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  Preordered sets and directed sets using extensible structures
      9.2  Posets and lattices using extensible structures
PART 10  BASIC ALGEBRAIC STRUCTURES
      10.1  Monoids
      10.2  Groups
      10.3  Rings
      10.4  Division rings and fields
      10.5  Left modules
      10.6  Vector spaces
      10.7  Ideals
      10.8  The complex numbers as an algebraic extensible structure
      10.9  Generalized pre-Hilbert and Hilbert spaces
PART 11  BASIC LINEAR ALGEBRA
      11.1  Vectors and free modules
      11.2  Associative algebras
      11.3  Abstract multivariate polynomials
      11.4  Matrices
      11.5  The determinant
      11.6  Polynomial matrices
      11.7  The characteristic polynomial
PART 12  BASIC TOPOLOGY
      12.1  Topology
      12.2  Filters and filter bases
      12.3  Uniform Structures and Spaces
      12.4  Metric spaces
      12.5  Metric subcomplex vector spaces
PART 13  BASIC REAL AND COMPLEX ANALYSIS
      13.1  Continuity
      13.2  Integrals
      13.3  Derivatives
PART 14  BASIC REAL AND COMPLEX FUNCTIONS
      14.1  Polynomials
      14.2  Sequences and series
      14.3  Basic trigonometry
      14.4  Basic number theory
PART 15  ELEMENTARY GEOMETRY
      15.1  Definition and Tarski's Axioms of Geometry
      15.2  Tarskian Geometry
      15.3  Properties of geometries
      15.4  Geometry in Hilbert spaces
PART 16  GRAPH THEORY
      16.1  Vertices and edges
      16.2  Undirected graphs
      16.3  Walks, paths and cycles
      16.4  Eulerian paths and the Konigsberg Bridge problem
      16.5  The Friendship Theorem
PART 17  GUIDES AND MISCELLANEA
      17.1  Guides (conventions, explanations, and examples)
      17.2  Humor
      17.3  (Future - to be reviewed and classified)
PART 18  COMPLEX TOPOLOGICAL VECTOR SPACES (DEPRECATED)
      18.1  Additional material on group theory (deprecated)
      18.2  Complex vector spaces
      18.3  Normed complex vector spaces
      18.4  Operators on complex vector spaces
      18.5  Inner product (pre-Hilbert) spaces
      18.6  Complex Banach spaces
      18.7  Complex Hilbert spaces
PART 19  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      19.1  Axiomatization of complex pre-Hilbert spaces
      19.2  Inner product and norms
      19.3  Cauchy sequences and completeness axiom
      19.4  Subspaces and projections
      19.5  Properties of Hilbert subspaces
      19.6  Operators on Hilbert spaces
      19.7  States on a Hilbert lattice and Godowski's equation
      19.8  Cover relation, atoms, exchange axiom, and modular symmetry
PART 20  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      ​20.1  Mathboxes for user contributions
      20.2  Mathbox for Stefan Allan
      20.3  Mathbox for Thierry Arnoux
      20.4  Mathbox for Jonathan Ben-Naim
      20.5  Mathbox for BTernaryTau
      20.6  Mathbox for Mario Carneiro
      20.7  Mathbox for Filip Cernatescu
      20.8  Mathbox for Paul Chapman
      20.9  Mathbox for Scott Fenton
      20.10  Mathbox for Jeff Hankins
      20.11  Mathbox for Anthony Hart
      20.12  Mathbox for Chen-Pang He
      20.13  Mathbox for Jeff Hoffman
      20.14  Mathbox for Asger C. Ipsen
      20.15  Mathbox for BJ
      20.16  Mathbox for Jim Kingdon
      20.17  Mathbox for ML
      20.18  Mathbox for Wolf Lammen
      20.19  Mathbox for Brendan Leahy
      20.20  Mathbox for Jeff Madsen
      20.21  Mathbox for Giovanni Mascellani
      20.22  Mathbox for Peter Mazsa
      20.23  Mathbox for Rodolfo Medina
      20.24  Mathbox for Norm Megill
      20.25  Mathbox for metakunt
      20.26  Mathbox for Steven Nguyen
      20.27  Mathbox for Igor Ieskov
      20.28  Mathbox for OpenAI
      20.29  Mathbox for Stefan O'Rear
      20.30  Mathbox for Jon Pennant
      20.31  Mathbox for Richard Penner
      20.32  Mathbox for Stanislas Polu
      20.33  Mathbox for Rohan Ridenour
      20.34  Mathbox for Steve Rodriguez
      20.35  Mathbox for Andrew Salmon
      20.36  Mathbox for Alan Sare
      20.37  Mathbox for Glauco Siliprandi
      20.38  Mathbox for Saveliy Skresanov
      20.39  Mathbox for Jarvin Udandy
      20.40  Mathbox for Adhemar
      20.41  Mathbox for Alexander van der Vekens
      20.42  Mathbox for Zhi Wang
      20.43  Mathbox for Emmett Weisz
      20.44  Mathbox for David A. Wheeler
      20.45  Mathbox for Kunhao Zheng

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

​​*PART 1  CLASSICAL FIRST-ORDER LOGIC WITH EQUALITY
      ​*1.1  Pre-logic
            *1.1.1  Inferences for assisting proof development   idi 1
      ​​*1.2  Propositional calculus
            1.2.1  Recursively define primitive wffs for propositional calculus   wn 3
            ​*1.2.2  The axioms of propositional calculus   ax-mp 5
            ​*1.2.3  Logical implication   mp2 9
            ​*1.2.4  Logical negation   con4 113
            ​*1.2.5  Logical equivalence   wb 209
            ​*1.2.6  Logical conjunction   wa 399
            ​*1.2.7  Logical disjunction   wo 844
            ​*1.2.8  Mixed connectives   jaao 952
            ​*1.2.9  The conditional operator for propositions   wif 1058
            ​*1.2.10  The weak deduction theorem for propositional calculus   elimh 1080
            1.2.11  Abbreviated conjunction and disjunction of three wff's   w3o 1083
            1.2.12  Logical "nand" (Sheffer stroke)   wnan 1482
            1.2.13  Logical "xor"   wxo 1502
            1.2.14  Logical "nor"   wnor 1521
            1.2.15  True and false constants   wal 1536
                  ​*1.2.15.1  Universal quantifier for use by df-tru   wal 1536
                  ​*1.2.15.2  Equality predicate for use by df-tru   cv 1537
                  1.2.15.3  The true constant   wtru 1539
                  1.2.15.4  The false constant   wfal 1550
            ​*1.2.16  Truth tables   truimtru 1561
                  1.2.16.1  Implication   truimtru 1561
                  1.2.16.2  Negation   nottru 1565
                  1.2.16.3  Equivalence   trubitru 1567
                  1.2.16.4  Conjunction   truantru 1571
                  1.2.16.5  Disjunction   truortru 1575
                  1.2.16.6  Alternative denial   trunantru 1579
                  1.2.16.7  Exclusive disjunction   truxortru 1583
                  1.2.16.8  Joint denial   trunortru 1587
            ​*1.2.17  Half adder and full adder in propositional calculus   whad 1594
                  1.2.17.1  Full adder: sum   whad 1594
                  1.2.17.2  Full adder: carry   wcad 1608
      ​1.3  Other axiomatizations related to classical propositional calculus
            ​*1.3.1  Minimal implicational calculus   minimp 1623
            ​*1.3.2  Implicational Calculus   impsingle 1629
            1.3.3  Derive the Lukasiewicz axioms from Meredith's sole axiom   meredith 1643
            1.3.4  Derive the standard axioms from the Lukasiewicz axioms   luklem1 1660
            ​*1.3.5  Derive Nicod's axiom from the standard axioms   nic-dfim 1671
            1.3.6  Derive the Lukasiewicz axioms from Nicod's axiom   nic-imp 1677
            1.3.7  Derive Nicod's Axiom from Lukasiewicz's First Sheffer Stroke Axiom   lukshef-ax1 1696
            1.3.8  Derive the Lukasiewicz Axioms from the Tarski-Bernays-Wajsberg Axioms   tbw-bijust 1700
            1.3.9  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's First CO Axiom   merco1 1715
            1.3.10  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's Second CO Axiom   merco2 1738
            1.3.11  Derive the Lukasiewicz axioms from the Russell-Bernays Axioms   rb-bijust 1751
            ​*1.3.12  Stoic logic non-modal portion (Chrysippus of Soli)   mptnan 1770
      ​​*1.4  Predicate calculus with equality: Tarski's system S2 (1 rule, 6 schemes)
            *1.4.1  Universal quantifier (continued); define "exists" and "not free"   wex 1781
                  1.4.1.1  Existential quantifier   wex 1781
                  1.4.1.2  Non-freeness predicate   wnf 1785
            1.4.2  Rule scheme ax-gen (Generalization)   ax-gen 1797
            1.4.3  Axiom scheme ax-4 (Quantified Implication)   ax-4 1811
                  ​*1.4.3.1  The empty domain of discourse   empty 1907
            1.4.4  Axiom scheme ax-5 (Distinctness) - first use of $d   ax-5 1911
            ​*1.4.5  Equality predicate (continued)   weq 1964
            1.4.6  Axiom scheme ax-6 (Existence)   ax-6 1970
            1.4.7  Axiom scheme ax-7 (Equality)   ax-7 2015
            1.4.8  Define proper substitution   sbjust 2068
            1.4.9  Membership predicate   wcel 2111
            1.4.10  Axiom scheme ax-8 (Left Equality for Binary Predicate)   ax-8 2113
            1.4.11  Axiom scheme ax-9 (Right Equality for Binary Predicate)   ax-9 2121
            ​*1.4.12  Logical redundancy of ax-10 , ax-11 , ax-12 , ax-13   ax6dgen 2129
      ​​*1.5  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes)
            1.5.1  Axiom scheme ax-10 (Quantified Negation)   ax-10 2142
            1.5.2  Axiom scheme ax-11 (Quantifier Commutation)   ax-11 2158
            1.5.3  Axiom scheme ax-12 (Substitution)   ax-12 2175
            1.5.4  Axiom scheme ax-13 (Quantified Equality)   ax-13 2379
      ​1.6  Uniqueness and unique existence
            1.6.1  Uniqueness: the at-most-one quantifier   wmo 2555
            1.6.2  Unique existence: the unique existential quantifier   weu 2587
      ​1.7  Other axiomatizations related to classical predicate calculus
            ​*1.7.1  Aristotelian logic: Assertic syllogisms   barbara 2684
            ​*1.7.2  Intuitionistic logic   axia1 2714
​​*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 2729
            2.1.2  Classes   cab 2735
                  2.1.2.1  Class abstractions   cab 2735
                  ​*2.1.2.2  Class equality   df-cleq 2750
                  2.1.2.3  Class membership   df-clel 2830
                  2.1.2.4  Elementary properties of class abstractions   abeq2 2884
            2.1.3  Class form not-free predicate   wnfc 2899
            2.1.4  Negated equality and membership   wne 2951
                  2.1.4.1  Negated equality   wne 2951
                  2.1.4.2  Negated membership   wnel 3055
            2.1.5  Restricted quantification   wral 3070
            2.1.6  The universal class   cvv 3409
            ​*2.1.7  Conditional equality (experimental)   wcdeq 3677
            2.1.8  Russell's Paradox   rru 3693
            2.1.9  Proper substitution of classes for sets   wsbc 3696
            2.1.10  Proper substitution of classes for sets into classes   csb 3805
            2.1.11  Define basic set operations and relations   cdif 3855
            2.1.12  Subclasses and subsets   df-ss 3875
            2.1.13  The difference, union, and intersection of two classes   dfdif3 4020
                  2.1.13.1  The difference of two classes   dfdif3 4020
                  2.1.13.2  The union of two classes   elun 4054
                  2.1.13.3  The intersection of two classes   elini 4098
                  2.1.13.4  The symmetric difference of two classes   csymdif 4146
                  2.1.13.5  Combinations of difference, union, and intersection of two classes   unabs 4159
                  2.1.13.6  Class abstractions with difference, union, and intersection of two classes   unab 4202
                  2.1.13.7  Restricted uniqueness with difference, union, and intersection   reuss2 4217
            2.1.14  The empty set   c0 4225
            ​*2.1.15  The conditional operator for classes   cif 4420
            ​*2.1.16  The weak deduction theorem for set theory   dedth 4478
            2.1.17  Power classes   cpw 4494
            2.1.18  Unordered and ordered pairs   snjust 4521
            2.1.19  The union of a class   cuni 4798
            2.1.20  The intersection of a class   cint 4838
            2.1.21  Indexed union and intersection   ciun 4883
            2.1.22  Disjointness   wdisj 4997
            2.1.23  Binary relations   wbr 5032
            2.1.24  Ordered-pair class abstractions (class builders)   copab 5094
            2.1.25  Functions in maps-to notation   cmpt 5112
            2.1.26  Transitive classes   wtr 5138
      ​2.2  ZF Set Theory - add the Axiom of Replacement
            2.2.1  Introduce the Axiom of Replacement   ax-rep 5156
            2.2.2  Derive the Axiom of Separation   axsepgfromrep 5167
            2.2.3  Derive the Null Set Axiom   axnulALT 5174
            2.2.4  Theorems requiring subset and intersection existence   nalset 5183
            2.2.5  Theorems requiring empty set existence   class2set 5222
      ​2.3  ZF Set Theory - add the Axiom of Power Sets
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 5234
            2.3.2  Derive the Axiom of Pairing   axprlem1 5292
            2.3.3  Ordered pair theorem   opnz 5333
            2.3.4  Ordered-pair class abstractions (cont.)   opabidw 5382
            2.3.5  Power class of union and intersection   pwin 5424
            2.3.6  The identity relation   cid 5429
            2.3.7  The membership relation (or epsilon relation)   cep 5434
            ​*2.3.8  Partial and total orderings   wpo 5441
            2.3.9  Founded and well-ordering relations   wfr 5480
            2.3.10  Relations   cxp 5522
            2.3.11  The Predecessor Class   cpred 6125
            2.3.12  Well-founded induction   tz6.26 6157
            2.3.13  Ordinals   word 6168
            2.3.14  Definite description binder (inverted iota)   cio 6292
            2.3.15  Functions   wfun 6329
            2.3.16  Cantor's Theorem   canth 7105
            2.3.17  Restricted iota (description binder)   crio 7107
            2.3.18  Operations   co 7150
                  2.3.18.1  Variable-to-class conversion for operations   caovclg 7336
            2.3.19  Maps-to notation   mpondm0 7382
            2.3.20  Function operation   cof 7403
            2.3.21  Proper subset relation   crpss 7446
      ​2.4  ZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 7459
            2.4.2  Ordinals (continued)   epweon 7496
            2.4.3  Transfinite induction   tfi 7567
            2.4.4  The natural numbers (i.e., finite ordinals)   com 7579
            2.4.5  Peano's postulates   peano1 7600
            2.4.6  Finite induction (for finite ordinals)   find 7606
            2.4.7  Relations and functions (cont.)   dmexg 7613
            2.4.8  First and second members of an ordered pair   c1st 7691
            ​*2.4.9  The support of functions   csupp 7835
            ​*2.4.10  Special maps-to operations   opeliunxp2f 7886
            2.4.11  Function transposition   ctpos 7901
            2.4.12  Curry and uncurry   ccur 7941
            2.4.13  Undefined values   cund 7948
            2.4.14  Well-founded recursion   cwrecs 7956
            2.4.15  Functions on ordinals; strictly monotone ordinal functions   iunon 7986
            2.4.16  "Strong" transfinite recursion   crecs 8017
            2.4.17  Recursive definition generator   crdg 8055
            2.4.18  Finite recursion   frfnom 8080
            2.4.19  Ordinal arithmetic   c1o 8105
            2.4.20  Natural number arithmetic   nna0 8240
            2.4.21  Equivalence relations and classes   wer 8296
            2.4.22  The mapping operation   cmap 8416
            2.4.23  Infinite Cartesian products   cixp 8479
            2.4.24  Equinumerosity   cen 8524
            2.4.25  Schroeder-Bernstein Theorem   sbthlem1 8649
            2.4.26  Equinumerosity (cont.)   xpf1o 8701
            2.4.27  Pigeonhole Principle   phplem1 8718
            2.4.28  Finite sets   dif1enlem 8731
            2.4.29  Finitely supported functions   cfsupp 8866
            2.4.30  Finite intersections   cfi 8907
            2.4.31  Hall's marriage theorem   marypha1lem 8930
            2.4.32  Supremum and infimum   csup 8937
            2.4.33  Ordinal isomorphism, Hartogs's theorem   coi 9006
            2.4.34  Hartogs function   char 9053
            2.4.35  Weak dominance   cwdom 9061
      ​2.5  ZF Set Theory - add the Axiom of Regularity
            2.5.1  Introduce the Axiom of Regularity   ax-reg 9089
            2.5.2  Axiom of Infinity equivalents   inf0 9117
      ​2.6  ZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-inf 9134
            2.6.2  Existence of omega (the set of natural numbers)   omex 9139
            2.6.3  Cantor normal form   ccnf 9157
            2.6.4  Transitive closure   trcl 9203
            2.6.5  Rank   cr1 9224
            2.6.6  Scott's trick; collection principle; Hilbert's epsilon   scottex 9347
            2.6.7  Disjoint union   cdju 9360
            2.6.8  Cardinal numbers   ccrd 9397
            2.6.9  Axiom of Choice equivalents   wac 9575
            ​*2.6.10  Cardinal number arithmetic   undjudom 9627
            2.6.11  The Ackermann bijection   ackbij2lem1 9679
            2.6.12  Cofinality (without Axiom of Choice)   cflem 9706
            2.6.13  Eight inequivalent definitions of finite set   sornom 9737
            2.6.14  Hereditarily size-limited sets without Choice   itunifval 9876
​​*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 9895
            3.1.2  Introduce the Axiom of Dependent Choice   ax-dc 9906
      ​3.2  ZFC Set Theory - add the Axiom of Choice
            3.2.1  Introduce the Axiom of Choice   ax-ac 9919
            3.2.2  AC equivalents: well-ordering, Zorn's lemma   numthcor 9954
            3.2.3  Cardinal number theorems using Axiom of Choice   cardval 10006
            3.2.4  Cardinal number arithmetic using Axiom of Choice   iunctb 10034
            3.2.5  Cofinality using the Axiom of Choice   alephreg 10042
      ​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 10080
            3.4.2  Derivation of the Axiom of Choice   gchaclem 10138
​​*PART 4  TG (TARSKI-GROTHENDIECK) SET THEORY
      ​4.1  Inaccessibles
            4.1.1  Weakly and strongly inaccessible cardinals   cwina 10142
            4.1.2  Weak universes   cwun 10160
            4.1.3  Tarski classes   ctsk 10208
            4.1.4  Grothendieck universes   cgru 10250
      ​4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
            4.2.1  Introduce the Tarski-Grothendieck Axiom   ax-groth 10283
            4.2.2  Derive the Power Set, Infinity and Choice Axioms   grothpw 10286
            4.2.3  Tarski map function   ctskm 10297
​​*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 10304
            5.1.2  Final derivation of real and complex number postulates   axaddf 10605
            5.1.3  Real and complex number postulates restated as axioms   ax-cnex 10631
      ​5.2  Derive the basic properties from the field axioms
            5.2.1  Some deductions from the field axioms for complex numbers   cnex 10656
            5.2.2  Infinity and the extended real number system   cpnf 10710
            5.2.3  Restate the ordering postulates with extended real "less than"   axlttri 10750
            5.2.4  Ordering on reals   lttr 10755
            5.2.5  Initial properties of the complex numbers   mul12 10843
      ​5.3  Real and complex numbers - basic operations
            5.3.1  Addition   add12 10895
            5.3.2  Subtraction   cmin 10908
            5.3.3  Multiplication   kcnktkm1cn 11109
            5.3.4  Ordering on reals (cont.)   gt0ne0 11143
            5.3.5  Reciprocals   ixi 11307
            5.3.6  Division   cdiv 11335
            5.3.7  Ordering on reals (cont.)   elimgt0 11516
            5.3.8  Completeness Axiom and Suprema   fimaxre 11622
            5.3.9  Imaginary and complex number properties   inelr 11664
            5.3.10  Function operation analogue theorems   ofsubeq0 11671
      ​5.4  Integer sets
            5.4.1  Positive integers (as a subset of complex numbers)   cn 11674
            5.4.2  Principle of mathematical induction   nnind 11692
            ​*5.4.3  Decimal representation of numbers   c2 11729
            ​*5.4.4  Some properties of specific numbers   neg1cn 11788
            5.4.5  Simple number properties   halfcl 11899
            5.4.6  The Archimedean property   nnunb 11930
            5.4.7  Nonnegative integers (as a subset of complex numbers)   cn0 11934
            ​*5.4.8  Extended nonnegative integers   cxnn0 12006
            5.4.9  Integers (as a subset of complex numbers)   cz 12020
            5.4.10  Decimal arithmetic   cdc 12137
            5.4.11  Upper sets of integers   cuz 12282
            5.4.12  Well-ordering principle for bounded-below sets of integers   uzwo3 12383
            5.4.13  Rational numbers (as a subset of complex numbers)   cq 12388
            5.4.14  Existence of the set of complex numbers   rpnnen1lem2 12417
      ​5.5  Order sets
            5.5.1  Positive reals (as a subset of complex numbers)   crp 12430
            5.5.2  Infinity and the extended real number system (cont.)   cxne 12545
            5.5.3  Supremum and infimum on the extended reals   xrsupexmnf 12739
            5.5.4  Real number intervals   cioo 12779
            5.5.5  Finite intervals of integers   cfz 12939
            ​*5.5.6  Finite intervals of nonnegative integers   elfz2nn0 13047
            5.5.7  Half-open integer ranges   cfzo 13082
      ​5.6  Elementary integer functions
            5.6.1  The floor and ceiling functions   cfl 13209
            5.6.2  The modulo (remainder) operation   cmo 13286
            5.6.3  Miscellaneous theorems about integers   om2uz0i 13364
            5.6.4  Strong induction over upper sets of integers   uzsinds 13404
            5.6.5  Finitely supported functions over the nonnegative integers   fsuppmapnn0fiublem 13407
            5.6.6  The infinite sequence builder "seq" - extension   cseq 13418
            5.6.7  Integer powers   cexp 13479
            5.6.8  Ordered pair theorem for nonnegative integers   nn0le2msqi 13677
            5.6.9  Factorial function   cfa 13683
            5.6.10  The binomial coefficient operation   cbc 13712
            5.6.11  The ` # ` (set size) function   chash 13740
                  5.6.11.1  Proper unordered pairs and triples (sets of size 2 and 3)   hashprlei 13878
                  5.6.11.2  Functions with a domain containing at least two different elements   fundmge2nop0 13902
                  5.6.11.3  Finite induction on the size of the first component of a binary relation   hashdifsnp1 13906
      ​​*5.7  Words over a set
            5.7.1  Definitions and basic theorems   cword 13913
            5.7.2  Last symbol of a word   clsw 13961
            5.7.3  Concatenations of words   cconcat 13969
            5.7.4  Singleton words   cs1 13996
            5.7.5  Concatenations with singleton words   ccatws1cl 14017
            5.7.6  Subwords/substrings   csubstr 14049
            5.7.7  Prefixes of a word   cpfx 14079
            5.7.8  Subwords of subwords   swrdswrdlem 14113
            5.7.9  Subwords and concatenations   pfxcctswrd 14119
            5.7.10  Subwords of concatenations   swrdccatfn 14133
            5.7.11  Splicing words (substring replacement)   csplice 14158
            5.7.12  Reversing words   creverse 14167
            5.7.13  Repeated symbol words   creps 14177
            ​*5.7.14  Cyclical shifts of words   ccsh 14197
            5.7.15  Mapping words by a function   wrdco 14240
            5.7.16  Longer string literals   cs2 14250
      ​​*5.8  Reflexive and transitive closures of relations
            5.8.1  The reflexive and transitive properties of relations   coss12d 14379
            5.8.2  Basic properties of closures   cleq1lem 14389
            5.8.3  Definitions and basic properties of transitive closures   ctcl 14392
            5.8.4  Exponentiation of relations   crelexp 14426
            5.8.5  Reflexive-transitive closure as an indexed union   crtrcl 14462
            ​*5.8.6  Principle of transitive induction.   relexpindlem 14470
      ​5.9  Elementary real and complex functions
            5.9.1  The "shift" operation   cshi 14473
            5.9.2  Signum (sgn or sign) function   csgn 14493
            5.9.3  Real and imaginary parts; conjugate   ccj 14503
            5.9.4  Square root; absolute value   csqrt 14640
      ​5.10  Elementary limits and convergence
            5.10.1  Superior limit (lim sup)   clsp 14875
            5.10.2  Limits   cli 14889
            5.10.3  Finite and infinite sums   csu 15090
            5.10.4  The binomial theorem   binomlem 15232
            5.10.5  The inclusion/exclusion principle   incexclem 15239
            5.10.6  Infinite sums (cont.)   isumshft 15242
            5.10.7  Miscellaneous converging and diverging sequences   divrcnv 15255
            5.10.8  Arithmetic series   arisum 15263
            5.10.9  Geometric series   expcnv 15267
            5.10.10  Ratio test for infinite series convergence   cvgrat 15287
            5.10.11  Mertens' theorem   mertenslem1 15288
            5.10.12  Finite and infinite products   prodf 15291
                  5.10.12.1  Product sequences   prodf 15291
                  5.10.12.2  Non-trivial convergence   ntrivcvg 15301
                  5.10.12.3  Complex products   cprod 15307
                  5.10.12.4  Finite products   fprod 15343
                  5.10.12.5  Infinite products   iprodclim 15400
            5.10.13  Falling and Rising Factorial   cfallfac 15406
            5.10.14  Bernoulli polynomials and sums of k-th powers   cbp 15448
      ​5.11  Elementary trigonometry
            5.11.1  The exponential, sine, and cosine functions   ce 15463
                  5.11.1.1  The circle constant (tau = 2 pi)   ctau 15603
            5.11.2  _e is irrational   eirrlem 15605
      ​5.12  Cardinality of real and complex number subsets
            5.12.1  Countability of integers and rationals   xpnnen 15612
            5.12.2  The reals are uncountable   rpnnen2lem1 15615
​​*PART 6  ELEMENTARY NUMBER THEORY
      ​6.1  Elementary properties of divisibility
            6.1.1  Irrationality of square root of 2   sqrt2irrlem 15649
            6.1.2  Some Number sets are chains of proper subsets   nthruc 15653
            6.1.3  The divides relation   cdvds 15655
            ​*6.1.4  Even and odd numbers   evenelz 15737
            6.1.5  The division algorithm   divalglem0 15794
            6.1.6  Bit sequences   cbits 15818
            6.1.7  The greatest common divisor operator   cgcd 15893
            6.1.8  Bézout's identity   bezoutlem1 15938
            6.1.9  Algorithms   nn0seqcvgd 15966
            6.1.10  Euclid's Algorithm   eucalgval2 15977
            ​*6.1.11  The least common multiple   clcm 15984
            ​*6.1.12  Coprimality and Euclid's lemma   coprmgcdb 16045
            6.1.13  Cancellability of congruences   congr 16060
      ​6.2  Elementary prime number theory
            ​*6.2.1  Elementary properties   cprime 16067
            ​*6.2.2  Coprimality and Euclid's lemma (cont.)   coprm 16107
            6.2.3  Properties of the canonical representation of a rational   cnumer 16128
            6.2.4  Euler's theorem   codz 16155
            6.2.5  Arithmetic modulo a prime number   modprm1div 16189
            6.2.6  Pythagorean Triples   coprimeprodsq 16200
            6.2.7  The prime count function   cpc 16228
            6.2.8  Pocklington's theorem   prmpwdvds 16295
            6.2.9  Infinite primes theorem   unbenlem 16299
            6.2.10  Sum of prime reciprocals   prmreclem1 16307
            6.2.11  Fundamental theorem of arithmetic   1arithlem1 16314
            6.2.12  Lagrange's four-square theorem   cgz 16320
            6.2.13  Van der Waerden's theorem   cvdwa 16356
            6.2.14  Ramsey's theorem   cram 16390
            ​*6.2.15  Primorial function   cprmo 16422
            ​*6.2.16  Prime gaps   prmgaplem1 16440
            6.2.17  Decimal arithmetic (cont.)   dec2dvds 16454
            6.2.18  Cyclical shifts of words (cont.)   cshwsidrepsw 16485
            6.2.19  Specific prime numbers   prmlem0 16497
            6.2.20  Very large primes   1259lem1 16522
​PART 7  BASIC STRUCTURES
      ​7.1  Extensible structures
            ​*7.1.1  Basic definitions   cstr 16537
            7.1.2  Slot definitions   cplusg 16623
            7.1.3  Definition of the structure product   crest 16752
            7.1.4  Definition of the structure quotient   cordt 16830
      ​7.2  Moore spaces
            7.2.1  Moore closures   mrcflem 16935
            7.2.2  Independent sets in a Moore system   mrisval 16959
            7.2.3  Algebraic closure systems   isacs 16980
​PART 8  BASIC CATEGORY THEORY
      ​8.1  Categories
            8.1.1  Categories   ccat 16993
            8.1.2  Opposite category   coppc 17039
            8.1.3  Monomorphisms and epimorphisms   cmon 17057
            8.1.4  Sections, inverses, isomorphisms   csect 17073
            ​*8.1.5  Isomorphic objects   ccic 17124
            8.1.6  Subcategories   cssc 17136
            8.1.7  Functors   cfunc 17183
            8.1.8  Full & faithful functors   cful 17231
            8.1.9  Natural transformations and the functor category   cnat 17270
            8.1.10  Initial, terminal and zero objects of a category   cinito 17307
      ​8.2  Arrows (disjointified hom-sets)
            8.2.1  Identity and composition for arrows   cida 17379
      ​8.3  Examples of categories
            8.3.1  The category of sets   csetc 17401
            8.3.2  The category of categories   ccatc 17420
            ​*8.3.3  The category of extensible structures   fncnvimaeqv 17436
      ​8.4  Categorical constructions
            8.4.1  Product of categories   cxpc 17484
            8.4.2  Functor evaluation   cevlf 17525
            8.4.3  Hom functor   chof 17564
​PART 9  BASIC ORDER THEORY
      ​9.1  Preordered sets and directed sets using extensible structures
      ​9.2  Posets and lattices using extensible structures
            9.2.1  Posets   cpo 17616
            9.2.2  Lattices   clat 17721
            9.2.3  The dual of an ordered set   codu 17804
            9.2.4  Subset order structures   cipo 17827
            9.2.5  Distributive lattices   latmass 17864
            9.2.6  Posets and lattices as relations   cps 17874
            9.2.7  Directed sets, nets   cdir 17904
​PART 10  BASIC ALGEBRAIC STRUCTURES
      ​10.1  Monoids
            ​*10.1.1  Magmas   cplusf 17915
            ​*10.1.2  Identity elements   mgmidmo 17936
            ​*10.1.3  Iterated sums in a magma   gsumvalx 17952
            ​*10.1.4  Semigroups   csgrp 17966
            ​*10.1.5  Definition and basic properties of monoids   cmnd 17977
            10.1.6  Monoid homomorphisms and submonoids   cmhm 18020
            ​*10.1.7  Iterated sums in a monoid   gsumvallem2 18064
            10.1.8  Free monoids   cfrmd 18078
                  ​*10.1.8.1  Monoid of endofunctions   cefmnd 18099
            10.1.9  Examples and counterexamples for magmas, semigroups and monoids   mgm2nsgrplem1 18149
      ​10.2  Groups
            10.2.1  Definition and basic properties   cgrp 18169
            ​*10.2.2  Group multiple operation   cmg 18291
            10.2.3  Subgroups and Quotient groups   csubg 18340
            ​*10.2.4  Cyclic monoids and groups   cycsubmel 18410
            10.2.5  Elementary theory of group homomorphisms   cghm 18422
            10.2.6  Isomorphisms of groups   cgim 18464
            10.2.7  Group actions   cga 18486
            10.2.8  Centralizers and centers   ccntz 18512
            10.2.9  The opposite group   coppg 18540
            10.2.10  Symmetric groups   csymg 18562
                  ​*10.2.10.1  Definition and basic properties   csymg 18562
                  10.2.10.2  Cayley's theorem   cayleylem1 18607
                  10.2.10.3  Permutations fixing one element   symgfix2 18611
                  ​*10.2.10.4  Transpositions in the symmetric group   cpmtr 18636
                  10.2.10.5  The sign of a permutation   cpsgn 18684
            10.2.11  p-Groups and Sylow groups; Sylow's theorems   cod 18719
            10.2.12  Direct products   clsm 18826
                  10.2.12.1  Direct products (extension)   smndlsmidm 18848
            10.2.13  Free groups   cefg 18899
            10.2.14  Abelian groups   ccmn 18973
                  10.2.14.1  Definition and basic properties   ccmn 18973
                  10.2.14.2  Cyclic groups   ccyg 19064
                  10.2.14.3  Group sum operation   gsumval3a 19091
                  10.2.14.4  Group sums over (ranges of) integers   fsfnn0gsumfsffz 19171
                  10.2.14.5  Internal direct products   cdprd 19183
                  10.2.14.6  The Fundamental Theorem of Abelian Groups   ablfacrplem 19255
            10.2.15  Simple groups   csimpg 19280
                  10.2.15.1  Definition and basic properties   csimpg 19280
                  10.2.15.2  Classification of abelian simple groups   ablsimpnosubgd 19294
      ​10.3  Rings
            10.3.1  Multiplicative Group   cmgp 19307
            10.3.2  Ring unit   cur 19319
                  10.3.2.1  Semirings   csrg 19323
                  ​*10.3.2.2  The binomial theorem for semirings   srgbinomlem1 19358
            10.3.3  Definition and basic properties of unital rings   crg 19365
            10.3.4  Opposite ring   coppr 19443
            10.3.5  Divisibility   cdsr 19459
            10.3.6  Ring primes   crpm 19533
            10.3.7  Ring homomorphisms   crh 19535
      ​10.4  Division rings and fields
            10.4.1  Definition and basic properties   cdr 19570
            10.4.2  Subrings of a ring   csubrg 19599
                  10.4.2.1  Sub-division rings   csdrg 19640
            10.4.3  Absolute value (abstract algebra)   cabv 19655
            10.4.4  Star rings   cstf 19682
      ​10.5  Left modules
            10.5.1  Definition and basic properties   clmod 19702
            10.5.2  Subspaces and spans in a left module   clss 19771
            10.5.3  Homomorphisms and isomorphisms of left modules   clmhm 19859
            10.5.4  Subspace sum; bases for a left module   clbs 19914
      ​10.6  Vector spaces
            10.6.1  Definition and basic properties   clvec 19942
      ​10.7  Ideals
            10.7.1  The subring algebra; ideals   csra 20008
            10.7.2  Two-sided ideals and quotient rings   c2idl 20072
            10.7.3  Principal ideal rings. Divisibility in the integers   clpidl 20082
            10.7.4  Nonzero rings and zero rings   cnzr 20098
            10.7.5  Left regular elements. More kinds of rings   crlreg 20120
      ​10.8  The complex numbers as an algebraic extensible structure
            10.8.1  Definition and basic properties   cpsmet 20150
            ​*10.8.2  Ring of integers   zring 20238
            10.8.3  Algebraic constructions based on the complex numbers   czrh 20269
            10.8.4  Signs as subgroup of the complex numbers   cnmsgnsubg 20342
            10.8.5  Embedding of permutation signs into a ring   zrhpsgnmhm 20349
            10.8.6  The ordered field of real numbers   crefld 20369
      ​10.9  Generalized pre-Hilbert and Hilbert spaces
            10.9.1  Definition and basic properties   cphl 20389
            10.9.2  Orthocomplements and closed subspaces   cocv 20425
            10.9.3  Orthogonal projection and orthonormal bases   cpj 20465
​​*PART 11  BASIC LINEAR ALGEBRA
      ​11.1  Vectors and free modules
            *11.1.1  Direct sum of left modules   cdsmm 20496
            ​*11.1.2  Free modules   cfrlm 20511
            ​*11.1.3  Standard basis (unit vectors)   cuvc 20547
            ​*11.1.4  Independent sets and families   clindf 20569
            11.1.5  Characterization of free modules   lmimlbs 20601
      ​11.2  Associative algebras
            11.2.1  Definition and basic properties   casa 20615
      ​11.3  Abstract multivariate polynomials
            11.3.1  Definition and basic properties   cmps 20666
            11.3.2  Polynomial evaluation   ces 20833
            11.3.3  Additional definitions for (multivariate) polynomials   cslv 20871
            ​*11.3.4  Univariate polynomials   cps1 20899
            11.3.5  Univariate polynomial evaluation   ces1 21032
      ​​*11.4  Matrices
            *11.4.1  The matrix multiplication   cmmul 21085
            ​*11.4.2  Square matrices   cmat 21107
            ​*11.4.3  The matrix algebra   matmulr 21138
            ​*11.4.4  Matrices of dimension 0 and 1   mat0dimbas0 21166
            ​*11.4.5  The subalgebras of diagonal and scalar matrices   cdmat 21188
            ​*11.4.6  Multiplication of a matrix with a "column vector"   cmvmul 21240
            11.4.7  Replacement functions for a square matrix   cmarrep 21256
            11.4.8  Submatrices   csubma 21276
      ​11.5  The determinant
            11.5.1  Definition and basic properties   cmdat 21284
            11.5.2  Determinants of 2 x 2 -matrices   m2detleiblem1 21324
            11.5.3  The matrix adjugate/adjunct   cmadu 21332
            ​*11.5.4  Laplace expansion of determinants (special case)   symgmatr01lem 21353
            11.5.5  Inverse matrix   invrvald 21376
            ​*11.5.6  Cramer's rule   slesolvec 21379
      ​​*11.6  Polynomial matrices
            11.6.1  Basic properties   pmatring 21392
            ​*11.6.2  Constant polynomial matrices   ccpmat 21403
            ​*11.6.3  Collecting coefficients of polynomial matrices   cdecpmat 21462
            ​*11.6.4  Ring isomorphism between polynomial matrices and polynomials over matrices   cpm2mp 21492
      ​​*11.7  The characteristic polynomial
            *11.7.1  Definition and basic properties   cchpmat 21526
            ​*11.7.2  The characteristic factor function G   fvmptnn04if 21549
            ​*11.7.3  The Cayley-Hamilton theorem   cpmadurid 21567
​PART 12  BASIC TOPOLOGY
      ​12.1  Topology
            ​*12.1.1  Topological spaces   ctop 21593
                  12.1.1.1  Topologies   ctop 21593
                  12.1.1.2  Topologies on sets   ctopon 21610
                  12.1.1.3  Topological spaces   ctps 21632
            12.1.2  Topological bases   ctb 21645
            12.1.3  Examples of topologies   distop 21695
            12.1.4  Closure and interior   ccld 21716
            12.1.5  Neighborhoods   cnei 21797
            12.1.6  Limit points and perfect sets   clp 21834
            12.1.7  Subspace topologies   restrcl 21857
            12.1.8  Order topology   ordtbaslem 21888
            12.1.9  Limits and continuity in topological spaces   ccn 21924
            12.1.10  Separated spaces: T0, T1, T2 (Hausdorff) ...   ct0 22006
            12.1.11  Compactness   ccmp 22086
            12.1.12  Bolzano-Weierstrass theorem   bwth 22110
            12.1.13  Connectedness   cconn 22111
            12.1.14  First- and second-countability   c1stc 22137
            12.1.15  Local topological properties   clly 22164
            12.1.16  Refinements   cref 22202
            12.1.17  Compactly generated spaces   ckgen 22233
            12.1.18  Product topologies   ctx 22260
            12.1.19  Continuous function-builders   cnmptid 22361
            12.1.20  Quotient maps and quotient topology   ckq 22393
            12.1.21  Homeomorphisms   chmeo 22453
      ​12.2  Filters and filter bases
            12.2.1  Filter bases   elmptrab 22527
            12.2.2  Filters   cfil 22545
            12.2.3  Ultrafilters   cufil 22599
            12.2.4  Filter limits   cfm 22633
            12.2.5  Extension by continuity   ccnext 22759
            12.2.6  Topological groups   ctmd 22770
            12.2.7  Infinite group sum on topological groups   ctsu 22826
            12.2.8  Topological rings, fields, vector spaces   ctrg 22856
      ​12.3  Uniform Structures and Spaces
            12.3.1  Uniform structures   cust 22900
            12.3.2  The topology induced by an uniform structure   cutop 22931
            12.3.3  Uniform Spaces   cuss 22954
            12.3.4  Uniform continuity   cucn 22976
            12.3.5  Cauchy filters in uniform spaces   ccfilu 22987
            12.3.6  Complete uniform spaces   ccusp 22998
      ​12.4  Metric spaces
            12.4.1  Pseudometric spaces   ispsmet 23006
            12.4.2  Basic metric space properties   cxms 23019
            12.4.3  Metric space balls   blfvalps 23085
            12.4.4  Open sets of a metric space   mopnval 23140
            12.4.5  Continuity in metric spaces   metcnp3 23242
            12.4.6  The uniform structure generated by a metric   metuval 23251
            12.4.7  Examples of metric spaces   dscmet 23274
            ​*12.4.8  Normed algebraic structures   cnm 23278
            12.4.9  Normed space homomorphisms (bounded linear operators)   cnmo 23407
            12.4.10  Topology on the reals   qtopbaslem 23460
            12.4.11  Topological definitions using the reals   cii 23576
            12.4.12  Path homotopy   chtpy 23668
            12.4.13  The fundamental group   cpco 23701
      ​12.5  Metric subcomplex vector spaces
            12.5.1  Subcomplex modules   cclm 23763
            ​*12.5.2  Subcomplex vector spaces   ccvs 23824
            ​*12.5.3  Normed subcomplex vector spaces   isncvsngp 23850
            12.5.4  Subcomplex pre-Hilbert space   ccph 23867
            12.5.5  Convergence and completeness   ccfil 23952
            12.5.6  Baire's Category Theorem   bcthlem1 24024
            12.5.7  Banach spaces and subcomplex Hilbert spaces   ccms 24032
                  12.5.7.1  The complete ordered field of the real numbers   retopn 24079
            12.5.8  Euclidean spaces   crrx 24083
            12.5.9  Minimizing Vector Theorem   minveclem1 24124
            12.5.10  Projection Theorem   pjthlem1 24137
​PART 13  BASIC REAL AND COMPLEX ANALYSIS
      ​13.1  Continuity
            13.1.1  Intermediate value theorem   pmltpclem1 24148
      ​13.2  Integrals
            13.2.1  Lebesgue measure   covol 24162
            13.2.2  Lebesgue integration   cmbf 24314
                  13.2.2.1  Lesbesgue integral   cmbf 24314
                  13.2.2.2  Lesbesgue directed integral   cdit 24545
      ​13.3  Derivatives
            13.3.1  Real and complex differentiation   climc 24561
                  13.3.1.1  Derivatives of functions of one complex or real variable   climc 24561
                  13.3.1.2  Results on real differentiation   dvferm1lem 24683
​PART 14  BASIC REAL AND COMPLEX FUNCTIONS
      ​14.1  Polynomials
            14.1.1  Polynomial degrees   cmdg 24750
            14.1.2  The division algorithm for univariate polynomials   cmn1 24825
            14.1.3  Elementary properties of complex polynomials   cply 24880
            14.1.4  The division algorithm for polynomials   cquot 24985
            14.1.5  Algebraic numbers   caa 25009
            14.1.6  Liouville's approximation theorem   aalioulem1 25027
      ​14.2  Sequences and series
            14.2.1  Taylor polynomials and Taylor's theorem   ctayl 25047
            14.2.2  Uniform convergence   culm 25070
            14.2.3  Power series   pserval 25104
      ​14.3  Basic trigonometry
            14.3.1  The exponential, sine, and cosine functions (cont.)   efcn 25137
            14.3.2  Properties of pi = 3.14159...   pilem1 25145
            14.3.3  Mapping of the exponential function   efgh 25232
            14.3.4  The natural logarithm on complex numbers   clog 25245
            ​*14.3.5  Logarithms to an arbitrary base   clogb 25449
            14.3.6  Theorems of Pythagoras, isosceles triangles, and intersecting chords   angval 25486
            14.3.7  Solutions of quadratic, cubic, and quartic equations   quad2 25524
            14.3.8  Inverse trigonometric functions   casin 25547
            14.3.9  The Birthday Problem   log2ublem1 25631
            14.3.10  Areas in R^2   carea 25640
            14.3.11  More miscellaneous converging sequences   rlimcnp 25650
            14.3.12  Inequality of arithmetic and geometric means   cvxcl 25669
            14.3.13  Euler-Mascheroni constant   cem 25676
            14.3.14  Zeta function   czeta 25697
            14.3.15  Gamma function   clgam 25700
      ​14.4  Basic number theory
            14.4.1  Wilson's theorem   wilthlem1 25752
            14.4.2  The Fundamental Theorem of Algebra   ftalem1 25757
            14.4.3  The Basel problem (ζ(2) = π²/6)   basellem1 25765
            14.4.4  Number-theoretical functions   ccht 25775
            14.4.5  Perfect Number Theorem   mersenne 25910
            14.4.6  Characters of Z/nZ   cdchr 25915
            14.4.7  Bertrand's postulate   bcctr 25958
            ​*14.4.8  Quadratic residues and the Legendre symbol   clgs 25977
            ​*14.4.9  Gauss' Lemma   gausslemma2dlem0a 26039
            14.4.10  Quadratic reciprocity   lgseisenlem1 26058
            14.4.11  All primes 4n+1 are the sum of two squares   2sqlem1 26100
            14.4.12  Chebyshev's Weak Prime Number Theorem, Dirichlet's Theorem   chebbnd1lem1 26152
            14.4.13  The Prime Number Theorem   mudivsum 26213
            14.4.14  Ostrowski's theorem   abvcxp 26298
​​*PART 15  ELEMENTARY GEOMETRY
      ​15.1  Definition and Tarski's Axioms of Geometry
            15.1.1  Justification for the congruence notation   tgjustf 26366
      ​15.2  Tarskian Geometry
            15.2.1  Congruence   tgcgrcomimp 26370
            15.2.2  Betweenness   tgbtwntriv2 26380
            15.2.3  Dimension   tglowdim1 26393
            15.2.4  Betweenness and Congruence   tgifscgr 26401
            15.2.5  Congruence of a series of points   ccgrg 26403
            15.2.6  Motions   cismt 26425
            15.2.7  Colinearity   tglng 26439
            15.2.8  Connectivity of betweenness   tgbtwnconn1lem1 26465
            15.2.9  Less-than relation in geometric congruences   cleg 26475
            15.2.10  Rays   chlg 26493
            15.2.11  Lines   btwnlng1 26512
            15.2.12  Point inversions   cmir 26545
            15.2.13  Right angles   crag 26586
            15.2.14  Half-planes   islnopp 26632
            15.2.15  Midpoints and Line Mirroring   cmid 26665
            15.2.16  Congruence of angles   ccgra 26700
            15.2.17  Angle Comparisons   cinag 26728
            15.2.18  Congruence Theorems   tgsas1 26747
            15.2.19  Equilateral triangles   ceqlg 26758
      ​15.3  Properties of geometries
            15.3.1  Isomorphisms between geometries   f1otrgds 26762
      ​15.4  Geometry in Hilbert spaces
            15.4.1  Geometry in the complex plane   cchhllem 26780
            15.4.2  Geometry in Euclidean spaces   cee 26781
                  15.4.2.1  Definition of the Euclidean space   cee 26781
                  15.4.2.2  Tarski's axioms for geometry for the Euclidean space   axdimuniq 26806
                  15.4.2.3  EE^n fulfills Tarski's Axioms   ceeng 26870
​​*PART 16  GRAPH THEORY
      ​*16.1  Vertices and edges
            16.1.1  The edge function extractor for extensible structures   cedgf 26881
            ​*16.1.2  Vertices and indexed edges   cvtx 26888
                  16.1.2.1  Definitions and basic properties   cvtx 26888
                  16.1.2.2  The vertices and edges of a graph represented as ordered pair   opvtxval 26895
                  16.1.2.3  The vertices and edges of a graph represented as extensible structure   funvtxdmge2val 26903
                  16.1.2.4  Representations of graphs without edges   snstrvtxval 26929
                  16.1.2.5  Degenerated cases of representations of graphs   vtxval0 26931
            16.1.3  Edges as range of the edge function   cedg 26939
      ​​*16.2  Undirected graphs
            16.2.1  Undirected hypergraphs   cuhgr 26948
            16.2.2  Undirected pseudographs and multigraphs   cupgr 26972
            ​*16.2.3  Loop-free graphs   umgrislfupgrlem 27014
            16.2.4  Edges as subsets of vertices of graphs   uhgredgiedgb 27018
            ​*16.2.5  Undirected simple graphs   cuspgr 27040
            16.2.6  Examples for graphs   usgr0e 27125
            16.2.7  Subgraphs   csubgr 27156
            16.2.8  Finite undirected simple graphs   cfusgr 27205
            16.2.9  Neighbors, complete graphs and universal vertices   cnbgr 27221
                  16.2.9.1  Neighbors   cnbgr 27221
                  16.2.9.2  Universal vertices   cuvtx 27274
                  16.2.9.3  Complete graphs   ccplgr 27298
            16.2.10  Vertex degree   cvtxdg 27354
            ​*16.2.11  Regular graphs   crgr 27444
      ​​*16.3  Walks, paths and cycles
            *16.3.1  Walks   cewlks 27484
            16.3.2  Walks for loop-free graphs   lfgrwlkprop 27576
            16.3.3  Trails   ctrls 27579
            16.3.4  Paths and simple paths   cpths 27600
            16.3.5  Closed walks   cclwlks 27658
            16.3.6  Circuits and cycles   ccrcts 27672
            ​*16.3.7  Walks as words   cwwlks 27710
            16.3.8  Walks/paths of length 2 (as length 3 strings)   2wlkdlem1 27810
            16.3.9  Walks in regular graphs   rusgrnumwwlkl1 27853
            ​*16.3.10  Closed walks as words   cclwwlk 27865
                  16.3.10.1  Closed walks as words   cclwwlk 27865
                  16.3.10.2  Closed walks of a fixed length as words   cclwwlkn 27908
                  16.3.10.3  Closed walks on a vertex of a fixed length as words   cclwwlknon 27971
            16.3.11  Examples for walks, trails and paths   0ewlk 27998
            16.3.12  Connected graphs   cconngr 28070
      ​16.4  Eulerian paths and the Konigsberg Bridge problem
            ​*16.4.1  Eulerian paths   ceupth 28081
            ​*16.4.2  The Königsberg Bridge problem   konigsbergvtx 28130
      ​16.5  The Friendship Theorem
            16.5.1  Friendship graphs - basics   cfrgr 28142
            16.5.2  The friendship theorem for small graphs   frgr1v 28155
            16.5.3  Theorems according to Mertzios and Unger   2pthfrgrrn 28166
            ​*16.5.4  Huneke's Proof of the Friendship Theorem   frgrncvvdeqlem1 28183
​PART 17  GUIDES AND MISCELLANEA
      ​17.1  Guides (conventions, explanations, and examples)
            ​*17.1.1  Conventions   conventions 28284
            17.1.2  Natural deduction   natded 28287
            ​*17.1.3  Natural deduction examples   ex-natded5.2 28288
            17.1.4  Definitional examples   ex-or 28305
            17.1.5  Other examples   aevdemo 28344
      ​17.2  Humor
            17.2.1  April Fool's theorem   avril1 28347
      ​17.3  (Future - to be reviewed and classified)
            17.3.1  Planar incidence geometry   cplig 28356
            ​*17.3.2  Aliases kept to prevent broken links   dummylink 28369
​​*PART 18  COMPLEX TOPOLOGICAL VECTOR SPACES (DEPRECATED)
      ​*18.1  Additional material on group theory (deprecated)
            18.1.1  Definitions and basic properties for groups   cgr 28371
            18.1.2  Abelian groups   cablo 28426
      ​18.2  Complex vector spaces
            18.2.1  Definition and basic properties   cvc 28440
            18.2.2  Examples of complex vector spaces   cnaddabloOLD 28463
      ​18.3  Normed complex vector spaces
            18.3.1  Definition and basic properties   cnv 28466
            18.3.2  Examples of normed complex vector spaces   cnnv 28559
            18.3.3  Induced metric of a normed complex vector space   imsval 28567
            18.3.4  Inner product   cdip 28582
            18.3.5  Subspaces   css 28603
      ​18.4  Operators on complex vector spaces
            18.4.1  Definitions and basic properties   clno 28622
      ​18.5  Inner product (pre-Hilbert) spaces
            18.5.1  Definition and basic properties   ccphlo 28694
            18.5.2  Examples of pre-Hilbert spaces   cncph 28701
            18.5.3  Properties of pre-Hilbert spaces   isph 28704
      ​18.6  Complex Banach spaces
            18.6.1  Definition and basic properties   ccbn 28744
            18.6.2  Examples of complex Banach spaces   cnbn 28751
            18.6.3  Uniform Boundedness Theorem   ubthlem1 28752
            18.6.4  Minimizing Vector Theorem   minvecolem1 28756
      ​18.7  Complex Hilbert spaces
            18.7.1  Definition and basic properties   chlo 28767
            18.7.2  Standard axioms for a complex Hilbert space   hlex 28780
            18.7.3  Examples of complex Hilbert spaces   cnchl 28798
            18.7.4  Hellinger-Toeplitz Theorem   htthlem 28799
​​*PART 19  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      ​19.1  Axiomatization of complex pre-Hilbert spaces
            19.1.1  Basic Hilbert space definitions   chba 28801
            19.1.2  Preliminary ZFC lemmas   df-hnorm 28850
            ​*19.1.3  Derive the Hilbert space axioms from ZFC set theory   axhilex-zf 28863
            ​*19.1.4  Introduce the vector space axioms for a Hilbert space   ax-hilex 28881
            19.1.5  Vector operations   hvmulex 28893
            19.1.6  Inner product postulates for a Hilbert space   ax-hfi 28961
      ​19.2  Inner product and norms
            19.2.1  Inner product   his5 28968
            19.2.2  Norms   dfhnorm2 29004
            19.2.3  Relate Hilbert space to normed complex vector spaces   hilablo 29042
            19.2.4  Bunjakovaskij-Cauchy-Schwarz inequality   bcsiALT 29061
      ​19.3  Cauchy sequences and completeness axiom
            19.3.1  Cauchy sequences and limits   hcau 29066
            19.3.2  Derivation of the completeness axiom from ZF set theory   hilmet 29076
            19.3.3  Completeness postulate for a Hilbert space   ax-hcompl 29084
            19.3.4  Relate Hilbert space to ZFC pre-Hilbert and Hilbert spaces   hhcms 29085
      ​19.4  Subspaces and projections
            19.4.1  Subspaces   df-sh 29089
            19.4.2  Closed subspaces   df-ch 29103
            19.4.3  Orthocomplements   df-oc 29134
            19.4.4  Subspace sum, span, lattice join, lattice supremum   df-shs 29190
            19.4.5  Projection theorem   pjhthlem1 29273
            19.4.6  Projectors   df-pjh 29277
      ​19.5  Properties of Hilbert subspaces
            19.5.1  Orthomodular law   omlsilem 29284
            19.5.2  Projectors (cont.)   pjhtheu2 29298
            19.5.3  Hilbert lattice operations   sh0le 29322
            19.5.4  Span (cont.) and one-dimensional subspaces   spansn0 29423
            19.5.5  Commutes relation for Hilbert lattice elements   df-cm 29465
            19.5.6  Foulis-Holland theorem   fh1 29500
            19.5.7  Quantum Logic Explorer axioms   qlax1i 29509
            19.5.8  Orthogonal subspaces   chscllem1 29519
            19.5.9  Orthoarguesian laws 5OA and 3OA   5oalem1 29536
            19.5.10  Projectors (cont.)   pjorthi 29551
            19.5.11  Mayet's equation E_3   mayete3i 29610
      ​19.6  Operators on Hilbert spaces
            ​*19.6.1  Operator sum, difference, and scalar multiplication   df-hosum 29612
            19.6.2  Zero and identity operators   df-h0op 29630
            19.6.3  Operations on Hilbert space operators   hoaddcl 29640
            19.6.4  Linear, continuous, bounded, Hermitian, unitary operators and norms   df-nmop 29721
            19.6.5  Linear and continuous functionals and norms   df-nmfn 29727
            19.6.6  Adjoint   df-adjh 29731
            19.6.7  Dirac bra-ket notation   df-bra 29732
            19.6.8  Positive operators   df-leop 29734
            19.6.9  Eigenvectors, eigenvalues, spectrum   df-eigvec 29735
            19.6.10  Theorems about operators and functionals   nmopval 29738
            19.6.11  Riesz lemma   riesz3i 29944
            19.6.12  Adjoints (cont.)   cnlnadjlem1 29949
            19.6.13  Quantum computation error bound theorem   unierri 29986
            19.6.14  Dirac bra-ket notation (cont.)   branmfn 29987
            19.6.15  Positive operators (cont.)   leopg 30004
            19.6.16  Projectors as operators   pjhmopi 30028
      ​19.7  States on a Hilbert lattice and Godowski's equation
            19.7.1  States on a Hilbert lattice   df-st 30093
            19.7.2  Godowski's equation   golem1 30153
      ​19.8  Cover relation, atoms, exchange axiom, and modular symmetry
            19.8.1  Covers relation; modular pairs   df-cv 30161
            19.8.2  Atoms   df-at 30220
            19.8.3  Superposition principle   superpos 30236
            19.8.4  Atoms, exchange and covering properties, atomicity   chcv1 30237
            19.8.5  Irreducibility   chirredlem1 30272
            19.8.6  Atoms (cont.)   atcvat3i 30278
            19.8.7  Modular symmetry   mdsymlem1 30285
​PART 20  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      ​20.1  Mathboxes for user contributions
            20.1.1  Mathbox guidelines   mathbox 30324
      ​20.2  Mathbox for Stefan Allan
      ​20.3  Mathbox for Thierry Arnoux
            20.3.1  Propositional Calculus - misc additions   bian1d 30329
            20.3.2  Predicate Calculus   sbc2iedf 30336
                  20.3.2.1  Predicate Calculus - misc additions   sbc2iedf 30336
                  20.3.2.2  Restricted quantification - misc additions   ralcom4f 30339
                  20.3.2.3  Equality   eqtrb 30344
                  20.3.2.4  Double restricted existential uniqueness quantification   opsbc2ie 30345
                  20.3.2.5  Double restricted existential uniqueness quantification syntax   w2reu 30347
                  20.3.2.6  Substitution (without distinct variables) - misc additions   sbceqbidf 30356
                  20.3.2.7  Existential "at most one" - misc additions   moel 30358
                  20.3.2.8  Existential uniqueness - misc additions   reuxfrdf 30361
                  20.3.2.9  Restricted "at most one" - misc additions   rmoxfrd 30363
            20.3.3  General Set Theory   dmrab 30366
                  20.3.3.1  Class abstractions (a.k.a. class builders)   dmrab 30366
                  20.3.3.2  Image Sets   abrexdomjm 30374
                  20.3.3.3  Set relations and operations - misc additions   elunsn 30380
                  20.3.3.4  Unordered pairs   eqsnd 30399
                  20.3.3.5  Conditional operator - misc additions   ifeqeqx 30407
                  20.3.3.6  Set union   uniinn0 30412
                  20.3.3.7  Indexed union - misc additions   cbviunf 30417
                  20.3.3.8  Indexed intersection - misc additions   iinabrex 30430
                  20.3.3.9  Disjointness - misc additions   disjnf 30431
            20.3.4  Relations and Functions   xpdisjres 30459
                  20.3.4.1  Relations - misc additions   xpdisjres 30459
                  20.3.4.2  Functions - misc additions   ac6sf2 30482
                  20.3.4.3  Operations - misc additions   mpomptxf 30540
                  20.3.4.4  Explicit Functions with one or two points as a domain   cosnopne 30551
                  20.3.4.5  Isomorphisms - misc. add.   gtiso 30557
                  20.3.4.6  Disjointness (additional proof requiring functions)   disjdsct 30559
                  20.3.4.7  First and second members of an ordered pair - misc additions   df1stres 30560
                  20.3.4.8  Supremum - misc additions   supssd 30568
                  20.3.4.9  Finite Sets   imafi2 30570
                  20.3.4.10  Countable Sets   snct 30572
            20.3.5  Real and Complex Numbers   creq0 30594
                  20.3.5.1  Complex operations - misc. additions   creq0 30594
                  20.3.5.2  Ordering on reals - misc additions   lt2addrd 30598
                  20.3.5.3  Extended reals - misc additions   xrlelttric 30599
                  20.3.5.4  Extended nonnegative integers - misc additions   xnn0gt0 30616
                  20.3.5.5  Real number intervals - misc additions   joiniooico 30619
                  20.3.5.6  Finite intervals of integers - misc additions   uzssico 30629
                  20.3.5.7  Half-open integer ranges - misc additions   iundisjfi 30641
                  20.3.5.8  The ` # ` (set size) function - misc additions   hashunif 30650
                  20.3.5.9  The greatest common divisor operator - misc. add   dvdszzq 30653
                  20.3.5.10  Integers   nnindf 30657
                  20.3.5.11  Decimal numbers   dfdec100 30668
            ​*20.3.6  Decimal expansion   cdp2 30669
                  ​*20.3.6.1  Decimal point   cdp 30686
                  20.3.6.2  Division in the extended real number system   cxdiv 30715
            20.3.7  Words over a set - misc additions   wrdfd 30734
                  20.3.7.1  Splicing words (substring replacement)   splfv3 30754
                  20.3.7.2  Cyclic shift of words   1cshid 30755
            20.3.8  Extensible Structures   ressplusf 30759
                  20.3.8.1  Structure restriction operator   ressplusf 30759
                  20.3.8.2  The opposite group   oppgle 30762
                  20.3.8.3  Posets   ressprs 30764
                  20.3.8.4  Complete lattices   clatp0cl 30780
                  20.3.8.5  Order Theory   cmnt 30782
                  20.3.8.6  Extended reals Structure - misc additions   ax-xrssca 30808
                  20.3.8.7  The extended nonnegative real numbers commutative monoid   xrge0base 30820
            20.3.9  Algebra   abliso 30831
                  20.3.9.1  Monoids Homomorphisms   abliso 30831
                  20.3.9.2  Finitely supported group sums - misc additions   gsumsubg 30832
                  20.3.9.3  Centralizers and centers - misc additions   cntzun 30846
                  20.3.9.4  Totally ordered monoids and groups   comnd 30849
                  20.3.9.5  The symmetric group   symgfcoeu 30877
                  20.3.9.6  Transpositions   pmtridf1o 30887
                  20.3.9.7  Permutation Signs   psgnid 30890
                  20.3.9.8  Permutation cycles   ctocyc 30899
                  20.3.9.9  The Alternating Group   evpmval 30938
                  20.3.9.10  Signum in an ordered monoid   csgns 30951
                  20.3.9.11  The Archimedean property for generic ordered algebraic structures   cinftm 30956
                  20.3.9.12  Semiring left modules   cslmd 30979
                  20.3.9.13  Simple groups   prmsimpcyc 31007
                  20.3.9.14  Rings - misc additions   rngurd 31008
                  20.3.9.15  Subfields   primefldchr 31019
                  20.3.9.16  Totally ordered rings and fields   corng 31020
                  20.3.9.17  Ring homomorphisms - misc additions   rhmdvdsr 31043
                  20.3.9.18  Scalar restriction operation   cresv 31049
                  20.3.9.19  The commutative ring of gaussian integers   gzcrng 31064
                  20.3.9.20  The archimedean ordered field of real numbers   reofld 31065
                  20.3.9.21  The quotient map and quotient modules   qusker 31070
                  20.3.9.22  The ring of integers modulo ` N `   znfermltl 31083
                  20.3.9.23  Independent sets and families   islinds5 31084
                  ​*20.3.9.24  Subgroup sum / Sumset / Minkowski sum   elgrplsmsn 31099
                  20.3.9.25  The quotient map   quslsm 31114
                  20.3.9.26  Ideals   intlidl 31123
                  20.3.9.27  Prime Ideals   cprmidl 31131
                  20.3.9.28  Maximal Ideals   cmxidl 31152
                  20.3.9.29  The semiring of ideals of a ring   cidlsrg 31166
                  20.3.9.30  Unique factorization domains   cufd 31182
                  20.3.9.31  Associative algebras   asclmulg 31187
                  20.3.9.32  Univariate Polynomials   fply1 31188
                  20.3.9.33  The subring algebra   sra1r 31192
                  20.3.9.34  Division Ring Extensions   drgext0g 31198
                  20.3.9.35  Vector Spaces   lvecdimfi 31204
                  20.3.9.36  Vector Space Dimension   cldim 31205
            20.3.10  Field Extensions   cfldext 31234
            20.3.11  Matrices   csmat 31264
                  20.3.11.1  Submatrices   csmat 31264
                  20.3.11.2  Matrix literals   clmat 31282
                  20.3.11.3  Laplace expansion of determinants   mdetpmtr1 31294
            20.3.12  Topology   ist0cld 31304
                  20.3.12.1  Open maps   txomap 31305
                  20.3.12.2  Topology of the unit circle   qtopt1 31306
                  20.3.12.3  Refinements   reff 31310
                  20.3.12.4  Open cover refinement property   ccref 31313
                  20.3.12.5  Lindelöf spaces   cldlf 31323
                  20.3.12.6  Paracompact spaces   cpcmp 31326
                  ​*20.3.12.7  Spectrum of a ring   crspec 31333
                  20.3.12.8  Pseudometrics   cmetid 31357
                  20.3.12.9  Continuity - misc additions   hauseqcn 31369
                  20.3.12.10  Topology of the closed unit interval   elunitge0 31370
                  20.3.12.11  Topology of ` ( RR X. RR ) `   unicls 31374
                  20.3.12.12  Order topology - misc. additions   cnvordtrestixx 31384
                  20.3.12.13  Continuity in topological spaces - misc. additions   mndpluscn 31397
                  20.3.12.14  Topology of the extended nonnegative real numbers ordered monoid   xrge0hmph 31403
                  20.3.12.15  Limits - misc additions   lmlim 31418
                  20.3.12.16  Univariate polynomials   pl1cn 31426
            20.3.13  Uniform Stuctures and Spaces   chcmp 31427
                  20.3.13.1  Hausdorff uniform completion   chcmp 31427
            20.3.14  Topology and algebraic structures   zringnm 31429
                  20.3.14.1  The norm on the ring of the integer numbers   zringnm 31429
                  20.3.14.2  Topological ` ZZ ` -modules   zlm0 31431
                  20.3.14.3  Canonical embedding of the field of the rational numbers into a division ring   cqqh 31441
                  20.3.14.4  Canonical embedding of the real numbers into a complete ordered field   crrh 31462
                  20.3.14.5  Embedding from the extended real numbers into a complete lattice   cxrh 31485
                  20.3.14.6  Canonical embeddings into the ordered field of the real numbers   zrhre 31488
                  ​*20.3.14.7  Topological Manifolds   cmntop 31491
            20.3.15  Real and complex functions   nexple 31496
                  20.3.15.1  Integer powers - misc. additions   nexple 31496
                  20.3.15.2  Indicator Functions   cind 31497
                  20.3.15.3  Extended sum   cesum 31514
            20.3.16  Mixed Function/Constant operation   cofc 31582
            20.3.17  Abstract measure   csiga 31595
                  20.3.17.1  Sigma-Algebra   csiga 31595
                  20.3.17.2  Generated sigma-Algebra   csigagen 31625
                  ​*20.3.17.3  lambda and pi-Systems, Rings of Sets   ispisys 31639
                  20.3.17.4  The Borel algebra on the real numbers   cbrsiga 31668
                  20.3.17.5  Product Sigma-Algebra   csx 31675
                  20.3.17.6  Measures   cmeas 31682
                  20.3.17.7  The counting measure   cntmeas 31713
                  20.3.17.8  The Lebesgue measure - misc additions   voliune 31716
                  20.3.17.9  The Dirac delta measure   cdde 31719
                  20.3.17.10  The 'almost everywhere' relation   cae 31724
                  20.3.17.11  Measurable functions   cmbfm 31736
                  20.3.17.12  Borel Algebra on ` ( RR X. RR ) `   br2base 31755
                  ​*20.3.17.13  Caratheodory's extension theorem   coms 31777
            20.3.18  Integration   itgeq12dv 31812
                  20.3.18.1  Lebesgue integral - misc additions   itgeq12dv 31812
                  20.3.18.2  Bochner integral   citgm 31813
            20.3.19  Euler's partition theorem   oddpwdc 31840
            20.3.20  Sequences defined by strong recursion   csseq 31869
            20.3.21  Fibonacci Numbers   cfib 31882
            20.3.22  Probability   cprb 31893
                  20.3.22.1  Probability Theory   cprb 31893
                  20.3.22.2  Conditional Probabilities   ccprob 31917
                  20.3.22.3  Real-valued Random Variables   crrv 31926
                  20.3.22.4  Preimage set mapping operator   corvc 31941
                  20.3.22.5  Distribution Functions   orvcelval 31954
                  20.3.22.6  Cumulative Distribution Functions   orvclteel 31958
                  20.3.22.7  Probabilities - example   coinfliplem 31964
                  20.3.22.8  Bertrand's Ballot Problem   ballotlemoex 31971
            20.3.23  Signum (sgn or sign) function - misc. additions   sgncl 32024
                  20.3.23.1  Operations on words   ccatmulgnn0dir 32040
            20.3.24  Polynomials with real coefficients - misc additions   plymul02 32044
            20.3.25  Descartes's rule of signs   signspval 32050
                  20.3.25.1  Sign changes in a word over real numbers   signspval 32050
                  20.3.25.2  Counting sign changes in a word over real numbers   signslema 32060
            20.3.26  Number Theory   efcld 32090
                  20.3.26.1  Representations of a number as sums of integers   crepr 32107
                  20.3.26.2  Vinogradov Trigonometric Sums and the Circle Method   cvts 32134
                  20.3.26.3  The Ternary Goldbach Conjecture: Final Statement   ax-hgt749 32143
            20.3.27  Elementary Geometry   cstrkg2d 32163
                  ​*20.3.27.1  Two-dimensional geometry   cstrkg2d 32163
                  20.3.27.2  Outer Five Segment (not used, no need to move to main)   cafs 32168
            ​*20.3.28  LeftPad Project   clpad 32173
      ​​*20.4  Mathbox for Jonathan Ben-Naim
            20.4.1  First-order logic and set theory   bnj170 32196
            20.4.2  Well founded induction and recursion   bnj110 32358
            20.4.3  The existence of a minimal element in certain classes   bnj69 32510
            20.4.4  Well-founded induction   bnj1204 32512
            20.4.5  Well-founded recursion, part 1 of 3   bnj60 32562
            20.4.6  Well-founded recursion, part 2 of 3   bnj1500 32568
            20.4.7  Well-founded recursion, part 3 of 3   bnj1522 32572
      ​20.5  Mathbox for BTernaryTau
            20.5.1  Acyclic graphs   cacycgr 32620
      ​20.6  Mathbox for Mario Carneiro
            20.6.1  Predicate calculus with all distinct variables   ax-7d 32637
            20.6.2  Miscellaneous stuff   quartfull 32643
            20.6.3  Derangements and the Subfactorial   deranglem 32644
            20.6.4  The Erdős-Szekeres theorem   erdszelem1 32669
            20.6.5  The Kuratowski closure-complement theorem   kur14lem1 32684
            20.6.6  Retracts and sections   cretr 32695
            20.6.7  Path-connected and simply connected spaces   cpconn 32697
            20.6.8  Covering maps   ccvm 32733
            20.6.9  Normal numbers   snmlff 32807
            20.6.10  Godel-sets of formulas - part 1   cgoe 32811
            20.6.11  Godel-sets of formulas - part 2   cgon 32910
            20.6.12  Models of ZF   cgze 32924
            ​*20.6.13  Metamath formal systems   cmcn 32938
            20.6.14  Grammatical formal systems   cm0s 33063
            20.6.15  Models of formal systems   cmuv 33083
            20.6.16  Splitting fields   citr 33105
            20.6.17  p-adic number fields   czr 33121
      ​​*20.7  Mathbox for Filip Cernatescu
      ​20.8  Mathbox for Paul Chapman
            20.8.1  Real and complex numbers (cont.)   climuzcnv 33145
            20.8.2  Miscellaneous theorems   elfzm12 33149
      ​20.9  Mathbox for Scott Fenton
            20.9.1  ZFC Axioms in primitive form   axextprim 33158
            20.9.2  Untangled classes   untelirr 33165
            20.9.3  Extra propositional calculus theorems   3orel2 33172
            20.9.4  Misc. Useful Theorems   nepss 33180
            20.9.5  Properties of real and complex numbers   sqdivzi 33208
            20.9.6  Infinite products   iprodefisumlem 33221
            20.9.7  Factorial limits   faclimlem1 33224
            20.9.8  Greatest common divisor and divisibility   gcd32 33230
            20.9.9  Properties of relationships   brtp 33232
            20.9.10  Properties of functions and mappings   funpsstri 33255
            20.9.11  Set induction (or epsilon induction)   setinds 33270
            20.9.12  Ordinal numbers   elpotr 33273
            20.9.13  Defined equality axioms   axextdfeq 33289
            20.9.14  Hypothesis builders   hbntg 33297
            20.9.15  (Trans)finite Recursion Theorems   tfisg 33302
            20.9.16  Transitive closure under a relationship   ctrpred 33303
            20.9.17  Founded Induction   frpomin 33325
            20.9.18  Ordering Cross Products, Part 2   xpord2lem 33344
            20.9.19  Ordering Ordinal Sequences   orderseqlem 33356
            20.9.20  Well-founded zero, successor, and limits   cwsuc 33359
            20.9.21  Founded Partial Recursion   cfrecs 33379
            20.9.22  Natural operations on ordinals   cnadd 33409
            20.9.23  Surreal Numbers   csur 33428
            20.9.24  Surreal Numbers: Ordering   sltsolem1 33463
            20.9.25  Surreal Numbers: Birthday Function   bdayfo 33465
            20.9.26  Surreal Numbers: Density   fvnobday 33466
            ​*20.9.27  Surreal Numbers: Full-Eta Property   bdayimaon 33481
            20.9.28  Surreal numbers - ordering theorems   csle 33532
            20.9.29  Surreal numbers - birthday theorems   bdayfun 33552
            20.9.30  Surreal numbers: Conway cuts   csslt 33560
            20.9.31  Surreal numbers - zero and one   c0s 33598
            20.9.32  Surreal numbers - cuts and options   cmade 33608
            20.9.33  Surreal numbers: Induction and recursion on one variable   cnorec 33664
            20.9.34  Surreal numbers: Induction and recursion on two variables   cnorec2 33675
            20.9.35  Surreal numbers - addition, negation, and subtraction   cadds 33687
            20.9.36  Quantifier-free definitions   ctxp 33703
            20.9.37  Alternate ordered pairs   caltop 33829
            20.9.38  Geometry in the Euclidean space   cofs 33855
                  20.9.38.1  Congruence properties   cofs 33855
                  20.9.38.2  Betweenness properties   btwntriv2 33885
                  20.9.38.3  Segment Transportation   ctransport 33902
                  20.9.38.4  Properties relating betweenness and congruence   cifs 33908
                  20.9.38.5  Connectivity of betweenness   btwnconn1lem1 33960
                  20.9.38.6  Segment less than or equal to   csegle 33979
                  20.9.38.7  Outside-of relationship   coutsideof 33992
                  20.9.38.8  Lines and Rays   cline2 34007
            20.9.39  Forward difference   cfwddif 34031
            20.9.40  Rank theorems   rankung 34039
            20.9.41  Hereditarily Finite Sets   chf 34045
      ​20.10  Mathbox for Jeff Hankins
            20.10.1  Miscellany   a1i14 34060
            20.10.2  Basic topological facts   topbnd 34084
            20.10.3  Topology of the real numbers   ivthALT 34095
            20.10.4  Refinements   cfne 34096
            20.10.5  Neighborhood bases determine topologies   neibastop1 34119
            20.10.6  Lattice structure of topologies   topmtcl 34123
            20.10.7  Filter bases   fgmin 34130
            20.10.8  Directed sets, nets   tailfval 34132
      ​20.11  Mathbox for Anthony Hart
            20.11.1  Propositional Calculus   tb-ax1 34143
            20.11.2  Predicate Calculus   nalfal 34163
            20.11.3  Miscellaneous single axioms   meran1 34171
            20.11.4  Connective Symmetry   negsym1 34177
      ​20.12  Mathbox for Chen-Pang He
            20.12.1  Ordinal topology   ontopbas 34188
      ​20.13  Mathbox for Jeff Hoffman
            20.13.1  Inferences for finite induction on generic function values   fveleq 34211
            20.13.2  gdc.mm   nnssi2 34215
      ​20.14  Mathbox for Asger C. Ipsen
            20.14.1  Continuous nowhere differentiable functions   dnival 34222
      ​​*20.15  Mathbox for BJ
            *20.15.1  Propositional calculus   bj-mp2c 34291
                  *20.15.1.1  Derived rules of inference   bj-mp2c 34291
                  ​*20.15.1.2  A syntactic theorem   bj-0 34293
                  20.15.1.3  Minimal implicational calculus   bj-a1k 34295
                  ​*20.15.1.4  Positive calculus   bj-syl66ib 34306
                  20.15.1.5  Implication and negation   bj-con2com 34312
                  ​*20.15.1.6  Disjunction   bj-jaoi1 34320
                  ​*20.15.1.7  Logical equivalence   bj-dfbi4 34322
                  20.15.1.8  The conditional operator for propositions   bj-consensus 34327
                  ​*20.15.1.9  Propositional calculus: miscellaneous   bj-imbi12 34332
            ​*20.15.2  Modal logic   bj-axdd2 34342
            ​*20.15.3  Provability logic   cprvb 34347
            ​*20.15.4  First-order logic   bj-genr 34356
                  20.15.4.1  Adding ax-gen   bj-genr 34356
                  20.15.4.2  Adding ax-4   bj-2alim 34360
                  20.15.4.3  Adding ax-5   bj-ax12wlem 34393
                  20.15.4.4  Equality and substitution   bj-ssbeq 34402
                  20.15.4.5  Adding ax-6   bj-spimvwt 34418
                  20.15.4.6  Adding ax-7   bj-cbvexw 34425
                  20.15.4.7  Membership predicate, ax-8 and ax-9   bj-ax89 34427
                  20.15.4.8  Adding ax-11   bj-alcomexcom 34430
                  20.15.4.9  Adding ax-12   axc11n11 34432
                  20.15.4.10  Nonfreeness   wnnf 34473
                  20.15.4.11  Adding ax-13   bj-axc10 34523
                  ​*20.15.4.12  Removing dependencies on ax-13 (and ax-11)   bj-axc10v 34533
                  ​*20.15.4.13  Distinct var metavariables   bj-hbaeb2 34559
                  ​*20.15.4.14  Around ~ equsal   bj-equsal1t 34563
                  ​*20.15.4.15  Some Principia Mathematica proofs   stdpc5t 34568
                  20.15.4.16  Alternate definition of substitution   bj-sbsb 34578
                  20.15.4.17  Lemmas for substitution   bj-sbf3 34580
                  20.15.4.18  Existential uniqueness   bj-eu3f 34583
                  ​*20.15.4.19  First-order logic: miscellaneous   bj-sblem1 34584
            20.15.5  Set theory   eliminable1 34600
                  ​*20.15.5.1  Eliminability of class terms   eliminable1 34600
                  ​*20.15.5.2  Classes without the axiom of extensionality   bj-denoteslem 34612
                  20.15.5.3  Characterization among sets versus among classes   elelb 34640
                  ​*20.15.5.4  The nonfreeness quantifier for classes   bj-nfcsym 34642
                  ​*20.15.5.5  Lemmas for class substitution   bj-sbeqALT 34643
                  20.15.5.6  Removing some axiom requirements and disjoint variable conditions   bj-exlimvmpi 34654
                  ​*20.15.5.7  Class abstractions   bj-unrab 34671
                  ​*20.15.5.8  Restricted nonfreeness   wrnf 34678
                  ​*20.15.5.9  Russell's paradox   bj-ru0 34680
                  20.15.5.10  Curry's paradox in set theory   currysetlem 34683
                  ​*20.15.5.11  Some disjointness results   bj-n0i 34689
                  ​*20.15.5.12  Complements on direct products   bj-xpimasn 34694
                  ​*20.15.5.13  "Singletonization" and tagging   bj-snsetex 34702
                  ​*20.15.5.14  Tuples of classes   bj-cproj 34729
                  ​*20.15.5.15  Set theory: elementary operations relative to a universe   bj-rcleqf 34764
                  ​*20.15.5.16  Set theory: miscellaneous   bj-clel3gALT 34769
                  ​*20.15.5.17  Evaluation   bj-evaleq 34789
                  20.15.5.18  Elementwise operations   celwise 34796
                  ​*20.15.5.19  Elementwise intersection (families of sets induced on a subset)   bj-rest00 34798
                  20.15.5.20  Moore collections (complements)   bj-raldifsn 34817
                  20.15.5.21  Maps-to notation for functions with three arguments   bj-0nelmpt 34833
                  ​*20.15.5.22  Currying   csethom 34839
                  ​*20.15.5.23  Setting components of extensible structures   cstrset 34851
            ​*20.15.6  Extended real and complex numbers, real and complex projective lines   bj-nfald 34854
                  20.15.6.1  Complements on class abstractions of ordered pairs and binary relations   bj-nfald 34854
                  ​*20.15.6.2  Identity relation (complements)   bj-opabssvv 34867
                  ​*20.15.6.3  Functionalized identity (diagonal in a Cartesian square)   cdiag2 34889
                  ​*20.15.6.4  Direct image and inverse image   cimdir 34895
                  ​*20.15.6.5  Extended numbers and projective lines as sets   cfractemp 34913
                  ​*20.15.6.6  Addition and opposite   caddcc 34954
                  ​*20.15.6.7  Order relation on the extended reals   cltxr 34958
                  ​*20.15.6.8  Argument, multiplication and inverse   carg 34960
                  20.15.6.9  The canonical bijection from the finite ordinals   ciomnn 34966
                  20.15.6.10  Divisibility   cnnbar 34977
            ​*20.15.7  Monoids   bj-smgrpssmgm 34985
                  ​*20.15.7.1  Finite sums in monoids   cfinsum 35000
            ​*20.15.8  Affine, Euclidean, and Cartesian geometry   bj-fvimacnv0 35003
                  *20.15.8.1  Real vector spaces   bj-fvimacnv0 35003
                  ​*20.15.8.2  Complex numbers (supplements)   bj-subcom 35024
                  ​*20.15.8.3  Barycentric coordinates   bj-bary1lem 35026
            20.15.9  Monoid of endomorphisms   cend 35029
      ​20.16  Mathbox for Jim Kingdon
                  20.16.0.1  Circle constant   taupilem3 35035
                  20.16.0.2  Number theory   dfgcd3 35040
                  20.16.0.3  Real numbers   irrdifflemf 35041
      ​20.17  Mathbox for ML
            20.17.1  Miscellaneous   csbdif 35044
            20.17.2  Cartesian exponentiation   cfinxp 35102
            20.17.3  Topology   iunctb2 35122
                  ​*20.17.3.1  Pi-base theorems   pibp16 35132
      ​20.18  Mathbox for Wolf Lammen
            20.18.1  1. Bootstrapping   wl-section-boot 35141
            20.18.2  Implication chains   wl-section-impchain 35165
            20.18.3  Theorems around the conditional operator   wl-ifp-ncond1 35183
            20.18.4  Alternative development of hadd, cadd   wl-df-3xor 35187
            20.18.5  An alternative axiom ~ ax-13   ax-wl-13v 35212
            20.18.6  Other stuff   wl-mps 35214
            20.18.7  1. Restricted Quantifiers   wl-ral 35298
      ​20.19  Mathbox for Brendan Leahy
      ​20.20  Mathbox for Jeff Madsen
            20.20.1  Logic and set theory   unirep 35453
            20.20.2  Real and complex numbers; integers   filbcmb 35480
            20.20.3  Sequences and sums   sdclem2 35482
            20.20.4  Topology   subspopn 35492
            20.20.5  Metric spaces   metf1o 35495
            20.20.6  Continuous maps and homeomorphisms   constcncf 35502
            20.20.7  Boundedness   ctotbnd 35506
            20.20.8  Isometries   cismty 35538
            20.20.9  Heine-Borel Theorem   heibor1lem 35549
            20.20.10  Banach Fixed Point Theorem   bfplem1 35562
            20.20.11  Euclidean space   crrn 35565
            20.20.12  Intervals (continued)   ismrer1 35578
            20.20.13  Operation properties   cass 35582
            20.20.14  Groups and related structures   cmagm 35588
            20.20.15  Group homomorphism and isomorphism   cghomOLD 35623
            20.20.16  Rings   crngo 35634
            20.20.17  Division Rings   cdrng 35688
            20.20.18  Ring homomorphisms   crnghom 35700
            20.20.19  Commutative rings   ccm2 35729
            20.20.20  Ideals   cidl 35747
            20.20.21  Prime rings and integral domains   cprrng 35786
            20.20.22  Ideal generators   cigen 35799
      ​20.21  Mathbox for Giovanni Mascellani
            ​*20.21.1  Tools for automatic proof building   efald2 35818
            ​*20.21.2  Tseitin axioms   fald 35869
            ​*20.21.3  Equality deductions   iuneq2f 35896
            ​*20.21.4  Miscellanea   orcomdd 35907
      ​20.22  Mathbox for Peter Mazsa
            20.22.1  Notations   cxrn 35914
            20.22.2  Preparatory theorems   el2v1 35952
            20.22.3  Range Cartesian product   df-xrn 36085
            20.22.4  Cosets by ` R `   df-coss 36121
            20.22.5  Relations   df-rels 36187
            20.22.6  Subset relations   df-ssr 36200
            20.22.7  Reflexivity   df-refs 36212
            20.22.8  Converse reflexivity   df-cnvrefs 36225
            20.22.9  Symmetry   df-syms 36240
            20.22.10  Reflexivity and symmetry   symrefref2 36261
            20.22.11  Transitivity   df-trs 36270
            20.22.12  Equivalence relations   df-eqvrels 36281
            20.22.13  Redundancy   df-redunds 36320
            20.22.14  Domain quotients   df-dmqss 36335
            20.22.15  Equivalence relations on domain quotients   df-ers 36359
            20.22.16  Functions   df-funss 36375
            20.22.17  Disjoints vs. converse functions   df-disjss 36398
      ​20.23  Mathbox for Rodolfo Medina
            20.23.1  Partitions   prtlem60 36451
      ​​*20.24  Mathbox for Norm Megill
            *20.24.1  Obsolete schemes ax-c4,c5,c7,c10,c11,c11n,c15,c9,c14,c16   ax-c5 36481
            ​*20.24.2  Rederive new axioms ax-4, ax-10, ax-6, ax-12, ax-13 from old   axc5 36491
            ​*20.24.3  Legacy theorems using obsolete axioms   ax5ALT 36505
            20.24.4  Experiments with weak deduction theorem   elimhyps 36559
            20.24.5  Miscellanea   cnaddcom 36570
            20.24.6  Atoms, hyperplanes, and covering in a left vector space (or module)   clsa 36572
            20.24.7  Functionals and kernels of a left vector space (or module)   clfn 36655
            20.24.8  Opposite rings and dual vector spaces   cld 36721
            20.24.9  Ortholattices and orthomodular lattices   cops 36770
            20.24.10  Atomic lattices with covering property   ccvr 36860
            20.24.11  Hilbert lattices   chlt 36948
            20.24.12  Projective geometries based on Hilbert lattices   clln 37089
            20.24.13  Construction of a vector space from a Hilbert lattice   cdlema1N 37389
            20.24.14  Construction of involution and inner product from a Hilbert lattice   clpoN 39078
      ​20.25  Mathbox for metakunt
            20.25.1  General helpful statements   leexp1ad 39560
            20.25.2  Some gcd and lcm results   12gcd5e1 39592
            20.25.3  Least common multiple inequality theorem   3factsumint1 39610
            20.25.4  Logarithm inequalities   3exp7 39642
            20.25.5  Miscellaneous results for AKS formalisation   intlewftc 39650
            20.25.6  Permutation results   metakunt1 39669
            20.25.7  Unused lemmas scheduled for deletion   andiff 39703
      ​20.26  Mathbox for Steven Nguyen
            20.26.1  Utility theorems   ioin9i8 39708
            20.26.2  Structures   nelsubginvcld 39749
            ​*20.26.3  Arithmetic theorems   c0exALT 39813
            20.26.4  Exponents and divisibility   oexpreposd 39845
            20.26.5  Real subtraction   cresub 39867
            ​*20.26.6  Projective spaces   cprjsp 39959
            20.26.7  Basic reductions for Fermat's Last Theorem   dffltz 39985
      ​20.27  Mathbox for Igor Ieskov
      ​20.28  Mathbox for OpenAI
      ​20.29  Mathbox for Stefan O'Rear
            20.29.1  Additional elementary logic and set theory   moxfr 40028
            20.29.2  Additional theory of functions   imaiinfv 40029
            20.29.3  Additional topology   elrfi 40030
            20.29.4  Characterization of closure operators. Kuratowski closure axioms   ismrcd1 40034
            20.29.5  Algebraic closure systems   cnacs 40038
            20.29.6  Miscellanea 1. Map utilities   constmap 40049
            20.29.7  Miscellanea for polynomials   mptfcl 40056
            20.29.8  Multivariate polynomials over the integers   cmzpcl 40057
            20.29.9  Miscellanea for Diophantine sets 1   coeq0i 40089
            20.29.10  Diophantine sets 1: definitions   cdioph 40091
            20.29.11  Diophantine sets 2 miscellanea   ellz1 40103
            20.29.12  Diophantine sets 2: union and intersection. Monotone Boolean algebra   diophin 40108
            20.29.13  Diophantine sets 3: construction   diophrex 40111
            20.29.14  Diophantine sets 4 miscellanea   2sbcrex 40120
            20.29.15  Diophantine sets 4: Quantification   rexrabdioph 40130
            20.29.16  Diophantine sets 5: Arithmetic sets   rabdiophlem1 40137
            20.29.17  Diophantine sets 6: reusability. renumbering of variables   eldioph4b 40147
            20.29.18  Pigeonhole Principle and cardinality helpers   fphpd 40152
            20.29.19  A non-closed set of reals is infinite   rencldnfilem 40156
            20.29.20  Lagrange's rational approximation theorem   irrapxlem1 40158
            20.29.21  Pell equations 1: A nontrivial solution always exists   pellexlem1 40165
            20.29.22  Pell equations 2: Algebraic number theory of the solution set   csquarenn 40172
            20.29.23  Pell equations 3: characterizing fundamental solution   infmrgelbi 40214
            ​*20.29.24  Logarithm laws generalized to an arbitrary base   reglogcl 40226
            20.29.25  Pell equations 4: the positive solution group is infinite cyclic   pellfund14 40234
            20.29.26  X and Y sequences 1: Definition and recurrence laws   crmx 40236
            20.29.27  Ordering and induction lemmas for the integers   monotuz 40277
            20.29.28  X and Y sequences 2: Order properties   rmxypos 40283
            20.29.29  Congruential equations   congtr 40301
            20.29.30  Alternating congruential equations   acongid 40311
            20.29.31  Additional theorems on integer divisibility   coprmdvdsb 40321
            20.29.32  X and Y sequences 3: Divisibility properties   jm2.18 40324
            20.29.33  X and Y sequences 4: Diophantine representability of Y   jm2.27a 40341
            20.29.34  X and Y sequences 5: Diophantine representability of X, ^, _C   rmxdiophlem 40351
            20.29.35  Uncategorized stuff not associated with a major project   setindtr 40360
            20.29.36  More equivalents of the Axiom of Choice   axac10 40369
            20.29.37  Finitely generated left modules   clfig 40406
            20.29.38  Noetherian left modules I   clnm 40414
            20.29.39  Addenda for structure powers   pwssplit4 40428
            20.29.40  Every set admits a group structure iff choice   unxpwdom3 40434
            20.29.41  Noetherian rings and left modules II   clnr 40448
            20.29.42  Hilbert's Basis Theorem   cldgis 40460
            20.29.43  Additional material on polynomials [DEPRECATED]   cmnc 40470
            20.29.44  Degree and minimal polynomial of algebraic numbers   cdgraa 40479
            20.29.45  Algebraic integers I   citgo 40496
            20.29.46  Endomorphism algebra   cmend 40514
            20.29.47  Cyclic groups and order   idomrootle 40534
            20.29.48  Cyclotomic polynomials   ccytp 40541
            20.29.49  Miscellaneous topology   fgraphopab 40549
      ​20.30  Mathbox for Jon Pennant
      ​20.31  Mathbox for Richard Penner
            20.31.1  Short Studies   ifpan123g 40562
                  20.31.1.1  Additional work on conditional logical operator   ifpan123g 40562
                  20.31.1.2  Sophisms   rp-fakeimass 40615
                  ​*20.31.1.3  Finite Sets   rp-isfinite5 40620
                  20.31.1.4  General Observations   intabssd 40622
                  20.31.1.5  Infinite Sets   pwelg 40654
                  ​*20.31.1.6  Finite intersection property   fipjust 40659
                  20.31.1.7  RP ADDTO: Subclasses and subsets   rababg 40668
                  20.31.1.8  RP ADDTO: The intersection of a class   elintabg 40669
                  20.31.1.9  RP ADDTO: Theorems requiring subset and intersection existence   elinintrab 40672
                  20.31.1.10  RP ADDTO: Relations   xpinintabd 40675
                  ​*20.31.1.11  RP ADDTO: Functions   elmapintab 40691
                  ​*20.31.1.12  RP ADDTO: Finite induction (for finite ordinals)   cnvcnvintabd 40695
                  20.31.1.13  RP ADDTO: First and second members of an ordered pair   elcnvlem 40696
                  20.31.1.14  RP ADDTO: The reflexive and transitive properties of relations   undmrnresiss 40699
                  20.31.1.15  RP ADDTO: Basic properties of closures   cleq2lem 40703
                  20.31.1.16  RP REPLACE: Definitions and basic properties of transitive closures   trcleq2lemRP 40725
                  ​*20.31.1.17  Additions for square root; absolute value   sqrtcvallem1 40726
            20.31.2  Additional statements on relations and subclasses   al3im 40742
                  20.31.2.1  Transitive relations (not to be confused with transitive classes).   trrelind 40761
                  20.31.2.2  Reflexive closures   crcl 40768
                  ​*20.31.2.3  Finite relationship composition.   relexp2 40773
                  20.31.2.4  Transitive closure of a relation   dftrcl3 40816
                  ​*20.31.2.5  Adapted from Frege   frege77d 40842
            ​*20.31.3  Propositions from _Begriffsschrift_   dfxor4 40862
                  *20.31.3.1  _Begriffsschrift_ Chapter I   dfxor4 40862
                  ​*20.31.3.2  _Begriffsschrift_ Notation hints   whe 40868
                  20.31.3.3  _Begriffsschrift_ Chapter II Implication   ax-frege1 40886
                  20.31.3.4  _Begriffsschrift_ Chapter II Implication and Negation   axfrege28 40925
                  ​*20.31.3.5  _Begriffsschrift_ Chapter II with logical equivalence   axfrege52a 40952
                  20.31.3.6  _Begriffsschrift_ Chapter II with equivalence of sets   axfrege52c 40983
                  ​*20.31.3.7  _Begriffsschrift_ Chapter II with equivalence of classes   frege53c 41010
                  ​*20.31.3.8  _Begriffsschrift_ Chapter III Properties hereditary in a sequence   dffrege69 41028
                  ​*20.31.3.9  _Begriffsschrift_ Chapter III Following in a sequence   dffrege76 41035
                  ​*20.31.3.10  _Begriffsschrift_ Chapter III Member of sequence   dffrege99 41058
                  ​*20.31.3.11  _Begriffsschrift_ Chapter III Single-valued procedures   dffrege115 41074
            ​*20.31.4  Exploring Topology via Seifert and Threlfall   enrelmap 41093
                  *20.31.4.1  Equinumerosity of sets of relations and maps   enrelmap 41093
                  ​*20.31.4.2  Generic Pseudoclosure Spaces, Pseudointerior Spaces, and Pseudoneighborhoods   or3or 41119
                  ​*20.31.4.3  Generic Neighborhood Spaces   gneispa 41228
            ​*20.31.5  Exploring Higher Homotopy via Kerodon   k0004lem1 41245
                  *20.31.5.1  Simplicial Sets   k0004lem1 41245
      ​20.32  Mathbox for Stanislas Polu
            20.32.1  IMO Problems   wwlemuld 41254
                  20.32.1.1  IMO 1972 B2   wwlemuld 41254
            ​*20.32.2  INT Inequalities Proof Generator   int-addcomd 41274
            ​*20.32.3  N-Digit Addition Proof Generator   unitadd 41296
            20.32.4  AM-GM (for k = 2,3,4)   gsumws3 41297
      ​20.33  Mathbox for Rohan Ridenour
            20.33.1  Misc   spALT 41302
            20.33.2  Monoid rings   cmnring 41314
            20.33.3  Shorter primitive equivalent of ax-groth   gru0eld 41332
                  20.33.3.1  Grothendieck universes are closed under collection   gru0eld 41332
                  20.33.3.2  Minimal universes   ismnu 41364
                  20.33.3.3  Primitive equivalent of ax-groth   expandan 41391
      ​20.34  Mathbox for Steve Rodriguez
            20.34.1  Miscellanea   nanorxor 41404
            20.34.2  Ratio test for infinite series convergence and divergence   dvgrat 41411
            20.34.3  Multiples   reldvds 41414
            20.34.4  Function operations   caofcan 41422
            20.34.5  Calculus   lhe4.4ex1a 41428
            20.34.6  The generalized binomial coefficient operation   cbcc 41435
            20.34.7  Binomial series   uzmptshftfval 41445
      ​20.35  Mathbox for Andrew Salmon
            20.35.1  Principia Mathematica * 10   pm10.12 41457
            20.35.2  Principia Mathematica * 11   2alanimi 41471
            20.35.3  Predicate Calculus   sbeqal1 41497
            20.35.4  Principia Mathematica * 13 and * 14   pm13.13a 41506
            20.35.5  Set Theory   elnev 41537
            20.35.6  Arithmetic   addcomgi 41555
            20.35.7  Geometry   cplusr 41556
      ​​*20.36  Mathbox for Alan Sare
            20.36.1  Auxiliary theorems for the Virtual Deduction tool   idiALT 41578
            20.36.2  Supplementary unification deductions   bi1imp 41582
            20.36.3  Conventional Metamath proofs, some derived from VD proofs   iidn3 41602
            20.36.4  What is Virtual Deduction?   wvd1 41670
            20.36.5  Virtual Deduction Theorems   df-vd1 41671
            20.36.6  Theorems proved using Virtual Deduction   trsspwALT 41919
            20.36.7  Theorems proved using Virtual Deduction with mmj2 assistance   simplbi2VD 41947
            20.36.8  Virtual Deduction transcriptions of textbook proofs   sb5ALTVD 42014
            20.36.9  Theorems proved using conjunction-form Virtual Deduction   elpwgdedVD 42018
            20.36.10  Theorems with a VD proof in conventional notation derived from a VD proof   suctrALT3 42025
            ​*20.36.11  Theorems with a proof in conventional notation derived from a VD proof   notnotrALT2 42028
      ​20.37  Mathbox for Glauco Siliprandi
            20.37.1  Miscellanea   evth2f 42039
            20.37.2  Functions   feq1dd 42184
            20.37.3  Ordering on real numbers - Real and complex numbers basic operations   sub2times 42295
            20.37.4  Real intervals   gtnelioc 42516
            20.37.5  Finite sums   fsumclf 42599
            20.37.6  Finite multiplication of numbers and finite multiplication of functions   fmul01 42610
            20.37.7  Limits   clim1fr1 42631
                  20.37.7.1  Inferior limit (lim inf)   clsi 42781
                  ​*20.37.7.2  Limits for sequences of extended real numbers   clsxlim 42848
            20.37.8  Trigonometry   coseq0 42894
            20.37.9  Continuous Functions   mulcncff 42900
            20.37.10  Derivatives   dvsinexp 42941
            20.37.11  Integrals   itgsin0pilem1 42980
            20.37.12  Stone Weierstrass theorem - real version   stoweidlem1 43031
            20.37.13  Wallis' product for π   wallispilem1 43095
            20.37.14  Stirling's approximation formula for ` n ` factorial   stirlinglem1 43104
            20.37.15  Dirichlet kernel   dirkerval 43121
            20.37.16  Fourier Series   fourierdlem1 43138
            20.37.17  e is transcendental   elaa2lem 43263
            20.37.18  n-dimensional Euclidean space   rrxtopn 43314
            20.37.19  Basic measure theory   csalg 43338
                  ​*20.37.19.1  σ-Algebras   csalg 43338
                  20.37.19.2  Sum of nonnegative extended reals   csumge0 43389
                  ​*20.37.19.3  Measures   cmea 43476
                  ​*20.37.19.4  Outer measures and Caratheodory's construction   come 43516
                  ​*20.37.19.5  Lebesgue measure on n-dimensional Real numbers   covoln 43563
                  ​*20.37.19.6  Measurable functions   csmblfn 43722
      ​20.38  Mathbox for Saveliy Skresanov
            20.38.1  Ceva's theorem   sigarval 43852
            20.38.2  Simple groups   simpcntrab 43872
      ​20.39  Mathbox for Jarvin Udandy
      ​20.40  Mathbox for Adhemar
            ​*20.40.1  Minimal implicational calculus   adh-minim 43982
      ​20.41  Mathbox for Alexander van der Vekens
            20.41.1  General auxiliary theorems (1)   eusnsn 44006
                  20.41.1.1  Unordered and ordered pairs - extension for singletons   eusnsn 44006
                  20.41.1.2  Unordered and ordered pairs - extension for unordered pairs   elprneb 44009
                  20.41.1.3  Unordered and ordered pairs - extension for ordered pairs   oppr 44010
                  20.41.1.4  Relations - extension   eubrv 44015
                  20.41.1.5  Definite description binder (inverted iota) - extension   iota0def 44018
                  20.41.1.6  Functions - extension   fveqvfvv 44020
            20.41.2  Alternative for Russell's definition of a description binder   caiota 44028
            20.41.3  Double restricted existential uniqueness   r19.32 44043
                  20.41.3.1  Restricted quantification (extension)   r19.32 44043
                  20.41.3.2  Restricted uniqueness and "at most one" quantification   reuf1odnf 44053
                  20.41.3.3  Analogs to Existential uniqueness (double quantification)   2reu3 44056
                  20.41.3.4  Additional theorems for double restricted existential uniqueness   2reu8i 44059
            ​*20.41.4  Alternative definitions of function and operation values   wdfat 44062
                  20.41.4.1  Restricted quantification (extension)   ralbinrald 44068
                  20.41.4.2  The universal class (extension)   nvelim 44069
                  20.41.4.3  Introduce the Axiom of Power Sets (extension)   alneu 44070
                  20.41.4.4  Predicate "defined at"   dfateq12d 44072
                  20.41.4.5  Alternative definition of the value of a function   dfafv2 44078
                  20.41.4.6  Alternative definition of the value of an operation   aoveq123d 44124
            ​*20.41.5  Alternative definitions of function values (2)   cafv2 44154
            20.41.6  General auxiliary theorems (2)   an4com24 44214
                  20.41.6.1  Logical conjunction - extension   an4com24 44214
                  20.41.6.2  Abbreviated conjunction and disjunction of three wff's - extension   3an4ancom24 44215
                  20.41.6.3  Negated membership (alternative)   cnelbr 44217
                  20.41.6.4  The empty set - extension   ralralimp 44224
                  20.41.6.5  Indexed union and intersection - extension   otiunsndisjX 44225
                  20.41.6.6  Functions - extension   fvifeq 44226
                  20.41.6.7  Maps-to notation - extension   fvmptrab 44238
                  20.41.6.8  Ordering on reals - extension   leltletr 44240
                  20.41.6.9  Subtraction - extension   cnambpcma 44241
                  20.41.6.10  Ordering on reals (cont.) - extension   leaddsuble 44244
                  20.41.6.11  Imaginary and complex number properties - extension   readdcnnred 44250
                  20.41.6.12  Nonnegative integers (as a subset of complex numbers) - extension   nn0resubcl 44255
                  20.41.6.13  Integers (as a subset of complex numbers) - extension   zgeltp1eq 44256
                  20.41.6.14  Decimal arithmetic - extension   1t10e1p1e11 44257
                  20.41.6.15  Upper sets of integers - extension   eluzge0nn0 44259
                  20.41.6.16  Infinity and the extended real number system (cont.) - extension   nltle2tri 44260
                  20.41.6.17  Finite intervals of integers - extension   ssfz12 44261
                  20.41.6.18  Half-open integer ranges - extension   fzopred 44269
                  20.41.6.19  The modulo (remainder) operation - extension   m1mod0mod1 44276
                  20.41.6.20  The infinite sequence builder "seq"   smonoord 44278
                  20.41.6.21  Finite and infinite sums - extension   fsummsndifre 44279
                  20.41.6.22  Extensible structures - extension   setsidel 44283
            ​*20.41.7  Preimages of function values   preimafvsnel 44286
            ​*20.41.8  Partitions of real intervals   ciccp 44320
            20.41.9  Shifting functions with an integer range domain   fargshiftfv 44346
            20.41.10  Words over a set (extension)   lswn0 44351
                  20.41.10.1  Last symbol of a word - extension   lswn0 44351
            20.41.11  Unordered pairs   wich 44352
                  20.41.11.1  Interchangeable setvar variables   wich 44352
                  20.41.11.2  Set of unordered pairs   sprid 44381
                  ​*20.41.11.3  Proper (unordered) pairs   prpair 44408
                  20.41.11.4  Set of proper unordered pairs   cprpr 44419
            20.41.12  Number theory (extension)   cfmtno 44434
                  ​*20.41.12.1  Fermat numbers   cfmtno 44434
                  ​*20.41.12.2  Mersenne primes   m2prm 44498
                  20.41.12.3  Proth's theorem   modexp2m1d 44519
                  20.41.12.4  Solutions of quadratic equations   quad1 44527
            ​*20.41.13  Even and odd numbers   ceven 44531
                  20.41.13.1  Definitions and basic properties   ceven 44531
                  20.41.13.2  Alternate definitions using the "divides" relation   dfeven2 44556
                  20.41.13.3  Alternate definitions using the "modulo" operation   dfeven3 44565
                  20.41.13.4  Alternate definitions using the "gcd" operation   iseven5 44571
                  20.41.13.5  Theorems of part 5 revised   zneoALTV 44576
                  20.41.13.6  Theorems of part 6 revised   odd2np1ALTV 44581
                  20.41.13.7  Theorems of AV's mathbox revised   0evenALTV 44595
                  20.41.13.8  Additional theorems   epoo 44610
                  20.41.13.9  Perfect Number Theorem (revised)   perfectALTVlem1 44628
            20.41.14  Number theory (extension 2)   cfppr 44631
                  ​*20.41.14.1  Fermat pseudoprimes   cfppr 44631
                  ​*20.41.14.2  Goldbach's conjectures   cgbe 44652
            20.41.15  Graph theory (extension)   cgrisom 44725
                  ​*20.41.15.1  Isomorphic graphs   cgrisom 44725
                  20.41.15.2  Loop-free graphs - extension   1hegrlfgr 44749
                  20.41.15.3  Walks - extension   cupwlks 44750
                  20.41.15.4  Edges of graphs expressed as sets of unordered pairs   upgredgssspr 44760
            20.41.16  Monoids (extension)   ovn0dmfun 44773
                  20.41.16.1  Auxiliary theorems   ovn0dmfun 44773
                  20.41.16.2  Magmas, Semigroups and Monoids (extension)   plusfreseq 44781
                  20.41.16.3  Magma homomorphisms and submagmas   cmgmhm 44786
                  20.41.16.4  Examples and counterexamples for magmas, semigroups and monoids (extension)   opmpoismgm 44816
                  20.41.16.5  Group sum operation (extension 1)   gsumsplit2f 44829
            ​*20.41.17  Magmas and internal binary operations (alternate approach)   ccllaw 44832
                  *20.41.17.1  Laws for internal binary operations   ccllaw 44832
                  ​*20.41.17.2  Internal binary operations   cintop 44845
                  20.41.17.3  Alternative definitions for magmas and semigroups   cmgm2 44864
            20.41.18  Categories (extension)   idfusubc0 44878
                  20.41.18.1  Subcategories (extension)   idfusubc0 44878
            20.41.19  Rings (extension)   lmod0rng 44881
                  20.41.19.1  Nonzero rings (extension)   lmod0rng 44881
                  ​*20.41.19.2  Non-unital rings ("rngs")   crng 44887
                  20.41.19.3  Rng homomorphisms   crngh 44898
                  20.41.19.4  Ring homomorphisms (extension)   rhmfn 44931
                  20.41.19.5  Ideals as non-unital rings   lidldomn1 44934
                  20.41.19.6  The non-unital ring of even integers   0even 44944
                  20.41.19.7  A constructed not unital ring   cznrnglem 44966
                  ​*20.41.19.8  The category of non-unital rings   crngc 44970
                  ​*20.41.19.9  The category of (unital) rings   cringc 45016
                  20.41.19.10  Subcategories of the category of rings   srhmsubclem1 45086
            20.41.20  Basic algebraic structures (extension)   opeliun2xp 45123
                  20.41.20.1  Auxiliary theorems   opeliun2xp 45123
                  20.41.20.2  The binomial coefficient operation (extension)   bcpascm1 45142
                  20.41.20.3  The ` ZZ `-module ` ZZ X. ZZ `   zlmodzxzlmod 45145
                  20.41.20.4  Group sum operation (extension 2)   mgpsumunsn 45152
                  20.41.20.5  Symmetric groups (extension)   exple2lt6 45155
                  20.41.20.6  Divisibility (extension)   invginvrid 45158
                  20.41.20.7  The support of functions (extension)   rmsupp0 45159
                  20.41.20.8  Finitely supported functions (extension)   rmsuppfi 45164
                  20.41.20.9  Left modules (extension)   lmodvsmdi 45173
                  20.41.20.10  Associative algebras (extension)   assaascl0 45175
                  20.41.20.11  Univariate polynomials (extension)   ply1vr1smo 45177
                  20.41.20.12  Univariate polynomials (examples)   linply1 45189
            20.41.21  Linear algebra (extension)   cdmatalt 45192
                  ​*20.41.21.1  The subalgebras of diagonal and scalar matrices (extension)   cdmatalt 45192
                  ​*20.41.21.2  Linear combinations   clinc 45200
                  ​*20.41.21.3  Linear independence   clininds 45236
                  20.41.21.4  Simple left modules and the ` ZZ `-module   lmod1lem1 45283
                  20.41.21.5  Differences between (left) modules and (left) vector spaces   lvecpsslmod 45303
            20.41.22  Complexity theory   suppdm 45306
                  20.41.22.1  Auxiliary theorems   suppdm 45306
                  20.41.22.2  The modulo (remainder) operation (extension)   fldivmod 45319
                  20.41.22.3  Even and odd integers   nn0onn0ex 45324
                  20.41.22.4  The natural logarithm on complex numbers (extension)   logcxp0 45336
                  20.41.22.5  Division of functions   cfdiv 45338
                  20.41.22.6  Upper bounds   cbigo 45348
                  20.41.22.7  Logarithm to an arbitrary base (extension)   rege1logbrege0 45359
                  ​*20.41.22.8  The binary logarithm   fldivexpfllog2 45366
                  20.41.22.9  Binary length   cblen 45370
                  ​*20.41.22.10  Digits   cdig 45396
                  20.41.22.11  Nonnegative integer as sum of its shifted digits   dignn0flhalflem1 45416
                  20.41.22.12  Algorithms for the multiplication of nonnegative integers   nn0mulfsum 45425
                  ​*20.41.22.13  N-ary functions   cnaryf 45427
                  ​*20.41.22.14  The Ackermann function   citco 45458
            20.41.23  Elementary geometry (extension)   fv1prop 45500
                  20.41.23.1  Auxiliary theorems   fv1prop 45500
                  20.41.23.2  Real euclidean space of dimension 2   rrx2pxel 45512
                  20.41.23.3  Spheres and lines in real Euclidean spaces   cline 45528
      ​20.42  Mathbox for Zhi Wang
            20.42.1  Propositional calculus   pm4.71da 45590
            20.42.2  ZF Set Theory - start with the Axiom of Extensionality   ralbidb 45596
                  20.42.2.1  Restricted quantification   ralbidb 45596
            ​*20.42.3  Topology   clduni 45599
                  20.42.3.1  Closure and interior   clduni 45599
                  20.42.3.2  Neighborhoods   opnneilem 45603
                  20.42.3.3  Separated spaces: T0, T1, T2 (Hausdorff) ...   isnrm4lem1 45610
      ​20.43  Mathbox for Emmett Weisz
            ​*20.43.1  Miscellaneous Theorems   nfintd 45616
            20.43.2  Set Recursion   csetrecs 45626
                  ​*20.43.2.1  Basic Properties of Set Recursion   csetrecs 45626
                  20.43.2.2  Examples and properties of set recursion   elsetrecslem 45641
            ​*20.43.3  Construction of Games and Surreal Numbers   cpg 45651
      ​​*20.44  Mathbox for David A. Wheeler
            20.44.1  Natural deduction   sbidd 45657
            ​*20.44.2  Greater than, greater than or equal to.   cge-real 45659
            ​*20.44.3  Hyperbolic trigonometric functions   csinh 45669
            ​*20.44.4  Reciprocal trigonometric functions (sec, csc, cot)   csec 45680
            ​*20.44.5  Identities for "if"   ifnmfalse 45702
            ​*20.44.6  Logarithms generalized to arbitrary base using ` logb `   logb2aval 45703
            ​*20.44.7  Logarithm laws generalized to an arbitrary base - log_   clog- 45704
            ​*20.44.8  Formally define notions such as reflexivity   wreflexive 45706
            ​*20.44.9  Algebra helpers   comraddi 45710
            ​*20.44.10  Algebra helper examples   i2linesi 45719
            ​*20.44.11  Formal methods "surprises"   alimp-surprise 45721
            ​*20.44.12  Allsome quantifier   walsi 45727
            ​*20.44.13  Miscellaneous   5m4e1 45738
            20.44.14  Theorems about algebraic numbers   aacllem 45742
      ​20.45  Mathbox for Kunhao Zheng
            20.45.1  Weighted AM-GM inequality   amgmwlem 45743