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  Existential uniqueness
      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  Presets 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  Abelian groups
      10.4  Rings
      10.5  Division rings and fields
      10.6  Left modules
      10.7  Vector spaces
      10.8  Ideals
      10.9  Associative algebras
      10.10  Abstract multivariate polynomials
      10.11  The complex numbers as an algebraic extensible structure
      10.12  Generalized pre-Hilbert and Hilbert spaces
PART 11  BASIC LINEAR ALGEBRA
      11.1  Vectors and free modules
      11.2  Matrices
      11.3  The determinant
      11.4  Polynomial matrices
      11.5  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 (USER'S 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 Mario Carneiro
      20.6  Mathbox for Filip Cernatescu
      20.7  Mathbox for Paul Chapman
      20.8  Mathbox for Scott Fenton
      20.9  Mathbox for Jeff Hankins
      20.10  Mathbox for Anthony Hart
      20.11  Mathbox for Chen-Pang He
      20.12  Mathbox for Jeff Hoffman
      20.13  Mathbox for Asger C. Ipsen
      20.14  Mathbox for BJ
      20.15  Mathbox for Jim Kingdon
      20.16  Mathbox for ML
      20.17  Mathbox for Wolf Lammen
      20.18  Mathbox for Brendan Leahy
      20.19  Mathbox for Jeff Madsen
      20.20  Mathbox for Giovanni Mascellani
      20.21  Mathbox for Peter Mazsa
      20.22  Mathbox for Rodolfo Medina
      20.23  Mathbox for Norm Megill
      20.24  Mathbox for OpenAI
      20.25  Mathbox for Stefan O'Rear
      20.26  Mathbox for Jon Pennant
      20.27  Mathbox for Richard Penner
      20.28  Mathbox for Stanislas Polu
      20.29  Mathbox for Steve Rodriguez
      20.30  Mathbox for Andrew Salmon
      20.31  Mathbox for Alan Sare
      20.32  Mathbox for Glauco Siliprandi
      20.33  Mathbox for Saveliy Skresanov
      20.34  Mathbox for Jarvin Udandy
      20.35  Mathbox for Alexander van der Vekens
      20.36  Mathbox for Emmett Weisz
      20.37  Mathbox for David A. Wheeler
      20.38  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   a1ii 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 112
            ​*1.2.5  Logical equivalence   wb 196
            ​*1.2.6  Logical disjunction and conjunction   wo 382
            ​*1.2.7  Miscellaneous theorems of propositional calculus   pm5.62 978
            ​*1.2.8  The conditional operator for propositions   wif 1032
            ​*1.2.9  The weak deduction theorem   elimh 1050
            1.2.10  Abbreviated conjunction and disjunction of three wff's   w3o 1053
            1.2.11  Logical 'nand' (Sheffer stroke)   wnan 1487
            1.2.12  Logical 'xor'   wxo 1504
            1.2.13  True and false constants   wal 1521
                  ​*1.2.13.1  Universal quantifier for use by df-tru   wal 1521
                  ​*1.2.13.2  Equality predicate for use by df-tru   cv 1522
                  1.2.13.3  Define the true and false constants   wtru 1524
            ​*1.2.14  Truth tables   truantru 1546
            ​*1.2.15  Half adder and full adder in propositional calculus   whad 1572
                  1.2.15.1  Full adder: sum   whad 1572
                  1.2.15.2  Full adder: carry   wcad 1585
      ​1.3  Other axiomatizations related to classical propositional calculus
            ​*1.3.1  Minimal implicational calculus   minimp 1600
            1.3.2  Derive the Lukasiewicz axioms from Meredith's sole axiom   meredith 1606
            1.3.3  Derive the standard axioms from the Lukasiewicz axioms   luklem1 1623
            ​*1.3.4  Derive Nicod's axiom from the standard axioms   nic-dfim 1634
            1.3.5  Derive the Lukasiewicz axioms from Nicod's axiom   nic-imp 1640
            1.3.6  Derive Nicod's Axiom from Lukasiewicz's First Sheffer Stroke Axiom   lukshef-ax1 1659
            1.3.7  Derive the Lukasiewicz Axioms from the Tarski-Bernays-Wajsberg Axioms   tbw-bijust 1663
            1.3.8  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's First CO Axiom   merco1 1678
            1.3.9  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's Second CO Axiom   merco2 1701
            1.3.10  Derive the Lukasiewicz axioms from the Russell-Bernays Axioms   rb-bijust 1714
            ​*1.3.11  Stoic logic non-modal portion (Chrysippus of Soli)   mptnan 1733
      ​​*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 1744
                  1.4.1.1  Existential quantifier   wex 1744
                  1.4.1.2  Non-freeness predicate   wnf 1748
            1.4.2  Rule scheme ax-gen (Generalization)   ax-gen 1762
            1.4.3  Axiom scheme ax-4 (Quantified Implication)   ax-4 1777
            1.4.4  Axiom scheme ax-5 (Distinctness) - first use of $d   ax-5 1879
            ​*1.4.5  Equality predicate (continued)   weq 1931
            1.4.6  Define proper substitution   wsb 1937
            1.4.7  Axiom scheme ax-6 (Existence)   ax-6 1945
            1.4.8  Axiom scheme ax-7 (Equality)   ax-7 1981
            1.4.9  Membership predicate   wcel 2030
            1.4.10  Axiom scheme ax-8 (Left Equality for Binary Predicate)   ax-8 2032
            1.4.11  Axiom scheme ax-9 (Right Equality for Binary Predicate)   ax-9 2039
            ​*1.4.12  Logical redundancy of ax-10 , ax-11 , ax-12 , ax-13   ax6dgen 2045
      ​​*1.5  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes)
            1.5.1  Axiom scheme ax-10 (Quantified Negation)   ax-10 2059
            1.5.2  Axiom scheme ax-11 (Quantifier Commutation)   ax-11 2074
            1.5.3  Axiom scheme ax-12 (Substitution)   ax-12 2087
            1.5.4  Axiom scheme ax-13 (Quantified Equality)   ax-13 2282
      ​1.6  Existential uniqueness
      ​1.7  Other axiomatizations related to classical predicate calculus
            ​*1.7.1  Aristotelian logic: Assertic syllogisms   barbara 2592
            ​*1.7.2  Intuitionistic logic   axia1 2616
​​*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 2631
            2.1.2  Class abstractions (a.k.a. class builders)   cab 2637
            2.1.3  Class form not-free predicate   wnfc 2780
            2.1.4  Negated equality and membership   wne 2823
                  2.1.4.1  Negated equality   wne 2823
                  2.1.4.2  Negated membership   wnel 2926
            2.1.5  Restricted quantification   wral 2941
            2.1.6  The universal class   cvv 3231
            ​*2.1.7  Conditional equality (experimental)   wcdeq 3451
            2.1.8  Russell's Paradox   ru 3467
            2.1.9  Proper substitution of classes for sets   wsbc 3468
            2.1.10  Proper substitution of classes for sets into classes   csb 3566
            2.1.11  Define basic set operations and relations   cdif 3604
            2.1.12  Subclasses and subsets   df-ss 3621
            2.1.13  The difference, union, and intersection of two classes   difeq1 3754
                  2.1.13.1  The difference of two classes   difeq1 3754
                  2.1.13.2  The union of two classes   elun 3786
                  2.1.13.3  The intersection of two classes   elin 3829
                  2.1.13.4  The symmetric difference of two classes   csymdif 3876
                  2.1.13.5  Combinations of difference, union, and intersection of two classes   unabs 3887
                  2.1.13.6  Class abstractions with difference, union, and intersection of two classes   unab 3927
                  2.1.13.7  Restricted uniqueness with difference, union, and intersection   reuss2 3940
            2.1.14  The empty set   c0 3948
            ​*2.1.15  "Weak deduction theorem" for set theory   cif 4119
            2.1.16  Power classes   cpw 4191
            2.1.17  Unordered and ordered pairs   snjust 4209
            2.1.18  The union of a class   cuni 4468
            2.1.19  The intersection of a class   cint 4507
            2.1.20  Indexed union and intersection   ciun 4552
            2.1.21  Disjointness   wdisj 4652
            2.1.22  Binary relations   wbr 4685
            2.1.23  Ordered-pair class abstractions (class builders)   copab 4745
            2.1.24  Functions in "maps-to" notation   cmpt 4762
            2.1.25  Transitive classes   wtr 4785
      ​2.2  ZF Set Theory - add the Axiom of Replacement
            2.2.1  Introduce the Axiom of Replacement   ax-rep 4804
            2.2.2  Derive the Axiom of Separation   axsep 4813
            2.2.3  Derive the Null Set Axiom   zfnuleu 4819
            2.2.4  Theorems requiring subset and intersection existence   nalset 4828
            2.2.5  Theorems requiring empty set existence   class2set 4862
      ​2.3  ZF Set Theory - add the Axiom of Power Sets
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 4873
            2.3.2  Derive the Axiom of Pairing   zfpair 4934
            2.3.3  Ordered pair theorem   opnz 4971
            2.3.4  Ordered-pair class abstractions (cont.)   opabid 5011
            2.3.5  Power class of union and intersection   pwin 5047
            2.3.6  The identity relation   cid 5052
            2.3.7  The membership (or epsilon) relation   cep 5057
            2.3.8  Partial and complete ordering   wpo 5062
            2.3.9  Founded and well-ordering relations   wfr 5099
            2.3.10  Relations   cxp 5141
            2.3.11  The Predecessor Class   cpred 5717
            2.3.12  Well-founded induction   tz6.26 5749
            2.3.13  Ordinals   word 5760
            2.3.14  Definite description binder (inverted iota)   cio 5887
            2.3.15  Functions   wfun 5920
            2.3.16  Cantor's Theorem   canth 6648
            2.3.17  Restricted iota (description binder)   crio 6650
            2.3.18  Operations   co 6690
            2.3.19  "Maps to" notation   mpt2ndm0 6917
            2.3.20  Function operation   cof 6937
            2.3.21  Proper subset relation   crpss 6978
      ​2.4  ZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 6991
            2.4.2  Ordinals (continued)   ordon 7024
            2.4.3  Transfinite induction   tfi 7095
            2.4.4  The natural numbers (i.e. finite ordinals)   com 7107
            2.4.5  Peano's postulates   peano1 7127
            2.4.6  Finite induction (for finite ordinals)   find 7133
            2.4.7  First and second members of an ordered pair   c1st 7208
            ​*2.4.8  The support of functions   csupp 7340
            ​*2.4.9  Special "Maps to" operations   opeliunxp2f 7381
            2.4.10  Function transposition   ctpos 7396
            2.4.11  Curry and uncurry   ccur 7436
            2.4.12  Undefined values   cund 7443
            2.4.13  Well-founded recursion   cwrecs 7451
            2.4.14  Functions on ordinals; strictly monotone ordinal functions   iunon 7481
            2.4.15  "Strong" transfinite recursion   crecs 7512
            2.4.16  Recursive definition generator   crdg 7550
            2.4.17  Finite recursion   frfnom 7575
            2.4.18  Ordinal arithmetic   c1o 7598
            2.4.19  Natural number arithmetic   nna0 7729
            2.4.20  Equivalence relations and classes   wer 7784
            2.4.21  The mapping operation   cmap 7899
            2.4.22  Infinite Cartesian products   cixp 7950
            2.4.23  Equinumerosity   cen 7994
            2.4.24  Schroeder-Bernstein Theorem   sbthlem1 8111
            2.4.25  Equinumerosity (cont.)   xpf1o 8163
            2.4.26  Pigeonhole Principle   phplem1 8180
            2.4.27  Finite sets   onomeneq 8191
            2.4.28  Finitely supported functions   cfsupp 8316
            2.4.29  Finite intersections   cfi 8357
            2.4.30  Hall's marriage theorem   marypha1lem 8380
            2.4.31  Supremum and infimum   csup 8387
            2.4.32  Ordinal isomorphism, Hartogs's theorem   coi 8455
            2.4.33  Hartogs function, order types, weak dominance   char 8502
      ​2.5  ZF Set Theory - add the Axiom of Regularity
            2.5.1  Introduce the Axiom of Regularity   ax-reg 8538
            2.5.2  Axiom of Infinity equivalents   inf0 8556
      ​2.6  ZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-inf 8573
            2.6.2  Existence of omega (the set of natural numbers)   omex 8578
            2.6.3  Cantor normal form   ccnf 8596
            2.6.4  Transitive closure   trcl 8642
            2.6.5  Rank   cr1 8663
            2.6.6  Scott's trick; collection principle; Hilbert's epsilon   scottex 8786
            2.6.7  Cardinal numbers   ccrd 8799
            2.6.8  Axiom of Choice equivalents   wac 8976
            2.6.9  Cardinal number arithmetic   ccda 9027
            2.6.10  The Ackermann bijection   ackbij2lem1 9079
            2.6.11  Cofinality (without Axiom of Choice)   cflem 9106
            2.6.12  Eight inequivalent definitions of finite set   sornom 9137
            2.6.13  Hereditarily size-limited sets without Choice   itunifval 9276
​​*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 9295
            3.1.2  Introduce the Axiom of Dependent Choice   ax-dc 9306
      ​3.2  ZFC Set Theory - add the Axiom of Choice
            3.2.1  Introduce the Axiom of Choice   ax-ac 9319
            3.2.2  AC equivalents: well-ordering, Zorn's lemma   numthcor 9354
            3.2.3  Cardinal number theorems using Axiom of Choice   cardval 9406
            3.2.4  Cardinal number arithmetic using Axiom of Choice   iunctb 9434
            3.2.5  Cofinality using Axiom of Choice   alephreg 9442
      ​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 9480
            3.4.2  Derivation of the Axiom of Choice   gchaclem 9538
​​*PART 4  TG (TARSKI-GROTHENDIECK) SET THEORY
      ​4.1  Inaccessibles
            4.1.1  Weakly and strongly inaccessible cardinals   cwina 9542
            4.1.2  Weak universes   cwun 9560
            4.1.3  Tarski classes   ctsk 9608
            4.1.4  Grothendieck universes   cgru 9650
      ​4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
            4.2.1  Introduce the Tarski-Grothendieck Axiom   ax-groth 9683
            4.2.2  Derive the Power Set, Infinity and Choice Axioms   grothpw 9686
            4.2.3  Tarski map function   ctskm 9697
​​*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 9704
            5.1.2  Final derivation of real and complex number postulates   axaddf 10004
            5.1.3  Real and complex number postulates restated as axioms   ax-cnex 10030
      ​5.2  Derive the basic properties from the field axioms
            5.2.1  Some deductions from the field axioms for complex numbers   cnex 10055
            5.2.2  Infinity and the extended real number system   cpnf 10109
            5.2.3  Restate the ordering postulates with extended real "less than"   axlttri 10147
            5.2.4  Ordering on reals   lttr 10152
            5.2.5  Initial properties of the complex numbers   mul12 10240
      ​5.3  Real and complex numbers - basic operations
            5.3.1  Addition   add12 10291
            5.3.2  Subtraction   cmin 10304
            5.3.3  Multiplication   kcnktkm1cn 10499
            5.3.4  Ordering on reals (cont.)   gt0ne0 10531
            5.3.5  Reciprocals   ixi 10694
            5.3.6  Division   cdiv 10722
            5.3.7  Ordering on reals (cont.)   elimgt0 10897
            5.3.8  Completeness Axiom and Suprema   fimaxre 11006
            5.3.9  Imaginary and complex number properties   inelr 11048
            5.3.10  Function operation analogue theorems   ofsubeq0 11055
      ​5.4  Integer sets
            5.4.1  Positive integers (as a subset of complex numbers)   cn 11058
            5.4.2  Principle of mathematical induction   nnind 11076
            ​*5.4.3  Decimal representation of numbers   c2 11108
            ​*5.4.4  Some properties of specific numbers   neg1cn 11162
            5.4.5  Simple number properties   halfcl 11295
            5.4.6  The Archimedean property   nnunb 11326
            5.4.7  Nonnegative integers (as a subset of complex numbers)   cn0 11330
            ​*5.4.8  Extended nonnegative integers   cxnn0 11401
            5.4.9  Integers (as a subset of complex numbers)   cz 11415
            5.4.10  Decimal arithmetic   cdc 11531
            5.4.11  Upper sets of integers   cuz 11725
            5.4.12  Well-ordering principle for bounded-below sets of integers   uzwo3 11821
            5.4.13  Rational numbers (as a subset of complex numbers)   cq 11826
            5.4.14  Existence of the set of complex numbers   rpnnen1lem2 11852
      ​5.5  Order sets
            5.5.1  Positive reals (as a subset of complex numbers)   crp 11870
            5.5.2  Infinity and the extended real number system (cont.)   cxne 11981
            5.5.3  Supremum and infimum on the extended reals   xrsupexmnf 12173
            5.5.4  Real number intervals   cioo 12213
            5.5.5  Finite intervals of integers   cfz 12364
            ​*5.5.6  Finite intervals of nonnegative integers   elfz2nn0 12469
            5.5.7  Half-open integer ranges   cfzo 12504
      ​5.6  Elementary integer functions
            5.6.1  The floor and ceiling functions   cfl 12631
            5.6.2  The modulo (remainder) operation   cmo 12708
            5.6.3  Miscellaneous theorems about integers   om2uz0i 12786
            5.6.4  Strong induction over upper sets of integers   uzsinds 12826
            5.6.5  Finitely supported functions over the nonnegative integers   fsuppmapnn0fiublem 12829
            5.6.6  The infinite sequence builder "seq" - extension   cseq 12841
            5.6.7  Integer powers   cexp 12900
            5.6.8  Ordered pair theorem for nonnegative integers   nn0le2msqi 13094
            5.6.9  Factorial function   cfa 13100
            5.6.10  The binomial coefficient operation   cbc 13129
            5.6.11  The ` # ` (set size) function   chash 13157
                  5.6.11.1  Proper unordered pairs and triples (sets of size 2 and 3)   hashprlei 13288
                  5.6.11.2  Functions with a domain containing at least two different elements   fundmge2nop0 13312
                  5.6.11.3  Finite induction on the size of the first component of a binary relation   brfi1indlem 13316
      ​​*5.7  Words over a set
            5.7.1  Definitions and basic theorems   cword 13323
            5.7.2  Last symbol of a word   lsw 13384
            5.7.3  Concatenations of words   ccatfn 13390
            5.7.4  Singleton words   ids1 13413
            5.7.5  Concatenations with singleton words   ccatws1cl 13433
            5.7.6  Subwords   swrdval 13462
            5.7.7  Subwords of subwords   swrdswrdlem 13505
            5.7.8  Subwords and concatenations   wrdcctswrd 13511
            5.7.9  Subwords of concatenations   swrdccatfn 13528
            5.7.10  Splicing words (substring replacement)   splval 13548
            5.7.11  Reversing words   revval 13555
            5.7.12  Repeated symbol words   reps 13563
            ​*5.7.13  Cyclical shifts of words   ccsh 13580
            5.7.14  Mapping words by a function   wrdco 13623
            5.7.15  Longer string literals   cs2 13632
      ​​*5.8  Reflexive and transitive closures of relations
            5.8.1  The reflexive and transitive properties of relations   coss12d 13757
            5.8.2  Basic properties of closures   cleq1lem 13767
            5.8.3  Definitions and basic properties of transitive closures   ctcl 13770
            5.8.4  Exponentiation of relations   crelexp 13804
            5.8.5  Reflexive-transitive closure as an indexed union   crtrcl 13839
            ​*5.8.6  Principle of transitive induction.   relexpindlem 13847
      ​5.9  Elementary real and complex functions
            5.9.1  The "shift" operation   cshi 13850
            5.9.2  Signum (sgn or sign) function   csgn 13870
            5.9.3  Real and imaginary parts; conjugate   ccj 13880
            5.9.4  Square root; absolute value   csqrt 14017
      ​5.10  Elementary limits and convergence
            5.10.1  Superior limit (lim sup)   clsp 14245
            5.10.2  Limits   cli 14259
            5.10.3  Finite and infinite sums   csu 14460
            5.10.4  The binomial theorem   binomlem 14605
            5.10.5  The inclusion/exclusion principle   incexclem 14612
            5.10.6  Infinite sums (cont.)   isumshft 14615
            5.10.7  Miscellaneous converging and diverging sequences   divrcnv 14628
            5.10.8  Arithmetic series   arisum 14636
            5.10.9  Geometric series   expcnv 14640
            5.10.10  Ratio test for infinite series convergence   cvgrat 14659
            5.10.11  Mertens' theorem   mertenslem1 14660
            5.10.12  Finite and infinite products   prodf 14663
                  5.10.12.1  Product sequences   prodf 14663
                  5.10.12.2  Non-trivial convergence   ntrivcvg 14673
                  5.10.12.3  Complex products   cprod 14679
                  5.10.12.4  Finite products   fprod 14715
                  5.10.12.5  Infinite products   iprodclim 14773
            5.10.13  Falling and Rising Factorial   cfallfac 14779
            5.10.14  Bernoulli polynomials and sums of k-th powers   cbp 14821
      ​5.11  Elementary trigonometry
            5.11.1  The exponential, sine, and cosine functions   ce 14836
            5.11.2  _e is irrational   eirrlem 14976
      ​5.12  Cardinality of real and complex number subsets
            5.12.1  Countability of integers and rationals   xpnnen 14983
            5.12.2  The reals are uncountable   rpnnen2lem1 14987
​​*PART 6  ELEMENTARY NUMBER THEORY
      ​6.1  Elementary properties of divisibility
            6.1.1  Irrationality of square root of 2   sqrt2irrlem 15021
            6.1.2  Some Number sets are chains of proper subsets   nthruc 15025
            6.1.3  The divides relation   cdvds 15027
            ​*6.1.4  Even and odd numbers   evenelz 15107
            6.1.5  The division algorithm   divalglem0 15163
            6.1.6  Bit sequences   cbits 15188
            6.1.7  The greatest common divisor operator   cgcd 15263
            6.1.8  Bézout's identity   bezoutlem1 15303
            6.1.9  Algorithms   nn0seqcvgd 15330
            6.1.10  Euclid's Algorithm   eucalgval2 15341
            ​*6.1.11  The least common multiple   clcm 15348
            ​*6.1.12  Coprimality and Euclid's lemma   coprmgcdb 15409
            6.1.13  Cancellability of congruences   congr 15425
      ​6.2  Elementary prime number theory
            ​*6.2.1  Elementary properties   cprime 15432
            ​*6.2.2  Coprimality and Euclid's lemma (cont.)   coprm 15470
            6.2.3  Properties of the canonical representation of a rational   cnumer 15488
            6.2.4  Euler's theorem   codz 15515
            6.2.5  Arithmetic modulo a prime number   modprm1div 15549
            6.2.6  Pythagorean Triples   coprimeprodsq 15560
            6.2.7  The prime count function   cpc 15588
            6.2.8  Pocklington's theorem   prmpwdvds 15655
            6.2.9  Infinite primes theorem   unbenlem 15659
            6.2.10  Sum of prime reciprocals   prmreclem1 15667
            6.2.11  Fundamental theorem of arithmetic   1arithlem1 15674
            6.2.12  Lagrange's four-square theorem   cgz 15680
            6.2.13  Van der Waerden's theorem   cvdwa 15716
            6.2.14  Ramsey's theorem   cram 15750
            ​*6.2.15  Primorial function   cprmo 15782
            ​*6.2.16  Prime gaps   prmgaplem1 15800
            6.2.17  Decimal arithmetic (cont.)   dec2dvds 15814
            6.2.18  Cyclical shifts of words (cont.)   cshwsidrepsw 15847
            6.2.19  Specific prime numbers   prmlem0 15859
            6.2.20  Very large primes   1259lem1 15885
​PART 7  BASIC STRUCTURES
      ​7.1  Extensible structures
            ​*7.1.1  Basic definitions   cstr 15900
            7.1.2  Slot definitions   cplusg 15988
            7.1.3  Definition of the structure product   crest 16128
            7.1.4  Definition of the structure quotient   cordt 16206
      ​7.2  Moore spaces
            7.2.1  Moore closures   mrcflem 16313
            7.2.2  Independent sets in a Moore system   mrisval 16337
            7.2.3  Algebraic closure systems   isacs 16359
​PART 8  BASIC CATEGORY THEORY
      ​8.1  Categories
            8.1.1  Categories   ccat 16372
            8.1.2  Opposite category   coppc 16418
            8.1.3  Monomorphisms and epimorphisms   cmon 16435
            8.1.4  Sections, inverses, isomorphisms   csect 16451
            ​*8.1.5  Isomorphic objects   ccic 16502
            8.1.6  Subcategories   cssc 16514
            8.1.7  Functors   cfunc 16561
            8.1.8  Full & faithful functors   cful 16609
            8.1.9  Natural transformations and the functor category   cnat 16648
            8.1.10  Initial, terminal and zero objects of a category   cinito 16685
      ​8.2  Arrows (disjointified hom-sets)
            8.2.1  Identity and composition for arrows   cida 16750
      ​8.3  Examples of categories
            8.3.1  The category of sets   csetc 16772
            8.3.2  The category of categories   ccatc 16791
            ​*8.3.3  The category of extensible structures   fncnvimaeqv 16807
      ​8.4  Categorical constructions
            8.4.1  Product of categories   cxpc 16855
            8.4.2  Functor evaluation   cevlf 16896
            8.4.3  Hom functor   chof 16935
​PART 9  BASIC ORDER THEORY
      ​9.1  Presets and directed sets using extensible structures
      ​9.2  Posets and lattices using extensible structures
            9.2.1  Posets   cpo 16987
            9.2.2  Lattices   clat 17092
            9.2.3  The dual of an ordered set   codu 17175
            9.2.4  Subset order structures   cipo 17198
            9.2.5  Distributive lattices   latmass 17235
            9.2.6  Posets and lattices as relations   cps 17245
            9.2.7  Directed sets, nets   cdir 17275
​PART 10  BASIC ALGEBRAIC STRUCTURES
      ​10.1  Monoids
            ​*10.1.1  Magmas   cplusf 17286
            ​*10.1.2  Identity elements   mgmidmo 17306
            ​*10.1.3  Ordered sums in a magma   gsumvalx 17317
            ​*10.1.4  Semigroups   csgrp 17330
            ​*10.1.5  Definition and basic properties of monoids   cmnd 17341
            10.1.6  Monoid homomorphisms and submonoids   cmhm 17380
            ​*10.1.7  Ordered sums in a monoid   gsumvallem2 17419
            10.1.8  Free monoids   cfrmd 17431
            10.1.9  Examples and counterexamples for magmas, semigroups and monoids   mgm2nsgrplem1 17452
      ​10.2  Groups
            10.2.1  Definition and basic properties   cgrp 17469
            ​*10.2.2  Group multiple operation   cmg 17587
            10.2.3  Subgroups and Quotient groups   csubg 17635
            10.2.4  Elementary theory of group homomorphisms   cghm 17704
            10.2.5  Isomorphisms of groups   cgim 17746
            10.2.6  Group actions   cga 17768
            10.2.7  Centralizers and centers   ccntz 17794
            10.2.8  The opposite group   coppg 17821
            10.2.9  Symmetric groups   csymg 17843
                  ​*10.2.9.1  Definition and basic properties   csymg 17843
                  10.2.9.2  Cayley's theorem   cayleylem1 17878
                  10.2.9.3  Permutations fixing one element   symgfix2 17882
                  ​*10.2.9.4  Transpositions in the symmetric group   cpmtr 17907
                  10.2.9.5  The sign of a permutation   cpsgn 17955
            10.2.10  p-Groups and Sylow groups; Sylow's theorems   cod 17990
            10.2.11  Direct products   clsm 18095
            10.2.12  Free groups   cefg 18165
      ​10.3  Abelian groups
            10.3.1  Definition and basic properties   ccmn 18239
            10.3.2  Cyclic groups   ccyg 18325
            10.3.3  Group sum operation   gsumval3a 18350
            10.3.4  Group sums over (ranges of) integers   fsfnn0gsumfsffz 18425
            10.3.5  Internal direct products   cdprd 18438
            10.3.6  The Fundamental Theorem of Abelian Groups   ablfacrplem 18510
      ​10.4  Rings
            10.4.1  Multiplicative Group   cmgp 18535
            10.4.2  Ring unit   cur 18547
                  10.4.2.1  Semirings   csrg 18551
                  ​*10.4.2.2  The binomial theorem for semirings   srgbinomlem1 18586
            10.4.3  Definition and basic properties of unital rings   crg 18593
            10.4.4  Opposite ring   coppr 18668
            10.4.5  Divisibility   cdsr 18684
            10.4.6  Ring primes   crpm 18758
            10.4.7  Ring homomorphisms   crh 18760
      ​10.5  Division rings and fields
            10.5.1  Definition and basic properties   cdr 18795
            10.5.2  Subrings of a ring   csubrg 18824
            10.5.3  Absolute value (abstract algebra)   cabv 18864
            10.5.4  Star rings   cstf 18891
      ​10.6  Left modules
            10.6.1  Definition and basic properties   clmod 18911
            10.6.2  Subspaces and spans in a left module   clss 18980
            10.6.3  Homomorphisms and isomorphisms of left modules   clmhm 19067
            10.6.4  Subspace sum; bases for a left module   clbs 19122
      ​10.7  Vector spaces
            10.7.1  Definition and basic properties   clvec 19150
      ​10.8  Ideals
            10.8.1  The subring algebra; ideals   csra 19216
            10.8.2  Two-sided ideals and quotient rings   c2idl 19279
            10.8.3  Principal ideal rings. Divisibility in the integers   clpidl 19289
            10.8.4  Nonzero rings and zero rings   cnzr 19305
            10.8.5  Left regular elements. More kinds of rings   crlreg 19327
      ​10.9  Associative algebras
            10.9.1  Definition and basic properties   casa 19357
      ​10.10  Abstract multivariate polynomials
            10.10.1  Definition and basic properties   cmps 19399
            10.10.2  Polynomial evaluation   ces 19552
            ​*10.10.3  Additional definitions for (multivariate) polynomials   cmhp 19585
            ​*10.10.4  Univariate polynomials   cps1 19593
            10.10.5  Univariate polynomial evaluation   ces1 19726
      ​10.11  The complex numbers as an algebraic extensible structure
            10.11.1  Definition and basic properties   cpsmet 19778
            ​*10.11.2  Ring of integers   zring 19866
            10.11.3  Algebraic constructions based on the complex numbers   czrh 19896
            10.11.4  Signs as subgroup of the complex numbers   cnmsgnsubg 19971
            10.11.5  Embedding of permutation signs into a ring   zrhpsgnmhm 19978
            10.11.6  The ordered field of real numbers   crefld 19998
      ​10.12  Generalized pre-Hilbert and Hilbert spaces
            10.12.1  Definition and basic properties   cphl 20017
            10.12.2  Orthocomplements and closed subspaces   cocv 20052
            10.12.3  Orthogonal projection and orthonormal bases   cpj 20092
​​*PART 11  BASIC LINEAR ALGEBRA
      ​11.1  Vectors and free modules
            *11.1.1  Direct sum of left modules   cdsmm 20123
            ​*11.1.2  Free modules   cfrlm 20138
            ​*11.1.3  Standard basis (unit vectors)   cuvc 20169
            ​*11.1.4  Independent sets and families   clindf 20191
            11.1.5  Characterization of free modules   lmimlbs 20223
      ​​*11.2  Matrices
            *11.2.1  The matrix multiplication   cmmul 20237
            ​*11.2.2  Square matrices   cmat 20261
            ​*11.2.3  The matrix algebra   matmulr 20292
            ​*11.2.4  Matrices of dimension 0 and 1   mat0dimbas0 20320
            ​*11.2.5  The subalgebras of diagonal and scalar matrices   cdmat 20342
            ​*11.2.6  Multiplication of a matrix with a "column vector"   cmvmul 20394
            11.2.7  Replacement functions for a square matrix   cmarrep 20410
            11.2.8  Submatrices   csubma 20430
      ​11.3  The determinant
            11.3.1  Definition and basic properties   cmdat 20438
            11.3.2  Determinants of 2 x 2 -matrices   m2detleiblem1 20478
            11.3.3  The matrix adjugate/adjunct   cmadu 20486
            ​*11.3.4  Laplace expansion of determinants (special case)   symgmatr01lem 20507
            11.3.5  Inverse matrix   invrvald 20530
            ​*11.3.6  Cramer's rule   slesolvec 20533
      ​​*11.4  Polynomial matrices
            11.4.1  Basic properties   pmatring 20546
            ​*11.4.2  Constant polynomial matrices   ccpmat 20556
            ​*11.4.3  Collecting coefficients of polynomial matrices   cdecpmat 20615
            ​*11.4.4  Ring isomorphism between polynomial matrices and polynomials over matrices   cpm2mp 20645
      ​​*11.5  The characteristic polynomial
            *11.5.1  Definition and basic properties   cchpmat 20679
            ​*11.5.2  The characteristic factor function G   fvmptnn04if 20702
            ​*11.5.3  The Cayley-Hamilton theorem   cpmadurid 20720
​PART 12  BASIC TOPOLOGY
      ​12.1  Topology
            ​*12.1.1  Topological spaces   ctop 20746
                  12.1.1.1  Topologies   ctop 20746
                  12.1.1.2  Topologies on sets   ctopon 20763
                  12.1.1.3  Topological spaces   ctps 20784
            12.1.2  Topological bases   ctb 20797
            12.1.3  Examples of topologies   distop 20847
            12.1.4  Closure and interior   ccld 20868
            12.1.5  Neighborhoods   cnei 20949
            12.1.6  Limit points and perfect sets   clp 20986
            12.1.7  Subspace topologies   restrcl 21009
            12.1.8  Order topology   ordtbaslem 21040
            12.1.9  Limits and continuity in topological spaces   ccn 21076
            12.1.10  Separated spaces: T0, T1, T2 (Hausdorff) ...   ct0 21158
            12.1.11  Compactness   ccmp 21237
            12.1.12  Bolzano-Weierstrass theorem   bwth 21261
            12.1.13  Connectedness   cconn 21262
            12.1.14  First- and second-countability   c1stc 21288
            12.1.15  Local topological properties   clly 21315
            12.1.16  Refinements   cref 21353
            12.1.17  Compactly generated spaces   ckgen 21384
            12.1.18  Product topologies   ctx 21411
            12.1.19  Continuous function-builders   cnmptid 21512
            12.1.20  Quotient maps and quotient topology   ckq 21544
            12.1.21  Homeomorphisms   chmeo 21604
      ​12.2  Filters and filter bases
            12.2.1  Filter bases   elmptrab 21678
            12.2.2  Filters   cfil 21696
            12.2.3  Ultrafilters   cufil 21750
            12.2.4  Filter limits   cfm 21784
            12.2.5  Extension by continuity   ccnext 21910
            12.2.6  Topological groups   ctmd 21921
            12.2.7  Infinite group sum on topological groups   ctsu 21976
            12.2.8  Topological rings, fields, vector spaces   ctrg 22006
      ​12.3  Uniform Structures and Spaces
            12.3.1  Uniform structures   cust 22050
            12.3.2  The topology induced by an uniform structure   cutop 22081
            12.3.3  Uniform Spaces   cuss 22104
            12.3.4  Uniform continuity   cucn 22126
            12.3.5  Cauchy filters in uniform spaces   ccfilu 22137
            12.3.6  Complete uniform spaces   ccusp 22148
      ​12.4  Metric spaces
            12.4.1  Pseudometric spaces   ispsmet 22156
            12.4.2  Basic metric space properties   cxme 22169
            12.4.3  Metric space balls   blfvalps 22235
            12.4.4  Open sets of a metric space   mopnval 22290
            12.4.5  Continuity in metric spaces   metcnp3 22392
            12.4.6  The uniform structure generated by a metric   metuval 22401
            12.4.7  Examples of metric spaces   dscmet 22424
            ​*12.4.8  Normed algebraic structures   cnm 22428
            12.4.9  Normed space homomorphisms (bounded linear operators)   cnmo 22556
            12.4.10  Topology on the reals   qtopbaslem 22609
            12.4.11  Topological definitions using the reals   cii 22725
            12.4.12  Path homotopy   chtpy 22813
            12.4.13  The fundamental group   cpco 22846
      ​12.5  Metric subcomplex vector spaces
            12.5.1  Subcomplex modules   cclm 22908
            ​*12.5.2  Subcomplex vector spaces   ccvs 22969
            ​*12.5.3  Normed subcomplex vector spaces   isncvsngp 22995
            12.5.4  Subcomplex pre-Hilbert space   ccph 23012
            12.5.5  Convergence and completeness   ccfil 23096
            12.5.6  Baire's Category Theorem   bcthlem1 23167
            12.5.7  Banach spaces and subcomplex Hilbert spaces   ccms 23175
                  12.5.7.1  The complete ordered field of the real numbers   retopn 23213
            12.5.8  Euclidean spaces   crrx 23217
            12.5.9  Minimizing Vector Theorem   minveclem1 23241
            12.5.10  Projection Theorem   pjthlem1 23254
​PART 13  BASIC REAL AND COMPLEX ANALYSIS
      ​13.1  Continuity
            13.1.1  Intermediate value theorem   pmltpclem1 23263
      ​13.2  Integrals
            13.2.1  Lebesgue measure   covol 23277
            13.2.2  Lebesgue integration   cmbf 23428
                  13.2.2.1  Lesbesgue integral   cmbf 23428
                  13.2.2.2  Lesbesgue directed integral   cdit 23655
      ​13.3  Derivatives
            13.3.1  Real and complex differentiation   climc 23671
                  13.3.1.1  Derivatives of functions of one complex or real variable   climc 23671
                  13.3.1.2  Results on real differentiation   dvferm1lem 23792
​PART 14  BASIC REAL AND COMPLEX FUNCTIONS
      ​14.1  Polynomials
            14.1.1  Polynomial degrees   cmdg 23858
            14.1.2  The division algorithm for univariate polynomials   cmn1 23930
            14.1.3  Elementary properties of complex polynomials   cply 23985
            14.1.4  The division algorithm for polynomials   cquot 24090
            14.1.5  Algebraic numbers   caa 24114
            14.1.6  Liouville's approximation theorem   aalioulem1 24132
      ​14.2  Sequences and series
            14.2.1  Taylor polynomials and Taylor's theorem   ctayl 24152
            14.2.2  Uniform convergence   culm 24175
            14.2.3  Power series   pserval 24209
      ​14.3  Basic trigonometry
            14.3.1  The exponential, sine, and cosine functions (cont.)   efcn 24242
            14.3.2  Properties of pi = 3.14159...   pilem1 24250
            14.3.3  Mapping of the exponential function   efgh 24332
            14.3.4  The natural logarithm on complex numbers   clog 24346
            ​*14.3.5  Logarithms to an arbitrary base   clogb 24547
            14.3.6  Theorems of Pythagoras, isosceles triangles, and intersecting chords   angval 24576
            14.3.7  Solutions of quadratic, cubic, and quartic equations   quad2 24611
            14.3.8  Inverse trigonometric functions   casin 24634
            14.3.9  The Birthday Problem   log2ublem1 24718
            14.3.10  Areas in R^2   carea 24727
            14.3.11  More miscellaneous converging sequences   rlimcnp 24737
            14.3.12  Inequality of arithmetic and geometric means   cvxcl 24756
            14.3.13  Euler-Mascheroni constant   cem 24763
            14.3.14  Zeta function   czeta 24784
            14.3.15  Gamma function   clgam 24787
      ​14.4  Basic number theory
            14.4.1  Wilson's theorem   wilthlem1 24839
            14.4.2  The Fundamental Theorem of Algebra   ftalem1 24844
            14.4.3  The Basel problem (ζ(2) = π²/6)   basellem1 24852
            14.4.4  Number-theoretical functions   ccht 24862
            14.4.5  Perfect Number Theorem   mersenne 24997
            14.4.6  Characters of Z/nZ   cdchr 25002
            14.4.7  Bertrand's postulate   bcctr 25045
            ​*14.4.8  Quadratic residues and the Legendre symbol   clgs 25064
            ​*14.4.9  Gauss' Lemma   gausslemma2dlem0a 25126
            14.4.10  Quadratic reciprocity   lgseisenlem1 25145
            14.4.11  All primes 4n+1 are the sum of two squares   2sqlem1 25187
            14.4.12  Chebyshev's Weak Prime Number Theorem, Dirichlet's Theorem   chebbnd1lem1 25203
            14.4.13  The Prime Number Theorem   mudivsum 25264
            14.4.14  Ostrowski's theorem   abvcxp 25349
​​*PART 15  ELEMENTARY GEOMETRY
      ​15.1  Definition and Tarski's Axioms of Geometry
      ​15.2  Tarskian Geometry
            15.2.1  Congruence   tgcgrcomimp 25417
            15.2.2  Betweenness   tgbtwntriv2 25427
            15.2.3  Dimension   tglowdim1 25440
            15.2.4  Betweenness and Congruence   tgifscgr 25448
            15.2.5  Congruence of a series of points   ccgrg 25450
            15.2.6  Motions   cismt 25472
            15.2.7  Colinearity   tglng 25486
            15.2.8  Connectivity of betweenness   tgbtwnconn1lem1 25512
            15.2.9  Less-than relation in geometric congruences   cleg 25522
            15.2.10  Rays   chlg 25540
            15.2.11  Lines   btwnlng1 25559
            15.2.12  Point inversions   cmir 25592
            15.2.13  Right angles   crag 25633
            15.2.14  Half-planes   islnopp 25676
            15.2.15  Midpoints and Line Mirroring   cmid 25709
            15.2.16  Congruence of angles   ccgra 25744
            15.2.17  Angle Comparisons   cinag 25771
            15.2.18  Congruence Theorems   tgsas1 25780
            15.2.19  Equilateral triangles   ceqlg 25790
      ​15.3  Properties of geometries
            15.3.1  Isomorphisms between geometries   f1otrgds 25794
      ​15.4  Geometry in Hilbert spaces
            15.4.1  Geometry in the complex plane   cchhllem 25812
            15.4.2  Geometry in Euclidean spaces   cee 25813
                  15.4.2.1  Definition of the Euclidean space   cee 25813
                  15.4.2.2  Tarski's axioms for geometry for the Euclidean space   axdimuniq 25838
                  15.4.2.3  EE^n fulfills Tarski's Axioms   ceeng 25902
​​*PART 16  GRAPH THEORY
      ​*16.1  Vertices and edges
            16.1.1  The edge function extractor for extensible structures   cedgf 25912
            ​*16.1.2  Vertices and indexed edges   cvtx 25919
                  16.1.2.1  Definitions and basic properties   cvtx 25919
                  16.1.2.2  The vertices and edges of a graph represented as ordered pair   opvtxval 25928
                  16.1.2.3  The vertices and edges of a graph represented as extensible structure   funvtxdmge2val 25936
                  16.1.2.4  Representations of graphs without edges   snstrvtxval 25974
                  16.1.2.5  Degenerated cases of representations of graphs   vtxval0 25976
            16.1.3  Edges as range of the edge function   cedg 25984
      ​​*16.2  Undirected graphs
            16.2.1  Undirected hypergraphs   cuhgr 25996
            16.2.2  Undirected pseudographs and multigraphs   cupgr 26020
            ​*16.2.3  Loop-free graphs   umgrislfupgrlem 26062
            16.2.4  Edges as subsets of vertices of graphs   uhgredgiedgb 26066
            ​*16.2.5  Undirected simple graphs   cuspgr 26088
            16.2.6  Examples for graphs   usgr0e 26173
            16.2.7  Subgraphs   csubgr 26204
            16.2.8  Finite undirected simple graphs   cfusgr 26253
            16.2.9  Neighbors, complete graphs and universal vertices   cnbgr 26269
                  16.2.9.1  Neighbors   cnbgr 26269
                  16.2.9.2  Universal vertices   cuvtx 26331
                  16.2.9.3  Complete graphs   ccplgr 26360
            16.2.10  Vertex degree   cvtxdg 26417
            ​*16.2.11  Regular graphs   crgr 26507
      ​​*16.3  Walks, paths and cycles
            *16.3.1  Walks   cewlks 26547
            16.3.2  Walks for loop-free graphs   lfgrwlkprop 26640
            16.3.3  Trails   ctrls 26643
            16.3.4  Paths and simple paths   cpths 26664
            16.3.5  Closed walks   cclwlks 26722
            16.3.6  Circuits and cycles   ccrcts 26735
            ​*16.3.7  Walks as words   cwwlks 26773
            16.3.8  Walks/paths of length 2 (as length 3 strings)   2wlkdlem1 26890
            16.3.9  Walks in regular graphs   rusgrnumwwlkl1 26935
            ​*16.3.10  Closed walks as words   cclwwlk 26949
                  16.3.10.1  Closed walks as words   cclwwlk 26949
                  16.3.10.2  Closed walks of a fixed length as words   cclwwlkn 26981
                  16.3.10.3  Closed walks on a vertex of a fixed length as words   cclwwlknon 27060
            16.3.11  Examples for walks, trails and paths   0ewlk 27092
            16.3.12  Connected graphs   cconngr 27164
      ​16.4  Eulerian paths and the Konigsberg Bridge problem
            ​*16.4.1  Eulerian paths   ceupth 27175
            ​*16.4.2  The Königsberg Bridge problem   konigsbergvtx 27224
      ​16.5  The Friendship Theorem
            16.5.1  Friendship graphs - basics   cfrgr 27236
            16.5.2  The friendship theorem for small graphs   frgr1v 27251
            16.5.3  Theorems according to Mertzios and Unger   2pthfrgrrn 27262
            ​*16.5.4  Huneke's Proof of the Friendship Theorem   frgrncvvdeqlem1 27279
​PART 17  GUIDES AND MISCELLANEA
      ​17.1  Guides (conventions, explanations, and examples)
            ​*17.1.1  Conventions   conventions 27387
            17.1.2  Natural deduction   natded 27390
            ​*17.1.3  Natural deduction examples   ex-natded5.2 27391
            17.1.4  Definitional examples   ex-or 27408
            17.1.5  Other examples   aevdemo 27447
      ​17.2  Humor
            17.2.1  April Fool's theorem   avril1 27449
      ​17.3  (Future - to be reviewed and classified)
            17.3.1  Planar incidence geometry   cplig 27456
            ​*17.3.2  Aliases kept to prevent broken links   dummylink 27469
​​*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 27471
            18.1.2  Abelian groups   cablo 27526
      ​18.2  Complex vector spaces
            18.2.1  Definition and basic properties   cvc 27541
            18.2.2  Examples of complex vector spaces   cnaddabloOLD 27564
      ​18.3  Normed complex vector spaces
            18.3.1  Definition and basic properties   cnv 27567
            18.3.2  Examples of normed complex vector spaces   cnnv 27660
            18.3.3  Induced metric of a normed complex vector space   imsval 27668
            18.3.4  Inner product   cdip 27683
            18.3.5  Subspaces   css 27704
      ​18.4  Operators on complex vector spaces
            18.4.1  Definitions and basic properties   clno 27723
      ​18.5  Inner product (pre-Hilbert) spaces
            18.5.1  Definition and basic properties   ccphlo 27795
            18.5.2  Examples of pre-Hilbert spaces   cncph 27802
            18.5.3  Properties of pre-Hilbert spaces   isph 27805
      ​18.6  Complex Banach spaces
            18.6.1  Definition and basic properties   ccbn 27846
            18.6.2  Examples of complex Banach spaces   cnbn 27853
            18.6.3  Uniform Boundedness Theorem   ubthlem1 27854
            18.6.4  Minimizing Vector Theorem   minvecolem1 27858
      ​18.7  Complex Hilbert spaces
            18.7.1  Definition and basic properties   chlo 27869
            18.7.2  Standard axioms for a complex Hilbert space   hlex 27882
            18.7.3  Examples of complex Hilbert spaces   cnchl 27900
            18.7.4  Subspaces   ssphl 27901
            18.7.5  Hellinger-Toeplitz Theorem   htthlem 27902
​​*PART 19  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      ​19.1  Axiomatization of complex pre-Hilbert spaces
            19.1.1  Basic Hilbert space definitions   chil 27904
            19.1.2  Preliminary ZFC lemmas   df-hnorm 27953
            ​*19.1.3  Derive the Hilbert space axioms from ZFC set theory   axhilex-zf 27966
            ​*19.1.4  Introduce the vector space axioms for a Hilbert space   ax-hilex 27984
            19.1.5  Vector operations   hvmulex 27996
            19.1.6  Inner product postulates for a Hilbert space   ax-hfi 28064
      ​19.2  Inner product and norms
            19.2.1  Inner product   his5 28071
            19.2.2  Norms   dfhnorm2 28107
            19.2.3  Relate Hilbert space to normed complex vector spaces   hilablo 28145
            19.2.4  Bunjakovaskij-Cauchy-Schwarz inequality   bcsiALT 28164
      ​19.3  Cauchy sequences and completeness axiom
            19.3.1  Cauchy sequences and limits   hcau 28169
            19.3.2  Derivation of the completeness axiom from ZF set theory   hilmet 28179
            19.3.3  Completeness postulate for a Hilbert space   ax-hcompl 28187
            19.3.4  Relate Hilbert space to ZFC pre-Hilbert and Hilbert spaces   hhcms 28188
      ​19.4  Subspaces and projections
            19.4.1  Subspaces   df-sh 28192
            19.4.2  Closed subspaces   df-ch 28206
            19.4.3  Orthocomplements   df-oc 28237
            19.4.4  Subspace sum, span, lattice join, lattice supremum   df-shs 28295
            19.4.5  Projection theorem   pjhthlem1 28378
            19.4.6  Projectors   df-pjh 28382
      ​19.5  Properties of Hilbert subspaces
            19.5.1  Orthomodular law   omlsilem 28389
            19.5.2  Projectors (cont.)   pjhtheu2 28403
            19.5.3  Hilbert lattice operations   sh0le 28427
            19.5.4  Span (cont.) and one-dimensional subspaces   spansn0 28528
            19.5.5  Commutes relation for Hilbert lattice elements   df-cm 28570
            19.5.6  Foulis-Holland theorem   fh1 28605
            19.5.7  Quantum Logic Explorer axioms   qlax1i 28614
            19.5.8  Orthogonal subspaces   chscllem1 28624
            19.5.9  Orthoarguesian laws 5OA and 3OA   5oalem1 28641
            19.5.10  Projectors (cont.)   pjorthi 28656
            19.5.11  Mayet's equation E_3   mayete3i 28715
      ​19.6  Operators on Hilbert spaces
            ​*19.6.1  Operator sum, difference, and scalar multiplication   df-hosum 28717
            19.6.2  Zero and identity operators   df-h0op 28735
            19.6.3  Operations on Hilbert space operators   hoaddcl 28745
            19.6.4  Linear, continuous, bounded, Hermitian, unitary operators and norms   df-nmop 28826
            19.6.5  Linear and continuous functionals and norms   df-nmfn 28832
            19.6.6  Adjoint   df-adjh 28836
            19.6.7  Dirac bra-ket notation   df-bra 28837
            19.6.8  Positive operators   df-leop 28839
            19.6.9  Eigenvectors, eigenvalues, spectrum   df-eigvec 28840
            19.6.10  Theorems about operators and functionals   nmopval 28843
            19.6.11  Riesz lemma   riesz3i 29049
            19.6.12  Adjoints (cont.)   cnlnadjlem1 29054
            19.6.13  Quantum computation error bound theorem   unierri 29091
            19.6.14  Dirac bra-ket notation (cont.)   branmfn 29092
            19.6.15  Positive operators (cont.)   leopg 29109
            19.6.16  Projectors as operators   pjhmopi 29133
      ​19.7  States on a Hilbert lattice and Godowski's equation
            19.7.1  States on a Hilbert lattice   df-st 29198
            19.7.2  Godowski's equation   golem1 29258
      ​19.8  Cover relation, atoms, exchange axiom, and modular symmetry
            19.8.1  Covers relation; modular pairs   df-cv 29266
            19.8.2  Atoms   df-at 29325
            19.8.3  Superposition principle   superpos 29341
            19.8.4  Atoms, exchange and covering properties, atomicity   chcv1 29342
            19.8.5  Irreducibility   chirredlem1 29377
            19.8.6  Atoms (cont.)   atcvat3i 29383
            19.8.7  Modular symmetry   mdsymlem1 29390
​PART 20  SUPPLEMENTARY MATERIAL (USER'S MATHBOXES)
      ​20.1  Mathboxes for user contributions
            20.1.1  Mathbox guidelines   mathbox 29429
      ​20.2  Mathbox for Stefan Allan
      ​20.3  Mathbox for Thierry Arnoux
            20.3.1  Propositional Calculus - misc additions   bian1d 29434
            20.3.2  Predicate Calculus   spc2ed 29440
                  20.3.2.1  Predicate Calculus - misc additions   spc2ed 29440
                  20.3.2.2  Restricted quantification - misc additions   ralcom4f 29444
                  20.3.2.3  Substitution (without distinct variables) - misc additions   sbceqbidf 29449
                  20.3.2.4  Existential "at most one" - misc additions   moel 29451
                  20.3.2.5  Existential uniqueness - misc additions   2reuswap2 29455
                  20.3.2.6  Restricted "at most one" - misc additions   rmoxfrdOLD 29459
            20.3.3  General Set Theory   difrab2 29465
                  20.3.3.1  Class abstractions (a.k.a. class builders)   difrab2 29465
                  20.3.3.2  Image Sets   abrexdomjm 29471
                  20.3.3.3  Set relations and operations - misc additions   rabss3d 29477
                  20.3.3.4  Unordered pairs   elpreq 29486
                  20.3.3.5  Conditional operator - misc additions   ifeqeqx 29487
                  20.3.3.6  Set union   uniinn0 29492
                  20.3.3.7  Indexed union - misc additions   cbviunf 29498
                  20.3.3.8  Disjointness - misc additions   disjnf 29510
            20.3.4  Relations and Functions   xpdisjres 29537
                  20.3.4.1  Relations - misc additions   xpdisjres 29537
                  20.3.4.2  Functions - misc additions   ac6sf2 29557
                  20.3.4.3  Operations - misc additions   mpt2mptxf 29605
                  20.3.4.4  Isomorphisms - misc. add.   gtiso 29606
                  20.3.4.5  Disjointness (additional proof requiring functions)   disjdsct 29608
                  20.3.4.6  First and second members of an ordered pair - misc additions   df1stres 29609
                  20.3.4.7  Supremum - misc additions   supssd 29615
                  20.3.4.8  Finite Sets   imafi2 29617
                  20.3.4.9  Countable Sets   snct 29619
            20.3.5  Real and Complex Numbers   addeq0 29638
                  20.3.5.1  Complex operations - misc. additions   addeq0 29638
                  20.3.5.2  Ordering on reals - misc additions   lt2addrd 29644
                  20.3.5.3  Extended reals - misc additions   xrlelttric 29645
                  20.3.5.4  Real number intervals - misc additions   joiniooico 29664
                  20.3.5.5  Finite intervals of integers - misc additions   uzssico 29674
                  20.3.5.6  Half-open integer ranges - misc additions   iundisjfi 29683
                  20.3.5.7  The ` # ` (set size) function - misc additions   hashunif 29690
                  20.3.5.8  The greatest common divisor operator - misc. add   numdenneg 29691
                  20.3.5.9  Integers   nnindf 29693
                  20.3.5.10  Decimal numbers   dfdec100 29704
            ​*20.3.6  Decimal expansion   cdp2 29705
                  ​*20.3.6.1  Decimal point   cdp 29723
                  20.3.6.2  Division in the extended real number system   cxdiv 29753
            20.3.7  Prime Number Theory   bhmafibid1 29772
                  20.3.7.1  Fermat's two square theorem   bhmafibid1 29772
            20.3.8  Extensible Structures   ressplusf 29778
                  20.3.8.1  Structure restriction operator   ressplusf 29778
                  20.3.8.2  The opposite group   oppgle 29781
                  20.3.8.3  Posets   ressprs 29783
                  20.3.8.4  Complete lattices   clatp0cl 29799
                  20.3.8.5  Extended reals Structure - misc additions   ax-xrssca 29801
                  20.3.8.6  The extended nonnegative real numbers commutative monoid   xrge0base 29813
            20.3.9  Algebra   abliso 29824
                  20.3.9.1  Monoids Homomorphisms   abliso 29824
                  20.3.9.2  Ordered monoids and groups   comnd 29825
                  20.3.9.3  Signum in an ordered monoid   csgns 29853
                  20.3.9.4  The Archimedean property for generic ordered algebraic structures   cinftm 29858
                  20.3.9.5  Semiring left modules   cslmd 29881
                  20.3.9.6  Finitely supported group sums - misc additions   gsumle 29907
                  20.3.9.7  Rings - misc additions   rngurd 29916
                  20.3.9.8  Ordered rings and fields   corng 29923
                  20.3.9.9  Ring homomorphisms - misc additions   rhmdvdsr 29946
                  20.3.9.10  Scalar restriction operation   cresv 29952
                  20.3.9.11  The commutative ring of gaussian integers   gzcrng 29967
                  20.3.9.12  The archimedean ordered field of real numbers   reofld 29968
            20.3.10  Matrices   symgfcoeu 29973
                  20.3.10.1  The symmetric group   symgfcoeu 29973
                  20.3.10.2  Permutation Signs   psgndmfi 29974
                  20.3.10.3  Transpositions   pmtridf1o 29984
                  20.3.10.4  Submatrices   csmat 29987
                  20.3.10.5  Matrix literals   clmat 30005
                  20.3.10.6  Laplace expansion of determinants   mdetpmtr1 30017
            20.3.11  Topology   fvproj 30027
                  20.3.11.1  Open maps   fvproj 30027
                  20.3.11.2  Topology of the unit circle   qtopt1 30030
                  20.3.11.3  Refinements   reff 30034
                  20.3.11.4  Open cover refinement property   ccref 30037
                  20.3.11.5  Lindelöf spaces   cldlf 30047
                  20.3.11.6  Paracompact spaces   cpcmp 30050
                  20.3.11.7  Pseudometrics   cmetid 30057
                  20.3.11.8  Continuity - misc additions   hauseqcn 30069
                  20.3.11.9  Topology of the closed unit   unitsscn 30070
                  20.3.11.10  Topology of ` ( RR X. RR ) `   unicls 30077
                  20.3.11.11  Order topology - misc. additions   cnvordtrestixx 30087
                  20.3.11.12  Continuity in topological spaces - misc. additions   mndpluscn 30100
                  20.3.11.13  Topology of the extended nonnegative real numbers ordered monoid   xrge0hmph 30106
                  20.3.11.14  Limits - misc additions   lmlim 30121
                  20.3.11.15  Univariate polynomials   pl1cn 30129
            20.3.12  Uniform Stuctures and Spaces   chcmp 30130
                  20.3.12.1  Hausdorff uniform completion   chcmp 30130
            20.3.13  Topology and algebraic structures   zringnm 30132
                  20.3.13.1  The norm on the ring of the integer numbers   zringnm 30132
                  20.3.13.2  Topological ` ZZ ` -modules   zlm0 30134
                  20.3.13.3  Canonical embedding of the field of the rational numbers into a division ring   cqqh 30144
                  20.3.13.4  Canonical embedding of the real numbers into a complete ordered field   crrh 30165
                  20.3.13.5  Embedding from the extended real numbers into a complete lattice   cxrh 30188
                  20.3.13.6  Canonical embeddings into the ordered field of the real numbers   zrhre 30191
                  ​*20.3.13.7  Topological Manifolds   cmntop 30194
            20.3.14  Real and complex functions   nexple 30199
                  20.3.14.1  Integer powers - misc. additions   nexple 30199
                  20.3.14.2  Indicator Functions   cind 30200
                  20.3.14.3  Extended sum   cesum 30217
            20.3.15  Mixed Function/Constant operation   cofc 30285
            20.3.16  Abstract measure   csiga 30298
                  20.3.16.1  Sigma-Algebra   csiga 30298
                  20.3.16.2  Generated sigma-Algebra   csigagen 30329
                  ​*20.3.16.3  lambda and pi-Systems, Rings of Sets   ispisys 30343
                  20.3.16.4  The Borel algebra on the real numbers   cbrsiga 30372
                  20.3.16.5  Product Sigma-Algebra   csx 30379
                  20.3.16.6  Measures   cmeas 30386
                  20.3.16.7  The counting measure   cntmeas 30417
                  20.3.16.8  The Lebesgue measure - misc additions   voliune 30420
                  20.3.16.9  The Dirac delta measure   cdde 30423
                  20.3.16.10  The 'almost everywhere' relation   cae 30428
                  20.3.16.11  Measurable functions   cmbfm 30440
                  20.3.16.12  Borel Algebra on ` ( RR X. RR ) `   br2base 30459
                  ​*20.3.16.13  Caratheodory's extension theorem   coms 30481
            20.3.17  Integration   itgeq12dv 30516
                  20.3.17.1  Lebesgue integral - misc additions   itgeq12dv 30516
                  20.3.17.2  Bochner integral   citgm 30517
            20.3.18  Euler's partition theorem   oddpwdc 30544
            20.3.19  Sequences defined by strong recursion   csseq 30573
            20.3.20  Fibonacci Numbers   cfib 30586
            20.3.21  Probability   cprb 30597
                  20.3.21.1  Probability Theory   cprb 30597
                  20.3.21.2  Conditional Probabilities   ccprob 30621
                  20.3.21.3  Real Valued Random Variables   crrv 30630
                  20.3.21.4  Preimage set mapping operator   corvc 30645
                  20.3.21.5  Distribution Functions   orvcelval 30658
                  20.3.21.6  Cumulative Distribution Functions   orvclteel 30662
                  20.3.21.7  Probabilities - example   coinfliplem 30668
                  20.3.21.8  Bertrand's Ballot Problem   ballotlemoex 30675
            20.3.22  Signum (sgn or sign) function - misc. additions   sgncl 30728
            20.3.23  Words over a set - misc additions   wrdfd 30744
                  20.3.23.1  Operations on words   ccatmulgnn0dir 30747
            20.3.24  Polynomials with real coefficients - misc additions   plymul02 30751
            20.3.25  Descartes's rule of signs   signspval 30757
                  20.3.25.1  Sign changes in a word over real numbers   signspval 30757
                  20.3.25.2  Counting sign changes in a word over real numbers   signslema 30767
            20.3.26  Number Theory   efcld 30797
                  20.3.26.1  Representations of a number as sums of integers   crepr 30814
                  20.3.26.2  Vinogradov Trigonometric Sums and the Circle Method   cvts 30841
                  20.3.26.3  The Ternary Goldbach Conjecture: Final Statement   ax-hgt749 30850
            20.3.27  Elementary Geometry   cstrkg2d 30870
                  ​*20.3.27.1  Two-dimension geometry   cstrkg2d 30870
                  20.3.27.2  Outer Five Segment (not used, no need to move to main)   cafs 30875
      ​​*20.4  Mathbox for Jonathan Ben-Naim
            20.4.1  First-order logic and set theory   bnj170 30892
            20.4.2  Well founded induction and recursion   bnj110 31054
            20.4.3  The existence of a minimal element in certain classes   bnj69 31204
            20.4.4  Well-founded induction   bnj1204 31206
            20.4.5  Well-founded recursion, part 1 of 3   bnj60 31256
            20.4.6  Well-founded recursion, part 2 of 3   bnj1500 31262
            20.4.7  Well-founded recursion, part 3 of 3   bnj1522 31266
      ​20.5  Mathbox for Mario Carneiro
            20.5.1  Predicate calculus with all distinct variables   ax-7d 31267
            20.5.2  Miscellaneous stuff   quartfull 31273
            20.5.3  Derangements and the Subfactorial   deranglem 31274
            20.5.4  The Erdős-Szekeres theorem   erdszelem1 31299
            20.5.5  The Kuratowski closure-complement theorem   kur14lem1 31314
            20.5.6  Retracts and sections   cretr 31325
            20.5.7  Path-connected and simply connected spaces   cpconn 31327
            20.5.8  Covering maps   ccvm 31363
            20.5.9  Normal numbers   snmlff 31437
            20.5.10  Godel-sets of formulas   cgoe 31441
            20.5.11  Models of ZF   cgze 31469
            ​*20.5.12  Metamath formal systems   cmcn 31483
            20.5.13  Grammatical formal systems   cm0s 31608
            20.5.14  Models of formal systems   cmuv 31626
            20.5.15  Splitting fields   citr 31648
            20.5.16  p-adic number fields   czr 31664
      ​​*20.6  Mathbox for Filip Cernatescu
      ​20.7  Mathbox for Paul Chapman
            20.7.1  Real and complex numbers (cont.)   climuzcnv 31691
            20.7.2  Miscellaneous theorems   elfzm12 31695
      ​20.8  Mathbox for Scott Fenton
            20.8.1  ZFC Axioms in primitive form   axextprim 31704
            20.8.2  Untangled classes   untelirr 31711
            20.8.3  Extra propositional calculus theorems   3orel2 31718
            20.8.4  Misc. Useful Theorems   nepss 31725
            20.8.5  Properties of real and complex numbers   sqdivzi 31736
            20.8.6  Infinite products   iprodefisumlem 31752
            20.8.7  Factorial limits   faclimlem1 31755
            20.8.8  Greatest common divisor and divisibility   pdivsq 31761
            20.8.9  Properties of relationships   brtp 31765
            20.8.10  Properties of functions and mappings   funpsstri 31789
            20.8.11  Epsilon induction   setinds 31807
            20.8.12  Ordinal numbers   elpotr 31810
            20.8.13  Defined equality axioms   axextdfeq 31827
            20.8.14  Hypothesis builders   hbntg 31835
            20.8.15  (Trans)finite Recursion Theorems   tfisg 31840
            20.8.16  Transitive closure under a relationship   ctrpred 31841
            20.8.17  Founded Induction   frpomin 31863
            20.8.18  Ordering Ordinal Sequences   orderseqlem 31877
            20.8.19  Well-founded zero, successor, and limits   cwsuc 31880
            20.8.20  Founded Recursion   cfrecs 31900
            20.8.21  Surreal Numbers   csur 31918
            20.8.22  Surreal Numbers: Ordering   sltsolem1 31951
            20.8.23  Surreal Numbers: Birthday Function   bdayfo 31953
            20.8.24  Surreal Numbers: Density   fvnobday 31954
            20.8.25  Surreal Numbers: Full-Eta Property   bdayimaon 31968
            20.8.26  Surreal numbers - ordering theorems   csle 31994
            20.8.27  Surreal numbers - birthday theorems   bdayfun 32013
            20.8.28  Surreal numbers: Conway cuts   csslt 32021
            20.8.29  Surreal numbers - cuts and options   cmade 32050
            20.8.30  Quantifier-free definitions   ctxp 32062
            20.8.31  Alternate ordered pairs   caltop 32188
            20.8.32  Geometry in the Euclidean space   cofs 32214
                  20.8.32.1  Congruence properties   cofs 32214
                  20.8.32.2  Betweenness properties   btwntriv2 32244
                  20.8.32.3  Segment Transportation   ctransport 32261
                  20.8.32.4  Properties relating betweenness and congruence   cifs 32267
                  20.8.32.5  Connectivity of betweenness   btwnconn1lem1 32319
                  20.8.32.6  Segment less than or equal to   csegle 32338
                  20.8.32.7  Outside-of relationship   coutsideof 32351
                  20.8.32.8  Lines and Rays   cline2 32366
            20.8.33  Forward difference   cfwddif 32390
            20.8.34  Rank theorems   rankung 32398
            20.8.35  Hereditarily Finite Sets   chf 32404
      ​20.9  Mathbox for Jeff Hankins
            20.9.1  Miscellany   a1i14 32419
            20.9.2  Basic topological facts   topbnd 32444
            20.9.3  Topology of the real numbers   ivthALT 32455
            20.9.4  Refinements   cfne 32456
            20.9.5  Neighborhood bases determine topologies   neibastop1 32479
            20.9.6  Lattice structure of topologies   topmtcl 32483
            20.9.7  Filter bases   fgmin 32490
            20.9.8  Directed sets, nets   tailfval 32492
      ​20.10  Mathbox for Anthony Hart
            20.10.1  Propositional Calculus   tb-ax1 32503
            20.10.2  Predicate Calculus   allt 32525
            20.10.3  Misc. Single Axiom Systems   meran1 32535
            20.10.4  Connective Symmetry   negsym1 32541
      ​20.11  Mathbox for Chen-Pang He
            20.11.1  Ordinal topology   ontopbas 32552
      ​20.12  Mathbox for Jeff Hoffman
            20.12.1  Inferences for finite induction on generic function values   fveleq 32575
            20.12.2  gdc.mm   nnssi2 32579
      ​20.13  Mathbox for Asger C. Ipsen
            20.13.1  Continuous nowhere differentiable functions   dnival 32586
      ​​*20.14  Mathbox for BJ
            *20.14.1  Propositional calculus   bj-mp2c 32656
                  *20.14.1.1  Derived rules of inference   bj-mp2c 32656
                  ​*20.14.1.2  A syntactic theorem   bj-0 32658
                  20.14.1.3  Minimal implicational calculus   bj-a1k 32660
                  20.14.1.4  Positive calculus   bj-orim2 32666
                  20.14.1.5  Implication and negation   pm4.81ALT 32671
                  ​*20.14.1.6  Disjunction   bj-jaoi1 32681
                  ​*20.14.1.7  Logical equivalence   bj-dfbi4 32683
                  20.14.1.8  The conditional operator for propositions   bj-consensus 32687
                  ​*20.14.1.9  Propositional calculus: miscellaneous   bj-imbi12 32692
            ​*20.14.2  Modal logic   bj-axdd2 32701
            ​*20.14.3  Provability logic   cprvb 32707
            ​*20.14.4  First-order logic   bj-genr 32716
                  20.14.4.1  Adding ax-gen   bj-genr 32716
                  20.14.4.2  Adding ax-4   bj-2alim 32719
                  20.14.4.3  Adding ax-5   bj-ax12wlem 32742
                  20.14.4.4  Equality and substitution   bj-ssbjust 32743
                  20.14.4.5  Adding ax-6   bj-extru 32779
                  20.14.4.6  Adding ax-7   bj-cbvexw 32789
                  20.14.4.7  Membership predicate, ax-8 and ax-9   bj-elequ2g 32791
                  20.14.4.8  Adding ax-11   bj-alcomexcom 32795
                  20.14.4.9  Adding ax-12   axc11n11 32797
                  20.14.4.10  Adding ax-13   bj-axc10 32832
                  ​*20.14.4.11  Removing dependencies on ax-13 (and ax-11)   bj-axc10v 32842
                  ​*20.14.4.12  Strengthenings of theorems of the main part   bj-sb3b 32929
                  ​*20.14.4.13  Distinct var metavariables   bj-hbaeb2 32930
                  ​*20.14.4.14  Around ~ equsal   bj-equsal1t 32934
                  ​*20.14.4.15  Some Principia Mathematica proofs   stdpc5t 32939
                  20.14.4.16  Alternate definition of substitution   bj-sbsb 32949
                  20.14.4.17  Lemmas for substitution   bj-sbf3 32951
                  20.14.4.18  Existential uniqueness   bj-eu3f 32954
                  ​*20.14.4.19  First-logic: miscellaneous   bj-sbidmOLD 32956
            20.14.5  Set theory   eliminable1 32965
                  ​*20.14.5.1  Eliminability of class terms   eliminable1 32965
                  ​*20.14.5.2  Classes without extensionality   bj-cleljustab 32972
                  ​*20.14.5.3  The class-form not-free predicate   bj-nfcsym 33011
                  ​*20.14.5.4  Proposal for the definitions of class membership and class equality   bj-ax8 33012
                  ​*20.14.5.5  Lemmas for class substitution   bj-sbeqALT 33020
                  20.14.5.6  Removing some dv conditions   bj-exlimmpi 33030
                  ​*20.14.5.7  Class abstractions   bj-unrab 33047
                  ​*20.14.5.8  Restricted non-freeness   wrnf 33055
                  ​*20.14.5.9  Russell's paradox   bj-ru0 33057
                  ​*20.14.5.10  Some disjointness results   bj-n0i 33060
                  ​*20.14.5.11  Complements on direct products   bj-xpimasn 33067
                  ​*20.14.5.12  "Singletonization" and tagging   bj-sels 33075
                  ​*20.14.5.13  Tuples of classes   bj-cproj 33103
                  ​*20.14.5.14  Set theory: miscellaneous   bj-disj2r 33138
                  20.14.5.15  Evaluation   bj-evaleq 33149
                  20.14.5.16  Elementwise operations   celwise 33157
                  ​*20.14.5.17  Elementwise intersection (families of sets induced on a subset)   bj-rest00 33159
                  20.14.5.18  Moore collections (complements)   bj-intss 33178
                  20.14.5.19  Maps-to notation for functions with three arguments   bj-0nelmpt 33194
                  ​*20.14.5.20  Currying   csethom 33200
                  ​*20.14.5.21  Setting components of extensible structures   cstrset 33212
            ​*20.14.6  Extended real and complex numbers, real and complex projective lines   bj-elid 33215
                  *20.14.6.1  Diagonal in a Cartesian square   bj-elid 33215
                  ​*20.14.6.2  Extended numbers and projective lines as sets   cinftyexpi 33223
                  ​*20.14.6.3  Addition and opposite   caddcc 33254
                  ​*20.14.6.4  Argument, multiplication and inverse   cprcpal 33258
            ​*20.14.7  Monoids   bj-cmnssmnd 33266
                  ​*20.14.7.1  Finite sums in monoids   cfinsum 33275
            ​*20.14.8  Affine, Euclidean, and Cartesian geometry   crrvec 33278
                  *20.14.8.1  Convex hull in real vector spaces   crrvec 33278
                  ​*20.14.8.2  Complex numbers (supplements)   bj-subcom 33284
                  ​*20.14.8.3  Barycentric coordinates   bj-bary1lem 33290
      ​20.15  Mathbox for Jim Kingdon
                  20.15.0.1  Circle constant   ctau 33293
                  20.15.0.2  Number theory   dfgcd3 33300
      ​20.16  Mathbox for ML
      ​20.17  Mathbox for Wolf Lammen
            20.17.1  1. Bootstrapping   wl-section-boot 33374
            20.17.2  Implication chains   wl-section-impchain 33398
            20.17.3  An alternative axiom ~ ax-13   ax-wl-13v 33416
            20.17.4  Other stuff   wl-jarri 33418
            20.17.5  1. Bootstrapping classes   wcel-wl 33503
      ​20.18  Mathbox for Brendan Leahy
      ​20.19  Mathbox for Jeff Madsen
            20.19.1  Logic and set theory   anim12da 33636
            20.19.2  Real and complex numbers; integers   filbcmb 33665
            20.19.3  Sequences and sums   sdclem2 33668
            20.19.4  Topology   subspopn 33678
            20.19.5  Metric spaces   metf1o 33681
            20.19.6  Continuous maps and homeomorphisms   constcncf 33688
            20.19.7  Boundedness   ctotbnd 33695
            20.19.8  Isometries   cismty 33727
            20.19.9  Heine-Borel Theorem   heibor1lem 33738
            20.19.10  Banach Fixed Point Theorem   bfplem1 33751
            20.19.11  Euclidean space   crrn 33754
            20.19.12  Intervals (continued)   ismrer1 33767
            20.19.13  Operation properties   cass 33771
            20.19.14  Groups and related structures   cmagm 33777
            20.19.15  Group homomorphism and isomorphism   cghomOLD 33812
            20.19.16  Rings   crngo 33823
            20.19.17  Division Rings   cdrng 33877
            20.19.18  Ring homomorphisms   crnghom 33889
            20.19.19  Commutative rings   ccm2 33918
            20.19.20  Ideals   cidl 33936
            20.19.21  Prime rings and integral domains   cprrng 33975
            20.19.22  Ideal generators   cigen 33988
      ​20.20  Mathbox for Giovanni Mascellani
            ​*20.20.1  Tools for automatic proof building   efald2 34007
            ​*20.20.2  Tseitin axioms   fald 34066
            ​*20.20.3  Equality deductions   iuneq2f 34093
            ​*20.20.4  Miscellanea   scottexf 34106
      ​20.21  Mathbox for Peter Mazsa
            20.21.1  Notations   cxrn 34112
            20.21.2  Preparatory theorems   elv 34126
            20.21.3  Range Cartesian product   df-xrn 34273
            20.21.4  Cosets by ` R `   df-coss 34309
            20.21.5  Relations   df-rels 34375
            20.21.6  Subset relations   df-ssr 34388
            20.21.7  Reflexivity   df-refs 34400
            20.21.8  Converse reflexivity   df-cnvrefs 34413
            20.21.9  Symmetry   df-syms 34428
            20.21.10  Reflexivity and symmetry   symrefref2 34447
      ​20.22  Mathbox for Rodolfo Medina
            20.22.1  Partitions   prtlem60 34456
      ​​*20.23  Mathbox for Norm Megill
            *20.23.1  Obsolete schemes ax-c4,c5,c7,c10,c11,c11n,c15,c9,c14,c16   ax-c5 34487
            ​*20.23.2  Rederive new axioms ax-4, ax-10, ax-6, ax-12, ax-13 from old   axc5 34497
            ​*20.23.3  Legacy theorems using obsolete axioms   ax5ALT 34511
            20.23.4  Experiments with weak deduction theorem   elimhyps 34565
            20.23.5  Miscellanea   cnaddcom 34577
            20.23.6  Atoms, hyperplanes, and covering in a left vector space (or module)   clsa 34579
            20.23.7  Functionals and kernels of a left vector space (or module)   clfn 34662
            20.23.8  Opposite rings and dual vector spaces   cld 34728
            20.23.9  Ortholattices and orthomodular lattices   cops 34777
            20.23.10  Atomic lattices with covering property   ccvr 34867
            20.23.11  Hilbert lattices   chlt 34955
            20.23.12  Projective geometries based on Hilbert lattices   clln 35095
            20.23.13  Construction of a vector space from a Hilbert lattice   cdlema1N 35395
            20.23.14  Construction of involution and inner product from a Hilbert lattice   clpoN 37086
      ​20.24  Mathbox for OpenAI
      ​20.25  Mathbox for Stefan O'Rear
            20.25.1  Additional elementary logic and set theory   moxfr 37572
            20.25.2  Additional theory of functions   imaiinfv 37573
            20.25.3  Additional topology   elrfi 37574
            20.25.4  Characterization of closure operators. Kuratowski closure axioms   ismrcd1 37578
            20.25.5  Algebraic closure systems   cnacs 37582
            20.25.6  Miscellanea 1. Map utilities   constmap 37593
            20.25.7  Miscellanea for polynomials   mptfcl 37600
            20.25.8  Multivariate polynomials over the integers   cmzpcl 37601
            20.25.9  Miscellanea for Diophantine sets 1   coeq0i 37633
            20.25.10  Diophantine sets 1: definitions   cdioph 37635
            20.25.11  Diophantine sets 2 miscellanea   ellz1 37647
            20.25.12  Diophantine sets 2: union and intersection. Monotone Boolean algebra   diophin 37653
            20.25.13  Diophantine sets 3: construction   diophrex 37656
            20.25.14  Diophantine sets 4 miscellanea   2sbcrex 37665
            20.25.15  Diophantine sets 4: Quantification   rexrabdioph 37675
            20.25.16  Diophantine sets 5: Arithmetic sets   rabdiophlem1 37682
            20.25.17  Diophantine sets 6: reusability. renumbering of variables   eldioph4b 37692
            20.25.18  Pigeonhole Principle and cardinality helpers   fphpd 37697
            20.25.19  A non-closed set of reals is infinite   rencldnfilem 37701
            20.25.20  Lagrange's rational approximation theorem   irrapxlem1 37703
            20.25.21  Pell equations 1: A nontrivial solution always exists   pellexlem1 37710
            20.25.22  Pell equations 2: Algebraic number theory of the solution set   csquarenn 37717
            20.25.23  Pell equations 3: characterizing fundamental solution   infmrgelbi 37759
            ​*20.25.24  Logarithm laws generalized to an arbitrary base   reglogcl 37771
            20.25.25  Pell equations 4: the positive solution group is infinite cyclic   pellfund14 37779
            20.25.26  X and Y sequences 1: Definition and recurrence laws   crmx 37781
            20.25.27  Ordering and induction lemmas for the integers   monotuz 37823
            20.25.28  X and Y sequences 2: Order properties   rmxypos 37831
            20.25.29  Congruential equations   congtr 37849
            20.25.30  Alternating congruential equations   acongid 37859
            20.25.31  Additional theorems on integer divisibility   coprmdvdsb 37869
            20.25.32  X and Y sequences 3: Divisibility properties   jm2.18 37872
            20.25.33  X and Y sequences 4: Diophantine representability of Y   jm2.27a 37889
            20.25.34  X and Y sequences 5: Diophantine representability of X, ^, _C   rmxdiophlem 37899
            20.25.35  Uncategorized stuff not associated with a major project   setindtr 37908
            20.25.36  More equivalents of the Axiom of Choice   axac10 37917
            20.25.37  Finitely generated left modules   clfig 37954
            20.25.38  Noetherian left modules I   clnm 37962
            20.25.39  Addenda for structure powers   pwssplit4 37976
            20.25.40  Every set admits a group structure iff choice   unxpwdom3 37982
            20.25.41  Noetherian rings and left modules II   clnr 37996
            20.25.42  Hilbert's Basis Theorem   cldgis 38008
            20.25.43  Additional material on polynomials [DEPRECATED]   cmnc 38018
            20.25.44  Degree and minimal polynomial of algebraic numbers   cdgraa 38027
            20.25.45  Algebraic integers I   citgo 38044
            20.25.46  Endomorphism algebra   cmend 38062
            20.25.47  Subfields   csdrg 38082
            20.25.48  Cyclic groups and order   idomrootle 38090
            20.25.49  Cyclotomic polynomials   ccytp 38097
            20.25.50  Miscellaneous topology   fgraphopab 38105
      ​20.26  Mathbox for Jon Pennant
      ​20.27  Mathbox for Richard Penner
            20.27.1  Short Studies   ifpan123g 38120
                  20.27.1.1  Additional work on conditional logical operator   ifpan123g 38120
                  20.27.1.2  Sophisms   rp-fakeimass 38174
                  ​*20.27.1.3  Finite Sets   rp-isfinite5 38180
                  20.27.1.4  Infinite Sets   pwelg 38182
                  ​*20.27.1.5  Finite intersection property   fipjust 38187
                  20.27.1.6  RP ADDTO: Subclasses and subsets   rababg 38196
                  20.27.1.7  RP ADDTO: The intersection of a class   elintabg 38197
                  20.27.1.8  RP ADDTO: Theorems requiring subset and intersection existence   elinintrab 38200
                  20.27.1.9  RP ADDTO: Relations   xpinintabd 38203
                  ​*20.27.1.10  RP ADDTO: Functions   elmapintab 38219
                  ​*20.27.1.11  RP ADDTO: Finite induction (for finite ordinals)   cnvcnvintabd 38223
                  20.27.1.12  RP ADDTO: First and second members of an ordered pair   elcnvlem 38224
                  20.27.1.13  RP ADDTO: The reflexive and transitive properties of relations   undmrnresiss 38227
                  20.27.1.14  RP ADDTO: Basic properties of closures   cleq2lem 38231
                  20.27.1.15  RP REPLACE: Definitions and basic properties of transitive closures   trcleq2lemRP 38254
            20.27.2  Additional statements on relations and subclasses   al3im 38255
                  20.27.2.1  Transitive relations (not to be confused with transitive classes).   trrelind 38274
                  20.27.2.2  Reflexive closures   crcl 38281
                  ​*20.27.2.3  Finite relationship composition.   relexp2 38286
                  20.27.2.4  Transitive closure of a relation   dftrcl3 38329
                  ​*20.27.2.5  Adapted from Frege   frege77d 38355
            ​*20.27.3  Propositions from _Begriffsschrift_   dfxor4 38375
                  *20.27.3.1  _Begriffsschrift_ Chapter I   dfxor4 38375
                  ​*20.27.3.2  _Begriffsschrift_ Notation hints   rp-imass 38382
                  20.27.3.3  _Begriffsschrift_ Chapter II Implication   ax-frege1 38401
                  20.27.3.4  _Begriffsschrift_ Chapter II Implication and Negation   axfrege28 38440
                  ​*20.27.3.5  _Begriffsschrift_ Chapter II with logical equivalence   axfrege52a 38467
                  20.27.3.6  _Begriffsschrift_ Chapter II with equivalence of sets   axfrege52c 38498
                  ​*20.27.3.7  _Begriffsschrift_ Chapter II with equivalence of classes   frege53c 38525
                  ​*20.27.3.8  _Begriffsschrift_ Chapter III Properties hereditary in a sequence   dffrege69 38543
                  ​*20.27.3.9  _Begriffsschrift_ Chapter III Following in a sequence   dffrege76 38550
                  ​*20.27.3.10  _Begriffsschrift_ Chapter III Member of sequence   dffrege99 38573
                  ​*20.27.3.11  _Begriffsschrift_ Chapter III Single-valued procedures   dffrege115 38589
            ​*20.27.4  Exploring Topology via Seifert And Threlfall   enrelmap 38608
                  *20.27.4.1  Equinumerosity of sets of relations and maps   enrelmap 38608
                  ​*20.27.4.2  Generic Pseudoclosure Spaces, Pseudointeror Spaces, and Pseudoneighborhoods   sscon34b 38634
                  ​*20.27.4.3  Generic Neighborhood Spaces   gneispa 38745
            ​*20.27.5  Exploring Higher Homotopy via Kerodon   k0004lem1 38762
                  *20.27.5.1  Simplicial Sets   k0004lem1 38762
      ​20.28  Mathbox for Stanislas Polu
            20.28.1  IMO Problems   wwlemuld 38771
                  20.28.1.1  IMO 1972 B2   wwlemuld 38771
            ​*20.28.2  INT Inequalities Proof Generator   int-addcomd 38793
            ​*20.28.3  N-Digit Addition Proof Generator   unitadd 38815
            20.28.4  AM-GM (for k = 2,3,4)   gsumws3 38816
      ​20.29  Mathbox for Steve Rodriguez
            20.29.1  Miscellanea   nanorxor 38821
            20.29.2  Ratio test for infinite series convergence and divergence   dvgrat 38828
            20.29.3  Multiples   reldvds 38831
            20.29.4  Function operations   caofcan 38839
            20.29.5  Calculus   lhe4.4ex1a 38845
            20.29.6  The generalized binomial coefficient operation   cbcc 38852
            20.29.7  Binomial series   uzmptshftfval 38862
      ​20.30  Mathbox for Andrew Salmon
            20.30.1  Principia Mathematica * 10   pm10.12 38874
            20.30.2  Principia Mathematica * 11   2alanimi 38888
            20.30.3  Predicate Calculus   sbeqal1 38915
            20.30.4  Principia Mathematica * 13 and * 14   pm13.13a 38925
            20.30.5  Set Theory   elnev 38956
            20.30.6  Arithmetic   addcomgi 38977
            20.30.7  Geometry   cplusr 38978
      ​​*20.31  Mathbox for Alan Sare
            20.31.1  Auxiliary theorems for the Virtual Deduction tool   idiALT 39000
            20.31.2  Supplementary unification deductions   bi1imp 39004
            20.31.3  Conventional Metamath proofs, some derived from VD proofs   iidn3 39024
            20.31.4  What is Virtual Deduction?   wvd1 39102
            20.31.5  Virtual Deduction Theorems   df-vd1 39103
            20.31.6  Theorems proved using Virtual Deduction   trsspwALT 39362
            20.31.7  Theorems proved using Virtual Deduction with mmj2 assistance   simplbi2VD 39395
            20.31.8  Virtual Deduction transcriptions of textbook proofs   sb5ALTVD 39463
            20.31.9  Theorems proved using conjunction-form Virtual Deduction   elpwgdedVD 39467
            20.31.10  Theorems with a VD proof in conventional notation derived from a VD proof   suctrALT3 39474
            ​*20.31.11  Theorems with a proof in conventional notation derived from a VD proof   notnotrALT2 39477
      ​20.32  Mathbox for Glauco Siliprandi
            20.32.1  Miscellanea   evth2f 39488
            20.32.2  Functions   unima 39660
            20.32.3  Ordering on real numbers - Real and complex numbers basic operations   sub2times 39799
            20.32.4  Real intervals   gtnelioc 40030
            20.32.5  Finite sums   fsumclf 40119
            20.32.6  Finite multiplication of numbers and finite multiplication of functions   fmul01 40130
            20.32.7  Limits   clim1fr1 40151
                  20.32.7.1  Inferior limit (lim inf)   clsi 40301
                  ​*20.32.7.2  Limits for sequences of extended real numbers   clsxlim 40362
            20.32.8  Trigonometry   coseq0 40393
            20.32.9  Continuous Functions   mulcncff 40399
            20.32.10  Derivatives   dvsinexp 40443
            20.32.11  Integrals   itgsin0pilem1 40483
            20.32.12  Stone Weierstrass theorem - real version   stoweidlem1 40536
            20.32.13  Wallis' product for π   wallispilem1 40600
            20.32.14  Stirling's approximation formula for ` n ` factorial   stirlinglem1 40609
            20.32.15  Dirichlet kernel   dirkerval 40626
            20.32.16  Fourier Series   fourierdlem1 40643
            20.32.17  e is transcendental   elaa2lem 40768
            20.32.18  n-dimensional Euclidean space   rrxtopn 40819
            20.32.19  Basic measure theory   csalg 40846
                  ​*20.32.19.1  σ-Algebras   csalg 40846
                  20.32.19.2  Sum of nonnegative extended reals   csumge0 40897
                  ​*20.32.19.3  Measures   cmea 40984
                  ​*20.32.19.4  Outer measures and Caratheodory's construction   come 41024
                  ​*20.32.19.5  Lebesgue measure on n-dimensional Real numbers   covoln 41071
                  ​*20.32.19.6  Measurable functions   csmblfn 41230
      ​20.33  Mathbox for Saveliy Skresanov
            20.33.1  Ceva's theorem   sigarval 41360
      ​20.34  Mathbox for Jarvin Udandy
      ​20.35  Mathbox for Alexander van der Vekens
            20.35.1  Double restricted existential uniqueness   r19.32 41488
                  20.35.1.1  Restricted quantification (extension)   r19.32 41488
                  20.35.1.2  The empty set (extension)   raaan2 41496
                  20.35.1.3  Restricted uniqueness and "at most one" quantification   rmoimi 41497
                  20.35.1.4  Analogs to Existential uniqueness (double quantification)   2reurex 41502
            ​*20.35.2  Alternative definitions of function and operation values   wdfat 41514
                  20.35.2.1  Restricted quantification (extension)   ralbinrald 41520
                  20.35.2.2  The universal class (extension)   nvelim 41521
                  20.35.2.3  Introduce the Axiom of Power Sets (extension)   alneu 41522
                  20.35.2.4  Relations (extension)   eldmressn 41524
                  20.35.2.5  Functions (extension)   fveqvfvv 41525
                  20.35.2.6  Predicate "defined at"   dfateq12d 41530
                  20.35.2.7  Alternative definition of the value of a function   dfafv2 41533
                  20.35.2.8  Alternative definition of the value of an operation   aoveq123d 41579
            20.35.3  General auxiliary theorems   an4com24 41609
                  20.35.3.1  Logical disjunction and conjunction - extension   an4com24 41609
                  20.35.3.2  Abbreviated conjunction and disjunction of three wff's - extension   3an4ancom24 41610
                  20.35.3.3  Negated membership (alternative)   cnelbr 41612
                  20.35.3.4  The empty set - extension   ralralimp 41619
                  20.35.3.5  Unordered and ordered pairs - extension   elprneb 41620
                  20.35.3.6  Indexed union and intersection - extension   otiunsndisjX 41621
                  20.35.3.7  Functions - extension   fvifeq 41622
                  20.35.3.8  "Maps to" notation - extension   fvmptrab 41631
                  20.35.3.9  Ordering on reals - extension   leltletr 41633
                  20.35.3.10  Subtraction - extension   cnambpcma 41634
                  20.35.3.11  Ordering on reals (cont.) - extension   leaddsuble 41636
                  20.35.3.12  Nonnegative integers (as a subset of complex numbers) - extension   nn0resubcl 41642
                  20.35.3.13  Integers (as a subset of complex numbers) - extension   zgeltp1eq 41643
                  20.35.3.14  Decimal arithmetic - extension   1t10e1p1e11 41644
                  20.35.3.15  Upper sets of integers - extension   eluzge0nn0 41647
                  20.35.3.16  Infinity and the extended real number system (cont.) - extension   nltle2tri 41648
                  20.35.3.17  Finite intervals of integers - extension   ssfz12 41649
                  20.35.3.18  Half-open integer ranges - extension   fzopred 41657
                  20.35.3.19  The modulo (remainder) operation - extension   m1mod0mod1 41664
                  20.35.3.20  The infinite sequence builder "seq"   smonoord 41666
                  20.35.3.21  Finite and infinite sums - extension   fsummsndifre 41667
                  20.35.3.22  Extensible structures - extension   setsidel 41671
            ​*20.35.4  Partitions of real intervals   ciccp 41674
            20.35.5  Shifting functions with an integer range domain   fargshiftfv 41700
            20.35.6  Words over a set (extension)   lswn0 41705
                  20.35.6.1  Last symbol of a word - extension   lswn0 41705
                  ​*20.35.6.2  Prefixes of a word   cpfx 41706
            20.35.7  Number theory (extension)   cfmtno 41764
                  ​*20.35.7.1  Fermat numbers   cfmtno 41764
                  ​*20.35.7.2  Mersenne primes   m2prm 41830
                  20.35.7.3  Proth's theorem   modexp2m1d 41854
            ​*20.35.8  Even and odd numbers   ceven 41862
                  20.35.8.1  Definitions and basic properties   ceven 41862
                  20.35.8.2  Alternate definitions using the "divides" relation   dfeven2 41887
                  20.35.8.3  Alternate definitions using the "modulo" operation   dfeven3 41895
                  20.35.8.4  Alternate definitions using the "gcd" operation   iseven5 41901
                  20.35.8.5  Theorems of part 5 revised   zneoALTV 41905
                  20.35.8.6  Theorems of part 6 revised   odd2np1ALTV 41910
                  20.35.8.7  Theorems of AV's mathbox revised   0evenALTV 41924
                  20.35.8.8  Additional theorems   epoo 41937
                  20.35.8.9  Perfect Number Theorem (revised)   perfectALTVlem1 41955
                  ​*20.35.8.10  Goldbach's conjectures   cgbe 41958
            20.35.9  Graph theory (extension)   1hegrlfgr 42038
                  20.35.9.1  Loop-free graphs - extension   1hegrlfgr 42038
                  20.35.9.2  Walks - extension   cupwlks 42039
            20.35.10  Set of unordered pairs   sprid 42049
            20.35.11  Monoids (extension)   ovn0dmfun 42089
                  20.35.11.1  Auxiliary theorems   ovn0dmfun 42089
                  20.35.11.2  Magmas and Semigroups (extension)   plusfreseq 42097
                  20.35.11.3  Magma homomorphisms and submagmas   cmgmhm 42102
                  20.35.11.4  Examples and counterexamples for magmas, semigroups and monoids (extension)   opmpt2ismgm 42132
            ​*20.35.12  Magmas and internal binary operations (alternate approach)   ccllaw 42144
                  *20.35.12.1  Laws for internal binary operations   ccllaw 42144
                  ​*20.35.12.2  Internal binary operations   cintop 42157
                  20.35.12.3  Alternative definitions for Magmas and Semigroups   cmgm2 42176
            20.35.13  Categories (extension)   idfusubc0 42190
                  20.35.13.1  Subcategories (extension)   idfusubc0 42190
            20.35.14  Rings (extension)   lmod0rng 42193
                  20.35.14.1  Nonzero rings (extension)   lmod0rng 42193
                  ​*20.35.14.2  Non-unital rings ("rngs")   crng 42199
                  20.35.14.3  Rng homomorphisms   crngh 42210
                  20.35.14.4  Ring homomorphisms (extension)   rhmfn 42243
                  20.35.14.5  Ideals as non-unital rings   lidldomn1 42246
                  20.35.14.6  The non-unital ring of even integers   0even 42256
                  20.35.14.7  A constructed not unital ring   cznrnglem 42278
                  ​*20.35.14.8  The category of non-unital rings   crngc 42282
                  ​*20.35.14.9  The category of (unital) rings   cringc 42328
                  20.35.14.10  Subcategories of the category of rings   srhmsubclem1 42398
            20.35.15  Basic algebraic structures (extension)   xpprsng 42435
                  20.35.15.1  Auxiliary theorems   xpprsng 42435
                  20.35.15.2  The binomial coefficient operation (extension)   bcpascm1 42454
                  20.35.15.3  The ` ZZ `-module ` ZZ X. ZZ `   zlmodzxzlmod 42457
                  20.35.15.4  Ordered group sum operation (extension)   gsumpr 42464
                  20.35.15.5  Symmetric groups (extension)   exple2lt6 42470
                  20.35.15.6  Divisibility (extension)   invginvrid 42473
                  20.35.15.7  The support of functions (extension)   rmsupp0 42474
                  20.35.15.8  Finitely supported functions (extension)   rmsuppfi 42479
                  20.35.15.9  Left modules (extension)   lmodvsmdi 42488
                  20.35.15.10  Associative algebras (extension)   ascl0 42490
                  20.35.15.11  Univariate polynomials (extension)   ply1vr1smo 42494
                  20.35.15.12  Univariate polynomials (examples)   linply1 42506
            20.35.16  Linear algebra (extension)   cdmatalt 42510
                  ​*20.35.16.1  The subalgebras of diagonal and scalar matrices (extension)   cdmatalt 42510
                  ​*20.35.16.2  Linear combinations   clinc 42518
                  ​*20.35.16.3  Linear independency   clininds 42554
                  20.35.16.4  Simple left modules and the ` ZZ `-module   lmod1lem1 42601
                  20.35.16.5  Differences between (left) modules and (left) vector spaces   lvecpsslmod 42621
            20.35.17  Complexity theory   offval0 42624
                  20.35.17.1  Auxiliary theorems   offval0 42624
                  20.35.17.2  The modulo (remainder) operation (extension)   fldivmod 42638
                  20.35.17.3  Even and odd integers   nn0onn0ex 42643
                  20.35.17.4  The natural logarithm on complex numbers (extension)   logcxp0 42654
                  20.35.17.5  Division of functions   cfdiv 42656
                  20.35.17.6  Upper bounds   cbigo 42666
                  20.35.17.7  Logarithm to an arbitrary base (extension)   rege1logbrege0 42677
                  ​*20.35.17.8  The binary logarithm   fldivexpfllog2 42684
                  20.35.17.9  Binary length   cblen 42688
                  ​*20.35.17.10  Digits   cdig 42714
                  20.35.17.11  Nonnegative integer as sum of its shifted digits   dignn0flhalflem1 42734
                  20.35.17.12  Algorithms for the multiplication of nonnegative integers   nn0mulfsum 42743
      ​20.36  Mathbox for Emmett Weisz
            ​*20.36.1  Miscellaneous Theorems   nfintd 42745
            20.36.2  Set Recursion   csetrecs 42755
                  ​*20.36.2.1  Basic Properties of Set Recursion   csetrecs 42755
                  20.36.2.2  Examples and properties of set recursion   elsetrecslem 42770
            ​*20.36.3  Construction of Games and Surreal Numbers   cpg 42780
      ​​*20.37  Mathbox for David A. Wheeler
            20.37.1  Natural deduction   19.8ad 42786
            ​*20.37.2  Greater than, greater than or equal to.   cge-real 42789
            ​*20.37.3  Hyperbolic trigonometric functions   csinh 42799
            ​*20.37.4  Reciprocal trigonometric functions (sec, csc, cot)   csec 42810
            ​*20.37.5  Identities for "if"   ifnmfalse 42832
            ​*20.37.6  Logarithms generalized to arbitrary base using ` logb `   logb2aval 42833
            ​*20.37.7  Logarithm laws generalized to an arbitrary base - log_   clog- 42834
            ​*20.37.8  Formally define terms such as Reflexivity   wreflexive 42836
            ​*20.37.9  Algebra helpers   comraddi 42840
            ​*20.37.10  Algebra helper examples   i2linesi 42852
            ​*20.37.11  Formal methods "surprises"   alimp-surprise 42854
            ​*20.37.12  Allsome quantifier   walsi 42860
            ​*20.37.13  Miscellaneous   5m4e1 42871
            20.37.14  Theorems about algebraic numbers   aacllem 42875
      ​20.38  Mathbox for Kunhao Zheng
            20.38.1  Weighted AM-GM inequality   amgmwlem 42876