TOC of Theorem List - Metamath Proof Explorer

Table of Contents Summary

PART 1  CLASSICAL FIRST-ORDER LOGIC WITH EQUALITY
      1.1  Pre-logic
      1.2  Propositional calculus
      1.3  Other axiomatizations related to classical propositional calculus
      1.4  Predicate calculus with equality: Tarski's system S2 (1 rule, 6 schemes)
      1.5  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes)
      1.6  Uniqueness and unique existence
      1.7  Other axiomatizations related to classical predicate calculus
PART 2  ZF (ZERMELO-FRAENKEL) SET THEORY
      2.1  ZF Set Theory - start with the Axiom of Extensionality
      2.2  ZF Set Theory - add the Axiom of Replacement
      2.3  ZF Set Theory - add the Axiom of Power Sets
      2.4  ZF Set Theory - add the Axiom of Union
      2.5  ZF Set Theory - add the Axiom of Regularity
      2.6  ZF Set Theory - add the Axiom of Infinity
PART 3  ZFC (ZERMELO-FRAENKEL WITH CHOICE) SET THEORY
      3.1  ZFC Set Theory - add Countable Choice and Dependent Choice
      3.2  ZFC Set Theory - add the Axiom of Choice
      3.3  ZFC Axioms with no distinct variable requirements
      3.4  The Generalized Continuum Hypothesis
PART 4  TG (TARSKI-GROTHENDIECK) SET THEORY
      4.1  Inaccessibles
      4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
PART 5  REAL AND COMPLEX NUMBERS
      5.1  Construction and axiomatization of real and complex numbers
      5.2  Derive the basic properties from the field axioms
      5.3  Real and complex numbers - basic operations
      5.4  Integer sets
      5.5  Order sets
      5.6  Elementary integer functions
      5.7  Words over a set
      5.8  Reflexive and transitive closures of relations
      5.9  Elementary real and complex functions
      5.10  Elementary limits and convergence
      5.11  Elementary trigonometry
      5.12  Cardinality of real and complex number subsets
PART 6  ELEMENTARY NUMBER THEORY
      6.1  Elementary properties of divisibility
      6.2  Elementary prime number theory
PART 7  BASIC STRUCTURES
      7.1  Extensible structures
      7.2  Moore spaces
PART 8  BASIC CATEGORY THEORY
      8.1  Categories
      8.2  Arrows (disjointified hom-sets)
      8.3  Examples of categories
      8.4  Categorical constructions
PART 9  BASIC ORDER THEORY
      9.1  Preordered sets and directed sets using extensible structures
      9.2  Posets and lattices using extensible structures
PART 10  BASIC ALGEBRAIC STRUCTURES
      10.1  Monoids
      10.2  Groups
      10.3  Rings
      10.4  Division rings and fields
      10.5  Left modules
      10.6  Vector spaces
      10.7  Ideals
      10.8  The complex numbers as an algebraic extensible structure
      10.9  Generalized pre-Hilbert and Hilbert spaces
PART 11  BASIC LINEAR ALGEBRA
      11.1  Vectors and free modules
      11.2  Associative algebras
      11.3  Abstract multivariate polynomials
      11.4  Matrices
      11.5  The determinant
      11.6  Polynomial matrices
      11.7  The characteristic polynomial
PART 12  BASIC TOPOLOGY
      12.1  Topology
      12.2  Filters and filter bases
      12.3  Uniform Structures and Spaces
      12.4  Metric spaces
      12.5  Metric subcomplex vector spaces
PART 13  BASIC REAL AND COMPLEX ANALYSIS
      13.1  Continuity
      13.2  Integrals
      13.3  Derivatives
PART 14  BASIC REAL AND COMPLEX FUNCTIONS
      14.1  Polynomials
      14.2  Sequences and series
      14.3  Basic trigonometry
      14.4  Basic number theory
PART 15  ELEMENTARY GEOMETRY
      15.1  Definition and Tarski's Axioms of Geometry
      15.2  Tarskian Geometry
      15.3  Properties of geometries
      15.4  Geometry in Hilbert spaces
PART 16  GRAPH THEORY
      16.1  Vertices and edges
      16.2  Undirected graphs
      16.3  Walks, paths and cycles
      16.4  Eulerian paths and the Konigsberg Bridge problem
      16.5  The Friendship Theorem
PART 17  GUIDES AND MISCELLANEA
      17.1  Guides (conventions, explanations, and examples)
      17.2  Humor
      17.3  (Future - to be reviewed and classified)
PART 18  COMPLEX TOPOLOGICAL VECTOR SPACES (DEPRECATED)
      18.1  Additional material on group theory (deprecated)
      18.2  Complex vector spaces
      18.3  Normed complex vector spaces
      18.4  Operators on complex vector spaces
      18.5  Inner product (pre-Hilbert) spaces
      18.6  Complex Banach spaces
      18.7  Complex Hilbert spaces
PART 19  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      19.1  Axiomatization of complex pre-Hilbert spaces
      19.2  Inner product and norms
      19.3  Cauchy sequences and completeness axiom
      19.4  Subspaces and projections
      19.5  Properties of Hilbert subspaces
      19.6  Operators on Hilbert spaces
      19.7  States on a Hilbert lattice and Godowski's equation
      19.8  Cover relation, atoms, exchange axiom, and modular symmetry
PART 20  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      ​20.1  Mathboxes for user contributions
      20.2  Mathbox for Stefan Allan
      20.3  Mathbox for Thierry Arnoux
      20.4  Mathbox for Jonathan Ben-Naim
      20.5  Mathbox for BTernaryTau
      20.6  Mathbox for Mario Carneiro
      20.7  Mathbox for Filip Cernatescu
      20.8  Mathbox for Paul Chapman
      20.9  Mathbox for Scott Fenton
      20.10  Mathbox for Jeff Hankins
      20.11  Mathbox for Anthony Hart
      20.12  Mathbox for Chen-Pang He
      20.13  Mathbox for Jeff Hoffman
      20.14  Mathbox for Asger C. Ipsen
      20.15  Mathbox for BJ
      20.16  Mathbox for Jim Kingdon
      20.17  Mathbox for ML
      20.18  Mathbox for Wolf Lammen
      20.19  Mathbox for Brendan Leahy
      20.20  Mathbox for Jeff Madsen
      20.21  Mathbox for Giovanni Mascellani
      20.22  Mathbox for Peter Mazsa
      20.23  Mathbox for Rodolfo Medina
      20.24  Mathbox for Norm Megill
      20.25  Mathbox for metakunt
      20.26  Mathbox for Steven Nguyen
      20.27  Mathbox for Igor Ieskov
      20.28  Mathbox for OpenAI
      20.29  Mathbox for Stefan O'Rear
      20.30  Mathbox for Jon Pennant
      20.31  Mathbox for Richard Penner
      20.32  Mathbox for Stanislas Polu
      20.33  Mathbox for Rohan Ridenour
      20.34  Mathbox for Steve Rodriguez
      20.35  Mathbox for Andrew Salmon
      20.36  Mathbox for Alan Sare
      20.37  Mathbox for Glauco Siliprandi
      20.38  Mathbox for Saveliy Skresanov
      20.39  Mathbox for Jarvin Udandy
      20.40  Mathbox for Adhemar
      20.41  Mathbox for Alexander van der Vekens
      20.42  Mathbox for Emmett Weisz
      20.43  Mathbox for David A. Wheeler
      20.44  Mathbox for Kunhao Zheng

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

​​*PART 1  CLASSICAL FIRST-ORDER LOGIC WITH EQUALITY
      ​*1.1  Pre-logic
            *1.1.1  Inferences for assisting proof development   idi 1
      ​​*1.2  Propositional calculus
            1.2.1  Recursively define primitive wffs for propositional calculus   wn 3
            ​*1.2.2  The axioms of propositional calculus   ax-mp 5
            ​*1.2.3  Logical implication   mp2 9
            ​*1.2.4  Logical negation   con4 113
            ​*1.2.5  Logical equivalence   wb 209
            ​*1.2.6  Logical conjunction   wa 399
            ​*1.2.7  Logical disjunction   wo 844
            ​*1.2.8  Mixed connectives   jaao 952
            ​*1.2.9  The conditional operator for propositions   wif 1058
            ​*1.2.10  The weak deduction theorem for propositional calculus   elimh 1080
            1.2.11  Abbreviated conjunction and disjunction of three wff's   w3o 1083
            1.2.12  Logical "nand" (Sheffer stroke)   wnan 1482
            1.2.13  Logical "xor"   wxo 1502
            1.2.14  Logical "nor"   wnor 1521
            1.2.15  True and false constants   wal 1536
                  ​*1.2.15.1  Universal quantifier for use by df-tru   wal 1536
                  ​*1.2.15.2  Equality predicate for use by df-tru   cv 1537
                  1.2.15.3  The true constant   wtru 1539
                  1.2.15.4  The false constant   wfal 1550
            ​*1.2.16  Truth tables   truimtru 1561
                  1.2.16.1  Implication   truimtru 1561
                  1.2.16.2  Negation   nottru 1565
                  1.2.16.3  Equivalence   trubitru 1567
                  1.2.16.4  Conjunction   truantru 1571
                  1.2.16.5  Disjunction   truortru 1575
                  1.2.16.6  Alternative denial   trunantru 1579
                  1.2.16.7  Exclusive disjunction   truxortru 1583
                  1.2.16.8  Joint denial   trunortru 1587
            ​*1.2.17  Half adder and full adder in propositional calculus   whad 1594
                  1.2.17.1  Full adder: sum   whad 1594
                  1.2.17.2  Full adder: carry   wcad 1608
      ​1.3  Other axiomatizations related to classical propositional calculus
            ​*1.3.1  Minimal implicational calculus   minimp 1623
            ​*1.3.2  Implicational Calculus   impsingle 1629
            1.3.3  Derive the Lukasiewicz axioms from Meredith's sole axiom   meredith 1643
            1.3.4  Derive the standard axioms from the Lukasiewicz axioms   luklem1 1660
            ​*1.3.5  Derive Nicod's axiom from the standard axioms   nic-dfim 1671
            1.3.6  Derive the Lukasiewicz axioms from Nicod's axiom   nic-imp 1677
            1.3.7  Derive Nicod's Axiom from Lukasiewicz's First Sheffer Stroke Axiom   lukshef-ax1 1696
            1.3.8  Derive the Lukasiewicz Axioms from the Tarski-Bernays-Wajsberg Axioms   tbw-bijust 1700
            1.3.9  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's First CO Axiom   merco1 1715
            1.3.10  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's Second CO Axiom   merco2 1738
            1.3.11  Derive the Lukasiewicz axioms from the Russell-Bernays Axioms   rb-bijust 1751
            ​*1.3.12  Stoic logic non-modal portion (Chrysippus of Soli)   mptnan 1770
      ​​*1.4  Predicate calculus with equality: Tarski's system S2 (1 rule, 6 schemes)
            *1.4.1  Universal quantifier (continued); define "exists" and "not free"   wex 1781
                  1.4.1.1  Existential quantifier   wex 1781
                  1.4.1.2  Non-freeness predicate   wnf 1785
            1.4.2  Rule scheme ax-gen (Generalization)   ax-gen 1797
            1.4.3  Axiom scheme ax-4 (Quantified Implication)   ax-4 1811
                  ​*1.4.3.1  The empty domain of discourse   empty 1907
            1.4.4  Axiom scheme ax-5 (Distinctness) - first use of $d   ax-5 1911
            ​*1.4.5  Equality predicate (continued)   weq 1964
            1.4.6  Axiom scheme ax-6 (Existence)   ax-6 1970
            1.4.7  Axiom scheme ax-7 (Equality)   ax-7 2015
            1.4.8  Define proper substitution   sbjust 2068
            1.4.9  Membership predicate   wcel 2111
            1.4.10  Axiom scheme ax-8 (Left Equality for Binary Predicate)   ax-8 2113
            1.4.11  Axiom scheme ax-9 (Right Equality for Binary Predicate)   ax-9 2121
            ​*1.4.12  Logical redundancy of ax-10 , ax-11 , ax-12 , ax-13   ax6dgen 2129
      ​​*1.5  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes)
            1.5.1  Axiom scheme ax-10 (Quantified Negation)   ax-10 2142
            1.5.2  Axiom scheme ax-11 (Quantifier Commutation)   ax-11 2158
            1.5.3  Axiom scheme ax-12 (Substitution)   ax-12 2175
            1.5.4  Axiom scheme ax-13 (Quantified Equality)   ax-13 2379
            ​*1.5.5  Alternate definition of substitution   sbimiALT 2553
      ​1.6  Uniqueness and unique existence
            1.6.1  Uniqueness: the at-most-one quantifier   wmo 2596
            1.6.2  Unique existence: the unique existential quantifier   weu 2628
      ​1.7  Other axiomatizations related to classical predicate calculus
            ​*1.7.1  Aristotelian logic: Assertic syllogisms   barbara 2725
            ​*1.7.2  Intuitionistic logic   axia1 2755
​​*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 2770
            2.1.2  Classes   cab 2776
                  2.1.2.1  Class abstractions   cab 2776
                  ​*2.1.2.2  Class equality   df-cleq 2791
                  2.1.2.3  Class membership   df-clel 2870
                  2.1.2.4  Elementary properties of class abstractions   abeq2 2922
            2.1.3  Class form not-free predicate   wnfc 2936
            2.1.4  Negated equality and membership   wne 2987
                  2.1.4.1  Negated equality   wne 2987
                  2.1.4.2  Negated membership   wnel 3091
            2.1.5  Restricted quantification   wral 3106
            2.1.6  The universal class   cvv 3441
            ​*2.1.7  Conditional equality (experimental)   wcdeq 3702
            2.1.8  Russell's Paradox   rru 3718
            2.1.9  Proper substitution of classes for sets   wsbc 3720
            2.1.10  Proper substitution of classes for sets into classes   csb 3828
            2.1.11  Define basic set operations and relations   cdif 3878
            2.1.12  Subclasses and subsets   df-ss 3898
            2.1.13  The difference, union, and intersection of two classes   dfdif3 4042
                  2.1.13.1  The difference of two classes   dfdif3 4042
                  2.1.13.2  The union of two classes   elun 4076
                  2.1.13.3  The intersection of two classes   elini 4120
                  2.1.13.4  The symmetric difference of two classes   csymdif 4168
                  2.1.13.5  Combinations of difference, union, and intersection of two classes   unabs 4181
                  2.1.13.6  Class abstractions with difference, union, and intersection of two classes   unab 4222
                  2.1.13.7  Restricted uniqueness with difference, union, and intersection   reuss2 4235
            2.1.14  The empty set   c0 4243
            ​*2.1.15  The conditional operator for classes   cif 4425
            ​*2.1.16  The weak deduction theorem for set theory   dedth 4481
            2.1.17  Power classes   cpw 4497
            2.1.18  Unordered and ordered pairs   snjust 4524
            2.1.19  The union of a class   cuni 4800
            2.1.20  The intersection of a class   cint 4838
            2.1.21  Indexed union and intersection   ciun 4881
            2.1.22  Disjointness   wdisj 4995
            2.1.23  Binary relations   wbr 5030
            2.1.24  Ordered-pair class abstractions (class builders)   copab 5092
            2.1.25  Functions in maps-to notation   cmpt 5110
            2.1.26  Transitive classes   wtr 5136
      ​2.2  ZF Set Theory - add the Axiom of Replacement
            2.2.1  Introduce the Axiom of Replacement   ax-rep 5154
            2.2.2  Derive the Axiom of Separation   axsepgfromrep 5165
            2.2.3  Derive the Null Set Axiom   axnulALT 5172
            2.2.4  Theorems requiring subset and intersection existence   nalset 5181
            2.2.5  Theorems requiring empty set existence   class2set 5219
      ​2.3  ZF Set Theory - add the Axiom of Power Sets
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 5231
            2.3.2  Derive the Axiom of Pairing   axprlem1 5289
            2.3.3  Ordered pair theorem   opnz 5330
            2.3.4  Ordered-pair class abstractions (cont.)   opabidw 5377
            2.3.5  Power class of union and intersection   pwin 5419
            2.3.6  The identity relation   cid 5424
            2.3.7  The membership relation (or epsilon relation)   cep 5429
            ​*2.3.8  Partial and total orderings   wpo 5436
            2.3.9  Founded and well-ordering relations   wfr 5475
            2.3.10  Relations   cxp 5517
            2.3.11  The Predecessor Class   cpred 6115
            2.3.12  Well-founded induction   tz6.26 6147
            2.3.13  Ordinals   word 6158
            2.3.14  Definite description binder (inverted iota)   cio 6281
            2.3.15  Functions   wfun 6318
            2.3.16  Cantor's Theorem   canth 7090
            2.3.17  Restricted iota (description binder)   crio 7092
            2.3.18  Operations   co 7135
                  2.3.18.1  Variable-to-class conversion for operations   caovclg 7320
            2.3.19  Maps-to notation   mpondm0 7366
            2.3.20  Function operation   cof 7387
            2.3.21  Proper subset relation   crpss 7428
      ​2.4  ZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 7441
            2.4.2  Ordinals (continued)   epweon 7477
            2.4.3  Transfinite induction   tfi 7548
            2.4.4  The natural numbers (i.e., finite ordinals)   com 7560
            2.4.5  Peano's postulates   peano1 7581
            2.4.6  Finite induction (for finite ordinals)   find 7587
            2.4.7  Relations and functions (cont.)   dmexg 7594
            2.4.8  First and second members of an ordered pair   c1st 7669
            ​*2.4.9  The support of functions   csupp 7813
            ​*2.4.10  Special maps-to operations   opeliunxp2f 7859
            2.4.11  Function transposition   ctpos 7874
            2.4.12  Curry and uncurry   ccur 7914
            2.4.13  Undefined values   cund 7921
            2.4.14  Well-founded recursion   cwrecs 7929
            2.4.15  Functions on ordinals; strictly monotone ordinal functions   iunon 7959
            2.4.16  "Strong" transfinite recursion   crecs 7990
            2.4.17  Recursive definition generator   crdg 8028
            2.4.18  Finite recursion   frfnom 8053
            2.4.19  Ordinal arithmetic   c1o 8078
            2.4.20  Natural number arithmetic   nna0 8213
            2.4.21  Equivalence relations and classes   wer 8269
            2.4.22  The mapping operation   cmap 8389
            2.4.23  Infinite Cartesian products   cixp 8444
            2.4.24  Equinumerosity   cen 8489
            2.4.25  Schroeder-Bernstein Theorem   sbthlem1 8611
            2.4.26  Equinumerosity (cont.)   xpf1o 8663
            2.4.27  Pigeonhole Principle   phplem1 8680
            2.4.28  Finite sets   onomeneq 8693
            2.4.29  Finitely supported functions   cfsupp 8817
            2.4.30  Finite intersections   cfi 8858
            2.4.31  Hall's marriage theorem   marypha1lem 8881
            2.4.32  Supremum and infimum   csup 8888
            2.4.33  Ordinal isomorphism, Hartogs's theorem   coi 8957
            2.4.34  Hartogs function   char 9004
            2.4.35  Weak dominance   cwdom 9012
      ​2.5  ZF Set Theory - add the Axiom of Regularity
            2.5.1  Introduce the Axiom of Regularity   ax-reg 9040
            2.5.2  Axiom of Infinity equivalents   inf0 9068
      ​2.6  ZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-inf 9085
            2.6.2  Existence of omega (the set of natural numbers)   omex 9090
            2.6.3  Cantor normal form   ccnf 9108
            2.6.4  Transitive closure   trcl 9154
            2.6.5  Rank   cr1 9175
            2.6.6  Scott's trick; collection principle; Hilbert's epsilon   scottex 9298
            2.6.7  Disjoint union   cdju 9311
            2.6.8  Cardinal numbers   ccrd 9348
            2.6.9  Axiom of Choice equivalents   wac 9526
            ​*2.6.10  Cardinal number arithmetic   undjudom 9578
            2.6.11  The Ackermann bijection   ackbij2lem1 9630
            2.6.12  Cofinality (without Axiom of Choice)   cflem 9657
            2.6.13  Eight inequivalent definitions of finite set   sornom 9688
            2.6.14  Hereditarily size-limited sets without Choice   itunifval 9827
​​*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 9846
            3.1.2  Introduce the Axiom of Dependent Choice   ax-dc 9857
      ​3.2  ZFC Set Theory - add the Axiom of Choice
            3.2.1  Introduce the Axiom of Choice   ax-ac 9870
            3.2.2  AC equivalents: well-ordering, Zorn's lemma   numthcor 9905
            3.2.3  Cardinal number theorems using Axiom of Choice   cardval 9957
            3.2.4  Cardinal number arithmetic using Axiom of Choice   iunctb 9985
            3.2.5  Cofinality using the Axiom of Choice   alephreg 9993
      ​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 10031
            3.4.2  Derivation of the Axiom of Choice   gchaclem 10089
​​*PART 4  TG (TARSKI-GROTHENDIECK) SET THEORY
      ​4.1  Inaccessibles
            4.1.1  Weakly and strongly inaccessible cardinals   cwina 10093
            4.1.2  Weak universes   cwun 10111
            4.1.3  Tarski classes   ctsk 10159
            4.1.4  Grothendieck universes   cgru 10201
      ​4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
            4.2.1  Introduce the Tarski-Grothendieck Axiom   ax-groth 10234
            4.2.2  Derive the Power Set, Infinity and Choice Axioms   grothpw 10237
            4.2.3  Tarski map function   ctskm 10248
​​*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 10255
            5.1.2  Final derivation of real and complex number postulates   axaddf 10556
            5.1.3  Real and complex number postulates restated as axioms   ax-cnex 10582
      ​5.2  Derive the basic properties from the field axioms
            5.2.1  Some deductions from the field axioms for complex numbers   cnex 10607
            5.2.2  Infinity and the extended real number system   cpnf 10661
            5.2.3  Restate the ordering postulates with extended real "less than"   axlttri 10701
            5.2.4  Ordering on reals   lttr 10706
            5.2.5  Initial properties of the complex numbers   mul12 10794
      ​5.3  Real and complex numbers - basic operations
            5.3.1  Addition   add12 10846
            5.3.2  Subtraction   cmin 10859
            5.3.3  Multiplication   kcnktkm1cn 11060
            5.3.4  Ordering on reals (cont.)   gt0ne0 11094
            5.3.5  Reciprocals   ixi 11258
            5.3.6  Division   cdiv 11286
            5.3.7  Ordering on reals (cont.)   elimgt0 11467
            5.3.8  Completeness Axiom and Suprema   fimaxre 11573
            5.3.9  Imaginary and complex number properties   inelr 11615
            5.3.10  Function operation analogue theorems   ofsubeq0 11622
      ​5.4  Integer sets
            5.4.1  Positive integers (as a subset of complex numbers)   cn 11625
            5.4.2  Principle of mathematical induction   nnind 11643
            ​*5.4.3  Decimal representation of numbers   c2 11680
            ​*5.4.4  Some properties of specific numbers   neg1cn 11739
            5.4.5  Simple number properties   halfcl 11850
            5.4.6  The Archimedean property   nnunb 11881
            5.4.7  Nonnegative integers (as a subset of complex numbers)   cn0 11885
            ​*5.4.8  Extended nonnegative integers   cxnn0 11955
            5.4.9  Integers (as a subset of complex numbers)   cz 11969
            5.4.10  Decimal arithmetic   cdc 12086
            5.4.11  Upper sets of integers   cuz 12231
            5.4.12  Well-ordering principle for bounded-below sets of integers   uzwo3 12331
            5.4.13  Rational numbers (as a subset of complex numbers)   cq 12336
            5.4.14  Existence of the set of complex numbers   rpnnen1lem2 12364
      ​5.5  Order sets
            5.5.1  Positive reals (as a subset of complex numbers)   crp 12377
            5.5.2  Infinity and the extended real number system (cont.)   cxne 12492
            5.5.3  Supremum and infimum on the extended reals   xrsupexmnf 12686
            5.5.4  Real number intervals   cioo 12726
            5.5.5  Finite intervals of integers   cfz 12885
            ​*5.5.6  Finite intervals of nonnegative integers   elfz2nn0 12993
            5.5.7  Half-open integer ranges   cfzo 13028
      ​5.6  Elementary integer functions
            5.6.1  The floor and ceiling functions   cfl 13155
            5.6.2  The modulo (remainder) operation   cmo 13232
            5.6.3  Miscellaneous theorems about integers   om2uz0i 13310
            5.6.4  Strong induction over upper sets of integers   uzsinds 13350
            5.6.5  Finitely supported functions over the nonnegative integers   fsuppmapnn0fiublem 13353
            5.6.6  The infinite sequence builder "seq" - extension   cseq 13364
            5.6.7  Integer powers   cexp 13425
            5.6.8  Ordered pair theorem for nonnegative integers   nn0le2msqi 13623
            5.6.9  Factorial function   cfa 13629
            5.6.10  The binomial coefficient operation   cbc 13658
            5.6.11  The ` # ` (set size) function   chash 13686
                  5.6.11.1  Proper unordered pairs and triples (sets of size 2 and 3)   hashprlei 13822
                  5.6.11.2  Functions with a domain containing at least two different elements   fundmge2nop0 13846
                  5.6.11.3  Finite induction on the size of the first component of a binary relation   hashdifsnp1 13850
      ​​*5.7  Words over a set
            5.7.1  Definitions and basic theorems   cword 13857
            5.7.2  Last symbol of a word   clsw 13905
            5.7.3  Concatenations of words   cconcat 13913
            5.7.4  Singleton words   cs1 13940
            5.7.5  Concatenations with singleton words   ccatws1cl 13961
            5.7.6  Subwords/substrings   csubstr 13993
            5.7.7  Prefixes of a word   cpfx 14023
            5.7.8  Subwords of subwords   swrdswrdlem 14057
            5.7.9  Subwords and concatenations   pfxcctswrd 14063
            5.7.10  Subwords of concatenations   swrdccatfn 14077
            5.7.11  Splicing words (substring replacement)   csplice 14102
            5.7.12  Reversing words   creverse 14111
            5.7.13  Repeated symbol words   creps 14121
            ​*5.7.14  Cyclical shifts of words   ccsh 14141
            5.7.15  Mapping words by a function   wrdco 14184
            5.7.16  Longer string literals   cs2 14194
      ​​*5.8  Reflexive and transitive closures of relations
            5.8.1  The reflexive and transitive properties of relations   coss12d 14323
            5.8.2  Basic properties of closures   cleq1lem 14333
            5.8.3  Definitions and basic properties of transitive closures   ctcl 14336
            5.8.4  Exponentiation of relations   crelexp 14370
            5.8.5  Reflexive-transitive closure as an indexed union   crtrcl 14406
            ​*5.8.6  Principle of transitive induction.   relexpindlem 14414
      ​5.9  Elementary real and complex functions
            5.9.1  The "shift" operation   cshi 14417
            5.9.2  Signum (sgn or sign) function   csgn 14437
            5.9.3  Real and imaginary parts; conjugate   ccj 14447
            5.9.4  Square root; absolute value   csqrt 14584
      ​5.10  Elementary limits and convergence
            5.10.1  Superior limit (lim sup)   clsp 14819
            5.10.2  Limits   cli 14833
            5.10.3  Finite and infinite sums   csu 15034
            5.10.4  The binomial theorem   binomlem 15176
            5.10.5  The inclusion/exclusion principle   incexclem 15183
            5.10.6  Infinite sums (cont.)   isumshft 15186
            5.10.7  Miscellaneous converging and diverging sequences   divrcnv 15199
            5.10.8  Arithmetic series   arisum 15207
            5.10.9  Geometric series   expcnv 15211
            5.10.10  Ratio test for infinite series convergence   cvgrat 15231
            5.10.11  Mertens' theorem   mertenslem1 15232
            5.10.12  Finite and infinite products   prodf 15235
                  5.10.12.1  Product sequences   prodf 15235
                  5.10.12.2  Non-trivial convergence   ntrivcvg 15245
                  5.10.12.3  Complex products   cprod 15251
                  5.10.12.4  Finite products   fprod 15287
                  5.10.12.5  Infinite products   iprodclim 15344
            5.10.13  Falling and Rising Factorial   cfallfac 15350
            5.10.14  Bernoulli polynomials and sums of k-th powers   cbp 15392
      ​5.11  Elementary trigonometry
            5.11.1  The exponential, sine, and cosine functions   ce 15407
                  5.11.1.1  The circle constant (tau = 2 pi)   ctau 15547
            5.11.2  _e is irrational   eirrlem 15549
      ​5.12  Cardinality of real and complex number subsets
            5.12.1  Countability of integers and rationals   xpnnen 15556
            5.12.2  The reals are uncountable   rpnnen2lem1 15559
​​*PART 6  ELEMENTARY NUMBER THEORY
      ​6.1  Elementary properties of divisibility
            6.1.1  Irrationality of square root of 2   sqrt2irrlem 15593
            6.1.2  Some Number sets are chains of proper subsets   nthruc 15597
            6.1.3  The divides relation   cdvds 15599
            ​*6.1.4  Even and odd numbers   evenelz 15677
            6.1.5  The division algorithm   divalglem0 15734
            6.1.6  Bit sequences   cbits 15758
            6.1.7  The greatest common divisor operator   cgcd 15833
            6.1.8  Bézout's identity   bezoutlem1 15877
            6.1.9  Algorithms   nn0seqcvgd 15904
            6.1.10  Euclid's Algorithm   eucalgval2 15915
            ​*6.1.11  The least common multiple   clcm 15922
            ​*6.1.12  Coprimality and Euclid's lemma   coprmgcdb 15983
            6.1.13  Cancellability of congruences   congr 15998
      ​6.2  Elementary prime number theory
            ​*6.2.1  Elementary properties   cprime 16005
            ​*6.2.2  Coprimality and Euclid's lemma (cont.)   coprm 16045
            6.2.3  Properties of the canonical representation of a rational   cnumer 16063
            6.2.4  Euler's theorem   codz 16090
            6.2.5  Arithmetic modulo a prime number   modprm1div 16124
            6.2.6  Pythagorean Triples   coprimeprodsq 16135
            6.2.7  The prime count function   cpc 16163
            6.2.8  Pocklington's theorem   prmpwdvds 16230
            6.2.9  Infinite primes theorem   unbenlem 16234
            6.2.10  Sum of prime reciprocals   prmreclem1 16242
            6.2.11  Fundamental theorem of arithmetic   1arithlem1 16249
            6.2.12  Lagrange's four-square theorem   cgz 16255
            6.2.13  Van der Waerden's theorem   cvdwa 16291
            6.2.14  Ramsey's theorem   cram 16325
            ​*6.2.15  Primorial function   cprmo 16357
            ​*6.2.16  Prime gaps   prmgaplem1 16375
            6.2.17  Decimal arithmetic (cont.)   dec2dvds 16389
            6.2.18  Cyclical shifts of words (cont.)   cshwsidrepsw 16419
            6.2.19  Specific prime numbers   prmlem0 16431
            6.2.20  Very large primes   1259lem1 16456
​PART 7  BASIC STRUCTURES
      ​7.1  Extensible structures
            ​*7.1.1  Basic definitions   cstr 16471
            7.1.2  Slot definitions   cplusg 16557
            7.1.3  Definition of the structure product   crest 16686
            7.1.4  Definition of the structure quotient   cordt 16764
      ​7.2  Moore spaces
            7.2.1  Moore closures   mrcflem 16869
            7.2.2  Independent sets in a Moore system   mrisval 16893
            7.2.3  Algebraic closure systems   isacs 16914
​PART 8  BASIC CATEGORY THEORY
      ​8.1  Categories
            8.1.1  Categories   ccat 16927
            8.1.2  Opposite category   coppc 16973
            8.1.3  Monomorphisms and epimorphisms   cmon 16990
            8.1.4  Sections, inverses, isomorphisms   csect 17006
            ​*8.1.5  Isomorphic objects   ccic 17057
            8.1.6  Subcategories   cssc 17069
            8.1.7  Functors   cfunc 17116
            8.1.8  Full & faithful functors   cful 17164
            8.1.9  Natural transformations and the functor category   cnat 17203
            8.1.10  Initial, terminal and zero objects of a category   cinito 17240
      ​8.2  Arrows (disjointified hom-sets)
            8.2.1  Identity and composition for arrows   cida 17305
      ​8.3  Examples of categories
            8.3.1  The category of sets   csetc 17327
            8.3.2  The category of categories   ccatc 17346
            ​*8.3.3  The category of extensible structures   fncnvimaeqv 17362
      ​8.4  Categorical constructions
            8.4.1  Product of categories   cxpc 17410
            8.4.2  Functor evaluation   cevlf 17451
            8.4.3  Hom functor   chof 17490
​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 17542
            9.2.2  Lattices   clat 17647
            9.2.3  The dual of an ordered set   codu 17730
            9.2.4  Subset order structures   cipo 17753
            9.2.5  Distributive lattices   latmass 17790
            9.2.6  Posets and lattices as relations   cps 17800
            9.2.7  Directed sets, nets   cdir 17830
​PART 10  BASIC ALGEBRAIC STRUCTURES
      ​10.1  Monoids
            ​*10.1.1  Magmas   cplusf 17841
            ​*10.1.2  Identity elements   mgmidmo 17862
            ​*10.1.3  Iterated sums in a magma   gsumvalx 17878
            ​*10.1.4  Semigroups   csgrp 17892
            ​*10.1.5  Definition and basic properties of monoids   cmnd 17903
            10.1.6  Monoid homomorphisms and submonoids   cmhm 17946
            ​*10.1.7  Iterated sums in a monoid   gsumvallem2 17990
            10.1.8  Free monoids   cfrmd 18004
                  ​*10.1.8.1  Monoid of endofunctions   cefmnd 18025
            10.1.9  Examples and counterexamples for magmas, semigroups and monoids   mgm2nsgrplem1 18075
      ​10.2  Groups
            10.2.1  Definition and basic properties   cgrp 18095
            ​*10.2.2  Group multiple operation   cmg 18216
            10.2.3  Subgroups and Quotient groups   csubg 18265
            ​*10.2.4  Cyclic monoids and groups   cycsubmel 18335
            10.2.5  Elementary theory of group homomorphisms   cghm 18347
            10.2.6  Isomorphisms of groups   cgim 18389
            10.2.7  Group actions   cga 18411
            10.2.8  Centralizers and centers   ccntz 18437
            10.2.9  The opposite group   coppg 18465
            10.2.10  Symmetric groups   csymg 18487
                  ​*10.2.10.1  Definition and basic properties   csymg 18487
                  10.2.10.2  Cayley's theorem   cayleylem1 18532
                  10.2.10.3  Permutations fixing one element   symgfix2 18536
                  ​*10.2.10.4  Transpositions in the symmetric group   cpmtr 18561
                  10.2.10.5  The sign of a permutation   cpsgn 18609
            10.2.11  p-Groups and Sylow groups; Sylow's theorems   cod 18644
            10.2.12  Direct products   clsm 18751
                  10.2.12.1  Direct products (extension)   smndlsmidm 18773
            10.2.13  Free groups   cefg 18824
            10.2.14  Abelian groups   ccmn 18898
                  10.2.14.1  Definition and basic properties   ccmn 18898
                  10.2.14.2  Cyclic groups   ccyg 18989
                  10.2.14.3  Group sum operation   gsumval3a 19016
                  10.2.14.4  Group sums over (ranges of) integers   fsfnn0gsumfsffz 19096
                  10.2.14.5  Internal direct products   cdprd 19108
                  10.2.14.6  The Fundamental Theorem of Abelian Groups   ablfacrplem 19180
            10.2.15  Simple groups   csimpg 19205
                  10.2.15.1  Definition and basic properties   csimpg 19205
                  10.2.15.2  Classification of abelian simple groups   ablsimpnosubgd 19219
      ​10.3  Rings
            10.3.1  Multiplicative Group   cmgp 19232
            10.3.2  Ring unit   cur 19244
                  10.3.2.1  Semirings   csrg 19248
                  ​*10.3.2.2  The binomial theorem for semirings   srgbinomlem1 19283
            10.3.3  Definition and basic properties of unital rings   crg 19290
            10.3.4  Opposite ring   coppr 19368
            10.3.5  Divisibility   cdsr 19384
            10.3.6  Ring primes   crpm 19458
            10.3.7  Ring homomorphisms   crh 19460
      ​10.4  Division rings and fields
            10.4.1  Definition and basic properties   cdr 19495
            10.4.2  Subrings of a ring   csubrg 19524
                  10.4.2.1  Sub-division rings   csdrg 19565
            10.4.3  Absolute value (abstract algebra)   cabv 19580
            10.4.4  Star rings   cstf 19607
      ​10.5  Left modules
            10.5.1  Definition and basic properties   clmod 19627
            10.5.2  Subspaces and spans in a left module   clss 19696
            10.5.3  Homomorphisms and isomorphisms of left modules   clmhm 19784
            10.5.4  Subspace sum; bases for a left module   clbs 19839
      ​10.6  Vector spaces
            10.6.1  Definition and basic properties   clvec 19867
      ​10.7  Ideals
            10.7.1  The subring algebra; ideals   csra 19933
            10.7.2  Two-sided ideals and quotient rings   c2idl 19997
            10.7.3  Principal ideal rings. Divisibility in the integers   clpidl 20007
            10.7.4  Nonzero rings and zero rings   cnzr 20023
            10.7.5  Left regular elements. More kinds of rings   crlreg 20045
      ​10.8  The complex numbers as an algebraic extensible structure
            10.8.1  Definition and basic properties   cpsmet 20075
            ​*10.8.2  Ring of integers   zring 20163
            10.8.3  Algebraic constructions based on the complex numbers   czrh 20193
            10.8.4  Signs as subgroup of the complex numbers   cnmsgnsubg 20266
            10.8.5  Embedding of permutation signs into a ring   zrhpsgnmhm 20273
            10.8.6  The ordered field of real numbers   crefld 20293
      ​10.9  Generalized pre-Hilbert and Hilbert spaces
            10.9.1  Definition and basic properties   cphl 20313
            10.9.2  Orthocomplements and closed subspaces   cocv 20349
            10.9.3  Orthogonal projection and orthonormal bases   cpj 20389
​​*PART 11  BASIC LINEAR ALGEBRA
      ​11.1  Vectors and free modules
            *11.1.1  Direct sum of left modules   cdsmm 20420
            ​*11.1.2  Free modules   cfrlm 20435
            ​*11.1.3  Standard basis (unit vectors)   cuvc 20471
            ​*11.1.4  Independent sets and families   clindf 20493
            11.1.5  Characterization of free modules   lmimlbs 20525
      ​11.2  Associative algebras
            11.2.1  Definition and basic properties   casa 20539
      ​11.3  Abstract multivariate polynomials
            11.3.1  Definition and basic properties   cmps 20589
            11.3.2  Polynomial evaluation   ces 20743
            11.3.3  Additional definitions for (multivariate) polynomials   cslv 20780
            ​*11.3.4  Univariate polynomials   cps1 20804
            11.3.5  Univariate polynomial evaluation   ces1 20937
      ​​*11.4  Matrices
            *11.4.1  The matrix multiplication   cmmul 20990
            ​*11.4.2  Square matrices   cmat 21012
            ​*11.4.3  The matrix algebra   matmulr 21043
            ​*11.4.4  Matrices of dimension 0 and 1   mat0dimbas0 21071
            ​*11.4.5  The subalgebras of diagonal and scalar matrices   cdmat 21093
            ​*11.4.6  Multiplication of a matrix with a "column vector"   cmvmul 21145
            11.4.7  Replacement functions for a square matrix   cmarrep 21161
            11.4.8  Submatrices   csubma 21181
      ​11.5  The determinant
            11.5.1  Definition and basic properties   cmdat 21189
            11.5.2  Determinants of 2 x 2 -matrices   m2detleiblem1 21229
            11.5.3  The matrix adjugate/adjunct   cmadu 21237
            ​*11.5.4  Laplace expansion of determinants (special case)   symgmatr01lem 21258
            11.5.5  Inverse matrix   invrvald 21281
            ​*11.5.6  Cramer's rule   slesolvec 21284
      ​​*11.6  Polynomial matrices
            11.6.1  Basic properties   pmatring 21297
            ​*11.6.2  Constant polynomial matrices   ccpmat 21308
            ​*11.6.3  Collecting coefficients of polynomial matrices   cdecpmat 21367
            ​*11.6.4  Ring isomorphism between polynomial matrices and polynomials over matrices   cpm2mp 21397
      ​​*11.7  The characteristic polynomial
            *11.7.1  Definition and basic properties   cchpmat 21431
            ​*11.7.2  The characteristic factor function G   fvmptnn04if 21454
            ​*11.7.3  The Cayley-Hamilton theorem   cpmadurid 21472
​PART 12  BASIC TOPOLOGY
      ​12.1  Topology
            ​*12.1.1  Topological spaces   ctop 21498
                  12.1.1.1  Topologies   ctop 21498
                  12.1.1.2  Topologies on sets   ctopon 21515
                  12.1.1.3  Topological spaces   ctps 21537
            12.1.2  Topological bases   ctb 21550
            12.1.3  Examples of topologies   distop 21600
            12.1.4  Closure and interior   ccld 21621
            12.1.5  Neighborhoods   cnei 21702
            12.1.6  Limit points and perfect sets   clp 21739
            12.1.7  Subspace topologies   restrcl 21762
            12.1.8  Order topology   ordtbaslem 21793
            12.1.9  Limits and continuity in topological spaces   ccn 21829
            12.1.10  Separated spaces: T0, T1, T2 (Hausdorff) ...   ct0 21911
            12.1.11  Compactness   ccmp 21991
            12.1.12  Bolzano-Weierstrass theorem   bwth 22015
            12.1.13  Connectedness   cconn 22016
            12.1.14  First- and second-countability   c1stc 22042
            12.1.15  Local topological properties   clly 22069
            12.1.16  Refinements   cref 22107
            12.1.17  Compactly generated spaces   ckgen 22138
            12.1.18  Product topologies   ctx 22165
            12.1.19  Continuous function-builders   cnmptid 22266
            12.1.20  Quotient maps and quotient topology   ckq 22298
            12.1.21  Homeomorphisms   chmeo 22358
      ​12.2  Filters and filter bases
            12.2.1  Filter bases   elmptrab 22432
            12.2.2  Filters   cfil 22450
            12.2.3  Ultrafilters   cufil 22504
            12.2.4  Filter limits   cfm 22538
            12.2.5  Extension by continuity   ccnext 22664
            12.2.6  Topological groups   ctmd 22675
            12.2.7  Infinite group sum on topological groups   ctsu 22731
            12.2.8  Topological rings, fields, vector spaces   ctrg 22761
      ​12.3  Uniform Structures and Spaces
            12.3.1  Uniform structures   cust 22805
            12.3.2  The topology induced by an uniform structure   cutop 22836
            12.3.3  Uniform Spaces   cuss 22859
            12.3.4  Uniform continuity   cucn 22881
            12.3.5  Cauchy filters in uniform spaces   ccfilu 22892
            12.3.6  Complete uniform spaces   ccusp 22903
      ​12.4  Metric spaces
            12.4.1  Pseudometric spaces   ispsmet 22911
            12.4.2  Basic metric space properties   cxms 22924
            12.4.3  Metric space balls   blfvalps 22990
            12.4.4  Open sets of a metric space   mopnval 23045
            12.4.5  Continuity in metric spaces   metcnp3 23147
            12.4.6  The uniform structure generated by a metric   metuval 23156
            12.4.7  Examples of metric spaces   dscmet 23179
            ​*12.4.8  Normed algebraic structures   cnm 23183
            12.4.9  Normed space homomorphisms (bounded linear operators)   cnmo 23311
            12.4.10  Topology on the reals   qtopbaslem 23364
            12.4.11  Topological definitions using the reals   cii 23480
            12.4.12  Path homotopy   chtpy 23572
            12.4.13  The fundamental group   cpco 23605
      ​12.5  Metric subcomplex vector spaces
            12.5.1  Subcomplex modules   cclm 23667
            ​*12.5.2  Subcomplex vector spaces   ccvs 23728
            ​*12.5.3  Normed subcomplex vector spaces   isncvsngp 23754
            12.5.4  Subcomplex pre-Hilbert space   ccph 23771
            12.5.5  Convergence and completeness   ccfil 23856
            12.5.6  Baire's Category Theorem   bcthlem1 23928
            12.5.7  Banach spaces and subcomplex Hilbert spaces   ccms 23936
                  12.5.7.1  The complete ordered field of the real numbers   retopn 23983
            12.5.8  Euclidean spaces   crrx 23987
            12.5.9  Minimizing Vector Theorem   minveclem1 24028
            12.5.10  Projection Theorem   pjthlem1 24041
​PART 13  BASIC REAL AND COMPLEX ANALYSIS
      ​13.1  Continuity
            13.1.1  Intermediate value theorem   pmltpclem1 24052
      ​13.2  Integrals
            13.2.1  Lebesgue measure   covol 24066
            13.2.2  Lebesgue integration   cmbf 24218
                  13.2.2.1  Lesbesgue integral   cmbf 24218
                  13.2.2.2  Lesbesgue directed integral   cdit 24449
      ​13.3  Derivatives
            13.3.1  Real and complex differentiation   climc 24465
                  13.3.1.1  Derivatives of functions of one complex or real variable   climc 24465
                  13.3.1.2  Results on real differentiation   dvferm1lem 24587
​PART 14  BASIC REAL AND COMPLEX FUNCTIONS
      ​14.1  Polynomials
            14.1.1  Polynomial degrees   cmdg 24654
            14.1.2  The division algorithm for univariate polynomials   cmn1 24726
            14.1.3  Elementary properties of complex polynomials   cply 24781
            14.1.4  The division algorithm for polynomials   cquot 24886
            14.1.5  Algebraic numbers   caa 24910
            14.1.6  Liouville's approximation theorem   aalioulem1 24928
      ​14.2  Sequences and series
            14.2.1  Taylor polynomials and Taylor's theorem   ctayl 24948
            14.2.2  Uniform convergence   culm 24971
            14.2.3  Power series   pserval 25005
      ​14.3  Basic trigonometry
            14.3.1  The exponential, sine, and cosine functions (cont.)   efcn 25038
            14.3.2  Properties of pi = 3.14159...   pilem1 25046
            14.3.3  Mapping of the exponential function   efgh 25133
            14.3.4  The natural logarithm on complex numbers   clog 25146
            ​*14.3.5  Logarithms to an arbitrary base   clogb 25350
            14.3.6  Theorems of Pythagoras, isosceles triangles, and intersecting chords   angval 25387
            14.3.7  Solutions of quadratic, cubic, and quartic equations   quad2 25425
            14.3.8  Inverse trigonometric functions   casin 25448
            14.3.9  The Birthday Problem   log2ublem1 25532
            14.3.10  Areas in R^2   carea 25541
            14.3.11  More miscellaneous converging sequences   rlimcnp 25551
            14.3.12  Inequality of arithmetic and geometric means   cvxcl 25570
            14.3.13  Euler-Mascheroni constant   cem 25577
            14.3.14  Zeta function   czeta 25598
            14.3.15  Gamma function   clgam 25601
      ​14.4  Basic number theory
            14.4.1  Wilson's theorem   wilthlem1 25653
            14.4.2  The Fundamental Theorem of Algebra   ftalem1 25658
            14.4.3  The Basel problem (ζ(2) = π²/6)   basellem1 25666
            14.4.4  Number-theoretical functions   ccht 25676
            14.4.5  Perfect Number Theorem   mersenne 25811
            14.4.6  Characters of Z/nZ   cdchr 25816
            14.4.7  Bertrand's postulate   bcctr 25859
            ​*14.4.8  Quadratic residues and the Legendre symbol   clgs 25878
            ​*14.4.9  Gauss' Lemma   gausslemma2dlem0a 25940
            14.4.10  Quadratic reciprocity   lgseisenlem1 25959
            14.4.11  All primes 4n+1 are the sum of two squares   2sqlem1 26001
            14.4.12  Chebyshev's Weak Prime Number Theorem, Dirichlet's Theorem   chebbnd1lem1 26053
            14.4.13  The Prime Number Theorem   mudivsum 26114
            14.4.14  Ostrowski's theorem   abvcxp 26199
​​*PART 15  ELEMENTARY GEOMETRY
      ​15.1  Definition and Tarski's Axioms of Geometry
            15.1.1  Justification for the congruence notation   tgjustf 26267
      ​15.2  Tarskian Geometry
            15.2.1  Congruence   tgcgrcomimp 26271
            15.2.2  Betweenness   tgbtwntriv2 26281
            15.2.3  Dimension   tglowdim1 26294
            15.2.4  Betweenness and Congruence   tgifscgr 26302
            15.2.5  Congruence of a series of points   ccgrg 26304
            15.2.6  Motions   cismt 26326
            15.2.7  Colinearity   tglng 26340
            15.2.8  Connectivity of betweenness   tgbtwnconn1lem1 26366
            15.2.9  Less-than relation in geometric congruences   cleg 26376
            15.2.10  Rays   chlg 26394
            15.2.11  Lines   btwnlng1 26413
            15.2.12  Point inversions   cmir 26446
            15.2.13  Right angles   crag 26487
            15.2.14  Half-planes   islnopp 26533
            15.2.15  Midpoints and Line Mirroring   cmid 26566
            15.2.16  Congruence of angles   ccgra 26601
            15.2.17  Angle Comparisons   cinag 26629
            15.2.18  Congruence Theorems   tgsas1 26648
            15.2.19  Equilateral triangles   ceqlg 26659
      ​15.3  Properties of geometries
            15.3.1  Isomorphisms between geometries   f1otrgds 26663
      ​15.4  Geometry in Hilbert spaces
            15.4.1  Geometry in the complex plane   cchhllem 26681
            15.4.2  Geometry in Euclidean spaces   cee 26682
                  15.4.2.1  Definition of the Euclidean space   cee 26682
                  15.4.2.2  Tarski's axioms for geometry for the Euclidean space   axdimuniq 26707
                  15.4.2.3  EE^n fulfills Tarski's Axioms   ceeng 26771
​​*PART 16  GRAPH THEORY
      ​*16.1  Vertices and edges
            16.1.1  The edge function extractor for extensible structures   cedgf 26782
            ​*16.1.2  Vertices and indexed edges   cvtx 26789
                  16.1.2.1  Definitions and basic properties   cvtx 26789
                  16.1.2.2  The vertices and edges of a graph represented as ordered pair   opvtxval 26796
                  16.1.2.3  The vertices and edges of a graph represented as extensible structure   funvtxdmge2val 26804
                  16.1.2.4  Representations of graphs without edges   snstrvtxval 26830
                  16.1.2.5  Degenerated cases of representations of graphs   vtxval0 26832
            16.1.3  Edges as range of the edge function   cedg 26840
      ​​*16.2  Undirected graphs
            16.2.1  Undirected hypergraphs   cuhgr 26849
            16.2.2  Undirected pseudographs and multigraphs   cupgr 26873
            ​*16.2.3  Loop-free graphs   umgrislfupgrlem 26915
            16.2.4  Edges as subsets of vertices of graphs   uhgredgiedgb 26919
            ​*16.2.5  Undirected simple graphs   cuspgr 26941
            16.2.6  Examples for graphs   usgr0e 27026
            16.2.7  Subgraphs   csubgr 27057
            16.2.8  Finite undirected simple graphs   cfusgr 27106
            16.2.9  Neighbors, complete graphs and universal vertices   cnbgr 27122
                  16.2.9.1  Neighbors   cnbgr 27122
                  16.2.9.2  Universal vertices   cuvtx 27175
                  16.2.9.3  Complete graphs   ccplgr 27199
            16.2.10  Vertex degree   cvtxdg 27255
            ​*16.2.11  Regular graphs   crgr 27345
      ​​*16.3  Walks, paths and cycles
            *16.3.1  Walks   cewlks 27385
            16.3.2  Walks for loop-free graphs   lfgrwlkprop 27477
            16.3.3  Trails   ctrls 27480
            16.3.4  Paths and simple paths   cpths 27501
            16.3.5  Closed walks   cclwlks 27559
            16.3.6  Circuits and cycles   ccrcts 27573
            ​*16.3.7  Walks as words   cwwlks 27611
            16.3.8  Walks/paths of length 2 (as length 3 strings)   2wlkdlem1 27711
            16.3.9  Walks in regular graphs   rusgrnumwwlkl1 27754
            ​*16.3.10  Closed walks as words   cclwwlk 27766
                  16.3.10.1  Closed walks as words   cclwwlk 27766
                  16.3.10.2  Closed walks of a fixed length as words   cclwwlkn 27809
                  16.3.10.3  Closed walks on a vertex of a fixed length as words   cclwwlknon 27872
            16.3.11  Examples for walks, trails and paths   0ewlk 27899
            16.3.12  Connected graphs   cconngr 27971
      ​16.4  Eulerian paths and the Konigsberg Bridge problem
            ​*16.4.1  Eulerian paths   ceupth 27982
            ​*16.4.2  The Königsberg Bridge problem   konigsbergvtx 28031
      ​16.5  The Friendship Theorem
            16.5.1  Friendship graphs - basics   cfrgr 28043
            16.5.2  The friendship theorem for small graphs   frgr1v 28056
            16.5.3  Theorems according to Mertzios and Unger   2pthfrgrrn 28067
            ​*16.5.4  Huneke's Proof of the Friendship Theorem   frgrncvvdeqlem1 28084
​PART 17  GUIDES AND MISCELLANEA
      ​17.1  Guides (conventions, explanations, and examples)
            ​*17.1.1  Conventions   conventions 28185
            17.1.2  Natural deduction   natded 28188
            ​*17.1.3  Natural deduction examples   ex-natded5.2 28189
            17.1.4  Definitional examples   ex-or 28206
            17.1.5  Other examples   aevdemo 28245
      ​17.2  Humor
            17.2.1  April Fool's theorem   avril1 28248
      ​17.3  (Future - to be reviewed and classified)
            17.3.1  Planar incidence geometry   cplig 28257
            ​*17.3.2  Aliases kept to prevent broken links   dummylink 28270
​​*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 28272
            18.1.2  Abelian groups   cablo 28327
      ​18.2  Complex vector spaces
            18.2.1  Definition and basic properties   cvc 28341
            18.2.2  Examples of complex vector spaces   cnaddabloOLD 28364
      ​18.3  Normed complex vector spaces
            18.3.1  Definition and basic properties   cnv 28367
            18.3.2  Examples of normed complex vector spaces   cnnv 28460
            18.3.3  Induced metric of a normed complex vector space   imsval 28468
            18.3.4  Inner product   cdip 28483
            18.3.5  Subspaces   css 28504
      ​18.4  Operators on complex vector spaces
            18.4.1  Definitions and basic properties   clno 28523
      ​18.5  Inner product (pre-Hilbert) spaces
            18.5.1  Definition and basic properties   ccphlo 28595
            18.5.2  Examples of pre-Hilbert spaces   cncph 28602
            18.5.3  Properties of pre-Hilbert spaces   isph 28605
      ​18.6  Complex Banach spaces
            18.6.1  Definition and basic properties   ccbn 28645
            18.6.2  Examples of complex Banach spaces   cnbn 28652
            18.6.3  Uniform Boundedness Theorem   ubthlem1 28653
            18.6.4  Minimizing Vector Theorem   minvecolem1 28657
      ​18.7  Complex Hilbert spaces
            18.7.1  Definition and basic properties   chlo 28668
            18.7.2  Standard axioms for a complex Hilbert space   hlex 28681
            18.7.3  Examples of complex Hilbert spaces   cnchl 28699
            18.7.4  Hellinger-Toeplitz Theorem   htthlem 28700
​​*PART 19  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      ​19.1  Axiomatization of complex pre-Hilbert spaces
            19.1.1  Basic Hilbert space definitions   chba 28702
            19.1.2  Preliminary ZFC lemmas   df-hnorm 28751
            ​*19.1.3  Derive the Hilbert space axioms from ZFC set theory   axhilex-zf 28764
            ​*19.1.4  Introduce the vector space axioms for a Hilbert space   ax-hilex 28782
            19.1.5  Vector operations   hvmulex 28794
            19.1.6  Inner product postulates for a Hilbert space   ax-hfi 28862
      ​19.2  Inner product and norms
            19.2.1  Inner product   his5 28869
            19.2.2  Norms   dfhnorm2 28905
            19.2.3  Relate Hilbert space to normed complex vector spaces   hilablo 28943
            19.2.4  Bunjakovaskij-Cauchy-Schwarz inequality   bcsiALT 28962
      ​19.3  Cauchy sequences and completeness axiom
            19.3.1  Cauchy sequences and limits   hcau 28967
            19.3.2  Derivation of the completeness axiom from ZF set theory   hilmet 28977
            19.3.3  Completeness postulate for a Hilbert space   ax-hcompl 28985
            19.3.4  Relate Hilbert space to ZFC pre-Hilbert and Hilbert spaces   hhcms 28986
      ​19.4  Subspaces and projections
            19.4.1  Subspaces   df-sh 28990
            19.4.2  Closed subspaces   df-ch 29004
            19.4.3  Orthocomplements   df-oc 29035
            19.4.4  Subspace sum, span, lattice join, lattice supremum   df-shs 29091
            19.4.5  Projection theorem   pjhthlem1 29174
            19.4.6  Projectors   df-pjh 29178
      ​19.5  Properties of Hilbert subspaces
            19.5.1  Orthomodular law   omlsilem 29185
            19.5.2  Projectors (cont.)   pjhtheu2 29199
            19.5.3  Hilbert lattice operations   sh0le 29223
            19.5.4  Span (cont.) and one-dimensional subspaces   spansn0 29324
            19.5.5  Commutes relation for Hilbert lattice elements   df-cm 29366
            19.5.6  Foulis-Holland theorem   fh1 29401
            19.5.7  Quantum Logic Explorer axioms   qlax1i 29410
            19.5.8  Orthogonal subspaces   chscllem1 29420
            19.5.9  Orthoarguesian laws 5OA and 3OA   5oalem1 29437
            19.5.10  Projectors (cont.)   pjorthi 29452
            19.5.11  Mayet's equation E_3   mayete3i 29511
      ​19.6  Operators on Hilbert spaces
            ​*19.6.1  Operator sum, difference, and scalar multiplication   df-hosum 29513
            19.6.2  Zero and identity operators   df-h0op 29531
            19.6.3  Operations on Hilbert space operators   hoaddcl 29541
            19.6.4  Linear, continuous, bounded, Hermitian, unitary operators and norms   df-nmop 29622
            19.6.5  Linear and continuous functionals and norms   df-nmfn 29628
            19.6.6  Adjoint   df-adjh 29632
            19.6.7  Dirac bra-ket notation   df-bra 29633
            19.6.8  Positive operators   df-leop 29635
            19.6.9  Eigenvectors, eigenvalues, spectrum   df-eigvec 29636
            19.6.10  Theorems about operators and functionals   nmopval 29639
            19.6.11  Riesz lemma   riesz3i 29845
            19.6.12  Adjoints (cont.)   cnlnadjlem1 29850
            19.6.13  Quantum computation error bound theorem   unierri 29887
            19.6.14  Dirac bra-ket notation (cont.)   branmfn 29888
            19.6.15  Positive operators (cont.)   leopg 29905
            19.6.16  Projectors as operators   pjhmopi 29929
      ​19.7  States on a Hilbert lattice and Godowski's equation
            19.7.1  States on a Hilbert lattice   df-st 29994
            19.7.2  Godowski's equation   golem1 30054
      ​19.8  Cover relation, atoms, exchange axiom, and modular symmetry
            19.8.1  Covers relation; modular pairs   df-cv 30062
            19.8.2  Atoms   df-at 30121
            19.8.3  Superposition principle   superpos 30137
            19.8.4  Atoms, exchange and covering properties, atomicity   chcv1 30138
            19.8.5  Irreducibility   chirredlem1 30173
            19.8.6  Atoms (cont.)   atcvat3i 30179
            19.8.7  Modular symmetry   mdsymlem1 30186
​PART 20  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      ​20.1  Mathboxes for user contributions
            20.1.1  Mathbox guidelines   mathbox 30225
      ​20.2  Mathbox for Stefan Allan
      ​20.3  Mathbox for Thierry Arnoux
            20.3.1  Propositional Calculus - misc additions   bian1d 30230
            20.3.2  Predicate Calculus   sbc2iedf 30237
                  20.3.2.1  Predicate Calculus - misc additions   sbc2iedf 30237
                  20.3.2.2  Restricted quantification - misc additions   ralcom4f 30240
                  20.3.2.3  Equality   eqtrb 30245
                  20.3.2.4  Double restricted existential uniqueness quantification   opsbc2ie 30246
                  20.3.2.5  Double restricted existential uniqueness quantification syntax   w2reu 30248
                  20.3.2.6  Substitution (without distinct variables) - misc additions   sbceqbidf 30257
                  20.3.2.7  Existential "at most one" - misc additions   moel 30259
                  20.3.2.8  Existential uniqueness - misc additions   reuxfrdf 30262
                  20.3.2.9  Restricted "at most one" - misc additions   rmoxfrd 30264
            20.3.3  General Set Theory   dmrab 30267
                  20.3.3.1  Class abstractions (a.k.a. class builders)   dmrab 30267
                  20.3.3.2  Image Sets   abrexdomjm 30275
                  20.3.3.3  Set relations and operations - misc additions   elunsn 30281
                  20.3.3.4  Unordered pairs   eqsnd 30301
                  20.3.3.5  Conditional operator - misc additions   ifeqeqx 30309
                  20.3.3.6  Set union   uniinn0 30314
                  20.3.3.7  Indexed union - misc additions   cbviunf 30319
                  20.3.3.8  Indexed intersection - misc additions   iinabrex 30332
                  20.3.3.9  Disjointness - misc additions   disjnf 30333
            20.3.4  Relations and Functions   xpdisjres 30361
                  20.3.4.1  Relations - misc additions   xpdisjres 30361
                  20.3.4.2  Functions - misc additions   ac6sf2 30384
                  20.3.4.3  Operations - misc additions   mpomptxf 30442
                  20.3.4.4  Explicit Functions with one or two points as a domain   brsnop 30453
                  20.3.4.5  Isomorphisms - misc. add.   gtiso 30460
                  20.3.4.6  Disjointness (additional proof requiring functions)   disjdsct 30462
                  20.3.4.7  First and second members of an ordered pair - misc additions   df1stres 30463
                  20.3.4.8  Supremum - misc additions   supssd 30471
                  20.3.4.9  Finite Sets   imafi2 30473
                  20.3.4.10  Countable Sets   snct 30475
            20.3.5  Real and Complex Numbers   creq0 30497
                  20.3.5.1  Complex operations - misc. additions   creq0 30497
                  20.3.5.2  Ordering on reals - misc additions   lt2addrd 30501
                  20.3.5.3  Extended reals - misc additions   xrlelttric 30502
                  20.3.5.4  Extended nonnegative integers - misc additions   xnn0gt0 30520
                  20.3.5.5  Real number intervals - misc additions   joiniooico 30523
                  20.3.5.6  Finite intervals of integers - misc additions   uzssico 30533
                  20.3.5.7  Half-open integer ranges - misc additions   iundisjfi 30545
                  20.3.5.8  The ` # ` (set size) function - misc additions   hashunif 30554
                  20.3.5.9  The greatest common divisor operator - misc. add   dvdszzq 30557
                  20.3.5.10  Integers   nnindf 30561
                  20.3.5.11  Decimal numbers   dfdec100 30572
            ​*20.3.6  Decimal expansion   cdp2 30573
                  ​*20.3.6.1  Decimal point   cdp 30590
                  20.3.6.2  Division in the extended real number system   cxdiv 30619
            20.3.7  Words over a set - misc additions   wrdfd 30638
                  20.3.7.1  Splicing words (substring replacement)   splfv3 30658
                  20.3.7.2  Cyclic shift of words   1cshid 30659
            20.3.8  Extensible Structures   ressplusf 30663
                  20.3.8.1  Structure restriction operator   ressplusf 30663
                  20.3.8.2  The opposite group   oppgle 30666
                  20.3.8.3  Posets   ressprs 30668
                  20.3.8.4  Complete lattices   clatp0cl 30684
                  20.3.8.5  Order Theory   cmnt 30686
                  20.3.8.6  Extended reals Structure - misc additions   ax-xrssca 30707
                  20.3.8.7  The extended nonnegative real numbers commutative monoid   xrge0base 30719
            20.3.9  Algebra   abliso 30730
                  20.3.9.1  Monoids Homomorphisms   abliso 30730
                  20.3.9.2  Finitely supported group sums - misc additions   gsumsubg 30731
                  20.3.9.3  Centralizers and centers - misc additions   cntzun 30745
                  20.3.9.4  Totally ordered monoids and groups   comnd 30748
                  20.3.9.5  The symmetric group   symgfcoeu 30776
                  20.3.9.6  Transpositions   pmtridf1o 30786
                  20.3.9.7  Permutation Signs   psgnid 30789
                  20.3.9.8  Permutation cycles   ctocyc 30798
                  20.3.9.9  The Alternating Group   evpmval 30837
                  20.3.9.10  Signum in an ordered monoid   csgns 30850
                  20.3.9.11  The Archimedean property for generic ordered algebraic structures   cinftm 30855
                  20.3.9.12  Semiring left modules   cslmd 30878
                  20.3.9.13  Simple groups   prmsimpcyc 30906
                  20.3.9.14  Rings - misc additions   rngurd 30907
                  20.3.9.15  Subfields   primefldchr 30918
                  20.3.9.16  Totally ordered rings and fields   corng 30919
                  20.3.9.17  Ring homomorphisms - misc additions   rhmdvdsr 30942
                  20.3.9.18  Scalar restriction operation   cresv 30948
                  20.3.9.19  The commutative ring of gaussian integers   gzcrng 30963
                  20.3.9.20  The archimedean ordered field of real numbers   reofld 30964
                  20.3.9.21  The quotient map and quotient modules   qusker 30969
                  20.3.9.22  Univariate Polynomials   fply1 30982
                  20.3.9.23  Independent sets and families   islinds5 30983
                  ​*20.3.9.24  Subgroup sum / Sumset / Minkowski sum   elgrplsmsn 30998
                  20.3.9.25  Ideals   intlidl 31010
                  20.3.9.26  Prime Ideals   cprmidl 31018
                  20.3.9.27  Maximal Ideals   cmxidl 31039
                  20.3.9.28  The semiring of ideals of a ring   cidlsrg 31053
                  20.3.9.29  Unique factorization domains   cufd 31069
                  20.3.9.30  The subring algebra   sra1r 31074
                  20.3.9.31  Division Ring Extensions   drgext0g 31080
                  20.3.9.32  Vector Spaces   lvecdimfi 31086
                  20.3.9.33  Vector Space Dimension   cldim 31087
            20.3.10  Field Extensions   cfldext 31116
            20.3.11  Matrices   csmat 31146
                  20.3.11.1  Submatrices   csmat 31146
                  20.3.11.2  Matrix literals   clmat 31164
                  20.3.11.3  Laplace expansion of determinants   mdetpmtr1 31176
            20.3.12  Topology   ist0cld 31186
                  20.3.12.1  Open maps   txomap 31187
                  20.3.12.2  Topology of the unit circle   qtopt1 31188
                  20.3.12.3  Refinements   reff 31192
                  20.3.12.4  Open cover refinement property   ccref 31195
                  20.3.12.5  Lindelöf spaces   cldlf 31205
                  20.3.12.6  Paracompact spaces   cpcmp 31208
                  ​*20.3.12.7  Spectrum of a ring   crspec 31215
                  20.3.12.8  Pseudometrics   cmetid 31239
                  20.3.12.9  Continuity - misc additions   hauseqcn 31251
                  20.3.12.10  Topology of the closed unit interval   elunitge0 31252
                  20.3.12.11  Topology of ` ( RR X. RR ) `   unicls 31256
                  20.3.12.12  Order topology - misc. additions   cnvordtrestixx 31266
                  20.3.12.13  Continuity in topological spaces - misc. additions   mndpluscn 31279
                  20.3.12.14  Topology of the extended nonnegative real numbers ordered monoid   xrge0hmph 31285
                  20.3.12.15  Limits - misc additions   lmlim 31300
                  20.3.12.16  Univariate polynomials   pl1cn 31308
            20.3.13  Uniform Stuctures and Spaces   chcmp 31309
                  20.3.13.1  Hausdorff uniform completion   chcmp 31309
            20.3.14  Topology and algebraic structures   zringnm 31311
                  20.3.14.1  The norm on the ring of the integer numbers   zringnm 31311
                  20.3.14.2  Topological ` ZZ ` -modules   zlm0 31313
                  20.3.14.3  Canonical embedding of the field of the rational numbers into a division ring   cqqh 31323
                  20.3.14.4  Canonical embedding of the real numbers into a complete ordered field   crrh 31344
                  20.3.14.5  Embedding from the extended real numbers into a complete lattice   cxrh 31367
                  20.3.14.6  Canonical embeddings into the ordered field of the real numbers   zrhre 31370
                  ​*20.3.14.7  Topological Manifolds   cmntop 31373
            20.3.15  Real and complex functions   nexple 31378
                  20.3.15.1  Integer powers - misc. additions   nexple 31378
                  20.3.15.2  Indicator Functions   cind 31379
                  20.3.15.3  Extended sum   cesum 31396
            20.3.16  Mixed Function/Constant operation   cofc 31464
            20.3.17  Abstract measure   csiga 31477
                  20.3.17.1  Sigma-Algebra   csiga 31477
                  20.3.17.2  Generated sigma-Algebra   csigagen 31507
                  ​*20.3.17.3  lambda and pi-Systems, Rings of Sets   ispisys 31521
                  20.3.17.4  The Borel algebra on the real numbers   cbrsiga 31550
                  20.3.17.5  Product Sigma-Algebra   csx 31557
                  20.3.17.6  Measures   cmeas 31564
                  20.3.17.7  The counting measure   cntmeas 31595
                  20.3.17.8  The Lebesgue measure - misc additions   voliune 31598
                  20.3.17.9  The Dirac delta measure   cdde 31601
                  20.3.17.10  The 'almost everywhere' relation   cae 31606
                  20.3.17.11  Measurable functions   cmbfm 31618
                  20.3.17.12  Borel Algebra on ` ( RR X. RR ) `   br2base 31637
                  ​*20.3.17.13  Caratheodory's extension theorem   coms 31659
            20.3.18  Integration   itgeq12dv 31694
                  20.3.18.1  Lebesgue integral - misc additions   itgeq12dv 31694
                  20.3.18.2  Bochner integral   citgm 31695
            20.3.19  Euler's partition theorem   oddpwdc 31722
            20.3.20  Sequences defined by strong recursion   csseq 31751
            20.3.21  Fibonacci Numbers   cfib 31764
            20.3.22  Probability   cprb 31775
                  20.3.22.1  Probability Theory   cprb 31775
                  20.3.22.2  Conditional Probabilities   ccprob 31799
                  20.3.22.3  Real-valued Random Variables   crrv 31808
                  20.3.22.4  Preimage set mapping operator   corvc 31823
                  20.3.22.5  Distribution Functions   orvcelval 31836
                  20.3.22.6  Cumulative Distribution Functions   orvclteel 31840
                  20.3.22.7  Probabilities - example   coinfliplem 31846
                  20.3.22.8  Bertrand's Ballot Problem   ballotlemoex 31853
            20.3.23  Signum (sgn or sign) function - misc. additions   sgncl 31906
                  20.3.23.1  Operations on words   ccatmulgnn0dir 31922
            20.3.24  Polynomials with real coefficients - misc additions   plymul02 31926
            20.3.25  Descartes's rule of signs   signspval 31932
                  20.3.25.1  Sign changes in a word over real numbers   signspval 31932
                  20.3.25.2  Counting sign changes in a word over real numbers   signslema 31942
            20.3.26  Number Theory   efcld 31972
                  20.3.26.1  Representations of a number as sums of integers   crepr 31989
                  20.3.26.2  Vinogradov Trigonometric Sums and the Circle Method   cvts 32016
                  20.3.26.3  The Ternary Goldbach Conjecture: Final Statement   ax-hgt749 32025
            20.3.27  Elementary Geometry   cstrkg2d 32045
                  ​*20.3.27.1  Two-dimensional geometry   cstrkg2d 32045
                  20.3.27.2  Outer Five Segment (not used, no need to move to main)   cafs 32050
            ​*20.3.28  LeftPad Project   clpad 32055
      ​​*20.4  Mathbox for Jonathan Ben-Naim
            20.4.1  First-order logic and set theory   bnj170 32078
            20.4.2  Well founded induction and recursion   bnj110 32240
            20.4.3  The existence of a minimal element in certain classes   bnj69 32392
            20.4.4  Well-founded induction   bnj1204 32394
            20.4.5  Well-founded recursion, part 1 of 3   bnj60 32444
            20.4.6  Well-founded recursion, part 2 of 3   bnj1500 32450
            20.4.7  Well-founded recursion, part 3 of 3   bnj1522 32454
      ​20.5  Mathbox for BTernaryTau
            20.5.1  Acyclic graphs   cacycgr 32502
      ​20.6  Mathbox for Mario Carneiro
            20.6.1  Predicate calculus with all distinct variables   ax-7d 32519
            20.6.2  Miscellaneous stuff   quartfull 32525
            20.6.3  Derangements and the Subfactorial   deranglem 32526
            20.6.4  The Erdős-Szekeres theorem   erdszelem1 32551
            20.6.5  The Kuratowski closure-complement theorem   kur14lem1 32566
            20.6.6  Retracts and sections   cretr 32577
            20.6.7  Path-connected and simply connected spaces   cpconn 32579
            20.6.8  Covering maps   ccvm 32615
            20.6.9  Normal numbers   snmlff 32689
            20.6.10  Godel-sets of formulas - part 1   cgoe 32693
            20.6.11  Godel-sets of formulas - part 2   cgon 32792
            20.6.12  Models of ZF   cgze 32806
            ​*20.6.13  Metamath formal systems   cmcn 32820
            20.6.14  Grammatical formal systems   cm0s 32945
            20.6.15  Models of formal systems   cmuv 32965
            20.6.16  Splitting fields   citr 32987
            20.6.17  p-adic number fields   czr 33003
      ​​*20.7  Mathbox for Filip Cernatescu
      ​20.8  Mathbox for Paul Chapman
            20.8.1  Real and complex numbers (cont.)   climuzcnv 33027
            20.8.2  Miscellaneous theorems   elfzm12 33031
      ​20.9  Mathbox for Scott Fenton
            20.9.1  ZFC Axioms in primitive form   axextprim 33040
            20.9.2  Untangled classes   untelirr 33047
            20.9.3  Extra propositional calculus theorems   3orel2 33054
            20.9.4  Misc. Useful Theorems   nepss 33061
            20.9.5  Properties of real and complex numbers   sqdivzi 33072
            20.9.6  Infinite products   iprodefisumlem 33085
            20.9.7  Factorial limits   faclimlem1 33088
            20.9.8  Greatest common divisor and divisibility   pdivsq 33094
            20.9.9  Properties of relationships   brtp 33098
            20.9.10  Properties of functions and mappings   funpsstri 33121
            20.9.11  Set induction (or epsilon induction)   setinds 33136
            20.9.12  Ordinal numbers   elpotr 33139
            20.9.13  Defined equality axioms   axextdfeq 33155
            20.9.14  Hypothesis builders   hbntg 33163
            20.9.15  (Trans)finite Recursion Theorems   tfisg 33168
            20.9.16  Transitive closure under a relationship   ctrpred 33169
            20.9.17  Founded Induction   frpomin 33191
            20.9.18  Ordering Ordinal Sequences   orderseqlem 33207
            20.9.19  Well-founded zero, successor, and limits   cwsuc 33210
            20.9.20  Founded Partial Recursion   cfrecs 33230
            20.9.21  Surreal Numbers   csur 33260
            20.9.22  Surreal Numbers: Ordering   sltsolem1 33293
            20.9.23  Surreal Numbers: Birthday Function   bdayfo 33295
            20.9.24  Surreal Numbers: Density   fvnobday 33296
            20.9.25  Surreal Numbers: Full-Eta Property   bdayimaon 33310
            20.9.26  Surreal numbers - ordering theorems   csle 33336
            20.9.27  Surreal numbers - birthday theorems   bdayfun 33355
            20.9.28  Surreal numbers: Conway cuts   csslt 33363
            20.9.29  Surreal numbers - cuts and options   cmade 33392
            20.9.30  Quantifier-free definitions   ctxp 33404
            20.9.31  Alternate ordered pairs   caltop 33530
            20.9.32  Geometry in the Euclidean space   cofs 33556
                  20.9.32.1  Congruence properties   cofs 33556
                  20.9.32.2  Betweenness properties   btwntriv2 33586
                  20.9.32.3  Segment Transportation   ctransport 33603
                  20.9.32.4  Properties relating betweenness and congruence   cifs 33609
                  20.9.32.5  Connectivity of betweenness   btwnconn1lem1 33661
                  20.9.32.6  Segment less than or equal to   csegle 33680
                  20.9.32.7  Outside-of relationship   coutsideof 33693
                  20.9.32.8  Lines and Rays   cline2 33708
            20.9.33  Forward difference   cfwddif 33732
            20.9.34  Rank theorems   rankung 33740
            20.9.35  Hereditarily Finite Sets   chf 33746
      ​20.10  Mathbox for Jeff Hankins
            20.10.1  Miscellany   a1i14 33761
            20.10.2  Basic topological facts   topbnd 33785
            20.10.3  Topology of the real numbers   ivthALT 33796
            20.10.4  Refinements   cfne 33797
            20.10.5  Neighborhood bases determine topologies   neibastop1 33820
            20.10.6  Lattice structure of topologies   topmtcl 33824
            20.10.7  Filter bases   fgmin 33831
            20.10.8  Directed sets, nets   tailfval 33833
      ​20.11  Mathbox for Anthony Hart
            20.11.1  Propositional Calculus   tb-ax1 33844
            20.11.2  Predicate Calculus   nalfal 33864
            20.11.3  Miscellaneous single axioms   meran1 33872
            20.11.4  Connective Symmetry   negsym1 33878
      ​20.12  Mathbox for Chen-Pang He
            20.12.1  Ordinal topology   ontopbas 33889
      ​20.13  Mathbox for Jeff Hoffman
            20.13.1  Inferences for finite induction on generic function values   fveleq 33912
            20.13.2  gdc.mm   nnssi2 33916
      ​20.14  Mathbox for Asger C. Ipsen
            20.14.1  Continuous nowhere differentiable functions   dnival 33923
      ​​*20.15  Mathbox for BJ
            *20.15.1  Propositional calculus   bj-mp2c 33992
                  *20.15.1.1  Derived rules of inference   bj-mp2c 33992
                  ​*20.15.1.2  A syntactic theorem   bj-0 33994
                  20.15.1.3  Minimal implicational calculus   bj-a1k 33996
                  ​*20.15.1.4  Positive calculus   bj-syl66ib 34003
                  20.15.1.5  Implication and negation   bj-con2com 34009
                  ​*20.15.1.6  Disjunction   bj-jaoi1 34017
                  ​*20.15.1.7  Logical equivalence   bj-dfbi4 34019
                  20.15.1.8  The conditional operator for propositions   bj-consensus 34024
                  ​*20.15.1.9  Propositional calculus: miscellaneous   bj-imbi12 34029
            ​*20.15.2  Modal logic   bj-axdd2 34039
            ​*20.15.3  Provability logic   cprvb 34044
            ​*20.15.4  First-order logic   bj-genr 34053
                  20.15.4.1  Adding ax-gen   bj-genr 34053
                  20.15.4.2  Adding ax-4   bj-2alim 34057
                  20.15.4.3  Adding ax-5   bj-ax12wlem 34090
                  20.15.4.4  Equality and substitution   bj-ssbeq 34099
                  20.15.4.5  Adding ax-6   bj-spimvwt 34115
                  20.15.4.6  Adding ax-7   bj-cbvexw 34122
                  20.15.4.7  Membership predicate, ax-8 and ax-9   bj-ax89 34124
                  20.15.4.8  Adding ax-11   bj-alcomexcom 34127
                  20.15.4.9  Adding ax-12   axc11n11 34129
                  20.15.4.10  Nonfreeness   wnnf 34170
                  20.15.4.11  Adding ax-13   bj-axc10 34220
                  ​*20.15.4.12  Removing dependencies on ax-13 (and ax-11)   bj-axc10v 34230
                  ​*20.15.4.13  Distinct var metavariables   bj-hbaeb2 34256
                  ​*20.15.4.14  Around ~ equsal   bj-equsal1t 34260
                  ​*20.15.4.15  Some Principia Mathematica proofs   stdpc5t 34265
                  20.15.4.16  Alternate definition of substitution   bj-sbsb 34275
                  20.15.4.17  Lemmas for substitution   bj-sbf3 34277
                  20.15.4.18  Existential uniqueness   bj-eu3f 34280
                  ​*20.15.4.19  First-order logic: miscellaneous   bj-sblem1 34281
            20.15.5  Set theory   eliminable1 34297
                  ​*20.15.5.1  Eliminability of class terms   eliminable1 34297
                  ​*20.15.5.2  Classes without the axiom of extensionality   bj-denoteslem 34309
                  20.15.5.3  Characterization among sets versus among classes   elelb 34337
                  ​*20.15.5.4  The nonfreeness quantifier for classes   bj-nfcsym 34339
                  ​*20.15.5.5  Proposal for the definitions of class membership and class equality   bj-ax9 34340
                  ​*20.15.5.6  Lemmas for class substitution   bj-sbeqALT 34341
                  20.15.5.7  Removing some axiom requirements and disjoint variable conditions   bj-exlimvmpi 34351
                  ​*20.15.5.8  Class abstractions   bj-unrab 34368
                  ​*20.15.5.9  Restricted nonfreeness   wrnf 34375
                  ​*20.15.5.10  Russell's paradox   bj-ru0 34377
                  20.15.5.11  Curry's paradox in set theory   currysetlem 34380
                  ​*20.15.5.12  Some disjointness results   bj-n0i 34386
                  ​*20.15.5.13  Complements on direct products   bj-xpimasn 34391
                  ​*20.15.5.14  "Singletonization" and tagging   bj-snsetex 34399
                  ​*20.15.5.15  Tuples of classes   bj-cproj 34426
                  ​*20.15.5.16  Set theory: elementary operations relative to a universe   bj-rcleqf 34461
                  ​*20.15.5.17  Set theory: miscellaneous   bj-pw0ALT 34466
                  ​*20.15.5.18  Evaluation   bj-evaleq 34487
                  20.15.5.19  Elementwise operations   celwise 34494
                  ​*20.15.5.20  Elementwise intersection (families of sets induced on a subset)   bj-rest00 34496
                  20.15.5.21  Moore collections (complements)   bj-raldifsn 34515
                  20.15.5.22  Maps-to notation for functions with three arguments   bj-0nelmpt 34531
                  ​*20.15.5.23  Currying   csethom 34537
                  ​*20.15.5.24  Setting components of extensible structures   cstrset 34549
            ​*20.15.6  Extended real and complex numbers, real and complex projective lines   bj-nfald 34552
                  20.15.6.1  Complements on class abstractions of ordered pairs and binary relations   bj-nfald 34552
                  ​*20.15.6.2  Identity relation (complements)   bj-opabssvv 34565
                  ​*20.15.6.3  Functionalized identity (diagonal in a Cartesian square)   cdiag2 34587
                  ​*20.15.6.4  Direct image and inverse image   cimdir 34593
                  ​*20.15.6.5  Extended numbers and projective lines as sets   cfractemp 34611
                  ​*20.15.6.6  Addition and opposite   caddcc 34652
                  ​*20.15.6.7  Order relation on the extended reals   cltxr 34656
                  ​*20.15.6.8  Argument, multiplication and inverse   carg 34658
                  20.15.6.9  The canonical bijection from the finite ordinals   ciomnn 34664
                  20.15.6.10  Divisibility   cnnbar 34675
            ​*20.15.7  Monoids   bj-smgrpssmgm 34683
                  ​*20.15.7.1  Finite sums in monoids   cfinsum 34698
            ​*20.15.8  Affine, Euclidean, and Cartesian geometry   bj-fvimacnv0 34701
                  *20.15.8.1  Real vector spaces   bj-fvimacnv0 34701
                  ​*20.15.8.2  Complex numbers (supplements)   bj-subcom 34722
                  ​*20.15.8.3  Barycentric coordinates   bj-bary1lem 34724
            20.15.9  Monoid of endomorphisms   cend 34727
      ​20.16  Mathbox for Jim Kingdon
                  20.16.0.1  Circle constant   taupilem3 34733
                  20.16.0.2  Number theory   dfgcd3 34738
                  20.16.0.3  Real numbers   irrdifflemf 34739
      ​20.17  Mathbox for ML
            20.17.1  Miscellaneous   csbdif 34742
            20.17.2  Cartesian exponentiation   cfinxp 34800
            20.17.3  Topology   iunctb2 34820
                  ​*20.17.3.1  Pi-base theorems   pibp16 34830
      ​20.18  Mathbox for Wolf Lammen
            20.18.1  1. Bootstrapping   wl-section-boot 34839
            20.18.2  Implication chains   wl-section-impchain 34863
            20.18.3  Theorems around the conditional operator   wl-ifp-ncond1 34881
            20.18.4  Alternative development of hadd, cadd   wl-df-3xor 34885
            20.18.5  An alternative axiom ~ ax-13   ax-wl-13v 34910
            20.18.6  Other stuff   wl-mps 34912
            20.18.7  1. Restricted Quantifiers   wl-ral 34996
      ​20.19  Mathbox for Brendan Leahy
      ​20.20  Mathbox for Jeff Madsen
            20.20.1  Logic and set theory   unirep 35151
            20.20.2  Real and complex numbers; integers   filbcmb 35178
            20.20.3  Sequences and sums   sdclem2 35180
            20.20.4  Topology   subspopn 35190
            20.20.5  Metric spaces   metf1o 35193
            20.20.6  Continuous maps and homeomorphisms   constcncf 35200
            20.20.7  Boundedness   ctotbnd 35204
            20.20.8  Isometries   cismty 35236
            20.20.9  Heine-Borel Theorem   heibor1lem 35247
            20.20.10  Banach Fixed Point Theorem   bfplem1 35260
            20.20.11  Euclidean space   crrn 35263
            20.20.12  Intervals (continued)   ismrer1 35276
            20.20.13  Operation properties   cass 35280
            20.20.14  Groups and related structures   cmagm 35286
            20.20.15  Group homomorphism and isomorphism   cghomOLD 35321
            20.20.16  Rings   crngo 35332
            20.20.17  Division Rings   cdrng 35386
            20.20.18  Ring homomorphisms   crnghom 35398
            20.20.19  Commutative rings   ccm2 35427
            20.20.20  Ideals   cidl 35445
            20.20.21  Prime rings and integral domains   cprrng 35484
            20.20.22  Ideal generators   cigen 35497
      ​20.21  Mathbox for Giovanni Mascellani
            ​*20.21.1  Tools for automatic proof building   efald2 35516
            ​*20.21.2  Tseitin axioms   fald 35567
            ​*20.21.3  Equality deductions   iuneq2f 35594
            ​*20.21.4  Miscellanea   orcomdd 35605
      ​20.22  Mathbox for Peter Mazsa
            20.22.1  Notations   cxrn 35612
            20.22.2  Preparatory theorems   el2v1 35650
            20.22.3  Range Cartesian product   df-xrn 35783
            20.22.4  Cosets by ` R `   df-coss 35819
            20.22.5  Relations   df-rels 35885
            20.22.6  Subset relations   df-ssr 35898
            20.22.7  Reflexivity   df-refs 35910
            20.22.8  Converse reflexivity   df-cnvrefs 35923
            20.22.9  Symmetry   df-syms 35938
            20.22.10  Reflexivity and symmetry   symrefref2 35959
            20.22.11  Transitivity   df-trs 35968
            20.22.12  Equivalence relations   df-eqvrels 35979
            20.22.13  Redundancy   df-redunds 36018
            20.22.14  Domain quotients   df-dmqss 36033
            20.22.15  Equivalence relations on domain quotients   df-ers 36057
            20.22.16  Functions   df-funss 36073
            20.22.17  Disjoints vs. converse functions   df-disjss 36096
      ​20.23  Mathbox for Rodolfo Medina
            20.23.1  Partitions   prtlem60 36149
      ​​*20.24  Mathbox for Norm Megill
            *20.24.1  Obsolete schemes ax-c4,c5,c7,c10,c11,c11n,c15,c9,c14,c16   ax-c5 36179
            ​*20.24.2  Rederive new axioms ax-4, ax-10, ax-6, ax-12, ax-13 from old   axc5 36189
            ​*20.24.3  Legacy theorems using obsolete axioms   ax5ALT 36203
            20.24.4  Experiments with weak deduction theorem   elimhyps 36257
            20.24.5  Miscellanea   cnaddcom 36268
            20.24.6  Atoms, hyperplanes, and covering in a left vector space (or module)   clsa 36270
            20.24.7  Functionals and kernels of a left vector space (or module)   clfn 36353
            20.24.8  Opposite rings and dual vector spaces   cld 36419
            20.24.9  Ortholattices and orthomodular lattices   cops 36468
            20.24.10  Atomic lattices with covering property   ccvr 36558
            20.24.11  Hilbert lattices   chlt 36646
            20.24.12  Projective geometries based on Hilbert lattices   clln 36787
            20.24.13  Construction of a vector space from a Hilbert lattice   cdlema1N 37087
            20.24.14  Construction of involution and inner product from a Hilbert lattice   clpoN 38776
      ​20.25  Mathbox for metakunt
            20.25.1  General helpful statements   leexp1ad 39258
            20.25.2  Some gcd and lcm results   12gcd5e1 39291
            20.25.3  Least common multiple inequality theorem   3factsumint1 39309
            20.25.4  Logarithm inequalities   3lexlogpow5ineq1 39341
            20.25.5  Miscellaneous results for AKS formalisation   intlewftc 39344
            20.25.6  Permutation results   metakunt1 39350
            20.25.7  Unused lemmas scheduled for deletion   andiff 39384
      ​20.26  Mathbox for Steven Nguyen
            20.26.1  Utility theorems   ioin9i8 39389
            ​*20.26.2  Arithmetic theorems   c0exALT 39460
            20.26.3  Exponents   oexpreposd 39487
            20.26.4  Real subtraction   cresub 39503
            ​*20.26.5  Projective spaces   cprjsp 39595
            20.26.6  Equivalent formulations of Fermat's Last Theorem   dffltz 39615
      ​20.27  Mathbox for Igor Ieskov
      ​20.28  Mathbox for OpenAI
      ​20.29  Mathbox for Stefan O'Rear
            20.29.1  Additional elementary logic and set theory   moxfr 39633
            20.29.2  Additional theory of functions   imaiinfv 39634
            20.29.3  Additional topology   elrfi 39635
            20.29.4  Characterization of closure operators. Kuratowski closure axioms   ismrcd1 39639
            20.29.5  Algebraic closure systems   cnacs 39643
            20.29.6  Miscellanea 1. Map utilities   constmap 39654
            20.29.7  Miscellanea for polynomials   mptfcl 39661
            20.29.8  Multivariate polynomials over the integers   cmzpcl 39662
            20.29.9  Miscellanea for Diophantine sets 1   coeq0i 39694
            20.29.10  Diophantine sets 1: definitions   cdioph 39696
            20.29.11  Diophantine sets 2 miscellanea   ellz1 39708
            20.29.12  Diophantine sets 2: union and intersection. Monotone Boolean algebra   diophin 39713
            20.29.13  Diophantine sets 3: construction   diophrex 39716
            20.29.14  Diophantine sets 4 miscellanea   2sbcrex 39725
            20.29.15  Diophantine sets 4: Quantification   rexrabdioph 39735
            20.29.16  Diophantine sets 5: Arithmetic sets   rabdiophlem1 39742
            20.29.17  Diophantine sets 6: reusability. renumbering of variables   eldioph4b 39752
            20.29.18  Pigeonhole Principle and cardinality helpers   fphpd 39757
            20.29.19  A non-closed set of reals is infinite   rencldnfilem 39761
            20.29.20  Lagrange's rational approximation theorem   irrapxlem1 39763
            20.29.21  Pell equations 1: A nontrivial solution always exists   pellexlem1 39770
            20.29.22  Pell equations 2: Algebraic number theory of the solution set   csquarenn 39777
            20.29.23  Pell equations 3: characterizing fundamental solution   infmrgelbi 39819
            ​*20.29.24  Logarithm laws generalized to an arbitrary base   reglogcl 39831
            20.29.25  Pell equations 4: the positive solution group is infinite cyclic   pellfund14 39839
            20.29.26  X and Y sequences 1: Definition and recurrence laws   crmx 39841
            20.29.27  Ordering and induction lemmas for the integers   monotuz 39882
            20.29.28  X and Y sequences 2: Order properties   rmxypos 39888
            20.29.29  Congruential equations   congtr 39906
            20.29.30  Alternating congruential equations   acongid 39916
            20.29.31  Additional theorems on integer divisibility   coprmdvdsb 39926
            20.29.32  X and Y sequences 3: Divisibility properties   jm2.18 39929
            20.29.33  X and Y sequences 4: Diophantine representability of Y   jm2.27a 39946
            20.29.34  X and Y sequences 5: Diophantine representability of X, ^, _C   rmxdiophlem 39956
            20.29.35  Uncategorized stuff not associated with a major project   setindtr 39965
            20.29.36  More equivalents of the Axiom of Choice   axac10 39974
            20.29.37  Finitely generated left modules   clfig 40011
            20.29.38  Noetherian left modules I   clnm 40019
            20.29.39  Addenda for structure powers   pwssplit4 40033
            20.29.40  Every set admits a group structure iff choice   unxpwdom3 40039
            20.29.41  Noetherian rings and left modules II   clnr 40053
            20.29.42  Hilbert's Basis Theorem   cldgis 40065
            20.29.43  Additional material on polynomials [DEPRECATED]   cmnc 40075
            20.29.44  Degree and minimal polynomial of algebraic numbers   cdgraa 40084
            20.29.45  Algebraic integers I   citgo 40101
            20.29.46  Endomorphism algebra   cmend 40119
            20.29.47  Cyclic groups and order   idomrootle 40139
            20.29.48  Cyclotomic polynomials   ccytp 40146
            20.29.49  Miscellaneous topology   fgraphopab 40154
      ​20.30  Mathbox for Jon Pennant
      ​20.31  Mathbox for Richard Penner
            20.31.1  Short Studies   ifpan123g 40167
                  20.31.1.1  Additional work on conditional logical operator   ifpan123g 40167
                  20.31.1.2  Sophisms   rp-fakeimass 40220
                  ​*20.31.1.3  Finite Sets   rp-isfinite5 40225
                  20.31.1.4  General Observations   intabssd 40227
                  20.31.1.5  Infinite Sets   pwelg 40259
                  ​*20.31.1.6  Finite intersection property   fipjust 40264
                  20.31.1.7  RP ADDTO: Subclasses and subsets   rababg 40273
                  20.31.1.8  RP ADDTO: The intersection of a class   elintabg 40274
                  20.31.1.9  RP ADDTO: Theorems requiring subset and intersection existence   elinintrab 40277
                  20.31.1.10  RP ADDTO: Relations   xpinintabd 40280
                  ​*20.31.1.11  RP ADDTO: Functions   elmapintab 40296
                  ​*20.31.1.12  RP ADDTO: Finite induction (for finite ordinals)   cnvcnvintabd 40300
                  20.31.1.13  RP ADDTO: First and second members of an ordered pair   elcnvlem 40301
                  20.31.1.14  RP ADDTO: The reflexive and transitive properties of relations   undmrnresiss 40304
                  20.31.1.15  RP ADDTO: Basic properties of closures   cleq2lem 40308
                  20.31.1.16  RP REPLACE: Definitions and basic properties of transitive closures   trcleq2lemRP 40330
                  ​*20.31.1.17  Additions for square root; absolute value   sqrtcvallem1 40331
            20.31.2  Additional statements on relations and subclasses   al3im 40347
                  20.31.2.1  Transitive relations (not to be confused with transitive classes).   trrelind 40366
                  20.31.2.2  Reflexive closures   crcl 40373
                  ​*20.31.2.3  Finite relationship composition.   relexp2 40378
                  20.31.2.4  Transitive closure of a relation   dftrcl3 40421
                  ​*20.31.2.5  Adapted from Frege   frege77d 40447
            ​*20.31.3  Propositions from _Begriffsschrift_   dfxor4 40467
                  *20.31.3.1  _Begriffsschrift_ Chapter I   dfxor4 40467
                  ​*20.31.3.2  _Begriffsschrift_ Notation hints   whe 40473
                  20.31.3.3  _Begriffsschrift_ Chapter II Implication   ax-frege1 40491
                  20.31.3.4  _Begriffsschrift_ Chapter II Implication and Negation   axfrege28 40530
                  ​*20.31.3.5  _Begriffsschrift_ Chapter II with logical equivalence   axfrege52a 40557
                  20.31.3.6  _Begriffsschrift_ Chapter II with equivalence of sets   axfrege52c 40588
                  ​*20.31.3.7  _Begriffsschrift_ Chapter II with equivalence of classes   frege53c 40615
                  ​*20.31.3.8  _Begriffsschrift_ Chapter III Properties hereditary in a sequence   dffrege69 40633
                  ​*20.31.3.9  _Begriffsschrift_ Chapter III Following in a sequence   dffrege76 40640
                  ​*20.31.3.10  _Begriffsschrift_ Chapter III Member of sequence   dffrege99 40663
                  ​*20.31.3.11  _Begriffsschrift_ Chapter III Single-valued procedures   dffrege115 40679
            ​*20.31.4  Exploring Topology via Seifert and Threlfall   enrelmap 40698
                  *20.31.4.1  Equinumerosity of sets of relations and maps   enrelmap 40698
                  ​*20.31.4.2  Generic Pseudoclosure Spaces, Pseudointerior Spaces, and Pseudoneighborhoods   or3or 40724
                  ​*20.31.4.3  Generic Neighborhood Spaces   gneispa 40833
            ​*20.31.5  Exploring Higher Homotopy via Kerodon   k0004lem1 40850
                  *20.31.5.1  Simplicial Sets   k0004lem1 40850
      ​20.32  Mathbox for Stanislas Polu
            20.32.1  IMO Problems   wwlemuld 40859
                  20.32.1.1  IMO 1972 B2   wwlemuld 40859
            ​*20.32.2  INT Inequalities Proof Generator   int-addcomd 40879
            ​*20.32.3  N-Digit Addition Proof Generator   unitadd 40901
            20.32.4  AM-GM (for k = 2,3,4)   gsumws3 40902
      ​20.33  Mathbox for Rohan Ridenour
            20.33.1  Misc   spALT 40907
            20.33.2  Monoid rings   cmnring 40919
            20.33.3  Shorter primitive equivalent of ax-groth   gru0eld 40937
                  20.33.3.1  Grothendieck universes are closed under collection   gru0eld 40937
                  20.33.3.2  Minimal universes   ismnu 40969
                  20.33.3.3  Primitive equivalent of ax-groth   expandan 40996
      ​20.34  Mathbox for Steve Rodriguez
            20.34.1  Miscellanea   nanorxor 41009
            20.34.2  Ratio test for infinite series convergence and divergence   dvgrat 41016
            20.34.3  Multiples   reldvds 41019
            20.34.4  Function operations   caofcan 41027
            20.34.5  Calculus   lhe4.4ex1a 41033
            20.34.6  The generalized binomial coefficient operation   cbcc 41040
            20.34.7  Binomial series   uzmptshftfval 41050
      ​20.35  Mathbox for Andrew Salmon
            20.35.1  Principia Mathematica * 10   pm10.12 41062
            20.35.2  Principia Mathematica * 11   2alanimi 41076
            20.35.3  Predicate Calculus   sbeqal1 41102
            20.35.4  Principia Mathematica * 13 and * 14   pm13.13a 41111
            20.35.5  Set Theory   elnev 41142
            20.35.6  Arithmetic   addcomgi 41160
            20.35.7  Geometry   cplusr 41161
      ​​*20.36  Mathbox for Alan Sare
            20.36.1  Auxiliary theorems for the Virtual Deduction tool   idiALT 41183
            20.36.2  Supplementary unification deductions   bi1imp 41187
            20.36.3  Conventional Metamath proofs, some derived from VD proofs   iidn3 41207
            20.36.4  What is Virtual Deduction?   wvd1 41275
            20.36.5  Virtual Deduction Theorems   df-vd1 41276
            20.36.6  Theorems proved using Virtual Deduction   trsspwALT 41524
            20.36.7  Theorems proved using Virtual Deduction with mmj2 assistance   simplbi2VD 41552
            20.36.8  Virtual Deduction transcriptions of textbook proofs   sb5ALTVD 41619
            20.36.9  Theorems proved using conjunction-form Virtual Deduction   elpwgdedVD 41623
            20.36.10  Theorems with a VD proof in conventional notation derived from a VD proof   suctrALT3 41630
            ​*20.36.11  Theorems with a proof in conventional notation derived from a VD proof   notnotrALT2 41633
      ​20.37  Mathbox for Glauco Siliprandi
            20.37.1  Miscellanea   evth2f 41644
            20.37.2  Functions   feq1dd 41791
            20.37.3  Ordering on real numbers - Real and complex numbers basic operations   sub2times 41905
            20.37.4  Real intervals   gtnelioc 42128
            20.37.5  Finite sums   fsumclf 42211
            20.37.6  Finite multiplication of numbers and finite multiplication of functions   fmul01 42222
            20.37.7  Limits   clim1fr1 42243
                  20.37.7.1  Inferior limit (lim inf)   clsi 42393
                  ​*20.37.7.2  Limits for sequences of extended real numbers   clsxlim 42460
            20.37.8  Trigonometry   coseq0 42506
            20.37.9  Continuous Functions   mulcncff 42512
            20.37.10  Derivatives   dvsinexp 42553
            20.37.11  Integrals   itgsin0pilem1 42592
            20.37.12  Stone Weierstrass theorem - real version   stoweidlem1 42643
            20.37.13  Wallis' product for π   wallispilem1 42707
            20.37.14  Stirling's approximation formula for ` n ` factorial   stirlinglem1 42716
            20.37.15  Dirichlet kernel   dirkerval 42733
            20.37.16  Fourier Series   fourierdlem1 42750
            20.37.17  e is transcendental   elaa2lem 42875
            20.37.18  n-dimensional Euclidean space   rrxtopn 42926
            20.37.19  Basic measure theory   csalg 42950
                  ​*20.37.19.1  σ-Algebras   csalg 42950
                  20.37.19.2  Sum of nonnegative extended reals   csumge0 43001
                  ​*20.37.19.3  Measures   cmea 43088
                  ​*20.37.19.4  Outer measures and Caratheodory's construction   come 43128
                  ​*20.37.19.5  Lebesgue measure on n-dimensional Real numbers   covoln 43175
                  ​*20.37.19.6  Measurable functions   csmblfn 43334
      ​20.38  Mathbox for Saveliy Skresanov
            20.38.1  Ceva's theorem   sigarval 43464
            20.38.2  Simple groups   simpcntrab 43484
      ​20.39  Mathbox for Jarvin Udandy
      ​20.40  Mathbox for Adhemar
            ​*20.40.1  Minimal implicational calculus   adh-minim 43594
      ​20.41  Mathbox for Alexander van der Vekens
            20.41.1  General auxiliary theorems (1)   eusnsn 43618
                  20.41.1.1  Unordered and ordered pairs - extension for singletons   eusnsn 43618
                  20.41.1.2  Unordered and ordered pairs - extension for unordered pairs   elprneb 43621
                  20.41.1.3  Unordered and ordered pairs - extension for ordered pairs   oppr 43622
                  20.41.1.4  Relations - extension   eubrv 43627
                  20.41.1.5  Definite description binder (inverted iota) - extension   iota0def 43630
                  20.41.1.6  Functions - extension   fveqvfvv 43632
            20.41.2  Alternative for Russell's definition of a description binder   caiota 43640
            20.41.3  Double restricted existential uniqueness   r19.32 43653
                  20.41.3.1  Restricted quantification (extension)   r19.32 43653
                  20.41.3.2  Restricted uniqueness and "at most one" quantification   reuf1odnf 43663
                  20.41.3.3  Analogs to Existential uniqueness (double quantification)   2reu3 43666
                  20.41.3.4  Additional theorems for double restricted existential uniqueness   2reu8i 43669
            ​*20.41.4  Alternative definitions of function and operation values   wdfat 43672
                  20.41.4.1  Restricted quantification (extension)   ralbinrald 43678
                  20.41.4.2  The universal class (extension)   nvelim 43679
                  20.41.4.3  Introduce the Axiom of Power Sets (extension)   alneu 43680
                  20.41.4.4  Predicate "defined at"   dfateq12d 43682
                  20.41.4.5  Alternative definition of the value of a function   dfafv2 43688
                  20.41.4.6  Alternative definition of the value of an operation   aoveq123d 43734
            ​*20.41.5  Alternative definitions of function values (2)   cafv2 43764
            20.41.6  General auxiliary theorems (2)   an4com24 43824
                  20.41.6.1  Logical conjunction - extension   an4com24 43824
                  20.41.6.2  Abbreviated conjunction and disjunction of three wff's - extension   3an4ancom24 43825
                  20.41.6.3  Negated membership (alternative)   cnelbr 43827
                  20.41.6.4  The empty set - extension   ralralimp 43834
                  20.41.6.5  Indexed union and intersection - extension   otiunsndisjX 43835
                  20.41.6.6  Functions - extension   fvifeq 43836
                  20.41.6.7  Maps-to notation - extension   fvmptrab 43848
                  20.41.6.8  Ordering on reals - extension   leltletr 43850
                  20.41.6.9  Subtraction - extension   cnambpcma 43851
                  20.41.6.10  Ordering on reals (cont.) - extension   leaddsuble 43854
                  20.41.6.11  Imaginary and complex number properties - extension   readdcnnred 43860
                  20.41.6.12  Nonnegative integers (as a subset of complex numbers) - extension   nn0resubcl 43865
                  20.41.6.13  Integers (as a subset of complex numbers) - extension   zgeltp1eq 43866
                  20.41.6.14  Decimal arithmetic - extension   1t10e1p1e11 43867
                  20.41.6.15  Upper sets of integers - extension   eluzge0nn0 43869
                  20.41.6.16  Infinity and the extended real number system (cont.) - extension   nltle2tri 43870
                  20.41.6.17  Finite intervals of integers - extension   ssfz12 43871
                  20.41.6.18  Half-open integer ranges - extension   fzopred 43879
                  20.41.6.19  The modulo (remainder) operation - extension   m1mod0mod1 43886
                  20.41.6.20  The infinite sequence builder "seq"   smonoord 43888
                  20.41.6.21  Finite and infinite sums - extension   fsummsndifre 43889
                  20.41.6.22  Extensible structures - extension   setsidel 43893
            ​*20.41.7  Preimages of function values   preimafvsnel 43896
            ​*20.41.8  Partitions of real intervals   ciccp 43930
            20.41.9  Shifting functions with an integer range domain   fargshiftfv 43956
            20.41.10  Words over a set (extension)   lswn0 43961
                  20.41.10.1  Last symbol of a word - extension   lswn0 43961
            20.41.11  Unordered pairs   wich 43962
                  20.41.11.1  Interchangeable setvar variables   wich 43962
                  20.41.11.2  Set of unordered pairs   sprid 43991
                  ​*20.41.11.3  Proper (unordered) pairs   prpair 44018
                  20.41.11.4  Set of proper unordered pairs   cprpr 44029
            20.41.12  Number theory (extension)   cfmtno 44044
                  ​*20.41.12.1  Fermat numbers   cfmtno 44044
                  ​*20.41.12.2  Mersenne primes   m2prm 44108
                  20.41.12.3  Proth's theorem   modexp2m1d 44130
                  20.41.12.4  Solutions of quadratic equations   quad1 44138
            ​*20.41.13  Even and odd numbers   ceven 44142
                  20.41.13.1  Definitions and basic properties   ceven 44142
                  20.41.13.2  Alternate definitions using the "divides" relation   dfeven2 44167
                  20.41.13.3  Alternate definitions using the "modulo" operation   dfeven3 44176
                  20.41.13.4  Alternate definitions using the "gcd" operation   iseven5 44182
                  20.41.13.5  Theorems of part 5 revised   zneoALTV 44187
                  20.41.13.6  Theorems of part 6 revised   odd2np1ALTV 44192
                  20.41.13.7  Theorems of AV's mathbox revised   0evenALTV 44206
                  20.41.13.8  Additional theorems   epoo 44221
                  20.41.13.9  Perfect Number Theorem (revised)   perfectALTVlem1 44239
            20.41.14  Number theory (extension 2)   cfppr 44242
                  ​*20.41.14.1  Fermat pseudoprimes   cfppr 44242
                  ​*20.41.14.2  Goldbach's conjectures   cgbe 44263
            20.41.15  Graph theory (extension)   cgrisom 44336
                  ​*20.41.15.1  Isomorphic graphs   cgrisom 44336
                  20.41.15.2  Loop-free graphs - extension   1hegrlfgr 44360
                  20.41.15.3  Walks - extension   cupwlks 44361
                  20.41.15.4  Edges of graphs expressed as sets of unordered pairs   upgredgssspr 44371
            20.41.16  Monoids (extension)   ovn0dmfun 44384
                  20.41.16.1  Auxiliary theorems   ovn0dmfun 44384
                  20.41.16.2  Magmas, Semigroups and Monoids (extension)   plusfreseq 44392
                  20.41.16.3  Magma homomorphisms and submagmas   cmgmhm 44397
                  20.41.16.4  Examples and counterexamples for magmas, semigroups and monoids (extension)   opmpoismgm 44427
                  20.41.16.5  Group sum operation (extension 1)   gsumsplit2f 44440
            ​*20.41.17  Magmas and internal binary operations (alternate approach)   ccllaw 44443
                  *20.41.17.1  Laws for internal binary operations   ccllaw 44443
                  ​*20.41.17.2  Internal binary operations   cintop 44456
                  20.41.17.3  Alternative definitions for magmas and semigroups   cmgm2 44475
            20.41.18  Categories (extension)   idfusubc0 44489
                  20.41.18.1  Subcategories (extension)   idfusubc0 44489
            20.41.19  Rings (extension)   lmod0rng 44492
                  20.41.19.1  Nonzero rings (extension)   lmod0rng 44492
                  ​*20.41.19.2  Non-unital rings ("rngs")   crng 44498
                  20.41.19.3  Rng homomorphisms   crngh 44509
                  20.41.19.4  Ring homomorphisms (extension)   rhmfn 44542
                  20.41.19.5  Ideals as non-unital rings   lidldomn1 44545
                  20.41.19.6  The non-unital ring of even integers   0even 44555
                  20.41.19.7  A constructed not unital ring   cznrnglem 44577
                  ​*20.41.19.8  The category of non-unital rings   crngc 44581
                  ​*20.41.19.9  The category of (unital) rings   cringc 44627
                  20.41.19.10  Subcategories of the category of rings   srhmsubclem1 44697
            20.41.20  Basic algebraic structures (extension)   opeliun2xp 44734
                  20.41.20.1  Auxiliary theorems   opeliun2xp 44734
                  20.41.20.2  The binomial coefficient operation (extension)   bcpascm1 44753
                  20.41.20.3  The ` ZZ `-module ` ZZ X. ZZ `   zlmodzxzlmod 44756
                  20.41.20.4  Group sum operation (extension 2)   mgpsumunsn 44763
                  20.41.20.5  Symmetric groups (extension)   exple2lt6 44766
                  20.41.20.6  Divisibility (extension)   invginvrid 44769
                  20.41.20.7  The support of functions (extension)   rmsupp0 44770
                  20.41.20.8  Finitely supported functions (extension)   rmsuppfi 44775
                  20.41.20.9  Left modules (extension)   lmodvsmdi 44784
                  20.41.20.10  Associative algebras (extension)   ascl1 44786
                  20.41.20.11  Univariate polynomials (extension)   ply1vr1smo 44789
                  20.41.20.12  Univariate polynomials (examples)   linply1 44801
            20.41.21  Linear algebra (extension)   cdmatalt 44805
                  ​*20.41.21.1  The subalgebras of diagonal and scalar matrices (extension)   cdmatalt 44805
                  ​*20.41.21.2  Linear combinations   clinc 44813
                  ​*20.41.21.3  Linear independence   clininds 44849
                  20.41.21.4  Simple left modules and the ` ZZ `-module   lmod1lem1 44896
                  20.41.21.5  Differences between (left) modules and (left) vector spaces   lvecpsslmod 44916
            20.41.22  Complexity theory   suppdm 44919
                  20.41.22.1  Auxiliary theorems   suppdm 44919
                  20.41.22.2  The modulo (remainder) operation (extension)   fldivmod 44932
                  20.41.22.3  Even and odd integers   nn0onn0ex 44937
                  20.41.22.4  The natural logarithm on complex numbers (extension)   logcxp0 44949
                  20.41.22.5  Division of functions   cfdiv 44951
                  20.41.22.6  Upper bounds   cbigo 44961
                  20.41.22.7  Logarithm to an arbitrary base (extension)   rege1logbrege0 44972
                  ​*20.41.22.8  The binary logarithm   fldivexpfllog2 44979
                  20.41.22.9  Binary length   cblen 44983
                  ​*20.41.22.10  Digits   cdig 45009
                  20.41.22.11  Nonnegative integer as sum of its shifted digits   dignn0flhalflem1 45029
                  20.41.22.12  Algorithms for the multiplication of nonnegative integers   nn0mulfsum 45038
                  ​*20.41.22.13  N-ary functions   cnaryf 45040
                  ​*20.41.22.14  The Ackermann function   citco 45071
            20.41.23  Elementary geometry (extension)   fv1prop 45113
                  20.41.23.1  Auxiliary theorems   fv1prop 45113
                  20.41.23.2  Real euclidean space of dimension 2   rrx2pxel 45125
                  20.41.23.3  Spheres and lines in real Euclidean spaces   cline 45141
      ​20.42  Mathbox for Emmett Weisz
            ​*20.42.1  Miscellaneous Theorems   nfintd 45203
            20.42.2  Set Recursion   csetrecs 45213
                  ​*20.42.2.1  Basic Properties of Set Recursion   csetrecs 45213
                  20.42.2.2  Examples and properties of set recursion   elsetrecslem 45228
            ​*20.42.3  Construction of Games and Surreal Numbers   cpg 45238
      ​​*20.43  Mathbox for David A. Wheeler
            20.43.1  Natural deduction   sbidd 45244
            ​*20.43.2  Greater than, greater than or equal to.   cge-real 45246
            ​*20.43.3  Hyperbolic trigonometric functions   csinh 45256
            ​*20.43.4  Reciprocal trigonometric functions (sec, csc, cot)   csec 45267
            ​*20.43.5  Identities for "if"   ifnmfalse 45289
            ​*20.43.6  Logarithms generalized to arbitrary base using ` logb `   logb2aval 45290
            ​*20.43.7  Logarithm laws generalized to an arbitrary base - log_   clog- 45291
            ​*20.43.8  Formally define terms such as Reflexivity   wreflexive 45293
            ​*20.43.9  Algebra helpers   comraddi 45297
            ​*20.43.10  Algebra helper examples   i2linesi 45306
            ​*20.43.11  Formal methods "surprises"   alimp-surprise 45308
            ​*20.43.12  Allsome quantifier   walsi 45314
            ​*20.43.13  Miscellaneous   5m4e1 45325
            20.43.14  Theorems about algebraic numbers   aacllem 45329
      ​20.44  Mathbox for Kunhao Zheng
            20.44.1  Weighted AM-GM inequality   amgmwlem 45330