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 383
            ​*1.2.7  Miscellaneous theorems of propositional calculus   pm5.62 958
            ​*1.2.8  The conditional operator for propositions   wif 1012
            ​*1.2.9  The weak deduction theorem   elimh 1030
            1.2.10  Abbreviated conjunction and disjunction of three wff's   w3o 1036
            1.2.11  Logical 'nand' (Sheffer stroke)   wnan 1447
            1.2.12  Logical 'xor'   wxo 1464
            1.2.13  True and false constants   wal 1481
                  ​*1.2.13.1  Universal quantifier for use by df-tru   wal 1481
                  ​*1.2.13.2  Equality predicate for use by df-tru   cv 1482
                  1.2.13.3  Define the true and false constants   wtru 1484
            ​*1.2.14  Truth tables   truantru 1506
            ​*1.2.15  Half adder and full adder in propositional calculus   whad 1532
                  1.2.15.1  Full adder: sum   whad 1532
                  1.2.15.2  Full adder: carry   wcad 1545
      ​1.3  Other axiomatizations related to classical propositional calculus
            ​*1.3.1  Minimal implicational calculus   minimp 1560
            1.3.2  Derive the Lukasiewicz axioms from Meredith's sole axiom   meredith 1566
            1.3.3  Derive the standard axioms from the Lukasiewicz axioms   luklem1 1583
            ​*1.3.4  Derive Nicod's axiom from the standard axioms   nic-dfim 1594
            1.3.5  Derive the Lukasiewicz axioms from Nicod's axiom   nic-imp 1600
            1.3.6  Derive Nicod's Axiom from Lukasiewicz's First Sheffer Stroke Axiom   lukshef-ax1 1619
            1.3.7  Derive the Lukasiewicz Axioms from the Tarski-Bernays-Wajsberg Axioms   tbw-bijust 1623
            1.3.8  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's First CO Axiom   merco1 1638
            1.3.9  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's Second CO Axiom   merco2 1661
            1.3.10  Derive the Lukasiewicz axioms from the Russell-Bernays Axioms   rb-bijust 1674
            ​*1.3.11  Stoic logic non-modal portion (Chrysippus of Soli)   mptnan 1693
      ​​*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 1704
                  1.4.1.1  Existential quantifier   wex 1704
                  1.4.1.2  Non-freeness predicate   wnf 1708
            1.4.2  Rule scheme ax-gen (Generalization)   ax-gen 1722
            1.4.3  Axiom scheme ax-4 (Quantified Implication)   ax-4 1737
            1.4.4  Axiom scheme ax-5 (Distinctness) - first use of $d   ax-5 1839
            ​*1.4.5  Equality predicate (continued)   weq 1874
            1.4.6  Define proper substitution   wsb 1880
            1.4.7  Axiom scheme ax-6 (Existence)   ax-6 1888
            1.4.8  Axiom scheme ax-7 (Equality)   ax-7 1935
            1.4.9  Membership predicate   wcel 1990
            1.4.10  Axiom scheme ax-8 (Left Equality for Binary Predicate)   ax-8 1992
            1.4.11  Axiom scheme ax-9 (Right Equality for Binary Predicate)   ax-9 1999
            ​*1.4.12  Logical redundancy of ax-10 , ax-11 , ax-12 , ax-13   ax6dgen 2005
      ​​*1.5  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes)
            1.5.1  Axiom scheme ax-10 (Quantified Negation)   ax-10 2019
            1.5.2  Axiom scheme ax-11 (Quantifier Commutation)   ax-11 2034
            1.5.3  Axiom scheme ax-12 (Substitution)   ax-12 2047
            1.5.4  Axiom scheme ax-13 (Quantified Equality)   ax-13 2246
      ​1.6  Existential uniqueness
      ​1.7  Other axiomatizations related to classical predicate calculus
            ​*1.7.1  Aristotelian logic: Assertic syllogisms   barbara 2563
            ​*1.7.2  Intuitionistic logic   axia1 2587
​​*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 2602
            2.1.2  Class abstractions (a.k.a. class builders)   cab 2608
            2.1.3  Class form not-free predicate   wnfc 2751
            2.1.4  Negated equality and membership   wne 2794
                  2.1.4.1  Negated equality   wne 2794
                  2.1.4.2  Negated membership   wnel 2897
            2.1.5  Restricted quantification   wral 2912
            2.1.6  The universal class   cvv 3200
            ​*2.1.7  Conditional equality (experimental)   wcdeq 3418
            2.1.8  Russell's Paradox   ru 3434
            2.1.9  Proper substitution of classes for sets   wsbc 3435
            2.1.10  Proper substitution of classes for sets into classes   csb 3533
            2.1.11  Define basic set operations and relations   cdif 3571
            2.1.12  Subclasses and subsets   df-ss 3588
            2.1.13  The difference, union, and intersection of two classes   difeq1 3721
                  2.1.13.1  The difference of two classes   difeq1 3721
                  2.1.13.2  The union of two classes   elun 3753
                  2.1.13.3  The intersection of two classes   elin 3796
                  2.1.13.4  The symmetric difference of two classes   csymdif 3843
                  2.1.13.5  Combinations of difference, union, and intersection of two classes   unabs 3854
                  2.1.13.6  Class abstractions with difference, union, and intersection of two classes   unab 3894
                  2.1.13.7  Restricted uniqueness with difference, union, and intersection   reuss2 3907
            2.1.14  The empty set   c0 3915
            ​*2.1.15  "Weak deduction theorem" for set theory   cif 4086
            2.1.16  Power classes   cpw 4158
            2.1.17  Unordered and ordered pairs   snjust 4176
            2.1.18  The union of a class   cuni 4436
            2.1.19  The intersection of a class   cint 4475
            2.1.20  Indexed union and intersection   ciun 4520
            2.1.21  Disjointness   wdisj 4620
            2.1.22  Binary relations   wbr 4653
            2.1.23  Ordered-pair class abstractions (class builders)   copab 4712
            2.1.24  Functions in "maps-to" notation   cmpt 4729
            2.1.25  Transitive classes   wtr 4752
      ​2.2  ZF Set Theory - add the Axiom of Replacement
            2.2.1  Introduce the Axiom of Replacement   ax-rep 4771
            2.2.2  Derive the Axiom of Separation   axsep 4780
            2.2.3  Derive the Null Set Axiom   zfnuleu 4786
            2.2.4  Theorems requiring subset and intersection existence   nalset 4795
            2.2.5  Theorems requiring empty set existence   class2set 4832
      ​2.3  ZF Set Theory - add the Axiom of Power Sets
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 4843
            2.3.2  Derive the Axiom of Pairing   zfpair 4904
            2.3.3  Ordered pair theorem   opnz 4942
            2.3.4  Ordered-pair class abstractions (cont.)   opabid 4982
            2.3.5  Power class of union and intersection   pwin 5018
            2.3.6  The identity relation   cid 5023
            2.3.7  The membership (or epsilon) relation   cep 5028
            2.3.8  Partial and complete ordering   wpo 5033
            2.3.9  Founded and well-ordering relations   wfr 5070
            2.3.10  Relations   cxp 5112
            2.3.11  The Predecessor Class   cpred 5679
            2.3.12  Well-founded induction   tz6.26 5711
            2.3.13  Ordinals   word 5722
            2.3.14  Definite description binder (inverted iota)   cio 5849
            2.3.15  Functions   wfun 5882
            2.3.16  Cantor's Theorem   canth 6608
            2.3.17  Restricted iota (description binder)   crio 6610
            2.3.18  Operations   co 6650
            2.3.19  "Maps to" notation   mpt2ndm0 6875
            2.3.20  Function operation   cof 6895
            2.3.21  Proper subset relation   crpss 6936
      ​2.4  ZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 6949
            2.4.2  Ordinals (continued)   ordon 6982
            2.4.3  Transfinite induction   tfi 7053
            2.4.4  The natural numbers (i.e. finite ordinals)   com 7065
            2.4.5  Peano's postulates   peano1 7085
            2.4.6  Finite induction (for finite ordinals)   find 7091
            2.4.7  First and second members of an ordered pair   c1st 7166
            ​*2.4.8  The support of functions   csupp 7295
            ​*2.4.9  Special "Maps to" operations   opeliunxp2f 7336
            2.4.10  Function transposition   ctpos 7351
            2.4.11  Curry and uncurry   ccur 7391
            2.4.12  Undefined values   cund 7398
            2.4.13  Well-founded recursion   cwrecs 7406
            2.4.14  Functions on ordinals; strictly monotone ordinal functions   iunon 7436
            2.4.15  "Strong" transfinite recursion   crecs 7467
            2.4.16  Recursive definition generator   crdg 7505
            2.4.17  Finite recursion   frfnom 7530
            2.4.18  Ordinal arithmetic   c1o 7553
            2.4.19  Natural number arithmetic   nna0 7684
            2.4.20  Equivalence relations and classes   wer 7739
            2.4.21  The mapping operation   cmap 7857
            2.4.22  Infinite Cartesian products   cixp 7908
            2.4.23  Equinumerosity   cen 7952
            2.4.24  Schroeder-Bernstein Theorem   sbthlem1 8070
            2.4.25  Equinumerosity (cont.)   xpf1o 8122
            2.4.26  Pigeonhole Principle   phplem1 8139
            2.4.27  Finite sets   onomeneq 8150
            2.4.28  Finitely supported functions   cfsupp 8275
            2.4.29  Finite intersections   cfi 8316
            2.4.30  Hall's marriage theorem   marypha1lem 8339
            2.4.31  Supremum and infimum   csup 8346
            2.4.32  Ordinal isomorphism, Hartogs's theorem   coi 8414
            2.4.33  Hartogs function, order types, weak dominance   char 8461
      ​2.5  ZF Set Theory - add the Axiom of Regularity
            2.5.1  Introduce the Axiom of Regularity   ax-reg 8497
            2.5.2  Axiom of Infinity equivalents   inf0 8518
      ​2.6  ZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-inf 8535
            2.6.2  Existence of omega (the set of natural numbers)   omex 8540
            2.6.3  Cantor normal form   ccnf 8558
            2.6.4  Transitive closure   trcl 8604
            2.6.5  Rank   cr1 8625
            2.6.6  Scott's trick; collection principle; Hilbert's epsilon   scottex 8748
            2.6.7  Cardinal numbers   ccrd 8761
            2.6.8  Axiom of Choice equivalents   wac 8938
            2.6.9  Cardinal number arithmetic   ccda 8989
            2.6.10  The Ackermann bijection   ackbij2lem1 9041
            2.6.11  Cofinality (without Axiom of Choice)   cflem 9068
            2.6.12  Eight inequivalent definitions of finite set   sornom 9099
            2.6.13  Hereditarily size-limited sets without Choice   itunifval 9238
​​*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 9257
            3.1.2  Introduce the Axiom of Dependent Choice   ax-dc 9268
      ​3.2  ZFC Set Theory - add the Axiom of Choice
            3.2.1  Introduce the Axiom of Choice   ax-ac 9281
            3.2.2  AC equivalents: well-ordering, Zorn's lemma   numthcor 9316
            3.2.3  Cardinal number theorems using Axiom of Choice   cardval 9368
            3.2.4  Cardinal number arithmetic using Axiom of Choice   iunctb 9396
            3.2.5  Cofinality using Axiom of Choice   alephreg 9404
      ​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 9442
            3.4.2  Derivation of the Axiom of Choice   gchaclem 9500
​​*PART 4  TG (TARSKI-GROTHENDIECK) SET THEORY
      ​4.1  Inaccessibles
            4.1.1  Weakly and strongly inaccessible cardinals   cwina 9504
            4.1.2  Weak universes   cwun 9522
            4.1.3  Tarski classes   ctsk 9570
            4.1.4  Grothendieck universes   cgru 9612
      ​4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
            4.2.1  Introduce the Tarski-Grothendieck Axiom   ax-groth 9645
            4.2.2  Derive the Power Set, Infinity and Choice Axioms   grothpw 9648
            4.2.3  Tarski map function   ctskm 9659
​​*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 9666
            5.1.2  Final derivation of real and complex number postulates   axaddf 9966
            5.1.3  Real and complex number postulates restated as axioms   ax-cnex 9992
      ​5.2  Derive the basic properties from the field axioms
            5.2.1  Some deductions from the field axioms for complex numbers   cnex 10017
            5.2.2  Infinity and the extended real number system   cpnf 10071
            5.2.3  Restate the ordering postulates with extended real "less than"   axlttri 10109
            5.2.4  Ordering on reals   lttr 10114
            5.2.5  Initial properties of the complex numbers   mul12 10202
      ​5.3  Real and complex numbers - basic operations
            5.3.1  Addition   add12 10253
            5.3.2  Subtraction   cmin 10266
            5.3.3  Multiplication   kcnktkm1cn 10461
            5.3.4  Ordering on reals (cont.)   gt0ne0 10493
            5.3.5  Reciprocals   ixi 10656
            5.3.6  Division   cdiv 10684
            5.3.7  Ordering on reals (cont.)   elimgt0 10859
            5.3.8  Completeness Axiom and Suprema   fimaxre 10968
            5.3.9  Imaginary and complex number properties   inelr 11010
            5.3.10  Function operation analogue theorems   ofsubeq0 11017
      ​5.4  Integer sets
            5.4.1  Positive integers (as a subset of complex numbers)   cn 11020
            5.4.2  Principle of mathematical induction   nnind 11038
            ​*5.4.3  Decimal representation of numbers   c2 11070
            ​*5.4.4  Some properties of specific numbers   neg1cn 11124
            5.4.5  Simple number properties   halfcl 11257
            5.4.6  The Archimedean property   nnunb 11288
            5.4.7  Nonnegative integers (as a subset of complex numbers)   cn0 11292
            ​*5.4.8  Extended nonnegative integers   cxnn0 11363
            5.4.9  Integers (as a subset of complex numbers)   cz 11377
            5.4.10  Decimal arithmetic   cdc 11493
            5.4.11  Upper sets of integers   cuz 11687
            5.4.12  Well-ordering principle for bounded-below sets of integers   uzwo3 11783
            5.4.13  Rational numbers (as a subset of complex numbers)   cq 11788
            5.4.14  Existence of the set of complex numbers   rpnnen1lem2 11814
      ​5.5  Order sets
            5.5.1  Positive reals (as a subset of complex numbers)   crp 11832
            5.5.2  Infinity and the extended real number system (cont.)   cxne 11943
            5.5.3  Supremum and infimum on the extended reals   xrsupexmnf 12135
            5.5.4  Real number intervals   cioo 12175
            5.5.5  Finite intervals of integers   cfz 12326
            ​*5.5.6  Finite intervals of nonnegative integers   elfz2nn0 12431
            5.5.7  Half-open integer ranges   cfzo 12465
      ​5.6  Elementary integer functions
            5.6.1  The floor and ceiling functions   cfl 12591
            5.6.2  The modulo (remainder) operation   cmo 12668
            5.6.3  Miscellaneous theorems about integers   om2uz0i 12746
            5.6.4  Strong induction over upper sets of integers   uzsinds 12786
            5.6.5  Finitely supported functions over the nonnegative integers   fsuppmapnn0fiublem 12789
            5.6.6  The infinite sequence builder "seq" - extension   cseq 12801
            5.6.7  Integer powers   cexp 12860
            5.6.8  Ordered pair theorem for nonnegative integers   nn0le2msqi 13054
            5.6.9  Factorial function   cfa 13060
            5.6.10  The binomial coefficient operation   cbc 13089
            5.6.11  The ` # ` (set size) function   chash 13117
                  5.6.11.1  Proper unordered pairs and triples (sets of size 2 and 3)   hashprlei 13250
                  5.6.11.2  Functions with a domain containing at least two different elements   fundmge2nop0 13274
                  5.6.11.3  Finite induction on the size of the first component of a binary relation   brfi1indlem 13278
      ​​*5.7  Words over a set
            5.7.1  Definitions and basic theorems   cword 13291
            5.7.2  Last symbol of a word   lsw 13351
            5.7.3  Concatenations of words   ccatfn 13357
            5.7.4  Singleton words   ids1 13377
            5.7.5  Concatenations with singleton words   ccatws1cl 13396
            5.7.6  Subwords   swrdval 13417
            5.7.7  Subwords of subwords   swrdswrdlem 13459
            5.7.8  Subwords and concatenations   wrdcctswrd 13465
            5.7.9  Subwords of concatenations   swrdccatfn 13482
            5.7.10  Splicing words (substring replacement)   splval 13502
            5.7.11  Reversing words   revval 13509
            5.7.12  Repeated symbol words   reps 13517
            ​*5.7.13  Cyclical shifts of words   ccsh 13534
            5.7.14  Mapping words by a function   wrdco 13577
            5.7.15  Longer string literals   cs2 13586
      ​​*5.8  Reflexive and transitive closures of relations
            5.8.1  The reflexive and transitive properties of relations   coss12d 13711
            5.8.2  Basic properties of closures   cleq1lem 13721
            5.8.3  Definitions and basic properties of transitive closures   ctcl 13724
            5.8.4  Exponentiation of relations   crelexp 13760
            5.8.5  Reflexive-transitive closure as an indexed union   crtrcl 13795
            ​*5.8.6  Principle of transitive induction.   relexpindlem 13803
      ​5.9  Elementary real and complex functions
            5.9.1  The "shift" operation   cshi 13806
            5.9.2  Signum (sgn or sign) function   csgn 13826
            5.9.3  Real and imaginary parts; conjugate   ccj 13836
            5.9.4  Square root; absolute value   csqrt 13973
      ​5.10  Elementary limits and convergence
            5.10.1  Superior limit (lim sup)   clsp 14201
            5.10.2  Limits   cli 14215
            5.10.3  Finite and infinite sums   csu 14416
            5.10.4  The binomial theorem   binomlem 14561
            5.10.5  The inclusion/exclusion principle   incexclem 14568
            5.10.6  Infinite sums (cont.)   isumshft 14571
            5.10.7  Miscellaneous converging and diverging sequences   divrcnv 14584
            5.10.8  Arithmetic series   arisum 14592
            5.10.9  Geometric series   expcnv 14596
            5.10.10  Ratio test for infinite series convergence   cvgrat 14615
            5.10.11  Mertens' theorem   mertenslem1 14616
            5.10.12  Finite and infinite products   prodf 14619
                  5.10.12.1  Product sequences   prodf 14619
                  5.10.12.2  Non-trivial convergence   ntrivcvg 14629
                  5.10.12.3  Complex products   cprod 14635
                  5.10.12.4  Finite products   fprod 14671
                  5.10.12.5  Infinite products   iprodclim 14729
            5.10.13  Falling and Rising Factorial   cfallfac 14735
            5.10.14  Bernoulli polynomials and sums of k-th powers   cbp 14777
      ​5.11  Elementary trigonometry
            5.11.1  The exponential, sine, and cosine functions   ce 14792
            5.11.2  _e is irrational   eirrlem 14932
      ​5.12  Cardinality of real and complex number subsets
            5.12.1  Countability of integers and rationals   xpnnen 14939
            5.12.2  The reals are uncountable   rpnnen2lem1 14943
​​*PART 6  ELEMENTARY NUMBER THEORY
      ​6.1  Elementary properties of divisibility
            6.1.1  Irrationality of square root of 2   sqrt2irrlem 14977
            6.1.2  Some Number sets are chains of proper subsets   nthruc 14981
            6.1.3  The divides relation   cdvds 14983
            ​*6.1.4  Even and odd numbers   evenelz 15060
            6.1.5  The division algorithm   divalglem0 15116
            6.1.6  Bit sequences   cbits 15141
            6.1.7  The greatest common divisor operator   cgcd 15216
            6.1.8  Bézout's identity   bezoutlem1 15256
            6.1.9  Algorithms   nn0seqcvgd 15283
            6.1.10  Euclid's Algorithm   eucalgval2 15294
            ​*6.1.11  The least common multiple   clcm 15301
            ​*6.1.12  Coprimality and Euclid's lemma   coprmgcdb 15362
            6.1.13  Cancellability of congruences   congr 15378
      ​6.2  Elementary prime number theory
            ​*6.2.1  Elementary properties   cprime 15385
            ​*6.2.2  Coprimality and Euclid's lemma (cont.)   coprm 15423
            6.2.3  Properties of the canonical representation of a rational   cnumer 15441
            6.2.4  Euler's theorem   codz 15468
            6.2.5  Arithmetic modulo a prime number   modprm1div 15502
            6.2.6  Pythagorean Triples   coprimeprodsq 15513
            6.2.7  The prime count function   cpc 15541
            6.2.8  Pocklington's theorem   prmpwdvds 15608
            6.2.9  Infinite primes theorem   unbenlem 15612
            6.2.10  Sum of prime reciprocals   prmreclem1 15620
            6.2.11  Fundamental theorem of arithmetic   1arithlem1 15627
            6.2.12  Lagrange's four-square theorem   cgz 15633
            6.2.13  Van der Waerden's theorem   cvdwa 15669
            6.2.14  Ramsey's theorem   cram 15703
            ​*6.2.15  Primorial function   cprmo 15735
            ​*6.2.16  Prime gaps   prmgaplem1 15753
            6.2.17  Decimal arithmetic (cont.)   dec2dvds 15767
            6.2.18  Cyclical shifts of words (cont.)   cshwsidrepsw 15800
            6.2.19  Specific prime numbers   prmlem0 15812
            6.2.20  Very large primes   1259lem1 15838
​PART 7  BASIC STRUCTURES
      ​7.1  Extensible structures
            ​*7.1.1  Basic definitions   cstr 15853
            7.1.2  Slot definitions   cplusg 15941
            7.1.3  Definition of the structure product   crest 16081
            7.1.4  Definition of the structure quotient   cordt 16159
      ​7.2  Moore spaces
            7.2.1  Moore closures   mrcflem 16266
            7.2.2  Independent sets in a Moore system   mrisval 16290
            7.2.3  Algebraic closure systems   isacs 16312
​PART 8  BASIC CATEGORY THEORY
      ​8.1  Categories
            8.1.1  Categories   ccat 16325
            8.1.2  Opposite category   coppc 16371
            8.1.3  Monomorphisms and epimorphisms   cmon 16388
            8.1.4  Sections, inverses, isomorphisms   csect 16404
            ​*8.1.5  Isomorphic objects   ccic 16455
            8.1.6  Subcategories   cssc 16467
            8.1.7  Functors   cfunc 16514
            8.1.8  Full & faithful functors   cful 16562
            8.1.9  Natural transformations and the functor category   cnat 16601
            8.1.10  Initial, terminal and zero objects of a category   cinito 16638
      ​8.2  Arrows (disjointified hom-sets)
            8.2.1  Identity and composition for arrows   cida 16703
      ​8.3  Examples of categories
            8.3.1  The category of sets   csetc 16725
            8.3.2  The category of categories   ccatc 16744
            ​*8.3.3  The category of extensible structures   fncnvimaeqv 16760
      ​8.4  Categorical constructions
            8.4.1  Product of categories   cxpc 16808
            8.4.2  Functor evaluation   cevlf 16849
            8.4.3  Hom functor   chof 16888
​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 16940
            9.2.2  Lattices   clat 17045
            9.2.3  The dual of an ordered set   codu 17128
            9.2.4  Subset order structures   cipo 17151
            9.2.5  Distributive lattices   latmass 17188
            9.2.6  Posets and lattices as relations   cps 17198
            9.2.7  Directed sets, nets   cdir 17228
​PART 10  BASIC ALGEBRAIC STRUCTURES
      ​10.1  Monoids
            ​*10.1.1  Magmas   cplusf 17239
            ​*10.1.2  Identity elements   mgmidmo 17259
            ​*10.1.3  Ordered sums in a magma   gsumvalx 17270
            ​*10.1.4  Semigroups   csgrp 17283
            ​*10.1.5  Definition and basic properties of monoids   cmnd 17294
            10.1.6  Monoid homomorphisms and submonoids   cmhm 17333
            ​*10.1.7  Ordered sums in a monoid   gsumvallem2 17372
            10.1.8  Free monoids   cfrmd 17384
            10.1.9  Examples and counterexamples for magmas, semigroups and monoids   mgm2nsgrplem1 17405
      ​10.2  Groups
            10.2.1  Definition and basic properties   cgrp 17422
            ​*10.2.2  Group multiple operation   cmg 17540
            10.2.3  Subgroups and Quotient groups   csubg 17588
            10.2.4  Elementary theory of group homomorphisms   cghm 17657
            10.2.5  Isomorphisms of groups   cgim 17699
            10.2.6  Group actions   cga 17722
            10.2.7  Centralizers and centers   ccntz 17748
            10.2.8  The opposite group   coppg 17775
            10.2.9  Symmetric groups   csymg 17797
                  ​*10.2.9.1  Definition and basic properties   csymg 17797
                  10.2.9.2  Cayley's theorem   cayleylem1 17832
                  10.2.9.3  Permutations fixing one element   symgfix2 17836
                  ​*10.2.9.4  Transpositions in the symmetric group   cpmtr 17861
                  10.2.9.5  The sign of a permutation   cpsgn 17909
            10.2.10  p-Groups and Sylow groups; Sylow's theorems   cod 17944
            10.2.11  Direct products   clsm 18049
            10.2.12  Free groups   cefg 18119
      ​10.3  Abelian groups
            10.3.1  Definition and basic properties   ccmn 18193
            10.3.2  Cyclic groups   ccyg 18279
            10.3.3  Group sum operation   gsumval3a 18304
            10.3.4  Group sums over (ranges of) integers   fsfnn0gsumfsffz 18379
            10.3.5  Internal direct products   cdprd 18392
            10.3.6  The Fundamental Theorem of Abelian Groups   ablfacrplem 18464
      ​10.4  Rings
            10.4.1  Multiplicative Group   cmgp 18489
            10.4.2  Ring unit   cur 18501
                  10.4.2.1  Semirings   csrg 18505
                  ​*10.4.2.2  The binomial theorem for semirings   srgbinomlem1 18540
            10.4.3  Definition and basic properties of unital rings   crg 18547
            10.4.4  Opposite ring   coppr 18622
            10.4.5  Divisibility   cdsr 18638
            10.4.6  Ring homomorphisms   crh 18712
      ​10.5  Division rings and fields
            10.5.1  Definition and basic properties   cdr 18747
            10.5.2  Subrings of a ring   csubrg 18776
            10.5.3  Absolute value (abstract algebra)   cabv 18816
            10.5.4  Star rings   cstf 18843
      ​10.6  Left modules
            10.6.1  Definition and basic properties   clmod 18863
            10.6.2  Subspaces and spans in a left module   clss 18932
            10.6.3  Homomorphisms and isomorphisms of left modules   clmhm 19019
            10.6.4  Subspace sum; bases for a left module   clbs 19074
      ​10.7  Vector spaces
            10.7.1  Definition and basic properties   clvec 19102
      ​10.8  Ideals
            10.8.1  The subring algebra; ideals   csra 19168
            10.8.2  Two-sided ideals and quotient rings   c2idl 19231
            10.8.3  Principal ideal rings. Divisibility in the integers   clpidl 19241
            10.8.4  Nonzero rings and zero rings   cnzr 19257
            10.8.5  Left regular elements. More kinds of rings   crlreg 19279
      ​10.9  Associative algebras
            10.9.1  Definition and basic properties   casa 19309
      ​10.10  Abstract multivariate polynomials
            10.10.1  Definition and basic properties   cmps 19351
            10.10.2  Polynomial evaluation   ces 19504
            ​*10.10.3  Additional definitions for (multivariate) polynomials   cmhp 19537
            ​*10.10.4  Univariate polynomials   cps1 19545
            10.10.5  Univariate polynomial evaluation   ces1 19678
      ​10.11  The complex numbers as an algebraic extensible structure
            10.11.1  Definition and basic properties   cpsmet 19730
            ​*10.11.2  Ring of integers   zring 19818
            10.11.3  Algebraic constructions based on the complex numbers   czrh 19848
            10.11.4  Signs as subgroup of the complex numbers   cnmsgnsubg 19923
            10.11.5  Embedding of permutation signs into a ring   zrhpsgnmhm 19930
            10.11.6  The ordered field of real numbers   crefld 19950
      ​10.12  Generalized pre-Hilbert and Hilbert spaces
            10.12.1  Definition and basic properties   cphl 19969
            10.12.2  Orthocomplements and closed subspaces   cocv 20004
            10.12.3  Orthogonal projection and orthonormal bases   cpj 20044
​​*PART 11  BASIC LINEAR ALGEBRA
      ​11.1  Vectors and free modules
            *11.1.1  Direct sum of left modules   cdsmm 20075
            ​*11.1.2  Free modules   cfrlm 20090
            ​*11.1.3  Standard basis (unit vectors)   cuvc 20121
            ​*11.1.4  Independent sets and families   clindf 20143
            11.1.5  Characterization of free modules   lmimlbs 20175
      ​​*11.2  Matrices
            *11.2.1  The matrix multiplication   cmmul 20189
            ​*11.2.2  Square matrices   cmat 20213
            ​*11.2.3  The matrix algebra   matmulr 20244
            ​*11.2.4  Matrices of dimension 0 and 1   mat0dimbas0 20272
            ​*11.2.5  The subalgebras of diagonal and scalar matrices   cdmat 20294
            ​*11.2.6  Multiplication of a matrix with a "column vector"   cmvmul 20346
            11.2.7  Replacement functions for a square matrix   cmarrep 20362
            11.2.8  Submatrices   csubma 20382
      ​11.3  The determinant
            11.3.1  Definition and basic properties   cmdat 20390
            11.3.2  Determinants of 2 x 2 -matrices   m2detleiblem1 20430
            11.3.3  The matrix adjugate/adjunct   cmadu 20438
            ​*11.3.4  Laplace expansion of determinants (special case)   symgmatr01lem 20459
            11.3.5  Inverse matrix   invrvald 20482
            ​*11.3.6  Cramer's rule   slesolvec 20485
      ​​*11.4  Polynomial matrices
            11.4.1  Basic properties   pmatring 20498
            ​*11.4.2  Constant polynomial matrices   ccpmat 20508
            ​*11.4.3  Collecting coefficients of polynomial matrices   cdecpmat 20567
            ​*11.4.4  Ring isomorphism between polynomial matrices and polynomials over matrices   cpm2mp 20597
      ​​*11.5  The characteristic polynomial
            *11.5.1  Definition and basic properties   cchpmat 20631
            ​*11.5.2  The characteristic factor function G   fvmptnn04if 20654
            ​*11.5.3  The Cayley-Hamilton theorem   cpmadurid 20672
​PART 12  BASIC TOPOLOGY
      ​12.1  Topology
            ​*12.1.1  Topological spaces   ctop 20698
                  12.1.1.1  Topologies   ctop 20698
                  12.1.1.2  Topologies on sets   ctopon 20715
                  12.1.1.3  Topological spaces   ctps 20736
            12.1.2  Topological bases   ctb 20749
            12.1.3  Examples of topologies   distop 20799
            12.1.4  Closure and interior   ccld 20820
            12.1.5  Neighborhoods   cnei 20901
            12.1.6  Limit points and perfect sets   clp 20938
            12.1.7  Subspace topologies   restrcl 20961
            12.1.8  Order topology   ordtbaslem 20992
            12.1.9  Limits and continuity in topological spaces   ccn 21028
            12.1.10  Separated spaces: T0, T1, T2 (Hausdorff) ...   ct0 21110
            12.1.11  Compactness   ccmp 21189
            12.1.12  Bolzano-Weierstrass theorem   bwth 21213
            12.1.13  Connectedness   cconn 21214
            12.1.14  First- and second-countability   c1stc 21240
            12.1.15  Local topological properties   clly 21267
            12.1.16  Refinements   cref 21305
            12.1.17  Compactly generated spaces   ckgen 21336
            12.1.18  Product topologies   ctx 21363
            12.1.19  Continuous function-builders   cnmptid 21464
            12.1.20  Quotient maps and quotient topology   ckq 21496
            12.1.21  Homeomorphisms   chmeo 21556
      ​12.2  Filters and filter bases
            12.2.1  Filter bases   elmptrab 21630
            12.2.2  Filters   cfil 21649
            12.2.3  Ultrafilters   cufil 21703
            12.2.4  Filter limits   cfm 21737
            12.2.5  Extension by continuity   ccnext 21863
            12.2.6  Topological groups   ctmd 21874
            12.2.7  Infinite group sum on topological groups   ctsu 21929
            12.2.8  Topological rings, fields, vector spaces   ctrg 21959
      ​12.3  Uniform Structures and Spaces
            12.3.1  Uniform structures   cust 22003
            12.3.2  The topology induced by an uniform structure   cutop 22034
            12.3.3  Uniform Spaces   cuss 22057
            12.3.4  Uniform continuity   cucn 22079
            12.3.5  Cauchy filters in uniform spaces   ccfilu 22090
            12.3.6  Complete uniform spaces   ccusp 22101
      ​12.4  Metric spaces
            12.4.1  Pseudometric spaces   ispsmet 22109
            12.4.2  Basic metric space properties   cxme 22122
            12.4.3  Metric space balls   blfvalps 22188
            12.4.4  Open sets of a metric space   mopnval 22243
            12.4.5  Continuity in metric spaces   metcnp3 22345
            12.4.6  The uniform structure generated by a metric   metuval 22354
            12.4.7  Examples of metric spaces   dscmet 22377
            ​*12.4.8  Normed algebraic structures   cnm 22381
            12.4.9  Normed space homomorphisms (bounded linear operators)   cnmo 22509
            12.4.10  Topology on the reals   qtopbaslem 22562
            12.4.11  Topological definitions using the reals   cii 22678
            12.4.12  Path homotopy   chtpy 22766
            12.4.13  The fundamental group   cpco 22800
      ​12.5  Metric subcomplex vector spaces
            12.5.1  Subcomplex modules   cclm 22862
            ​*12.5.2  Subcomplex vector spaces   ccvs 22923
            ​*12.5.3  Normed subcomplex vector spaces   isncvsngp 22949
            12.5.4  Subcomplex pre-Hilbert space   ccph 22966
            12.5.5  Convergence and completeness   ccfil 23050
            12.5.6  Baire's Category Theorem   bcthlem1 23121
            12.5.7  Banach spaces and subcomplex Hilbert spaces   ccms 23129
                  12.5.7.1  The complete ordered field of the real numbers   retopn 23167
            12.5.8  Euclidean spaces   crrx 23171
            12.5.9  Minimizing Vector Theorem   minveclem1 23195
            12.5.10  Projection Theorem   pjthlem1 23208
​PART 13  BASIC REAL AND COMPLEX ANALYSIS
      ​13.1  Continuity
            13.1.1  Intermediate value theorem   pmltpclem1 23217
      ​13.2  Integrals
            13.2.1  Lebesgue measure   covol 23231
            13.2.2  Lebesgue integration   cmbf 23383
                  13.2.2.1  Lesbesgue integral   cmbf 23383
                  13.2.2.2  Lesbesgue directed integral   cdit 23610
      ​13.3  Derivatives
            13.3.1  Real and complex differentiation   climc 23626
                  13.3.1.1  Derivatives of functions of one complex or real variable   climc 23626
                  13.3.1.2  Results on real differentiation   dvferm1lem 23747
​PART 14  BASIC REAL AND COMPLEX FUNCTIONS
      ​14.1  Polynomials
            14.1.1  Polynomial degrees   cmdg 23813
            14.1.2  The division algorithm for univariate polynomials   cmn1 23885
            14.1.3  Elementary properties of complex polynomials   cply 23940
            14.1.4  The division algorithm for polynomials   cquot 24045
            14.1.5  Algebraic numbers   caa 24069
            14.1.6  Liouville's approximation theorem   aalioulem1 24087
      ​14.2  Sequences and series
            14.2.1  Taylor polynomials and Taylor's theorem   ctayl 24107
            14.2.2  Uniform convergence   culm 24130
            14.2.3  Power series   pserval 24164
      ​14.3  Basic trigonometry
            14.3.1  The exponential, sine, and cosine functions (cont.)   efcn 24197
            14.3.2  Properties of pi = 3.14159...   pilem1 24205
            14.3.3  Mapping of the exponential function   efgh 24287
            14.3.4  The natural logarithm on complex numbers   clog 24301
            ​*14.3.5  Logarithms to an arbitrary base   clogb 24502
            14.3.6  Theorems of Pythagoras, isosceles triangles, and intersecting chords   angval 24531
            14.3.7  Solutions of quadratic, cubic, and quartic equations   quad2 24566
            14.3.8  Inverse trigonometric functions   casin 24589
            14.3.9  The Birthday Problem   log2ublem1 24673
            14.3.10  Areas in R^2   carea 24682
            14.3.11  More miscellaneous converging sequences   rlimcnp 24692
            14.3.12  Inequality of arithmetic and geometric means   cvxcl 24711
            14.3.13  Euler-Mascheroni constant   cem 24718
            14.3.14  Zeta function   czeta 24739
            14.3.15  Gamma function   clgam 24742
      ​14.4  Basic number theory
            14.4.1  Wilson's theorem   wilthlem1 24794
            14.4.2  The Fundamental Theorem of Algebra   ftalem1 24799
            14.4.3  The Basel problem (ζ(2) = π²/6)   basellem1 24807
            14.4.4  Number-theoretical functions   ccht 24817
            14.4.5  Perfect Number Theorem   mersenne 24952
            14.4.6  Characters of Z/nZ   cdchr 24957
            14.4.7  Bertrand's postulate   bcctr 25000
            ​*14.4.8  Quadratic residues and the Legendre symbol   clgs 25019
            ​*14.4.9  Gauss' Lemma   gausslemma2dlem0a 25081
            14.4.10  Quadratic reciprocity   lgseisenlem1 25100
            14.4.11  All primes 4n+1 are the sum of two squares   2sqlem1 25142
            14.4.12  Chebyshev's Weak Prime Number Theorem, Dirichlet's Theorem   chebbnd1lem1 25158
            14.4.13  The Prime Number Theorem   mudivsum 25219
            14.4.14  Ostrowski's theorem   abvcxp 25304
​​*PART 15  ELEMENTARY GEOMETRY
      ​15.1  Definition and Tarski's Axioms of Geometry
      ​15.2  Tarskian Geometry
            15.2.1  Congruence   tgcgrcomimp 25372
            15.2.2  Betweenness   tgbtwntriv2 25382
            15.2.3  Dimension   tglowdim1 25395
            15.2.4  Betweenness and Congruence   tgifscgr 25403
            15.2.5  Congruence of a series of points   ccgrg 25405
            15.2.6  Motions   cismt 25427
            15.2.7  Colinearity   tglng 25441
            15.2.8  Connectivity of betweenness   tgbtwnconn1lem1 25467
            15.2.9  Less-than relation in geometric congruences   cleg 25477
            15.2.10  Rays   chlg 25495
            15.2.11  Lines   btwnlng1 25514
            15.2.12  Point inversions   cmir 25547
            15.2.13  Right angles   crag 25588
            15.2.14  Half-planes   islnopp 25631
            15.2.15  Midpoints and Line Mirroring   cmid 25664
            15.2.16  Congruence of angles   ccgra 25699
            15.2.17  Angle Comparisons   cinag 25726
            15.2.18  Congruence Theorems   tgsas1 25735
            15.2.19  Equilateral triangles   ceqlg 25745
      ​15.3  Properties of geometries
            15.3.1  Isomorphisms between geometries   f1otrgds 25749
      ​15.4  Geometry in Hilbert spaces
            15.4.1  Geometry in the complex plane   cchhllem 25767
            15.4.2  Geometry in Euclidean spaces   cee 25768
                  15.4.2.1  Definition of the Euclidean space   cee 25768
                  15.4.2.2  Tarski's axioms for geometry for the Euclidean space   axdimuniq 25793
                  15.4.2.3  EE^n fulfills Tarski's Axioms   ceeng 25857
​​*PART 16  GRAPH THEORY
      ​*16.1  Vertices and edges
            16.1.1  The edge function extractor for extensible structures   cedgf 25867
            ​*16.1.2  Vertices and indexed edges   cvtx 25874
                  16.1.2.1  Definitions and basic properties   cvtx 25874
                  16.1.2.2  The vertices and edges of a graph represented as ordered pair   opvtxval 25883
                  16.1.2.3  The vertices and edges of a graph represented as extensible structure   funvtxdmge2val 25891
                  16.1.2.4  Representations of graphs without edges   snstrvtxval 25929
                  16.1.2.5  Degenerated cases of representations of graphs   vtxval0 25931
            16.1.3  Edges as range of the edge function   cedg 25939
      ​​*16.2  Undirected graphs
            16.2.1  Undirected hypergraphs   cuhgr 25951
            16.2.2  Undirected pseudographs and multigraphs   cupgr 25975
            ​*16.2.3  Loop-free graphs   umgrislfupgrlem 26017
            16.2.4  Edges as subsets of vertices of graphs   uhgredgiedgb 26021
            ​*16.2.5  Undirected simple graphs   cuspgr 26043
            16.2.6  Examples for graphs   usgr0e 26128
            16.2.7  Subgraphs   csubgr 26159
            16.2.8  Finite undirected simple graphs   cfusgr 26208
            16.2.9  Neighbors, complete graphs and universal vertices   cnbgr 26224
                  16.2.9.1  Definitions   cnbgr 26224
                  16.2.9.2  Neighbors   nbgrcl 26233
                  16.2.9.3  Universal vertices   uvtxaval 26287
                  16.2.9.4  Complete graphs   iscplgr 26310
            16.2.10  Vertex degree   cvtxdg 26361
            ​*16.2.11  Regular graphs   crgr 26451
      ​16.3  Walks, paths and cycles
            ​*16.3.1  Walks   cewlks 26491
            16.3.2  Walks for loop-free graphs   lfgrwlkprop 26584
            16.3.3  Trails   ctrls 26587
            16.3.4  Paths and simple paths   cpths 26608
            16.3.5  Closed walks   cclwlks 26666
            16.3.6  Circuits and cycles   ccrcts 26679
            ​*16.3.7  Walks as words   cwwlks 26717
            16.3.8  Walks/paths of length 2 (as length 3 strings)   2wlkdlem1 26821
            16.3.9  Walks in regular graphs   rusgrnumwwlkl1 26863
            ​*16.3.10  Closed walks as words   cclwwlks 26875
            16.3.11  Examples for walks, trails and paths   0ewlk 26975
            16.3.12  Connected graphs   cconngr 27046
      ​16.4  Eulerian paths and the Konigsberg Bridge problem
            ​*16.4.1  Eulerian paths   ceupth 27057
            ​*16.4.2  The Königsberg Bridge problem   konigsbergvtx 27106
      ​16.5  The Friendship Theorem
            16.5.1  Friendship graphs - basics   cfrgr 27120
            16.5.2  The friendship theorem for small graphs   frgr1v 27135
            16.5.3  Theorems according to Mertzios and Unger   2pthfrgrrn 27146
            ​*16.5.4  Huneke's Proof of the Friendship Theorem   frgrncvvdeqlem1 27163
​PART 17  GUIDES AND MISCELLANEA
      ​17.1  Guides (conventions, explanations, and examples)
            ​*17.1.1  Conventions   conventions 27258
            17.1.2  Natural deduction   natded 27260
            ​*17.1.3  Natural deduction examples   ex-natded5.2 27261
            17.1.4  Definitional examples   ex-or 27278
            17.1.5  Other examples   aevdemo 27317
      ​17.2  Humor
            17.2.1  April Fool's theorem   avril1 27319
      ​17.3  (Future - to be reviewed and classified)
            17.3.1  Planar incidence geometry   cplig 27326
            17.3.2  Algebra preliminaries   crpm 27339
            ​*17.3.3  Aliases kept to prevent broken links   dummylink 27341
​​*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 27343
            18.1.2  Abelian groups   cablo 27398
      ​18.2  Complex vector spaces
            18.2.1  Definition and basic properties   cvc 27413
            18.2.2  Examples of complex vector spaces   cnaddabloOLD 27436
      ​18.3  Normed complex vector spaces
            18.3.1  Definition and basic properties   cnv 27439
            18.3.2  Examples of normed complex vector spaces   cnnv 27532
            18.3.3  Induced metric of a normed complex vector space   imsval 27540
            18.3.4  Inner product   cdip 27555
            18.3.5  Subspaces   css 27576
      ​18.4  Operators on complex vector spaces
            18.4.1  Definitions and basic properties   clno 27595
      ​18.5  Inner product (pre-Hilbert) spaces
            18.5.1  Definition and basic properties   ccphlo 27667
            18.5.2  Examples of pre-Hilbert spaces   cncph 27674
            18.5.3  Properties of pre-Hilbert spaces   isph 27677
      ​18.6  Complex Banach spaces
            18.6.1  Definition and basic properties   ccbn 27718
            18.6.2  Examples of complex Banach spaces   cnbn 27725
            18.6.3  Uniform Boundedness Theorem   ubthlem1 27726
            18.6.4  Minimizing Vector Theorem   minvecolem1 27730
      ​18.7  Complex Hilbert spaces
            18.7.1  Definition and basic properties   chlo 27741
            18.7.2  Standard axioms for a complex Hilbert space   hlex 27754
            18.7.3  Examples of complex Hilbert spaces   cnchl 27772
            18.7.4  Subspaces   ssphl 27773
            18.7.5  Hellinger-Toeplitz Theorem   htthlem 27774
​​*PART 19  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      ​19.1  Axiomatization of complex pre-Hilbert spaces
            19.1.1  Basic Hilbert space definitions   chil 27776
            19.1.2  Preliminary ZFC lemmas   df-hnorm 27825
            ​*19.1.3  Derive the Hilbert space axioms from ZFC set theory   axhilex-zf 27838
            ​*19.1.4  Introduce the vector space axioms for a Hilbert space   ax-hilex 27856
            19.1.5  Vector operations   hvmulex 27868
            19.1.6  Inner product postulates for a Hilbert space   ax-hfi 27936
      ​19.2  Inner product and norms
            19.2.1  Inner product   his5 27943
            19.2.2  Norms   dfhnorm2 27979
            19.2.3  Relate Hilbert space to normed complex vector spaces   hilablo 28017
            19.2.4  Bunjakovaskij-Cauchy-Schwarz inequality   bcsiALT 28036
      ​19.3  Cauchy sequences and completeness axiom
            19.3.1  Cauchy sequences and limits   hcau 28041
            19.3.2  Derivation of the completeness axiom from ZF set theory   hilmet 28051
            19.3.3  Completeness postulate for a Hilbert space   ax-hcompl 28059
            19.3.4  Relate Hilbert space to ZFC pre-Hilbert and Hilbert spaces   hhcms 28060
      ​19.4  Subspaces and projections
            19.4.1  Subspaces   df-sh 28064
            19.4.2  Closed subspaces   df-ch 28078
            19.4.3  Orthocomplements   df-oc 28109
            19.4.4  Subspace sum, span, lattice join, lattice supremum   df-shs 28167
            19.4.5  Projection theorem   pjhthlem1 28250
            19.4.6  Projectors   df-pjh 28254
      ​19.5  Properties of Hilbert subspaces
            19.5.1  Orthomodular law   omlsilem 28261
            19.5.2  Projectors (cont.)   pjhtheu2 28275
            19.5.3  Hilbert lattice operations   sh0le 28299
            19.5.4  Span (cont.) and one-dimensional subspaces   spansn0 28400
            19.5.5  Commutes relation for Hilbert lattice elements   df-cm 28442
            19.5.6  Foulis-Holland theorem   fh1 28477
            19.5.7  Quantum Logic Explorer axioms   qlax1i 28486
            19.5.8  Orthogonal subspaces   chscllem1 28496
            19.5.9  Orthoarguesian laws 5OA and 3OA   5oalem1 28513
            19.5.10  Projectors (cont.)   pjorthi 28528
            19.5.11  Mayet's equation E_3   mayete3i 28587
      ​19.6  Operators on Hilbert spaces
            ​*19.6.1  Operator sum, difference, and scalar multiplication   df-hosum 28589
            19.6.2  Zero and identity operators   df-h0op 28607
            19.6.3  Operations on Hilbert space operators   hoaddcl 28617
            19.6.4  Linear, continuous, bounded, Hermitian, unitary operators and norms   df-nmop 28698
            19.6.5  Linear and continuous functionals and norms   df-nmfn 28704
            19.6.6  Adjoint   df-adjh 28708
            19.6.7  Dirac bra-ket notation   df-bra 28709
            19.6.8  Positive operators   df-leop 28711
            19.6.9  Eigenvectors, eigenvalues, spectrum   df-eigvec 28712
            19.6.10  Theorems about operators and functionals   nmopval 28715
            19.6.11  Riesz lemma   riesz3i 28921
            19.6.12  Adjoints (cont.)   cnlnadjlem1 28926
            19.6.13  Quantum computation error bound theorem   unierri 28963
            19.6.14  Dirac bra-ket notation (cont.)   branmfn 28964
            19.6.15  Positive operators (cont.)   leopg 28981
            19.6.16  Projectors as operators   pjhmopi 29005
      ​19.7  States on a Hilbert lattice and Godowski's equation
            19.7.1  States on a Hilbert lattice   df-st 29070
            19.7.2  Godowski's equation   golem1 29130
      ​19.8  Cover relation, atoms, exchange axiom, and modular symmetry
            19.8.1  Covers relation; modular pairs   df-cv 29138
            19.8.2  Atoms   df-at 29197
            19.8.3  Superposition principle   superpos 29213
            19.8.4  Atoms, exchange and covering properties, atomicity   chcv1 29214
            19.8.5  Irreducibility   chirredlem1 29249
            19.8.6  Atoms (cont.)   atcvat3i 29255
            19.8.7  Modular symmetry   mdsymlem1 29262
​PART 20  SUPPLEMENTARY MATERIAL (USER'S MATHBOXES)
      ​20.1  Mathboxes for user contributions
            20.1.1  Mathbox guidelines   mathbox 29301
      ​20.2  Mathbox for Stefan Allan
      ​20.3  Mathbox for Thierry Arnoux
            20.3.1  Propositional Calculus - misc additions   bian1d 29306
            20.3.2  Predicate Calculus   spc2ed 29312
                  20.3.2.1  Predicate Calculus - misc additions   spc2ed 29312
                  20.3.2.2  Restricted quantification - misc additions   ralcom4f 29316
                  20.3.2.3  Substitution (without distinct variables) - misc additions   sbceqbidf 29321
                  20.3.2.4  Existential "at most one" - misc additions   moel 29323
                  20.3.2.5  Existential uniqueness - misc additions   2reuswap2 29328
                  20.3.2.6  Restricted "at most one" - misc additions   rmoxfrdOLD 29332
            20.3.3  General Set Theory   rabrab 29338
                  20.3.3.1  Class abstractions (a.k.a. class builders)   rabrab 29338
                  20.3.3.2  Image Sets   abrexdomjm 29345
                  20.3.3.3  Set relations and operations - misc additions   rabss3d 29351
                  20.3.3.4  Unordered pairs   elpreq 29360
                  20.3.3.5  Conditional operator - misc additions   ifeqeqx 29361
                  20.3.3.6  Set union   uniinn0 29366
                  20.3.3.7  Indexed union - misc additions   cbviunf 29372
                  20.3.3.8  Disjointness - misc additions   disjnf 29384
            20.3.4  Relations and Functions   xpdisjres 29411
                  20.3.4.1  Relations - misc additions   xpdisjres 29411
                  20.3.4.2  Functions - misc additions   ac6sf2 29429
                  20.3.4.3  Operations - misc additions   mpt2mptxf 29477
                  20.3.4.4  Isomorphisms - misc. add.   gtiso 29478
                  20.3.4.5  Disjointness (additional proof requiring functions)   disjdsct 29480
                  20.3.4.6  First and second members of an ordered pair - misc additions   df1stres 29481
                  20.3.4.7  Supremum - misc additions   supssd 29487
                  20.3.4.8  Finite Sets   imafi2 29489
                  20.3.4.9  Countable Sets   snct 29491
            20.3.5  Real and Complex Numbers   addeq0 29510
                  20.3.5.1  Complex operations - misc. additions   addeq0 29510
                  20.3.5.2  Ordering on reals - misc additions   lt2addrd 29516
                  20.3.5.3  Extended reals - misc additions   xrlelttric 29517
                  20.3.5.4  Real number intervals - misc additions   joiniooico 29536
                  20.3.5.5  Finite intervals of integers - misc additions   uzssico 29546
                  20.3.5.6  Half-open integer ranges - misc additions   iundisjfi 29555
                  20.3.5.7  The ` # ` (set size) function - misc additions   hashunif 29562
                  20.3.5.8  The greatest common divisor operator - misc. add   numdenneg 29563
                  20.3.5.9  Integers   nnindf 29565
                  20.3.5.10  Decimal numbers   dfdec100 29576
            ​*20.3.6  Decimal expansion   cdp2 29577
                  ​*20.3.6.1  Decimal point   cdp 29595
                  20.3.6.2  Division in the extended real number system   cxdiv 29625
            20.3.7  Prime Number Theory   bhmafibid1 29644
                  20.3.7.1  Fermat's two square theorem   bhmafibid1 29644
            20.3.8  Extensible Structures   ressplusf 29650
                  20.3.8.1  Structure restriction operator   ressplusf 29650
                  20.3.8.2  The opposite group   oppgle 29653
                  20.3.8.3  Posets   ressprs 29655
                  20.3.8.4  Complete lattices   clatp0cl 29671
                  20.3.8.5  Extended reals Structure - misc additions   ax-xrssca 29673
                  20.3.8.6  The extended nonnegative real numbers commutative monoid   xrge0base 29685
            20.3.9  Algebra   abliso 29696
                  20.3.9.1  Monoids Homomorphisms   abliso 29696
                  20.3.9.2  Ordered monoids and groups   comnd 29697
                  20.3.9.3  Signum in an ordered monoid   csgns 29725
                  20.3.9.4  The Archimedean property for generic ordered algebraic structures   cinftm 29730
                  20.3.9.5  Semiring left modules   cslmd 29753
                  20.3.9.6  Finitely supported group sums - misc additions   gsumle 29779
                  20.3.9.7  Rings - misc additions   rngurd 29788
                  20.3.9.8  Ordered rings and fields   corng 29795
                  20.3.9.9  Ring homomorphisms - misc additions   rhmdvdsr 29818
                  20.3.9.10  Scalar restriction operation   cresv 29824
                  20.3.9.11  The commutative ring of gaussian integers   gzcrng 29839
                  20.3.9.12  The archimedean ordered field of real numbers   reofld 29840
            20.3.10  Matrices   symgfcoeu 29845
                  20.3.10.1  The symmetric group   symgfcoeu 29845
                  20.3.10.2  Permutation Signs   psgndmfi 29846
                  20.3.10.3  Transpositions   pmtridf1o 29856
                  20.3.10.4  Submatrices   csmat 29859
                  20.3.10.5  Matrix literals   clmat 29877
                  20.3.10.6  Laplace expansion of determinants   mdetpmtr1 29889
            20.3.11  Topology   fvproj 29899
                  20.3.11.1  Open maps   fvproj 29899
                  20.3.11.2  Topology of the unit circle   qtopt1 29902
                  20.3.11.3  Refinements   reff 29906
                  20.3.11.4  Open cover refinement property   ccref 29909
                  20.3.11.5  Lindelöf spaces   cldlf 29919
                  20.3.11.6  Paracompact spaces   cpcmp 29922
                  20.3.11.7  Pseudometrics   cmetid 29929
                  20.3.11.8  Continuity - misc additions   hauseqcn 29941
                  20.3.11.9  Topology of the closed unit   unitsscn 29942
                  20.3.11.10  Topology of ` ( RR X. RR ) `   unicls 29949
                  20.3.11.11  Order topology - misc. additions   cnvordtrestixx 29959
                  20.3.11.12  Continuity in topological spaces - misc. additions   mndpluscn 29972
                  20.3.11.13  Topology of the extended nonnegative real numbers ordered monoid   xrge0hmph 29978
                  20.3.11.14  Limits - misc additions   lmlim 29993
                  20.3.11.15  Univariate polynomials   pl1cn 30001
            20.3.12  Uniform Stuctures and Spaces   chcmp 30002
                  20.3.12.1  Hausdorff uniform completion   chcmp 30002
            20.3.13  Topology and algebraic structures   zringnm 30004
                  20.3.13.1  The norm on the ring of the integer numbers   zringnm 30004
                  20.3.13.2  Topological ` ZZ ` -modules   zlm0 30006
                  20.3.13.3  Canonical embedding of the field of the rational numbers into a division ring   cqqh 30016
                  20.3.13.4  Canonical embedding of the real numbers into a complete ordered field   crrh 30037
                  20.3.13.5  Embedding from the extended real numbers into a complete lattice   cxrh 30060
                  20.3.13.6  Canonical embeddings into the ordered field of the real numbers   zrhre 30063
                  ​*20.3.13.7  Topological Manifolds   cmntop 30066
            20.3.14  Real and complex functions   nexple 30071
                  20.3.14.1  Integer powers - misc. additions   nexple 30071
                  20.3.14.2  Indicator Functions   cind 30072
                  20.3.14.3  Extended sum   cesum 30089
            20.3.15  Mixed Function/Constant operation   cofc 30157
            20.3.16  Abstract measure   csiga 30170
                  20.3.16.1  Sigma-Algebra   csiga 30170
                  20.3.16.2  Generated sigma-Algebra   csigagen 30201
                  ​*20.3.16.3  lambda and pi-Systems, Rings of Sets   ispisys 30215
                  20.3.16.4  The Borel algebra on the real numbers   cbrsiga 30244
                  20.3.16.5  Product Sigma-Algebra   csx 30251
                  20.3.16.6  Measures   cmeas 30258
                  20.3.16.7  The counting measure   cntmeas 30289
                  20.3.16.8  The Lebesgue measure - misc additions   voliune 30292
                  20.3.16.9  The Dirac delta measure   cdde 30295
                  20.3.16.10  The 'almost everywhere' relation   cae 30300
                  20.3.16.11  Measurable functions   cmbfm 30312
                  20.3.16.12  Borel Algebra on ` ( RR X. RR ) `   br2base 30331
                  ​*20.3.16.13  Caratheodory's extension theorem   coms 30353
            20.3.17  Integration   itgeq12dv 30388
                  20.3.17.1  Lebesgue integral - misc additions   itgeq12dv 30388
                  20.3.17.2  Bochner integral   citgm 30389
            20.3.18  Euler's partition theorem   oddpwdc 30416
            20.3.19  Sequences defined by strong recursion   csseq 30445
            20.3.20  Fibonacci Numbers   cfib 30458
            20.3.21  Probability   cprb 30469
                  20.3.21.1  Probability Theory   cprb 30469
                  20.3.21.2  Conditional Probabilities   ccprob 30493
                  20.3.21.3  Real Valued Random Variables   crrv 30502
                  20.3.21.4  Preimage set mapping operator   corvc 30517
                  20.3.21.5  Distribution Functions   orvcelval 30530
                  20.3.21.6  Cumulative Distribution Functions   orvclteel 30534
                  20.3.21.7  Probabilities - example   coinfliplem 30540
                  20.3.21.8  Bertrand's Ballot Problem   ballotlemoex 30547
            20.3.22  Signum (sgn or sign) function - misc. additions   sgncl 30600
            20.3.23  Words over a set - misc additions   wrdfd 30616
                  20.3.23.1  Operations on words   ccatmulgnn0dir 30619
            20.3.24  Polynomials with real coefficients - misc additions   plymul02 30623
            20.3.25  Descartes's rule of signs   signspval 30629
                  20.3.25.1  Sign changes in a word over real numbers   signspval 30629
                  20.3.25.2  Counting sign changes in a word over real numbers   signslema 30639
            20.3.26  Number Theory   efcld 30669
                  20.3.26.1  Representations of a number as sums of integers   crepr 30686
                  20.3.26.2  Vinogradov Trigonometric Sums and the Circle Method   cvts 30713
                  20.3.26.3  The Ternary Goldbach Conjecture: Final Statement   ax-hgt749 30722
            20.3.27  Elementary Geometry   cstrkg2d 30742
                  ​*20.3.27.1  Two-dimension geometry   cstrkg2d 30742
                  20.3.27.2  Outer Five Segment (not used, no need to move to main)   cafs 30747
      ​​*20.4  Mathbox for Jonathan Ben-Naim
            20.4.1  First-order logic and set theory   bnj170 30764
            20.4.2  Well founded induction and recursion   bnj110 30928
            20.4.3  The existence of a minimal element in certain classes   bnj69 31078
            20.4.4  Well-founded induction   bnj1204 31080
            20.4.5  Well-founded recursion, part 1 of 3   bnj60 31130
            20.4.6  Well-founded recursion, part 2 of 3   bnj1500 31136
            20.4.7  Well-founded recursion, part 3 of 3   bnj1522 31140
      ​20.5  Mathbox for Mario Carneiro
            20.5.1  Predicate calculus with all distinct variables   ax-7d 31141
            20.5.2  Miscellaneous stuff   quartfull 31147
            20.5.3  Derangements and the Subfactorial   deranglem 31148
            20.5.4  The Erdős-Szekeres theorem   erdszelem1 31173
            20.5.5  The Kuratowski closure-complement theorem   kur14lem1 31188
            20.5.6  Retracts and sections   cretr 31199
            20.5.7  Path-connected and simply connected spaces   cpconn 31201
            20.5.8  Covering maps   ccvm 31237
            20.5.9  Normal numbers   snmlff 31311
            20.5.10  Godel-sets of formulas   cgoe 31315
            20.5.11  Models of ZF   cgze 31343
            ​*20.5.12  Metamath formal systems   cmcn 31357
            20.5.13  Grammatical formal systems   cm0s 31482
            20.5.14  Models of formal systems   cmuv 31500
            20.5.15  Splitting fields   citr 31522
            20.5.16  p-adic number fields   czr 31538
      ​​*20.6  Mathbox for Filip Cernatescu
      ​20.7  Mathbox for Paul Chapman
            20.7.1  Real and complex numbers (cont.)   climuzcnv 31565
            20.7.2  Miscellaneous theorems   elfzm12 31569
      ​20.8  Mathbox for Scott Fenton
            20.8.1  ZFC Axioms in primitive form   axextprim 31578
            20.8.2  Untangled classes   untelirr 31585
            20.8.3  Extra propositional calculus theorems   3orel2 31592
            20.8.4  Misc. Useful Theorems   nepss 31599
            20.8.5  Properties of real and complex numbers   sqdivzi 31610
            20.8.6  Infinite products   iprodefisumlem 31626
            20.8.7  Factorial limits   faclimlem1 31629
            20.8.8  Greatest common divisor and divisibility   pdivsq 31635
            20.8.9  Properties of relationships   brtp 31639
            20.8.10  Properties of functions and mappings   funpsstri 31663
            20.8.11  Epsilon induction   setinds 31683
            20.8.12  Ordinal numbers   elpotr 31686
            20.8.13  Defined equality axioms   axextdfeq 31703
            20.8.14  Hypothesis builders   hbntg 31711
            20.8.15  (Trans)finite Recursion Theorems   tfisg 31716
            20.8.16  Transitive closure under a relationship   ctrpred 31717
            20.8.17  Founded Induction   frmin 31739
            20.8.18  Ordering Ordinal Sequences   orderseqlem 31749
            20.8.19  Well-founded zero, successor, and limits   cwsuc 31752
            20.8.20  Founded Recursion   frr3g 31779
            20.8.21  Surreal Numbers   csur 31793
            20.8.22  Surreal Numbers: Ordering   sltsolem1 31826
            20.8.23  Surreal Numbers: Birthday Function   bdayfo 31828
            20.8.24  Surreal Numbers: Density   fvnobday 31829
            20.8.25  Surreal Numbers: Full-Eta Property   bdayimaon 31843
            20.8.26  Surreal numbers - ordering theorems   csle 31869
            20.8.27  Surreal numbers - birthday theorems   bdayfun 31888
            20.8.28  Surreal numbers: Conway cuts   csslt 31896
            20.8.29  Surreal numbers - cuts and options   cmade 31925
            20.8.30  Quantifier-free definitions   ctxp 31937
            20.8.31  Alternate ordered pairs   caltop 32063
            20.8.32  Geometry in the Euclidean space   cofs 32089
                  20.8.32.1  Congruence properties   cofs 32089
                  20.8.32.2  Betweenness properties   btwntriv2 32119
                  20.8.32.3  Segment Transportation   ctransport 32136
                  20.8.32.4  Properties relating betweenness and congruence   cifs 32142
                  20.8.32.5  Connectivity of betweenness   btwnconn1lem1 32194
                  20.8.32.6  Segment less than or equal to   csegle 32213
                  20.8.32.7  Outside-of relationship   coutsideof 32226
                  20.8.32.8  Lines and Rays   cline2 32241
            20.8.33  Forward difference   cfwddif 32265
            20.8.34  Rank theorems   rankung 32273
            20.8.35  Hereditarily Finite Sets   chf 32279
      ​20.9  Mathbox for Jeff Hankins
            20.9.1  Miscellany   a1i14 32294
            20.9.2  Basic topological facts   topbnd 32319
            20.9.3  Topology of the real numbers   ivthALT 32330
            20.9.4  Refinements   cfne 32331
            20.9.5  Neighborhood bases determine topologies   neibastop1 32354
            20.9.6  Lattice structure of topologies   topmtcl 32358
            20.9.7  Filter bases   fgmin 32365
            20.9.8  Directed sets, nets   tailfval 32367
      ​20.10  Mathbox for Anthony Hart
            20.10.1  Propositional Calculus   tb-ax1 32378
            20.10.2  Predicate Calculus   allt 32400
            20.10.3  Misc. Single Axiom Systems   meran1 32410
            20.10.4  Connective Symmetry   negsym1 32416
      ​20.11  Mathbox for Chen-Pang He
            20.11.1  Ordinal topology   ontopbas 32427
      ​20.12  Mathbox for Jeff Hoffman
            20.12.1  Inferences for finite induction on generic function values   fveleq 32450
            20.12.2  gdc.mm   nnssi2 32454
      ​20.13  Mathbox for Asger C. Ipsen
            20.13.1  Continuous nowhere differentiable functions   dnival 32461
      ​​*20.14  Mathbox for BJ
            *20.14.1  Propositional calculus   bj-mp2c 32531
                  *20.14.1.1  Derived rules of inference   bj-mp2c 32531
                  ​*20.14.1.2  A syntactic theorem   bj-0 32533
                  20.14.1.3  Minimal implicational calculus   bj-a1k 32535
                  20.14.1.4  Positive calculus   bj-orim2 32541
                  20.14.1.5  Implication and negation   pm4.81ALT 32546
                  ​*20.14.1.6  Disjunction   bj-jaoi1 32556
                  ​*20.14.1.7  Logical equivalence   bj-dfbi4 32558
                  20.14.1.8  The conditional operator for propositions   bj-consensus 32562
                  ​*20.14.1.9  Propositional calculus: miscellaneous   bj-imbi12 32567
            ​*20.14.2  Modal logic   bj-axdd2 32576
            ​*20.14.3  Provability logic   cprvb 32582
            ​*20.14.4  First-order logic   bj-genr 32591
                  20.14.4.1  Adding ax-gen   bj-genr 32591
                  20.14.4.2  Adding ax-4   bj-2alim 32594
                  20.14.4.3  Adding ax-5   bj-ax12wlem 32617
                  20.14.4.4  Equality and substitution   bj-ssbjust 32618
                  20.14.4.5  Adding ax-6   bj-extru 32654
                  20.14.4.6  Adding ax-7   bj-cbvexw 32664
                  20.14.4.7  Membership predicate, ax-8 and ax-9   bj-elequ2g 32666
                  20.14.4.8  Adding ax-11   bj-alcomexcom 32670
                  20.14.4.9  Adding ax-12   axc11n11 32672
                  20.14.4.10  Adding ax-13   bj-axc10 32707
                  ​*20.14.4.11  Removing dependencies on ax-13 (and ax-11)   bj-axc10v 32717
                  ​*20.14.4.12  Strengthenings of theorems of the main part   bj-sb3b 32804
                  ​*20.14.4.13  Distinct var metavariables   bj-hbaeb2 32805
                  ​*20.14.4.14  Around ~ equsal   bj-equsal1t 32809
                  ​*20.14.4.15  Some Principia Mathematica proofs   stdpc5t 32814
                  20.14.4.16  Alternate definition of substitution   bj-sbsb 32824
                  20.14.4.17  Lemmas for substitution   bj-sbf3 32826
                  20.14.4.18  Existential uniqueness   bj-eu3f 32829
                  ​*20.14.4.19  First-logic: miscellaneous   bj-sbidmOLD 32831
            20.14.5  Set theory   eliminable1 32840
                  ​*20.14.5.1  Eliminability of class terms   eliminable1 32840
                  ​*20.14.5.2  Classes without extensionality   bj-cleljustab 32847
                  ​*20.14.5.3  The class-form not-free predicate   bj-nfcsym 32886
                  ​*20.14.5.4  Proposal for the definitions of class membership and class equality   bj-ax8 32887
                  ​*20.14.5.5  Lemmas for class substitution   bj-sbeqALT 32895
                  20.14.5.6  Removing some dv conditions   bj-exlimmpi 32905
                  ​*20.14.5.7  Class abstractions   bj-unrab 32922
                  ​*20.14.5.8  Restricted non-freeness   wrnf 32930
                  ​*20.14.5.9  Russell's paradox   bj-ru0 32932
                  ​*20.14.5.10  Some disjointness results   bj-n0i 32935
                  ​*20.14.5.11  Complements on direct products   bj-xpimasn 32942
                  ​*20.14.5.12  "Singletonization" and tagging   bj-sels 32950
                  ​*20.14.5.13  Tuples of classes   bj-cproj 32978
                  ​*20.14.5.14  Set theory: miscellaneous   bj-disj2r 33013
                  20.14.5.15  Evaluation   bj-evaleq 33024
                  20.14.5.16  Elementwise operations   celwise 33032
                  ​*20.14.5.17  Elementwise intersection (families of sets induced on a subset)   bj-rest00 33034
                  20.14.5.18  Moore collections (complements)   bj-intss 33053
                  20.14.5.19  Maps-to notation for functions with three arguments   bj-0nelmpt 33069
                  ​*20.14.5.20  Currying   csethom 33075
            ​*20.14.6  Extended real and complex numbers, real and complex projective lines   bj-elid 33085
                  *20.14.6.1  Diagonal in a Cartesian square   bj-elid 33085
                  ​*20.14.6.2  Extended numbers and projective lines as sets   cinftyexpi 33093
                  ​*20.14.6.3  Addition and opposite   caddcc 33124
                  ​*20.14.6.4  Argument, multiplication and inverse   cprcpal 33128
            ​*20.14.7  Monoids   bj-cmnssmnd 33136
                  ​*20.14.7.1  Finite sums in monoids   cfinsum 33145
            ​*20.14.8  Affine, Euclidean, and Cartesian geometry   crrvec 33148
                  *20.14.8.1  Convex hull in real vector spaces   crrvec 33148
                  ​*20.14.8.2  Complex numbers (supplements)   bj-subcom 33154
                  ​*20.14.8.3  Barycentric coordinates   bj-bary1lem 33160
      ​20.15  Mathbox for Jim Kingdon
                  20.15.0.1  Circle constant   ctau 33163
                  20.15.0.2  Number theory   dfgcd3 33170
      ​20.16  Mathbox for ML
      ​20.17  Mathbox for Wolf Lammen
            20.17.1  1. Bootstrapping   wl-section-boot 33244
            20.17.2  Implication chains   wl-section-impchain 33268
            20.17.3  An alternative axiom ~ ax-13   ax-wl-13v 33286
            20.17.4  Other stuff   wl-jarri 33288
            20.17.5  1. Bootstrapping classes   wcel-wl 33373
      ​20.18  Mathbox for Brendan Leahy
      ​20.19  Mathbox for Jeff Madsen
            20.19.1  Logic and set theory   anim12da 33506
            20.19.2  Real and complex numbers; integers   filbcmb 33535
            20.19.3  Sequences and sums   sdclem2 33538
            20.19.4  Topology   subspopn 33548
            20.19.5  Metric spaces   metf1o 33551
            20.19.6  Continuous maps and homeomorphisms   constcncf 33558
            20.19.7  Boundedness   ctotbnd 33565
            20.19.8  Isometries   cismty 33597
            20.19.9  Heine-Borel Theorem   heibor1lem 33608
            20.19.10  Banach Fixed Point Theorem   bfplem1 33621
            20.19.11  Euclidean space   crrn 33624
            20.19.12  Intervals (continued)   ismrer1 33637
            20.19.13  Operation properties   cass 33641
            20.19.14  Groups and related structures   cmagm 33647
            20.19.15  Group homomorphism and isomorphism   cghomOLD 33682
            20.19.16  Rings   crngo 33693
            20.19.17  Division Rings   cdrng 33747
            20.19.18  Ring homomorphisms   crnghom 33759
            20.19.19  Commutative rings   ccm2 33788
            20.19.20  Ideals   cidl 33806
            20.19.21  Prime rings and integral domains   cprrng 33845
            20.19.22  Ideal generators   cigen 33858
      ​20.20  Mathbox for Giovanni Mascellani
            ​*20.20.1  Tools for automatic proof building   efald2 33877
            ​*20.20.2  Tseitin axioms   fald 33936
            ​*20.20.3  Equality deductions   iuneq2f 33963
            ​*20.20.4  Miscellanea   scottexf 33976
      ​20.21  Mathbox for Peter Mazsa
            20.21.1  Notations   cxrn 33982
            20.21.2  Preparatory theorems   elv 33983
            20.21.3  Range Cartesian product   df-xrn 34134
      ​20.22  Mathbox for Rodolfo Medina
            20.22.1  Partitions   prtlem60 34137
      ​​*20.23  Mathbox for Norm Megill
            *20.23.1  Obsolete schemes ax-c4,c5,c7,c10,c11,c11n,c15,c9,c14,c16   ax-c5 34168
            ​*20.23.2  Rederive new axioms ax-4, ax-10, ax-6, ax-12, ax-13 from old   axc5 34178
            ​*20.23.3  Legacy theorems using obsolete axioms   ax5ALT 34192
            20.23.4  Experiments with weak deduction theorem   elimhyps 34247
            20.23.5  Miscellanea   cnaddcom 34259
            20.23.6  Atoms, hyperplanes, and covering in a left vector space (or module)   clsa 34261
            20.23.7  Functionals and kernels of a left vector space (or module)   clfn 34344
            20.23.8  Opposite rings and dual vector spaces   cld 34410
            20.23.9  Ortholattices and orthomodular lattices   cops 34459
            20.23.10  Atomic lattices with covering property   ccvr 34549
            20.23.11  Hilbert lattices   chlt 34637
            20.23.12  Projective geometries based on Hilbert lattices   clln 34777
            20.23.13  Construction of a vector space from a Hilbert lattice   cdlema1N 35077
            20.23.14  Construction of involution and inner product from a Hilbert lattice   clpoN 36769
      ​20.24  Mathbox for OpenAI
      ​20.25  Mathbox for Stefan O'Rear
            20.25.1  Additional elementary logic and set theory   moxfr 37255
            20.25.2  Additional theory of functions   imaiinfv 37256
            20.25.3  Additional topology   elrfi 37257
            20.25.4  Characterization of closure operators. Kuratowski closure axioms   ismrcd1 37261
            20.25.5  Algebraic closure systems   cnacs 37265
            20.25.6  Miscellanea 1. Map utilities   constmap 37276
            20.25.7  Miscellanea for polynomials   mptfcl 37283
            20.25.8  Multivariate polynomials over the integers   cmzpcl 37284
            20.25.9  Miscellanea for Diophantine sets 1   coeq0i 37316
            20.25.10  Diophantine sets 1: definitions   cdioph 37318
            20.25.11  Diophantine sets 2 miscellanea   ellz1 37330
            20.25.12  Diophantine sets 2: union and intersection. Monotone Boolean algebra   diophin 37336
            20.25.13  Diophantine sets 3: construction   diophrex 37339
            20.25.14  Diophantine sets 4 miscellanea   2sbcrex 37348
            20.25.15  Diophantine sets 4: Quantification   rexrabdioph 37358
            20.25.16  Diophantine sets 5: Arithmetic sets   rabdiophlem1 37365
            20.25.17  Diophantine sets 6: reusability. renumbering of variables   eldioph4b 37375
            20.25.18  Pigeonhole Principle and cardinality helpers   fphpd 37380
            20.25.19  A non-closed set of reals is infinite   rencldnfilem 37384
            20.25.20  Lagrange's rational approximation theorem   irrapxlem1 37386
            20.25.21  Pell equations 1: A nontrivial solution always exists   pellexlem1 37393
            20.25.22  Pell equations 2: Algebraic number theory of the solution set   csquarenn 37400
            20.25.23  Pell equations 3: characterizing fundamental solution   infmrgelbi 37442
            ​*20.25.24  Logarithm laws generalized to an arbitrary base   reglogcl 37454
            20.25.25  Pell equations 4: the positive solution group is infinite cyclic   pellfund14 37462
            20.25.26  X and Y sequences 1: Definition and recurrence laws   crmx 37464
            20.25.27  Ordering and induction lemmas for the integers   monotuz 37506
            20.25.28  X and Y sequences 2: Order properties   rmxypos 37514
            20.25.29  Congruential equations   congtr 37532
            20.25.30  Alternating congruential equations   acongid 37542
            20.25.31  Additional theorems on integer divisibility   coprmdvdsb 37552
            20.25.32  X and Y sequences 3: Divisibility properties   jm2.18 37555
            20.25.33  X and Y sequences 4: Diophantine representability of Y   jm2.27a 37572
            20.25.34  X and Y sequences 5: Diophantine representability of X, ^, _C   rmxdiophlem 37582
            20.25.35  Uncategorized stuff not associated with a major project   setindtr 37591
            20.25.36  More equivalents of the Axiom of Choice   axac10 37600
            20.25.37  Finitely generated left modules   clfig 37637
            20.25.38  Noetherian left modules I   clnm 37645
            20.25.39  Addenda for structure powers   pwssplit4 37659
            20.25.40  Every set admits a group structure iff choice   unxpwdom3 37665
            20.25.41  Noetherian rings and left modules II   clnr 37679
            20.25.42  Hilbert's Basis Theorem   cldgis 37691
            20.25.43  Additional material on polynomials [DEPRECATED]   cmnc 37701
            20.25.44  Degree and minimal polynomial of algebraic numbers   cdgraa 37710
            20.25.45  Algebraic integers I   citgo 37727
            20.25.46  Endomorphism algebra   cmend 37745
            20.25.47  Subfields   csdrg 37765
            20.25.48  Cyclic groups and order   idomrootle 37773
            20.25.49  Cyclotomic polynomials   ccytp 37780
            20.25.50  Miscellaneous topology   fgraphopab 37788
      ​20.26  Mathbox for Jon Pennant
      ​20.27  Mathbox for Richard Penner
            20.27.1  Short Studies   ifpan123g 37803
                  20.27.1.1  Additional work on conditional logical operator   ifpan123g 37803
                  20.27.1.2  Sophisms   rp-fakeimass 37857
                  ​*20.27.1.3  Finite Sets   rp-isfinite5 37863
                  20.27.1.4  Infinite Sets   pwelg 37865
                  ​*20.27.1.5  Finite intersection property   fipjust 37870
                  20.27.1.6  RP ADDTO: Subclasses and subsets   rababg 37879
                  20.27.1.7  RP ADDTO: The intersection of a class   elintabg 37880
                  20.27.1.8  RP ADDTO: Theorems requiring subset and intersection existence   elinintrab 37883
                  20.27.1.9  RP ADDTO: Relations   xpinintabd 37886
                  ​*20.27.1.10  RP ADDTO: Functions   elmapintab 37902
                  ​*20.27.1.11  RP ADDTO: Finite induction (for finite ordinals)   cnvcnvintabd 37906
                  20.27.1.12  RP ADDTO: First and second members of an ordered pair   elcnvlem 37907
                  20.27.1.13  RP ADDTO: The reflexive and transitive properties of relations   undmrnresiss 37910
                  20.27.1.14  RP ADDTO: Basic properties of closures   cleq2lem 37914
                  20.27.1.15  RP REPLACE: Definitions and basic properties of transitive closures   trcleq2lemRP 37937
            20.27.2  Additional statements on relations and subclasses   al3im 37938
                  20.27.2.1  Transitive relations (not to be confused with transitive classes).   trrelind 37957
                  20.27.2.2  Reflexive closures   crcl 37964
                  ​*20.27.2.3  Finite relationship composition.   relexp2 37969
                  20.27.2.4  Transitive closure of a relation   dftrcl3 38012
                  ​*20.27.2.5  Adapted from Frege   frege77d 38038
            ​*20.27.3  Propositions from _Begriffsschrift_   dfxor4 38058
                  *20.27.3.1  _Begriffsschrift_ Chapter I   dfxor4 38058
                  ​*20.27.3.2  _Begriffsschrift_ Notation hints   rp-imass 38065
                  20.27.3.3  _Begriffsschrift_ Chapter II Implication   ax-frege1 38084
                  20.27.3.4  _Begriffsschrift_ Chapter II Implication and Negation   axfrege28 38123
                  ​*20.27.3.5  _Begriffsschrift_ Chapter II with logical equivalence   axfrege52a 38150
                  20.27.3.6  _Begriffsschrift_ Chapter II with equivalence of sets   axfrege52c 38181
                  20.27.3.7  _Begriffsschrift_ Chapter II with equivalence of classes (where they are sets)   frege53c 38208
                  ​*20.27.3.8  _Begriffsschrift_ Chapter III Properties hereditary in a sequence   dffrege69 38226
                  ​*20.27.3.9  _Begriffsschrift_ Chapter III Following in a sequence   dffrege76 38233
                  ​*20.27.3.10  _Begriffsschrift_ Chapter III Member of sequence   dffrege99 38256
                  ​*20.27.3.11  _Begriffsschrift_ Chapter III Single-valued procedures   dffrege115 38272
            ​*20.27.4  Exploring Topology via Seifert And Threlfall   enrelmap 38291
                  *20.27.4.1  Equinumerosity of sets of relations and maps   enrelmap 38291
                  ​*20.27.4.2  Generic Pseudoclosure Spaces, Pseudointeror Spaces, and Pseudoneighborhoods   sscon34b 38317
                  ​*20.27.4.3  Generic Neighborhood Spaces   gneispa 38428
            ​*20.27.5  Exploring Higher Homotopy via Kerodon   k0004lem1 38445
                  *20.27.5.1  Simplicial Sets   k0004lem1 38445
      ​20.28  Mathbox for Stanislas Polu
            20.28.1  IMO Problems   wwlemuld 38454
                  20.28.1.1  IMO 1972 B2   wwlemuld 38454
            ​*20.28.2  INT Inequalities Proof Generator   int-addcomd 38476
            ​*20.28.3  N-Digit Addition Proof Generator   unitadd 38498
            20.28.4  AM-GM (for k = 2,3,4)   gsumws3 38499
      ​20.29  Mathbox for Steve Rodriguez
            20.29.1  Miscellanea   nanorxor 38504
            20.29.2  Ratio test for infinite series convergence and divergence   dvgrat 38511
            20.29.3  Multiples   reldvds 38514
            20.29.4  Function operations   caofcan 38522
            20.29.5  Calculus   lhe4.4ex1a 38528
            20.29.6  The generalized binomial coefficient operation   cbcc 38535
            20.29.7  Binomial series   uzmptshftfval 38545
      ​20.30  Mathbox for Andrew Salmon
            20.30.1  Principia Mathematica * 10   pm10.12 38557
            20.30.2  Principia Mathematica * 11   2alanimi 38571
            20.30.3  Predicate Calculus   sbeqal1 38598
            20.30.4  Principia Mathematica * 13 and * 14   pm13.13a 38608
            20.30.5  Set Theory   elnev 38639
            20.30.6  Arithmetic   addcomgi 38660
            20.30.7  Geometry   cplusr 38661
      ​​*20.31  Mathbox for Alan Sare
            20.31.1  Auxiliary theorems for the Virtual Deduction tool   idiALT 38683
            20.31.2  Supplementary unification deductions   bi1imp 38687
            20.31.3  Conventional Metamath proofs, some derived from VD proofs   iidn3 38707
            20.31.4  What is Virtual Deduction?   wvd1 38785
            20.31.5  Virtual Deduction Theorems   df-vd1 38786
            20.31.6  Theorems proved using Virtual Deduction   trsspwALT 39045
            20.31.7  Theorems proved using Virtual Deduction with mmj2 assistance   simplbi2VD 39081
            20.31.8  Virtual Deduction transcriptions of textbook proofs   sb5ALTVD 39149
            20.31.9  Theorems proved using conjunction-form Virtual Deduction   elpwgdedVD 39153
            20.31.10  Theorems with a VD proof in conventional notation derived from a VD proof   suctrALT3 39160
            ​*20.31.11  Theorems with a proof in conventional notation derived from a VD proof   notnotrALT2 39163
      ​20.32  Mathbox for Glauco Siliprandi
            20.32.1  Miscellanea   evth2f 39174
            20.32.2  Functions   unima 39346
            20.32.3  Ordering on real numbers - Real and complex numbers basic operations   sub2times 39485
            20.32.4  Real intervals   gtnelioc 39712
            20.32.5  Finite sums   fsumclf 39801
            20.32.6  Finite multiplication of numbers and finite multiplication of functions   fmul01 39812
            20.32.7  Limits   clim1fr1 39833
                  20.32.7.1  Inferior limit (lim inf)   clsi 39983
                  ​*20.32.7.2  Limits for sequences of extended real numbers   clsxlim 40044
            20.32.8  Trigonometry   coseq0 40075
            20.32.9  Continuous Functions   mulcncff 40081
            20.32.10  Derivatives   dvsinexp 40125
            20.32.11  Integrals   itgsin0pilem1 40165
            20.32.12  Stone Weierstrass theorem - real version   stoweidlem1 40218
            20.32.13  Wallis' product for π   wallispilem1 40282
            20.32.14  Stirling's approximation formula for ` n ` factorial   stirlinglem1 40291
            20.32.15  Dirichlet kernel   dirkerval 40308
            20.32.16  Fourier Series   fourierdlem1 40325
            20.32.17  e is transcendental   elaa2lem 40450
            20.32.18  n-dimensional Euclidean space   rrxtopn 40501
            20.32.19  Basic measure theory   csalg 40528
                  ​*20.32.19.1  σ-Algebras   csalg 40528
                  20.32.19.2  Sum of nonnegative extended reals   csumge0 40579
                  ​*20.32.19.3  Measures   cmea 40666
                  ​*20.32.19.4  Outer measures and Caratheodory's construction   come 40703
                  ​*20.32.19.5  Lebesgue measure on n-dimensional Real numbers   covoln 40750
                  ​*20.32.19.6  Measurable functions   csmblfn 40909
      ​20.33  Mathbox for Saveliy Skresanov
            20.33.1  Ceva's theorem   sigarval 41039
      ​20.34  Mathbox for Jarvin Udandy
      ​20.35  Mathbox for Alexander van der Vekens
            20.35.1  Double restricted existential uniqueness   r19.32 41167
                  20.35.1.1  Restricted quantification (extension)   r19.32 41167
                  20.35.1.2  The empty set (extension)   raaan2 41175
                  20.35.1.3  Restricted uniqueness and "at most one" quantification   rmoimi 41176
                  20.35.1.4  Analogs to Existential uniqueness (double quantification)   2reurex 41181
            ​*20.35.2  Alternative definitions of function and operation values   wdfat 41193
                  20.35.2.1  Restricted quantification (extension)   ralbinrald 41199
                  20.35.2.2  The universal class (extension)   nvelim 41200
                  20.35.2.3  Introduce the Axiom of Power Sets (extension)   alneu 41201
                  20.35.2.4  Relations (extension)   eldmressn 41203
                  20.35.2.5  Functions (extension)   fveqvfvv 41204
                  20.35.2.6  Predicate "defined at"   dfateq12d 41209
                  20.35.2.7  Alternative definition of the value of a function   dfafv2 41212
                  20.35.2.8  Alternative definition of the value of an operation   aoveq123d 41258
            20.35.3  General auxiliary theorems   cnelbr 41288
                  20.35.3.1  Negated membership (alternative)   cnelbr 41288
                  20.35.3.2  The empty set - extension   ralralimp 41295
                  20.35.3.3  Unordered and ordered pairs - extension   elprneb 41296
                  20.35.3.4  Indexed union and intersection - extension   otiunsndisjX 41298
                  20.35.3.5  Functions - extension   fvifeq 41299
                  20.35.3.6  Ordering on reals - extension   leltletr 41308
                  20.35.3.7  Subtraction - extension   cnambpcma 41309
                  20.35.3.8  Ordering on reals (cont.) - extension   leaddsuble 41311
                  20.35.3.9  Nonnegative integers (as a subset of complex numbers) - extension   nn0resubcl 41317
                  20.35.3.10  Integers (as a subset of complex numbers) - extension   zgeltp1eq 41318
                  20.35.3.11  Decimal arithmetic - extension   1t10e1p1e11 41319
                  20.35.3.12  Upper sets of integers - extension   eluzge0nn0 41322
                  20.35.3.13  Infinity and the extended real number system (cont.) - extension   nltle2tri 41323
                  20.35.3.14  Finite intervals of integers - extension   ssfz12 41324
                  20.35.3.15  Half-open integer ranges - extension   fzopred 41332
                  20.35.3.16  The modulo (remainder) operation - extension   m1mod0mod1 41339
                  20.35.3.17  The infinite sequence builder "seq"   smonoord 41341
                  20.35.3.18  Finite and infinite sums - extension   fsummsndifre 41342
                  20.35.3.19  Extensible structures - extension   setsidel 41346
            ​*20.35.4  Partitions of real intervals   ciccp 41349
            20.35.5  Shifting functions with an integer range domain   fargshiftfv 41375
            20.35.6  Words over a set (extension)   lswn0 41380
                  20.35.6.1  Last symbol of a word - extension   lswn0 41380
                  ​*20.35.6.2  Prefixes of a word   cpfx 41381
            20.35.7  Number theory (extension)   cfmtno 41439
                  ​*20.35.7.1  Fermat numbers   cfmtno 41439
                  ​*20.35.7.2  Mersenne primes   m2prm 41505
                  20.35.7.3  Proth's theorem   modexp2m1d 41529
            ​*20.35.8  Even and odd numbers   ceven 41537
                  20.35.8.1  Definitions and basic properties   ceven 41537
                  20.35.8.2  Alternate definitions using the "divides" relation   dfeven2 41562
                  20.35.8.3  Alternate definitions using the "modulo" operation   dfeven3 41570
                  20.35.8.4  Alternate definitions using the "gcd" operation   iseven5 41576
                  20.35.8.5  Theorems of part 5 revised   zneoALTV 41580
                  20.35.8.6  Theorems of part 6 revised   odd2np1ALTV 41585
                  20.35.8.7  Theorems of AV's mathbox revised   0evenALTV 41599
                  20.35.8.8  Additional theorems   epoo 41612
                  20.35.8.9  Perfect Number Theorem (revised)   perfectALTVlem1 41630
                  ​*20.35.8.10  Goldbach's conjectures   cgbe 41633
            20.35.9  Graph theory (extension)   1hegrlfgr 41713
                  20.35.9.1  Loop-free graphs - extension   1hegrlfgr 41713
                  20.35.9.2  Walks - extension   cupwlks 41714
            20.35.10  Set of unordered pairs   sprid 41724
            20.35.11  Monoids (extension)   ovn0dmfun 41764
                  20.35.11.1  Auxiliary theorems   ovn0dmfun 41764
                  20.35.11.2  Magmas and Semigroups (extension)   plusfreseq 41772
                  20.35.11.3  Magma homomorphisms and submagmas   cmgmhm 41777
                  20.35.11.4  Examples and counterexamples for magmas, semigroups and monoids (extension)   opmpt2ismgm 41807
            ​*20.35.12  Magmas and internal binary operations (alternate approach)   ccllaw 41819
                  *20.35.12.1  Laws for internal binary operations   ccllaw 41819
                  ​*20.35.12.2  Internal binary operations   cintop 41832
                  20.35.12.3  Alternative definitions for Magmas and Semigroups   cmgm2 41851
            20.35.13  Categories (extension)   idfusubc0 41865
                  20.35.13.1  Subcategories (extension)   idfusubc0 41865
            20.35.14  Rings (extension)   lmod0rng 41868
                  20.35.14.1  Nonzero rings (extension)   lmod0rng 41868
                  ​*20.35.14.2  Non-unital rings ("rngs")   crng 41874
                  20.35.14.3  Rng homomorphisms   crngh 41885
                  20.35.14.4  Ring homomorphisms (extension)   rhmfn 41918
                  20.35.14.5  Ideals as non-unital rings   lidldomn1 41921
                  20.35.14.6  The non-unital ring of even integers   0even 41931
                  20.35.14.7  A constructed not unital ring   cznrnglem 41953
                  ​*20.35.14.8  The category of non-unital rings   crngc 41957
                  ​*20.35.14.9  The category of (unital) rings   cringc 42003
                  20.35.14.10  Subcategories of the category of rings   srhmsubclem1 42073
            20.35.15  Basic algebraic structures (extension)   xpprsng 42110
                  20.35.15.1  Auxiliary theorems   xpprsng 42110
                  20.35.15.2  The binomial coefficient operation (extension)   bcpascm1 42129
                  20.35.15.3  The ` ZZ `-module ` ZZ X. ZZ `   zlmodzxzlmod 42132
                  20.35.15.4  Ordered group sum operation (extension)   gsumpr 42139
                  20.35.15.5  Symmetric groups (extension)   exple2lt6 42145
                  20.35.15.6  Divisibility (extension)   invginvrid 42148
                  20.35.15.7  The support of functions (extension)   rmsupp0 42149
                  20.35.15.8  Finitely supported functions (extension)   rmsuppfi 42154
                  20.35.15.9  Left modules (extension)   lmodvsmdi 42163
                  20.35.15.10  Associative algebras (extension)   ascl0 42165
                  20.35.15.11  Univariate polynomials (extension)   ply1vr1smo 42169
                  20.35.15.12  Univariate polynomials (examples)   linply1 42181
            20.35.16  Linear algebra (extension)   cdmatalt 42185
                  ​*20.35.16.1  The subalgebras of diagonal and scalar matrices (extension)   cdmatalt 42185
                  ​*20.35.16.2  Linear combinations   clinc 42193
                  ​*20.35.16.3  Linear independency   clininds 42229
                  20.35.16.4  Simple left modules and the ` ZZ `-module   lmod1lem1 42276
                  20.35.16.5  Differences between (left) modules and (left) vector spaces   lvecpsslmod 42296
            20.35.17  Complexity theory   offval0 42299
                  20.35.17.1  Auxiliary theorems   offval0 42299
                  20.35.17.2  The modulo (remainder) operation (extension)   fldivmod 42313
                  20.35.17.3  Even and odd integers   nn0onn0ex 42318
                  20.35.17.4  The natural logarithm on complex numbers (extension)   logcxp0 42329
                  20.35.17.5  Division of functions   cfdiv 42331
                  20.35.17.6  Upper bounds   cbigo 42341
                  20.35.17.7  Logarithm to an arbitrary base (extension)   rege1logbrege0 42352
                  ​*20.35.17.8  The binary logarithm   fldivexpfllog2 42359
                  20.35.17.9  Binary length   cblen 42363
                  ​*20.35.17.10  Digits   cdig 42389
                  20.35.17.11  Nonnegative integer as sum of its shifted digits   dignn0flhalflem1 42409
                  20.35.17.12  Algorithms for the multiplication of nonnegative integers   nn0mulfsum 42418
      ​20.36  Mathbox for Emmett Weisz
            ​*20.36.1  Miscellaneous Theorems   nfintd 42420
            20.36.2  Set Recursion   csetrecs 42430
                  ​*20.36.2.1  Basic Properties of Set Recursion   csetrecs 42430
                  20.36.2.2  Examples and properties of set recursion   elsetrecslem 42444
            ​*20.36.3  Construction of Games and Surreal Numbers   cpg 42452
      ​​*20.37  Mathbox for David A. Wheeler
            20.37.1  Natural deduction   19.8ad 42458
            ​*20.37.2  Greater than, greater than or equal to.   cge-real 42461
            ​*20.37.3  Hyperbolic trigonometric functions   csinh 42471
            ​*20.37.4  Reciprocal trigonometric functions (sec, csc, cot)   csec 42482
            ​*20.37.5  Identities for "if"   ifnmfalse 42504
            ​*20.37.6  Logarithms generalized to arbitrary base using ` logb `   logb2aval 42505
            ​*20.37.7  Logarithm laws generalized to an arbitrary base - log_   clog- 42506
            ​*20.37.8  Formally define terms such as Reflexivity   wreflexive 42508
            ​*20.37.9  Algebra helpers   comraddi 42512
            ​*20.37.10  Algebra helper examples   i2linesi 42524
            ​*20.37.11  Formal methods "surprises"   alimp-surprise 42526
            ​*20.37.12  Allsome quantifier   walsi 42532
            ​*20.37.13  Miscellaneous   5m4e1 42543
            20.37.14  AA theorems   aacllem 42547
      ​20.38  Mathbox for Kunhao Zheng
            20.38.1  Weighted AM-GM Inequality   amgmwlem 42548