TOC of Theorem List - Metamath Proof Explorer

Table of Contents Summary

PART 1  CLASSICAL FIRST-ORDER LOGIC WITH EQUALITY
      1.1  Pre-logic
      1.2  Propositional calculus
      1.3  Other axiomatizations related to classical propositional calculus
      1.4  Predicate calculus with equality: Tarski's system S2 (1 rule, 6 schemes)
      1.5  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes)
      1.6  Uniqueness and unique existence
      1.7  Other axiomatizations related to classical predicate calculus
PART 2  ZF (ZERMELO-FRAENKEL) SET THEORY
      2.1  ZF Set Theory - start with the Axiom of Extensionality
      2.2  ZF Set Theory - add the Axiom of Replacement
      2.3  ZF Set Theory - add the Axiom of Power Sets
      2.4  ZF Set Theory - add the Axiom of Union
      2.5  ZF Set Theory - add the Axiom of Regularity
      2.6  ZF Set Theory - add the Axiom of Infinity
PART 3  ZFC (ZERMELO-FRAENKEL WITH CHOICE) SET THEORY
      3.1  ZFC Set Theory - add Countable Choice and Dependent Choice
      3.2  ZFC Set Theory - add the Axiom of Choice
      3.3  ZFC Axioms with no distinct variable requirements
      3.4  The Generalized Continuum Hypothesis
PART 4  TG (TARSKI-GROTHENDIECK) SET THEORY
      4.1  Inaccessibles
      4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
PART 5  REAL AND COMPLEX NUMBERS
      5.1  Construction and axiomatization of real and complex numbers
      5.2  Derive the basic properties from the field axioms
      5.3  Real and complex numbers - basic operations
      5.4  Integer sets
      5.5  Order sets
      5.6  Elementary integer functions
      5.7  Words over a set
      5.8  Reflexive and transitive closures of relations
      5.9  Elementary real and complex functions
      5.10  Elementary limits and convergence
      5.11  Elementary trigonometry
      5.12  Cardinality of real and complex number subsets
PART 6  ELEMENTARY NUMBER THEORY
      6.1  Elementary properties of divisibility
      6.2  Elementary prime number theory
PART 7  BASIC STRUCTURES
      7.1  Extensible structures
      7.2  Moore spaces
PART 8  BASIC CATEGORY THEORY
      8.1  Categories
      8.2  Arrows (disjointified hom-sets)
      8.3  Examples of categories
      8.4  Categorical constructions
PART 9  BASIC ORDER THEORY
      9.1  Preordered sets and directed sets using extensible structures
      9.2  Posets and lattices using extensible structures
PART 10  BASIC ALGEBRAIC STRUCTURES
      10.1  Monoids
      10.2  Groups
      10.3  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 Steven Nguyen
      20.25  Mathbox for OpenAI
      20.26  Mathbox for Stefan O'Rear
      20.27  Mathbox for Jon Pennant
      20.28  Mathbox for Richard Penner
      20.29  Mathbox for Stanislas Polu
      20.30  Mathbox for Steve Rodriguez
      20.31  Mathbox for Andrew Salmon
      20.32  Mathbox for Alan Sare
      20.33  Mathbox for Glauco Siliprandi
      20.34  Mathbox for Saveliy Skresanov
      20.35  Mathbox for Jarvin Udandy
      20.36  Mathbox for Alexander van der Vekens
      20.37  Mathbox for Emmett Weisz
      20.38  Mathbox for David A. Wheeler
      20.39  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 113
            ​*1.2.5  Logical equivalence   wb 198
            ​*1.2.6  Logical conjunction   wa 386
            ​*1.2.7  Logical disjunction   wo 878
            ​*1.2.8  Mixed connectives   jaao 982
            ​*1.2.9  The conditional operator for propositions   wif 1089
            ​*1.2.10  The weak deduction theorem for propositional calculus   elimh 1107
            1.2.11  Abbreviated conjunction and disjunction of three wff's   w3o 1110
            1.2.12  Logical "nand" (Sheffer stroke)   wnan 1612
            1.2.13  Logical "xor"   wxo 1637
            1.2.14  True and false constants   wal 1654
                  ​*1.2.14.1  Universal quantifier for use by df-tru   wal 1654
                  ​*1.2.14.2  Equality predicate for use by df-tru   cv 1655
                  1.2.14.3  The true constant   wtru 1657
                  1.2.14.4  The false constant   wfal 1669
            ​*1.2.15  Truth tables   truimtru 1680
                  1.2.15.1  Implication   truimtru 1680
                  1.2.15.2  Negation   nottru 1684
                  1.2.15.3  Equivalence   trubitru 1686
                  1.2.15.4  Conjunction   truantru 1690
                  1.2.15.5  Disjunction   truortru 1694
                  1.2.15.6  Alternative denial   trunantru 1698
                  1.2.15.7  Exclusive disjunction   truxortru 1702
            ​*1.2.16  Half adder and full adder in propositional calculus   whad 1706
                  1.2.16.1  Full adder: sum   whad 1706
                  1.2.16.2  Full adder: carry   wcad 1719
      ​1.3  Other axiomatizations related to classical propositional calculus
            ​*1.3.1  Minimal implicational calculus   minimp 1734
            1.3.2  Derive the Lukasiewicz axioms from Meredith's sole axiom   meredith 1740
            1.3.3  Derive the standard axioms from the Lukasiewicz axioms   luklem1 1757
            ​*1.3.4  Derive Nicod's axiom from the standard axioms   nic-dfim 1768
            1.3.5  Derive the Lukasiewicz axioms from Nicod's axiom   nic-imp 1774
            1.3.6  Derive Nicod's Axiom from Lukasiewicz's First Sheffer Stroke Axiom   lukshef-ax1 1793
            1.3.7  Derive the Lukasiewicz Axioms from the Tarski-Bernays-Wajsberg Axioms   tbw-bijust 1797
            1.3.8  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's First CO Axiom   merco1 1812
            1.3.9  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's Second CO Axiom   merco2 1835
            1.3.10  Derive the Lukasiewicz axioms from the Russell-Bernays Axioms   rb-bijust 1848
            ​*1.3.11  Stoic logic non-modal portion (Chrysippus of Soli)   mptnan 1867
      ​​*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 1878
                  1.4.1.1  Existential quantifier   wex 1878
                  1.4.1.2  Non-freeness predicate   wnf 1882
            1.4.2  Rule scheme ax-gen (Generalization)   ax-gen 1894
            1.4.3  Axiom scheme ax-4 (Quantified Implication)   ax-4 1908
            1.4.4  Axiom scheme ax-5 (Distinctness) - first use of $d   ax-5 2009
            ​*1.4.5  Equality predicate (continued)   weq 2061
            1.4.6  Define proper substitution   wsb 2067
            1.4.7  Axiom scheme ax-6 (Existence)   ax-6 2075
            1.4.8  Axiom scheme ax-7 (Equality)   ax-7 2112
            1.4.9  Membership predicate   wcel 2164
            1.4.10  Axiom scheme ax-8 (Left Equality for Binary Predicate)   ax-8 2166
            1.4.11  Axiom scheme ax-9 (Right Equality for Binary Predicate)   ax-9 2173
            ​*1.4.12  Logical redundancy of ax-10 , ax-11 , ax-12 , ax-13   ax6dgen 2179
      ​​*1.5  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes)
            1.5.1  Axiom scheme ax-10 (Quantified Negation)   ax-10 2192
            1.5.2  Axiom scheme ax-11 (Quantifier Commutation)   ax-11 2207
            1.5.3  Axiom scheme ax-12 (Substitution)   ax-12 2220
            1.5.4  Axiom scheme ax-13 (Quantified Equality)   ax-13 2389
      ​1.6  Uniqueness and unique existence
            1.6.1  Uniqueness: the at-most-one quantifier   wmo 2603
            1.6.2  Unique existence: the unique existential quantifier   weu 2639
      ​1.7  Other axiomatizations related to classical predicate calculus
            ​*1.7.1  Aristotelian logic: Assertic syllogisms   barbara 2746
            ​*1.7.2  Intuitionistic logic   axia1 2788
​​*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 2803
            2.1.2  Class abstractions (a.k.a. class builders)   cab 2811
            2.1.3  Class form not-free predicate   wnfc 2956
            2.1.4  Negated equality and membership   wne 2999
                  2.1.4.1  Negated equality   wne 2999
                  2.1.4.2  Negated membership   wnel 3102
            2.1.5  Restricted quantification   wral 3117
            2.1.6  The universal class   cvv 3414
            ​*2.1.7  Conditional equality (experimental)   wcdeq 3645
            2.1.8  Russell's Paradox   ru 3661
            2.1.9  Proper substitution of classes for sets   wsbc 3662
            2.1.10  Proper substitution of classes for sets into classes   csb 3757
            2.1.11  Define basic set operations and relations   cdif 3795
            2.1.12  Subclasses and subsets   df-ss 3812
            2.1.13  The difference, union, and intersection of two classes   dfdif3 3949
                  2.1.13.1  The difference of two classes   dfdif3 3949
                  2.1.13.2  The union of two classes   elun 3982
                  2.1.13.3  The intersection of two classes   elin 4025
                  2.1.13.4  The symmetric difference of two classes   csymdif 4071
                  2.1.13.5  Combinations of difference, union, and intersection of two classes   unabs 4086
                  2.1.13.6  Class abstractions with difference, union, and intersection of two classes   unab 4125
                  2.1.13.7  Restricted uniqueness with difference, union, and intersection   reuss2 4138
            2.1.14  The empty set   c0 4146
            ​*2.1.15  The conditional operator for classes   cif 4308
            ​*2.1.16  The weak deduction theorem for set theory   dedth 4364
            2.1.17  Power classes   cpw 4380
            2.1.18  Unordered and ordered pairs   snjust 4398
            2.1.19  The union of a class   cuni 4660
            2.1.20  The intersection of a class   cint 4699
            2.1.21  Indexed union and intersection   ciun 4742
            2.1.22  Disjointness   wdisj 4843
            2.1.23  Binary relations   wbr 4875
            2.1.24  Ordered-pair class abstractions (class builders)   copab 4937
            2.1.25  Functions in maps-to notation   cmpt 4954
            2.1.26  Transitive classes   wtr 4977
      ​2.2  ZF Set Theory - add the Axiom of Replacement
            2.2.1  Introduce the Axiom of Replacement   ax-rep 4996
            2.2.2  Derive the Axiom of Separation   axsep 5006
            2.2.3  Derive the Null Set Axiom   zfnuleuOLD 5012
            2.2.4  Theorems requiring subset and intersection existence   nalset 5022
            2.2.5  Theorems requiring empty set existence   class2set 5056
      ​2.3  ZF Set Theory - add the Axiom of Power Sets
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 5067
            2.3.2  Derive the Axiom of Pairing   zfpair 5127
            2.3.3  Ordered pair theorem   opnz 5164
            2.3.4  Ordered-pair class abstractions (cont.)   opabid 5210
            2.3.5  Power class of union and intersection   pwin 5246
            2.3.6  The identity relation   cid 5251
            2.3.7  The membership relation (or epsilon relation)   cep 5256
            ​*2.3.8  Partial and complete ordering   wpo 5263
            2.3.9  Founded and well-ordering relations   wfr 5302
            2.3.10  Relations   cxp 5344
            2.3.11  The Predecessor Class   cpred 5923
            2.3.12  Well-founded induction   tz6.26 5955
            2.3.13  Ordinals   word 5966
            2.3.14  Definite description binder (inverted iota)   cio 6088
            2.3.15  Functions   wfun 6121
            2.3.16  Cantor's Theorem   canth 6868
            2.3.17  Restricted iota (description binder)   crio 6870
            2.3.18  Operations   co 6910
            2.3.19  Maps-to notation   mpt2ndm0 7140
            2.3.20  Function operation   cof 7160
            2.3.21  Proper subset relation   crpss 7201
      ​2.4  ZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 7214
            2.4.2  Ordinals (continued)   epweon 7248
            2.4.3  Transfinite induction   tfi 7319
            2.4.4  The natural numbers (i.e., finite ordinals)   com 7331
            2.4.5  Peano's postulates   peano1 7351
            2.4.6  Finite induction (for finite ordinals)   find 7357
            2.4.7  First and second members of an ordered pair   c1st 7431
            ​*2.4.8  The support of functions   csupp 7564
            ​*2.4.9  Special maps-to operations   opeliunxp2f 7606
            2.4.10  Function transposition   ctpos 7621
            2.4.11  Curry and uncurry   ccur 7661
            2.4.12  Undefined values   cund 7668
            2.4.13  Well-founded recursion   cwrecs 7676
            2.4.14  Functions on ordinals; strictly monotone ordinal functions   iunon 7707
            2.4.15  "Strong" transfinite recursion   crecs 7738
            2.4.16  Recursive definition generator   crdg 7776
            2.4.17  Finite recursion   frfnom 7801
            2.4.18  Ordinal arithmetic   c1o 7824
            2.4.19  Natural number arithmetic   nna0 7956
            2.4.20  Equivalence relations and classes   wer 8011
            2.4.21  The mapping operation   cmap 8127
            2.4.22  Infinite Cartesian products   cixp 8181
            2.4.23  Equinumerosity   cen 8225
            2.4.24  Schroeder-Bernstein Theorem   sbthlem1 8345
            2.4.25  Equinumerosity (cont.)   xpf1o 8397
            2.4.26  Pigeonhole Principle   phplem1 8414
            2.4.27  Finite sets   onomeneq 8425
            2.4.28  Finitely supported functions   cfsupp 8550
            2.4.29  Finite intersections   cfi 8591
            2.4.30  Hall's marriage theorem   marypha1lem 8614
            2.4.31  Supremum and infimum   csup 8621
            2.4.32  Ordinal isomorphism, Hartogs's theorem   coi 8690
            2.4.33  Hartogs function, order types, weak dominance   char 8737
      ​2.5  ZF Set Theory - add the Axiom of Regularity
            2.5.1  Introduce the Axiom of Regularity   ax-reg 8773
            2.5.2  Axiom of Infinity equivalents   inf0 8802
      ​2.6  ZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-inf 8819
            2.6.2  Existence of omega (the set of natural numbers)   omex 8824
            2.6.3  Cantor normal form   ccnf 8842
            2.6.4  Transitive closure   trcl 8888
            2.6.5  Rank   cr1 8909
            2.6.6  Scott's trick; collection principle; Hilbert's epsilon   scottex 9032
            2.6.7  Disjoint union   cdju 9045
            2.6.8  Cardinal numbers   ccrd 9081
            2.6.9  Axiom of Choice equivalents   wac 9258
            2.6.10  Cardinal number arithmetic   ccda 9311
            2.6.11  The Ackermann bijection   ackbij2lem1 9363
            2.6.12  Cofinality (without Axiom of Choice)   cflem 9390
            2.6.13  Eight inequivalent definitions of finite set   sornom 9421
            2.6.14  Hereditarily size-limited sets without Choice   itunifval 9560
​​*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 9579
            3.1.2  Introduce the Axiom of Dependent Choice   ax-dc 9590
      ​3.2  ZFC Set Theory - add the Axiom of Choice
            3.2.1  Introduce the Axiom of Choice   ax-ac 9603
            3.2.2  AC equivalents: well-ordering, Zorn's lemma   numthcor 9638
            3.2.3  Cardinal number theorems using Axiom of Choice   cardval 9690
            3.2.4  Cardinal number arithmetic using Axiom of Choice   iunctb 9718
            3.2.5  Cofinality using the Axiom of Choice   alephreg 9726
      ​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 9764
            3.4.2  Derivation of the Axiom of Choice   gchaclem 9822
​​*PART 4  TG (TARSKI-GROTHENDIECK) SET THEORY
      ​4.1  Inaccessibles
            4.1.1  Weakly and strongly inaccessible cardinals   cwina 9826
            4.1.2  Weak universes   cwun 9844
            4.1.3  Tarski classes   ctsk 9892
            4.1.4  Grothendieck universes   cgru 9934
      ​4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
            4.2.1  Introduce the Tarski-Grothendieck Axiom   ax-groth 9967
            4.2.2  Derive the Power Set, Infinity and Choice Axioms   grothpw 9970
            4.2.3  Tarski map function   ctskm 9981
​​*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 9988
            5.1.2  Final derivation of real and complex number postulates   axaddf 10289
            5.1.3  Real and complex number postulates restated as axioms   ax-cnex 10315
      ​5.2  Derive the basic properties from the field axioms
            5.2.1  Some deductions from the field axioms for complex numbers   cnex 10340
            5.2.2  Infinity and the extended real number system   cpnf 10395
            5.2.3  Restate the ordering postulates with extended real "less than"   axlttri 10435
            5.2.4  Ordering on reals   lttr 10440
            5.2.5  Initial properties of the complex numbers   mul12 10528
      ​5.3  Real and complex numbers - basic operations
            5.3.1  Addition   add12 10579
            5.3.2  Subtraction   cmin 10592
            5.3.3  Multiplication   kcnktkm1cn 10792
            5.3.4  Ordering on reals (cont.)   gt0ne0 10824
            5.3.5  Reciprocals   ixi 10988
            5.3.6  Division   cdiv 11016
            5.3.7  Ordering on reals (cont.)   elimgt0 11196
            5.3.8  Completeness Axiom and Suprema   fimaxre 11305
            5.3.9  Imaginary and complex number properties   inelr 11347
            5.3.10  Function operation analogue theorems   ofsubeq0 11354
      ​5.4  Integer sets
            5.4.1  Positive integers (as a subset of complex numbers)   cn 11357
            5.4.2  Principle of mathematical induction   nnind 11377
            ​*5.4.3  Decimal representation of numbers   c2 11413
            ​*5.4.4  Some properties of specific numbers   neg1cn 11479
            5.4.5  Simple number properties   halfcl 11590
            5.4.6  The Archimedean property   nnunb 11621
            5.4.7  Nonnegative integers (as a subset of complex numbers)   cn0 11625
            ​*5.4.8  Extended nonnegative integers   cxnn0 11697
            5.4.9  Integers (as a subset of complex numbers)   cz 11711
            5.4.10  Decimal arithmetic   cdc 11828
            5.4.11  Upper sets of integers   cuz 11975
            5.4.12  Well-ordering principle for bounded-below sets of integers   uzwo3 12073
            5.4.13  Rational numbers (as a subset of complex numbers)   cq 12078
            5.4.14  Existence of the set of complex numbers   rpnnen1lem2 12106
      ​5.5  Order sets
            5.5.1  Positive reals (as a subset of complex numbers)   crp 12119
            5.5.2  Infinity and the extended real number system (cont.)   cxne 12236
            5.5.3  Supremum and infimum on the extended reals   xrsupexmnf 12430
            5.5.4  Real number intervals   cioo 12470
            5.5.5  Finite intervals of integers   cfz 12626
            ​*5.5.6  Finite intervals of nonnegative integers   elfz2nn0 12732
            5.5.7  Half-open integer ranges   cfzo 12767
      ​5.6  Elementary integer functions
            5.6.1  The floor and ceiling functions   cfl 12893
            5.6.2  The modulo (remainder) operation   cmo 12970
            5.6.3  Miscellaneous theorems about integers   om2uz0i 13048
            5.6.4  Strong induction over upper sets of integers   uzsinds 13088
            5.6.5  Finitely supported functions over the nonnegative integers   fsuppmapnn0fiublem 13091
            5.6.6  The infinite sequence builder "seq" - extension   cseq 13102
            5.6.7  Integer powers   cexp 13161
            5.6.8  Ordered pair theorem for nonnegative integers   nn0le2msqi 13354
            5.6.9  Factorial function   cfa 13360
            5.6.10  The binomial coefficient operation   cbc 13389
            5.6.11  The ` # ` (set size) function   chash 13417
                  5.6.11.1  Proper unordered pairs and triples (sets of size 2 and 3)   hashprlei 13546
                  5.6.11.2  Functions with a domain containing at least two different elements   fundmge2nop0 13570
                  5.6.11.3  Finite induction on the size of the first component of a binary relation   hashdifsnp1 13574
      ​​*5.7  Words over a set
            5.7.1  Definitions and basic theorems   cword 13581
            5.7.2  Last symbol of a word   clsw 13629
            5.7.3  Concatenations of words   cconcat 13637
            5.7.4  Singleton words   cs1 13662
            5.7.5  Concatenations with singleton words   ccatws1cl 13683
            5.7.6  Subwords/substrings   csubstr 13707
            5.7.7  Prefixes of a word   cpfx 13756
            5.7.8  Subwords of subwords   swrdswrdlem 13790
            5.7.9  Subwords and concatenations   pfxcctswrd 13800
            5.7.10  Subwords of concatenations   swrdccatfn 13827
            5.7.11  Splicing words (substring replacement)   csplice 13862
            5.7.12  Reversing words   creverse 13881
            5.7.13  Repeated symbol words   creps 13891
            ​*5.7.14  Cyclical shifts of words   ccsh 13911
            5.7.15  Mapping words by a function   wrdco 13959
            5.7.16  Longer string literals   cs2 13969
      ​​*5.8  Reflexive and transitive closures of relations
            5.8.1  The reflexive and transitive properties of relations   coss12d 14097
            5.8.2  Basic properties of closures   cleq1lem 14107
            5.8.3  Definitions and basic properties of transitive closures   ctcl 14110
            5.8.4  Exponentiation of relations   crelexp 14144
            5.8.5  Reflexive-transitive closure as an indexed union   crtrcl 14179
            ​*5.8.6  Principle of transitive induction.   relexpindlem 14187
      ​5.9  Elementary real and complex functions
            5.9.1  The "shift" operation   cshi 14190
            5.9.2  Signum (sgn or sign) function   csgn 14210
            5.9.3  Real and imaginary parts; conjugate   ccj 14220
            5.9.4  Square root; absolute value   csqrt 14357
      ​5.10  Elementary limits and convergence
            5.10.1  Superior limit (lim sup)   clsp 14585
            5.10.2  Limits   cli 14599
            5.10.3  Finite and infinite sums   csu 14800
            5.10.4  The binomial theorem   binomlem 14942
            5.10.5  The inclusion/exclusion principle   incexclem 14949
            5.10.6  Infinite sums (cont.)   isumshft 14952
            5.10.7  Miscellaneous converging and diverging sequences   divrcnv 14965
            5.10.8  Arithmetic series   arisum 14973
            5.10.9  Geometric series   expcnv 14977
            5.10.10  Ratio test for infinite series convergence   cvgrat 14995
            5.10.11  Mertens' theorem   mertenslem1 14996
            5.10.12  Finite and infinite products   prodf 14999
                  5.10.12.1  Product sequences   prodf 14999
                  5.10.12.2  Non-trivial convergence   ntrivcvg 15009
                  5.10.12.3  Complex products   cprod 15015
                  5.10.12.4  Finite products   fprod 15051
                  5.10.12.5  Infinite products   iprodclim 15108
            5.10.13  Falling and Rising Factorial   cfallfac 15114
            5.10.14  Bernoulli polynomials and sums of k-th powers   cbp 15156
      ​5.11  Elementary trigonometry
            5.11.1  The exponential, sine, and cosine functions   ce 15171
                  5.11.1.1  The circle constant (tau = 2 pi)   ctau 15311
            5.11.2  _e is irrational   eirrlem 15313
      ​5.12  Cardinality of real and complex number subsets
            5.12.1  Countability of integers and rationals   xpnnen 15320
            5.12.2  The reals are uncountable   rpnnen2lem1 15324
​​*PART 6  ELEMENTARY NUMBER THEORY
      ​6.1  Elementary properties of divisibility
            6.1.1  Irrationality of square root of 2   sqrt2irrlem 15358
            6.1.2  Some Number sets are chains of proper subsets   nthruc 15362
            6.1.3  The divides relation   cdvds 15364
            ​*6.1.4  Even and odd numbers   evenelz 15441
            6.1.5  The division algorithm   divalglem0 15497
            6.1.6  Bit sequences   cbits 15521
            6.1.7  The greatest common divisor operator   cgcd 15596
            6.1.8  Bézout's identity   bezoutlem1 15636
            6.1.9  Algorithms   nn0seqcvgd 15663
            6.1.10  Euclid's Algorithm   eucalgval2 15674
            ​*6.1.11  The least common multiple   clcm 15681
            ​*6.1.12  Coprimality and Euclid's lemma   coprmgcdb 15742
            6.1.13  Cancellability of congruences   congr 15757
      ​6.2  Elementary prime number theory
            ​*6.2.1  Elementary properties   cprime 15764
            ​*6.2.2  Coprimality and Euclid's lemma (cont.)   coprm 15801
            6.2.3  Properties of the canonical representation of a rational   cnumer 15819
            6.2.4  Euler's theorem   codz 15846
            6.2.5  Arithmetic modulo a prime number   modprm1div 15880
            6.2.6  Pythagorean Triples   coprimeprodsq 15891
            6.2.7  The prime count function   cpc 15919
            6.2.8  Pocklington's theorem   prmpwdvds 15986
            6.2.9  Infinite primes theorem   unbenlem 15990
            6.2.10  Sum of prime reciprocals   prmreclem1 15998
            6.2.11  Fundamental theorem of arithmetic   1arithlem1 16005
            6.2.12  Lagrange's four-square theorem   cgz 16011
            6.2.13  Van der Waerden's theorem   cvdwa 16047
            6.2.14  Ramsey's theorem   cram 16081
            ​*6.2.15  Primorial function   cprmo 16113
            ​*6.2.16  Prime gaps   prmgaplem1 16131
            6.2.17  Decimal arithmetic (cont.)   dec2dvds 16145
            6.2.18  Cyclical shifts of words (cont.)   cshwsidrepsw 16173
            6.2.19  Specific prime numbers   prmlem0 16185
            6.2.20  Very large primes   1259lem1 16210
​PART 7  BASIC STRUCTURES
      ​7.1  Extensible structures
            ​*7.1.1  Basic definitions   cstr 16225
            7.1.2  Slot definitions   cplusg 16312
            7.1.3  Definition of the structure product   crest 16441
            7.1.4  Definition of the structure quotient   cordt 16519
      ​7.2  Moore spaces
            7.2.1  Moore closures   mrcflem 16626
            7.2.2  Independent sets in a Moore system   mrisval 16650
            7.2.3  Algebraic closure systems   isacs 16671
​PART 8  BASIC CATEGORY THEORY
      ​8.1  Categories
            8.1.1  Categories   ccat 16684
            8.1.2  Opposite category   coppc 16730
            8.1.3  Monomorphisms and epimorphisms   cmon 16747
            8.1.4  Sections, inverses, isomorphisms   csect 16763
            ​*8.1.5  Isomorphic objects   ccic 16814
            8.1.6  Subcategories   cssc 16826
            8.1.7  Functors   cfunc 16873
            8.1.8  Full & faithful functors   cful 16921
            8.1.9  Natural transformations and the functor category   cnat 16960
            8.1.10  Initial, terminal and zero objects of a category   cinito 16997
      ​8.2  Arrows (disjointified hom-sets)
            8.2.1  Identity and composition for arrows   cida 17062
      ​8.3  Examples of categories
            8.3.1  The category of sets   csetc 17084
            8.3.2  The category of categories   ccatc 17103
            ​*8.3.3  The category of extensible structures   fncnvimaeqv 17119
      ​8.4  Categorical constructions
            8.4.1  Product of categories   cxpc 17168
            8.4.2  Functor evaluation   cevlf 17209
            8.4.3  Hom functor   chof 17248
​PART 9  BASIC ORDER THEORY
      ​9.1  Preordered sets and directed sets using extensible structures
      ​9.2  Posets and lattices using extensible structures
            9.2.1  Posets   cpo 17300
            9.2.2  Lattices   clat 17405
            9.2.3  The dual of an ordered set   codu 17488
            9.2.4  Subset order structures   cipo 17511
            9.2.5  Distributive lattices   latmass 17548
            9.2.6  Posets and lattices as relations   cps 17558
            9.2.7  Directed sets, nets   cdir 17588
​PART 10  BASIC ALGEBRAIC STRUCTURES
      ​10.1  Monoids
            ​*10.1.1  Magmas   cplusf 17599
            ​*10.1.2  Identity elements   mgmidmo 17619
            ​*10.1.3  Ordered sums in a magma   gsumvalx 17630
            ​*10.1.4  Semigroups   csgrp 17643
            ​*10.1.5  Definition and basic properties of monoids   cmnd 17654
            10.1.6  Monoid homomorphisms and submonoids   cmhm 17693
            ​*10.1.7  Ordered sums in a monoid   gsumvallem2 17732
            10.1.8  Free monoids   cfrmd 17745
            10.1.9  Examples and counterexamples for magmas, semigroups and monoids   mgm2nsgrplem1 17766
      ​10.2  Groups
            10.2.1  Definition and basic properties   cgrp 17783
            ​*10.2.2  Group multiple operation   cmg 17901
            10.2.3  Subgroups and Quotient groups   csubg 17946
            10.2.4  Elementary theory of group homomorphisms   cghm 18015
            10.2.5  Isomorphisms of groups   cgim 18057
            10.2.6  Group actions   cga 18079
            10.2.7  Centralizers and centers   ccntz 18105
            10.2.8  The opposite group   coppg 18132
            10.2.9  Symmetric groups   csymg 18154
                  ​*10.2.9.1  Definition and basic properties   csymg 18154
                  10.2.9.2  Cayley's theorem   cayleylem1 18189
                  10.2.9.3  Permutations fixing one element   symgfix2 18193
                  ​*10.2.9.4  Transpositions in the symmetric group   cpmtr 18218
                  10.2.9.5  The sign of a permutation   cpsgn 18266
            10.2.10  p-Groups and Sylow groups; Sylow's theorems   cod 18302
            10.2.11  Direct products   clsm 18407
            10.2.12  Free groups   cefg 18477
      ​10.3  Abelian groups
            10.3.1  Definition and basic properties   ccmn 18553
            10.3.2  Cyclic groups   ccyg 18639
            10.3.3  Group sum operation   gsumval3a 18664
            10.3.4  Group sums over (ranges of) integers   fsfnn0gsumfsffz 18739
            10.3.5  Internal direct products   cdprd 18753
            10.3.6  The Fundamental Theorem of Abelian Groups   ablfacrplem 18825
      ​10.4  Rings
            10.4.1  Multiplicative Group   cmgp 18850
            10.4.2  Ring unit   cur 18862
                  10.4.2.1  Semirings   csrg 18866
                  ​*10.4.2.2  The binomial theorem for semirings   srgbinomlem1 18901
            10.4.3  Definition and basic properties of unital rings   crg 18908
            10.4.4  Opposite ring   coppr 18983
            10.4.5  Divisibility   cdsr 18999
            10.4.6  Ring primes   crpm 19073
            10.4.7  Ring homomorphisms   crh 19075
      ​10.5  Division rings and fields
            10.5.1  Definition and basic properties   cdr 19110
            10.5.2  Subrings of a ring   csubrg 19139
            10.5.3  Absolute value (abstract algebra)   cabv 19179
            10.5.4  Star rings   cstf 19206
      ​10.6  Left modules
            10.6.1  Definition and basic properties   clmod 19226
            10.6.2  Subspaces and spans in a left module   clss 19295
            10.6.3  Homomorphisms and isomorphisms of left modules   clmhm 19385
            10.6.4  Subspace sum; bases for a left module   clbs 19440
      ​10.7  Vector spaces
            10.7.1  Definition and basic properties   clvec 19468
      ​10.8  Ideals
            10.8.1  The subring algebra; ideals   csra 19536
            10.8.2  Two-sided ideals and quotient rings   c2idl 19599
            10.8.3  Principal ideal rings. Divisibility in the integers   clpidl 19609
            10.8.4  Nonzero rings and zero rings   cnzr 19625
            10.8.5  Left regular elements. More kinds of rings   crlreg 19647
      ​10.9  Associative algebras
            10.9.1  Definition and basic properties   casa 19677
      ​10.10  Abstract multivariate polynomials
            10.10.1  Definition and basic properties   cmps 19719
            10.10.2  Polynomial evaluation   ces 19871
            ​*10.10.3  Additional definitions for (multivariate) polynomials   cmhp 19904
            ​*10.10.4  Univariate polynomials   cps1 19912
            10.10.5  Univariate polynomial evaluation   ces1 20045
      ​10.11  The complex numbers as an algebraic extensible structure
            10.11.1  Definition and basic properties   cpsmet 20097
            ​*10.11.2  Ring of integers   zring 20185
            10.11.3  Algebraic constructions based on the complex numbers   czrh 20215
            10.11.4  Signs as subgroup of the complex numbers   cnmsgnsubg 20289
            10.11.5  Embedding of permutation signs into a ring   zrhpsgnmhm 20296
            10.11.6  The ordered field of real numbers   crefld 20318
      ​10.12  Generalized pre-Hilbert and Hilbert spaces
            10.12.1  Definition and basic properties   cphl 20338
            10.12.2  Orthocomplements and closed subspaces   cocv 20374
            10.12.3  Orthogonal projection and orthonormal bases   cpj 20414
​​*PART 11  BASIC LINEAR ALGEBRA
      ​11.1  Vectors and free modules
            *11.1.1  Direct sum of left modules   cdsmm 20445
            ​*11.1.2  Free modules   cfrlm 20460
            ​*11.1.3  Standard basis (unit vectors)   cuvc 20495
            ​*11.1.4  Independent sets and families   clindf 20517
            11.1.5  Characterization of free modules   lmimlbs 20549
      ​​*11.2  Matrices
            *11.2.1  The matrix multiplication   cmmul 20563
            ​*11.2.2  Square matrices   cmat 20587
            ​*11.2.3  The matrix algebra   matmulr 20618
            ​*11.2.4  Matrices of dimension 0 and 1   mat0dimbas0 20647
            ​*11.2.5  The subalgebras of diagonal and scalar matrices   cdmat 20669
            ​*11.2.6  Multiplication of a matrix with a "column vector"   cmvmul 20721
            11.2.7  Replacement functions for a square matrix   cmarrep 20737
            11.2.8  Submatrices   csubma 20757
      ​11.3  The determinant
            11.3.1  Definition and basic properties   cmdat 20765
            11.3.2  Determinants of 2 x 2 -matrices   m2detleiblem1 20805
            11.3.3  The matrix adjugate/adjunct   cmadu 20813
            ​*11.3.4  Laplace expansion of determinants (special case)   symgmatr01lem 20835
            11.3.5  Inverse matrix   invrvald 20858
            ​*11.3.6  Cramer's rule   slesolvec 20861
      ​​*11.4  Polynomial matrices
            11.4.1  Basic properties   pmatring 20875
            ​*11.4.2  Constant polynomial matrices   ccpmat 20885
            ​*11.4.3  Collecting coefficients of polynomial matrices   cdecpmat 20944
            ​*11.4.4  Ring isomorphism between polynomial matrices and polynomials over matrices   cpm2mp 20974
      ​​*11.5  The characteristic polynomial
            *11.5.1  Definition and basic properties   cchpmat 21008
            ​*11.5.2  The characteristic factor function G   fvmptnn04if 21031
            ​*11.5.3  The Cayley-Hamilton theorem   cpmadurid 21049
​PART 12  BASIC TOPOLOGY
      ​12.1  Topology
            ​*12.1.1  Topological spaces   ctop 21075
                  12.1.1.1  Topologies   ctop 21075
                  12.1.1.2  Topologies on sets   ctopon 21092
                  12.1.1.3  Topological spaces   ctps 21114
            12.1.2  Topological bases   ctb 21127
            12.1.3  Examples of topologies   distop 21177
            12.1.4  Closure and interior   ccld 21198
            12.1.5  Neighborhoods   cnei 21279
            12.1.6  Limit points and perfect sets   clp 21316
            12.1.7  Subspace topologies   restrcl 21339
            12.1.8  Order topology   ordtbaslem 21370
            12.1.9  Limits and continuity in topological spaces   ccn 21406
            12.1.10  Separated spaces: T0, T1, T2 (Hausdorff) ...   ct0 21488
            12.1.11  Compactness   ccmp 21567
            12.1.12  Bolzano-Weierstrass theorem   bwth 21591
            12.1.13  Connectedness   cconn 21592
            12.1.14  First- and second-countability   c1stc 21618
            12.1.15  Local topological properties   clly 21645
            12.1.16  Refinements   cref 21683
            12.1.17  Compactly generated spaces   ckgen 21714
            12.1.18  Product topologies   ctx 21741
            12.1.19  Continuous function-builders   cnmptid 21842
            12.1.20  Quotient maps and quotient topology   ckq 21874
            12.1.21  Homeomorphisms   chmeo 21934
      ​12.2  Filters and filter bases
            12.2.1  Filter bases   elmptrab 22008
            12.2.2  Filters   cfil 22026
            12.2.3  Ultrafilters   cufil 22080
            12.2.4  Filter limits   cfm 22114
            12.2.5  Extension by continuity   ccnext 22240
            12.2.6  Topological groups   ctmd 22251
            12.2.7  Infinite group sum on topological groups   ctsu 22306
            12.2.8  Topological rings, fields, vector spaces   ctrg 22336
      ​12.3  Uniform Structures and Spaces
            12.3.1  Uniform structures   cust 22380
            12.3.2  The topology induced by an uniform structure   cutop 22411
            12.3.3  Uniform Spaces   cuss 22434
            12.3.4  Uniform continuity   cucn 22456
            12.3.5  Cauchy filters in uniform spaces   ccfilu 22467
            12.3.6  Complete uniform spaces   ccusp 22478
      ​12.4  Metric spaces
            12.4.1  Pseudometric spaces   ispsmet 22486
            12.4.2  Basic metric space properties   cxms 22499
            12.4.3  Metric space balls   blfvalps 22565
            12.4.4  Open sets of a metric space   mopnval 22620
            12.4.5  Continuity in metric spaces   metcnp3 22722
            12.4.6  The uniform structure generated by a metric   metuval 22731
            12.4.7  Examples of metric spaces   dscmet 22754
            ​*12.4.8  Normed algebraic structures   cnm 22758
            12.4.9  Normed space homomorphisms (bounded linear operators)   cnmo 22886
            12.4.10  Topology on the reals   qtopbaslem 22939
            12.4.11  Topological definitions using the reals   cii 23055
            12.4.12  Path homotopy   chtpy 23143
            12.4.13  The fundamental group   cpco 23176
      ​12.5  Metric subcomplex vector spaces
            12.5.1  Subcomplex modules   cclm 23238
            ​*12.5.2  Subcomplex vector spaces   ccvs 23299
            ​*12.5.3  Normed subcomplex vector spaces   isncvsngp 23325
            12.5.4  Subcomplex pre-Hilbert space   ccph 23342
            12.5.5  Convergence and completeness   ccfil 23427
            12.5.6  Baire's Category Theorem   bcthlem1 23499
            12.5.7  Banach spaces and subcomplex Hilbert spaces   ccms 23507
                  12.5.7.1  The complete ordered field of the real numbers   retopn 23554
            12.5.8  Euclidean spaces   crrx 23558
            12.5.9  Minimizing Vector Theorem   minveclem1 23599
            12.5.10  Projection Theorem   pjthlem1 23612
​PART 13  BASIC REAL AND COMPLEX ANALYSIS
      ​13.1  Continuity
            13.1.1  Intermediate value theorem   pmltpclem1 23621
      ​13.2  Integrals
            13.2.1  Lebesgue measure   covol 23635
            13.2.2  Lebesgue integration   cmbf 23787
                  13.2.2.1  Lesbesgue integral   cmbf 23787
                  13.2.2.2  Lesbesgue directed integral   cdit 24016
      ​13.3  Derivatives
            13.3.1  Real and complex differentiation   climc 24032
                  13.3.1.1  Derivatives of functions of one complex or real variable   climc 24032
                  13.3.1.2  Results on real differentiation   dvferm1lem 24153
​PART 14  BASIC REAL AND COMPLEX FUNCTIONS
      ​14.1  Polynomials
            14.1.1  Polynomial degrees   cmdg 24219
            14.1.2  The division algorithm for univariate polynomials   cmn1 24291
            14.1.3  Elementary properties of complex polynomials   cply 24346
            14.1.4  The division algorithm for polynomials   cquot 24451
            14.1.5  Algebraic numbers   caa 24475
            14.1.6  Liouville's approximation theorem   aalioulem1 24493
      ​14.2  Sequences and series
            14.2.1  Taylor polynomials and Taylor's theorem   ctayl 24513
            14.2.2  Uniform convergence   culm 24536
            14.2.3  Power series   pserval 24570
      ​14.3  Basic trigonometry
            14.3.1  The exponential, sine, and cosine functions (cont.)   efcn 24603
            14.3.2  Properties of pi = 3.14159...   pilem1 24611
            14.3.3  Mapping of the exponential function   efgh 24694
            14.3.4  The natural logarithm on complex numbers   clog 24707
            ​*14.3.5  Logarithms to an arbitrary base   clogb 24911
            14.3.6  Theorems of Pythagoras, isosceles triangles, and intersecting chords   angval 24948
            14.3.7  Solutions of quadratic, cubic, and quartic equations   quad2 24986
            14.3.8  Inverse trigonometric functions   casin 25009
            14.3.9  The Birthday Problem   log2ublem1 25093
            14.3.10  Areas in R^2   carea 25102
            14.3.11  More miscellaneous converging sequences   rlimcnp 25112
            14.3.12  Inequality of arithmetic and geometric means   cvxcl 25131
            14.3.13  Euler-Mascheroni constant   cem 25138
            14.3.14  Zeta function   czeta 25159
            14.3.15  Gamma function   clgam 25162
      ​14.4  Basic number theory
            14.4.1  Wilson's theorem   wilthlem1 25214
            14.4.2  The Fundamental Theorem of Algebra   ftalem1 25219
            14.4.3  The Basel problem (ζ(2) = π²/6)   basellem1 25227
            14.4.4  Number-theoretical functions   ccht 25237
            14.4.5  Perfect Number Theorem   mersenne 25372
            14.4.6  Characters of Z/nZ   cdchr 25377
            14.4.7  Bertrand's postulate   bcctr 25420
            ​*14.4.8  Quadratic residues and the Legendre symbol   clgs 25439
            ​*14.4.9  Gauss' Lemma   gausslemma2dlem0a 25501
            14.4.10  Quadratic reciprocity   lgseisenlem1 25520
            14.4.11  All primes 4n+1 are the sum of two squares   2sqlem1 25562
            14.4.12  Chebyshev's Weak Prime Number Theorem, Dirichlet's Theorem   chebbnd1lem1 25578
            14.4.13  The Prime Number Theorem   mudivsum 25639
            14.4.14  Ostrowski's theorem   abvcxp 25724
​​*PART 15  ELEMENTARY GEOMETRY
      ​15.1  Definition and Tarski's Axioms of Geometry
            15.1.1  Justification for the congruence notation   tgjustf 25792
      ​15.2  Tarskian Geometry
            15.2.1  Congruence   tgcgrcomimp 25796
            15.2.2  Betweenness   tgbtwntriv2 25806
            15.2.3  Dimension   tglowdim1 25819
            15.2.4  Betweenness and Congruence   tgifscgr 25827
            15.2.5  Congruence of a series of points   ccgrg 25829
            15.2.6  Motions   cismt 25851
            15.2.7  Colinearity   tglng 25865
            15.2.8  Connectivity of betweenness   tgbtwnconn1lem1 25891
            15.2.9  Less-than relation in geometric congruences   cleg 25901
            15.2.10  Rays   chlg 25919
            15.2.11  Lines   btwnlng1 25938
            15.2.12  Point inversions   cmir 25971
            15.2.13  Right angles   crag 26012
            15.2.14  Half-planes   islnopp 26055
            15.2.15  Midpoints and Line Mirroring   cmid 26088
            15.2.16  Congruence of angles   ccgra 26123
            15.2.17  Angle Comparisons   cinag 26151
            15.2.18  Congruence Theorems   tgsas1 26160
            15.2.19  Equilateral triangles   ceqlg 26171
      ​15.3  Properties of geometries
            15.3.1  Isomorphisms between geometries   f1otrgds 26175
      ​15.4  Geometry in Hilbert spaces
            15.4.1  Geometry in the complex plane   cchhllem 26193
            15.4.2  Geometry in Euclidean spaces   cee 26194
                  15.4.2.1  Definition of the Euclidean space   cee 26194
                  15.4.2.2  Tarski's axioms for geometry for the Euclidean space   axdimuniq 26219
                  15.4.2.3  EE^n fulfills Tarski's Axioms   ceeng 26283
​​*PART 16  GRAPH THEORY
      ​*16.1  Vertices and edges
            16.1.1  The edge function extractor for extensible structures   cedgf 26294
            ​*16.1.2  Vertices and indexed edges   cvtx 26301
                  16.1.2.1  Definitions and basic properties   cvtx 26301
                  16.1.2.2  The vertices and edges of a graph represented as ordered pair   opvtxval 26308
                  16.1.2.3  The vertices and edges of a graph represented as extensible structure   funvtxdmge2val 26316
                  16.1.2.4  Representations of graphs without edges   snstrvtxval 26342
                  16.1.2.5  Degenerated cases of representations of graphs   vtxval0 26344
            16.1.3  Edges as range of the edge function   cedg 26352
      ​​*16.2  Undirected graphs
            16.2.1  Undirected hypergraphs   cuhgr 26361
            16.2.2  Undirected pseudographs and multigraphs   cupgr 26385
            ​*16.2.3  Loop-free graphs   umgrislfupgrlem 26427
            16.2.4  Edges as subsets of vertices of graphs   uhgredgiedgb 26431
            ​*16.2.5  Undirected simple graphs   cuspgr 26454
            16.2.6  Examples for graphs   usgr0e 26540
            16.2.7  Subgraphs   csubgr 26571
            16.2.8  Finite undirected simple graphs   cfusgr 26620
            16.2.9  Neighbors, complete graphs and universal vertices   cnbgr 26636
                  16.2.9.1  Neighbors   cnbgr 26636
                  16.2.9.2  Universal vertices   cuvtx 26690
                  16.2.9.3  Complete graphs   ccplgr 26714
            16.2.10  Vertex degree   cvtxdg 26770
            ​*16.2.11  Regular graphs   crgr 26860
      ​​*16.3  Walks, paths and cycles
            *16.3.1  Walks   cewlks 26900
            16.3.2  Walks for loop-free graphs   lfgrwlkprop 26995
            16.3.3  Trails   ctrls 26998
            16.3.4  Paths and simple paths   cpths 27021
            16.3.5  Closed walks   cclwlks 27079
            16.3.6  Circuits and cycles   ccrcts 27093
            ​*16.3.7  Walks as words   cwwlks 27131
            16.3.8  Walks/paths of length 2 (as length 3 strings)   2wlkdlem1 27261
            16.3.9  Walks in regular graphs   rusgrnumwwlkl1 27304
            ​*16.3.10  Closed walks as words   cclwwlk 27317
                  16.3.10.1  Closed walks as words   cclwwlk 27317
                  16.3.10.2  Closed walks of a fixed length as words   cclwwlkn 27369
                  16.3.10.3  Closed walks on a vertex of a fixed length as words   cclwwlknon 27458
            16.3.11  Examples for walks, trails and paths   0ewlk 27486
            16.3.12  Connected graphs   cconngr 27558
      ​16.4  Eulerian paths and the Konigsberg Bridge problem
            ​*16.4.1  Eulerian paths   ceupth 27569
            ​*16.4.2  The Königsberg Bridge problem   konigsbergvtx 27621
      ​16.5  The Friendship Theorem
            16.5.1  Friendship graphs - basics   cfrgr 27633
            16.5.2  The friendship theorem for small graphs   frgr1v 27648
            16.5.3  Theorems according to Mertzios and Unger   2pthfrgrrn 27659
            ​*16.5.4  Huneke's Proof of the Friendship Theorem   frgrncvvdeqlem1 27676
​PART 17  GUIDES AND MISCELLANEA
      ​17.1  Guides (conventions, explanations, and examples)
            ​*17.1.1  Conventions   conventions 27811
            17.1.2  Natural deduction   natded 27814
            ​*17.1.3  Natural deduction examples   ex-natded5.2 27815
            17.1.4  Definitional examples   ex-or 27832
            17.1.5  Other examples   aevdemo 27871
      ​17.2  Humor
            17.2.1  April Fool's theorem   avril1 27873
      ​17.3  (Future - to be reviewed and classified)
            17.3.1  Planar incidence geometry   cplig 27880
            ​*17.3.2  Aliases kept to prevent broken links   dummylink 27893
​​*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 27895
            18.1.2  Abelian groups   cablo 27950
      ​18.2  Complex vector spaces
            18.2.1  Definition and basic properties   cvc 27964
            18.2.2  Examples of complex vector spaces   cnaddabloOLD 27987
      ​18.3  Normed complex vector spaces
            18.3.1  Definition and basic properties   cnv 27990
            18.3.2  Examples of normed complex vector spaces   cnnv 28083
            18.3.3  Induced metric of a normed complex vector space   imsval 28091
            18.3.4  Inner product   cdip 28106
            18.3.5  Subspaces   css 28127
      ​18.4  Operators on complex vector spaces
            18.4.1  Definitions and basic properties   clno 28146
      ​18.5  Inner product (pre-Hilbert) spaces
            18.5.1  Definition and basic properties   ccphlo 28218
            18.5.2  Examples of pre-Hilbert spaces   cncph 28225
            18.5.3  Properties of pre-Hilbert spaces   isph 28228
      ​18.6  Complex Banach spaces
            18.6.1  Definition and basic properties   ccbn 28269
            18.6.2  Examples of complex Banach spaces   cnbn 28276
            18.6.3  Uniform Boundedness Theorem   ubthlem1 28277
            18.6.4  Minimizing Vector Theorem   minvecolem1 28281
      ​18.7  Complex Hilbert spaces
            18.7.1  Definition and basic properties   chlo 28292
            18.7.2  Standard axioms for a complex Hilbert space   hlex 28305
            18.7.3  Examples of complex Hilbert spaces   cnchl 28323
            18.7.4  Subspaces   ssphlOLD 28324
            18.7.5  Hellinger-Toeplitz Theorem   htthlem 28325
​​*PART 19  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      ​19.1  Axiomatization of complex pre-Hilbert spaces
            19.1.1  Basic Hilbert space definitions   chba 28327
            19.1.2  Preliminary ZFC lemmas   df-hnorm 28376
            ​*19.1.3  Derive the Hilbert space axioms from ZFC set theory   axhilex-zf 28389
            ​*19.1.4  Introduce the vector space axioms for a Hilbert space   ax-hilex 28407
            19.1.5  Vector operations   hvmulex 28419
            19.1.6  Inner product postulates for a Hilbert space   ax-hfi 28487
      ​19.2  Inner product and norms
            19.2.1  Inner product   his5 28494
            19.2.2  Norms   dfhnorm2 28530
            19.2.3  Relate Hilbert space to normed complex vector spaces   hilablo 28568
            19.2.4  Bunjakovaskij-Cauchy-Schwarz inequality   bcsiALT 28587
      ​19.3  Cauchy sequences and completeness axiom
            19.3.1  Cauchy sequences and limits   hcau 28592
            19.3.2  Derivation of the completeness axiom from ZF set theory   hilmet 28602
            19.3.3  Completeness postulate for a Hilbert space   ax-hcompl 28610
            19.3.4  Relate Hilbert space to ZFC pre-Hilbert and Hilbert spaces   hhcms 28611
      ​19.4  Subspaces and projections
            19.4.1  Subspaces   df-sh 28615
            19.4.2  Closed subspaces   df-ch 28629
            19.4.3  Orthocomplements   df-oc 28660
            19.4.4  Subspace sum, span, lattice join, lattice supremum   df-shs 28718
            19.4.5  Projection theorem   pjhthlem1 28801
            19.4.6  Projectors   df-pjh 28805
      ​19.5  Properties of Hilbert subspaces
            19.5.1  Orthomodular law   omlsilem 28812
            19.5.2  Projectors (cont.)   pjhtheu2 28826
            19.5.3  Hilbert lattice operations   sh0le 28850
            19.5.4  Span (cont.) and one-dimensional subspaces   spansn0 28951
            19.5.5  Commutes relation for Hilbert lattice elements   df-cm 28993
            19.5.6  Foulis-Holland theorem   fh1 29028
            19.5.7  Quantum Logic Explorer axioms   qlax1i 29037
            19.5.8  Orthogonal subspaces   chscllem1 29047
            19.5.9  Orthoarguesian laws 5OA and 3OA   5oalem1 29064
            19.5.10  Projectors (cont.)   pjorthi 29079
            19.5.11  Mayet's equation E_3   mayete3i 29138
      ​19.6  Operators on Hilbert spaces
            ​*19.6.1  Operator sum, difference, and scalar multiplication   df-hosum 29140
            19.6.2  Zero and identity operators   df-h0op 29158
            19.6.3  Operations on Hilbert space operators   hoaddcl 29168
            19.6.4  Linear, continuous, bounded, Hermitian, unitary operators and norms   df-nmop 29249
            19.6.5  Linear and continuous functionals and norms   df-nmfn 29255
            19.6.6  Adjoint   df-adjh 29259
            19.6.7  Dirac bra-ket notation   df-bra 29260
            19.6.8  Positive operators   df-leop 29262
            19.6.9  Eigenvectors, eigenvalues, spectrum   df-eigvec 29263
            19.6.10  Theorems about operators and functionals   nmopval 29266
            19.6.11  Riesz lemma   riesz3i 29472
            19.6.12  Adjoints (cont.)   cnlnadjlem1 29477
            19.6.13  Quantum computation error bound theorem   unierri 29514
            19.6.14  Dirac bra-ket notation (cont.)   branmfn 29515
            19.6.15  Positive operators (cont.)   leopg 29532
            19.6.16  Projectors as operators   pjhmopi 29556
      ​19.7  States on a Hilbert lattice and Godowski's equation
            19.7.1  States on a Hilbert lattice   df-st 29621
            19.7.2  Godowski's equation   golem1 29681
      ​19.8  Cover relation, atoms, exchange axiom, and modular symmetry
            19.8.1  Covers relation; modular pairs   df-cv 29689
            19.8.2  Atoms   df-at 29748
            19.8.3  Superposition principle   superpos 29764
            19.8.4  Atoms, exchange and covering properties, atomicity   chcv1 29765
            19.8.5  Irreducibility   chirredlem1 29800
            19.8.6  Atoms (cont.)   atcvat3i 29806
            19.8.7  Modular symmetry   mdsymlem1 29813
​PART 20  SUPPLEMENTARY MATERIAL (USER'S MATHBOXES)
      ​20.1  Mathboxes for user contributions
            20.1.1  Mathbox guidelines   mathbox 29852
      ​20.2  Mathbox for Stefan Allan
      ​20.3  Mathbox for Thierry Arnoux
            20.3.1  Propositional Calculus - misc additions   bian1d 29857
            20.3.2  Predicate Calculus   spc2ed 29863
                  20.3.2.1  Predicate Calculus - misc additions   spc2ed 29863
                  20.3.2.2  Restricted quantification - misc additions   ralcom4f 29867
                  20.3.2.3  Substitution (without distinct variables) - misc additions   sbceqbidf 29872
                  20.3.2.4  Existential "at most one" - misc additions   moel 29874
                  20.3.2.5  Existential uniqueness - misc additions   2reuswap2 29877
                  20.3.2.6  Restricted "at most one" - misc additions   rmoxfrd 29881
            20.3.3  General Set Theory   difrab2 29883
                  20.3.3.1  Class abstractions (a.k.a. class builders)   difrab2 29883
                  20.3.3.2  Image Sets   abrexdomjm 29889
                  20.3.3.3  Set relations and operations - misc additions   rabss3d 29895
                  20.3.3.4  Unordered pairs   elpreq 29904
                  20.3.3.5  Conditional operator - misc additions   ifeqeqx 29905
                  20.3.3.6  Set union   uniinn0 29910
                  20.3.3.7  Indexed union - misc additions   cbviunf 29916
                  20.3.3.8  Disjointness - misc additions   disjnf 29927
            20.3.4  Relations and Functions   xpdisjres 29954
                  20.3.4.1  Relations - misc additions   xpdisjres 29954
                  20.3.4.2  Functions - misc additions   ac6sf2 29974
                  20.3.4.3  Operations - misc additions   mpt2mptxf 30021
                  20.3.4.4  Isomorphisms - misc. add.   gtiso 30022
                  20.3.4.5  Disjointness (additional proof requiring functions)   disjdsct 30024
                  20.3.4.6  First and second members of an ordered pair - misc additions   df1stres 30025
                  20.3.4.7  Supremum - misc additions   supssd 30031
                  20.3.4.8  Finite Sets   imafi2 30033
                  20.3.4.9  Countable Sets   snct 30035
            20.3.5  Real and Complex Numbers   subeqxfrd 30054
                  20.3.5.1  Complex operations - misc. additions   subeqxfrd 30054
                  20.3.5.2  Ordering on reals - misc additions   lt2addrd 30059
                  20.3.5.3  Extended reals - misc additions   xrlelttric 30060
                  20.3.5.4  Real number intervals - misc additions   joiniooico 30079
                  20.3.5.5  Finite intervals of integers - misc additions   uzssico 30089
                  20.3.5.6  Half-open integer ranges - misc additions   iundisjfi 30098
                  20.3.5.7  The ` # ` (set size) function - misc additions   hashunif 30105
                  20.3.5.8  The greatest common divisor operator - misc. add   numdenneg 30106
                  20.3.5.9  Integers   nnindf 30108
                  20.3.5.10  Decimal numbers   dfdec100 30119
            ​*20.3.6  Decimal expansion   cdp2 30120
                  ​*20.3.6.1  Decimal point   cdp 30137
                  20.3.6.2  Division in the extended real number system   cxdiv 30166
            20.3.7  Prime Number Theory   bhmafibid1 30185
                  20.3.7.1  Fermat's two square theorem   bhmafibid1 30185
            20.3.8  Extensible Structures   ressplusf 30191
                  20.3.8.1  Structure restriction operator   ressplusf 30191
                  20.3.8.2  The opposite group   oppgle 30194
                  20.3.8.3  Posets   ressprs 30196
                  20.3.8.4  Complete lattices   clatp0cl 30212
                  20.3.8.5  Extended reals Structure - misc additions   ax-xrssca 30214
                  20.3.8.6  The extended nonnegative real numbers commutative monoid   xrge0base 30226
            20.3.9  Algebra   abliso 30237
                  20.3.9.1  Monoids Homomorphisms   abliso 30237
                  20.3.9.2  Ordered monoids and groups   comnd 30238
                  20.3.9.3  Signum in an ordered monoid   csgns 30266
                  20.3.9.4  The Archimedean property for generic ordered algebraic structures   cinftm 30271
                  20.3.9.5  Semiring left modules   cslmd 30294
                  20.3.9.6  Finitely supported group sums - misc additions   gsumle 30320
                  20.3.9.7  Rings - misc additions   rngurd 30329
                  20.3.9.8  Ordered rings and fields   corng 30336
                  20.3.9.9  Ring homomorphisms - misc additions   rhmdvdsr 30359
                  20.3.9.10  Scalar restriction operation   cresv 30365
                  20.3.9.11  The commutative ring of gaussian integers   gzcrng 30380
                  20.3.9.12  The archimedean ordered field of real numbers   reofld 30381
            20.3.10  Matrices   symgfcoeu 30386
                  20.3.10.1  The symmetric group   symgfcoeu 30386
                  20.3.10.2  Permutation Signs   psgndmfi 30387
                  20.3.10.3  Transpositions   pmtridf1o 30397
                  20.3.10.4  Submatrices   csmat 30400
                  20.3.10.5  Matrix literals   clmat 30418
                  20.3.10.6  Laplace expansion of determinants   mdetpmtr1 30430
            20.3.11  Topology   fvproj 30440
                  20.3.11.1  Open maps   fvproj 30440
                  20.3.11.2  Topology of the unit circle   qtopt1 30443
                  20.3.11.3  Refinements   reff 30447
                  20.3.11.4  Open cover refinement property   ccref 30450
                  20.3.11.5  Lindelöf spaces   cldlf 30460
                  20.3.11.6  Paracompact spaces   cpcmp 30463
                  20.3.11.7  Pseudometrics   cmetid 30470
                  20.3.11.8  Continuity - misc additions   hauseqcn 30482
                  20.3.11.9  Topology of the closed unit interval   unitsscn 30483
                  20.3.11.10  Topology of ` ( RR X. RR ) `   unicls 30490
                  20.3.11.11  Order topology - misc. additions   cnvordtrestixx 30500
                  20.3.11.12  Continuity in topological spaces - misc. additions   mndpluscn 30513
                  20.3.11.13  Topology of the extended nonnegative real numbers ordered monoid   xrge0hmph 30519
                  20.3.11.14  Limits - misc additions   lmlim 30534
                  20.3.11.15  Univariate polynomials   pl1cn 30542
            20.3.12  Uniform Stuctures and Spaces   chcmp 30543
                  20.3.12.1  Hausdorff uniform completion   chcmp 30543
            20.3.13  Topology and algebraic structures   zringnm 30545
                  20.3.13.1  The norm on the ring of the integer numbers   zringnm 30545
                  20.3.13.2  Topological ` ZZ ` -modules   zlm0 30547
                  20.3.13.3  Canonical embedding of the field of the rational numbers into a division ring   cqqh 30557
                  20.3.13.4  Canonical embedding of the real numbers into a complete ordered field   crrh 30578
                  20.3.13.5  Embedding from the extended real numbers into a complete lattice   cxrh 30601
                  20.3.13.6  Canonical embeddings into the ordered field of the real numbers   zrhre 30604
                  ​*20.3.13.7  Topological Manifolds   cmntop 30607
            20.3.14  Real and complex functions   nexple 30612
                  20.3.14.1  Integer powers - misc. additions   nexple 30612
                  20.3.14.2  Indicator Functions   cind 30613
                  20.3.14.3  Extended sum   cesum 30630
            20.3.15  Mixed Function/Constant operation   cofc 30698
            20.3.16  Abstract measure   csiga 30711
                  20.3.16.1  Sigma-Algebra   csiga 30711
                  20.3.16.2  Generated sigma-Algebra   csigagen 30742
                  ​*20.3.16.3  lambda and pi-Systems, Rings of Sets   ispisys 30756
                  20.3.16.4  The Borel algebra on the real numbers   cbrsiga 30785
                  20.3.16.5  Product Sigma-Algebra   csx 30792
                  20.3.16.6  Measures   cmeas 30799
                  20.3.16.7  The counting measure   cntmeas 30830
                  20.3.16.8  The Lebesgue measure - misc additions   voliune 30833
                  20.3.16.9  The Dirac delta measure   cdde 30836
                  20.3.16.10  The 'almost everywhere' relation   cae 30841
                  20.3.16.11  Measurable functions   cmbfm 30853
                  20.3.16.12  Borel Algebra on ` ( RR X. RR ) `   br2base 30872
                  ​*20.3.16.13  Caratheodory's extension theorem   coms 30894
            20.3.17  Integration   itgeq12dv 30929
                  20.3.17.1  Lebesgue integral - misc additions   itgeq12dv 30929
                  20.3.17.2  Bochner integral   citgm 30930
            20.3.18  Euler's partition theorem   oddpwdc 30957
            20.3.19  Sequences defined by strong recursion   csseq 30986
            20.3.20  Fibonacci Numbers   cfib 31000
            20.3.21  Probability   cprb 31011
                  20.3.21.1  Probability Theory   cprb 31011
                  20.3.21.2  Conditional Probabilities   ccprob 31035
                  20.3.21.3  Real-valued Random Variables   crrv 31044
                  20.3.21.4  Preimage set mapping operator   corvc 31059
                  20.3.21.5  Distribution Functions   orvcelval 31072
                  20.3.21.6  Cumulative Distribution Functions   orvclteel 31076
                  20.3.21.7  Probabilities - example   coinfliplem 31082
                  20.3.21.8  Bertrand's Ballot Problem   ballotlemoex 31089
            20.3.22  Signum (sgn or sign) function - misc. additions   sgncl 31142
            20.3.23  Words over a set - misc additions   wrdfd 31158
                  20.3.23.1  Operations on words   ccatmulgnn0dir 31162
            20.3.24  Polynomials with real coefficients - misc additions   plymul02 31166
            20.3.25  Descartes's rule of signs   signspval 31172
                  20.3.25.1  Sign changes in a word over real numbers   signspval 31172
                  20.3.25.2  Counting sign changes in a word over real numbers   signslema 31182
            20.3.26  Number Theory   efcld 31214
                  20.3.26.1  Representations of a number as sums of integers   crepr 31231
                  20.3.26.2  Vinogradov Trigonometric Sums and the Circle Method   cvts 31258
                  20.3.26.3  The Ternary Goldbach Conjecture: Final Statement   ax-hgt749 31267
            20.3.27  Elementary Geometry   cstrkg2d 31287
                  ​*20.3.27.1  Two-dimension geometry   cstrkg2d 31287
                  20.3.27.2  Outer Five Segment (not used, no need to move to main)   cafs 31292
      ​​*20.4  Mathbox for Jonathan Ben-Naim
            20.4.1  First-order logic and set theory   bnj170 31309
            20.4.2  Well founded induction and recursion   bnj110 31470
            20.4.3  The existence of a minimal element in certain classes   bnj69 31620
            20.4.4  Well-founded induction   bnj1204 31622
            20.4.5  Well-founded recursion, part 1 of 3   bnj60 31672
            20.4.6  Well-founded recursion, part 2 of 3   bnj1500 31678
            20.4.7  Well-founded recursion, part 3 of 3   bnj1522 31682
      ​20.5  Mathbox for Mario Carneiro
            20.5.1  Predicate calculus with all distinct variables   ax-7d 31683
            20.5.2  Miscellaneous stuff   quartfull 31689
            20.5.3  Derangements and the Subfactorial   deranglem 31690
            20.5.4  The Erdős-Szekeres theorem   erdszelem1 31715
            20.5.5  The Kuratowski closure-complement theorem   kur14lem1 31730
            20.5.6  Retracts and sections   cretr 31741
            20.5.7  Path-connected and simply connected spaces   cpconn 31743
            20.5.8  Covering maps   ccvm 31779
            20.5.9  Normal numbers   snmlff 31853
            20.5.10  Godel-sets of formulas   cgoe 31857
            20.5.11  Models of ZF   cgze 31885
            ​*20.5.12  Metamath formal systems   cmcn 31899
            20.5.13  Grammatical formal systems   cm0s 32024
            20.5.14  Models of formal systems   cmuv 32042
            20.5.15  Splitting fields   citr 32064
            20.5.16  p-adic number fields   czr 32080
      ​​*20.6  Mathbox for Filip Cernatescu
      ​20.7  Mathbox for Paul Chapman
            20.7.1  Real and complex numbers (cont.)   climuzcnv 32105
            20.7.2  Miscellaneous theorems   elfzm12 32109
      ​20.8  Mathbox for Scott Fenton
            20.8.1  ZFC Axioms in primitive form   axextprim 32118
            20.8.2  Untangled classes   untelirr 32125
            20.8.3  Extra propositional calculus theorems   3orel2 32132
            20.8.4  Misc. Useful Theorems   nepss 32139
            20.8.5  Properties of real and complex numbers   sqdivzi 32150
            20.8.6  Infinite products   iprodefisumlem 32164
            20.8.7  Factorial limits   faclimlem1 32167
            20.8.8  Greatest common divisor and divisibility   pdivsq 32173
            20.8.9  Properties of relationships   brtp 32177
            20.8.10  Properties of functions and mappings   funpsstri 32200
            20.8.11  Epsilon induction   setinds 32216
            20.8.12  Ordinal numbers   elpotr 32219
            20.8.13  Defined equality axioms   axextdfeq 32236
            20.8.14  Hypothesis builders   hbntg 32244
            20.8.15  (Trans)finite Recursion Theorems   tfisg 32249
            20.8.16  Transitive closure under a relationship   ctrpred 32250
            20.8.17  Founded Induction   frpomin 32272
            20.8.18  Ordering Ordinal Sequences   orderseqlem 32286
            20.8.19  Well-founded zero, successor, and limits   cwsuc 32289
            20.8.20  Founded Recursion   cfrecs 32309
            20.8.21  Surreal Numbers   csur 32327
            20.8.22  Surreal Numbers: Ordering   sltsolem1 32360
            20.8.23  Surreal Numbers: Birthday Function   bdayfo 32362
            20.8.24  Surreal Numbers: Density   fvnobday 32363
            20.8.25  Surreal Numbers: Full-Eta Property   bdayimaon 32377
            20.8.26  Surreal numbers - ordering theorems   csle 32403
            20.8.27  Surreal numbers - birthday theorems   bdayfun 32422
            20.8.28  Surreal numbers: Conway cuts   csslt 32430
            20.8.29  Surreal numbers - cuts and options   cmade 32459
            20.8.30  Quantifier-free definitions   ctxp 32471
            20.8.31  Alternate ordered pairs   caltop 32597
            20.8.32  Geometry in the Euclidean space   cofs 32623
                  20.8.32.1  Congruence properties   cofs 32623
                  20.8.32.2  Betweenness properties   btwntriv2 32653
                  20.8.32.3  Segment Transportation   ctransport 32670
                  20.8.32.4  Properties relating betweenness and congruence   cifs 32676
                  20.8.32.5  Connectivity of betweenness   btwnconn1lem1 32728
                  20.8.32.6  Segment less than or equal to   csegle 32747
                  20.8.32.7  Outside-of relationship   coutsideof 32760
                  20.8.32.8  Lines and Rays   cline2 32775
            20.8.33  Forward difference   cfwddif 32799
            20.8.34  Rank theorems   rankung 32807
            20.8.35  Hereditarily Finite Sets   chf 32813
      ​20.9  Mathbox for Jeff Hankins
            20.9.1  Miscellany   a1i14 32828
            20.9.2  Basic topological facts   topbnd 32852
            20.9.3  Topology of the real numbers   ivthALT 32863
            20.9.4  Refinements   cfne 32864
            20.9.5  Neighborhood bases determine topologies   neibastop1 32887
            20.9.6  Lattice structure of topologies   topmtcl 32891
            20.9.7  Filter bases   fgmin 32898
            20.9.8  Directed sets, nets   tailfval 32900
      ​20.10  Mathbox for Anthony Hart
            20.10.1  Propositional Calculus   tb-ax1 32911
            20.10.2  Predicate Calculus   nalfal 32931
            20.10.3  Miscellaneous single axioms   meran1 32938
            20.10.4  Connective Symmetry   negsym1 32944
      ​20.11  Mathbox for Chen-Pang He
            20.11.1  Ordinal topology   ontopbas 32955
      ​20.12  Mathbox for Jeff Hoffman
            20.12.1  Inferences for finite induction on generic function values   fveleq 32978
            20.12.2  gdc.mm   nnssi2 32982
      ​20.13  Mathbox for Asger C. Ipsen
            20.13.1  Continuous nowhere differentiable functions   dnival 32989
      ​​*20.14  Mathbox for BJ
            *20.14.1  Propositional calculus   bj-mp2c 33058
                  *20.14.1.1  Derived rules of inference   bj-mp2c 33058
                  ​*20.14.1.2  A syntactic theorem   bj-0 33060
                  20.14.1.3  Minimal implicational calculus   bj-a1k 33062
                  20.14.1.4  Positive calculus   bj-syl66ib 33066
                  20.14.1.5  Implication and negation   bj-con2com 33071
                  ​*20.14.1.6  Disjunction   bj-jaoi1 33079
                  ​*20.14.1.7  Logical equivalence   bj-dfbi4 33081
                  20.14.1.8  The conditional operator for propositions   bj-consensus 33086
                  ​*20.14.1.9  Propositional calculus: miscellaneous   bj-imbi12 33091
            ​*20.14.2  Modal logic   bj-axdd2 33100
            ​*20.14.3  Provability logic   cprvb 33106
            ​*20.14.4  First-order logic   bj-genr 33115
                  20.14.4.1  Adding ax-gen   bj-genr 33115
                  20.14.4.2  Adding ax-4   bj-2alim 33118
                  20.14.4.3  Adding ax-5   bj-ax12wlem 33144
                  20.14.4.4  Equality and substitution   bj-ssbjustlem 33153
                  20.14.4.5  Adding ax-6   bj-alequexv 33189
                  20.14.4.6  Adding ax-7   bj-cbvexw 33198
                  20.14.4.7  Membership predicate, ax-8 and ax-9   bj-elequ2g 33200
                  20.14.4.8  Adding ax-11   bj-alcomexcom 33204
                  20.14.4.9  Adding ax-12   axc11n11 33206
                  20.14.4.10  Adding ax-13   bj-axc10 33240
                  ​*20.14.4.11  Removing dependencies on ax-13 (and ax-11)   bj-axc10v 33250
                  ​*20.14.4.12  Distinct var metavariables   bj-hbaeb2 33325
                  ​*20.14.4.13  Around ~ equsal   bj-equsal1t 33329
                  ​*20.14.4.14  Some Principia Mathematica proofs   stdpc5t 33334
                  20.14.4.15  Alternate definition of substitution   bj-sbsb 33344
                  20.14.4.16  Lemmas for substitution   bj-sbf3 33346
                  20.14.4.17  Existential uniqueness   bj-eu3f 33349
                  ​*20.14.4.18  First-order logic: miscellaneous   bj-moeub 33350
            20.14.5  Set theory   eliminable1 33360
                  ​*20.14.5.1  Eliminability of class terms   eliminable1 33360
                  ​*20.14.5.2  Classes without extensionality   bj-cleljustab 33367
                  20.14.5.3  Characterization among sets versus among classes   elelb 33401
                  ​*20.14.5.4  The nonfreeness quantifier for classes   bj-nfcsym 33402
                  ​*20.14.5.5  Proposal for the definitions of class membership and class equality   bj-ax8 33403
                  ​*20.14.5.6  Lemmas for class substitution   bj-sbeqALT 33411
                  20.14.5.7  Removing some axiom requirements and disjoint variable conditions   bj-exlimvmpi 33421
                  ​*20.14.5.8  Class abstractions   bj-unrab 33441
                  ​*20.14.5.9  Restricted non-freeness   wrnf 33448
                  ​*20.14.5.10  Russell's paradox   bj-ru0 33450
                  ​*20.14.5.11  Some disjointness results   bj-n0i 33453
                  ​*20.14.5.12  Complements on direct products   bj-xpimasn 33459
                  ​*20.14.5.13  "Singletonization" and tagging   bj-sels 33467
                  ​*20.14.5.14  Tuples of classes   bj-cproj 33495
                  ​*20.14.5.15  Set theory: miscellaneous   bj-disj2r 33530
                  20.14.5.16  Evaluation   bj-evaleq 33542
                  20.14.5.17  Elementwise operations   celwise 33550
                  ​*20.14.5.18  Elementwise intersection (families of sets induced on a subset)   bj-rest00 33552
                  20.14.5.19  Moore collections (complements)   bj-intss 33571
                  20.14.5.20  Maps-to notation for functions with three arguments   bj-0nelmpt 33587
                  ​*20.14.5.21  Currying   csethom 33593
                  ​*20.14.5.22  Setting components of extensible structures   cstrset 33605
            ​*20.14.6  Extended real and complex numbers, real and complex projective lines   bj-elid 33608
                  *20.14.6.1  Identity relation (complements)   bj-elid 33608
                  ​*20.14.6.2  Diagonal in a Cartesian square   cdiag2 33612
                  ​*20.14.6.3  Extended numbers and projective lines as sets   cfractemp 33617
                  ​*20.14.6.4  Addition and opposite   caddcc 33659
                  ​*20.14.6.5  Order relation on the extended reals   cltxr 33663
                  ​*20.14.6.6  Argument, multiplication and inverse   carg 33665
                  20.14.6.7  The canonical bijection from the finite ordinals   ciomnn 33671
            ​*20.14.7  Monoids   bj-cmnssmnd 33683
                  ​*20.14.7.1  Finite sums in monoids   cfinsum 33692
            ​*20.14.8  Affine, Euclidean, and Cartesian geometry   crrvec 33695
                  *20.14.8.1  Convex hull in real vector spaces   crrvec 33695
                  ​*20.14.8.2  Complex numbers (supplements)   bj-subcom 33701
                  ​*20.14.8.3  Barycentric coordinates   bj-bary1lem 33703
      ​20.15  Mathbox for Jim Kingdon
                  20.15.0.1  Circle constant   taupilem3 33706
                  20.15.0.2  Number theory   dfgcd3 33711
      ​20.16  Mathbox for ML
            ​*20.16.1  Cantor normal form up to epsilon 0   cnfin0 33780
      ​20.17  Mathbox for Wolf Lammen
            20.17.1  1. Bootstrapping   wl-section-boot 33786
            20.17.2  Implication chains   wl-section-impchain 33810
            20.17.3  An alternative axiom ~ ax-13   ax-wl-13v 33828
            20.17.4  Other stuff   wl-mps 33830
            20.17.5  1. Bootstrapping classes   wcel-wl 33911
      ​20.18  Mathbox for Brendan Leahy
      ​20.19  Mathbox for Jeff Madsen
            20.19.1  Logic and set theory   anim12da 34044
            20.19.2  Real and complex numbers; integers   filbcmb 34073
            20.19.3  Sequences and sums   sdclem2 34075
            20.19.4  Topology   subspopn 34085
            20.19.5  Metric spaces   metf1o 34088
            20.19.6  Continuous maps and homeomorphisms   constcncf 34095
            20.19.7  Boundedness   ctotbnd 34102
            20.19.8  Isometries   cismty 34134
            20.19.9  Heine-Borel Theorem   heibor1lem 34145
            20.19.10  Banach Fixed Point Theorem   bfplem1 34158
            20.19.11  Euclidean space   crrn 34161
            20.19.12  Intervals (continued)   ismrer1 34174
            20.19.13  Operation properties   cass 34178
            20.19.14  Groups and related structures   cmagm 34184
            20.19.15  Group homomorphism and isomorphism   cghomOLD 34219
            20.19.16  Rings   crngo 34230
            20.19.17  Division Rings   cdrng 34284
            20.19.18  Ring homomorphisms   crnghom 34296
            20.19.19  Commutative rings   ccm2 34325
            20.19.20  Ideals   cidl 34343
            20.19.21  Prime rings and integral domains   cprrng 34382
            20.19.22  Ideal generators   cigen 34395
      ​20.20  Mathbox for Giovanni Mascellani
            ​*20.20.1  Tools for automatic proof building   efald2 34414
            ​*20.20.2  Tseitin axioms   fald 34471
            ​*20.20.3  Equality deductions   iuneq2f 34498
            ​*20.20.4  Miscellanea   orcomdd 34511
      ​20.21  Mathbox for Peter Mazsa
            20.21.1  Notations   cxrn 34518
            20.21.2  Preparatory theorems   el2v 34541
            20.21.3  Range Cartesian product   df-xrn 34676
            20.21.4  Cosets by ` R `   df-coss 34712
            20.21.5  Relations   df-rels 34778
            20.21.6  Subset relations   df-ssr 34791
            20.21.7  Reflexivity   df-refs 34803
            20.21.8  Converse reflexivity   df-cnvrefs 34816
            20.21.9  Symmetry   df-syms 34831
            20.21.10  Reflexivity and symmetry   symrefref2 34852
            20.21.11  Transitivity   df-trs 34861
            20.21.12  Equivalence relations   df-eqvrels 34872
            20.21.13  Redundancy   df-reds 34907
      ​20.22  Mathbox for Rodolfo Medina
            20.22.1  Partitions   prtlem60 34922
      ​​*20.23  Mathbox for Norm Megill
            *20.23.1  Obsolete schemes ax-c4,c5,c7,c10,c11,c11n,c15,c9,c14,c16   ax-c5 34953
            ​*20.23.2  Rederive new axioms ax-4, ax-10, ax-6, ax-12, ax-13 from old   axc5 34963
            ​*20.23.3  Legacy theorems using obsolete axioms   ax5ALT 34977
            20.23.4  Experiments with weak deduction theorem   elimhyps 35031
            20.23.5  Miscellanea   cnaddcom 35042
            20.23.6  Atoms, hyperplanes, and covering in a left vector space (or module)   clsa 35044
            20.23.7  Functionals and kernels of a left vector space (or module)   clfn 35127
            20.23.8  Opposite rings and dual vector spaces   cld 35193
            20.23.9  Ortholattices and orthomodular lattices   cops 35242
            20.23.10  Atomic lattices with covering property   ccvr 35332
            20.23.11  Hilbert lattices   chlt 35420
            20.23.12  Projective geometries based on Hilbert lattices   clln 35561
            20.23.13  Construction of a vector space from a Hilbert lattice   cdlema1N 35861
            20.23.14  Construction of involution and inner product from a Hilbert lattice   clpoN 37550
      ​20.24  Mathbox for Steven Nguyen
            20.24.1  Utility theorems   ioin9i8 38032
            20.24.2  Russell's paradox   cbvabvw 38041
            ​*20.24.3  Arithmetic theorems   c0exALT 38044
            20.24.4  Real subtraction   cresub 38064
            20.24.5  Equivalent formulations of Fermat's Last Theorem   dffltz 38092
      ​20.25  Mathbox for OpenAI
      ​20.26  Mathbox for Stefan O'Rear
            20.26.1  Additional elementary logic and set theory   moxfr 38094
            20.26.2  Additional theory of functions   imaiinfv 38095
            20.26.3  Additional topology   elrfi 38096
            20.26.4  Characterization of closure operators. Kuratowski closure axioms   ismrcd1 38100
            20.26.5  Algebraic closure systems   cnacs 38104
            20.26.6  Miscellanea 1. Map utilities   constmap 38115
            20.26.7  Miscellanea for polynomials   mptfcl 38122
            20.26.8  Multivariate polynomials over the integers   cmzpcl 38123
            20.26.9  Miscellanea for Diophantine sets 1   coeq0i 38155
            20.26.10  Diophantine sets 1: definitions   cdioph 38157
            20.26.11  Diophantine sets 2 miscellanea   ellz1 38169
            20.26.12  Diophantine sets 2: union and intersection. Monotone Boolean algebra   diophin 38175
            20.26.13  Diophantine sets 3: construction   diophrex 38178
            20.26.14  Diophantine sets 4 miscellanea   2sbcrex 38187
            20.26.15  Diophantine sets 4: Quantification   rexrabdioph 38197
            20.26.16  Diophantine sets 5: Arithmetic sets   rabdiophlem1 38204
            20.26.17  Diophantine sets 6: reusability. renumbering of variables   eldioph4b 38214
            20.26.18  Pigeonhole Principle and cardinality helpers   fphpd 38219
            20.26.19  A non-closed set of reals is infinite   rencldnfilem 38223
            20.26.20  Lagrange's rational approximation theorem   irrapxlem1 38225
            20.26.21  Pell equations 1: A nontrivial solution always exists   pellexlem1 38232
            20.26.22  Pell equations 2: Algebraic number theory of the solution set   csquarenn 38239
            20.26.23  Pell equations 3: characterizing fundamental solution   infmrgelbi 38281
            ​*20.26.24  Logarithm laws generalized to an arbitrary base   reglogcl 38293
            20.26.25  Pell equations 4: the positive solution group is infinite cyclic   pellfund14 38301
            20.26.26  X and Y sequences 1: Definition and recurrence laws   crmx 38303
            20.26.27  Ordering and induction lemmas for the integers   monotuz 38344
            20.26.28  X and Y sequences 2: Order properties   rmxypos 38352
            20.26.29  Congruential equations   congtr 38370
            20.26.30  Alternating congruential equations   acongid 38380
            20.26.31  Additional theorems on integer divisibility   coprmdvdsb 38390
            20.26.32  X and Y sequences 3: Divisibility properties   jm2.18 38393
            20.26.33  X and Y sequences 4: Diophantine representability of Y   jm2.27a 38410
            20.26.34  X and Y sequences 5: Diophantine representability of X, ^, _C   rmxdiophlem 38420
            20.26.35  Uncategorized stuff not associated with a major project   setindtr 38429
            20.26.36  More equivalents of the Axiom of Choice   axac10 38438
            20.26.37  Finitely generated left modules   clfig 38475
            20.26.38  Noetherian left modules I   clnm 38483
            20.26.39  Addenda for structure powers   pwssplit4 38497
            20.26.40  Every set admits a group structure iff choice   unxpwdom3 38503
            20.26.41  Noetherian rings and left modules II   clnr 38517
            20.26.42  Hilbert's Basis Theorem   cldgis 38529
            20.26.43  Additional material on polynomials [DEPRECATED]   cmnc 38539
            20.26.44  Degree and minimal polynomial of algebraic numbers   cdgraa 38548
            20.26.45  Algebraic integers I   citgo 38565
            20.26.46  Endomorphism algebra   cmend 38583
            20.26.47  Subfields   csdrg 38603
            20.26.48  Cyclic groups and order   idomrootle 38611
            20.26.49  Cyclotomic polynomials   ccytp 38618
            20.26.50  Miscellaneous topology   fgraphopab 38626
      ​20.27  Mathbox for Jon Pennant
      ​20.28  Mathbox for Richard Penner
            20.28.1  Short Studies   ifpan123g 38640
                  20.28.1.1  Additional work on conditional logical operator   ifpan123g 38640
                  20.28.1.2  Sophisms   rp-fakeimass 38694
                  ​*20.28.1.3  Finite Sets   rp-isfinite5 38699
                  20.28.1.4  Infinite Sets   pwelg 38701
                  ​*20.28.1.5  Finite intersection property   fipjust 38706
                  20.28.1.6  RP ADDTO: Subclasses and subsets   rababg 38715
                  20.28.1.7  RP ADDTO: The intersection of a class   elintabg 38716
                  20.28.1.8  RP ADDTO: Theorems requiring subset and intersection existence   elinintrab 38719
                  20.28.1.9  RP ADDTO: Relations   xpinintabd 38722
                  ​*20.28.1.10  RP ADDTO: Functions   elmapintab 38738
                  ​*20.28.1.11  RP ADDTO: Finite induction (for finite ordinals)   cnvcnvintabd 38742
                  20.28.1.12  RP ADDTO: First and second members of an ordered pair   elcnvlem 38743
                  20.28.1.13  RP ADDTO: The reflexive and transitive properties of relations   undmrnresiss 38746
                  20.28.1.14  RP ADDTO: Basic properties of closures   cleq2lem 38750
                  20.28.1.15  RP REPLACE: Definitions and basic properties of transitive closures   trcleq2lemRP 38773
            20.28.2  Additional statements on relations and subclasses   al3im 38774
                  20.28.2.1  Transitive relations (not to be confused with transitive classes).   trrelind 38793
                  20.28.2.2  Reflexive closures   crcl 38800
                  ​*20.28.2.3  Finite relationship composition.   relexp2 38805
                  20.28.2.4  Transitive closure of a relation   dftrcl3 38848
                  ​*20.28.2.5  Adapted from Frege   frege77d 38874
            ​*20.28.3  Propositions from _Begriffsschrift_   dfxor4 38894
                  *20.28.3.1  _Begriffsschrift_ Chapter I   dfxor4 38894
                  ​*20.28.3.2  _Begriffsschrift_ Notation hints   rp-imass 38900
                  20.28.3.3  _Begriffsschrift_ Chapter II Implication   ax-frege1 38919
                  20.28.3.4  _Begriffsschrift_ Chapter II Implication and Negation   axfrege28 38958
                  ​*20.28.3.5  _Begriffsschrift_ Chapter II with logical equivalence   axfrege52a 38985
                  20.28.3.6  _Begriffsschrift_ Chapter II with equivalence of sets   axfrege52c 39016
                  ​*20.28.3.7  _Begriffsschrift_ Chapter II with equivalence of classes   frege53c 39043
                  ​*20.28.3.8  _Begriffsschrift_ Chapter III Properties hereditary in a sequence   dffrege69 39061
                  ​*20.28.3.9  _Begriffsschrift_ Chapter III Following in a sequence   dffrege76 39068
                  ​*20.28.3.10  _Begriffsschrift_ Chapter III Member of sequence   dffrege99 39091
                  ​*20.28.3.11  _Begriffsschrift_ Chapter III Single-valued procedures   dffrege115 39107
            ​*20.28.4  Exploring Topology via Seifert and Threlfall   enrelmap 39126
                  *20.28.4.1  Equinumerosity of sets of relations and maps   enrelmap 39126
                  ​*20.28.4.2  Generic Pseudoclosure Spaces, Pseudointerior Spaces, and Pseudoneighborhoods   sscon34b 39152
                  ​*20.28.4.3  Generic Neighborhood Spaces   gneispa 39263
            ​*20.28.5  Exploring Higher Homotopy via Kerodon   k0004lem1 39280
                  *20.28.5.1  Simplicial Sets   k0004lem1 39280
      ​20.29  Mathbox for Stanislas Polu
            20.29.1  IMO Problems   wwlemuld 39289
                  20.29.1.1  IMO 1972 B2   wwlemuld 39289
            ​*20.29.2  INT Inequalities Proof Generator   int-addcomd 39311
            ​*20.29.3  N-Digit Addition Proof Generator   unitadd 39333
            20.29.4  AM-GM (for k = 2,3,4)   gsumws3 39334
      ​20.30  Mathbox for Steve Rodriguez
            20.30.1  Miscellanea   nanorxor 39339
            20.30.2  Ratio test for infinite series convergence and divergence   dvgrat 39346
            20.30.3  Multiples   reldvds 39349
            20.30.4  Function operations   caofcan 39357
            20.30.5  Calculus   lhe4.4ex1a 39363
            20.30.6  The generalized binomial coefficient operation   cbcc 39370
            20.30.7  Binomial series   uzmptshftfval 39380
      ​20.31  Mathbox for Andrew Salmon
            20.31.1  Principia Mathematica * 10   pm10.12 39392
            20.31.2  Principia Mathematica * 11   2alanimi 39406
            20.31.3  Predicate Calculus   sbeqal1 39433
            20.31.4  Principia Mathematica * 13 and * 14   pm13.13a 39442
            20.31.5  Set Theory   elnev 39473
            20.31.6  Arithmetic   addcomgi 39493
            20.31.7  Geometry   cplusr 39494
      ​​*20.32  Mathbox for Alan Sare
            20.32.1  Auxiliary theorems for the Virtual Deduction tool   idiALT 39516
            20.32.2  Supplementary unification deductions   bi1imp 39520
            20.32.3  Conventional Metamath proofs, some derived from VD proofs   iidn3 39540
            20.32.4  What is Virtual Deduction?   wvd1 39608
            20.32.5  Virtual Deduction Theorems   df-vd1 39609
            20.32.6  Theorems proved using Virtual Deduction   trsspwALT 39867
            20.32.7  Theorems proved using Virtual Deduction with mmj2 assistance   simplbi2VD 39895
            20.32.8  Virtual Deduction transcriptions of textbook proofs   sb5ALTVD 39962
            20.32.9  Theorems proved using conjunction-form Virtual Deduction   elpwgdedVD 39966
            20.32.10  Theorems with a VD proof in conventional notation derived from a VD proof   suctrALT3 39973
            ​*20.32.11  Theorems with a proof in conventional notation derived from a VD proof   notnotrALT2 39976
      ​20.33  Mathbox for Glauco Siliprandi
            20.33.1  Miscellanea   evth2f 39987
            20.33.2  Functions   unima 40150
            20.33.3  Ordering on real numbers - Real and complex numbers basic operations   sub2times 40279
            20.33.4  Real intervals   gtnelioc 40505
            20.33.5  Finite sums   fsumclf 40590
            20.33.6  Finite multiplication of numbers and finite multiplication of functions   fmul01 40601
            20.33.7  Limits   clim1fr1 40622
                  20.33.7.1  Inferior limit (lim inf)   clsi 40772
                  ​*20.33.7.2  Limits for sequences of extended real numbers   clsxlim 40833
            20.33.8  Trigonometry   coseq0 40864
            20.33.9  Continuous Functions   mulcncff 40870
            20.33.10  Derivatives   dvsinexp 40914
            20.33.11  Integrals   itgsin0pilem1 40954
            20.33.12  Stone Weierstrass theorem - real version   stoweidlem1 41006
            20.33.13  Wallis' product for π   wallispilem1 41070
            20.33.14  Stirling's approximation formula for ` n ` factorial   stirlinglem1 41079
            20.33.15  Dirichlet kernel   dirkerval 41096
            20.33.16  Fourier Series   fourierdlem1 41113
            20.33.17  e is transcendental   elaa2lem 41238
            20.33.18  n-dimensional Euclidean space   rrxtopn 41289
            20.33.19  Basic measure theory   csalg 41313
                  ​*20.33.19.1  σ-Algebras   csalg 41313
                  20.33.19.2  Sum of nonnegative extended reals   csumge0 41364
                  ​*20.33.19.3  Measures   cmea 41451
                  ​*20.33.19.4  Outer measures and Caratheodory's construction   come 41491
                  ​*20.33.19.5  Lebesgue measure on n-dimensional Real numbers   covoln 41538
                  ​*20.33.19.6  Measurable functions   csmblfn 41697
      ​20.34  Mathbox for Saveliy Skresanov
            20.34.1  Ceva's theorem   sigarval 41827
      ​20.35  Mathbox for Jarvin Udandy
      ​20.36  Mathbox for Alexander van der Vekens
            20.36.1  General auxiliary theorems (1)   raaan2 41955
                  20.36.1.1  The empty set - extension   raaan2 41955
                  20.36.1.2  Unordered and ordered pairs - extension for singletons   eusnsn 41956
                  20.36.1.3  Unordered and ordered pairs - extension for unordered pairs   elprneb 41959
                  20.36.1.4  Relations - extension   eubrv 41960
                  20.36.1.5  Definite description binder (inverted iota) - extension   iota0def 41963
                  20.36.1.6  Functions - extension   fveqvfvv 41965
            20.36.2  Alternative for Russell's definition of a description binder   caiota 41974
            20.36.3  Double restricted existential uniqueness   r19.32 41987
                  20.36.3.1  Restricted quantification (extension)   r19.32 41987
                  20.36.3.2  Restricted uniqueness and "at most one" quantification   rmoimi 41995
                  20.36.3.3  Analogs to Existential uniqueness (double quantification)   2reurex 42000
            ​*20.36.4  Alternative definitions of function and operation values   wdfat 42012
                  20.36.4.1  Restricted quantification (extension)   ralbinrald 42018
                  20.36.4.2  The universal class (extension)   nvelim 42019
                  20.36.4.3  Introduce the Axiom of Power Sets (extension)   alneu 42020
                  20.36.4.4  Predicate "defined at"   dfateq12d 42022
                  20.36.4.5  Alternative definition of the value of a function   dfafv2 42028
                  20.36.4.6  Alternative definition of the value of an operation   aoveq123d 42074
            ​*20.36.5  Alternative definitions of function values (2)   cafv2 42104
            20.36.6  General auxiliary theorems (2)   an4com24 42164
                  20.36.6.1  Logical conjunction - extension   an4com24 42164
                  20.36.6.2  Abbreviated conjunction and disjunction of three wff's - extension   3an4ancom24 42165
                  20.36.6.3  Negated membership (alternative)   cnelbr 42167
                  20.36.6.4  Subclasses and subsets - extension   dfss7 42174
                  20.36.6.5  The empty set - extension   ralralimp 42175
                  20.36.6.6  Indexed union and intersection - extension   otiunsndisjX 42176
                  20.36.6.7  Functions - extension   fvifeq 42177
                  20.36.6.8  Maps-to notation - extension   fvmptrab 42189
                  20.36.6.9  Ordering on reals - extension   leltletr 42191
                  20.36.6.10  Subtraction - extension   cnambpcma 42192
                  20.36.6.11  Ordering on reals (cont.) - extension   leaddsuble 42194
                  20.36.6.12  Nonnegative integers (as a subset of complex numbers) - extension   nn0resubcl 42200
                  20.36.6.13  Integers (as a subset of complex numbers) - extension   zgeltp1eq 42201
                  20.36.6.14  Decimal arithmetic - extension   1t10e1p1e11 42202
                  20.36.6.15  Upper sets of integers - extension   eluzge0nn0 42204
                  20.36.6.16  Infinity and the extended real number system (cont.) - extension   nltle2tri 42205
                  20.36.6.17  Finite intervals of integers - extension   ssfz12 42206
                  20.36.6.18  Half-open integer ranges - extension   fzopred 42214
                  20.36.6.19  The modulo (remainder) operation - extension   m1mod0mod1 42221
                  20.36.6.20  The infinite sequence builder "seq"   smonoord 42223
                  20.36.6.21  Finite and infinite sums - extension   fsummsndifre 42224
                  20.36.6.22  Extensible structures - extension   setsidel 42228
            ​*20.36.7  Partitions of real intervals   ciccp 42231
            20.36.8  Shifting functions with an integer range domain   fargshiftfv 42257
            20.36.9  Words over a set (extension)   lswn0 42262
                  20.36.9.1  Last symbol of a word - extension   lswn0 42262
            ​*20.36.10  Proper (unordered) pairs   prpair 42263
            20.36.11  Number theory (extension)   cfmtno 42283
                  ​*20.36.11.1  Fermat numbers   cfmtno 42283
                  ​*20.36.11.2  Mersenne primes   m2prm 42349
                  20.36.11.3  Proth's theorem   modexp2m1d 42373
            ​*20.36.12  Even and odd numbers   ceven 42381
                  20.36.12.1  Definitions and basic properties   ceven 42381
                  20.36.12.2  Alternate definitions using the "divides" relation   dfeven2 42406
                  20.36.12.3  Alternate definitions using the "modulo" operation   dfeven3 42414
                  20.36.12.4  Alternate definitions using the "gcd" operation   iseven5 42420
                  20.36.12.5  Theorems of part 5 revised   zneoALTV 42424
                  20.36.12.6  Theorems of part 6 revised   odd2np1ALTV 42429
                  20.36.12.7  Theorems of AV's mathbox revised   0evenALTV 42443
                  20.36.12.8  Additional theorems   epoo 42456
                  20.36.12.9  Perfect Number Theorem (revised)   perfectALTVlem1 42474
                  ​*20.36.12.10  Goldbach's conjectures   cgbe 42477
            20.36.13  Graph theory (extension)   cgrisom 42550
                  ​*20.36.13.1  Isomorphic graphs   cgrisom 42550
                  20.36.13.2  Loop-free graphs - extension   1hegrlfgr 42574
                  20.36.13.3  Walks - extension   cupwlks 42575
            20.36.14  Set of unordered pairs   sprid 42585
            20.36.15  Monoids (extension)   ovn0dmfun 42625
                  20.36.15.1  Auxiliary theorems   ovn0dmfun 42625
                  20.36.15.2  Magmas and Semigroups (extension)   plusfreseq 42633
                  20.36.15.3  Magma homomorphisms and submagmas   cmgmhm 42638
                  20.36.15.4  Examples and counterexamples for magmas, semigroups and monoids (extension)   opmpt2ismgm 42668
            ​*20.36.16  Magmas and internal binary operations (alternate approach)   ccllaw 42680
                  *20.36.16.1  Laws for internal binary operations   ccllaw 42680
                  ​*20.36.16.2  Internal binary operations   cintop 42693
                  20.36.16.3  Alternative definitions for Magmas and Semigroups   cmgm2 42712
            20.36.17  Categories (extension)   idfusubc0 42726
                  20.36.17.1  Subcategories (extension)   idfusubc0 42726
            20.36.18  Rings (extension)   lmod0rng 42729
                  20.36.18.1  Nonzero rings (extension)   lmod0rng 42729
                  ​*20.36.18.2  Non-unital rings ("rngs")   crng 42735
                  20.36.18.3  Rng homomorphisms   crngh 42746
                  20.36.18.4  Ring homomorphisms (extension)   rhmfn 42779
                  20.36.18.5  Ideals as non-unital rings   lidldomn1 42782
                  20.36.18.6  The non-unital ring of even integers   0even 42792
                  20.36.18.7  A constructed not unital ring   cznrnglem 42814
                  ​*20.36.18.8  The category of non-unital rings   crngc 42818
                  ​*20.36.18.9  The category of (unital) rings   cringc 42864
                  20.36.18.10  Subcategories of the category of rings   srhmsubclem1 42934
            20.36.19  Basic algebraic structures (extension)   xpprsng 42971
                  20.36.19.1  Auxiliary theorems   xpprsng 42971
                  20.36.19.2  The binomial coefficient operation (extension)   bcpascm1 42990
                  20.36.19.3  The ` ZZ `-module ` ZZ X. ZZ `   zlmodzxzlmod 42993
                  20.36.19.4  Ordered group sum operation (extension)   gsumpr 43000
                  20.36.19.5  Symmetric groups (extension)   exple2lt6 43006
                  20.36.19.6  Divisibility (extension)   invginvrid 43009
                  20.36.19.7  The support of functions (extension)   rmsupp0 43010
                  20.36.19.8  Finitely supported functions (extension)   rmsuppfi 43015
                  20.36.19.9  Left modules (extension)   lmodvsmdi 43024
                  20.36.19.10  Associative algebras (extension)   ascl0 43026
                  20.36.19.11  Univariate polynomials (extension)   ply1vr1smo 43030
                  20.36.19.12  Univariate polynomials (examples)   linply1 43042
            20.36.20  Linear algebra (extension)   cdmatalt 43046
                  ​*20.36.20.1  The subalgebras of diagonal and scalar matrices (extension)   cdmatalt 43046
                  ​*20.36.20.2  Linear combinations   clinc 43054
                  ​*20.36.20.3  Linear independence   clininds 43090
                  20.36.20.4  Simple left modules and the ` ZZ `-module   lmod1lem1 43137
                  20.36.20.5  Differences between (left) modules and (left) vector spaces   lvecpsslmod 43157
            20.36.21  Complexity theory   offval0 43160
                  20.36.21.1  Auxiliary theorems   offval0 43160
                  20.36.21.2  The modulo (remainder) operation (extension)   fldivmod 43174
                  20.36.21.3  Even and odd integers   nn0onn0ex 43179
                  20.36.21.4  The natural logarithm on complex numbers (extension)   logcxp0 43190
                  20.36.21.5  Division of functions   cfdiv 43192
                  20.36.21.6  Upper bounds   cbigo 43202
                  20.36.21.7  Logarithm to an arbitrary base (extension)   rege1logbrege0 43213
                  ​*20.36.21.8  The binary logarithm   fldivexpfllog2 43220
                  20.36.21.9  Binary length   cblen 43224
                  ​*20.36.21.10  Digits   cdig 43250
                  20.36.21.11  Nonnegative integer as sum of its shifted digits   dignn0flhalflem1 43270
                  20.36.21.12  Algorithms for the multiplication of nonnegative integers   nn0mulfsum 43279
            20.36.22  Elementary geometry (extension)   fv1prop 43281
                  20.36.22.1  Auxiliary theorems   fv1prop 43281
                  20.36.22.2  Spheres and lines in real Euclidean spaces   cline 43291
      ​20.37  Mathbox for Emmett Weisz
            ​*20.37.1  Miscellaneous Theorems   nfintd 43325
            20.37.2  Set Recursion   csetrecs 43335
                  ​*20.37.2.1  Basic Properties of Set Recursion   csetrecs 43335
                  20.37.2.2  Examples and properties of set recursion   elsetrecslem 43350
            ​*20.37.3  Construction of Games and Surreal Numbers   cpg 43360
      ​​*20.38  Mathbox for David A. Wheeler
            20.38.1  Natural deduction   19.8ad 43366
            ​*20.38.2  Greater than, greater than or equal to.   cge-real 43369
            ​*20.38.3  Hyperbolic trigonometric functions   csinh 43379
            ​*20.38.4  Reciprocal trigonometric functions (sec, csc, cot)   csec 43390
            ​*20.38.5  Identities for "if"   ifnmfalse 43412
            ​*20.38.6  Logarithms generalized to arbitrary base using ` logb `   logb2aval 43413
            ​*20.38.7  Logarithm laws generalized to an arbitrary base - log_   clog- 43414
            ​*20.38.8  Formally define terms such as Reflexivity   wreflexive 43416
            ​*20.38.9  Algebra helpers   comraddi 43420
            ​*20.38.10  Algebra helper examples   i2linesi 43430
            ​*20.38.11  Formal methods "surprises"   alimp-surprise 43432
            ​*20.38.12  Allsome quantifier   walsi 43438
            ​*20.38.13  Miscellaneous   5m4e1 43449
            20.38.14  Theorems about algebraic numbers   aacllem 43453
      ​20.39  Mathbox for Kunhao Zheng
            20.39.1  Weighted AM-GM inequality   amgmwlem 43454