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 880
            ​*1.2.8  Mixed connectives   jaao 984
            ​*1.2.9  The conditional operator for propositions   wif 1091
            ​*1.2.10  The weak deduction theorem for propositional calculus   elimh 1109
            1.2.11  Abbreviated conjunction and disjunction of three wff's   w3o 1112
            1.2.12  Logical "nand" (Sheffer stroke)   wnan 1614
            1.2.13  Logical "xor"   wxo 1639
            1.2.14  True and false constants   wal 1656
                  ​*1.2.14.1  Universal quantifier for use by df-tru   wal 1656
                  ​*1.2.14.2  Equality predicate for use by df-tru   cv 1657
                  1.2.14.3  The true constant   wtru 1659
                  1.2.14.4  The false constant   wfal 1671
            ​*1.2.15  Truth tables   truimtru 1682
                  1.2.15.1  Implication   truimtru 1682
                  1.2.15.2  Negation   nottru 1686
                  1.2.15.3  Equivalence   trubitru 1688
                  1.2.15.4  Conjunction   truantru 1692
                  1.2.15.5  Disjunction   truortru 1696
                  1.2.15.6  Alternative denial   trunantru 1700
                  1.2.15.7  Exclusive disjunction   truxortru 1704
            ​*1.2.16  Half adder and full adder in propositional calculus   whad 1708
                  1.2.16.1  Full adder: sum   whad 1708
                  1.2.16.2  Full adder: carry   wcad 1721
      ​1.3  Other axiomatizations related to classical propositional calculus
            ​*1.3.1  Minimal implicational calculus   minimp 1736
            1.3.2  Derive the Lukasiewicz axioms from Meredith's sole axiom   meredith 1742
            1.3.3  Derive the standard axioms from the Lukasiewicz axioms   luklem1 1759
            ​*1.3.4  Derive Nicod's axiom from the standard axioms   nic-dfim 1770
            1.3.5  Derive the Lukasiewicz axioms from Nicod's axiom   nic-imp 1776
            1.3.6  Derive Nicod's Axiom from Lukasiewicz's First Sheffer Stroke Axiom   lukshef-ax1 1795
            1.3.7  Derive the Lukasiewicz Axioms from the Tarski-Bernays-Wajsberg Axioms   tbw-bijust 1799
            1.3.8  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's First CO Axiom   merco1 1814
            1.3.9  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's Second CO Axiom   merco2 1837
            1.3.10  Derive the Lukasiewicz axioms from the Russell-Bernays Axioms   rb-bijust 1850
            ​*1.3.11  Stoic logic non-modal portion (Chrysippus of Soli)   mptnan 1869
      ​​*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 1880
                  1.4.1.1  Existential quantifier   wex 1880
                  1.4.1.2  Non-freeness predicate   wnf 1884
            1.4.2  Rule scheme ax-gen (Generalization)   ax-gen 1896
            1.4.3  Axiom scheme ax-4 (Quantified Implication)   ax-4 1910
            1.4.4  Axiom scheme ax-5 (Distinctness) - first use of $d   ax-5 2011
            ​*1.4.5  Equality predicate (continued)   weq 2063
            1.4.6  Define proper substitution   wsb 2069
            1.4.7  Axiom scheme ax-6 (Existence)   ax-6 2077
            1.4.8  Axiom scheme ax-7 (Equality)   ax-7 2114
            1.4.9  Membership predicate   wcel 2166
            1.4.10  Axiom scheme ax-8 (Left Equality for Binary Predicate)   ax-8 2168
            1.4.11  Axiom scheme ax-9 (Right Equality for Binary Predicate)   ax-9 2175
            ​*1.4.12  Logical redundancy of ax-10 , ax-11 , ax-12 , ax-13   ax6dgen 2181
      ​​*1.5  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes)
            1.5.1  Axiom scheme ax-10 (Quantified Negation)   ax-10 2194
            1.5.2  Axiom scheme ax-11 (Quantifier Commutation)   ax-11 2209
            1.5.3  Axiom scheme ax-12 (Substitution)   ax-12 2222
            1.5.4  Axiom scheme ax-13 (Quantified Equality)   ax-13 2391
      ​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 3947
                  2.1.13.1  The difference of two classes   dfdif3 3947
                  2.1.13.2  The union of two classes   elun 3980
                  2.1.13.3  The intersection of two classes   elin 4023
                  2.1.13.4  The symmetric difference of two classes   csymdif 4069
                  2.1.13.5  Combinations of difference, union, and intersection of two classes   unabs 4084
                  2.1.13.6  Class abstractions with difference, union, and intersection of two classes   unab 4123
                  2.1.13.7  Restricted uniqueness with difference, union, and intersection   reuss2 4136
            2.1.14  The empty set   c0 4144
            ​*2.1.15  The conditional operator for classes   cif 4306
            ​*2.1.16  The weak deduction theorem for set theory   dedth 4362
            2.1.17  Power classes   cpw 4378
            2.1.18  Unordered and ordered pairs   snjust 4396
            2.1.19  The union of a class   cuni 4658
            2.1.20  The intersection of a class   cint 4697
            2.1.21  Indexed union and intersection   ciun 4740
            2.1.22  Disjointness   wdisj 4841
            2.1.23  Binary relations   wbr 4873
            2.1.24  Ordered-pair class abstractions (class builders)   copab 4935
            2.1.25  Functions in maps-to notation   cmpt 4952
            2.1.26  Transitive classes   wtr 4975
      ​2.2  ZF Set Theory - add the Axiom of Replacement
            2.2.1  Introduce the Axiom of Replacement   ax-rep 4994
            2.2.2  Derive the Axiom of Separation   axsep 5004
            2.2.3  Derive the Null Set Axiom   zfnuleuOLD 5010
            2.2.4  Theorems requiring subset and intersection existence   nalset 5020
            2.2.5  Theorems requiring empty set existence   class2set 5054
      ​2.3  ZF Set Theory - add the Axiom of Power Sets
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 5065
            2.3.2  Derive the Axiom of Pairing   zfpair 5125
            2.3.3  Ordered pair theorem   opnz 5162
            2.3.4  Ordered-pair class abstractions (cont.)   opabid 5208
            2.3.5  Power class of union and intersection   pwin 5244
            2.3.6  The identity relation   cid 5249
            2.3.7  The membership relation (or epsilon relation)   cep 5254
            ​*2.3.8  Partial and complete ordering   wpo 5261
            2.3.9  Founded and well-ordering relations   wfr 5298
            2.3.10  Relations   cxp 5340
            2.3.11  The Predecessor Class   cpred 5919
            2.3.12  Well-founded induction   tz6.26 5951
            2.3.13  Ordinals   word 5962
            2.3.14  Definite description binder (inverted iota)   cio 6084
            2.3.15  Functions   wfun 6117
            2.3.16  Cantor's Theorem   canth 6863
            2.3.17  Restricted iota (description binder)   crio 6865
            2.3.18  Operations   co 6905
            2.3.19  Maps-to notation   mpt2ndm0 7135
            2.3.20  Function operation   cof 7155
            2.3.21  Proper subset relation   crpss 7196
      ​2.4  ZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 7209
            2.4.2  Ordinals (continued)   epweon 7243
            2.4.3  Transfinite induction   tfi 7314
            2.4.4  The natural numbers (i.e., finite ordinals)   com 7326
            2.4.5  Peano's postulates   peano1 7346
            2.4.6  Finite induction (for finite ordinals)   find 7352
            2.4.7  First and second members of an ordered pair   c1st 7426
            ​*2.4.8  The support of functions   csupp 7559
            ​*2.4.9  Special maps-to operations   opeliunxp2f 7601
            2.4.10  Function transposition   ctpos 7616
            2.4.11  Curry and uncurry   ccur 7656
            2.4.12  Undefined values   cund 7663
            2.4.13  Well-founded recursion   cwrecs 7671
            2.4.14  Functions on ordinals; strictly monotone ordinal functions   iunon 7702
            2.4.15  "Strong" transfinite recursion   crecs 7733
            2.4.16  Recursive definition generator   crdg 7771
            2.4.17  Finite recursion   frfnom 7796
            2.4.18  Ordinal arithmetic   c1o 7819
            2.4.19  Natural number arithmetic   nna0 7951
            2.4.20  Equivalence relations and classes   wer 8006
            2.4.21  The mapping operation   cmap 8122
            2.4.22  Infinite Cartesian products   cixp 8175
            2.4.23  Equinumerosity   cen 8219
            2.4.24  Schroeder-Bernstein Theorem   sbthlem1 8339
            2.4.25  Equinumerosity (cont.)   xpf1o 8391
            2.4.26  Pigeonhole Principle   phplem1 8408
            2.4.27  Finite sets   onomeneq 8419
            2.4.28  Finitely supported functions   cfsupp 8544
            2.4.29  Finite intersections   cfi 8585
            2.4.30  Hall's marriage theorem   marypha1lem 8608
            2.4.31  Supremum and infimum   csup 8615
            2.4.32  Ordinal isomorphism, Hartogs's theorem   coi 8683
            2.4.33  Hartogs function, order types, weak dominance   char 8730
      ​2.5  ZF Set Theory - add the Axiom of Regularity
            2.5.1  Introduce the Axiom of Regularity   ax-reg 8766
            2.5.2  Axiom of Infinity equivalents   inf0 8795
      ​2.6  ZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-inf 8812
            2.6.2  Existence of omega (the set of natural numbers)   omex 8817
            2.6.3  Cantor normal form   ccnf 8835
            2.6.4  Transitive closure   trcl 8881
            2.6.5  Rank   cr1 8902
            2.6.6  Scott's trick; collection principle; Hilbert's epsilon   scottex 9025
            2.6.7  Disjoint union   cdju 9038
            2.6.8  Cardinal numbers   ccrd 9074
            2.6.9  Axiom of Choice equivalents   wac 9251
            2.6.10  Cardinal number arithmetic   ccda 9304
            2.6.11  The Ackermann bijection   ackbij2lem1 9356
            2.6.12  Cofinality (without Axiom of Choice)   cflem 9383
            2.6.13  Eight inequivalent definitions of finite set   sornom 9414
            2.6.14  Hereditarily size-limited sets without Choice   itunifval 9553
​​*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 9572
            3.1.2  Introduce the Axiom of Dependent Choice   ax-dc 9583
      ​3.2  ZFC Set Theory - add the Axiom of Choice
            3.2.1  Introduce the Axiom of Choice   ax-ac 9596
            3.2.2  AC equivalents: well-ordering, Zorn's lemma   numthcor 9631
            3.2.3  Cardinal number theorems using Axiom of Choice   cardval 9683
            3.2.4  Cardinal number arithmetic using Axiom of Choice   iunctb 9711
            3.2.5  Cofinality using the Axiom of Choice   alephreg 9719
      ​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 9757
            3.4.2  Derivation of the Axiom of Choice   gchaclem 9815
​​*PART 4  TG (TARSKI-GROTHENDIECK) SET THEORY
      ​4.1  Inaccessibles
            4.1.1  Weakly and strongly inaccessible cardinals   cwina 9819
            4.1.2  Weak universes   cwun 9837
            4.1.3  Tarski classes   ctsk 9885
            4.1.4  Grothendieck universes   cgru 9927
      ​4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
            4.2.1  Introduce the Tarski-Grothendieck Axiom   ax-groth 9960
            4.2.2  Derive the Power Set, Infinity and Choice Axioms   grothpw 9963
            4.2.3  Tarski map function   ctskm 9974
​​*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 9981
            5.1.2  Final derivation of real and complex number postulates   axaddf 10282
            5.1.3  Real and complex number postulates restated as axioms   ax-cnex 10308
      ​5.2  Derive the basic properties from the field axioms
            5.2.1  Some deductions from the field axioms for complex numbers   cnex 10333
            5.2.2  Infinity and the extended real number system   cpnf 10388
            5.2.3  Restate the ordering postulates with extended real "less than"   axlttri 10428
            5.2.4  Ordering on reals   lttr 10433
            5.2.5  Initial properties of the complex numbers   mul12 10521
      ​5.3  Real and complex numbers - basic operations
            5.3.1  Addition   add12 10572
            5.3.2  Subtraction   cmin 10585
            5.3.3  Multiplication   kcnktkm1cn 10785
            5.3.4  Ordering on reals (cont.)   gt0ne0 10817
            5.3.5  Reciprocals   ixi 10981
            5.3.6  Division   cdiv 11009
            5.3.7  Ordering on reals (cont.)   elimgt0 11189
            5.3.8  Completeness Axiom and Suprema   fimaxre 11298
            5.3.9  Imaginary and complex number properties   inelr 11340
            5.3.10  Function operation analogue theorems   ofsubeq0 11347
      ​5.4  Integer sets
            5.4.1  Positive integers (as a subset of complex numbers)   cn 11350
            5.4.2  Principle of mathematical induction   nnind 11370
            ​*5.4.3  Decimal representation of numbers   c2 11406
            ​*5.4.4  Some properties of specific numbers   neg1cn 11472
            5.4.5  Simple number properties   halfcl 11583
            5.4.6  The Archimedean property   nnunb 11614
            5.4.7  Nonnegative integers (as a subset of complex numbers)   cn0 11618
            ​*5.4.8  Extended nonnegative integers   cxnn0 11690
            5.4.9  Integers (as a subset of complex numbers)   cz 11704
            5.4.10  Decimal arithmetic   cdc 11821
            5.4.11  Upper sets of integers   cuz 11968
            5.4.12  Well-ordering principle for bounded-below sets of integers   uzwo3 12066
            5.4.13  Rational numbers (as a subset of complex numbers)   cq 12071
            5.4.14  Existence of the set of complex numbers   rpnnen1lem2 12099
      ​5.5  Order sets
            5.5.1  Positive reals (as a subset of complex numbers)   crp 12112
            5.5.2  Infinity and the extended real number system (cont.)   cxne 12229
            5.5.3  Supremum and infimum on the extended reals   xrsupexmnf 12423
            5.5.4  Real number intervals   cioo 12463
            5.5.5  Finite intervals of integers   cfz 12619
            ​*5.5.6  Finite intervals of nonnegative integers   elfz2nn0 12725
            5.5.7  Half-open integer ranges   cfzo 12760
      ​5.6  Elementary integer functions
            5.6.1  The floor and ceiling functions   cfl 12886
            5.6.2  The modulo (remainder) operation   cmo 12963
            5.6.3  Miscellaneous theorems about integers   om2uz0i 13041
            5.6.4  Strong induction over upper sets of integers   uzsinds 13081
            5.6.5  Finitely supported functions over the nonnegative integers   fsuppmapnn0fiublem 13084
            5.6.6  The infinite sequence builder "seq" - extension   cseq 13095
            5.6.7  Integer powers   cexp 13154
            5.6.8  Ordered pair theorem for nonnegative integers   nn0le2msqi 13347
            5.6.9  Factorial function   cfa 13353
            5.6.10  The binomial coefficient operation   cbc 13382
            5.6.11  The ` # ` (set size) function   chash 13410
                  5.6.11.1  Proper unordered pairs and triples (sets of size 2 and 3)   hashprlei 13539
                  5.6.11.2  Functions with a domain containing at least two different elements   fundmge2nop0 13563
                  5.6.11.3  Finite induction on the size of the first component of a binary relation   hashdifsnp1 13567
      ​​*5.7  Words over a set
            5.7.1  Definitions and basic theorems   cword 13574
            5.7.2  Last symbol of a word   clsw 13622
            5.7.3  Concatenations of words   cconcat 13630
            5.7.4  Singleton words   cs1 13655
            5.7.5  Concatenations with singleton words   ccatws1cl 13676
            5.7.6  Subwords/substrings   csubstr 13700
            5.7.7  Prefixes of a word   cpfx 13749
            5.7.8  Subwords of subwords   swrdswrdlem 13783
            5.7.9  Subwords and concatenations   pfxcctswrd 13793
            5.7.10  Subwords of concatenations   swrdccatfn 13820
            5.7.11  Splicing words (substring replacement)   csplice 13855
            5.7.12  Reversing words   creverse 13874
            5.7.13  Repeated symbol words   creps 13884
            ​*5.7.14  Cyclical shifts of words   ccsh 13904
            5.7.15  Mapping words by a function   wrdco 13952
            5.7.16  Longer string literals   cs2 13962
      ​​*5.8  Reflexive and transitive closures of relations
            5.8.1  The reflexive and transitive properties of relations   coss12d 14090
            5.8.2  Basic properties of closures   cleq1lem 14100
            5.8.3  Definitions and basic properties of transitive closures   ctcl 14103
            5.8.4  Exponentiation of relations   crelexp 14137
            5.8.5  Reflexive-transitive closure as an indexed union   crtrcl 14172
            ​*5.8.6  Principle of transitive induction.   relexpindlem 14180
      ​5.9  Elementary real and complex functions
            5.9.1  The "shift" operation   cshi 14183
            5.9.2  Signum (sgn or sign) function   csgn 14203
            5.9.3  Real and imaginary parts; conjugate   ccj 14213
            5.9.4  Square root; absolute value   csqrt 14350
      ​5.10  Elementary limits and convergence
            5.10.1  Superior limit (lim sup)   clsp 14578
            5.10.2  Limits   cli 14592
            5.10.3  Finite and infinite sums   csu 14793
            5.10.4  The binomial theorem   binomlem 14935
            5.10.5  The inclusion/exclusion principle   incexclem 14942
            5.10.6  Infinite sums (cont.)   isumshft 14945
            5.10.7  Miscellaneous converging and diverging sequences   divrcnv 14958
            5.10.8  Arithmetic series   arisum 14966
            5.10.9  Geometric series   expcnv 14970
            5.10.10  Ratio test for infinite series convergence   cvgrat 14988
            5.10.11  Mertens' theorem   mertenslem1 14989
            5.10.12  Finite and infinite products   prodf 14992
                  5.10.12.1  Product sequences   prodf 14992
                  5.10.12.2  Non-trivial convergence   ntrivcvg 15002
                  5.10.12.3  Complex products   cprod 15008
                  5.10.12.4  Finite products   fprod 15044
                  5.10.12.5  Infinite products   iprodclim 15101
            5.10.13  Falling and Rising Factorial   cfallfac 15107
            5.10.14  Bernoulli polynomials and sums of k-th powers   cbp 15149
      ​5.11  Elementary trigonometry
            5.11.1  The exponential, sine, and cosine functions   ce 15164
                  5.11.1.1  The circle constant (tau = 2 pi)   ctau 15304
            5.11.2  _e is irrational   eirrlem 15306
      ​5.12  Cardinality of real and complex number subsets
            5.12.1  Countability of integers and rationals   xpnnen 15313
            5.12.2  The reals are uncountable   rpnnen2lem1 15317
​​*PART 6  ELEMENTARY NUMBER THEORY
      ​6.1  Elementary properties of divisibility
            6.1.1  Irrationality of square root of 2   sqrt2irrlem 15351
            6.1.2  Some Number sets are chains of proper subsets   nthruc 15355
            6.1.3  The divides relation   cdvds 15357
            ​*6.1.4  Even and odd numbers   evenelz 15434
            6.1.5  The division algorithm   divalglem0 15490
            6.1.6  Bit sequences   cbits 15514
            6.1.7  The greatest common divisor operator   cgcd 15589
            6.1.8  Bézout's identity   bezoutlem1 15629
            6.1.9  Algorithms   nn0seqcvgd 15656
            6.1.10  Euclid's Algorithm   eucalgval2 15667
            ​*6.1.11  The least common multiple   clcm 15674
            ​*6.1.12  Coprimality and Euclid's lemma   coprmgcdb 15735
            6.1.13  Cancellability of congruences   congr 15750
      ​6.2  Elementary prime number theory
            ​*6.2.1  Elementary properties   cprime 15757
            ​*6.2.2  Coprimality and Euclid's lemma (cont.)   coprm 15794
            6.2.3  Properties of the canonical representation of a rational   cnumer 15812
            6.2.4  Euler's theorem   codz 15839
            6.2.5  Arithmetic modulo a prime number   modprm1div 15873
            6.2.6  Pythagorean Triples   coprimeprodsq 15884
            6.2.7  The prime count function   cpc 15912
            6.2.8  Pocklington's theorem   prmpwdvds 15979
            6.2.9  Infinite primes theorem   unbenlem 15983
            6.2.10  Sum of prime reciprocals   prmreclem1 15991
            6.2.11  Fundamental theorem of arithmetic   1arithlem1 15998
            6.2.12  Lagrange's four-square theorem   cgz 16004
            6.2.13  Van der Waerden's theorem   cvdwa 16040
            6.2.14  Ramsey's theorem   cram 16074
            ​*6.2.15  Primorial function   cprmo 16106
            ​*6.2.16  Prime gaps   prmgaplem1 16124
            6.2.17  Decimal arithmetic (cont.)   dec2dvds 16138
            6.2.18  Cyclical shifts of words (cont.)   cshwsidrepsw 16166
            6.2.19  Specific prime numbers   prmlem0 16178
            6.2.20  Very large primes   1259lem1 16203
​PART 7  BASIC STRUCTURES
      ​7.1  Extensible structures
            ​*7.1.1  Basic definitions   cstr 16218
            7.1.2  Slot definitions   cplusg 16305
            7.1.3  Definition of the structure product   crest 16434
            7.1.4  Definition of the structure quotient   cordt 16512
      ​7.2  Moore spaces
            7.2.1  Moore closures   mrcflem 16619
            7.2.2  Independent sets in a Moore system   mrisval 16643
            7.2.3  Algebraic closure systems   isacs 16664
​PART 8  BASIC CATEGORY THEORY
      ​8.1  Categories
            8.1.1  Categories   ccat 16677
            8.1.2  Opposite category   coppc 16723
            8.1.3  Monomorphisms and epimorphisms   cmon 16740
            8.1.4  Sections, inverses, isomorphisms   csect 16756
            ​*8.1.5  Isomorphic objects   ccic 16807
            8.1.6  Subcategories   cssc 16819
            8.1.7  Functors   cfunc 16866
            8.1.8  Full & faithful functors   cful 16914
            8.1.9  Natural transformations and the functor category   cnat 16953
            8.1.10  Initial, terminal and zero objects of a category   cinito 16990
      ​8.2  Arrows (disjointified hom-sets)
            8.2.1  Identity and composition for arrows   cida 17055
      ​8.3  Examples of categories
            8.3.1  The category of sets   csetc 17077
            8.3.2  The category of categories   ccatc 17096
            ​*8.3.3  The category of extensible structures   fncnvimaeqv 17112
      ​8.4  Categorical constructions
            8.4.1  Product of categories   cxpc 17161
            8.4.2  Functor evaluation   cevlf 17202
            8.4.3  Hom functor   chof 17241
​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 17293
            9.2.2  Lattices   clat 17398
            9.2.3  The dual of an ordered set   codu 17481
            9.2.4  Subset order structures   cipo 17504
            9.2.5  Distributive lattices   latmass 17541
            9.2.6  Posets and lattices as relations   cps 17551
            9.2.7  Directed sets, nets   cdir 17581
​PART 10  BASIC ALGEBRAIC STRUCTURES
      ​10.1  Monoids
            ​*10.1.1  Magmas   cplusf 17592
            ​*10.1.2  Identity elements   mgmidmo 17612
            ​*10.1.3  Ordered sums in a magma   gsumvalx 17623
            ​*10.1.4  Semigroups   csgrp 17636
            ​*10.1.5  Definition and basic properties of monoids   cmnd 17647
            10.1.6  Monoid homomorphisms and submonoids   cmhm 17686
            ​*10.1.7  Ordered sums in a monoid   gsumvallem2 17725
            10.1.8  Free monoids   cfrmd 17738
            10.1.9  Examples and counterexamples for magmas, semigroups and monoids   mgm2nsgrplem1 17759
      ​10.2  Groups
            10.2.1  Definition and basic properties   cgrp 17776
            ​*10.2.2  Group multiple operation   cmg 17894
            10.2.3  Subgroups and Quotient groups   csubg 17939
            10.2.4  Elementary theory of group homomorphisms   cghm 18008
            10.2.5  Isomorphisms of groups   cgim 18050
            10.2.6  Group actions   cga 18072
            10.2.7  Centralizers and centers   ccntz 18098
            10.2.8  The opposite group   coppg 18125
            10.2.9  Symmetric groups   csymg 18147
                  ​*10.2.9.1  Definition and basic properties   csymg 18147
                  10.2.9.2  Cayley's theorem   cayleylem1 18182
                  10.2.9.3  Permutations fixing one element   symgfix2 18186
                  ​*10.2.9.4  Transpositions in the symmetric group   cpmtr 18211
                  10.2.9.5  The sign of a permutation   cpsgn 18259
            10.2.10  p-Groups and Sylow groups; Sylow's theorems   cod 18295
            10.2.11  Direct products   clsm 18400
            10.2.12  Free groups   cefg 18470
      ​10.3  Abelian groups
            10.3.1  Definition and basic properties   ccmn 18546
            10.3.2  Cyclic groups   ccyg 18632
            10.3.3  Group sum operation   gsumval3a 18657
            10.3.4  Group sums over (ranges of) integers   fsfnn0gsumfsffz 18732
            10.3.5  Internal direct products   cdprd 18746
            10.3.6  The Fundamental Theorem of Abelian Groups   ablfacrplem 18818
      ​10.4  Rings
            10.4.1  Multiplicative Group   cmgp 18843
            10.4.2  Ring unit   cur 18855
                  10.4.2.1  Semirings   csrg 18859
                  ​*10.4.2.2  The binomial theorem for semirings   srgbinomlem1 18894
            10.4.3  Definition and basic properties of unital rings   crg 18901
            10.4.4  Opposite ring   coppr 18976
            10.4.5  Divisibility   cdsr 18992
            10.4.6  Ring primes   crpm 19066
            10.4.7  Ring homomorphisms   crh 19068
      ​10.5  Division rings and fields
            10.5.1  Definition and basic properties   cdr 19103
            10.5.2  Subrings of a ring   csubrg 19132
            10.5.3  Absolute value (abstract algebra)   cabv 19172
            10.5.4  Star rings   cstf 19199
      ​10.6  Left modules
            10.6.1  Definition and basic properties   clmod 19219
            10.6.2  Subspaces and spans in a left module   clss 19288
            10.6.3  Homomorphisms and isomorphisms of left modules   clmhm 19378
            10.6.4  Subspace sum; bases for a left module   clbs 19433
      ​10.7  Vector spaces
            10.7.1  Definition and basic properties   clvec 19461
      ​10.8  Ideals
            10.8.1  The subring algebra; ideals   csra 19529
            10.8.2  Two-sided ideals and quotient rings   c2idl 19592
            10.8.3  Principal ideal rings. Divisibility in the integers   clpidl 19602
            10.8.4  Nonzero rings and zero rings   cnzr 19618
            10.8.5  Left regular elements. More kinds of rings   crlreg 19640
      ​10.9  Associative algebras
            10.9.1  Definition and basic properties   casa 19670
      ​10.10  Abstract multivariate polynomials
            10.10.1  Definition and basic properties   cmps 19712
            10.10.2  Polynomial evaluation   ces 19864
            ​*10.10.3  Additional definitions for (multivariate) polynomials   cmhp 19897
            ​*10.10.4  Univariate polynomials   cps1 19905
            10.10.5  Univariate polynomial evaluation   ces1 20038
      ​10.11  The complex numbers as an algebraic extensible structure
            10.11.1  Definition and basic properties   cpsmet 20090
            ​*10.11.2  Ring of integers   zring 20178
            10.11.3  Algebraic constructions based on the complex numbers   czrh 20208
            10.11.4  Signs as subgroup of the complex numbers   cnmsgnsubg 20282
            10.11.5  Embedding of permutation signs into a ring   zrhpsgnmhm 20289
            10.11.6  The ordered field of real numbers   crefld 20311
      ​10.12  Generalized pre-Hilbert and Hilbert spaces
            10.12.1  Definition and basic properties   cphl 20331
            10.12.2  Orthocomplements and closed subspaces   cocv 20367
            10.12.3  Orthogonal projection and orthonormal bases   cpj 20407
​​*PART 11  BASIC LINEAR ALGEBRA
      ​11.1  Vectors and free modules
            *11.1.1  Direct sum of left modules   cdsmm 20438
            ​*11.1.2  Free modules   cfrlm 20453
            ​*11.1.3  Standard basis (unit vectors)   cuvc 20488
            ​*11.1.4  Independent sets and families   clindf 20510
            11.1.5  Characterization of free modules   lmimlbs 20542
      ​​*11.2  Matrices
            *11.2.1  The matrix multiplication   cmmul 20556
            ​*11.2.2  Square matrices   cmat 20580
            ​*11.2.3  The matrix algebra   matmulr 20611
            ​*11.2.4  Matrices of dimension 0 and 1   mat0dimbas0 20640
            ​*11.2.5  The subalgebras of diagonal and scalar matrices   cdmat 20662
            ​*11.2.6  Multiplication of a matrix with a "column vector"   cmvmul 20714
            11.2.7  Replacement functions for a square matrix   cmarrep 20730
            11.2.8  Submatrices   csubma 20750
      ​11.3  The determinant
            11.3.1  Definition and basic properties   cmdat 20758
            11.3.2  Determinants of 2 x 2 -matrices   m2detleiblem1 20798
            11.3.3  The matrix adjugate/adjunct   cmadu 20806
            ​*11.3.4  Laplace expansion of determinants (special case)   symgmatr01lem 20828
            11.3.5  Inverse matrix   invrvald 20851
            ​*11.3.6  Cramer's rule   slesolvec 20854
      ​​*11.4  Polynomial matrices
            11.4.1  Basic properties   pmatring 20868
            ​*11.4.2  Constant polynomial matrices   ccpmat 20878
            ​*11.4.3  Collecting coefficients of polynomial matrices   cdecpmat 20937
            ​*11.4.4  Ring isomorphism between polynomial matrices and polynomials over matrices   cpm2mp 20967
      ​​*11.5  The characteristic polynomial
            *11.5.1  Definition and basic properties   cchpmat 21001
            ​*11.5.2  The characteristic factor function G   fvmptnn04if 21024
            ​*11.5.3  The Cayley-Hamilton theorem   cpmadurid 21042
​PART 12  BASIC TOPOLOGY
      ​12.1  Topology
            ​*12.1.1  Topological spaces   ctop 21068
                  12.1.1.1  Topologies   ctop 21068
                  12.1.1.2  Topologies on sets   ctopon 21085
                  12.1.1.3  Topological spaces   ctps 21107
            12.1.2  Topological bases   ctb 21120
            12.1.3  Examples of topologies   distop 21170
            12.1.4  Closure and interior   ccld 21191
            12.1.5  Neighborhoods   cnei 21272
            12.1.6  Limit points and perfect sets   clp 21309
            12.1.7  Subspace topologies   restrcl 21332
            12.1.8  Order topology   ordtbaslem 21363
            12.1.9  Limits and continuity in topological spaces   ccn 21399
            12.1.10  Separated spaces: T0, T1, T2 (Hausdorff) ...   ct0 21481
            12.1.11  Compactness   ccmp 21560
            12.1.12  Bolzano-Weierstrass theorem   bwth 21584
            12.1.13  Connectedness   cconn 21585
            12.1.14  First- and second-countability   c1stc 21611
            12.1.15  Local topological properties   clly 21638
            12.1.16  Refinements   cref 21676
            12.1.17  Compactly generated spaces   ckgen 21707
            12.1.18  Product topologies   ctx 21734
            12.1.19  Continuous function-builders   cnmptid 21835
            12.1.20  Quotient maps and quotient topology   ckq 21867
            12.1.21  Homeomorphisms   chmeo 21927
      ​12.2  Filters and filter bases
            12.2.1  Filter bases   elmptrab 22001
            12.2.2  Filters   cfil 22019
            12.2.3  Ultrafilters   cufil 22073
            12.2.4  Filter limits   cfm 22107
            12.2.5  Extension by continuity   ccnext 22233
            12.2.6  Topological groups   ctmd 22244
            12.2.7  Infinite group sum on topological groups   ctsu 22299
            12.2.8  Topological rings, fields, vector spaces   ctrg 22329
      ​12.3  Uniform Structures and Spaces
            12.3.1  Uniform structures   cust 22373
            12.3.2  The topology induced by an uniform structure   cutop 22404
            12.3.3  Uniform Spaces   cuss 22427
            12.3.4  Uniform continuity   cucn 22449
            12.3.5  Cauchy filters in uniform spaces   ccfilu 22460
            12.3.6  Complete uniform spaces   ccusp 22471
      ​12.4  Metric spaces
            12.4.1  Pseudometric spaces   ispsmet 22479
            12.4.2  Basic metric space properties   cxms 22492
            12.4.3  Metric space balls   blfvalps 22558
            12.4.4  Open sets of a metric space   mopnval 22613
            12.4.5  Continuity in metric spaces   metcnp3 22715
            12.4.6  The uniform structure generated by a metric   metuval 22724
            12.4.7  Examples of metric spaces   dscmet 22747
            ​*12.4.8  Normed algebraic structures   cnm 22751
            12.4.9  Normed space homomorphisms (bounded linear operators)   cnmo 22879
            12.4.10  Topology on the reals   qtopbaslem 22932
            12.4.11  Topological definitions using the reals   cii 23048
            12.4.12  Path homotopy   chtpy 23136
            12.4.13  The fundamental group   cpco 23169
      ​12.5  Metric subcomplex vector spaces
            12.5.1  Subcomplex modules   cclm 23231
            ​*12.5.2  Subcomplex vector spaces   ccvs 23292
            ​*12.5.3  Normed subcomplex vector spaces   isncvsngp 23318
            12.5.4  Subcomplex pre-Hilbert space   ccph 23335
            12.5.5  Convergence and completeness   ccfil 23420
            12.5.6  Baire's Category Theorem   bcthlem1 23492
            12.5.7  Banach spaces and subcomplex Hilbert spaces   ccms 23500
                  12.5.7.1  The complete ordered field of the real numbers   retopn 23547
            12.5.8  Euclidean spaces   crrx 23551
            12.5.9  Minimizing Vector Theorem   minveclem1 23592
            12.5.10  Projection Theorem   pjthlem1 23605
​PART 13  BASIC REAL AND COMPLEX ANALYSIS
      ​13.1  Continuity
            13.1.1  Intermediate value theorem   pmltpclem1 23614
      ​13.2  Integrals
            13.2.1  Lebesgue measure   covol 23628
            13.2.2  Lebesgue integration   cmbf 23780
                  13.2.2.1  Lesbesgue integral   cmbf 23780
                  13.2.2.2  Lesbesgue directed integral   cdit 24009
      ​13.3  Derivatives
            13.3.1  Real and complex differentiation   climc 24025
                  13.3.1.1  Derivatives of functions of one complex or real variable   climc 24025
                  13.3.1.2  Results on real differentiation   dvferm1lem 24146
​PART 14  BASIC REAL AND COMPLEX FUNCTIONS
      ​14.1  Polynomials
            14.1.1  Polynomial degrees   cmdg 24212
            14.1.2  The division algorithm for univariate polynomials   cmn1 24284
            14.1.3  Elementary properties of complex polynomials   cply 24339
            14.1.4  The division algorithm for polynomials   cquot 24444
            14.1.5  Algebraic numbers   caa 24468
            14.1.6  Liouville's approximation theorem   aalioulem1 24486
      ​14.2  Sequences and series
            14.2.1  Taylor polynomials and Taylor's theorem   ctayl 24506
            14.2.2  Uniform convergence   culm 24529
            14.2.3  Power series   pserval 24563
      ​14.3  Basic trigonometry
            14.3.1  The exponential, sine, and cosine functions (cont.)   efcn 24596
            14.3.2  Properties of pi = 3.14159...   pilem1 24604
            14.3.3  Mapping of the exponential function   efgh 24687
            14.3.4  The natural logarithm on complex numbers   clog 24700
            ​*14.3.5  Logarithms to an arbitrary base   clogb 24904
            14.3.6  Theorems of Pythagoras, isosceles triangles, and intersecting chords   angval 24941
            14.3.7  Solutions of quadratic, cubic, and quartic equations   quad2 24979
            14.3.8  Inverse trigonometric functions   casin 25002
            14.3.9  The Birthday Problem   log2ublem1 25086
            14.3.10  Areas in R^2   carea 25095
            14.3.11  More miscellaneous converging sequences   rlimcnp 25105
            14.3.12  Inequality of arithmetic and geometric means   cvxcl 25124
            14.3.13  Euler-Mascheroni constant   cem 25131
            14.3.14  Zeta function   czeta 25152
            14.3.15  Gamma function   clgam 25155
      ​14.4  Basic number theory
            14.4.1  Wilson's theorem   wilthlem1 25207
            14.4.2  The Fundamental Theorem of Algebra   ftalem1 25212
            14.4.3  The Basel problem (ζ(2) = π²/6)   basellem1 25220
            14.4.4  Number-theoretical functions   ccht 25230
            14.4.5  Perfect Number Theorem   mersenne 25365
            14.4.6  Characters of Z/nZ   cdchr 25370
            14.4.7  Bertrand's postulate   bcctr 25413
            ​*14.4.8  Quadratic residues and the Legendre symbol   clgs 25432
            ​*14.4.9  Gauss' Lemma   gausslemma2dlem0a 25494
            14.4.10  Quadratic reciprocity   lgseisenlem1 25513
            14.4.11  All primes 4n+1 are the sum of two squares   2sqlem1 25555
            14.4.12  Chebyshev's Weak Prime Number Theorem, Dirichlet's Theorem   chebbnd1lem1 25571
            14.4.13  The Prime Number Theorem   mudivsum 25632
            14.4.14  Ostrowski's theorem   abvcxp 25717
​​*PART 15  ELEMENTARY GEOMETRY
      ​15.1  Definition and Tarski's Axioms of Geometry
            15.1.1  Justification for the congruence notation   tgjustf 25785
      ​15.2  Tarskian Geometry
            15.2.1  Congruence   tgcgrcomimp 25789
            15.2.2  Betweenness   tgbtwntriv2 25799
            15.2.3  Dimension   tglowdim1 25812
            15.2.4  Betweenness and Congruence   tgifscgr 25820
            15.2.5  Congruence of a series of points   ccgrg 25822
            15.2.6  Motions   cismt 25844
            15.2.7  Colinearity   tglng 25858
            15.2.8  Connectivity of betweenness   tgbtwnconn1lem1 25884
            15.2.9  Less-than relation in geometric congruences   cleg 25894
            15.2.10  Rays   chlg 25912
            15.2.11  Lines   btwnlng1 25931
            15.2.12  Point inversions   cmir 25964
            15.2.13  Right angles   crag 26005
            15.2.14  Half-planes   islnopp 26048
            15.2.15  Midpoints and Line Mirroring   cmid 26081
            15.2.16  Congruence of angles   ccgra 26116
            15.2.17  Angle Comparisons   cinag 26144
            15.2.18  Congruence Theorems   tgsas1 26153
            15.2.19  Equilateral triangles   ceqlg 26164
      ​15.3  Properties of geometries
            15.3.1  Isomorphisms between geometries   f1otrgds 26168
      ​15.4  Geometry in Hilbert spaces
            15.4.1  Geometry in the complex plane   cchhllem 26186
            15.4.2  Geometry in Euclidean spaces   cee 26187
                  15.4.2.1  Definition of the Euclidean space   cee 26187
                  15.4.2.2  Tarski's axioms for geometry for the Euclidean space   axdimuniq 26212
                  15.4.2.3  EE^n fulfills Tarski's Axioms   ceeng 26276
​​*PART 16  GRAPH THEORY
      ​*16.1  Vertices and edges
            16.1.1  The edge function extractor for extensible structures   cedgf 26287
            ​*16.1.2  Vertices and indexed edges   cvtx 26294
                  16.1.2.1  Definitions and basic properties   cvtx 26294
                  16.1.2.2  The vertices and edges of a graph represented as ordered pair   opvtxval 26301
                  16.1.2.3  The vertices and edges of a graph represented as extensible structure   funvtxdmge2val 26309
                  16.1.2.4  Representations of graphs without edges   snstrvtxval 26335
                  16.1.2.5  Degenerated cases of representations of graphs   vtxval0 26337
            16.1.3  Edges as range of the edge function   cedg 26345
      ​​*16.2  Undirected graphs
            16.2.1  Undirected hypergraphs   cuhgr 26354
            16.2.2  Undirected pseudographs and multigraphs   cupgr 26378
            ​*16.2.3  Loop-free graphs   umgrislfupgrlem 26420
            16.2.4  Edges as subsets of vertices of graphs   uhgredgiedgb 26424
            ​*16.2.5  Undirected simple graphs   cuspgr 26447
            16.2.6  Examples for graphs   usgr0e 26533
            16.2.7  Subgraphs   csubgr 26564
            16.2.8  Finite undirected simple graphs   cfusgr 26613
            16.2.9  Neighbors, complete graphs and universal vertices   cnbgr 26629
                  16.2.9.1  Neighbors   cnbgr 26629
                  16.2.9.2  Universal vertices   cuvtx 26683
                  16.2.9.3  Complete graphs   ccplgr 26707
            16.2.10  Vertex degree   cvtxdg 26763
            ​*16.2.11  Regular graphs   crgr 26853
      ​​*16.3  Walks, paths and cycles
            *16.3.1  Walks   cewlks 26893
            16.3.2  Walks for loop-free graphs   lfgrwlkprop 26988
            16.3.3  Trails   ctrls 26991
            16.3.4  Paths and simple paths   cpths 27014
            16.3.5  Closed walks   cclwlks 27072
            16.3.6  Circuits and cycles   ccrcts 27086
            ​*16.3.7  Walks as words   cwwlks 27124
            16.3.8  Walks/paths of length 2 (as length 3 strings)   2wlkdlem1 27254
            16.3.9  Walks in regular graphs   rusgrnumwwlkl1 27297
            ​*16.3.10  Closed walks as words   cclwwlk 27310
                  16.3.10.1  Closed walks as words   cclwwlk 27310
                  16.3.10.2  Closed walks of a fixed length as words   cclwwlkn 27362
                  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 27490
            16.3.12  Connected graphs   cconngr 27562
      ​16.4  Eulerian paths and the Konigsberg Bridge problem
            ​*16.4.1  Eulerian paths   ceupth 27573
            ​*16.4.2  The Königsberg Bridge problem   konigsbergvtx 27625
      ​16.5  The Friendship Theorem
            16.5.1  Friendship graphs - basics   cfrgr 27637
            16.5.2  The friendship theorem for small graphs   frgr1v 27652
            16.5.3  Theorems according to Mertzios and Unger   2pthfrgrrn 27663
            ​*16.5.4  Huneke's Proof of the Friendship Theorem   frgrncvvdeqlem1 27680
​PART 17  GUIDES AND MISCELLANEA
      ​17.1  Guides (conventions, explanations, and examples)
            ​*17.1.1  Conventions   conventions 27815
            17.1.2  Natural deduction   natded 27818
            ​*17.1.3  Natural deduction examples   ex-natded5.2 27819
            17.1.4  Definitional examples   ex-or 27836
            17.1.5  Other examples   aevdemo 27875
      ​17.2  Humor
            17.2.1  April Fool's theorem   avril1 27877
      ​17.3  (Future - to be reviewed and classified)
            17.3.1  Planar incidence geometry   cplig 27884
            ​*17.3.2  Aliases kept to prevent broken links   dummylink 27897
​​*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 27899
            18.1.2  Abelian groups   cablo 27954
      ​18.2  Complex vector spaces
            18.2.1  Definition and basic properties   cvc 27968
            18.2.2  Examples of complex vector spaces   cnaddabloOLD 27991
      ​18.3  Normed complex vector spaces
            18.3.1  Definition and basic properties   cnv 27994
            18.3.2  Examples of normed complex vector spaces   cnnv 28087
            18.3.3  Induced metric of a normed complex vector space   imsval 28095
            18.3.4  Inner product   cdip 28110
            18.3.5  Subspaces   css 28131
      ​18.4  Operators on complex vector spaces
            18.4.1  Definitions and basic properties   clno 28150
      ​18.5  Inner product (pre-Hilbert) spaces
            18.5.1  Definition and basic properties   ccphlo 28222
            18.5.2  Examples of pre-Hilbert spaces   cncph 28229
            18.5.3  Properties of pre-Hilbert spaces   isph 28232
      ​18.6  Complex Banach spaces
            18.6.1  Definition and basic properties   ccbn 28273
            18.6.2  Examples of complex Banach spaces   cnbn 28280
            18.6.3  Uniform Boundedness Theorem   ubthlem1 28281
            18.6.4  Minimizing Vector Theorem   minvecolem1 28285
      ​18.7  Complex Hilbert spaces
            18.7.1  Definition and basic properties   chlo 28296
            18.7.2  Standard axioms for a complex Hilbert space   hlex 28309
            18.7.3  Examples of complex Hilbert spaces   cnchl 28327
            18.7.4  Subspaces   ssphlOLD 28328
            18.7.5  Hellinger-Toeplitz Theorem   htthlem 28329
​​*PART 19  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      ​19.1  Axiomatization of complex pre-Hilbert spaces
            19.1.1  Basic Hilbert space definitions   chba 28331
            19.1.2  Preliminary ZFC lemmas   df-hnorm 28380
            ​*19.1.3  Derive the Hilbert space axioms from ZFC set theory   axhilex-zf 28393
            ​*19.1.4  Introduce the vector space axioms for a Hilbert space   ax-hilex 28411
            19.1.5  Vector operations   hvmulex 28423
            19.1.6  Inner product postulates for a Hilbert space   ax-hfi 28491
      ​19.2  Inner product and norms
            19.2.1  Inner product   his5 28498
            19.2.2  Norms   dfhnorm2 28534
            19.2.3  Relate Hilbert space to normed complex vector spaces   hilablo 28572
            19.2.4  Bunjakovaskij-Cauchy-Schwarz inequality   bcsiALT 28591
      ​19.3  Cauchy sequences and completeness axiom
            19.3.1  Cauchy sequences and limits   hcau 28596
            19.3.2  Derivation of the completeness axiom from ZF set theory   hilmet 28606
            19.3.3  Completeness postulate for a Hilbert space   ax-hcompl 28614
            19.3.4  Relate Hilbert space to ZFC pre-Hilbert and Hilbert spaces   hhcms 28615
      ​19.4  Subspaces and projections
            19.4.1  Subspaces   df-sh 28619
            19.4.2  Closed subspaces   df-ch 28633
            19.4.3  Orthocomplements   df-oc 28664
            19.4.4  Subspace sum, span, lattice join, lattice supremum   df-shs 28722
            19.4.5  Projection theorem   pjhthlem1 28805
            19.4.6  Projectors   df-pjh 28809
      ​19.5  Properties of Hilbert subspaces
            19.5.1  Orthomodular law   omlsilem 28816
            19.5.2  Projectors (cont.)   pjhtheu2 28830
            19.5.3  Hilbert lattice operations   sh0le 28854
            19.5.4  Span (cont.) and one-dimensional subspaces   spansn0 28955
            19.5.5  Commutes relation for Hilbert lattice elements   df-cm 28997
            19.5.6  Foulis-Holland theorem   fh1 29032
            19.5.7  Quantum Logic Explorer axioms   qlax1i 29041
            19.5.8  Orthogonal subspaces   chscllem1 29051
            19.5.9  Orthoarguesian laws 5OA and 3OA   5oalem1 29068
            19.5.10  Projectors (cont.)   pjorthi 29083
            19.5.11  Mayet's equation E_3   mayete3i 29142
      ​19.6  Operators on Hilbert spaces
            ​*19.6.1  Operator sum, difference, and scalar multiplication   df-hosum 29144
            19.6.2  Zero and identity operators   df-h0op 29162
            19.6.3  Operations on Hilbert space operators   hoaddcl 29172
            19.6.4  Linear, continuous, bounded, Hermitian, unitary operators and norms   df-nmop 29253
            19.6.5  Linear and continuous functionals and norms   df-nmfn 29259
            19.6.6  Adjoint   df-adjh 29263
            19.6.7  Dirac bra-ket notation   df-bra 29264
            19.6.8  Positive operators   df-leop 29266
            19.6.9  Eigenvectors, eigenvalues, spectrum   df-eigvec 29267
            19.6.10  Theorems about operators and functionals   nmopval 29270
            19.6.11  Riesz lemma   riesz3i 29476
            19.6.12  Adjoints (cont.)   cnlnadjlem1 29481
            19.6.13  Quantum computation error bound theorem   unierri 29518
            19.6.14  Dirac bra-ket notation (cont.)   branmfn 29519
            19.6.15  Positive operators (cont.)   leopg 29536
            19.6.16  Projectors as operators   pjhmopi 29560
      ​19.7  States on a Hilbert lattice and Godowski's equation
            19.7.1  States on a Hilbert lattice   df-st 29625
            19.7.2  Godowski's equation   golem1 29685
      ​19.8  Cover relation, atoms, exchange axiom, and modular symmetry
            19.8.1  Covers relation; modular pairs   df-cv 29693
            19.8.2  Atoms   df-at 29752
            19.8.3  Superposition principle   superpos 29768
            19.8.4  Atoms, exchange and covering properties, atomicity   chcv1 29769
            19.8.5  Irreducibility   chirredlem1 29804
            19.8.6  Atoms (cont.)   atcvat3i 29810
            19.8.7  Modular symmetry   mdsymlem1 29817
​PART 20  SUPPLEMENTARY MATERIAL (USER'S MATHBOXES)
      ​20.1  Mathboxes for user contributions
            20.1.1  Mathbox guidelines   mathbox 29856
      ​20.2  Mathbox for Stefan Allan
      ​20.3  Mathbox for Thierry Arnoux
            20.3.1  Propositional Calculus - misc additions   bian1d 29861
            20.3.2  Predicate Calculus   spc2ed 29867
                  20.3.2.1  Predicate Calculus - misc additions   spc2ed 29867
                  20.3.2.2  Restricted quantification - misc additions   ralcom4f 29871
                  20.3.2.3  Substitution (without distinct variables) - misc additions   sbceqbidf 29876
                  20.3.2.4  Existential "at most one" - misc additions   moel 29878
                  20.3.2.5  Existential uniqueness - misc additions   2reuswap2 29881
                  20.3.2.6  Restricted "at most one" - misc additions   rmoxfrd 29885
            20.3.3  General Set Theory   difrab2 29887
                  20.3.3.1  Class abstractions (a.k.a. class builders)   difrab2 29887
                  20.3.3.2  Image Sets   abrexdomjm 29893
                  20.3.3.3  Set relations and operations - misc additions   rabss3d 29899
                  20.3.3.4  Unordered pairs   elpreq 29908
                  20.3.3.5  Conditional operator - misc additions   ifeqeqx 29909
                  20.3.3.6  Set union   uniinn0 29914
                  20.3.3.7  Indexed union - misc additions   cbviunf 29920
                  20.3.3.8  Disjointness - misc additions   disjnf 29931
            20.3.4  Relations and Functions   xpdisjres 29958
                  20.3.4.1  Relations - misc additions   xpdisjres 29958
                  20.3.4.2  Functions - misc additions   ac6sf2 29978
                  20.3.4.3  Operations - misc additions   mpt2mptxf 30025
                  20.3.4.4  Isomorphisms - misc. add.   gtiso 30026
                  20.3.4.5  Disjointness (additional proof requiring functions)   disjdsct 30028
                  20.3.4.6  First and second members of an ordered pair - misc additions   df1stres 30029
                  20.3.4.7  Supremum - misc additions   supssd 30035
                  20.3.4.8  Finite Sets   imafi2 30037
                  20.3.4.9  Countable Sets   snct 30039
            20.3.5  Real and Complex Numbers   subeqxfrd 30058
                  20.3.5.1  Complex operations - misc. additions   subeqxfrd 30058
                  20.3.5.2  Ordering on reals - misc additions   lt2addrd 30063
                  20.3.5.3  Extended reals - misc additions   xrlelttric 30064
                  20.3.5.4  Real number intervals - misc additions   joiniooico 30083
                  20.3.5.5  Finite intervals of integers - misc additions   uzssico 30093
                  20.3.5.6  Half-open integer ranges - misc additions   iundisjfi 30102
                  20.3.5.7  The ` # ` (set size) function - misc additions   hashunif 30109
                  20.3.5.8  The greatest common divisor operator - misc. add   numdenneg 30110
                  20.3.5.9  Integers   nnindf 30112
                  20.3.5.10  Decimal numbers   dfdec100 30123
            ​*20.3.6  Decimal expansion   cdp2 30124
                  ​*20.3.6.1  Decimal point   cdp 30141
                  20.3.6.2  Division in the extended real number system   cxdiv 30170
            20.3.7  Prime Number Theory   bhmafibid1 30189
                  20.3.7.1  Fermat's two square theorem   bhmafibid1 30189
            20.3.8  Extensible Structures   ressplusf 30195
                  20.3.8.1  Structure restriction operator   ressplusf 30195
                  20.3.8.2  The opposite group   oppgle 30198
                  20.3.8.3  Posets   ressprs 30200
                  20.3.8.4  Complete lattices   clatp0cl 30216
                  20.3.8.5  Extended reals Structure - misc additions   ax-xrssca 30218
                  20.3.8.6  The extended nonnegative real numbers commutative monoid   xrge0base 30230
            20.3.9  Algebra   abliso 30241
                  20.3.9.1  Monoids Homomorphisms   abliso 30241
                  20.3.9.2  Ordered monoids and groups   comnd 30242
                  20.3.9.3  Signum in an ordered monoid   csgns 30270
                  20.3.9.4  The Archimedean property for generic ordered algebraic structures   cinftm 30275
                  20.3.9.5  Semiring left modules   cslmd 30298
                  20.3.9.6  Finitely supported group sums - misc additions   gsumle 30324
                  20.3.9.7  Rings - misc additions   rngurd 30333
                  20.3.9.8  Ordered rings and fields   corng 30340
                  20.3.9.9  Ring homomorphisms - misc additions   rhmdvdsr 30363
                  20.3.9.10  Scalar restriction operation   cresv 30369
                  20.3.9.11  The commutative ring of gaussian integers   gzcrng 30384
                  20.3.9.12  The archimedean ordered field of real numbers   reofld 30385
            20.3.10  Matrices   symgfcoeu 30390
                  20.3.10.1  The symmetric group   symgfcoeu 30390
                  20.3.10.2  Permutation Signs   psgndmfi 30391
                  20.3.10.3  Transpositions   pmtridf1o 30401
                  20.3.10.4  Submatrices   csmat 30404
                  20.3.10.5  Matrix literals   clmat 30422
                  20.3.10.6  Laplace expansion of determinants   mdetpmtr1 30434
            20.3.11  Topology   fvproj 30444
                  20.3.11.1  Open maps   fvproj 30444
                  20.3.11.2  Topology of the unit circle   qtopt1 30447
                  20.3.11.3  Refinements   reff 30451
                  20.3.11.4  Open cover refinement property   ccref 30454
                  20.3.11.5  Lindelöf spaces   cldlf 30464
                  20.3.11.6  Paracompact spaces   cpcmp 30467
                  20.3.11.7  Pseudometrics   cmetid 30474
                  20.3.11.8  Continuity - misc additions   hauseqcn 30486
                  20.3.11.9  Topology of the closed unit interval   unitsscn 30487
                  20.3.11.10  Topology of ` ( RR X. RR ) `   unicls 30494
                  20.3.11.11  Order topology - misc. additions   cnvordtrestixx 30504
                  20.3.11.12  Continuity in topological spaces - misc. additions   mndpluscn 30517
                  20.3.11.13  Topology of the extended nonnegative real numbers ordered monoid   xrge0hmph 30523
                  20.3.11.14  Limits - misc additions   lmlim 30538
                  20.3.11.15  Univariate polynomials   pl1cn 30546
            20.3.12  Uniform Stuctures and Spaces   chcmp 30547
                  20.3.12.1  Hausdorff uniform completion   chcmp 30547
            20.3.13  Topology and algebraic structures   zringnm 30549
                  20.3.13.1  The norm on the ring of the integer numbers   zringnm 30549
                  20.3.13.2  Topological ` ZZ ` -modules   zlm0 30551
                  20.3.13.3  Canonical embedding of the field of the rational numbers into a division ring   cqqh 30561
                  20.3.13.4  Canonical embedding of the real numbers into a complete ordered field   crrh 30582
                  20.3.13.5  Embedding from the extended real numbers into a complete lattice   cxrh 30605
                  20.3.13.6  Canonical embeddings into the ordered field of the real numbers   zrhre 30608
                  ​*20.3.13.7  Topological Manifolds   cmntop 30611
            20.3.14  Real and complex functions   nexple 30616
                  20.3.14.1  Integer powers - misc. additions   nexple 30616
                  20.3.14.2  Indicator Functions   cind 30617
                  20.3.14.3  Extended sum   cesum 30634
            20.3.15  Mixed Function/Constant operation   cofc 30702
            20.3.16  Abstract measure   csiga 30715
                  20.3.16.1  Sigma-Algebra   csiga 30715
                  20.3.16.2  Generated sigma-Algebra   csigagen 30746
                  ​*20.3.16.3  lambda and pi-Systems, Rings of Sets   ispisys 30760
                  20.3.16.4  The Borel algebra on the real numbers   cbrsiga 30789
                  20.3.16.5  Product Sigma-Algebra   csx 30796
                  20.3.16.6  Measures   cmeas 30803
                  20.3.16.7  The counting measure   cntmeas 30834
                  20.3.16.8  The Lebesgue measure - misc additions   voliune 30837
                  20.3.16.9  The Dirac delta measure   cdde 30840
                  20.3.16.10  The 'almost everywhere' relation   cae 30845
                  20.3.16.11  Measurable functions   cmbfm 30857
                  20.3.16.12  Borel Algebra on ` ( RR X. RR ) `   br2base 30876
                  ​*20.3.16.13  Caratheodory's extension theorem   coms 30898
            20.3.17  Integration   itgeq12dv 30933
                  20.3.17.1  Lebesgue integral - misc additions   itgeq12dv 30933
                  20.3.17.2  Bochner integral   citgm 30934
            20.3.18  Euler's partition theorem   oddpwdc 30961
            20.3.19  Sequences defined by strong recursion   csseq 30990
            20.3.20  Fibonacci Numbers   cfib 31004
            20.3.21  Probability   cprb 31015
                  20.3.21.1  Probability Theory   cprb 31015
                  20.3.21.2  Conditional Probabilities   ccprob 31039
                  20.3.21.3  Real-valued Random Variables   crrv 31048
                  20.3.21.4  Preimage set mapping operator   corvc 31063
                  20.3.21.5  Distribution Functions   orvcelval 31076
                  20.3.21.6  Cumulative Distribution Functions   orvclteel 31080
                  20.3.21.7  Probabilities - example   coinfliplem 31086
                  20.3.21.8  Bertrand's Ballot Problem   ballotlemoex 31093
            20.3.22  Signum (sgn or sign) function - misc. additions   sgncl 31146
            20.3.23  Words over a set - misc additions   wrdfd 31162
                  20.3.23.1  Operations on words   ccatmulgnn0dir 31166
            20.3.24  Polynomials with real coefficients - misc additions   plymul02 31170
            20.3.25  Descartes's rule of signs   signspval 31176
                  20.3.25.1  Sign changes in a word over real numbers   signspval 31176
                  20.3.25.2  Counting sign changes in a word over real numbers   signslema 31186
            20.3.26  Number Theory   efcld 31218
                  20.3.26.1  Representations of a number as sums of integers   crepr 31235
                  20.3.26.2  Vinogradov Trigonometric Sums and the Circle Method   cvts 31262
                  20.3.26.3  The Ternary Goldbach Conjecture: Final Statement   ax-hgt749 31271
            20.3.27  Elementary Geometry   cstrkg2d 31291
                  ​*20.3.27.1  Two-dimension geometry   cstrkg2d 31291
                  20.3.27.2  Outer Five Segment (not used, no need to move to main)   cafs 31296
      ​​*20.4  Mathbox for Jonathan Ben-Naim
            20.4.1  First-order logic and set theory   bnj170 31313
            20.4.2  Well founded induction and recursion   bnj110 31474
            20.4.3  The existence of a minimal element in certain classes   bnj69 31624
            20.4.4  Well-founded induction   bnj1204 31626
            20.4.5  Well-founded recursion, part 1 of 3   bnj60 31676
            20.4.6  Well-founded recursion, part 2 of 3   bnj1500 31682
            20.4.7  Well-founded recursion, part 3 of 3   bnj1522 31686
      ​20.5  Mathbox for Mario Carneiro
            20.5.1  Predicate calculus with all distinct variables   ax-7d 31687
            20.5.2  Miscellaneous stuff   quartfull 31693
            20.5.3  Derangements and the Subfactorial   deranglem 31694
            20.5.4  The Erdős-Szekeres theorem   erdszelem1 31719
            20.5.5  The Kuratowski closure-complement theorem   kur14lem1 31734
            20.5.6  Retracts and sections   cretr 31745
            20.5.7  Path-connected and simply connected spaces   cpconn 31747
            20.5.8  Covering maps   ccvm 31783
            20.5.9  Normal numbers   snmlff 31857
            20.5.10  Godel-sets of formulas   cgoe 31861
            20.5.11  Models of ZF   cgze 31889
            ​*20.5.12  Metamath formal systems   cmcn 31903
            20.5.13  Grammatical formal systems   cm0s 32028
            20.5.14  Models of formal systems   cmuv 32046
            20.5.15  Splitting fields   citr 32068
            20.5.16  p-adic number fields   czr 32084
      ​​*20.6  Mathbox for Filip Cernatescu
      ​20.7  Mathbox for Paul Chapman
            20.7.1  Real and complex numbers (cont.)   climuzcnv 32109
            20.7.2  Miscellaneous theorems   elfzm12 32113
      ​20.8  Mathbox for Scott Fenton
            20.8.1  ZFC Axioms in primitive form   axextprim 32122
            20.8.2  Untangled classes   untelirr 32129
            20.8.3  Extra propositional calculus theorems   3orel2 32136
            20.8.4  Misc. Useful Theorems   nepss 32143
            20.8.5  Properties of real and complex numbers   sqdivzi 32154
            20.8.6  Infinite products   iprodefisumlem 32168
            20.8.7  Factorial limits   faclimlem1 32171
            20.8.8  Greatest common divisor and divisibility   pdivsq 32177
            20.8.9  Properties of relationships   brtp 32181
            20.8.10  Properties of functions and mappings   funpsstri 32205
            20.8.11  Epsilon induction   setinds 32221
            20.8.12  Ordinal numbers   elpotr 32224
            20.8.13  Defined equality axioms   axextdfeq 32241
            20.8.14  Hypothesis builders   hbntg 32249
            20.8.15  (Trans)finite Recursion Theorems   tfisg 32254
            20.8.16  Transitive closure under a relationship   ctrpred 32255
            20.8.17  Founded Induction   frpomin 32277
            20.8.18  Ordering Ordinal Sequences   orderseqlem 32291
            20.8.19  Well-founded zero, successor, and limits   cwsuc 32294
            20.8.20  Founded Recursion   cfrecs 32314
            20.8.21  Surreal Numbers   csur 32332
            20.8.22  Surreal Numbers: Ordering   sltsolem1 32365
            20.8.23  Surreal Numbers: Birthday Function   bdayfo 32367
            20.8.24  Surreal Numbers: Density   fvnobday 32368
            20.8.25  Surreal Numbers: Full-Eta Property   bdayimaon 32382
            20.8.26  Surreal numbers - ordering theorems   csle 32408
            20.8.27  Surreal numbers - birthday theorems   bdayfun 32427
            20.8.28  Surreal numbers: Conway cuts   csslt 32435
            20.8.29  Surreal numbers - cuts and options   cmade 32464
            20.8.30  Quantifier-free definitions   ctxp 32476
            20.8.31  Alternate ordered pairs   caltop 32602
            20.8.32  Geometry in the Euclidean space   cofs 32628
                  20.8.32.1  Congruence properties   cofs 32628
                  20.8.32.2  Betweenness properties   btwntriv2 32658
                  20.8.32.3  Segment Transportation   ctransport 32675
                  20.8.32.4  Properties relating betweenness and congruence   cifs 32681
                  20.8.32.5  Connectivity of betweenness   btwnconn1lem1 32733
                  20.8.32.6  Segment less than or equal to   csegle 32752
                  20.8.32.7  Outside-of relationship   coutsideof 32765
                  20.8.32.8  Lines and Rays   cline2 32780
            20.8.33  Forward difference   cfwddif 32804
            20.8.34  Rank theorems   rankung 32812
            20.8.35  Hereditarily Finite Sets   chf 32818
      ​20.9  Mathbox for Jeff Hankins
            20.9.1  Miscellany   a1i14 32833
            20.9.2  Basic topological facts   topbnd 32857
            20.9.3  Topology of the real numbers   ivthALT 32868
            20.9.4  Refinements   cfne 32869
            20.9.5  Neighborhood bases determine topologies   neibastop1 32892
            20.9.6  Lattice structure of topologies   topmtcl 32896
            20.9.7  Filter bases   fgmin 32903
            20.9.8  Directed sets, nets   tailfval 32905
      ​20.10  Mathbox for Anthony Hart
            20.10.1  Propositional Calculus   tb-ax1 32916
            20.10.2  Predicate Calculus   nalfal 32936
            20.10.3  Miscellaneous single axioms   meran1 32943
            20.10.4  Connective Symmetry   negsym1 32949
      ​20.11  Mathbox for Chen-Pang He
            20.11.1  Ordinal topology   ontopbas 32960
      ​20.12  Mathbox for Jeff Hoffman
            20.12.1  Inferences for finite induction on generic function values   fveleq 32983
            20.12.2  gdc.mm   nnssi2 32987
      ​20.13  Mathbox for Asger C. Ipsen
            20.13.1  Continuous nowhere differentiable functions   dnival 32994
      ​​*20.14  Mathbox for BJ
            *20.14.1  Propositional calculus   bj-mp2c 33063
                  *20.14.1.1  Derived rules of inference   bj-mp2c 33063
                  ​*20.14.1.2  A syntactic theorem   bj-0 33065
                  20.14.1.3  Minimal implicational calculus   bj-a1k 33067
                  20.14.1.4  Positive calculus   bj-syl66ib 33071
                  20.14.1.5  Implication and negation   bj-con2com 33076
                  ​*20.14.1.6  Disjunction   bj-jaoi1 33084
                  ​*20.14.1.7  Logical equivalence   bj-dfbi4 33086
                  20.14.1.8  The conditional operator for propositions   bj-consensus 33091
                  ​*20.14.1.9  Propositional calculus: miscellaneous   bj-imbi12 33096
            ​*20.14.2  Modal logic   bj-axdd2 33105
            ​*20.14.3  Provability logic   cprvb 33111
            ​*20.14.4  First-order logic   bj-genr 33120
                  20.14.4.1  Adding ax-gen   bj-genr 33120
                  20.14.4.2  Adding ax-4   bj-2alim 33123
                  20.14.4.3  Adding ax-5   bj-ax12wlem 33147
                  20.14.4.4  Equality and substitution   bj-ssbjustlem 33154
                  20.14.4.5  Adding ax-6   bj-alequexv 33190
                  20.14.4.6  Adding ax-7   bj-cbvexw 33199
                  20.14.4.7  Membership predicate, ax-8 and ax-9   bj-elequ2g 33201
                  20.14.4.8  Adding ax-11   bj-alcomexcom 33205
                  20.14.4.9  Adding ax-12   axc11n11 33207
                  20.14.4.10  Adding ax-13   bj-axc10 33241
                  ​*20.14.4.11  Removing dependencies on ax-13 (and ax-11)   bj-axc10v 33251
                  ​*20.14.4.12  Distinct var metavariables   bj-hbaeb2 33329
                  ​*20.14.4.13  Around ~ equsal   bj-equsal1t 33333
                  ​*20.14.4.14  Some Principia Mathematica proofs   stdpc5t 33338
                  20.14.4.15  Alternate definition of substitution   bj-sbsb 33348
                  20.14.4.16  Lemmas for substitution   bj-sbf3 33350
                  20.14.4.17  Existential uniqueness   bj-eu3f 33353
                  ​*20.14.4.18  First-order logic: miscellaneous   bj-moeub 33354
            20.14.5  Set theory   eliminable1 33364
                  ​*20.14.5.1  Eliminability of class terms   eliminable1 33364
                  ​*20.14.5.2  Classes without extensionality   bj-cleljustab 33371
                  20.14.5.3  Characterization among sets versus among classes   elelb 33406
                  ​*20.14.5.4  The nonfreeness quantifier for classes   bj-nfcsym 33407
                  ​*20.14.5.5  Proposal for the definitions of class membership and class equality   bj-ax8 33408
                  ​*20.14.5.6  Lemmas for class substitution   bj-sbeqALT 33416
                  20.14.5.7  Removing some axiom requirements and disjoint variable conditions   bj-exlimvmpi 33426
                  ​*20.14.5.8  Class abstractions   bj-unrab 33446
                  ​*20.14.5.9  Restricted non-freeness   wrnf 33453
                  ​*20.14.5.10  Russell's paradox   bj-ru0 33455
                  ​*20.14.5.11  Some disjointness results   bj-n0i 33458
                  ​*20.14.5.12  Complements on direct products   bj-xpimasn 33464
                  ​*20.14.5.13  "Singletonization" and tagging   bj-sels 33472
                  ​*20.14.5.14  Tuples of classes   bj-cproj 33500
                  ​*20.14.5.15  Set theory: miscellaneous   bj-disj2r 33535
                  20.14.5.16  Evaluation   bj-evaleq 33547
                  20.14.5.17  Elementwise operations   celwise 33555
                  ​*20.14.5.18  Elementwise intersection (families of sets induced on a subset)   bj-rest00 33557
                  20.14.5.19  Moore collections (complements)   bj-intss 33576
                  20.14.5.20  Maps-to notation for functions with three arguments   bj-0nelmpt 33592
                  ​*20.14.5.21  Currying   csethom 33598
                  ​*20.14.5.22  Setting components of extensible structures   cstrset 33610
            ​*20.14.6  Extended real and complex numbers, real and complex projective lines   bj-elid 33613
                  *20.14.6.1  Identity relation (complements)   bj-elid 33613
                  ​*20.14.6.2  Diagonal in a Cartesian square   cdiag2 33617
                  ​*20.14.6.3  Extended numbers and projective lines as sets   cfractemp 33622
                  ​*20.14.6.4  Addition and opposite   caddcc 33664
                  ​*20.14.6.5  Order relation on the extended reals   cltxr 33668
                  ​*20.14.6.6  Argument, multiplication and inverse   carg 33670
                  20.14.6.7  The canonical bijection from the finite ordinals   ciomnn 33676
            ​*20.14.7  Monoids   bj-cmnssmnd 33688
                  ​*20.14.7.1  Finite sums in monoids   cfinsum 33697
            ​*20.14.8  Affine, Euclidean, and Cartesian geometry   crrvec 33700
                  *20.14.8.1  Convex hull in real vector spaces   crrvec 33700
                  ​*20.14.8.2  Complex numbers (supplements)   bj-subcom 33706
                  ​*20.14.8.3  Barycentric coordinates   bj-bary1lem 33708
      ​20.15  Mathbox for Jim Kingdon
                  20.15.0.1  Circle constant   taupilem3 33711
                  20.15.0.2  Number theory   dfgcd3 33716
      ​20.16  Mathbox for ML
            ​*20.16.1  Cantor normal form up to epsilon 0   cnfin0 33785
      ​20.17  Mathbox for Wolf Lammen
            20.17.1  1. Bootstrapping   wl-section-boot 33791
            20.17.2  Implication chains   wl-section-impchain 33815
            20.17.3  An alternative axiom ~ ax-13   ax-wl-13v 33833
            20.17.4  Other stuff   wl-mps 33835
            20.17.5  1. Bootstrapping classes   wcel-wl 33916
      ​20.18  Mathbox for Brendan Leahy
      ​20.19  Mathbox for Jeff Madsen
            20.19.1  Logic and set theory   anim12da 34049
            20.19.2  Real and complex numbers; integers   filbcmb 34078
            20.19.3  Sequences and sums   sdclem2 34080
            20.19.4  Topology   subspopn 34090
            20.19.5  Metric spaces   metf1o 34093
            20.19.6  Continuous maps and homeomorphisms   constcncf 34100
            20.19.7  Boundedness   ctotbnd 34107
            20.19.8  Isometries   cismty 34139
            20.19.9  Heine-Borel Theorem   heibor1lem 34150
            20.19.10  Banach Fixed Point Theorem   bfplem1 34163
            20.19.11  Euclidean space   crrn 34166
            20.19.12  Intervals (continued)   ismrer1 34179
            20.19.13  Operation properties   cass 34183
            20.19.14  Groups and related structures   cmagm 34189
            20.19.15  Group homomorphism and isomorphism   cghomOLD 34224
            20.19.16  Rings   crngo 34235
            20.19.17  Division Rings   cdrng 34289
            20.19.18  Ring homomorphisms   crnghom 34301
            20.19.19  Commutative rings   ccm2 34330
            20.19.20  Ideals   cidl 34348
            20.19.21  Prime rings and integral domains   cprrng 34387
            20.19.22  Ideal generators   cigen 34400
      ​20.20  Mathbox for Giovanni Mascellani
            ​*20.20.1  Tools for automatic proof building   efald2 34419
            ​*20.20.2  Tseitin axioms   fald 34476
            ​*20.20.3  Equality deductions   iuneq2f 34503
            ​*20.20.4  Miscellanea   orcomdd 34516
      ​20.21  Mathbox for Peter Mazsa
            20.21.1  Notations   cxrn 34523
            20.21.2  Preparatory theorems   el2v 34546
            20.21.3  Range Cartesian product   df-xrn 34681
            20.21.4  Cosets by ` R `   df-coss 34717
            20.21.5  Relations   df-rels 34783
            20.21.6  Subset relations   df-ssr 34796
            20.21.7  Reflexivity   df-refs 34808
            20.21.8  Converse reflexivity   df-cnvrefs 34821
            20.21.9  Symmetry   df-syms 34836
            20.21.10  Reflexivity and symmetry   symrefref2 34857
            20.21.11  Transitivity   df-trs 34866
            20.21.12  Equivalence relations   df-eqvrels 34877
            20.21.13  Redundancy   df-reds 34912
      ​20.22  Mathbox for Rodolfo Medina
            20.22.1  Partitions   prtlem60 34927
      ​​*20.23  Mathbox for Norm Megill
            *20.23.1  Obsolete schemes ax-c4,c5,c7,c10,c11,c11n,c15,c9,c14,c16   ax-c5 34958
            ​*20.23.2  Rederive new axioms ax-4, ax-10, ax-6, ax-12, ax-13 from old   axc5 34968
            ​*20.23.3  Legacy theorems using obsolete axioms   ax5ALT 34982
            20.23.4  Experiments with weak deduction theorem   elimhyps 35036
            20.23.5  Miscellanea   cnaddcom 35047
            20.23.6  Atoms, hyperplanes, and covering in a left vector space (or module)   clsa 35049
            20.23.7  Functionals and kernels of a left vector space (or module)   clfn 35132
            20.23.8  Opposite rings and dual vector spaces   cld 35198
            20.23.9  Ortholattices and orthomodular lattices   cops 35247
            20.23.10  Atomic lattices with covering property   ccvr 35337
            20.23.11  Hilbert lattices   chlt 35425
            20.23.12  Projective geometries based on Hilbert lattices   clln 35566
            20.23.13  Construction of a vector space from a Hilbert lattice   cdlema1N 35866
            20.23.14  Construction of involution and inner product from a Hilbert lattice   clpoN 37555
      ​20.24  Mathbox for Steven Nguyen
            20.24.1  Utility theorems   ioin9i8 38037
            20.24.2  Russell's paradox   cbvabvw 38046
            ​*20.24.3  Arithmetic theorems   c0exALT 38049
            20.24.4  Real subtraction   cresub 38069
            20.24.5  Equivalent formulations of Fermat's Last Theorem   dffltz 38097
      ​20.25  Mathbox for OpenAI
      ​20.26  Mathbox for Stefan O'Rear
            20.26.1  Additional elementary logic and set theory   moxfr 38099
            20.26.2  Additional theory of functions   imaiinfv 38100
            20.26.3  Additional topology   elrfi 38101
            20.26.4  Characterization of closure operators. Kuratowski closure axioms   ismrcd1 38105
            20.26.5  Algebraic closure systems   cnacs 38109
            20.26.6  Miscellanea 1. Map utilities   constmap 38120
            20.26.7  Miscellanea for polynomials   mptfcl 38127
            20.26.8  Multivariate polynomials over the integers   cmzpcl 38128
            20.26.9  Miscellanea for Diophantine sets 1   coeq0i 38160
            20.26.10  Diophantine sets 1: definitions   cdioph 38162
            20.26.11  Diophantine sets 2 miscellanea   ellz1 38174
            20.26.12  Diophantine sets 2: union and intersection. Monotone Boolean algebra   diophin 38180
            20.26.13  Diophantine sets 3: construction   diophrex 38183
            20.26.14  Diophantine sets 4 miscellanea   2sbcrex 38192
            20.26.15  Diophantine sets 4: Quantification   rexrabdioph 38202
            20.26.16  Diophantine sets 5: Arithmetic sets   rabdiophlem1 38209
            20.26.17  Diophantine sets 6: reusability. renumbering of variables   eldioph4b 38219
            20.26.18  Pigeonhole Principle and cardinality helpers   fphpd 38224
            20.26.19  A non-closed set of reals is infinite   rencldnfilem 38228
            20.26.20  Lagrange's rational approximation theorem   irrapxlem1 38230
            20.26.21  Pell equations 1: A nontrivial solution always exists   pellexlem1 38237
            20.26.22  Pell equations 2: Algebraic number theory of the solution set   csquarenn 38244
            20.26.23  Pell equations 3: characterizing fundamental solution   infmrgelbi 38286
            ​*20.26.24  Logarithm laws generalized to an arbitrary base   reglogcl 38298
            20.26.25  Pell equations 4: the positive solution group is infinite cyclic   pellfund14 38306
            20.26.26  X and Y sequences 1: Definition and recurrence laws   crmx 38308
            20.26.27  Ordering and induction lemmas for the integers   monotuz 38349
            20.26.28  X and Y sequences 2: Order properties   rmxypos 38357
            20.26.29  Congruential equations   congtr 38375
            20.26.30  Alternating congruential equations   acongid 38385
            20.26.31  Additional theorems on integer divisibility   coprmdvdsb 38395
            20.26.32  X and Y sequences 3: Divisibility properties   jm2.18 38398
            20.26.33  X and Y sequences 4: Diophantine representability of Y   jm2.27a 38415
            20.26.34  X and Y sequences 5: Diophantine representability of X, ^, _C   rmxdiophlem 38425
            20.26.35  Uncategorized stuff not associated with a major project   setindtr 38434
            20.26.36  More equivalents of the Axiom of Choice   axac10 38443
            20.26.37  Finitely generated left modules   clfig 38480
            20.26.38  Noetherian left modules I   clnm 38488
            20.26.39  Addenda for structure powers   pwssplit4 38502
            20.26.40  Every set admits a group structure iff choice   unxpwdom3 38508
            20.26.41  Noetherian rings and left modules II   clnr 38522
            20.26.42  Hilbert's Basis Theorem   cldgis 38534
            20.26.43  Additional material on polynomials [DEPRECATED]   cmnc 38544
            20.26.44  Degree and minimal polynomial of algebraic numbers   cdgraa 38553
            20.26.45  Algebraic integers I   citgo 38570
            20.26.46  Endomorphism algebra   cmend 38588
            20.26.47  Subfields   csdrg 38608
            20.26.48  Cyclic groups and order   idomrootle 38616
            20.26.49  Cyclotomic polynomials   ccytp 38623
            20.26.50  Miscellaneous topology   fgraphopab 38631
      ​20.27  Mathbox for Jon Pennant
      ​20.28  Mathbox for Richard Penner
            20.28.1  Short Studies   ifpan123g 38645
                  20.28.1.1  Additional work on conditional logical operator   ifpan123g 38645
                  20.28.1.2  Sophisms   rp-fakeimass 38699
                  ​*20.28.1.3  Finite Sets   rp-isfinite5 38704
                  20.28.1.4  Infinite Sets   pwelg 38706
                  ​*20.28.1.5  Finite intersection property   fipjust 38711
                  20.28.1.6  RP ADDTO: Subclasses and subsets   rababg 38720
                  20.28.1.7  RP ADDTO: The intersection of a class   elintabg 38721
                  20.28.1.8  RP ADDTO: Theorems requiring subset and intersection existence   elinintrab 38724
                  20.28.1.9  RP ADDTO: Relations   xpinintabd 38727
                  ​*20.28.1.10  RP ADDTO: Functions   elmapintab 38743
                  ​*20.28.1.11  RP ADDTO: Finite induction (for finite ordinals)   cnvcnvintabd 38747
                  20.28.1.12  RP ADDTO: First and second members of an ordered pair   elcnvlem 38748
                  20.28.1.13  RP ADDTO: The reflexive and transitive properties of relations   undmrnresiss 38751
                  20.28.1.14  RP ADDTO: Basic properties of closures   cleq2lem 38755
                  20.28.1.15  RP REPLACE: Definitions and basic properties of transitive closures   trcleq2lemRP 38778
            20.28.2  Additional statements on relations and subclasses   al3im 38779
                  20.28.2.1  Transitive relations (not to be confused with transitive classes).   trrelind 38798
                  20.28.2.2  Reflexive closures   crcl 38805
                  ​*20.28.2.3  Finite relationship composition.   relexp2 38810
                  20.28.2.4  Transitive closure of a relation   dftrcl3 38853
                  ​*20.28.2.5  Adapted from Frege   frege77d 38879
            ​*20.28.3  Propositions from _Begriffsschrift_   dfxor4 38899
                  *20.28.3.1  _Begriffsschrift_ Chapter I   dfxor4 38899
                  ​*20.28.3.2  _Begriffsschrift_ Notation hints   rp-imass 38905
                  20.28.3.3  _Begriffsschrift_ Chapter II Implication   ax-frege1 38924
                  20.28.3.4  _Begriffsschrift_ Chapter II Implication and Negation   axfrege28 38963
                  ​*20.28.3.5  _Begriffsschrift_ Chapter II with logical equivalence   axfrege52a 38990
                  20.28.3.6  _Begriffsschrift_ Chapter II with equivalence of sets   axfrege52c 39021
                  ​*20.28.3.7  _Begriffsschrift_ Chapter II with equivalence of classes   frege53c 39048
                  ​*20.28.3.8  _Begriffsschrift_ Chapter III Properties hereditary in a sequence   dffrege69 39066
                  ​*20.28.3.9  _Begriffsschrift_ Chapter III Following in a sequence   dffrege76 39073
                  ​*20.28.3.10  _Begriffsschrift_ Chapter III Member of sequence   dffrege99 39096
                  ​*20.28.3.11  _Begriffsschrift_ Chapter III Single-valued procedures   dffrege115 39112
            ​*20.28.4  Exploring Topology via Seifert and Threlfall   enrelmap 39131
                  *20.28.4.1  Equinumerosity of sets of relations and maps   enrelmap 39131
                  ​*20.28.4.2  Generic Pseudoclosure Spaces, Pseudointerior Spaces, and Pseudoneighborhoods   sscon34b 39157
                  ​*20.28.4.3  Generic Neighborhood Spaces   gneispa 39268
            ​*20.28.5  Exploring Higher Homotopy via Kerodon   k0004lem1 39285
                  *20.28.5.1  Simplicial Sets   k0004lem1 39285
      ​20.29  Mathbox for Stanislas Polu
            20.29.1  IMO Problems   wwlemuld 39294
                  20.29.1.1  IMO 1972 B2   wwlemuld 39294
            ​*20.29.2  INT Inequalities Proof Generator   int-addcomd 39316
            ​*20.29.3  N-Digit Addition Proof Generator   unitadd 39338
            20.29.4  AM-GM (for k = 2,3,4)   gsumws3 39339
      ​20.30  Mathbox for Steve Rodriguez
            20.30.1  Miscellanea   nanorxor 39344
            20.30.2  Ratio test for infinite series convergence and divergence   dvgrat 39351
            20.30.3  Multiples   reldvds 39354
            20.30.4  Function operations   caofcan 39362
            20.30.5  Calculus   lhe4.4ex1a 39368
            20.30.6  The generalized binomial coefficient operation   cbcc 39375
            20.30.7  Binomial series   uzmptshftfval 39385
      ​20.31  Mathbox for Andrew Salmon
            20.31.1  Principia Mathematica * 10   pm10.12 39397
            20.31.2  Principia Mathematica * 11   2alanimi 39411
            20.31.3  Predicate Calculus   sbeqal1 39438
            20.31.4  Principia Mathematica * 13 and * 14   pm13.13a 39447
            20.31.5  Set Theory   elnev 39478
            20.31.6  Arithmetic   addcomgi 39498
            20.31.7  Geometry   cplusr 39499
      ​​*20.32  Mathbox for Alan Sare
            20.32.1  Auxiliary theorems for the Virtual Deduction tool   idiALT 39521
            20.32.2  Supplementary unification deductions   bi1imp 39525
            20.32.3  Conventional Metamath proofs, some derived from VD proofs   iidn3 39545
            20.32.4  What is Virtual Deduction?   wvd1 39613
            20.32.5  Virtual Deduction Theorems   df-vd1 39614
            20.32.6  Theorems proved using Virtual Deduction   trsspwALT 39872
            20.32.7  Theorems proved using Virtual Deduction with mmj2 assistance   simplbi2VD 39900
            20.32.8  Virtual Deduction transcriptions of textbook proofs   sb5ALTVD 39967
            20.32.9  Theorems proved using conjunction-form Virtual Deduction   elpwgdedVD 39971
            20.32.10  Theorems with a VD proof in conventional notation derived from a VD proof   suctrALT3 39978
            ​*20.32.11  Theorems with a proof in conventional notation derived from a VD proof   notnotrALT2 39981
      ​20.33  Mathbox for Glauco Siliprandi
            20.33.1  Miscellanea   evth2f 39992
            20.33.2  Functions   unima 40155
            20.33.3  Ordering on real numbers - Real and complex numbers basic operations   sub2times 40285
            20.33.4  Real intervals   gtnelioc 40511
            20.33.5  Finite sums   fsumclf 40596
            20.33.6  Finite multiplication of numbers and finite multiplication of functions   fmul01 40607
            20.33.7  Limits   clim1fr1 40628
                  20.33.7.1  Inferior limit (lim inf)   clsi 40778
                  ​*20.33.7.2  Limits for sequences of extended real numbers   clsxlim 40839
            20.33.8  Trigonometry   coseq0 40870
            20.33.9  Continuous Functions   mulcncff 40876
            20.33.10  Derivatives   dvsinexp 40920
            20.33.11  Integrals   itgsin0pilem1 40960
            20.33.12  Stone Weierstrass theorem - real version   stoweidlem1 41012
            20.33.13  Wallis' product for π   wallispilem1 41076
            20.33.14  Stirling's approximation formula for ` n ` factorial   stirlinglem1 41085
            20.33.15  Dirichlet kernel   dirkerval 41102
            20.33.16  Fourier Series   fourierdlem1 41119
            20.33.17  e is transcendental   elaa2lem 41244
            20.33.18  n-dimensional Euclidean space   rrxtopn 41295
            20.33.19  Basic measure theory   csalg 41319
                  ​*20.33.19.1  σ-Algebras   csalg 41319
                  20.33.19.2  Sum of nonnegative extended reals   csumge0 41370
                  ​*20.33.19.3  Measures   cmea 41457
                  ​*20.33.19.4  Outer measures and Caratheodory's construction   come 41497
                  ​*20.33.19.5  Lebesgue measure on n-dimensional Real numbers   covoln 41544
                  ​*20.33.19.6  Measurable functions   csmblfn 41703
      ​20.34  Mathbox for Saveliy Skresanov
            20.34.1  Ceva's theorem   sigarval 41833
      ​20.35  Mathbox for Jarvin Udandy
      ​20.36  Mathbox for Alexander van der Vekens
            20.36.1  General auxiliary theorems (1)   raaan2 41961
                  20.36.1.1  The empty set - extension   raaan2 41961
                  20.36.1.2  Unordered and ordered pairs - extension for singletons   eusnsn 41962
                  20.36.1.3  Unordered and ordered pairs - extension for unordered pairs   elprneb 41965
                  20.36.1.4  Relations - extension   eubrv 41966
                  20.36.1.5  Definite description binder (inverted iota) - extension   iota0def 41969
                  20.36.1.6  Functions - extension   fveqvfvv 41971
            20.36.2  Alternative for Russell's definition of a description binder   caiota 41980
            20.36.3  Double restricted existential uniqueness   r19.32 41993
                  20.36.3.1  Restricted quantification (extension)   r19.32 41993
                  20.36.3.2  Restricted uniqueness and "at most one" quantification   rmoimi 42001
                  20.36.3.3  Analogs to Existential uniqueness (double quantification)   2reurex 42006
            ​*20.36.4  Alternative definitions of function and operation values   wdfat 42018
                  20.36.4.1  Restricted quantification (extension)   ralbinrald 42024
                  20.36.4.2  The universal class (extension)   nvelim 42025
                  20.36.4.3  Introduce the Axiom of Power Sets (extension)   alneu 42026
                  20.36.4.4  Predicate "defined at"   dfateq12d 42028
                  20.36.4.5  Alternative definition of the value of a function   dfafv2 42034
                  20.36.4.6  Alternative definition of the value of an operation   aoveq123d 42080
            ​*20.36.5  Alternative definitions of function values (2)   cafv2 42110
            20.36.6  General auxiliary theorems (2)   an4com24 42170
                  20.36.6.1  Logical conjunction - extension   an4com24 42170
                  20.36.6.2  Abbreviated conjunction and disjunction of three wff's - extension   3an4ancom24 42171
                  20.36.6.3  Negated membership (alternative)   cnelbr 42173
                  20.36.6.4  Subclasses and subsets - extension   dfss7 42180
                  20.36.6.5  The empty set - extension   ralralimp 42181
                  20.36.6.6  Indexed union and intersection - extension   otiunsndisjX 42182
                  20.36.6.7  Functions - extension   fvifeq 42183
                  20.36.6.8  Maps-to notation - extension   fvmptrab 42195
                  20.36.6.9  Ordering on reals - extension   leltletr 42197
                  20.36.6.10  Subtraction - extension   cnambpcma 42198
                  20.36.6.11  Ordering on reals (cont.) - extension   leaddsuble 42200
                  20.36.6.12  Nonnegative integers (as a subset of complex numbers) - extension   nn0resubcl 42206
                  20.36.6.13  Integers (as a subset of complex numbers) - extension   zgeltp1eq 42207
                  20.36.6.14  Decimal arithmetic - extension   1t10e1p1e11 42208
                  20.36.6.15  Upper sets of integers - extension   eluzge0nn0 42210
                  20.36.6.16  Infinity and the extended real number system (cont.) - extension   nltle2tri 42211
                  20.36.6.17  Finite intervals of integers - extension   ssfz12 42212
                  20.36.6.18  Half-open integer ranges - extension   fzopred 42220
                  20.36.6.19  The modulo (remainder) operation - extension   m1mod0mod1 42227
                  20.36.6.20  The infinite sequence builder "seq"   smonoord 42229
                  20.36.6.21  Finite and infinite sums - extension   fsummsndifre 42230
                  20.36.6.22  Extensible structures - extension   setsidel 42234
            ​*20.36.7  Partitions of real intervals   ciccp 42237
            20.36.8  Shifting functions with an integer range domain   fargshiftfv 42263
            20.36.9  Words over a set (extension)   lswn0 42268
                  20.36.9.1  Last symbol of a word - extension   lswn0 42268
            20.36.10  Number theory (extension)   cfmtno 42269
                  ​*20.36.10.1  Fermat numbers   cfmtno 42269
                  ​*20.36.10.2  Mersenne primes   m2prm 42335
                  20.36.10.3  Proth's theorem   modexp2m1d 42359
            ​*20.36.11  Even and odd numbers   ceven 42367
                  20.36.11.1  Definitions and basic properties   ceven 42367
                  20.36.11.2  Alternate definitions using the "divides" relation   dfeven2 42392
                  20.36.11.3  Alternate definitions using the "modulo" operation   dfeven3 42400
                  20.36.11.4  Alternate definitions using the "gcd" operation   iseven5 42406
                  20.36.11.5  Theorems of part 5 revised   zneoALTV 42410
                  20.36.11.6  Theorems of part 6 revised   odd2np1ALTV 42415
                  20.36.11.7  Theorems of AV's mathbox revised   0evenALTV 42429
                  20.36.11.8  Additional theorems   epoo 42442
                  20.36.11.9  Perfect Number Theorem (revised)   perfectALTVlem1 42460
                  ​*20.36.11.10  Goldbach's conjectures   cgbe 42463
            20.36.12  Graph theory (extension)   cgrisom 42536
                  ​*20.36.12.1  Isomorphic graphs   cgrisom 42536
                  20.36.12.2  Loop-free graphs - extension   1hegrlfgr 42560
                  20.36.12.3  Walks - extension   cupwlks 42561
            20.36.13  Set of unordered pairs   sprid 42571
            20.36.14  Monoids (extension)   ovn0dmfun 42611
                  20.36.14.1  Auxiliary theorems   ovn0dmfun 42611
                  20.36.14.2  Magmas and Semigroups (extension)   plusfreseq 42619
                  20.36.14.3  Magma homomorphisms and submagmas   cmgmhm 42624
                  20.36.14.4  Examples and counterexamples for magmas, semigroups and monoids (extension)   opmpt2ismgm 42654
            ​*20.36.15  Magmas and internal binary operations (alternate approach)   ccllaw 42666
                  *20.36.15.1  Laws for internal binary operations   ccllaw 42666
                  ​*20.36.15.2  Internal binary operations   cintop 42679
                  20.36.15.3  Alternative definitions for Magmas and Semigroups   cmgm2 42698
            20.36.16  Categories (extension)   idfusubc0 42712
                  20.36.16.1  Subcategories (extension)   idfusubc0 42712
            20.36.17  Rings (extension)   lmod0rng 42715
                  20.36.17.1  Nonzero rings (extension)   lmod0rng 42715
                  ​*20.36.17.2  Non-unital rings ("rngs")   crng 42721
                  20.36.17.3  Rng homomorphisms   crngh 42732
                  20.36.17.4  Ring homomorphisms (extension)   rhmfn 42765
                  20.36.17.5  Ideals as non-unital rings   lidldomn1 42768
                  20.36.17.6  The non-unital ring of even integers   0even 42778
                  20.36.17.7  A constructed not unital ring   cznrnglem 42800
                  ​*20.36.17.8  The category of non-unital rings   crngc 42804
                  ​*20.36.17.9  The category of (unital) rings   cringc 42850
                  20.36.17.10  Subcategories of the category of rings   srhmsubclem1 42920
            20.36.18  Basic algebraic structures (extension)   xpprsng 42957
                  20.36.18.1  Auxiliary theorems   xpprsng 42957
                  20.36.18.2  The binomial coefficient operation (extension)   bcpascm1 42976
                  20.36.18.3  The ` ZZ `-module ` ZZ X. ZZ `   zlmodzxzlmod 42979
                  20.36.18.4  Ordered group sum operation (extension)   gsumpr 42986
                  20.36.18.5  Symmetric groups (extension)   exple2lt6 42992
                  20.36.18.6  Divisibility (extension)   invginvrid 42995
                  20.36.18.7  The support of functions (extension)   rmsupp0 42996
                  20.36.18.8  Finitely supported functions (extension)   rmsuppfi 43001
                  20.36.18.9  Left modules (extension)   lmodvsmdi 43010
                  20.36.18.10  Associative algebras (extension)   ascl0 43012
                  20.36.18.11  Univariate polynomials (extension)   ply1vr1smo 43016
                  20.36.18.12  Univariate polynomials (examples)   linply1 43028
            20.36.19  Linear algebra (extension)   cdmatalt 43032
                  ​*20.36.19.1  The subalgebras of diagonal and scalar matrices (extension)   cdmatalt 43032
                  ​*20.36.19.2  Linear combinations   clinc 43040
                  ​*20.36.19.3  Linear independence   clininds 43076
                  20.36.19.4  Simple left modules and the ` ZZ `-module   lmod1lem1 43123
                  20.36.19.5  Differences between (left) modules and (left) vector spaces   lvecpsslmod 43143
            20.36.20  Complexity theory   offval0 43146
                  20.36.20.1  Auxiliary theorems   offval0 43146
                  20.36.20.2  The modulo (remainder) operation (extension)   fldivmod 43160
                  20.36.20.3  Even and odd integers   nn0onn0ex 43165
                  20.36.20.4  The natural logarithm on complex numbers (extension)   logcxp0 43176
                  20.36.20.5  Division of functions   cfdiv 43178
                  20.36.20.6  Upper bounds   cbigo 43188
                  20.36.20.7  Logarithm to an arbitrary base (extension)   rege1logbrege0 43199
                  ​*20.36.20.8  The binary logarithm   fldivexpfllog2 43206
                  20.36.20.9  Binary length   cblen 43210
                  ​*20.36.20.10  Digits   cdig 43236
                  20.36.20.11  Nonnegative integer as sum of its shifted digits   dignn0flhalflem1 43256
                  20.36.20.12  Algorithms for the multiplication of nonnegative integers   nn0mulfsum 43265
            20.36.21  Elementary geometry (extension)   fv1prop 43267
                  20.36.21.1  Auxiliary theorems   fv1prop 43267
                  20.36.21.2  Spheres and lines in real Euclidean spaces   cline 43278
      ​20.37  Mathbox for Emmett Weisz
            ​*20.37.1  Miscellaneous Theorems   nfintd 43315
            20.37.2  Set Recursion   csetrecs 43325
                  ​*20.37.2.1  Basic Properties of Set Recursion   csetrecs 43325
                  20.37.2.2  Examples and properties of set recursion   elsetrecslem 43340
            ​*20.37.3  Construction of Games and Surreal Numbers   cpg 43350
      ​​*20.38  Mathbox for David A. Wheeler
            20.38.1  Natural deduction   19.8ad 43356
            ​*20.38.2  Greater than, greater than or equal to.   cge-real 43359
            ​*20.38.3  Hyperbolic trigonometric functions   csinh 43369
            ​*20.38.4  Reciprocal trigonometric functions (sec, csc, cot)   csec 43380
            ​*20.38.5  Identities for "if"   ifnmfalse 43402
            ​*20.38.6  Logarithms generalized to arbitrary base using ` logb `   logb2aval 43403
            ​*20.38.7  Logarithm laws generalized to an arbitrary base - log_   clog- 43404
            ​*20.38.8  Formally define terms such as Reflexivity   wreflexive 43406
            ​*20.38.9  Algebra helpers   comraddi 43410
            ​*20.38.10  Algebra helper examples   i2linesi 43420
            ​*20.38.11  Formal methods "surprises"   alimp-surprise 43422
            ​*20.38.12  Allsome quantifier   walsi 43428
            ​*20.38.13  Miscellaneous   5m4e1 43439
            20.38.14  Theorems about algebraic numbers   aacllem 43443
      ​20.39  Mathbox for Kunhao Zheng
            20.39.1  Weighted AM-GM inequality   amgmwlem 43444