs   cupgr 29658
            ​*17.2.3  Loop-free graphs   umgrislfupgrlem 29700
            17.2.4  Edges as subsets of vertices of graphs   uhgredgiedgb 29704
            ​*17.2.5  Undirected simple graphs   cuspgr 29729
            17.2.6  Examples for graphs   usgr0e 29817
            17.2.7  Subgraphs   csubgr 29848
            17.2.8  Finite undirected simple graphs   cfusgr 29897
            17.2.9  Neighbors, complete graphs and universal vertices   cnbgr 29913
                  17.2.9.1  Neighbors   cnbgr 29913
                  17.2.9.2  Universal vertices   cuvtx 29966
                  17.2.9.3  Complete graphs   ccplgr 29990
            17.2.10  Vertex degree   cvtxdg 30046
            ​*17.2.11  Regular graphs   crgr 30136
      ​​*17.3  Walks, paths and cycles
            *17.3.1  Walks   cewlks 30176
            17.3.2  Walks for loop-free graphs   lfgrwlkprop 30270
            17.3.3  Trails   ctrls 30273
            17.3.4  Paths and simple paths   cpths 30295
            17.3.5  Closed walks   cclwlks 30357
            17.3.6  Circuits and cycles   ccrcts 30371
            ​*17.3.7  Walks as words   cwwlks 30414
            17.3.8  Walks/paths of length 2 (as length 3 strings)   2wlkdlem1 30514
            17.3.9  Walks in regular graphs   rusgrnumwwlkl1 30560
            ​*17.3.10  Closed walks as words   cclwwlk 30572
                  17.3.10.1  Closed walks as words   cclwwlk 30572
                  17.3.10.2  Closed walks of a fixed length as words   cclwwlkn 30615
                  17.3.10.3  Closed walks on a vertex of a fixed length as words   cclwwlknon 30678
            17.3.11  Examples for walks, trails and paths   0ewlk 30705
            17.3.12  Acyclic graphs   cacycgr 30748
            17.3.13  Connected graphs   cconngr 30787
      ​17.4  Eulerian paths and the Konigsberg Bridge problem
            ​*17.4.1  Eulerian paths   ceupth 30798
            ​*17.4.2  The Königsberg Bridge problem   konigsbergvtx 30847
      ​17.5  The Friendship Theorem
            17.5.1  Friendship graphs - basics   cfrgr 30859
            17.5.2  The friendship theorem for small graphs   frgr1v 30872
            17.5.3  Theorems according to Mertzios and Unger   2pthfrgrrn 30883
            ​*17.5.4  Huneke's Proof of the Friendship Theorem   frgrncvvdeqlem1 30900
​PART 18  GUIDES AND MISCELLANEA
      ​18.1  Guides (conventions, explanations, and examples)
            ​*18.1.1  Conventions   conventions 31001
            18.1.2  Natural deduction   natded 31004
            ​*18.1.3  Natural deduction examples   ex-natded5.2 31005
            18.1.4  Definitional examples   ex-or 31022
            18.1.5  Other examples   aevdemo 31061
      ​18.2  Humor
            18.2.1  April Fool's theorem   avril1 31064
      ​18.3  (Future - to be reviewed and classified)
            18.3.1  Planar incidence geometry   cplig 31076
            ​*18.3.2  Aliases kept to prevent broken links   dummylink 31089
​​*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 31091
            19.1.2  Abelian groups   cablo 31146
      ​19.2  Complex vector spaces
            19.2.1  Definition and basic properties   cvc 31160
            19.2.2  Examples of complex vector spaces   cnaddabloOLD 31183
      ​19.3  Normed complex vector spaces
            19.3.1  Definition and basic properties   cnv 31186
            19.3.2  Examples of normed complex vector spaces   cnnv 31279
            19.3.3  Induced metric of a normed complex vector space   imsval 31287
            19.3.4  Inner product   cdip 31302
            19.3.5  Subspaces   css 31323
      ​19.4  Operators on complex vector spaces
            19.4.1  Definitions and basic properties   clno 31342
      ​19.5  Inner product (pre-Hilbert) spaces
            19.5.1  Definition and basic properties   ccphlo 31414
            19.5.2  Examples of pre-Hilbert spaces   cncph 31421
            19.5.3  Properties of pre-Hilbert spaces   isph 31424
      ​19.6  Complex Banach spaces
            19.6.1  Definition and basic properties   ccbn 31464
            19.6.2  Examples of complex Banach spaces   cnbn 31471
            19.6.3  Uniform Boundedness Theorem   ubthlem1 31472
            19.6.4  Minimizing Vector Theorem   minvecolem1 31476
      ​19.7  Complex Hilbert spaces
            19.7.1  Definition and basic properties   chlo 31487
            19.7.2  Standard axioms for a complex Hilbert space   hlex 31500
            19.7.3  Examples of complex Hilbert spaces   cnchl 31518
            19.7.4  Hellinger-Toeplitz Theorem   htthlem 31519
​​*PART 20  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      ​20.1  Axiomatization of complex pre-Hilbert spaces
            20.1.1  Basic Hilbert space definitions   chba 31521
            20.1.2  Preliminary ZFC lemmas   df-hnorm 31570
            ​*20.1.3  Derive the Hilbert space axioms from ZFC set theory   axhilex-zf 31583
            ​*20.1.4  Introduce the vector space axioms for a Hilbert space   ax-hilex 31601
            20.1.5  Vector operations   hvmulex 31613
            20.1.6  Inner product postulates for a Hilbert space   ax-hfi 31681
      ​20.2  Inner product and norms
            20.2.1  Inner product   his5 31688
            20.2.2  Norms   dfhnorm2 31724
            20.2.3  Relate Hilbert space to normed complex vector spaces   hilablo 31762
            20.2.4  Bunjakovaskij-Cauchy-Schwarz inequality   bcsiALT 31781
      ​20.3  Cauchy sequences and completeness axiom
            20.3.1  Cauchy sequences and limits   hcau 31786
            20.3.2  Derivation of the completeness axiom from ZF set theory   hilmet 31796
            20.3.3  Completeness postulate for a Hilbert space   ax-hcompl 31804
            20.3.4  Relate Hilbert space to ZFC pre-Hilbert and Hilbert spaces   hhcms 31805
      ​20.4  Subspaces and projections
            20.4.1  Subspaces   df-sh 31809
            20.4.2  Closed subspaces   df-ch 31823
            20.4.3  Orthocomplements   df-oc 31854
            20.4.4  Subspace sum, span, lattice join, lattice supremum   df-shs 31910
            20.4.5  Projection theorem   pjhthlem1 31993
            20.4.6  Projectors   df-pjh 31997
      ​20.5  Properties of Hilbert subspaces
            20.5.1  Orthomodular law   omlsilem 32004
            20.5.2  Projectors (cont.)   pjhtheu2 32018
            20.5.3  Hilbert lattice operations   sh0le 32042
            20.5.4  Span (cont.) and one-dimensional subspaces   spansn0 32143
            20.5.5  Commutes relation for Hilbert lattice elements   df-cm 32185
            20.5.6  Foulis-Holland theorem   fh1 32220
            20.5.7  Quantum Logic Explorer axioms   qlax1i 32229
            20.5.8  Orthogonal subspaces   chscllem1 32239
            20.5.9  Orthoarguesian laws 5OA and 3OA   5oalem1 32256
            20.5.10  Projectors (cont.)   pjorthi 32271
            20.5.11  Mayet's equation E_3   mayete3i 32330
      ​20.6  Operators on Hilbert spaces
            ​*20.6.1  Operator sum, difference, and scalar multiplication   df-hosum 32332
            20.6.2  Zero and identity operators   df-h0op 32350
            20.6.3  Operations on Hilbert space operators   hoaddcl 32360
            20.6.4  Linear, continuous, bounded, Hermitian, unitary operators and norms   df-nmop 32441
            20.6.5  Linear and continuous functionals and norms   df-nmfn 32447
            20.6.6  Adjoint   df-adjh 32451
            20.6.7  Dirac bra-ket notation   df-bra 32452
            20.6.8  Positive operators   df-leop 32454
            20.6.9  Eigenvectors, eigenvalues, spectrum   df-eigvec 32455
            20.6.10  Theorems about operators and functionals   nmopval 32458
            20.6.11  Riesz lemma   riesz3i 32664
            20.6.12  Adjoints (cont.)   cnlnadjlem1 32669
            20.6.13  Quantum computation error bound theorem   unierri 32706
            20.6.14  Dirac bra-ket notation (cont.)   branmfn 32707
            20.6.15  Positive operators (cont.)   leopg 32724
            20.6.16  Projectors as operators   pjhmopi 32748
      ​20.7  States on a Hilbert lattice and Godowski's equation
            20.7.1  States on a Hilbert lattice   df-st 32813
            20.7.2  Godowski's equation   golem1 32873
      ​20.8  Cover relation, atoms, exchange axiom, and modular symmetry
            20.8.1  Covers relation; modular pairs   df-cv 32881
            20.8.2  Atoms   df-at 32940
            20.8.3  Superposition principle   superpos 32956
            20.8.4  Atoms, exchange and covering properties, atomicity   chcv1 32957
            20.8.5  Irreducibility   chirredlem1 32992
            20.8.6  Atoms (cont.)   atcvat3i 32998
            20.8.7  Modular symmetry   mdsymlem1 33005
​PART 21  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      ​21.1  Mathboxes for user contributions
            21.1.1  Mathbox guidelines   mathbox 33044
      ​21.2  Mathbox for Stefan Allan
      ​21.3  Mathbox for Thierry Arnoux
            21.3.1  Propositional Calculus - misc additions   ad11antr 33049
            21.3.2  Predicate Calculus   sbc2iedf 33062
                  21.3.2.1  Predicate Calculus - misc additions   sbc2iedf 33062
                  21.3.2.2  Restricted quantification - misc additions   ralcom4f 33064
                  21.3.2.3  Equality   eqtrb 33070
                  21.3.2.4  Double restricted existential uniqueness quantification   opsbc2ie 33072
                  21.3.2.5  Double restricted existential uniqueness quantification syntax   w2reu 33074
                  21.3.2.6  Substitution (without distinct variables) - misc additions   sbceqbidf 33083
                  21.3.2.7  Existential "at most one" - misc additions   mo5f 33085
                  21.3.2.8  Existential uniqueness - misc additions   reuxfrdf 33087
                  21.3.2.9  Restricted "at most one" - misc additions   rmoxfrd 33089
                  21.3.2.10  Restricted iota (description binder)   riotaeqbidva 33092
            21.3.3  General Set Theory   dmrab 33093
                  21.3.3.1  Class abstractions (a.k.a. class builders)   dmrab 33093
                  21.3.3.2  Image Sets   abrexdomjm 33103
                  21.3.3.3  Set relations and operations - misc additions   nelun 33109
                  21.3.3.4  Unordered pairs   elpreq 33124
                  21.3.3.5  Unordered triples   tpssg 33133
                  21.3.3.6  Conditional operator - misc additions   ifeqeqx 33138
                  21.3.3.7  Set union   uniinn0 33147
                  21.3.3.8  Indexed union - misc additions   cbviunf 33150
                  21.3.3.9  Indexed intersection - misc additions   iinabrex 33163
                  21.3.3.10  Disjointness - misc additions   disjnf 33164
            21.3.4  Relations and Functions   xpdisjres 33192
                  21.3.4.1  Relations - misc additions   xpdisjres 33192
                  21.3.4.2  Functions - misc additions   fconst7v 33214
                  21.3.4.3  Operations - misc additions   mpomptxf 33272
                  21.3.4.4  Support of a function   suppovss 33274
                  21.3.4.5  Explicit Functions with one or two points as a domain   cosnopne 33287
                  21.3.4.6  Isomorphisms - misc. additions   gtiso 33294
                  21.3.4.7  Disjointness (additional proof requiring functions)   disjdsct 33296
                  21.3.4.8  First and second members of an ordered pair - misc additions   df1stres 33297
                  21.3.4.9  Countable Sets   snct 33305
            21.3.5  Real and Complex Numbers   sgnval2 33327
                  21.3.5.1  Complex operations - misc. additions   creq0 33328
                  21.3.5.2  Ordering on reals - misc additions   lt2addrd 33342
                  21.3.5.3  Extended reals - misc additions   nn0mnfxrd 33343
                  21.3.5.4  Extended nonnegative integers - misc additions   xnn0gt0 33361
                  21.3.5.5  Real number intervals - misc additions   joiniooico 33366
                  21.3.5.6  Finite intervals of integers - misc additions   uzssico 33376
                  21.3.5.7  Half-open integer ranges - misc additions   iundisjfi 33388
                  21.3.5.8  The ` # ` (set size) function - misc additions   hashunif 33398
                  21.3.5.9  The greatest common divisor operator - misc. additions   elq2 33403
                  21.3.5.10  Integers   nn0split01 33409
                  21.3.5.11  Decimal numbers   dfdec100 33421
            21.3.6  Real and complex functions   sgnsgn 33422
                  21.3.6.1  Signum (sgn or sign) function - misc. additions   sgnsgn 33422
                  21.3.6.2  Integer powers - misc. additions   nexple 33424
                  21.3.6.3  Indicator Functions (continued)   indsumin 33428
            ​*21.3.7  Decimal expansion   cdp2 33437
                  ​*21.3.7.1  Decimal point   cdp 33454
                  21.3.7.2  Division in the extended real number system   cxdiv 33483
            21.3.8  Words over a set - misc additions   wrdres 33502
                  21.3.8.1  Splicing words (substring replacement)   splfv3 33519
                  21.3.8.2  Cyclic shift of words   1cshid 33520
            21.3.9  Extensible Structures   ressplusf 33524
                  21.3.9.1  Structure restriction operator   ressplusf 33524
                  21.3.9.2  Posets   ressprs 33527
                  21.3.9.3  Complete lattices   clatp0cl 33537
                  21.3.9.4  Order Theory   cmnt 33539
                  21.3.9.5  Extended reals Structure - misc additions   ax-xrssca 33565
                  21.3.9.6  The extended nonnegative real numbers commutative monoid   xrge00 33575
            21.3.10  Algebra   mndcld 33583
                  21.3.10.1  Monoids   mndcld 33583
                  21.3.10.2  Monoids Homomorphisms   abliso 33596
                  21.3.10.3  Groups - misc additions   grpidcld 33600
                  21.3.10.4  Abelian Groups - misc additions   ablcomd 33606
                  21.3.10.5  Finitely supported group sums - misc additions   gsumsubg 33607
                  21.3.10.6  Group or monoid sums over words   gsumwun 33637
                  21.3.10.7  Centralizers and centers - misc additions   cntzun 33640
                  21.3.10.8  The symmetric group   symgfcoeu 33643
                  21.3.10.9  Transpositions   pmtridf1o 33655
                  21.3.10.10  Permutation Signs   psgnid 33658
                  21.3.10.11  Permutation cycles   ctocyc 33667
                  21.3.10.12  The Alternating Group   evpmval 33706
                  21.3.10.13  Signum in an ordered monoid   csgns 33719
                  21.3.10.14  Fixed points   cfxp 33724
                  21.3.10.15  The Archimedean property for generic ordered algebraic structures   cinftm 33737
                  21.3.10.16  Semiring left modules   cslmd 33761
                  21.3.10.17  Simple groups   prmsimpcyc 33789
                  21.3.10.18  Rings - misc additions   ringrngd 33790
                  21.3.10.19  Subrings generated by a set   elrgspnlem1 33803
                  21.3.10.20  The zero ring   irrednzr 33811
                  21.3.10.21  Localization of rings   cerl 33814
                  21.3.10.22  Integral Domains   domnmuln0rd 33838
                  21.3.10.23  Euclidean Domains   ceuf 33852
                  21.3.10.24  Division Rings   rndrhmcl 33858
                  21.3.10.25  The field of rational numbers   qfld 33859
                  21.3.10.26  Subfields   subsdrg 33860
                  21.3.10.27  Field of fractions   cfrac 33864
                  21.3.10.28  Field extensions generated by a set   cfldgen 33872
                  21.3.10.29  Ring homomorphisms - misc additions   rhmdvd 33885
                  21.3.10.30  Scalar restriction operation   cresv 33887
                  21.3.10.31  The commutative ring of gaussian integers   gzcrng 33902
                  21.3.10.32  The archimedean ordered field of real numbers   cnfldfld 33903
                  21.3.10.33  The quotient map and quotient modules   qusker 33910
                  21.3.10.34  The ring of integers modulo ` N `   znfermltl 33922
                  21.3.10.35  Independent sets and families   islinds5 33923
                  21.3.10.36  Ring associates, ring units   dvdsruassoi 33939
                  ​*21.3.10.37  Subgroup sum / Sumset / Minkowski sum   elgrplsmsn 33945
                  21.3.10.38  The quotient map   quslsm 33956
                  21.3.10.39  Ideals   intlidl 33970
                  21.3.10.40  Maximal Ideals   cmxidl 33984
                  21.3.10.41  Local rings   drnglring 34024
                  21.3.10.42  The semiring of ideals of a ring   cidlsrg 34032
                  21.3.10.43  Prime Elements   rprmval 34048
                  21.3.10.44  Unique factorization domains   cufd 34070
                  21.3.10.45  The ring of integers   zringidom 34083
                  21.3.10.46  Associative Algebra   assaassd 34087
                  21.3.10.47  Univariate Polynomials   0ringmon1p 34089
                  21.3.10.48  Polynomial quotient and polynomial remainder   q1pdir 34135
                  21.3.10.49  Multivariate Polynomials   psrbasfsupp 34143
                  21.3.10.50  The ring of symmetric polynomials   csply 34187
                  21.3.10.51  The subring algebra   sra1r 34213
                  21.3.10.52  Division Ring Extensions   drgext0g 34222
                  21.3.10.53  Vector Spaces   lvecdimfi 34228
                  21.3.10.54  Vector Space Dimension   cldim 34231
            21.3.11  Field Extensions   cfldext 34270
                  21.3.11.1  Algebraic numbers   cirng 34315
                  21.3.11.2  Algebraic extensions   calgext 34327
                  21.3.11.3  Minimal polynomials   cminply 34331
                  21.3.11.4  Quadratic Field Extensions   rtelextdg2lem 34358
                  21.3.11.5  Towers of quadratic extentions   fldext2chn 34360
            ​*21.3.12  Constructible Numbers   cconstr 34361
                  21.3.12.1  Impossible constructions   2sqr3minply 34412
            21.3.13  Matrices   csmat 34425
                  21.3.13.1  Submatrices   csmat 34425
                  21.3.13.2  Matrix literals   clmat 34443
                  21.3.13.3  Laplace expansion of determinants   mdetpmtr1 34455
            21.3.14  Topology   ist0cld 34465
                  21.3.14.1  Open maps   txomap 34466
                  21.3.14.2  Topology of the unit circle   qtopt1 34467
                  21.3.14.3  Refinements   reff 34471
                  21.3.14.4  Open cover refinement property   ccref 34474
                  21.3.14.5  Lindelöf spaces   cldlf 34484
                  21.3.14.6  Paracompact spaces   cpcmp 34487
                  ​*21.3.14.7  Spectrum of a ring   crspec 34494
                  21.3.14.8  Pseudometrics   cmetid 34518
                  21.3.14.9  Continuity - misc additions   hauseqcn 34530
                  21.3.14.10  Topology of the closed unit interval   elunitge0 34531
                  21.3.14.11  Topology of ` ( RR X. RR ) `   unicls 34535
                  21.3.14.12  Order topology - misc. additions   cnvordtrestixx 34545
                  21.3.14.13  Continuity in topological spaces - misc. additions   mndpluscn 34558
                  21.3.14.14  Topology of the extended nonnegative real numbers ordered monoid   xrge0hmph 34564
                  21.3.14.15  Limits - misc additions   lmlim 34579
                  21.3.14.16  Univariate polynomials   pl1cn 34587
            21.3.15  Uniform Stuctures and Spaces   chcmp 34588
                  21.3.15.1  Hausdorff uniform completion   chcmp 34588
            21.3.16  Topology and algebraic structures   zringnm 34590
                  21.3.16.1  The norm on the ring of the integer numbers   zringnm 34590
                  21.3.16.2  Topological ` ZZ ` -modules   zlm0 34592
                  21.3.16.3  Canonical embedding of the field of the rational numbers into a division ring   cqqh 34602
                  21.3.16.4  Canonical embedding of the real numbers into a complete ordered field   crrh 34625
                  21.3.16.5  Embedding from the extended real numbers into a complete lattice   cxrh 34648
                  21.3.16.6  Canonical embeddings into the ordered field of the real numbers   zrhre 34651
                  ​*21.3.16.7  Topological Manifolds   cmntop 34654
                  21.3.16.8  Extended sum   cesum 34659
            21.3.17  Mixed Function/Constant operation   cofc 34727
            21.3.18  Abstract measure   csiga 34740
                  21.3.18.1  Sigma-Algebra   csiga 34740
                  21.3.18.2  Generated sigma-Algebra   csigagen 34771
                  ​*21.3.18.3  lambda and pi-Systems, Rings of Sets   ispisys 34785
                  21.3.18.4  The Borel algebra on the real numbers   cbrsiga 34814
                  21.3.18.5  Product Sigma-Algebra   csx 34821
                  21.3.18.6  Measures   cmeas 34828
                  21.3.18.7  The counting measure   cntmeas 34859
                  21.3.18.8  The Lebesgue measure - misc additions   voliune 34862
                  21.3.18.9  The Dirac delta measure   cdde 34865
                  21.3.18.10  The 'almost everywhere' relation   cae 34870
                  21.3.18.11  Measurable functions   cmbfm 34882
                  21.3.18.12  Borel Algebra on ` ( RR X. RR ) `   br2base 34901
                  ​*21.3.18.13  Caratheodory's extension theorem   coms 34923
            21.3.19  Integration   itgeq12dv 34958
                  21.3.19.1  Lebesgue integral - misc additions   itgeq12dv 34958
                  21.3.19.2  Bochner integral   citgm 34959
            21.3.20  Euler's partition theorem   oddpwdc 34986
            21.3.21  Sequences defined by strong recursion   csseq 35015
            21.3.22  Fibonacci Numbers   cfib 35028
            21.3.23  Probability   cprb 35039
                  21.3.23.1  Probability Theory   cprb 35039
                  21.3.23.2  Conditional Probabilities   ccprob 35063
                  21.3.23.3  Real-valued Random Variables   crrv 35072
                  21.3.23.4  Preimage set mapping operator   corvc 35088
                  21.3.23.5  Distribution Functions   orvcelval 35101
                  21.3.23.6  Cumulative Distribution Functions   orvclteel 35105
                  21.3.23.7  Probabilities - example   coinfliplem 35111
                  21.3.23.8  Bertrand's Ballot Problem   ballotlemoex 35118
            21.3.24  Signum (sgn or sign) function - misc. additions   fzssfzo 35171
                  21.3.24.1  Operations on words   ccatmulgnn0dir 35174
            21.3.25  Polynomials with real coefficients - misc additions   plyrecld 35178
            21.3.26  Descartes's rule of signs   signspval 35181
                  21.3.26.1  Sign changes in a word over real numbers   signspval 35181
                  21.3.26.2  Counting sign changes in a word over real numbers   signslema 35191
            21.3.27  Number Theory   iblidicc 35221
                  21.3.27.1  Representations of a number as sums of integers   crepr 35237
                  21.3.27.2  Vinogradov Trigonometric Sums and the Circle Method   cvts 35264
                  21.3.27.3  The Ternary Goldbach Conjecture: Final Statement   ax-hgt749 35273
            21.3.28  Elementary Geometry   cstrkg2d 35293
                  ​*21.3.28.1  Two-dimensional geometry   cstrkg2d 35293
                  21.3.28.2  Morley's Miracle   cgranbtwn 35298
                  21.3.28.3  Outer Five Segment (not used, no need to move to main)   cafs 35301
            ​*21.3.29  LeftPad Project   clpad 35306
      ​​*21.4  Mathbox for Jonathan Ben-Naim
            21.4.1  First-order logic and set theory   bnj170 35329
            21.4.2  Well founded induction and recursion   bnj110 35488
            21.4.3  The existence of a minimal element in certain classes   bnj69 35640
            21.4.4  Well-founded induction   bnj1204 35642
            21.4.5  Well-founded recursion, part 1 of 3   bnj60 35692
            21.4.6  Well-founded recursion, part 2 of 3   bnj1500 35698
            21.4.7  Well-founded recursion, part 3 of 3   bnj1522 35702
      ​21.5  Mathbox for BTernaryTau
            21.5.1  First-order logic   nfan1c 35703
                  21.5.1.1  Auxiliary axiom schemes   nfan1c 35703
            21.5.2  ZF set theory   inv2 35709
                  21.5.2.1  Ordinals 5 through 9   c5o 35761
                  21.5.2.2  Finitism   prcinf 35781
                  21.5.2.3  Introduce ax-regs   ax-regs 35794
                  21.5.2.4  Derive ax-regs   axregs 35807
                  21.5.2.5  ZFC axioms with reduced distinct variable conditions   axsepg2 35808
                  21.5.2.6  Cardinality without the Axiom of Choice   ckard 35817
                  21.5.2.7  Gödel operations   ccnv2 35841
                  21.5.2.8  The constructible universe   clexo 35863
                  21.5.2.9  Global choice   gblacfnacd 35881
            21.5.3  Real and complex numbers   zltp1ne 35900
            21.5.4  Graph theory   cplgredgex 35905
                  21.5.4.1  Acyclic graphs   acycgr0v 35913
      ​21.6  Mathbox for Mario Carneiro
            21.6.1  Predicate calculus with all distinct variables   ax-7d 35924
            21.6.2  Miscellaneous stuff   quartfull 35930
            21.6.3  Derangements and the Subfactorial   deranglem 35931
            21.6.4  The Erdős-Szekeres theorem   erdszelem1 35956
            21.6.5  The Kuratowski closure-complement theorem   kur14lem1 35971
            21.6.6  Retracts and sections   cretr 35982
            21.6.7  Path-connected and simply connected spaces   cpconn 35984
            21.6.8  Covering maps   ccvm 36020
            21.6.9  Normal numbers   snmlff 36094
            21.6.10  Godel-sets of formulas - part 1   cgoe 36098
            21.6.11  Godel-sets of formulas - part 2   cgon 36197
            21.6.12  Models of ZF   cgze 36211
            ​*21.6.13  Metamath formal systems   cmcn 36225
            21.6.14  Grammatical formal systems   cm0s 36350
            21.6.15  Models of formal systems   cmuv 36370
            21.6.16  Splitting fields   ccpms 36392
            21.6.17  p-adic number fields   czr 36412
      ​​*21.7  Mathbox for Filip Cernatescu
      ​21.8  Mathbox for Paul Chapman
            21.8.1  Real and complex numbers (cont.)   climuzcnv 36436
            21.8.2  Miscellaneous theorems   elfzm12 36440
      ​21.9  Mathbox for Hongxiu Chen
      ​21.10  Mathbox for Adrian Ducourtial
            21.10.1  Propositional calculus   currybi 36453
            21.10.2  Clone theory   ccloneop 36460
      ​21.11  Mathbox for Scott Fenton
            21.11.1  ZFC Axioms in primitive form   axextprim 36466
            21.11.2  Untangled classes   untelirr 36473
            21.11.3  Extra propositional calculus theorems   3jaodd 36480
            21.11.4  Misc. Useful Theorems   nepss 36483
            21.11.5  Properties of real and complex numbers   sqdivzi 36493
            21.11.6  Infinite products   iprodefisumlem 36505
            21.11.7  Factorial limits   faclimlem1 36508
            21.11.8  Greatest common divisor and divisibility   gcd32 36514
            21.11.9  Properties of relationships   dftr6 36516
            21.11.10  Properties of functions and mappings   funpsstri 36531
            21.11.11  Ordinal numbers   elpotr 36543
            21.11.12  Defined equality axioms   axextdfeq 36559
            21.11.13  Hypothesis builders   hbntg 36567
            21.11.14  Well-founded zero, successor, and limits   cwsuc 36572
            21.11.15  Quantifier-free definitions   ctxp 36592
            21.11.16  Alternate ordered pairs   caltop 36721
            21.11.17  Geometry in the Euclidean space   cofs 36747
                  21.11.17.1  Congruence properties   cofs 36747
                  21.11.17.2  Betweenness properties   btwntriv2 36777
                  21.11.17.3  Segment Transportation   ctransport 36794
                  21.11.17.4  Properties relating betweenness and congruence   cifs 36800
                  21.11.17.5  Connectivity of betweenness   btwnconn1lem1 36852
                  21.11.17.6  Segment less than or equal to   csegle 36871
                  21.11.17.7  Outside-of relationship   coutsideof 36884
                  21.11.17.8  Lines and Rays   cline2 36899
            21.11.18  Forward difference   cfwddif 36923
            21.11.19  Rank theorems   rank0 36931
            21.11.20  Hereditarily Finite Sets   hftr 36933
            21.11.21  Natural ordinal operations   cnmul 36936
      ​21.12  Mathbox for Gino Giotto
            21.12.1  Equality theorems   rmoeqi 36976
                  21.12.1.1  Inference versions   rmoeqi 36976
                  21.12.1.2  Deduction versions   rmoeqdv 37001
            21.12.2  Change bound variables   in-ax8 37013
                  21.12.2.1  Change bound variables and domains   cbvralvw2 37015
                  21.12.2.2  Change bound variables, deduction versions   cbvmodavw 37039
                  21.12.2.3  Change bound variables and domains, deduction versions   cbvrmodavw2 37072
            21.12.3  Study of ax-mulf usage   mpomulnzcnf 37088
      ​21.13  Mathbox for Jeff Hankins
            21.13.1  Miscellany   a1i14 37089
            21.13.2  Basic topological facts   topbnd 37112
            21.13.3  Topology of the real numbers   ivthALT 37123
            21.13.4  Refinements   cfne 37124
            21.13.5  Neighborhood bases determine topologies   neibastop1 37147
            21.13.6  Lattice structure of topologies   topmtcl 37151
            21.13.7  Filter bases   fgmin 37158
            21.13.8  Directed sets, nets   tailfval 37160
      ​21.14  Mathbox for Anthony Hart
            21.14.1  Propositional Calculus   tb-ax1 37171
            21.14.2  Predicate Calculus   nalfal 37191
            21.14.3  Miscellaneous single axioms   meran1 37199
            21.14.4  Connective Symmetry   negsym1 37205
      ​21.15  Mathbox for Chen-Pang He
            21.15.1  Ordinal topology   ontopbas 37216
      ​21.16  Mathbox for Jeff Hoffman
            21.16.1  Inferences for finite induction on generic function values   fveleq 37239
            21.16.2  gdc.mm   nnssi2 37243
      ​21.17  Mathbox for Matthew House
            21.17.1  Relations on well-ordered indexed unions   weiunval 37250
            21.17.2  Axiom of Transitive Containment   axtco 37259
            21.17.3  Transitive closure of a class   tr0elw 37272
            ​*21.17.4  Stronger axioms of regularity   mh-setind 37324
            21.17.5  Short axioms written in primitive symbols   mh-inf3f1 37329
      ​21.18  Mathbox for Asger C. Ipsen
            21.18.1  Continuous nowhere differentiable functions   dnival 37337
      ​​*21.19  Mathbox for BJ
            *21.19.1  Propositional calculus   bj-mp2c 37406
                  *21.19.1.1  Derived rules of inference   bj-mp2c 37406
                  ​*21.19.1.2  A syntactic theorem   bj-0 37408
                  ​*21.19.1.3  Minimal implicational calculus   bj-poni 37410
                  ​*21.19.1.4  Positive calculus   bj-bisimpl 37422
                  ​*21.19.1.5  Implication and negation   bj-con2com 37430
                  ​*21.19.1.6  Disjunction   bj-jaoi1 37441
                  ​*21.19.1.7  Logical equivalence   bj-dfbi4 37443
                  21.19.1.8  The conditional operator for propositions   bj-consensus 37448
                  ​*21.19.1.9  Propositional calculus: miscellaneous   bj-imbi12 37453
            ​*21.19.2  Modal logic   bj-axdd2 37462
            ​*21.19.3  Provability logic   cprvb 37467
            ​*21.19.4  First-order logic   bj-exexalal 37476
                  21.19.4.1  Universal and existential quantifiers, nonfreeness predicate   bj-exexalal 37476
                  21.19.4.2  Adding ax-gen   bj-genr 37477
                  21.19.4.3  Adding ax-4   bj-almp 37481
                  21.19.4.4  Adding ax-5   bj-spvw 37534
                  21.19.4.5  Equality and substitution   bj-df-sb 37549
                  21.19.4.6  Adding ax-6   bj-spim0 37568
                  21.19.4.7  Adding ax-7   bj-cbvexw 37576
                  21.19.4.8  Membership predicate, ax-8 and ax-9   bj-ax89 37578
                  21.19.4.9  Adding ax-11   bj-alcomexcom 37580
                  21.19.4.10  Adding ax-12   axc11n11 37584
                  ​*21.19.4.11  Really adding ax-12   bj-substax12 37626
                  21.19.4.12  Nonfreeness   wnnf 37628
                  21.19.4.13  Adding ax-13   bj-axc10 37695
                  ​*21.19.4.14  Removing dependencies on ax-13 (and ax-11)   bj-axc10v 37705
                  ​*21.19.4.15  Distinct var metavariables   bj-hbaeb2 37730
                  ​*21.19.4.16  Around ~ equsal   bj-equsal1t 37734
                  ​*21.19.4.17  Some Principia Mathematica proofs   stdpc5t 37739
                  21.19.4.18  Alternate definition of substitution   bj-sbsb 37749
                  21.19.4.19  Lemmas for substitution   bj-sbf3 37751
                  21.19.4.20  Existential uniqueness   bj-eu3f 37753
                  ​*21.19.4.21  First-order logic: miscellaneous   bj-sblem1 37754
            21.19.5  Set theory   eliminable1 37771
                  ​*21.19.5.1  Eliminability of class terms   eliminable1 37771
                  ​*21.19.5.2  Classes without the axiom of extensionality   bj-denoteslem 37783
                  21.19.5.3  Characterization among sets versus among classes   elelb 37809
                  ​*21.19.5.4  The nonfreeness quantifier for classes   bj-nfcsym 37811
                  ​*21.19.5.5  Lemmas for class substitution   bj-sbeqALT 37812
                  21.19.5.6  Removing some axiom requirements and disjoint variable conditions   bj-exlimvmpi 37823
                  ​*21.19.5.7  Class abstractions   bj-elabd2ALT 37838
                  21.19.5.8  Generalized class abstractions   bj-cgab 37846
                  ​*21.19.5.9  Restricted nonfreeness   wrnf 37854
                  ​*21.19.5.10  Russell's paradox   bj-ru1 37856
                  21.19.5.11  Curry's paradox in set theory   currysetlem 37858
                  ​*21.19.5.12  Some disjointness results   bj-n0i 37864
                  ​*21.19.5.13  Complements on direct products   bj-xpimasn 37868
                  ​*21.19.5.14  "Singletonization" and tagging   bj-snsetex 37876
                  ​*21.19.5.15  Tuples of classes   bj-cproj 37903
                  ​*21.19.5.16  Set theory: elementary operations relative to a universe   bj-rcleqf 37938
                  ​*21.19.5.17  Axioms for finite unions   bj-abex 37943
                  21.19.5.18  Relations, functions, composition: supplements   elco 37960
                  ​*21.19.5.19  Set theory: miscellaneous   eleq2w2ALT 37962
                  ​*21.19.5.20  Axioms of separation and replacement   bj-axnul 37988
                  ​*21.19.5.21  Evaluation at a class   bj-evaleq 37992
                  21.19.5.22  Elementwise operations   celwise 38000
                  ​*21.19.5.23  Elementwise intersection (families of sets induced on a subset)   bj-rest00 38002
                  21.19.5.24  Moore collections (complements)   bj-raldifsn 38021
                  21.19.5.25  Maps-to notation for functions with three arguments   bj-0nelmpt 38037
                  ​*21.19.5.26  Currying   csethom 38041
                  ​*21.19.5.27  Setting components of extensible structures   cstrset 38053
            ​*21.19.6  Extended real and complex numbers, real and complex projective lines   bj-nfald 38056
                  21.19.6.1  Complements on class abstractions of ordered pairs and binary relations   bj-nfald 38056
                  ​*21.19.6.2  Identity relation (complements)   bj-opabssvv 38071
                  ​*21.19.6.3  Functionalized identity (diagonal in a Cartesian square)   cdiag2 38093
                  ​*21.19.6.4  Direct image and inverse image   cimdir 38099
                  ​*21.19.6.5  Extended numbers and projective lines as sets   cfractemp 38117
                  ​*21.19.6.6  Addition and opposite   caddcc 38158
                  ​*21.19.6.7  Order relation on the extended reals   cltxr 38162
                  ​*21.19.6.8  Argument, multiplication and inverse   carg 38164
                  21.19.6.9  The canonical bijection from the finite ordinals   ciomnn 38170
                  21.19.6.10  Divisibility   cnnbar 38181
            ​*21.19.7  Monoids   bj-smgrpssmgm 38189
                  ​*21.19.7.1  Finite sums in monoids   cfinsum 38204
            ​*21.19.8  Affine, Euclidean, and Cartesian geometry   bj-fvimacnv0 38207
                  *21.19.8.1  Real vector spaces   bj-fvimacnv0 38207
                  ​*21.19.8.2  Complex numbers (supplements)   bj-subcom 38229
                  ​*21.19.8.3  Barycentric coordinates   bj-bary1lem 38231
            21.19.9  Monoid of endomorphisms   cend 38234
      ​21.20  Mathbox for Jim Kingdon
            21.20.1  Circle constant   taupilem3 38240
            21.20.2  Number theory   dfgcd3 38245
            21.20.3  Real numbers   irrdifflemf 38246
      ​21.21  Mathbox for ML
            21.21.1  Miscellaneous   csbrecsg 38251
            21.21.2  Cartesian exponentiation   cfinxp 38306
            21.21.3  Topology   iunctb2 38326
                  ​*21.21.3.1  Pi-base theorems   pibp16 38336
      ​21.22  Mathbox for Wolf Lammen
            21.22.1  1. Bootstrapping   wl-section-boot 38345
            21.22.2  Implication chains   wl-section-impchain 38369
            21.22.3  Theorems around the conditional operator   wl-ifp-ncond1 38387
            21.22.4  Alternative development of hadd, cadd   wl-df-3xor 38391
            21.22.5  An alternative axiom ~ ax-13   ax-wl-13v 38416
            21.22.6  Bootstrapping set theory with classes   wl-cleq-0 38418
            21.22.7  Other stuff   wl-mps 38439
      ​21.23  Mathbox for Brendan Leahy
      ​21.24  Mathbox for Thomas van Maaren
            ​*21.24.1  The language of Propositional Calculus.   cpropvar 38633
      ​21.25  Mathbox for Jeff Madsen
            21.25.1  Logic and set theory   unirep 38648
            21.25.2  Real and complex numbers; integers   filbcmb 38674
            21.25.3  Sequences and sums   sdclem2 38676
            21.25.4  Topology   subspopn 38686
            21.25.5  Metric spaces   metf1o 38689
            21.25.6  Continuous maps and homeomorphisms   constcncf 38696
            21.25.7  Boundedness   ctotbnd 38700
            21.25.8  Isometries   cismty 38732
            21.25.9  Heine-Borel Theorem   heibor1lem 38743
            21.25.10  Banach Fixed Point Theorem   bfplem1 38756
            21.25.11  Euclidean space   crrn 38759
            21.25.12  Intervals (continued)   ismrer1 38772
            ​*21.25.13  Operation properties   cass 38776
            21.25.14  Groups and related structures   cmagm 38782
            21.25.15  Group homomorphism and isomorphism   cghomOLD 38817
            21.25.16  Rings   crngo 38828
            21.25.17  Division Rings   cdrng 38882
            21.25.18  Ring homomorphisms   crngohom 38894
            21.25.19  Commutative rings   ccm2 38923
            21.25.20  Ideals   cidl 38941
            21.25.21  Prime rings and integral domains   cprrng 38980
            21.25.22  Ideal generators   cigen 38993
      ​21.26  Mathbox for Giovanni Mascellani
            ​*21.26.1  Tools for automatic proof building   efald2 39012
            ​*21.26.2  Tseitin axioms   fald 39061
            ​*21.26.3  Equality deductions   iuneq2f 39088
            ​*21.26.4  Miscellanea   orcomdd 39099
      ​21.27  Mathbox for Peter Mazsa
            21.27.1  Notations   cxrn 39106
            21.27.2  Preparatory theorems   el2v1 39161
            21.27.3  Range Cartesian product   df-xrn 39312
            21.27.4  Relations   df-rels 39372
            21.27.5  Quotient map (coset map)   df-qmap 39378
            21.27.6  Lifts, shifts, successor, and predecessor   df-adjliftmap 39387
            21.27.7  Cosets by ` R `   df-coss 39433
            21.27.8  Subset relations   df-ssr 39510
            21.27.9  Reflexivity   df-refs 39522
            21.27.10  Converse reflexivity   df-cnvrefs 39537
            21.27.11  Symmetry   df-syms 39554
            21.27.12  Reflexivity and symmetry   symrefref2 39579
            21.27.13  Transitivity   df-trs 39588
            21.27.14  Equivalence relations   df-eqvrels 39600
            21.27.15  Redundancy   df-redunds 39639
            21.27.16  Domain quotients   df-dmqss 39654
            21.27.17  Equivalence relations on domain quotients   df-ers 39680
            21.27.18  Functions   df-funss 39697
            21.27.19  Disjoints vs. converse functions   df-disjss 39720
            21.27.20  Antisymmetry   df-antisymrel 39795
            21.27.21  Partitions: disjoints on domain quotients   df-parts 39800
            21.27.22  Partition-Equivalence Theorems   disjim 39816
            21.27.23  Type-safe Partition-Equivalence: PetParts, PetErs, Pet2Parts, Pet2Ers   df-petparts 39900
      ​21.28  Mathbox for Rodolfo Medina
            21.28.1  Partitions   prtlem60 39910
      ​​*21.29  Mathbox for Norm Megill
            *21.29.1  Obsolete schemes ax-c4,c5,c7,c10,c11,c11n,c15,c9,c14,c16   ax-c5 39940
            ​*21.29.2  Rederive new axioms ax-4, ax-10, ax-6, ax-12, ax-13 from old   axc5 39950
            ​*21.29.3  Legacy theorems using obsolete axioms   ax5ALT 39964
            21.29.4  Experiments with weak deduction theorem   elimhyps 40018
            21.29.5  Miscellanea   cnaddcom 40029
            21.29.6  Atoms, hyperplanes, and covering in a left vector space (or module)   clsa 40031
            21.29.7  Functionals and kernels of a left vector space (or module)   clfn 40114
            21.29.8  Opposite rings and dual vector spaces   cld 40180
            21.29.9  Ortholattices and orthomodular lattices   cops 40229
            21.29.10  Atomic lattices with covering property   ccvr 40319
            21.29.11  Hilbert lattices   chlt 40407
            21.29.12  Projective geometries based on Hilbert lattices   clln 40548
            21.29.13  Construction of a vector space from a Hilbert lattice   cdlema1N 40848
            21.29.14  Construction of involution and inner product from a Hilbert lattice   clpoN 42537
      ​21.30  Mathbox for metakunt
            21.30.1  Commutative Semiring   ccsrg 43019
            21.30.2  General helpful statements   rhmzrhval 43022
            21.30.3  Some gcd and lcm results   12gcd5e1 43053
            21.30.4  Least common multiple inequality theorem   3factsumint1 43071
            21.30.5  Logarithm inequalities   3exp7 43103
            21.30.6  Miscellaneous results for AKS formalisation   intlewftc 43111
            21.30.7  Sticks and stones   sticksstones1 43196
            21.30.8  Continuation AKS   aks6d1c6lem1 43220
      ​21.31  Mathbox for Luke Murphy
            21.31.1  Solutions of quadratic equations   quadfac 43255
            21.31.2  April Fool's theorem   25or6to4 43256
      ​21.32  Mathbox for Steven Nguyen
            21.32.1  Utility theorems   jarrii 43257
            ​*21.32.2  Arithmetic theorems   c0exALT 43303
            21.32.3  Exponents and divisibility   oexpreposd 43379
            21.32.4  Trigonometry and Calculus   tanhalfpim 43400
            ​*21.32.5  Independence of ax-mulcom   cresub 43416
            21.32.6  Structures   sn-base0 43559
            ​*21.32.7  Projective spaces   cprjsp 43629
            21.32.8  Basic reductions for Fermat's Last Theorem   dffltz 43670
            ​*21.32.9  Exemplar theorems   iddii 43675
                  ​*21.32.9.1  Standard replacements of ax-10 , ax-11 , ax-12   nfa1w 43686
      ​21.33  Mathbox for Igor Ieskov
      ​21.34  Mathbox for OpenAI
      ​21.35  Mathbox for Stefan O'Rear
            21.35.1  Additional elementary logic and set theory   moxfr 43702
            21.35.2  Additional theory of functions   imaiinfv 43703
            21.35.3  Additional topology   elrfi 43704
            21.35.4  Characterization of closure operators. Kuratowski closure axioms   ismrcd1 43708
            21.35.5  Algebraic closure systems   cnacs 43712
            21.35.6  Miscellanea 1. Map utilities   constmap 43723
            21.35.7  Miscellanea for polynomials   mptfcl 43730
            21.35.8  Multivariate polynomials over the integers   cmzpcl 43731
            21.35.9  Miscellanea for Diophantine sets 1   coeq0i 43763
            21.35.10  Diophantine sets 1: definitions   cdioph 43765
            21.35.11  Diophantine sets 2 miscellanea   ellz1 43777
            21.35.12  Diophantine sets 2: union and intersection. Monotone Boolean algebra   diophin 43782
            21.35.13  Diophantine sets 3: construction   diophrex 43785
            21.35.14  Diophantine sets 4 miscellanea   2sbcrex 43794
            21.35.15  Diophantine sets 4: Quantification   rexrabdioph 43800
            21.35.16  Diophantine sets 5: Arithmetic sets   rabdiophlem1 43807
            21.35.17  Diophantine sets 6: reusability. renumbering of variables   eldioph4b 43817
            21.35.18  Pigeonhole Principle and cardinality helpers   fphpd 43822
            21.35.19  A non-closed set of reals is infinite   rencldnfilem 43826
            21.35.20  Lagrange's rational approximation theorem   irrapxlem1 43828
            21.35.21  Pell equations 1: A nontrivial solution always exists   pellexlem1 43835
            21.35.22  Pell equations 2: Algebraic number theory of the solution set   csquarenn 43842
            21.35.23  Pell equations 3: characterizing fundamental solution   infmrgelbi 43884
            ​*21.35.24  Logarithm laws generalized to an arbitrary base   reglogcl 43896
            21.35.25  Pell equations 4: the positive solution group is infinite cyclic   pellfund14 43904
            21.35.26  X and Y sequences 1: Definition and recurrence laws   crmx 43906
            21.35.27  Ordering and induction lemmas for the integers   monotuz 43947
            21.35.28  X and Y sequences 2: Order properties   rmxypos 43953
            21.35.29  Congruential equations   congtr 43971
            21.35.30  Alternating congruential equations   acongid 43981
            21.35.31  Additional theorems on integer divisibility   coprmdvdsb 43991
            21.35.32  X and Y sequences 3: Divisibility properties   jm2.18 43994
            21.35.33  X and Y sequences 4: Diophantine representability of Y   jm2.27a 44011
            21.35.34  X and Y sequences 5: Diophantine representability of X, ^, _C   rmxdiophlem 44021
            21.35.35  Uncategorized stuff not associated with a major project   setindtr 44030
            21.35.36  More equivalents of the Axiom of Choice   axac10 44039
            21.35.37  Finitely generated left modules   clfig 44068
            21.35.38  Noetherian left modules I   clnm 44076
            21.35.39  Addenda for structure powers   pwssplit4 44090
            21.35.40  Every set admits a group structure iff choice   unxpwdom3 44096
            21.35.41  Noetherian rings and left modules II   clnr 44110
            21.35.42  Hilbert's Basis Theorem   cldgis 44122
            21.35.43  Additional material on polynomials [DEPRECATED]   cmnc 44132
            21.35.44  Degree and minimal polynomial of algebraic numbers   cdgraa 44141
            21.35.45  Algebraic integers I   citgo 44158
            21.35.46  Endomorphism algebra   cmend 44172
            21.35.47  Cyclic groups and order   idomodle 44192
            21.35.48  Cyclotomic polynomials   ccytp 44198
            21.35.49  Miscellaneous topology   fgraphopab 44204
      ​21.36  Mathbox for Noam Pasman
      ​21.37  Mathbox for Jon Pennant
      ​21.38  Mathbox for Richard Penner
            21.38.1  Set Theory and Ordinal Numbers   uniel 44218
            21.38.2  Natural addition of Cantor normal forms   oawordex2 44327
            21.38.3  Surreal Contributions   abeqabi 44408
            21.38.4  Short Studies   nlimsuc 44441
                  21.38.4.1  Additional work on conditional logical operator   ifpan123g 44459
                  21.38.4.2  Sophisms   rp-fakeimass 44512
                  ​*21.38.4.3  Finite Sets   rp-isfinite5 44517
                  21.38.4.4  General Observations   intabssd 44519
                  21.38.4.5  Infinite Sets   pwelg 44560
                  ​*21.38.4.6  Finite intersection property   fipjust 44565
                  21.38.4.7  RP ADDTO: Subclasses and subsets   rababg 44574
                  21.38.4.8  RP ADDTO: The intersection of a class   elinintab 44575
                  21.38.4.9  RP ADDTO: Theorems requiring subset and intersection existence   elinintrab 44577
                  21.38.4.10  RP ADDTO: Relations   xpinintabd 44580
                  ​*21.38.4.11  RP ADDTO: Functions   elmapintab 44595
                  ​*21.38.4.12  RP ADDTO: Finite induction (for finite ordinals)   cnvcnvintabd 44599
                  21.38.4.13  RP ADDTO: First and second members of an ordered pair   elcnvlem 44600
                  21.38.4.14  RP ADDTO: The reflexive and transitive properties of relations   undmrnresiss 44603
                  21.38.4.15  RP ADDTO: Basic properties of closures   cleq2lem 44607
                  21.38.4.16  RP REPLACE: Definitions and basic properties of transitive closures   trcleq2lemRP 44629
                  ​*21.38.4.17  Additions for square root; absolute value   sqrtcvallem1 44630
            21.38.5  Additional statements on relations and subclasses   al3im 44646
                  21.38.5.1  Transitive relations (not to be confused with transitive classes)   trrelind 44664
                  21.38.5.2  Reflexive closures   crcl 44671
                  ​*21.38.5.3  Finite relationship composition   relexp2 44676
                  21.38.5.4  Transitive closure of a relation   dftrcl3 44719
                  ​*21.38.5.5  Adapted from Frege   frege77d 44745
            ​*21.38.6  Propositions from _Begriffsschrift_   dfxor4 44765
                  *21.38.6.1  _Begriffsschrift_ Chapter I   dfxor4 44765
                  ​*21.38.6.2  _Begriffsschrift_ Notation hints   whe 44771
                  21.38.6.3  _Begriffsschrift_ Chapter II Implication   ax-frege1 44789
                  21.38.6.4  _Begriffsschrift_ Chapter II Implication and Negation   axfrege28 44828
                  ​*21.38.6.5  _Begriffsschrift_ Chapter II with logical equivalence   axfrege52a 44855
                  21.38.6.6  _Begriffsschrift_ Chapter II with equivalence of sets   axfrege52c 44886
                  ​*21.38.6.7  _Begriffsschrift_ Chapter II with equivalence of classes   frege53c 44913
                  ​*21.38.6.8  _Begriffsschrift_ Chapter III Properties hereditary in a sequence   dffrege69 44931
                  ​*21.38.6.9  _Begriffsschrift_ Chapter III Following in a sequence   dffrege76 44938
                  ​*21.38.6.10  _Begriffsschrift_ Chapter III Member of sequence   dffrege99 44961
                  ​*21.38.6.11  _Begriffsschrift_ Chapter III Single-valued procedures   dffrege115 44977
            ​*21.38.7  Exploring Topology via Seifert and Threlfall   enrelmap 44996
                  *21.38.7.1  Equinumerosity of sets of relations and maps   enrelmap 44996
                  ​*21.38.7.2  Generic Pseudoclosure Spaces, Pseudointerior Spaces, and Pseudoneighborhoods   or3or 45022
                  ​*21.38.7.3  Generic Neighborhood Spaces   gneispa 45129
            ​*21.38.8  Exploring Higher Homotopy via Kerodon   k0004lem1 45146
                  *21.38.8.1  Simplicial Sets   k0004lem1 45146
      ​21.39  Mathbox for Stanislas Polu
            21.39.1  IMO Problems   wwlemuld 45155
                  21.39.1.1  IMO 1972 B2   wwlemuld 45155
            ​*21.39.2  INT Inequalities Proof Generator   int-addcomd 45172
            ​*21.39.3  N-Digit Addition Proof Generator   unitadd 45194
            21.39.4  AM-GM (for k = 2,3,4)   gsumws3 45195
      ​21.40  Mathbox for Rohan Ridenour
            21.40.1  Misc   spALT 45200
            21.40.2  Monoid rings   cmnring 45208
            21.40.3  Shorter primitive equivalent of ax-groth   gru0eld 45226
                  21.40.3.1  Grothendieck universes are closed under collection   gru0eld 45226
                  21.40.3.2  Minimal universes   ismnu 45244
                  21.40.3.3  Primitive equivalent of ax-groth   expandan 45271
      ​21.41  Mathbox for Steve Rodriguez
            21.41.1  Miscellanea   nanorxor 45288
            21.41.2  Ratio test for infinite series convergence and divergence   dvgrat 45295
            21.41.3  Multiples   reldvds 45298
            21.41.4  Function operations   caofcan 45306
            21.41.5  Calculus   lhe4.4ex1a 45312
            21.41.6  The generalized binomial coefficient operation   cbcc 45319
            21.41.7  Binomial series   uzmptshftfval 45329
      ​21.42  Mathbox for Andrew Salmon
            21.42.1  Principia Mathematica * 10   pm10.12 45341
            21.42.2  Principia Mathematica * 11   2alanimi 45355
            21.42.3  Predicate Calculus   sbeqal1 45381
            21.42.4  Principia Mathematica * 13 and * 14   pm13.13a 45390
            21.42.5  Set Theory   elnev 45420
            21.42.6  Arithmetic   addcomgi 45437
            21.42.7  Geometry   cplusr 45438
      ​​*21.43  Mathbox for Alan Sare
            21.43.1  Auxiliary theorems for the Virtual Deduction tool   idiALT 45460
            21.43.2  Supplementary unification deductions   bi1imp 45464
            21.43.3  Conventional Metamath proofs, some derived from VD proofs   iidn3 45483
            21.43.4  What is Virtual Deduction?   wvd1 45551
            21.43.5  Virtual Deduction Theorems   df-vd1 45552
            21.43.6  Theorems proved using Virtual Deduction   trsspwALT 45799
            21.43.7  Theorems proved using Virtual Deduction with mmj2 assistance   simplbi2VD 45827
            21.43.8  Virtual Deduction transcriptions of textbook proofs   sb5ALTVD 45894
            21.43.9  Theorems proved using conjunction-form Virtual Deduction   elpwgdedVD 45898
            21.43.10  Theorems with a VD proof in conventional notation derived from a VD proof   suctrALT3 45905
            ​*21.43.11  Theorems with a proof in conventional notation derived from a VD proof   notnotrALT2 45908
      ​21.44  Mathbox for Eric Schmidt
            21.44.1  Miscellany   rspesbcd 45919
            21.44.2  Study of dfbi1ALT   dfbi1ALTa 45928
            21.44.3  Relation-preserving functions   wrelp 45931
            21.44.4  Orbits   orbitex 45944
            21.44.5  Well-founded sets   trwf 45948
            21.44.6  Absoluteness in transitive models   ralabso 45957
            21.44.7  Lemmas for showing axioms hold in models   traxext 45966
            21.44.8  The class of well-founded sets is a model for ZFC   wfaxext 45982
            21.44.9  Permutation models   brpermmodel 45992
            21.44.10  Isomorphism of finite ordinals and non-negative integers   hashnna 46008
            21.44.11  Hereditarily finite sets   hfpr 46015
            21.44.12  Hereditarily finite structures   chfstruct 46022
      ​21.45  Mathbox for Glauco Siliprandi
            21.45.1  Miscellanea   evth2f 46031
            21.45.2  Functions   fnresdmss 46182
            21.45.3  Ordering on real numbers - Real and complex numbers basic operations   sub2times 46288
            21.45.4  Real intervals   gtnelioc 46502
            21.45.5  Finite sums   fsummulc1f 46582
            21.45.6  Finite multiplication of numbers and finite multiplication of functions   fmul01 46591
            21.45.7  Limits   clim1fr1 46612
                  21.45.7.1  Inferior limit (lim inf)   clsi 46760
                  ​*21.45.7.2  Limits for sequences of extended real numbers   clsxlim 46827
            21.45.8  Trigonometry   coseq0 46873
            21.45.9  Continuous Functions   mulcncff 46879
            21.45.10  Derivatives   dvsinexp 46920
            21.45.11  Integrals   itgsin0pilem1 46959
            21.45.12  Stone Weierstrass theorem - real version   stoweidlem1 47010
            21.45.13  Wallis' product for π   wallispilem1 47074
            21.45.14  Stirling's approximation formula for ` n ` factorial   stirlinglem1 47083
            21.45.15  Dirichlet kernel   dirkerval 47100
            21.45.16  Fourier Series   fourierdlem1 47117
            21.45.17  e is transcendental   elaa2lem 47242
            21.45.18  n-dimensional Euclidean space   rrxtopn 47293
            21.45.19  Basic measure theory   csalg 47317
                  ​*21.45.19.1  σ-Algebras   csalg 47317
                  21.45.19.2  Sum of nonnegative extended reals   csumge0 47371
                  ​*21.45.19.3  Measures   cmea 47458
                  ​*21.45.19.4  Outer measures and Caratheodory's construction   come 47498
                  ​*21.45.19.5  Lebesgue measure on n-dimensional Real numbers   covoln 47545
                  ​*21.45.19.6  Measurable functions   csmblfn 47704
      ​21.46  Mathbox for Saveliy Skresanov
            21.46.1  Ceva's theorem   sigarval 47859
            21.46.2  Simple groups   simpcntrab 47879
      ​21.47  Mathbox for Ender Ting
            21.47.1  Interesting facts   et-ltneverrefl 47880
            21.47.2  Increasing sequences and subsequences   ormklocald 47885
            21.47.3  Scratchpad for number theory   evenwodadd 47910
            21.47.4  Scratchpad for math on real numbers   squeezedltsq 47911
            21.47.5  Turing Machine Finite Reach theorem   tmachlem-extapes 47943
      ​21.48  Mathbox for Jarvin Udandy
      ​21.49  Mathbox for Adhemar
            ​*21.49.1  Minimal implicational calculus   adh-minim 48070
      ​21.50  Mathbox for Alexander van der Vekens
            21.50.1  General auxiliary theorems (1)   n0nsn2el 48094
                  21.50.1.1  Unordered and ordered pairs - extension for singletons   n0nsn2el 48094
                  21.50.1.2  Unordered and ordered pairs - extension for unordered pairs   elprneb 48098
                  21.50.1.3  Unordered and ordered pairs - extension for ordered pairs   oppr 48099
                  21.50.1.4  Relations - extension   eubrv 48104
                  21.50.1.5  Definite description binder (inverted iota) - extension   iota0def 48107
                  21.50.1.6  Functions - extension   fveqvfvv 48109
            21.50.2  Alternative for Russell's definition of a description binder   caiota 48152
            21.50.3  Double restricted existential uniqueness   r19.32 48167
                  21.50.3.1  Restricted quantification (extension)   r19.32 48167
                  21.50.3.2  Restricted uniqueness and "at most one" quantification   reuf1odnf 48176
                  21.50.3.3  Analogs to Existential uniqueness (double quantification)   2reu3 48179
                  21.50.3.4  Additional theorems for double restricted existential uniqueness   2reu8i 48182
            ​*21.50.4  Alternative definitions of function and operation values   wdfat 48185
                  21.50.4.1  Restricted quantification (extension)   ralbinrald 48191
                  21.50.4.2  The universal class (extension)   nvelim 48192
                  21.50.4.3  Introduce the Axiom of Power Sets (extension)   alneu 48193
                  21.50.4.4  Predicate "defined at"   dfateq12d 48195
                  21.50.4.5  Alternative definition of the value of a function   dfafv2 48201
                  21.50.4.6  Alternative definition of the value of an operation   aoveq123d 48247
            ​*21.50.5  Alternative definitions of function values (2)   cafv2 48277
            21.50.6  General auxiliary theorems (2)   an4com24 48337
                  21.50.6.1  Logical conjunction - extension   an4com24 48337
                  21.50.6.2  Abbreviated conjunction and disjunction of three wff's - extension   3an4ancom24 48338
                  21.50.6.3  Negated membership (alternative)   cnelbr 48340
                  21.50.6.4  The empty set - extension   ralralimp 48347
                  21.50.6.5  Indexed union and intersection - extension   otiunsndisjX 48348
                  21.50.6.6  Functions - extension   fvifeq 48349
                  21.50.6.7  Maps-to notation - extension   fvmptrab 48361
                  21.50.6.8  Subtraction - extension   cnambpcma 48363
                  21.50.6.9  Ordering on reals (cont.) - extension   leaddsuble 48366
                  21.50.6.10  Imaginary and complex number properties - extension   readdcnnred 48372
                  21.50.6.11  Nonnegative integers (as a subset of complex numbers) - extension   nn0resubcl 48377
                  21.50.6.12  Integers (as a subset of complex numbers) - extension   zgeltp1eq 48378
                  21.50.6.13  Decimal arithmetic - extension   1t10e1p1e11 48379
                  21.50.6.14  Upper sets of integers - extension   eluzge0nn0 48381
                  21.50.6.15  Infinity and the extended real number system (cont.) - extension   nltle2tri 48382
                  21.50.6.16  Finite intervals of integers - extension   ssfz12 48383
                  21.50.6.17  Half-open integer ranges - extension   fzopred 48392
                  21.50.6.18  The floor and ceiling functions - extension   2ltceilhalf 48401
                  21.50.6.19  The modulo (remainder) operation - extension   fldivmod 48413
                  21.50.6.20  The infinite sequence builder "seq"   smonoord 48446
                  21.50.6.21  Integer powers - extension   2timesltsq 48447
                  21.50.6.22  Finite and infinite sums - extension   fsummsndifre 48449
                  21.50.6.23  The divides relation - extension   nndivides2 48453
                  21.50.6.24  Extensible structures - extension   setsidel 48457
            ​*21.50.7  Preimages of function values   preimafvsnel 48460
            ​*21.50.8  Partitions of real intervals   ciccp 48494
            21.50.9  Shifting functions with an integer range domain   fargshiftfv 48520
            21.50.10  Words over a set (extension)   lswn0 48525
                  21.50.10.1  Last symbol of a word - extension   lswn0 48525
            21.50.11  Unordered pairs   wich 48526
                  21.50.11.1  Interchangeable setvar variables   wich 48526
                  21.50.11.2  Set of unordered pairs   sprid 48555
                  ​*21.50.11.3  Proper (unordered) pairs   prpair 48582
                  21.50.11.4  Set of proper unordered pairs   cprpr 48593
            21.50.12  Number theory (extension)   nprmmul1 48608
                  21.50.12.1  Properties of non-prime numbers   nprmmul1 48608
                  ​*21.50.12.2  Fermat numbers   cfmtno 48611
                  ​*21.50.12.3  Mersenne primes   m2prm 48675
                  21.50.12.4  Proth's theorem   modexp2m1d 48696
                  21.50.12.5  The prime-counting function according to Ján Mináč   nprmdvdsfacm1lem1 48704
                  21.50.12.6  Solutions of quadratic equations   quad1 48717
            ​*21.50.13  Even and odd numbers   ceven 48721
                  21.50.13.1  Definitions and basic properties   ceven 48721
                  21.50.13.2  Alternate definitions using the "divides" relation   dfeven2 48746
                  21.50.13.3  Alternate definitions using the "modulo" operation   dfeven3 48755
                  21.50.13.4  Alternate definitions using the "gcd" operation   iseven5 48761
                  21.50.13.5  Theorems of part 5 revised   zneoALTV 48766
                  21.50.13.6  Theorems of part 6 revised   odd2np1ALTV 48771
                  21.50.13.7  Theorems of AV's mathbox revised   0evenALTV 48785
                  21.50.13.8  Additional theorems   epoo 48800
                  21.50.13.9  Perfect Number Theorem (revised)   perfectALTVlem1 48818
            21.50.14  Number theory (extension 2)   cfppr 48821
                  ​*21.50.14.1  Fermat pseudoprimes   cfppr 48821
                  ​*21.50.14.2  Goldbach's conjectures   cgbe 48842
            21.50.15  Graph theory (extension)   cclnbgr 48915
                  21.50.15.1  Closed neighborhood of a vertex   cclnbgr 48915
                  ​*21.50.15.2  Semiclosed and semiopen neighborhoods (experimental)   dfsclnbgr2 48943
                  21.50.15.3  Induced subgraphs   cisubgr 48957
                  ​*21.50.15.4  Isomorphisms of graphs   cgrisom 48971
                  ​*21.50.15.5  Triangles in graphs   cgrtri 49034
                  ​*21.50.15.6  Star graphs   cstgr 49048
                  ​*21.50.15.7  Local isomorphisms of graphs   cgrlim 49073
                  ​*21.50.15.8  Generalized Petersen graphs   cgpg 49137
                  21.50.15.9  Loop-free graphs - extension   1hegrlfgr 49229
                  21.50.15.10  Walks - extension   cupwlks 49230
                  21.50.15.11  Edges of graphs expressed as sets of unordered pairs   upgredgssspr 49240
            21.50.16  Monoids (extension)   ovn0dmfun 49253
                  21.50.16.1  Auxiliary theorems   ovn0dmfun 49253
                  21.50.16.2  Magmas, Semigroups and Monoids (extension)   plusfreseq 49260
                  21.50.16.3  Examples and counterexamples for magmas, semigroups and monoids (extension)   opmpoismgm 49263
                  21.50.16.4  Group sum operation (extension 1)   gsumsplit2f 49276
            ​*21.50.17  Magmas and internal binary operations (alternate approach)   ccllaw 49279
                  *21.50.17.1  Laws for internal binary operations   ccllaw 49279
                  ​*21.50.17.2  Internal binary operations   cintop 49292
                  21.50.17.3  Alternative definitions for magmas and semigroups   cmgm2 49311
            21.50.18  Rings (extension)   lmod0rng 49325
                  21.50.18.1  Nonzero rings (extension)   lmod0rng 49325
                  21.50.18.2  Ideals as non-unital rings   lidldomn1 49327
                  21.50.18.3  The non-unital ring of even integers   0even 49333
                  21.50.18.4  A constructed not unital ring   cznrnglem 49355
                  ​*21.50.18.5  The category of non-unital rings (alternate definition)   crngcALTV 49359
                  ​*21.50.18.6  The category of (unital) rings (alternate definition)   cringcALTV 49383
            ​*21.50.19  Prime rings (and integral domains)   cprmrng 49430
            21.50.20  Basic algebraic structures (extension)   eliunxp2 49445
                  21.50.20.1  Auxiliary theorems   eliunxp2 49445
                  21.50.20.2  The binomial coefficient operation (extension)   bcpascm1 49462
                  21.50.20.3  The ` ZZ `-module ` ZZ X. ZZ `   zlmodzxzlmod 49465
                  21.50.20.4  Group sum operation (extension 2)   mgpsumunsn 49472
                  21.50.20.5  Symmetric groups (extension)   exple2lt6 49475
                  21.50.20.6  Divisibility (extension)   invginvrid 49478
                  21.50.20.7  The support of functions (extension)   rmsupp0 49479
                  21.50.20.8  Finitely supported functions (extension)   rmsuppfi 49483
                  21.50.20.9  Left modules (extension)   lmodvsmdi 49490
                  21.50.20.10  Associative algebras (extension)   assaascl0 49492
                  21.50.20.11  Univariate polynomials (extension)   ply1vr1smo 49494
                  21.50.20.12  Univariate polynomials (examples)   linply1 49504
            21.50.21  Linear algebra (extension)   cdmatalt 49507
                  ​*21.50.21.1  The subalgebras of diagonal and scalar matrices (extension)   cdmatalt 49507
                  ​*21.50.21.2  Linear combinations   clinc 49515
                  ​*21.50.21.3  Linear independence   clininds 49551
                  21.50.21.4  Simple left modules and the ` ZZ `-module   lmod1lem1 49598
                  21.50.21.5  Differences between (left) modules and (left) vector spaces   lvecpsslmod 49618
            21.50.22  Complexity theory   suppdm 49621
                  21.50.22.1  Auxiliary theorems   suppdm 49621
                  21.50.22.2  Even and odd integers   nn0onn0ex 49634
                  21.50.22.3  The natural logarithm on complex numbers (extension)   logcxp0 49646
                  21.50.22.4  Division of functions   cfdiv 49648
                  21.50.22.5  Upper bounds   cbigo 49658
                  21.50.22.6  Logarithm to an arbitrary base (extension)   rege1logbrege0 49669
                  ​*21.50.22.7  The binary logarithm   fldivexpfllog2 49676
                  21.50.22.8  Binary length   cblen 49680
                  ​*21.50.22.9  Digits   cdig 49706
                  21.50.22.10  Nonnegative integer as sum of its shifted digits   dignn0flhalflem1 49726
                  21.50.22.11  Algorithms for the multiplication of nonnegative integers   nn0mulfsum 49735
                  ​*21.50.22.12  N-ary functions   cnaryf 49737
                  ​*21.50.22.13  The Ackermann function   citco 49768
            21.50.23  Elementary geometry (extension)   fv1prop 49810
                  21.50.23.1  Auxiliary theorems   fv1prop 49810
                  21.50.23.2  Real euclidean space of dimension 2   rrx2pxel 49822
                  21.50.23.3  Spheres and lines in real Euclidean spaces   cline 49838
      ​21.51  Mathbox for Zhi Wang
            21.51.1  Propositional calculus   imbi12d2 49900
            21.51.2  Predicate calculus with equality   dtrucor3 49908
                  21.51.2.1  Axiom scheme ax-5 (Distinctness)   dtrucor3 49908
            21.51.3  ZF Set Theory - start with the Axiom of Extensionality   ralbidb 49909
                  21.51.3.1  Restricted quantification   ralbidb 49909
                  21.51.3.2  The universal class   reuxfr1dd 49916
                  21.51.3.3  The empty set   ssdisjd 49917
                  21.51.3.4  Unordered and ordered pairs   vsn 49921
                  21.51.3.5  The union of a class   unilbss 49927
                  21.51.3.6  Indexed union and intersection   iuneq0 49928
            21.51.4  ZF Set Theory - add the Axiom of Replacement   inpw 49934
                  21.51.4.1  Theorems requiring subset and intersection existence   inpw 49934
            21.51.5  ZF Set Theory - add the Axiom of Power Sets   opth1neg 49935
                  21.51.5.1  Ordered pair theorem   opth1neg 49935
                  21.51.5.2  Ordered-pair class abstractions (cont.)   brab2dd 49937
                  21.51.5.3  Relations   iinxp 49940
                  21.51.5.4  Functions   mof0 49947
                  21.51.5.5  Operations   ovsng 49967
            21.51.6  ZF Set Theory - add the Axiom of Union   fonex 49976
                  21.51.6.1  Relations and functions (cont.)   fonex 49976
                  21.51.6.2  First and second members of an ordered pair   eloprab1st2nd 49977
                  21.51.6.3  Function transposition   resinsnlem 49978
                  21.51.6.4  Infinite Cartesian products   ixpv 49997
                  21.51.6.5  Equinumerosity   fvconst0ci 49998
            21.51.7  Order sets   iccin 50003
                  21.51.7.1  Real number intervals   iccin 50003
            21.51.8  Extensible structures   slotresfo 50006
                  21.51.8.1  Basic definitions   slotresfo 50006
            21.51.9  Moore spaces   mreuniss 50007
            ​*21.51.10  Topology   clduni 50008
                  21.51.10.1  Closure and interior   clduni 50008
                  21.51.10.2  Neighborhoods   neircl 50012
                  21.51.10.3  Subspace topologies   restcls2lem 50020
                  21.51.10.4  Limits and continuity in topological spaces   cnneiima 50024
                  21.51.10.5  Topological definitions using the reals   iooii 50025
                  21.51.10.6  Separated sets   sepnsepolem1 50029
                  21.51.10.7  Separated spaces: T0, T1, T2 (Hausdorff) ...   isnrm4 50038
            21.51.11  Preordered sets and directed sets using extensible structures   isprsd 50062
            21.51.12  Posets and lattices using extensible structures   lubeldm2 50063
                  21.51.12.1  Posets   lubeldm2 50063
                  21.51.12.2  Lattices   toslat 50089
                  21.51.12.3  Subset order structures   intubeu 50091
            21.51.13  Rings   elmgpcntrd 50112
                  21.51.13.1  Multiplicative Group   elmgpcntrd 50112
            21.51.14  Associative algebras   asclelbasALT 50113
                  21.51.14.1  Definition and basic properties   asclelbasALT 50113
            21.51.15  Categories   homf0 50116
                  21.51.15.1  Categories   homf0 50116
                  21.51.15.2  Opposite category   oppccatb 50123
                  21.51.15.3  Monomorphisms and epimorphisms   idmon 50127
                  21.51.15.4  Sections, inverses, isomorphisms   sectrcl 50129
                  21.51.15.5  Isomorphic objects   cicfn 50149
                  21.51.15.6  Subcategories   dmdm 50160
                  21.51.15.7  Functors   reldmfunc 50182
                  21.51.15.8  Opposite functors   coppf 50229
                  21.51.15.9  Full & faithful functors   imasubc 50258
                  21.51.15.10  Universal property   upciclem1 50273
                  21.51.15.11  Natural transformations and the functor category   isnatd 50330
                  21.51.15.12  Initial, terminal and zero objects of a category   initoo2 50339
                  21.51.15.13  Product of categories   reldmxpc 50353
                  21.51.15.14  Swap functors   cswapf 50366
                  21.51.15.15  Functor evaluation   oppc1stflem 50394
                  21.51.15.16  Transposed curry functors   cofuswapfcl 50400
                  21.51.15.17  Constant functors   diag1 50411
                  21.51.15.18  Functor composition bifunctors   fucofulem1 50417
                  21.51.15.19  Post-composition functors   postcofval 50471
                  21.51.15.20  Pre-composition functors   precofvallem 50473
            21.51.16  Examples of categories   catcrcl 50502
                  21.51.16.1  The category of categories   catcrcl 50502
                  21.51.16.2  Thin categories   cthinc 50524
                  21.51.16.3  Terminal categories   ctermc 50579
                  21.51.16.4  Preordered sets as thin categories   cprstc 50656
                  21.51.16.5  Monoids as categories   cmndtc 50684
                  21.51.16.6  Categories with at most one object and at most two morphisms   2arwcatlem1 50702
            21.51.17  Kan extensions and related concepts   clan 50712
                  21.51.17.1  Kan extensions   clan 50712
                  21.51.17.2  Limits and colimits   clmd 50750
      ​21.52  Mathbox for Emmett Weisz
            ​*21.52.1  Miscellaneous Theorems   nfintd 50780
            21.52.2  Examples and properties of set recursion   setrecseq 50787
            ​*21.52.3  Construction of Games and Surreal Numbers   cpg 50801
      ​​*21.53  Mathbox for David A. Wheeler
            21.53.1  Natural deduction   sbidd 50810
            ​*21.53.2  Greater than, greater than or equal to   cge-real 50812
            ​*21.53.3  Hyperbolic trigonometric functions   csinh 50822
            ​*21.53.4  Reciprocal trigonometric functions (sec, csc, cot)   csec 50833
            ​*21.53.5  Identities for "if"   ifnmfalse 50858
            ​*21.53.6  Logarithms generalized to arbitrary base using ` logb `   logb2aval 50859
            ​*21.53.7  Logarithm laws generalized to an arbitrary base - log_   clog- 50860
            ​*21.53.8  Formally define notions such as reflexivity   wreflexive 50862
            ​*21.53.9  Algebra helpers   mvlraddi 50866
            ​*21.53.10  Algebra helper examples   i2linesi 50873
            ​*21.53.11  Formal methods "surprises"   alimp-surprise 50875
            ​*21.53.12  Allsome quantifier   wals 50881
            ​*21.53.13  Allsome one quantifier   walseu 50914
            ​*21.53.14  Miscellaneous   5m4e1 50934
            21.53.15  Theorems about algebraic numbers   aacllem 50938
      ​21.54  Mathbox for Mingli Yuan
      ​21.55  Mathbox for Jiamin Zhao
            21.55.1  Cross product and scalar triple product in RR^3   1ne3 50940
            21.55.2  Veronese map and linear dependence   cveronese 50967
      ​21.56  Mathbox for Kunhao Zheng
            21.56.1  Weighted AM-GM inequality   amgmwlem 50986