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  Dual of an order structure
      9.2  Preordered sets and directed sets
      9.3  Partially ordered sets (posets)
      9.4  Totally ordered sets (tosets)
      9.5  Lattices
      9.6  Posets, directed sets, and lattices as relations
      9.7  Chains
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  Subring algebras and 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  SURREAL NUMBERS
      15.1  Sign sequence representation and Alling's axioms
      15.2  Initial consequences of Alling's axioms
      15.3  Conway cut representation
      15.4  Induction and recursion
      15.5  Surreal arithmetic
      15.6  Subsystems of surreals
PART 16  ELEMENTARY GEOMETRY
      16.1  Definition and Tarski's Axioms of Geometry
      16.2  Tarskian Geometry
      16.3  Properties of geometries
      16.4  Geometry in Hilbert spaces
PART 17  GRAPH THEORY
      17.1  Vertices and edges
      17.2  Undirected graphs
      17.3  Walks, paths and cycles
      17.4  Eulerian paths and the Konigsberg Bridge problem
      17.5  The Friendship Theorem
PART 18  GUIDES AND MISCELLANEA
      18.1  Guides (conventions, explanations, and examples)
      18.2  Humor
      18.3  (Future - to be reviewed and classified)
PART 19  COMPLEX TOPOLOGICAL VECTOR SPACES (DEPRECATED)
      19.1  Additional material on group theory (deprecated)
      19.2  Complex vector spaces
      19.3  Normed complex vector spaces
      19.4  Operators on complex vector spaces
      19.5  Inner product (pre-Hilbert) spaces
      19.6  Complex Banach spaces
      19.7  Complex Hilbert spaces
PART 20  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      20.1  Axiomatization of complex pre-Hilbert spaces
      20.2  Inner product and norms
      20.3  Cauchy sequences and completeness axiom
      20.4  Subspaces and projections
      20.5  Properties of Hilbert subspaces
      20.6  Operators on Hilbert spaces
      20.7  States on a Hilbert lattice and Godowski's equation
      20.8  Cover relation, atoms, exchange axiom, and modular symmetry
PART 21  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      ​21.1  Mathboxes for user contributions
      21.2  Mathbox for Stefan Allan
      21.3  Mathbox for Thierry Arnoux
      21.4  Mathbox for Jonathan Ben-Naim
      21.5  Mathbox for BTernaryTau
      21.6  Mathbox for Mario Carneiro
      21.7  Mathbox for Filip Cernatescu
      21.8  Mathbox for Paul Chapman
      21.9  Mathbox for Hongxiu Chen
      21.10  Mathbox for Adrian Ducourtial
      21.11  Mathbox for Scott Fenton
      21.12  Mathbox for Gino Giotto
      21.13  Mathbox for Jeff Hankins
      21.14  Mathbox for Anthony Hart
      21.15  Mathbox for Chen-Pang He
      21.16  Mathbox for Jeff Hoffman
      21.17  Mathbox for Matthew House
      21.18  Mathbox for Asger C. Ipsen
      21.19  Mathbox for BJ
      21.20  Mathbox for Jim Kingdon
      21.21  Mathbox for ML
      21.22  Mathbox for Wolf Lammen
      21.23  Mathbox for Brendan Leahy
      21.24  Mathbox for Jeff Madsen
      21.25  Mathbox for Giovanni Mascellani
      21.26  Mathbox for Peter Mazsa
      21.27  Mathbox for Rodolfo Medina
      21.28  Mathbox for Norm Megill
      21.29  Mathbox for metakunt
      21.30  Mathbox for Steven Nguyen
      21.31  Mathbox for Igor Ieskov
      21.32  Mathbox for OpenAI
      21.33  Mathbox for Stefan O'Rear
      21.34  Mathbox for Noam Pasman
      21.35  Mathbox for Jon Pennant
      21.36  Mathbox for Richard Penner
      21.37  Mathbox for Stanislas Polu
      21.38  Mathbox for Rohan Ridenour
      21.39  Mathbox for Steve Rodriguez
      21.40  Mathbox for Andrew Salmon
      21.41  Mathbox for Alan Sare
      21.42  Mathbox for Eric Schmidt
      21.43  Mathbox for Glauco Siliprandi
      21.44  Mathbox for Saveliy Skresanov
      21.45  Mathbox for Ender Ting
      21.46  Mathbox for Jarvin Udandy
      21.47  Mathbox for Adhemar
      21.48  Mathbox for Alexander van der Vekens
      21.49  Mathbox for Zhi Wang
      21.50  Mathbox for Emmett Weisz
      21.51  Mathbox for David A. Wheeler
      21.52  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 208
            ​*1.2.6  Logical conjunction   wa 399
            ​*1.2.7  Logical disjunction   wo 858
            ​*1.2.8  Mixed connectives   jaao 967
            ​*1.2.9  The conditional operator for propositions   wif 1073
            ​*1.2.10  The weak deduction theorem for propositional calculus   elimh 1093
            1.2.11  Abbreviated conjunction and disjunction of three wff's   w3o 1096
            1.2.12  Logical "nand" (Sheffer stroke)   wnan 1510
            1.2.13  Logical "xor"   wxo 1530
            1.2.14  Logical "nor"   wnor 1547
            1.2.15  True and false constants   wal 1557
                  ​*1.2.15.1  Universal quantifier for use by df-tru   wal 1557
                  ​*1.2.15.2  Equality predicate for use by df-tru   cv 1558
                  1.2.15.3  The true constant   wtru 1560
                  1.2.15.4  The false constant   wfal 1571
            ​*1.2.16  Truth tables   truimtru 1582
                  1.2.16.1  Implication   truimtru 1582
                  1.2.16.2  Negation   nottru 1586
                  1.2.16.3  Equivalence   trubitru 1588
                  1.2.16.4  Conjunction   truantru 1592
                  1.2.16.5  Disjunction   truortru 1596
                  1.2.16.6  Alternative denial   trunantru 1600
                  1.2.16.7  Exclusive disjunction   truxortru 1604
                  1.2.16.8  Joint denial   trunortru 1608
            ​*1.2.17  Half adder and full adder in propositional calculus   whad 1612
                  1.2.17.1  Full adder: sum   whad 1612
                  1.2.17.2  Full adder: carry   wcad 1625
      ​1.3  Other axiomatizations related to classical propositional calculus
            ​*1.3.1  Minimal implicational calculus   minimp 1640
            ​*1.3.2  Implicational Calculus   impsingle 1646
            1.3.3  Derive the Lukasiewicz axioms from Meredith's sole axiom   meredith 1660
            1.3.4  Derive the standard axioms from the Lukasiewicz axioms   luklem1 1677
            ​*1.3.5  Derive Nicod's axiom from the standard axioms   nic-dfim 1688
            1.3.6  Derive the Lukasiewicz axioms from Nicod's axiom   nic-imp 1694
            1.3.7  Derive Nicod's Axiom from Lukasiewicz's First Sheffer Stroke Axiom   lukshef-ax1 1713
            1.3.8  Derive the Lukasiewicz Axioms from the Tarski-Bernays-Wajsberg Axioms   tbw-bijust 1717
            1.3.9  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's First CO Axiom   merco1 1732
            1.3.10  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's Second CO Axiom   merco2 1755
            1.3.11  Derive the Lukasiewicz axioms from the Russell-Bernays Axioms   rb-bijust 1768
            ​*1.3.12  Stoic logic non-modal portion (Chrysippus of Soli)   mptnan 1787
      ​​*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 1798
                  1.4.1.1  Existential quantifier   wex 1798
                  1.4.1.2  Nonfreeness predicate   wnf 1802
            1.4.2  Rule scheme ax-gen (Generalization)   ax-gen 1814
            1.4.3  Axiom scheme ax-4 (Quantified Implication)   ax-4 1828
                  ​*1.4.3.1  The empty domain of discourse   empty 1925
            1.4.4  Axiom scheme ax-5 (Distinctness) - first use of $d   ax-5 1929
            ​*1.4.5  Equality predicate (continued)   weq 1981
            1.4.6  Axiom scheme ax-6 (Existence)   ax-6 1986
            1.4.7  Axiom scheme ax-7 (Equality)   ax-7 2027
            1.4.8  Define proper substitution   justify-df 2084
            1.4.9  Membership predicate   wcel 2141
            1.4.10  Axiom scheme ax-8 (Left Equality for Binary Predicate)   ax-8 2143
            1.4.11  Axiom scheme ax-9 (Right Equality for Binary Predicate)   ax-9 2151
            ​*1.4.12  Logical redundancy of ax-10 , ax-11 , ax-12 , ax-13   ax6dgen 2161
      ​​*1.5  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes)
            1.5.1  Axiom scheme ax-10 (Quantified Negation)   ax-10 2174
            1.5.2  Axiom scheme ax-11 (Quantifier Commutation)   ax-11 2190
            1.5.3  Axiom scheme ax-12 (Substitution)   ax-12 2211
            1.5.4  Axiom scheme ax-13 (Quantified Equality)   ax-13 2402
      ​1.6  Uniqueness and unique existence
            1.6.1  Uniqueness: the at-most-one quantifier   wmo 2563
            1.6.2  Unique existence: the unique existential quantifier   weu 2594
      ​1.7  Other axiomatizations related to classical predicate calculus
            ​*1.7.1  Aristotelian logic: Assertic syllogisms   barbara 2688
            ​*1.7.2  Intuitionistic logic   axia1 2718
​​*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 2733
            2.1.2  Classes   cab 2739
                  2.1.2.1  Class abstractions   cab 2739
                  ​*2.1.2.2  Class equality   df-cleq 2753
                  2.1.2.3  Class membership   df-clel 2836
                  2.1.2.4  Elementary properties of class abstractions   eqabdv 2894
            2.1.3  Class form not-free predicate   wnfc 2908
            2.1.4  Negated equality and membership   wne 2956
                  2.1.4.1  Negated equality   wne 2956
                  2.1.4.2  Negated membership   wnel 3060
            2.1.5  Restricted quantification   wral 3075
                  2.1.5.1  Restricted universal and existential quantification   wral 3075
                  2.1.5.2  Restricted existential uniqueness and at-most-one quantifier   wreu 3364
                  2.1.5.3  Restricted class abstraction   crab 3413
            2.1.6  The universal class   cvv 3453
            ​*2.1.7  Conditional equality (experimental)   wcdeq 3725
            2.1.8  Russell's Paradox   rru 3741
            2.1.9  Proper substitution of classes for sets   wsbc 3744
            2.1.10  Proper substitution of classes for sets into classes   csb 3852
            2.1.11  Define basic set operations and relations   cdif 3901
            2.1.12  Subclasses and subsets   df-ss 3921
            2.1.13  The difference, union, and intersection of two classes   dfdif3 4071
                  2.1.13.1  The difference of two classes   dfdif3 4071
                  2.1.13.2  The union of two classes   elun 4106
                  2.1.13.3  The intersection of two classes   elini 4151
                  2.1.13.4  The symmetric difference of two classes   csymdif 4204
                  2.1.13.5  Combinations of difference, union, and intersection of two classes   unabs 4217
                  2.1.13.6  Class abstractions with difference, union, and intersection of two classes   unabw 4259
                  2.1.13.7  Restricted uniqueness with difference, union, and intersection   reuun2 4277
            2.1.14  The empty set   c0 4285
            ​*2.1.15  The conditional operator for classes   cif 4479
            ​*2.1.16  The weak deduction theorem for set theory   dedth 4538
            2.1.17  Power classes   cpw 4554
            2.1.18  Unordered and ordered pairs   snjust 4580
            2.1.19  The union of a class   cuni 4864
            2.1.20  The intersection of a class   cint 4904
            2.1.21  Indexed union and intersection   ciun 4948
            2.1.22  Disjointness   wdisj 5066
            2.1.23  Binary relations   wbr 5099
            2.1.24  Ordered-pair class abstractions (class builders)   copab 5161
            2.1.25  Functions in maps-to notation   cmpt 5180
            2.1.26  Transitive classes   wtr 5206
      ​2.2  ZF Set Theory - add the Axiom of Replacement
            2.2.1  Introduce the Axiom of Replacement   ax-rep 5226
            2.2.2  Derive the Axiom of Separation   axsepgfromrep 5243
            2.2.3  Derive the Null Set Axiom   axnulALT 5253
            2.2.4  Theorems requiring subset and intersection existence   exnelv 5262
            2.2.5  Theorems requiring empty set existence   class2set 5310
      ​2.3  ZF Set Theory - add the Axiom of Power Sets
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 5321
            2.3.2  Derive the Axiom of Pairing   axprlem1 5379
            2.3.3  Ordered pair theorem   opnz 5440
            2.3.4  Ordered-pair class abstractions (cont.)   opabidw 5493
            2.3.5  Power class of union and intersection   pwin 5536
            2.3.6  The identity relation   cid 5539
            2.3.7  The membership relation (or epsilon relation)   cep 5544
            ​*2.3.8  Partial and total orderings   wpo 5551
            2.3.9  Founded and well-ordering relations   wfr 5595
            2.3.10  Relations   cxp 5643
            2.3.11  The Predecessor Class   cpred 6281
            2.3.12  Well-founded induction (variant)   frpomin 6321
            2.3.13  Well-ordered induction   tz6.26 6328
            2.3.14  Ordinals   word 6339
            2.3.15  Definite description binder (inverted iota)   cio 6469
            2.3.16  Functions   wfun 6509
            2.3.17  Cantor's Theorem   canth 7344
            2.3.18  Restricted iota (description binder)   crio 7346
            2.3.19  Operations   co 7390
                  2.3.19.1  Variable-to-class conversion for operations   caovclg 7582
            2.3.20  Maps-to notation   mpondm0 7630
            2.3.21  Function operation   cof 7652
            2.3.22  Proper subset relation   crpss 7699
      ​2.4  ZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 7712
            2.4.2  Ordinals (continued)   epweon 7752
            2.4.3  Transfinite induction   tfi 7827
            2.4.4  The natural numbers (i.e., finite ordinals)   com 7840
            2.4.5  Peano's postulates   peano1 7863
            2.4.6  Finite induction (for finite ordinals)   find 7870
            2.4.7  Relations and functions (cont.)   dmexg 7876
            2.4.8  First and second members of an ordered pair   c1st 7962
            2.4.9  Induction on Cartesian products   frpoins3xpg 8113
            2.4.10  Ordering on Cartesian products   xpord2lem 8115
            2.4.11  Ordering Ordinal Sequences   orderseqlem 8130
            ​*2.4.12  The support of functions   csupp 8133
            ​*2.4.13  Special maps-to operations   opeliunxp2f 8183
            2.4.14  Function transposition   ctpos 8198
            2.4.15  Curry and uncurry   ccur 8238
            2.4.16  Undefined values   cund 8245
            2.4.17  Well-founded recursion   cfrecs 8254
            2.4.18  Well-ordered recursion   cwrecs 8285
            2.4.19  Functions on ordinals; strictly monotone ordinal functions   iunon 8303
            2.4.20  "Strong" transfinite recursion   crecs 8334
            2.4.21  Recursive definition generator   crdg 8373
            2.4.22  Finite recursion   frfnom 8399
            2.4.23  Ordinal arithmetic   c1o 8423
            2.4.24  Natural number arithmetic   nna0 8567
            2.4.25  Natural addition   cnadd 8628
            2.4.26  Equivalence relations and classes   wer 8668
            2.4.27  The mapping operation   cmap 8801
            2.4.28  Infinite Cartesian products   cixp 8873
            2.4.29  Equinumerosity   cen 8918
            2.4.30  Schroeder-Bernstein Theorem   sbthlem1 9053
            2.4.31  Equinumerosity (cont.)   xpf1o 9105
            2.4.32  Finite sets   dif1enlem 9122
            2.4.33  Pigeonhole Principle   phplem1 9166
            2.4.34  Finite sets (cont.)   onomeneq 9176
            2.4.35  Finitely supported functions   cfsupp 9302
            2.4.36  Finite intersections   cfi 9351
            2.4.37  Hall's marriage theorem   marypha1lem 9374
            2.4.38  Supremum and infimum   csup 9381
            2.4.39  Ordinal isomorphism, Hartogs's theorem   coi 9452
            2.4.40  Hartogs function   char 9499
            2.4.41  Weak dominance   cwdom 9507
      ​2.5  ZF Set Theory - add the Axiom of Regularity
            2.5.1  Introduce the Axiom of Regularity   ax-reg 9535
            2.5.2  Axiom of Infinity equivalents   inf0 9571
      ​2.6  ZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-inf 9588
            2.6.2  Existence of omega (the set of natural numbers)   omex 9593
            2.6.3  Cantor normal form   ccnf 9611
            2.6.4  Transitive closure of a relation   cttrcl 9657
            2.6.5  Transitive closure   trcl 9678
            2.6.6  Set induction (or epsilon induction)   setind 9697
            2.6.7  Well-Founded Induction   frmin 9702
            2.6.8  Well-Founded Recursion   frr3g 9709
            2.6.9  Rank   cr1 9715
            2.6.10  Scott's trick; collection principle; Hilbert's epsilon   scottex 9838
            2.6.11  Disjoint union   cdju 9851
            2.6.12  Cardinal numbers   ccrd 9888
            2.6.13  Axiom of Choice equivalents   wac 10066
            ​*2.6.14  Cardinal number arithmetic   undjudom 10119
            2.6.15  The Ackermann bijection   ackbij2lem1 10169
            2.6.16  Cofinality (without Axiom of Choice)   cflem 10196
            2.6.17  Eight inequivalent definitions of finite set   sornom 10229
            2.6.18  Hereditarily size-limited sets without Choice   itunifval 10368
​​*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 10387
            3.1.2  Introduce the Axiom of Dependent Choice   ax-dc 10398
      ​3.2  ZFC Set Theory - add the Axiom of Choice
            3.2.1  Introduce the Axiom of Choice   ax-ac 10411
            3.2.2  AC equivalents: well-ordering, Zorn's lemma   numthcor 10446
            3.2.3  Cardinal number theorems using Axiom of Choice   cardval 10498
            3.2.4  Cardinal number arithmetic using Axiom of Choice   iunctb 10527
            3.2.5  Cofinality using the Axiom of Choice   alephreg 10535
      ​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 10573
            3.4.2  Derivation of the Axiom of Choice   gchaclem 10631
​​*PART 4  TG (TARSKI-GROTHENDIECK) SET THEORY
      ​4.1  Inaccessibles
            4.1.1  Weakly and strongly inaccessible cardinals   cwina 10635
            4.1.2  Weak universes   cwun 10653
            4.1.3  Tarski classes   ctsk 10701
            4.1.4  Grothendieck universes   cgru 10743
      ​4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
            4.2.1  Introduce the Tarski-Grothendieck Axiom   ax-groth 10776
            4.2.2  Derive the Power Set, Infinity and Choice Axioms   grothpw 10779
            4.2.3  Tarski map function   ctskm 10790
​​*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 10797
            5.1.2  Final derivation of real and complex number postulates   axaddf 11098
            5.1.3  Real and complex number postulates restated as axioms   ax-cnex 11124
      ​5.2  Derive the basic properties from the field axioms
            5.2.1  Some deductions from the field axioms for complex numbers   cnex 11149
            5.2.2  Infinity and the extended real number system   cpnf 11208
            5.2.3  Restate the ordering postulates with extended real "less than"   axlttri 11249
            5.2.4  Ordering on reals   lttr 11254
            5.2.5  Initial properties of the complex numbers   mul12 11343
      ​5.3  Real and complex numbers - basic operations
            5.3.1  Addition   add12 11396
            5.3.2  Subtraction   cmin 11409
            5.3.3  Multiplication   kcnktkm1cn 11613
            5.3.4  Ordering on reals (cont.)   gt0ne0 11647
            5.3.5  Reciprocals   ixi 11811
            5.3.6  Division   cdiv 11839
            5.3.7  Ordering on reals (cont.)   elimgt0 12024
            5.3.8  Completeness Axiom and Suprema   fimaxre 12131
            5.3.9  Imaginary and complex number properties   neg1cn 12175
            5.3.10  Function operation analogue theorems   ofsubeq0 12187
            ​*5.3.11  Indicator Functions   cind 12190
      ​5.4  Integer sets
            5.4.1  Positive integers (as a subset of complex numbers)   cn 12205
            5.4.2  Principle of mathematical induction   nnind 12223
            ​*5.4.3  Decimal representation of numbers   c2 12267
            ​*5.4.4  Some properties of specific numbers   1pneg1e0 12330
            5.4.5  Simple number properties   halfcl 12442
            5.4.6  The Archimedean property   nnunb 12472
            5.4.7  Nonnegative integers (as a subset of complex numbers)   cn0 12476
            ​*5.4.8  Extended nonnegative integers   cxnn0 12549
            5.4.9  Integers (as a subset of complex numbers)   cz 12563
            5.4.10  Decimal arithmetic   cdc 12683
            5.4.11  Upper sets of integers   cuz 12834
            5.4.12  Well-ordering principle for bounded-below sets of integers   uzwo3 12939
            5.4.13  Rational numbers (as a subset of complex numbers)   cq 12944
            5.4.14  Existence of the set of complex numbers   rpnnen1lem2 12973
      ​5.5  Order sets
            5.5.1  Positive reals (as a subset of complex numbers)   crp 12988
            5.5.2  Infinity and the extended real number system (cont.)   cxne 13106
            5.5.3  Supremum and infimum on the extended reals   xrsupexmnf 13303
            5.5.4  Real number intervals   cioo 13344
            5.5.5  Finite intervals of integers   cfz 13507
            ​*5.5.6  Finite intervals of nonnegative integers   elfz2nn0 13618
            5.5.7  Half-open integer ranges   cfzo 13654
      ​5.6  Elementary integer functions
            5.6.1  The floor and ceiling functions   cfl 13795
            5.6.2  The modulo (remainder) operation   cmo 13874
            5.6.3  Miscellaneous theorems about integers   om2uz0i 13955
            5.6.4  Strong induction over upper sets of integers   uzsinds 13995
            5.6.5  Finitely supported functions over the nonnegative integers   fsuppmapnn0fiublem 13998
            5.6.6  The infinite sequence builder "seq" - extension   cseq 14009
            5.6.7  Integer powers   cexp 14069
            5.6.8  Ordered pair theorem for nonnegative integers   nn0le2msqi 14275
            5.6.9  Factorial function   cfa 14281
            5.6.10  The binomial coefficient operation   cbc 14310
            5.6.11  The ` # ` (set size) function   chash 14338
                  5.6.11.1  Proper unordered pairs and triples (sets of size 2 and 3)   hashprlei 14476
                  5.6.11.2  Functions with a domain containing at least two different elements   fundmge2nop0 14510
                  5.6.11.3  Finite induction on the size of the first component of a binary relation   hashdifsnp1 14514
      ​​*5.7  Words over a set
            5.7.1  Definitions and basic theorems   cword 14521
            5.7.2  Last symbol of a word   clsw 14570
            5.7.3  Concatenations of words   cconcat 14578
            5.7.4  Singleton words   cs1 14604
            5.7.5  Concatenations with singleton words   ccatws1cl 14625
            5.7.6  Subwords/substrings   csubstr 14649
            5.7.7  Prefixes of a word   cpfx 14679
            5.7.8  Subwords of subwords   swrdswrdlem 14712
            5.7.9  Subwords and concatenations   pfxcctswrd 14718
            5.7.10  Subwords of concatenations   swrdccatfn 14732
            5.7.11  Splicing words (substring replacement)   csplice 14757
            5.7.12  Reversing words   creverse 14766
            5.7.13  Repeated symbol words   creps 14776
            ​*5.7.14  Cyclical shifts of words   ccsh 14796
            5.7.15  Mapping words by a function   wrdco 14839
            5.7.16  Longer string literals   cs2 14849
      ​​*5.8  Reflexive and transitive closures of relations
            5.8.1  The reflexive and transitive properties of relations   coss12d 14980
            5.8.2  Basic properties of closures   cleq1lem 14990
            5.8.3  Definitions and basic properties of transitive closures   ctcl 14993
            5.8.4  Exponentiation of relations   crelexp 15027
            5.8.5  Reflexive-transitive closure as an indexed union   crtrcl 15063
            ​*5.8.6  Principle of transitive induction   relexpindlem 15071
      ​5.9  Elementary real and complex functions
            5.9.1  The "shift" operation   cshi 15074
            5.9.2  Signum (sgn or sign) function   csgn 15094
            5.9.3  Real and imaginary parts; conjugate   ccj 15104
            5.9.4  Square root; absolute value   csqrt 15241
      ​5.10  Elementary limits and convergence
            5.10.1  Superior limit (lim sup)   clsp 15478
            5.10.2  Limits   cli 15492
            5.10.3  Finite and infinite sums   csu 15694
            5.10.4  The binomial theorem   binomlem 15840
            5.10.5  The inclusion/exclusion principle   incexclem 15847
            5.10.6  Infinite sums (cont.)   isumshft 15850
            5.10.7  Miscellaneous converging and diverging sequences   divrcnv 15863
            5.10.8  Arithmetic series   arisum 15871
            5.10.9  Geometric series   expcnv 15875
            5.10.10  Ratio test for infinite series convergence   cvgrat 15894
            5.10.11  Mertens' theorem   mertenslem1 15895
            5.10.12  Finite and infinite products   prodf 15898
                  5.10.12.1  Product sequences   prodf 15898
                  5.10.12.2  Non-trivial convergence   ntrivcvg 15908
                  5.10.12.3  Complex products   cprod 15914
                  5.10.12.4  Finite products   fprod 15952
                  5.10.12.5  Infinite products   iprodclim 16009
            5.10.13  Falling and Rising Factorial   cfallfac 16015
            5.10.14  Bernoulli polynomials and sums of k-th powers   cbp 16057
      ​5.11  Elementary trigonometry
            5.11.1  The exponential, sine, and cosine functions   ce 16072
                  5.11.1.1  The circle constant (tau = 2 pi)   ctau 16215
            5.11.2  _e is irrational   eirrlem 16217
      ​5.12  Cardinality of real and complex number subsets
            5.12.1  Countability of integers and rationals   xpnnen 16224
            5.12.2  The reals are uncountable   rpnnen2lem1 16227
​​*PART 6  ELEMENTARY NUMBER THEORY
      ​6.1  Elementary properties of divisibility
            6.1.1  Irrationality of square root of 2   sqrt2irrlem 16261
            6.1.2  Some Number sets are chains of proper subsets   nthruc 16265
            6.1.3  The divides relation   cdvds 16267
            ​*6.1.4  Even and odd numbers   evenelz 16351
            6.1.5  The division algorithm   divalglem0 16408
            6.1.6  Bit sequences   cbits 16434
            6.1.7  The greatest common divisor operator   cgcd 16509
            6.1.8  Bézout's identity   bezoutlem1 16554
            6.1.9  Algorithms   nn0seqcvgd 16585
            6.1.10  Euclid's Algorithm   eucalgval2 16596
            ​*6.1.11  The least common multiple   clcm 16603
            ​*6.1.12  Coprimality and Euclid's lemma   coprmgcdb 16664
            6.1.13  Cancellability of congruences   congr 16679
      ​6.2  Elementary prime number theory
            ​*6.2.1  Elementary properties   cprime 16686
            ​*6.2.2  Coprimality and Euclid's lemma (cont.)   coprm 16727
            6.2.3  Properties of the canonical representation of a rational   cnumer 16749
            6.2.4  Euler's theorem   codz 16779
            6.2.5  Arithmetic modulo a prime number   modprm1div 16814
            6.2.6  Pythagorean Triples   coprimeprodsq 16825
            6.2.7  The prime count function   cpc 16853
            6.2.8  Pocklington's theorem   prmpwdvds 16921
            6.2.9  Infinite primes theorem   unbenlem 16925
            6.2.10  Sum of prime reciprocals   prmreclem1 16933
            6.2.11  Fundamental theorem of arithmetic   1arithlem1 16940
            6.2.12  Lagrange's four-square theorem   cgz 16946
            6.2.13  Van der Waerden's theorem   cvdwa 16982
            6.2.14  Ramsey's theorem   cram 17016
            ​*6.2.15  Primorial function   cprmo 17048
            ​*6.2.16  Prime gaps   prmgaplem1 17066
            6.2.17  Decimal arithmetic (cont.)   dec2dvds 17080
            6.2.18  Cyclical shifts of words (cont.)   cshwsidrepsw 17110
            6.2.19  Specific prime numbers   prmlem0 17122
            6.2.20  Very large primes   1259lem1 17148
​PART 7  BASIC STRUCTURES
      ​7.1  Extensible structures
            ​*7.1.1  Basic definitions   cstr 17163
                  7.1.1.1  Extensible structures as structures with components   cstr 17163
                  7.1.1.2  Substitution of components   csts 17180
                  7.1.1.3  Slots   cslot 17198
                  ​*7.1.1.4  Structure component indices   cnx 17210
                  7.1.1.5  Base sets   cbs 17226
                  7.1.1.6  Base set restrictions   cress 17247
            7.1.2  Slot definitions   cplusg 17267
            7.1.3  Definition of the structure product   crest 17430
            7.1.4  Definition of the structure quotient   cordt 17510
      ​7.2  Moore spaces
            7.2.1  Moore closures   mrcflem 17619
            7.2.2  Independent sets in a Moore system   mrisval 17643
            7.2.3  Algebraic closure systems   isacs 17664
​PART 8  BASIC CATEGORY THEORY
      ​8.1  Categories
            8.1.1  Categories   ccat 17677
            8.1.2  Opposite category   coppc 17724
            8.1.3  Monomorphisms and epimorphisms   cmon 17742
            8.1.4  Sections, inverses, isomorphisms   csect 17758
            ​*8.1.5  Isomorphic objects   ccic 17809
            8.1.6  Subcategories   cssc 17821
            8.1.7  Functors   cfunc 17868
            8.1.8  Full & faithful functors   cful 17918
            8.1.9  Natural transformations and the functor category   cnat 17958
            8.1.10  Initial, terminal and zero objects of a category   cinito 17995
      ​8.2  Arrows (disjointified hom-sets)
            8.2.1  Identity and composition for arrows   cida 18067
      ​8.3  Examples of categories
            8.3.1  The category of sets   csetc 18089
            8.3.2  The category of categories   ccatc 18112
            ​*8.3.3  The category of extensible structures   fncnvimaeqv 18133
      ​8.4  Categorical constructions
            8.4.1  Product of categories   cxpc 18181
            8.4.2  Functor evaluation   cevlf 18222
            8.4.3  Hom functor   chof 18261
​PART 9  BASIC ORDER THEORY
      ​9.1  Dual of an order structure
      ​9.2  Preordered sets and directed sets
      ​9.3  Partially ordered sets (posets)
      ​9.4  Totally ordered sets (tosets)
      ​9.5  Lattices
            9.5.1  Lattices   clat 18444
            9.5.2  Complete lattices   ccla 18511
            9.5.3  Distributive lattices   cdlat 18533
            9.5.4  Subset order structures   cipo 18540
      ​9.6  Posets, directed sets, and lattices as relations
            ​*9.6.1  Posets and lattices as relations   cps 18577
            9.6.2  Directed sets, nets   cdir 18607
      ​9.7  Chains
​PART 10  BASIC ALGEBRAIC STRUCTURES
      ​10.1  Monoids
            ​*10.1.1  Magmas   cplusf 18652
            ​*10.1.2  Identity elements   mgmidmo 18675
            ​*10.1.3  Iterated sums in a magma   gsumvalx 18691
            10.1.4  Magma homomorphisms and submagmas   cmgmhm 18705
            ​*10.1.5  Semigroups   csgrp 18733
            ​*10.1.6  Definition and basic properties of monoids   cmnd 18749
            10.1.7  Monoid homomorphisms and submonoids   cmhm 18796
            ​*10.1.8  Iterated sums in a monoid   gsumvallem2 18849
            10.1.9  Free monoids   cfrmd 18862
                  ​*10.1.9.1  Monoid of endofunctions   cefmnd 18883
            10.1.10  Examples and counterexamples for magmas, semigroups and monoids   mgm2nsgrplem1 18936
      ​10.2  Groups
            10.2.1  Definition and basic properties   cgrp 18956
            ​*10.2.2  Group multiple operation   cmg 19090
            10.2.3  Subgroups and Quotient groups   csubg 19143
            ​*10.2.4  Cyclic monoids and groups   cycsubmel 19222
            10.2.5  Elementary theory of group homomorphisms   cghm 19234
            10.2.6  Isomorphisms of groups   cgim 19278
                  10.2.6.1  The first isomorphism theorem of groups   ghmqusnsglem1 19301
            10.2.7  Group actions   cga 19310
            10.2.8  Centralizers and centers   ccntz 19336
            10.2.9  The opposite group   coppg 19366
            10.2.10  Symmetric groups   csymg 19390
                  ​*10.2.10.1  Definition and basic properties   csymg 19390
                  10.2.10.2  Cayley's theorem   cayleylem1 19433
                  10.2.10.3  Permutations fixing one element   symgfix2 19437
                  ​*10.2.10.4  Transpositions in the symmetric group   cpmtr 19462
                  10.2.10.5  The sign of a permutation   cpsgn 19510
            10.2.11  p-Groups and Sylow groups; Sylow's theorems   cod 19545
            10.2.12  Direct products   clsm 19655
                  10.2.12.1  Direct products (extension)   smndlsmidm 19677
            10.2.13  Free groups   cefg 19727
            10.2.14  Abelian groups   ccmn 19801
                  10.2.14.1  Definition and basic properties   ccmn 19801
                  10.2.14.2  Cyclic groups   ccyg 19898
                  10.2.14.3  Group sum operation   gsumval3a 19924
                  10.2.14.4  Group sums over (ranges of) integers   fsfnn0gsumfsffz 20004
                  10.2.14.5  Internal direct products   cdprd 20016
                  10.2.14.6  The Fundamental Theorem of Abelian Groups   ablfacrplem 20088
            10.2.15  Simple groups   csimpg 20113
                  10.2.15.1  Definition and basic properties   csimpg 20113
                  10.2.15.2  Classification of abelian simple groups   ablsimpnosubgd 20127
            10.2.16  Totally ordered monoids and groups   comnd 20140
      ​10.3  Rings
            10.3.1  Multiplicative Group   cmgp 20167
            ​*10.3.2  Non-unital rings ("rngs")   crng 20179
            ​*10.3.3  Ring unity (multiplicative identity)   cur 20208
            10.3.4  Semirings   csrg 20213
                  ​*10.3.4.1  The binomial theorem for semirings   srgbinomlem1 20253
            10.3.5  Unital rings   crg 20260
            10.3.6  Opposite ring   coppr 20362
            10.3.7  Divisibility   cdsr 20380
            10.3.8  Ring primes   crpm 20458
            10.3.9  Homomorphisms of non-unital rings   crnghm 20460
            10.3.10  Ring homomorphisms   crh 20495
            10.3.11  Nonzero rings and zero rings   cnzr 20539
            10.3.12  Local rings   clring 20565
            10.3.13  Subrings   csubrng 20572
                  10.3.13.1  Subrings of non-unital rings   csubrng 20572
                  10.3.13.2  Subrings of unital rings   csubrg 20596
                  10.3.13.3  Subrings generated by a subset   crgspn 20637
            10.3.14  Categories of rings   crngc 20643
                  ​*10.3.14.1  The category of non-unital rings   crngc 20643
                  ​*10.3.14.2  The category of (unital) rings   cringc 20672
                  10.3.14.3  Subcategories of the category of rings   srhmsubclem1 20704
            10.3.15  Left regular elements and domains   crlreg 20718
      ​10.4  Division rings and fields
            10.4.1  Definition and basic properties   cdr 20756
            10.4.2  Sub-division rings   csdrg 20813
            10.4.3  Absolute value (abstract algebra)   cabv 20835
            10.4.4  Star rings   cstf 20864
            10.4.5  Totally ordered rings and fields   corng 20884
      ​10.5  Left modules
            10.5.1  Definition and basic properties   clmod 20905
            10.5.2  Subspaces and spans in a left module   clss 20976
            10.5.3  Homomorphisms and isomorphisms of left modules   clmhm 21064
            10.5.4  Subspace sum; bases for a left module   clbs 21119
      ​10.6  Vector spaces
            10.6.1  Definition and basic properties   clvec 21147
      ​10.7  Subring algebras and ideals
            10.7.1  Subring algebras   csra 21216
            ​*10.7.2  Left ideals and spans   clidl 21254
            10.7.3  Two-sided ideals and quotient rings   c2idl 21297
                  ​*10.7.3.1  Condition for a non-unital ring to be unital   rngqiprng1elbas 21334
            10.7.4  Principal ideal rings. Divisibility in the integers   clpidl 21368
            10.7.5  Principal ideal domains   cpid 21384
      ​10.8  The complex numbers as an algebraic extensible structure
            10.8.1  Definition and basic properties   cpsmet 21386
            ​*10.8.2  Ring of integers   czring 21476
                  ​*10.8.2.1  Example for a condition for a non-unital ring to be unital   pzriprnglem1 21511
            10.8.3  Algebraic constructions based on the complex numbers   czrh 21529
            10.8.4  Signs as subgroup of the complex numbers   cnmsgnsubg 21607
            10.8.5  Embedding of permutation signs into a ring   zrhpsgnmhm 21614
            10.8.6  The ordered field of real numbers   crefld 21634
      ​10.9  Generalized pre-Hilbert and Hilbert spaces
            10.9.1  Definition and basic properties   cphl 21654
            10.9.2  Orthocomplements and closed subspaces   cocv 21690
            10.9.3  Orthogonal projection and orthonormal bases   cpj 21730
​​*PART 11  BASIC LINEAR ALGEBRA
      ​11.1  Vectors and free modules
            *11.1.1  Direct sum of left modules   cdsmm 21761
            ​*11.1.2  Free modules   cfrlm 21776
            ​*11.1.3  Standard basis (unit vectors)   cuvc 21812
            ​*11.1.4  Independent sets and families   clindf 21834
            11.1.5  Characterization of free modules   lmimlbs 21866
      ​11.2  Associative algebras
            11.2.1  Definition and basic properties   casa 21880
      ​11.3  Abstract multivariate polynomials
            11.3.1  Definition and basic properties   cmps 21934
            11.3.2  Polynomial evaluation   ces 22103
            11.3.3  The "variable selection" function   cslv 22147
            11.3.4  Additional definitions for (multivariate) polynomials   cmhp 22176
            ​*11.3.5  Univariate polynomials   cps1 22215
            11.3.6  Univariate polynomial evaluation   ces1 22354
                  11.3.6.1  Specialization of polynomial evaluation as a ring homomorphism   evls1scafv 22407
      ​​*11.4  Matrices
            *11.4.1  The matrix multiplication   cmmul 22428
            ​*11.4.2  Square matrices   cmat 22445
            ​*11.4.3  The matrix algebra   matmulr 22476
            ​*11.4.4  Matrices of dimension 0 and 1   mat0dimbas0 22504
            ​*11.4.5  The subalgebras of diagonal and scalar matrices   cdmat 22526
            ​*11.4.6  Multiplication of a matrix with a "column vector"   cmvmul 22578
            11.4.7  Replacement functions for a square matrix   cmarrep 22594
            11.4.8  Submatrices   csubma 22614
      ​11.5  The determinant
            11.5.1  Definition and basic properties   cmdat 22622
            11.5.2  Determinants of 2 x 2 -matrices   m2detleiblem1 22662
            11.5.3  The matrix adjugate/adjunct   cmadu 22670
            ​*11.5.4  Laplace expansion of determinants (special case)   symgmatr01lem 22691
            11.5.5  Inverse matrix   invrvald 22714
            ​*11.5.6  Cramer's rule   slesolvec 22717
      ​​*11.6  Polynomial matrices
            11.6.1  Basic properties   pmatring 22730
            ​*11.6.2  Constant polynomial matrices   ccpmat 22741
            ​*11.6.3  Collecting coefficients of polynomial matrices   cdecpmat 22800
            ​*11.6.4  Ring isomorphism between polynomial matrices and polynomials over matrices   cpm2mp 22830
      ​​*11.7  The characteristic polynomial
            *11.7.1  Definition and basic properties   cchpmat 22864
            ​*11.7.2  The characteristic factor function G   fvmptnn04if 22887
            ​*11.7.3  The Cayley-Hamilton theorem   cpmadurid 22905
​PART 12  BASIC TOPOLOGY
      ​12.1  Topology
            ​*12.1.1  Topological spaces   ctop 22931
                  12.1.1.1  Topologies   ctop 22931
                  12.1.1.2  Topologies on sets   ctopon 22948
                  12.1.1.3  Topological spaces   ctps 22970
            12.1.2  Topological bases   ctb 22983
            12.1.3  Examples of topologies   distop 23033
            12.1.4  Closure and interior   ccld 23054
            12.1.5  Neighborhoods   cnei 23135
            12.1.6  Limit points and perfect sets   clp 23172
            12.1.7  Subspace topologies   restrcl 23195
            12.1.8  Order topology   ordtbaslem 23226
            12.1.9  Limits and continuity in topological spaces   ccn 23262
            12.1.10  Separated spaces: T0, T1, T2 (Hausdorff) ...   ct0 23344
            12.1.11  Compactness   ccmp 23424
            12.1.12  Bolzano-Weierstrass theorem   bwth 23448
            12.1.13  Connectedness   cconn 23449
            12.1.14  First- and second-countability   c1stc 23475
            12.1.15  Local topological properties   clly 23502
            12.1.16  Refinements   cref 23540
            12.1.17  Compactly generated spaces   ckgen 23571
            12.1.18  Product topologies   ctx 23598
            12.1.19  Continuous function-builders   cnmptid 23699
            12.1.20  Quotient maps and quotient topology   ckq 23731
            12.1.21  Homeomorphisms   chmeo 23791
      ​12.2  Filters and filter bases
            12.2.1  Filter bases   elmptrab 23865
            12.2.2  Filters   cfil 23883
            12.2.3  Ultrafilters   cufil 23937
            12.2.4  Filter limits   cfm 23971
            12.2.5  Extension by continuity   ccnext 24097
            12.2.6  Topological groups   ctmd 24108
            12.2.7  Infinite group sum on topological groups   ctsu 24164
            12.2.8  Topological rings, fields, vector spaces   ctrg 24194
      ​12.3  Uniform Structures and Spaces
            12.3.1  Uniform structures   cust 24238
            12.3.2  The topology induced by an uniform structure   cutop 24268
            12.3.3  Uniform Spaces   cuss 24291
            12.3.4  Uniform continuity   cucn 24312
            12.3.5  Cauchy filters in uniform spaces   ccfilu 24323
            12.3.6  Complete uniform spaces   ccusp 24334
      ​12.4  Metric spaces
            12.4.1  Pseudometric spaces   ispsmet 24342
            12.4.2  Basic metric space properties   cxms 24355
            12.4.3  Metric space balls   blfvalps 24421
            12.4.4  Open sets of a metric space   mopnval 24476
            12.4.5  Continuity in metric spaces   metcnp3 24578
            12.4.6  The uniform structure generated by a metric   metuval 24587
            12.4.7  Examples of metric spaces   dscmet 24610
            ​*12.4.8  Normed algebraic structures   cnm 24614
            12.4.9  Normed space homomorphisms (bounded linear operators)   cnmo 24743
            12.4.10  Topology on the reals   qtopbaslem 24796
            12.4.11  Topological definitions using the reals   cii 24915
            12.4.12  Path homotopy   chtpy 25007
            12.4.13  The fundamental group   cpco 25040
      ​12.5  Metric subcomplex vector spaces
            12.5.1  Subcomplex modules   cclm 25102
            ​*12.5.2  Subcomplex vector spaces   ccvs 25163
            ​*12.5.3  Normed subcomplex vector spaces   isncvsngp 25189
            12.5.4  Subcomplex pre-Hilbert spaces   ccph 25206
            12.5.5  Convergence and completeness   ccfil 25292
            12.5.6  Baire's Category Theorem   bcthlem1 25364
            12.5.7  Banach spaces and subcomplex Hilbert spaces   ccms 25372
                  12.5.7.1  The complete ordered field of the real numbers   retopn 25419
            12.5.8  Euclidean spaces   crrx 25423
            12.5.9  Minimizing Vector Theorem   minveclem1 25464
            12.5.10  Projection Theorem   pjthlem1 25477
​PART 13  BASIC REAL AND COMPLEX ANALYSIS
      ​13.1  Continuity
            13.1.1  Intermediate value theorem   pmltpclem1 25488
      ​13.2  Integrals
            13.2.1  Lebesgue measure   covol 25502
            13.2.2  Lebesgue integration   cmbf 25654
                  13.2.2.1  Lesbesgue integral   cmbf 25654
                  13.2.2.2  Lesbesgue directed integral   cdit 25886
      ​13.3  Derivatives
            13.3.1  Real and complex differentiation   climc 25902
                  13.3.1.1  Derivatives of functions of one complex or real variable   climc 25902
                  13.3.1.2  Results on real differentiation   dvferm1lem 26024
​PART 14  BASIC REAL AND COMPLEX FUNCTIONS
      ​14.1  Polynomials
            14.1.1  Polynomial degrees   cmdg 26091
            14.1.2  The division algorithm for univariate polynomials   cmn1 26164
            14.1.3  Elementary properties of complex polynomials   cply 26222
            14.1.4  The division algorithm for polynomials   cquot 26329
            14.1.5  Algebraic numbers   caa 26353
            14.1.6  Liouville's approximation theorem   aalioulem1 26371
      ​14.2  Sequences and series
            14.2.1  Taylor polynomials and Taylor's theorem   ctayl 26391
            14.2.2  Uniform convergence   culm 26414
            14.2.3  Power series   pserval 26448
      ​14.3  Basic trigonometry
            14.3.1  The exponential, sine, and cosine functions (cont.)   efcn 26481
            14.3.2  Properties of pi = 3.14159...   pilem1 26489
            14.3.3  Mapping of the exponential function   efgh 26581
            14.3.4  The natural logarithm on complex numbers   clog 26594
            ​*14.3.5  Logarithms to an arbitrary base   clogb 26804
            14.3.6  Theorems of Pythagoras, isosceles triangles, and intersecting chords   angval 26841
            14.3.7  Solutions of quadratic, cubic, and quartic equations   quad2 26879
            14.3.8  Inverse trigonometric functions   casin 26902
            14.3.9  The Birthday Problem   log2ublem1 26986
            14.3.10  Areas in R^2   carea 26995
            14.3.11  More miscellaneous converging sequences   rlimcnp 27005
            14.3.12  Inequality of arithmetic and geometric means   cvxcl 27024
            14.3.13  Euler-Mascheroni constant   cem 27031
            14.3.14  Zeta function   czeta 27052
            14.3.15  Gamma function   clgam 27055
      ​14.4  Basic number theory
            14.4.1  Wilson's theorem   wilthlem1 27107
            14.4.2  The Fundamental Theorem of Algebra   ftalem1 27112
            14.4.3  The Basel problem (ζ(2) = π²/6)   basellem1 27120
            14.4.4  Number-theoretical functions   ccht 27130
            14.4.5  Perfect Number Theorem   mersenne 27266
            14.4.6  Characters of Z/nZ   cdchr 27271
            14.4.7  Bertrand's postulate   bcctr 27314
            ​*14.4.8  Quadratic residues and the Legendre symbol   clgs 27333
            ​*14.4.9  Gauss' Lemma   gausslemma2dlem0a 27395
            14.4.10  Quadratic reciprocity   lgseisenlem1 27414
            14.4.11  All primes 4n+1 are the sum of two squares   2sqlem1 27456
            14.4.12  Chebyshev's Weak Prime Number Theorem, Dirichlet's Theorem   chebbnd1lem1 27508
            14.4.13  The Prime Number Theorem   mudivsum 27569
            14.4.14  Ostrowski's theorem   abvcxp 27654
​PART 15  SURREAL NUMBERS
      ​​*15.1  Sign sequence representation and Alling's axioms
            15.1.1  Definitions and initial properties   csur 27679
            15.1.2  Ordering   ltssolem1 27714
            15.1.3  Birthday Function   bdayfo 27716
            15.1.4  Density   fvnobday 27717
            ​*15.1.5  Full-Eta Property   bdayimaon 27732
      ​15.2  Initial consequences of Alling's axioms
            15.2.1  Ordering Theorems   cles 27783
            15.2.2  Birthday Theorems   bdayfun 27815
      ​​*15.3  Conway cut representation
            15.3.1  Conway cuts   cslts 27825
            15.3.2  Zero and One   c0s 27873
            15.3.3  Cuts and Options   cmade 27890
            15.3.4  Cofinality and coinitiality   cofslts 27986
      ​15.4  Induction and recursion
            15.4.1  Induction and recursion on one variable   cnorec 28005
            15.4.2  Induction and recursion on two variables   cnorec2 28016
      ​15.5  Surreal arithmetic
            15.5.1  Addition   cadds 28027
            15.5.2  Negation and Subtraction   cnegs 28087
            15.5.3  Multiplication   cmuls 28174
            15.5.4  Division   cdivs 28255
            15.5.5  Absolute value   cabss 28305
      ​15.6  Subsystems of surreals
            15.6.1  Ordinal numbers   cons 28319
            15.6.2  Surreal recursive sequences   cseqs 28351
            15.6.3  Natural numbers   cn0s 28380
            15.6.4  Integers   czs 28446
            15.6.5  Dyadic fractions   c2s 28478
            15.6.6  Real numbers   creno 28557
​​*PART 16  ELEMENTARY GEOMETRY
      ​16.1  Definition and Tarski's Axioms of Geometry
            16.1.1  Justification for the congruence notation   tgjustf 28617
      ​16.2  Tarskian Geometry
            16.2.1  Congruence   tgcgrcomimp 28621
            16.2.2  Betweenness   tgbtwntriv2 28631
            16.2.3  Dimension   tglowdim1 28644
            16.2.4  Betweenness and Congruence   tgifscgr 28652
            16.2.5  Congruence of a series of points   ccgrg 28654
            16.2.6  Motions   cismt 28676
            16.2.7  Colinearity   tglng 28690
            16.2.8  Connectivity of betweenness   tgbtwnconn1lem1 28716
            16.2.9  Less-than relation in geometric congruences   cleg 28726
            16.2.10  Rays   chlg 28744
            16.2.11  Lines   btwnlng1 28763
            16.2.12  Point inversions   cmir 28796
            16.2.13  Right angles   crag 28837
            16.2.14  Half-planes   islnopp 28883
            16.2.15  Midpoints and Line Mirroring   cmid 28916
            16.2.16  Congruence of angles   ccgra 28951
            16.2.17  Angle Comparisons   cinag 28979
            16.2.18  Congruence Theorems   tgsas1 28998
            16.2.19  Equilateral triangles   ceqlg 29009
      ​16.3  Properties of geometries
            16.3.1  Isomorphisms between geometries   f1otrgds 29013
      ​16.4  Geometry in Hilbert spaces
            16.4.1  Geometry in the complex plane   cchhllem 29031
            16.4.2  Geometry in Euclidean spaces   cee 29032
                  16.4.2.1  Definition of the Euclidean space   cee 29032
                  16.4.2.2  Tarski's axioms for geometry for the Euclidean space   axdimuniq 29058
                  16.4.2.3  EE^n fulfills Tarski's Axioms   ceeng 29122
​​*PART 17  GRAPH THEORY
      ​*17.1  Vertices and edges
            17.1.1  The edge function extractor for extensible structures   cedgf 29133
            ​*17.1.2  Vertices and indexed edges   cvtx 29141
                  17.1.2.1  Definitions and basic properties   cvtx 29141
                  17.1.2.2  The vertices and edges of a graph represented as ordered pair   opvtxval 29148
                  17.1.2.3  The vertices and edges of a graph represented as extensible structure   funvtxdmge2val 29156
                  17.1.2.4  Representations of graphs without edges   snstrvtxval 29182
                  17.1.2.5  Degenerated cases of representations of graphs   vtxval0 29184
            17.1.3  Edges as range of the edge function   cedg 29192
      ​​*17.2  Undirected graphs
            17.2.1  Undirected hypergraphs   cuhgr 29201
            17.2.2  Undirected pseudographs and multigraphs   cupgr 29225
            ​*17.2.3  Loop-free graphs   umgrislfupgrlem 29267
            17.2.4  Edges as subsets of vertices of graphs   uhgredgiedgb 29271
            ​*17.2.5  Undirected simple graphs   cuspgr 29293
            17.2.6  Examples for graphs   usgr0e 29381
            17.2.7  Subgraphs   csubgr 29412
            17.2.8  Finite undirected simple graphs   cfusgr 29461
            17.2.9  Neighbors, complete graphs and universal vertices   cnbgr 29477
                  17.2.9.1  Neighbors   cnbgr 29477
                  17.2.9.2  Universal vertices   cuvtx 29530
                  17.2.9.3  Complete graphs   ccplgr 29554
            17.2.10  Vertex degree   cvtxdg 29610
            ​*17.2.11  Regular graphs   crgr 29700
      ​​*17.3  Walks, paths and cycles
            *17.3.1  Walks   cewlks 29740
            17.3.2  Walks for loop-free graphs   lfgrwlkprop 29830
            17.3.3  Trails   ctrls 29833
            17.3.4  Paths and simple paths   cpths 29854
            17.3.5  Closed walks   cclwlks 29914
            17.3.6  Circuits and cycles   ccrcts 29928
            ​*17.3.7  Walks as words   cwwlks 29969
            17.3.8  Walks/paths of length 2 (as length 3 strings)   2wlkdlem1 30069
            17.3.9  Walks in regular graphs   rusgrnumwwlkl1 30115
            ​*17.3.10  Closed walks as words   cclwwlk 30127
                  17.3.10.1  Closed walks as words   cclwwlk 30127
                  17.3.10.2  Closed walks of a fixed length as words   cclwwlkn 30170
                  17.3.10.3  Closed walks on a vertex of a fixed length as words   cclwwlknon 30233
            17.3.11  Examples for walks, trails and paths   0ewlk 30260
            17.3.12  Connected graphs   cconngr 30332
      ​17.4  Eulerian paths and the Konigsberg Bridge problem
            ​*17.4.1  Eulerian paths   ceupth 30343
            ​*17.4.2  The Königsberg Bridge problem   konigsbergvtx 30392
      ​17.5  The Friendship Theorem
            17.5.1  Friendship graphs - basics   cfrgr 30404
            17.5.2  The friendship theorem for small graphs   frgr1v 30417
            17.5.3  Theorems according to Mertzios and Unger   2pthfrgrrn 30428
            ​*17.5.4  Huneke's Proof of the Friendship Theorem   frgrncvvdeqlem1 30445
​PART 18  GUIDES AND MISCELLANEA
      ​18.1  Guides (conventions, explanations, and examples)
            ​*18.1.1  Conventions   conventions 30546
            18.1.2  Natural deduction   natded 30549
            ​*18.1.3  Natural deduction examples   ex-natded5.2 30550
            18.1.4  Definitional examples   ex-or 30567
            18.1.5  Other examples   aevdemo 30606
      ​18.2  Humor
            18.2.1  April Fool's theorem   avril1 30609
      ​18.3  (Future - to be reviewed and classified)
            18.3.1  Planar incidence geometry   cplig 30621
            ​*18.3.2  Aliases kept to prevent broken links   dummylink 30634
​​*PART 19  COMPLEX TOPOLOGICAL VECTOR SPACES (DEPRECATED)
      ​*19.1  Additional material on group theory (deprecated)
            19.1.1  Definitions and basic properties for groups   cgr 30636
            19.1.2  Abelian groups   cablo 30691
      ​19.2  Complex vector spaces
            19.2.1  Definition and basic properties   cvc 30705
            19.2.2  Examples of complex vector spaces   cnaddabloOLD 30728
      ​19.3  Normed complex vector spaces
            19.3.1  Definition and basic properties   cnv 30731
            19.3.2  Examples of normed complex vector spaces   cnnv 30824
            19.3.3  Induced metric of a normed complex vector space   imsval 30832
            19.3.4  Inner product   cdip 30847
            19.3.5  Subspaces   css 30868
      ​19.4  Operators on complex vector spaces
            19.4.1  Definitions and basic properties   clno 30887
      ​19.5  Inner product (pre-Hilbert) spaces
            19.5.1  Definition and basic properties   ccphlo 30959
            19.5.2  Examples of pre-Hilbert spaces   cncph 30966
            19.5.3  Properties of pre-Hilbert spaces   isph 30969
      ​19.6  Complex Banach spaces
            19.6.1  Definition and basic properties   ccbn 31009
            19.6.2  Examples of complex Banach spaces   cnbn 31016
            19.6.3  Uniform Boundedness Theorem   ubthlem1 31017
            19.6.4  Minimizing Vector Theorem   minvecolem1 31021
      ​19.7  Complex Hilbert spaces
            19.7.1  Definition and basic properties   chlo 31032
            19.7.2  Standard axioms for a complex Hilbert space   hlex 31045
            19.7.3  Examples of complex Hilbert spaces   cnchl 31063
            19.7.4  Hellinger-Toeplitz Theorem   htthlem 31064
​​*PART 20  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      ​20.1  Axiomatization of complex pre-Hilbert spaces
            20.1.1  Basic Hilbert space definitions   chba 31066
            20.1.2  Preliminary ZFC lemmas   df-hnorm 31115
            ​*20.1.3  Derive the Hilbert space axioms from ZFC set theory   axhilex-zf 31128
            ​*20.1.4  Introduce the vector space axioms for a Hilbert space   ax-hilex 31146
            20.1.5  Vector operations   hvmulex 31158
            20.1.6  Inner product postulates for a Hilbert space   ax-hfi 31226
      ​20.2  Inner product and norms
            20.2.1  Inner product   his5 31233
            20.2.2  Norms   dfhnorm2 31269
            20.2.3  Relate Hilbert space to normed complex vector spaces   hilablo 31307
            20.2.4  Bunjakovaskij-Cauchy-Schwarz inequality   bcsiALT 31326
      ​20.3  Cauchy sequences and completeness axiom
            20.3.1  Cauchy sequences and limits   hcau 31331
            20.3.2  Derivation of the completeness axiom from ZF set theory   hilmet 31341
            20.3.3  Completeness postulate for a Hilbert space   ax-hcompl 31349
            20.3.4  Relate Hilbert space to ZFC pre-Hilbert and Hilbert spaces   hhcms 31350
      ​20.4  Subspaces and projections
            20.4.1  Subspaces   df-sh 31354
            20.4.2  Closed subspaces   df-ch 31368
            20.4.3  Orthocomplements   df-oc 31399
            20.4.4  Subspace sum, span, lattice join, lattice supremum   df-shs 31455
            20.4.5  Projection theorem   pjhthlem1 31538
            20.4.6  Projectors   df-pjh 31542
      ​20.5  Properties of Hilbert subspaces
            20.5.1  Orthomodular law   omlsilem 31549
            20.5.2  Projectors (cont.)   pjhtheu2 31563
            20.5.3  Hilbert lattice operations   sh0le 31587
            20.5.4  Span (cont.) and one-dimensional subspaces   spansn0 31688
            20.5.5  Commutes relation for Hilbert lattice elements   df-cm 31730
            20.5.6  Foulis-Holland theorem   fh1 31765
            20.5.7  Quantum Logic Explorer axioms   qlax1i 31774
            20.5.8  Orthogonal subspaces   chscllem1 31784
            20.5.9  Orthoarguesian laws 5OA and 3OA   5oalem1 31801
            20.5.10  Projectors (cont.)   pjorthi 31816
            20.5.11  Mayet's equation E_3   mayete3i 31875
      ​20.6  Operators on Hilbert spaces
            ​*20.6.1  Operator sum, difference, and scalar multiplication   df-hosum 31877
            20.6.2  Zero and identity operators   df-h0op 31895
            20.6.3  Operations on Hilbert space operators   hoaddcl 31905
            20.6.4  Linear, continuous, bounded, Hermitian, unitary operators and norms   df-nmop 31986
            20.6.5  Linear and continuous functionals and norms   df-nmfn 31992
            20.6.6  Adjoint   df-adjh 31996
            20.6.7  Dirac bra-ket notation   df-bra 31997
            20.6.8  Positive operators   df-leop 31999
            20.6.9  Eigenvectors, eigenvalues, spectrum   df-eigvec 32000
            20.6.10  Theorems about operators and functionals   nmopval 32003
            20.6.11  Riesz lemma   riesz3i 32209
            20.6.12  Adjoints (cont.)   cnlnadjlem1 32214
            20.6.13  Quantum computation error bound theorem   unierri 32251
            20.6.14  Dirac bra-ket notation (cont.)   branmfn 32252
            20.6.15  Positive operators (cont.)   leopg 32269
            20.6.16  Projectors as operators   pjhmopi 32293
      ​20.7  States on a Hilbert lattice and Godowski's equation
            20.7.1  States on a Hilbert lattice   df-st 32358
            20.7.2  Godowski's equation   golem1 32418
      ​20.8  Cover relation, atoms, exchange axiom, and modular symmetry
            20.8.1  Covers relation; modular pairs   df-cv 32426
            20.8.2  Atoms   df-at 32485
            20.8.3  Superposition principle   superpos 32501
            20.8.4  Atoms, exchange and covering properties, atomicity   chcv1 32502
            20.8.5  Irreducibility   chirredlem1 32537
            20.8.6  Atoms (cont.)   atcvat3i 32543
            20.8.7  Modular symmetry   mdsymlem1 32550
​PART 21  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      ​21.1  Mathboxes for user contributions
            21.1.1  Mathbox guidelines   mathbox 32589
      ​21.2  Mathbox for Stefan Allan
      ​21.3  Mathbox for Thierry Arnoux
            21.3.1  Propositional Calculus - misc additions   ad11antr 32594
            21.3.2  Predicate Calculus   sbc2iedf 32610
                  21.3.2.1  Predicate Calculus - misc additions   sbc2iedf 32610
                  21.3.2.2  Restricted quantification - misc additions   ralcom4f 32612
                  21.3.2.3  Equality   eqtrb 32619
                  21.3.2.4  Double restricted existential uniqueness quantification   opsbc2ie 32621
                  21.3.2.5  Double restricted existential uniqueness quantification syntax   w2reu 32623
                  21.3.2.6  Substitution (without distinct variables) - misc additions   sbceqbidf 32632
                  21.3.2.7  Existential "at most one" - misc additions   mo5f 32634
                  21.3.2.8  Existential uniqueness - misc additions   reuxfrdf 32636
                  21.3.2.9  Restricted "at most one" - misc additions   rmoxfrd 32638
                  21.3.2.10  Restricted iota (description binder)   riotaeqbidva 32641
            21.3.3  General Set Theory   dmrab 32642
                  21.3.3.1  Class abstractions (a.k.a. class builders)   dmrab 32642
                  21.3.3.2  Image Sets   abrexdomjm 32653
                  21.3.3.3  Set relations and operations - misc additions   nelun 32659
                  21.3.3.4  Unordered pairs   elpreq 32674
                  21.3.3.5  Unordered triples   tpssg 32683
                  21.3.3.6  Conditional operator - misc additions   ifeqeqx 32688
                  21.3.3.7  Set union   uniinn0 32697
                  21.3.3.8  Indexed union - misc additions   cbviunf 32702
                  21.3.3.9  Indexed intersection - misc additions   iinabrex 32716
                  21.3.3.10  Disjointness - misc additions   disjnf 32717
            21.3.4  Relations and Functions   xpdisjres 32745
                  21.3.4.1  Relations - misc additions   xpdisjres 32745
                  21.3.4.2  Functions - misc additions   fconst7v 32770
                  21.3.4.3  Operations - misc additions   mpomptxf 32828
                  21.3.4.4  The mapping operation   elmaprd 32830
                  21.3.4.5  Support of a function   suppovss 32831
                  21.3.4.6  Explicit Functions with one or two points as a domain   cosnopne 32844
                  21.3.4.7  Isomorphisms - misc. additions   gtiso 32851
                  21.3.4.8  Disjointness (additional proof requiring functions)   disjdsct 32853
                  21.3.4.9  First and second members of an ordered pair - misc additions   df1stres 32854
                  21.3.4.10  Countable Sets   snct 32862
            21.3.5  Real and Complex Numbers   sgnval2 32885
                  21.3.5.1  Complex operations - misc. additions   creq0 32886
                  21.3.5.2  Ordering on reals - misc additions   lt2addrd 32900
                  21.3.5.3  Extended reals - misc additions   nn0mnfxrd 32901
                  21.3.5.4  Extended nonnegative integers - misc additions   xnn0gt0 32919
                  21.3.5.5  Real number intervals - misc additions   joiniooico 32924
                  21.3.5.6  Finite intervals of integers - misc additions   uzssico 32934
                  21.3.5.7  Half-open integer ranges - misc additions   iundisjfi 32946
                  21.3.5.8  The ` # ` (set size) function - misc additions   hashunif 32956
                  21.3.5.9  The greatest common divisor operator - misc. additions   elq2 32962
                  21.3.5.10  Integers   nn0split01 32968
                  21.3.5.11  Decimal numbers   dfdec100 32980
            21.3.6  Real and complex functions   sgncl 32981
                  21.3.6.1  Signum (sgn or sign) function - misc. additions   sgncl 32981
                  21.3.6.2  Integer powers - misc. additions   nexple 32994
                  21.3.6.3  Indicator Functions (continued)   indsumin 32998
            ​*21.3.7  Decimal expansion   cdp2 33007
                  ​*21.3.7.1  Decimal point   cdp 33024
                  21.3.7.2  Division in the extended real number system   cxdiv 33053
            21.3.8  Words over a set - misc additions   wrdres 33072
                  21.3.8.1  Splicing words (substring replacement)   splfv3 33095
                  21.3.8.2  Cyclic shift of words   1cshid 33096
            21.3.9  Extensible Structures   ressplusf 33100
                  21.3.9.1  Structure restriction operator   ressplusf 33100
                  21.3.9.2  Posets   ressprs 33103
                  21.3.9.3  Complete lattices   clatp0cl 33113
                  21.3.9.4  Order Theory   cmnt 33115
                  21.3.9.5  Extended reals Structure - misc additions   ax-xrssca 33141
                  21.3.9.6  The extended nonnegative real numbers commutative monoid   xrge00 33151
            21.3.10  Algebra   mndcld 33159
                  21.3.10.1  Monoids   mndcld 33159
                  21.3.10.2  Monoids Homomorphisms   abliso 33173
                  21.3.10.3  Groups - misc additions   grpidcld 33177
                  21.3.10.4  Abelian Groups - misc additions   ablcomd 33184
                  21.3.10.5  Finitely supported group sums - misc additions   gsumsubg 33185
                  21.3.10.6  Group or monoid sums over words   gsumwun 33215
                  21.3.10.7  Centralizers and centers - misc additions   cntzun 33218
                  21.3.10.8  The symmetric group   symgfcoeu 33221
                  21.3.10.9  Transpositions   pmtridf1o 33233
                  21.3.10.10  Permutation Signs   psgnid 33236
                  21.3.10.11  Permutation cycles   ctocyc 33245
                  21.3.10.12  The Alternating Group   evpmval 33284
                  21.3.10.13  Signum in an ordered monoid   csgns 33297
                  21.3.10.14  Fixed points   cfxp 33302
                  21.3.10.15  The Archimedean property for generic ordered algebraic structures   cinftm 33315
                  21.3.10.16  Semiring left modules   cslmd 33339
                  21.3.10.17  Simple groups   prmsimpcyc 33367
                  21.3.10.18  Rings - misc additions   ringrngd 33368
                  21.3.10.19  Subrings generated by a set   elrgspnlem1 33382
                  21.3.10.20  The zero ring   irrednzr 33390
                  21.3.10.21  Localization of rings   cerl 33393
                  21.3.10.22  Integral Domains   domnmuln0rd 33417
                  21.3.10.23  Euclidean Domains   ceuf 33434
                  21.3.10.24  Division Rings   ringinveu 33440
                  21.3.10.25  The field of rational numbers   qfld 33443
                  21.3.10.26  Subfields   subsdrg 33444
                  21.3.10.27  Field of fractions   cfrac 33448
                  21.3.10.28  Field extensions generated by a set   cfldgen 33456
                  21.3.10.29  Ring homomorphisms - misc additions   rhmdvd 33469
                  21.3.10.30  Scalar restriction operation   cresv 33471
                  21.3.10.31  The commutative ring of gaussian integers   gzcrng 33486
                  21.3.10.32  The archimedean ordered field of real numbers   cnfldfld 33487
                  21.3.10.33  The quotient map and quotient modules   qusker 33494
                  21.3.10.34  The ring of integers modulo ` N `   znfermltl 33511
                  21.3.10.35  Independent sets and families   islinds5 33512
                  21.3.10.36  Ring associates, ring units   dvdsruassoi 33529
                  ​*21.3.10.37  Subgroup sum / Sumset / Minkowski sum   elgrplsmsn 33535
                  21.3.10.38  The quotient map   quslsm 33550
                  21.3.10.39  Ideals   lidlmcld 33564
                  21.3.10.40  Prime Ideals   cprmidl 33580
                  21.3.10.41  Maximal Ideals   cmxidl 33606
                  21.3.10.42  Local rings   drnglring 33647
                  21.3.10.43  The semiring of ideals of a ring   cidlsrg 33655
                  21.3.10.44  Prime Elements   rprmval 33671
                  21.3.10.45  Unique factorization domains   cufd 33693
                  21.3.10.46  The ring of integers   zringidom 33706
                  21.3.10.47  Associative Algebra   assaassd 33710
                  21.3.10.48  Univariate Polynomials   0ringmon1p 33712
                  21.3.10.49  Polynomial quotient and polynomial remainder   q1pdir 33758
                  21.3.10.50  Multivariate Polynomials   psrbasfsupp 33767
                  21.3.10.51  The ring of symmetric polynomials   csply 33811
                  21.3.10.52  The subring algebra   sra1r 33837
                  21.3.10.53  Division Ring Extensions   drgext0g 33846
                  21.3.10.54  Vector Spaces   lvecdimfi 33852
                  21.3.10.55  Vector Space Dimension   cldim 33855
            21.3.11  Field Extensions   cfldext 33894
                  21.3.11.1  Algebraic numbers   cirng 33939
                  21.3.11.2  Algebraic extensions   calgext 33951
                  21.3.11.3  Minimal polynomials   cminply 33955
                  21.3.11.4  Quadratic Field Extensions   rtelextdg2lem 33982
                  21.3.11.5  Towers of quadratic extentions   fldext2chn 33984
            ​*21.3.12  Constructible Numbers   cconstr 33985
                  21.3.12.1  Impossible constructions   2sqr3minply 34036
            21.3.13  Matrices   csmat 34049
                  21.3.13.1  Submatrices   csmat 34049
                  21.3.13.2  Matrix literals   clmat 34067
                  21.3.13.3  Laplace expansion of determinants   mdetpmtr1 34079
            21.3.14  Topology   ist0cld 34089
                  21.3.14.1  Open maps   txomap 34090
                  21.3.14.2  Topology of the unit circle   qtopt1 34091
                  21.3.14.3  Refinements   reff 34095
                  21.3.14.4  Open cover refinement property   ccref 34098
                  21.3.14.5  Lindelöf spaces   cldlf 34108
                  21.3.14.6  Paracompact spaces   cpcmp 34111
                  ​*21.3.14.7  Spectrum of a ring   crspec 34118
                  21.3.14.8  Pseudometrics   cmetid 34142
                  21.3.14.9  Continuity - misc additions   hauseqcn 34154
                  21.3.14.10  Topology of the closed unit interval   elunitge0 34155
                  21.3.14.11  Topology of ` ( RR X. RR ) `   unicls 34159
                  21.3.14.12  Order topology - misc. additions   cnvordtrestixx 34169
                  21.3.14.13  Continuity in topological spaces - misc. additions   mndpluscn 34182
                  21.3.14.14  Topology of the extended nonnegative real numbers ordered monoid   xrge0hmph 34188
                  21.3.14.15  Limits - misc additions   lmlim 34203
                  21.3.14.16  Univariate polynomials   pl1cn 34211
            21.3.15  Uniform Stuctures and Spaces   chcmp 34212
                  21.3.15.1  Hausdorff uniform completion   chcmp 34212
            21.3.16  Topology and algebraic structures   zringnm 34214
                  21.3.16.1  The norm on the ring of the integer numbers   zringnm 34214
                  21.3.16.2  Topological ` ZZ ` -modules   zlm0 34216
                  21.3.16.3  Canonical embedding of the field of the rational numbers into a division ring   cqqh 34226
                  21.3.16.4  Canonical embedding of the real numbers into a complete ordered field   crrh 34249
                  21.3.16.5  Embedding from the extended real numbers into a complete lattice   cxrh 34272
                  21.3.16.6  Canonical embeddings into the ordered field of the real numbers   zrhre 34275
                  ​*21.3.16.7  Topological Manifolds   cmntop 34278
                  21.3.16.8  Extended sum   cesum 34283
            21.3.17  Mixed Function/Constant operation   cofc 34351
            21.3.18  Abstract measure   csiga 34364
                  21.3.18.1  Sigma-Algebra   csiga 34364
                  21.3.18.2  Generated sigma-Algebra   csigagen 34394
                  ​*21.3.18.3  lambda and pi-Systems, Rings of Sets   ispisys 34408
                  21.3.18.4  The Borel algebra on the real numbers   cbrsiga 34437
                  21.3.18.5  Product Sigma-Algebra   csx 34444
                  21.3.18.6  Measures   cmeas 34451
                  21.3.18.7  The counting measure   cntmeas 34482
                  21.3.18.8  The Lebesgue measure - misc additions   voliune 34485
                  21.3.18.9  The Dirac delta measure   cdde 34488
                  21.3.18.10  The 'almost everywhere' relation   cae 34493
                  21.3.18.11  Measurable functions   cmbfm 34505
                  21.3.18.12  Borel Algebra on ` ( RR X. RR ) `   br2base 34525
                  ​*21.3.18.13  Caratheodory's extension theorem   coms 34547
            21.3.19  Integration   itgeq12dv 34582
                  21.3.19.1  Lebesgue integral - misc additions   itgeq12dv 34582
                  21.3.19.2  Bochner integral   citgm 34583
            21.3.20  Euler's partition theorem   oddpwdc 34610
            21.3.21  Sequences defined by strong recursion   csseq 34639
            21.3.22  Fibonacci Numbers   cfib 34652
            21.3.23  Probability   cprb 34663
                  21.3.23.1  Probability Theory   cprb 34663
                  21.3.23.2  Conditional Probabilities   ccprob 34687
                  21.3.23.3  Real-valued Random Variables   crrv 34696
                  21.3.23.4  Preimage set mapping operator   corvc 34712
                  21.3.23.5  Distribution Functions   orvcelval 34725
                  21.3.23.6  Cumulative Distribution Functions   orvclteel 34729
                  21.3.23.7  Probabilities - example   coinfliplem 34735
                  21.3.23.8  Bertrand's Ballot Problem   ballotlemoex 34742
            21.3.24  Signum (sgn or sign) function - misc. additions   fzssfzo 34795
                  21.3.24.1  Operations on words   ccatmulgnn0dir 34798
            21.3.25  Polynomials with real coefficients - misc additions   plymul02 34802
            21.3.26  Descartes's rule of signs   signspval 34808
                  21.3.26.1  Sign changes in a word over real numbers   signspval 34808
                  21.3.26.2  Counting sign changes in a word over real numbers   signslema 34818
            21.3.27  Number Theory   iblidicc 34848
                  21.3.27.1  Representations of a number as sums of integers   crepr 34864
                  21.3.27.2  Vinogradov Trigonometric Sums and the Circle Method   cvts 34891
                  21.3.27.3  The Ternary Goldbach Conjecture: Final Statement   ax-hgt749 34900
            21.3.28  Elementary Geometry   cstrkg2d 34920
                  ​*21.3.28.1  Two-dimensional geometry   cstrkg2d 34920
                  21.3.28.2  Morley's Miracle   cgranbtwn 34925
                  21.3.28.3  Outer Five Segment (not used, no need to move to main)   cafs 34928
            ​*21.3.29  LeftPad Project   clpad 34933
      ​​*21.4  Mathbox for Jonathan Ben-Naim
            21.4.1  First-order logic and set theory   bnj170 34956
            21.4.2  Well founded induction and recursion   bnj110 35115
            21.4.3  The existence of a minimal element in certain classes   bnj69 35267
            21.4.4  Well-founded induction   bnj1204 35269
            21.4.5  Well-founded recursion, part 1 of 3   bnj60 35319
            21.4.6  Well-founded recursion, part 2 of 3   bnj1500 35325
            21.4.7  Well-founded recursion, part 3 of 3   bnj1522 35329
      ​21.5  Mathbox for BTernaryTau
            21.5.1  First-order logic   nfan1c 35330
                  21.5.1.1  Auxiliary axiom schemes   nfan1c 35330
            21.5.2  ZF set theory   exdifsn 35336
                  21.5.2.1  Finitism   prcinf 35369
                  21.5.2.2  Introduce ax-regs   ax-regs 35382
                  21.5.2.3  Derive ax-regs   axregs 35395
                  21.5.2.4  ZFC axioms with reduced distinct variable conditions   axsepg2 35396
                  21.5.2.5  Global choice   gblacfnacd 35405
            21.5.3  Real and complex numbers   zltp1ne 35413
            21.5.4  Graph theory   lfuhgr 35421
                  21.5.4.1  Acyclic graphs   cacycgr 35445
      ​21.6  Mathbox for Mario Carneiro
            21.6.1  Predicate calculus with all distinct variables   ax-7d 35462
            21.6.2  Miscellaneous stuff   quartfull 35468
            21.6.3  Derangements and the Subfactorial   deranglem 35469
            21.6.4  The Erdős-Szekeres theorem   erdszelem1 35494
            21.6.5  The Kuratowski closure-complement theorem   kur14lem1 35509
            21.6.6  Retracts and sections   cretr 35520
            21.6.7  Path-connected and simply connected spaces   cpconn 35522
            21.6.8  Covering maps   ccvm 35558
            21.6.9  Normal numbers   snmlff 35632
            21.6.10  Godel-sets of formulas - part 1   cgoe 35636
            21.6.11  Godel-sets of formulas - part 2   cgon 35735
            21.6.12  Models of ZF   cgze 35749
            ​*21.6.13  Metamath formal systems   cmcn 35763
            21.6.14  Grammatical formal systems   cm0s 35888
            21.6.15  Models of formal systems   cmuv 35908
            21.6.16  Splitting fields   ccpms 35930
            21.6.17  p-adic number fields   czr 35950
      ​​*21.7  Mathbox for Filip Cernatescu
      ​21.8  Mathbox for Paul Chapman
            21.8.1  Real and complex numbers (cont.)   climuzcnv 35974
            21.8.2  Miscellaneous theorems   elfzm12 35978
      ​21.9  Mathbox for Hongxiu Chen
      ​21.10  Mathbox for Adrian Ducourtial
            21.10.1  Propositional calculus   currybi 35991
            21.10.2  Clone theory   ccloneop 35998
      ​21.11  Mathbox for Scott Fenton
            21.11.1  ZFC Axioms in primitive form   axextprim 36004
            21.11.2  Untangled classes   untelirr 36011
            21.11.3  Extra propositional calculus theorems   3jaodd 36018
            21.11.4  Misc. Useful Theorems   nepss 36021
            21.11.5  Properties of real and complex numbers   sqdivzi 36031
            21.11.6  Infinite products   iprodefisumlem 36043
            21.11.7  Factorial limits   faclimlem1 36046
            21.11.8  Greatest common divisor and divisibility   gcd32 36052
            21.11.9  Properties of relationships   dftr6 36054
            21.11.10  Properties of functions and mappings   funpsstri 36069
            21.11.11  Ordinal numbers   elpotr 36082
            21.11.12  Defined equality axioms   axextdfeq 36098
            21.11.13  Hypothesis builders   hbntg 36106
            21.11.14  Well-founded zero, successor, and limits   cwsuc 36111
            21.11.15  Quantifier-free definitions   ctxp 36131
            21.11.16  Alternate ordered pairs   caltop 36259
            21.11.17  Geometry in the Euclidean space   cofs 36285
                  21.11.17.1  Congruence properties   cofs 36285
                  21.11.17.2  Betweenness properties   btwntriv2 36315
                  21.11.17.3  Segment Transportation   ctransport 36332
                  21.11.17.4  Properties relating betweenness and congruence   cifs 36338
                  21.11.17.5  Connectivity of betweenness   btwnconn1lem1 36390
                  21.11.17.6  Segment less than or equal to   csegle 36409
                  21.11.17.7  Outside-of relationship   coutsideof 36422
                  21.11.17.8  Lines and Rays   cline2 36437
            21.11.18  Forward difference   cfwddif 36461
            21.11.19  Rank theorems   rankung 36469
            21.11.20  Hereditarily Finite Sets   chf 36475
            21.11.21  Natural ordinal operations   cnmul 36490
      ​21.12  Mathbox for Gino Giotto
            21.12.1  Equality theorems   rmoeqi 36500
                  21.12.1.1  Inference versions   rmoeqi 36500
                  21.12.1.2  Deduction versions   rmoeqdv 36525
            21.12.2  Change bound variables   in-ax8 36537
                  21.12.2.1  Change bound variables and domains   cbvralvw2 36539
                  21.12.2.2  Change bound variables, deduction versions   cbvmodavw 36563
                  21.12.2.3  Change bound variables and domains, deduction versions   cbvrmodavw2 36596
            21.12.3  Study of ax-mulf usage   mpomulnzcnf 36612
      ​21.13  Mathbox for Jeff Hankins
            21.13.1  Miscellany   a1i14 36613
            21.13.2  Basic topological facts   topbnd 36637
            21.13.3  Topology of the real numbers   ivthALT 36648
            21.13.4  Refinements   cfne 36649
            21.13.5  Neighborhood bases determine topologies   neibastop1 36672
            21.13.6  Lattice structure of topologies   topmtcl 36676
            21.13.7  Filter bases   fgmin 36683
            21.13.8  Directed sets, nets   tailfval 36685
      ​21.14  Mathbox for Anthony Hart
            21.14.1  Propositional Calculus   tb-ax1 36696
            21.14.2  Predicate Calculus   nalfal 36716
            21.14.3  Miscellaneous single axioms   meran1 36724
            21.14.4  Connective Symmetry   negsym1 36730
      ​21.15  Mathbox for Chen-Pang He
            21.15.1  Ordinal topology   ontopbas 36741
      ​21.16  Mathbox for Jeff Hoffman
            21.16.1  Inferences for finite induction on generic function values   fveleq 36764
            21.16.2  gdc.mm   nnssi2 36768
      ​21.17  Mathbox for Matthew House
            21.17.1  Relations on well-ordered indexed unions   weiunval 36775
            21.17.2  Axiom of Transitive Containment   axtco 36784
            21.17.3  Transitive closure of a class   tr0elw 36797
            ​*21.17.4  Stronger axioms of regularity   mh-setind 36849
            21.17.5  Short axioms written in primitive symbols   mh-inf3f1 36854
      ​21.18  Mathbox for Asger C. Ipsen
            21.18.1  Continuous nowhere differentiable functions   dnival 36862
      ​​*21.19  Mathbox for BJ
            *21.19.1  Propositional calculus   bj-mp2c 36931
                  *21.19.1.1  Derived rules of inference   bj-mp2c 36931
                  ​*21.19.1.2  A syntactic theorem   bj-0 36933
                  ​*21.19.1.3  Minimal implicational calculus   bj-a1k 36935
                  ​*21.19.1.4  Positive calculus   bj-bisimpl 36948
                  ​*21.19.1.5  Implication and negation   bj-con2com 36956
                  ​*21.19.1.6  Disjunction   bj-jaoi1 36967
                  ​*21.19.1.7  Logical equivalence   bj-dfbi4 36969
                  21.19.1.8  The conditional operator for propositions   bj-consensus 36974
                  ​*21.19.1.9  Propositional calculus: miscellaneous   bj-imbi12 36979
            ​*21.19.2  Modal logic   bj-axdd2 36988
            ​*21.19.3  Provability logic   cprvb 36993
            ​*21.19.4  First-order logic   bj-exexalal 37002
                  21.19.4.1  Universal and existential quantifiers, nonfreeness predicate   bj-exexalal 37002
                  21.19.4.2  Adding ax-gen   bj-genr 37003
                  21.19.4.3  Adding ax-4   bj-almp 37007
                  21.19.4.4  Adding ax-5   bj-spvw 37060
                  21.19.4.5  Equality and substitution   bj-df-sb 37075
                  21.19.4.6  Adding ax-6   bj-spim0 37094
                  21.19.4.7  Adding ax-7   bj-cbvexw 37102
                  21.19.4.8  Membership predicate, ax-8 and ax-9   bj-ax89 37104
                  21.19.4.9  Adding ax-11   bj-alcomexcom 37106
                  21.19.4.10  Adding ax-12   axc11n11 37110
                  ​*21.19.4.11  Really adding ax-12   bj-substax12 37152
                  21.19.4.12  Nonfreeness   wnnf 37154
                  21.19.4.13  Adding ax-13   bj-axc10 37221
                  ​*21.19.4.14  Removing dependencies on ax-13 (and ax-11)   bj-axc10v 37231
                  ​*21.19.4.15  Distinct var metavariables   bj-hbaeb2 37256
                  ​*21.19.4.16  Around ~ equsal   bj-equsal1t 37260
                  ​*21.19.4.17  Some Principia Mathematica proofs   stdpc5t 37265
                  21.19.4.18  Alternate definition of substitution   bj-sbsb 37275
                  21.19.4.19  Lemmas for substitution   bj-sbf3 37277
                  21.19.4.20  Existential uniqueness   bj-eu3f 37279
                  ​*21.19.4.21  First-order logic: miscellaneous   bj-sblem1 37280
            21.19.5  Set theory   eliminable1 37297
                  ​*21.19.5.1  Eliminability of class terms   eliminable1 37297
                  ​*21.19.5.2  Classes without the axiom of extensionality   bj-denoteslem 37309
                  21.19.5.3  Characterization among sets versus among classes   elelb 37335
                  ​*21.19.5.4  The nonfreeness quantifier for classes   bj-nfcsym 37337
                  ​*21.19.5.5  Lemmas for class substitution   bj-sbeqALT 37338
                  21.19.5.6  Removing some axiom requirements and disjoint variable conditions   bj-exlimvmpi 37349
                  ​*21.19.5.7  Class abstractions   bj-elabd2ALT 37363
                  21.19.5.8  Generalized class abstractions   bj-cgab 37371
                  ​*21.19.5.9  Restricted nonfreeness   wrnf 37379
                  ​*21.19.5.10  Russell's paradox   bj-ru1 37381
                  21.19.5.11  Curry's paradox in set theory   currysetlem 37383
                  ​*21.19.5.12  Some disjointness results   bj-n0i 37389
                  ​*21.19.5.13  Complements on direct products   bj-xpimasn 37393
                  ​*21.19.5.14  "Singletonization" and tagging   bj-snsetex 37401
                  ​*21.19.5.15  Tuples of classes   bj-cproj 37428
                  ​*21.19.5.16  Set theory: elementary operations relative to a universe   bj-rcleqf 37463
                  ​*21.19.5.17  Axioms for finite unions   bj-abex 37468
                  ​*21.19.5.18  Set theory: miscellaneous   eleq2w2ALT 37485
                  ​*21.19.5.19  Axioms of separation and replacement   bj-axnul 37510
                  ​*21.19.5.20  Evaluation at a class   bj-evaleq 37514
                  21.19.5.21  Elementwise operations   celwise 37522
                  ​*21.19.5.22  Elementwise intersection (families of sets induced on a subset)   bj-rest00 37524
                  21.19.5.23  Moore collections (complements)   bj-raldifsn 37543
                  21.19.5.24  Maps-to notation for functions with three arguments   bj-0nelmpt 37559
                  ​*21.19.5.25  Currying   csethom 37565
                  ​*21.19.5.26  Setting components of extensible structures   cstrset 37577
            ​*21.19.6  Extended real and complex numbers, real and complex projective lines   bj-nfald 37580
                  21.19.6.1  Complements on class abstractions of ordered pairs and binary relations   bj-nfald 37580
                  ​*21.19.6.2  Identity relation (complements)   bj-opabssvv 37595
                  ​*21.19.6.3  Functionalized identity (diagonal in a Cartesian square)   cdiag2 37617
                  ​*21.19.6.4  Direct image and inverse image   cimdir 37623
                  ​*21.19.6.5  Extended numbers and projective lines as sets   cfractemp 37641
                  ​*21.19.6.6  Addition and opposite   caddcc 37682
                  ​*21.19.6.7  Order relation on the extended reals   cltxr 37686
                  ​*21.19.6.8  Argument, multiplication and inverse   carg 37688
                  21.19.6.9  The canonical bijection from the finite ordinals   ciomnn 37694
                  21.19.6.10  Divisibility   cnnbar 37705
            ​*21.19.7  Monoids   bj-smgrpssmgm 37713
                  ​*21.19.7.1  Finite sums in monoids   cfinsum 37728
            ​*21.19.8  Affine, Euclidean, and Cartesian geometry   bj-fvimacnv0 37731
                  *21.19.8.1  Real vector spaces   bj-fvimacnv0 37731
                  ​*21.19.8.2  Complex numbers (supplements)   bj-subcom 37753
                  ​*21.19.8.3  Barycentric coordinates   bj-bary1lem 37755
            21.19.9  Monoid of endomorphisms   cend 37758
      ​21.20  Mathbox for Jim Kingdon
            21.20.1  Circle constant   taupilem3 37764
            21.20.2  Number theory   dfgcd3 37769
            21.20.3  Real numbers   irrdifflemf 37770
      ​21.21  Mathbox for ML
            21.21.1  Miscellaneous   csbrecsg 37775
            21.21.2  Cartesian exponentiation   cfinxp 37830
            21.21.3  Topology   iunctb2 37850
                  ​*21.21.3.1  Pi-base theorems   pibp16 37860
      ​21.22  Mathbox for Wolf Lammen
            21.22.1  1. Bootstrapping   wl-section-boot 37869
            21.22.2  Implication chains   wl-section-impchain 37893
            21.22.3  Theorems around the conditional operator   wl-ifp-ncond1 37911
            21.22.4  Alternative development of hadd, cadd   wl-df-3xor 37915
            21.22.5  An alternative axiom ~ ax-13   ax-wl-13v 37940
            21.22.6  Bootstrapping set theory with classes   wl-cleq-0 37942
            21.22.7  Other stuff   wl-mps 37963
      ​21.23  Mathbox for Brendan Leahy
      ​21.24  Mathbox for Jeff Madsen
            21.24.1  Logic and set theory   unirep 38166
            21.24.2  Real and complex numbers; integers   filbcmb 38192
            21.24.3  Sequences and sums   sdclem2 38194
            21.24.4  Topology   subspopn 38204
            21.24.5  Metric spaces   metf1o 38207
            21.24.6  Continuous maps and homeomorphisms   constcncf 38214
            21.24.7  Boundedness   ctotbnd 38218
            21.24.8  Isometries   cismty 38250
            21.24.9  Heine-Borel Theorem   heibor1lem 38261
            21.24.10  Banach Fixed Point Theorem   bfplem1 38274
            21.24.11  Euclidean space   crrn 38277
            21.24.12  Intervals (continued)   ismrer1 38290
            21.24.13  Operation properties   cass 38294
            21.24.14  Groups and related structures   cmagm 38300
            21.24.15  Group homomorphism and isomorphism   cghomOLD 38335
            21.24.16  Rings   crngo 38346
            21.24.17  Division Rings   cdrng 38400
            21.24.18  Ring homomorphisms   crngohom 38412
            21.24.19  Commutative rings   ccm2 38441
            21.24.20  Ideals   cidl 38459
            21.24.21  Prime rings and integral domains   cprrng 38498
            21.24.22  Ideal generators   cigen 38511
      ​21.25  Mathbox for Giovanni Mascellani
            ​*21.25.1  Tools for automatic proof building   efald2 38530
            ​*21.25.2  Tseitin axioms   fald 38581
            ​*21.25.3  Equality deductions   iuneq2f 38608
            ​*21.25.4  Miscellanea   orcomdd 38619
      ​21.26  Mathbox for Peter Mazsa
            21.26.1  Notations   cxrn 38626
            21.26.2  Preparatory theorems   el2v1 38681
            21.26.3  Range Cartesian product   df-xrn 38832
            21.26.4  Relations   df-rels 38892
            21.26.5  Quotient map (coset map)   df-qmap 38898
            21.26.6  Lifts, shifts, successor, and predecessor   df-adjliftmap 38907
            21.26.7  Cosets by ` R `   df-coss 38953
            21.26.8  Subset relations   df-ssr 39030
            21.26.9  Reflexivity   df-refs 39042
            21.26.10  Converse reflexivity   df-cnvrefs 39057
            21.26.11  Symmetry   df-syms 39074
            21.26.12  Reflexivity and symmetry   symrefref2 39099
            21.26.13  Transitivity   df-trs 39108
            21.26.14  Equivalence relations   df-eqvrels 39120
            21.26.15  Redundancy   df-redunds 39159
            21.26.16  Domain quotients   df-dmqss 39174
            21.26.17  Equivalence relations on domain quotients   df-ers 39200
            21.26.18  Functions   df-funss 39217
            21.26.19  Disjoints vs. converse functions   df-disjss 39240
            21.26.20  Antisymmetry   df-antisymrel 39315
            21.26.21  Partitions: disjoints on domain quotients   df-parts 39320
            21.26.22  Partition-Equivalence Theorems   disjim 39336
            21.26.23  Type-safe Partition-Equivalence: PetParts, PetErs, Pet2Parts, Pet2Ers   df-petparts 39420
      ​21.27  Mathbox for Rodolfo Medina
            21.27.1  Partitions   prtlem60 39430
      ​​*21.28  Mathbox for Norm Megill
            *21.28.1  Obsolete schemes ax-c4,c5,c7,c10,c11,c11n,c15,c9,c14,c16   ax-c5 39460
            ​*21.28.2  Rederive new axioms ax-4, ax-10, ax-6, ax-12, ax-13 from old   axc5 39470
            ​*21.28.3  Legacy theorems using obsolete axioms   ax5ALT 39484
            21.28.4  Experiments with weak deduction theorem   elimhyps 39538
            21.28.5  Miscellanea   cnaddcom 39549
            21.28.6  Atoms, hyperplanes, and covering in a left vector space (or module)   clsa 39551
            21.28.7  Functionals and kernels of a left vector space (or module)   clfn 39634
            21.28.8  Opposite rings and dual vector spaces   cld 39700
            21.28.9  Ortholattices and orthomodular lattices   cops 39749
            21.28.10  Atomic lattices with covering property   ccvr 39839
            21.28.11  Hilbert lattices   chlt 39927
            21.28.12  Projective geometries based on Hilbert lattices   clln 40068
            21.28.13  Construction of a vector space from a Hilbert lattice   cdlema1N 40368
            21.28.14  Construction of involution and inner product from a Hilbert lattice   clpoN 42057
      ​21.29  Mathbox for metakunt
            21.29.1  Commutative Semiring   ccsrg 42539
            21.29.2  General helpful statements   rhmzrhval 42542
            21.29.3  Some gcd and lcm results   12gcd5e1 42573
            21.29.4  Least common multiple inequality theorem   3factsumint1 42591
            21.29.5  Logarithm inequalities   3exp7 42623
            21.29.6  Miscellaneous results for AKS formalisation   intlewftc 42631
            21.29.7  Sticks and stones   sticksstones1 42716
            21.29.8  Continuation AKS   aks6d1c6lem1 42740
      ​21.30  Mathbox for Steven Nguyen
            21.30.1  Utility theorems   jarrii 42775
            ​*21.30.2  Arithmetic theorems   c0exALT 42821
            21.30.3  Exponents and divisibility   oexpreposd 42884
            21.30.4  Trigonometry and Calculus   tanhalfpim 42911
            ​*21.30.5  Independence of ax-mulcom   cresub 42927
            21.30.6  Structures   sn-base0 43070
            ​*21.30.7  Projective spaces   cprjsp 43136
            21.30.8  Basic reductions for Fermat's Last Theorem   dffltz 43169
            ​*21.30.9  Exemplar theorems   iddii 43199
                  ​*21.30.9.1  Standard replacements of ax-10 , ax-11 , ax-12   nfa1w 43210
      ​21.31  Mathbox for Igor Ieskov
      ​21.32  Mathbox for OpenAI
      ​21.33  Mathbox for Stefan O'Rear
            21.33.1  Additional elementary logic and set theory   moxfr 43226
            21.33.2  Additional theory of functions   imaiinfv 43227
            21.33.3  Additional topology   elrfi 43228
            21.33.4  Characterization of closure operators. Kuratowski closure axioms   ismrcd1 43232
            21.33.5  Algebraic closure systems   cnacs 43236
            21.33.6  Miscellanea 1. Map utilities   constmap 43247
            21.33.7  Miscellanea for polynomials   mptfcl 43254
            21.33.8  Multivariate polynomials over the integers   cmzpcl 43255
            21.33.9  Miscellanea for Diophantine sets 1   coeq0i 43287
            21.33.10  Diophantine sets 1: definitions   cdioph 43289
            21.33.11  Diophantine sets 2 miscellanea   ellz1 43301
            21.33.12  Diophantine sets 2: union and intersection. Monotone Boolean algebra   diophin 43306
            21.33.13  Diophantine sets 3: construction   diophrex 43309
            21.33.14  Diophantine sets 4 miscellanea   2sbcrex 43318
            21.33.15  Diophantine sets 4: Quantification   rexrabdioph 43324
            21.33.16  Diophantine sets 5: Arithmetic sets   rabdiophlem1 43331
            21.33.17  Diophantine sets 6: reusability. renumbering of variables   eldioph4b 43341
            21.33.18  Pigeonhole Principle and cardinality helpers   fphpd 43346
            21.33.19  A non-closed set of reals is infinite   rencldnfilem 43350
            21.33.20  Lagrange's rational approximation theorem   irrapxlem1 43352
            21.33.21  Pell equations 1: A nontrivial solution always exists   pellexlem1 43359
            21.33.22  Pell equations 2: Algebraic number theory of the solution set   csquarenn 43366
            21.33.23  Pell equations 3: characterizing fundamental solution   infmrgelbi 43408
            ​*21.33.24  Logarithm laws generalized to an arbitrary base   reglogcl 43420
            21.33.25  Pell equations 4: the positive solution group is infinite cyclic   pellfund14 43428
            21.33.26  X and Y sequences 1: Definition and recurrence laws   crmx 43430
            21.33.27  Ordering and induction lemmas for the integers   monotuz 43471
            21.33.28  X and Y sequences 2: Order properties   rmxypos 43477
            21.33.29  Congruential equations   congtr 43495
            21.33.30  Alternating congruential equations   acongid 43505
            21.33.31  Additional theorems on integer divisibility   coprmdvdsb 43515
            21.33.32  X and Y sequences 3: Divisibility properties   jm2.18 43518
            21.33.33  X and Y sequences 4: Diophantine representability of Y   jm2.27a 43535
            21.33.34  X and Y sequences 5: Diophantine representability of X, ^, _C   rmxdiophlem 43545
            21.33.35  Uncategorized stuff not associated with a major project   setindtr 43554
            21.33.36  More equivalents of the Axiom of Choice   axac10 43563
            21.33.37  Finitely generated left modules   clfig 43597
            21.33.38  Noetherian left modules I   clnm 43605
            21.33.39  Addenda for structure powers   pwssplit4 43619
            21.33.40  Every set admits a group structure iff choice   unxpwdom3 43625
            21.33.41  Noetherian rings and left modules II   clnr 43639
            21.33.42  Hilbert's Basis Theorem   cldgis 43651
            21.33.43  Additional material on polynomials [DEPRECATED]   cmnc 43661
            21.33.44  Degree and minimal polynomial of algebraic numbers   cdgraa 43670
            21.33.45  Algebraic integers I   citgo 43687
            21.33.46  Endomorphism algebra   cmend 43701
            21.33.47  Cyclic groups and order   idomodle 43721
            21.33.48  Cyclotomic polynomials   ccytp 43727
            21.33.49  Miscellaneous topology   fgraphopab 43733
      ​21.34  Mathbox for Noam Pasman
      ​21.35  Mathbox for Jon Pennant
      ​21.36  Mathbox for Richard Penner
            21.36.1  Set Theory and Ordinal Numbers   uniel 43747
            21.36.2  Natural addition of Cantor normal forms   oawordex2 43856
            21.36.3  Surreal Contributions   abeqabi 43937
            21.36.4  Short Studies   nlimsuc 43970
                  21.36.4.1  Additional work on conditional logical operator   ifpan123g 43988
                  21.36.4.2  Sophisms   rp-fakeimass 44041
                  ​*21.36.4.3  Finite Sets   rp-isfinite5 44046
                  21.36.4.4  General Observations   intabssd 44048
                  21.36.4.5  Infinite Sets   pwelg 44089
                  ​*21.36.4.6  Finite intersection property   fipjust 44094
                  21.36.4.7  RP ADDTO: Subclasses and subsets   rababg 44103
                  21.36.4.8  RP ADDTO: The intersection of a class   elinintab 44104
                  21.36.4.9  RP ADDTO: Theorems requiring subset and intersection existence   elinintrab 44106
                  21.36.4.10  RP ADDTO: Relations   xpinintabd 44109
                  ​*21.36.4.11  RP ADDTO: Functions   elmapintab 44125
                  ​*21.36.4.12  RP ADDTO: Finite induction (for finite ordinals)   cnvcnvintabd 44129
                  21.36.4.13  RP ADDTO: First and second members of an ordered pair   elcnvlem 44130
                  21.36.4.14  RP ADDTO: The reflexive and transitive properties of relations   undmrnresiss 44133
                  21.36.4.15  RP ADDTO: Basic properties of closures   cleq2lem 44137
                  21.36.4.16  RP REPLACE: Definitions and basic properties of transitive closures   trcleq2lemRP 44159
                  ​*21.36.4.17  Additions for square root; absolute value   sqrtcvallem1 44160
            21.36.5  Additional statements on relations and subclasses   al3im 44176
                  21.36.5.1  Transitive relations (not to be confused with transitive classes)   trrelind 44194
                  21.36.5.2  Reflexive closures   crcl 44201
                  ​*21.36.5.3  Finite relationship composition   relexp2 44206
                  21.36.5.4  Transitive closure of a relation   dftrcl3 44249
                  ​*21.36.5.5  Adapted from Frege   frege77d 44275
            ​*21.36.6  Propositions from _Begriffsschrift_   dfxor4 44295
                  *21.36.6.1  _Begriffsschrift_ Chapter I   dfxor4 44295
                  ​*21.36.6.2  _Begriffsschrift_ Notation hints   whe 44301
                  21.36.6.3  _Begriffsschrift_ Chapter II Implication   ax-frege1 44319
                  21.36.6.4  _Begriffsschrift_ Chapter II Implication and Negation   axfrege28 44358
                  ​*21.36.6.5  _Begriffsschrift_ Chapter II with logical equivalence   axfrege52a 44385
                  21.36.6.6  _Begriffsschrift_ Chapter II with equivalence of sets   axfrege52c 44416
                  ​*21.36.6.7  _Begriffsschrift_ Chapter II with equivalence of classes   frege53c 44443
                  ​*21.36.6.8  _Begriffsschrift_ Chapter III Properties hereditary in a sequence   dffrege69 44461
                  ​*21.36.6.9  _Begriffsschrift_ Chapter III Following in a sequence   dffrege76 44468
                  ​*21.36.6.10  _Begriffsschrift_ Chapter III Member of sequence   dffrege99 44491
                  ​*21.36.6.11  _Begriffsschrift_ Chapter III Single-valued procedures   dffrege115 44507
            ​*21.36.7  Exploring Topology via Seifert and Threlfall   enrelmap 44526
                  *21.36.7.1  Equinumerosity of sets of relations and maps   enrelmap 44526
                  ​*21.36.7.2  Generic Pseudoclosure Spaces, Pseudointerior Spaces, and Pseudoneighborhoods   or3or 44552
                  ​*21.36.7.3  Generic Neighborhood Spaces   gneispa 44659
            ​*21.36.8  Exploring Higher Homotopy via Kerodon   k0004lem1 44676
                  *21.36.8.1  Simplicial Sets   k0004lem1 44676
      ​21.37  Mathbox for Stanislas Polu
            21.37.1  IMO Problems   wwlemuld 44685
                  21.37.1.1  IMO 1972 B2   wwlemuld 44685
            ​*21.37.2  INT Inequalities Proof Generator   int-addcomd 44702
            ​*21.37.3  N-Digit Addition Proof Generator   unitadd 44724
            21.37.4  AM-GM (for k = 2,3,4)   gsumws3 44725
      ​21.38  Mathbox for Rohan Ridenour
            21.38.1  Misc   spALT 44730
            21.38.2  Monoid rings   cmnring 44740
            21.38.3  Shorter primitive equivalent of ax-groth   gru0eld 44758
                  21.38.3.1  Grothendieck universes are closed under collection   gru0eld 44758
                  21.38.3.2  Minimal universes   ismnu 44790
                  21.38.3.3  Primitive equivalent of ax-groth   expandan 44817
      ​21.39  Mathbox for Steve Rodriguez
            21.39.1  Miscellanea   nanorxor 44834
            21.39.2  Ratio test for infinite series convergence and divergence   dvgrat 44841
            21.39.3  Multiples   reldvds 44844
            21.39.4  Function operations   caofcan 44852
            21.39.5  Calculus   lhe4.4ex1a 44858
            21.39.6  The generalized binomial coefficient operation   cbcc 44865
            21.39.7  Binomial series   uzmptshftfval 44875
      ​21.40  Mathbox for Andrew Salmon
            21.40.1  Principia Mathematica * 10   pm10.12 44887
            21.40.2  Principia Mathematica * 11   2alanimi 44901
            21.40.3  Predicate Calculus   sbeqal1 44927
            21.40.4  Principia Mathematica * 13 and * 14   pm13.13a 44936
            21.40.5  Set Theory   elnev 44966
            21.40.6  Arithmetic   addcomgi 44984
            21.40.7  Geometry   cplusr 44985
      ​​*21.41  Mathbox for Alan Sare
            21.41.1  Auxiliary theorems for the Virtual Deduction tool   idiALT 45007
            21.41.2  Supplementary unification deductions   bi1imp 45011
            21.41.3  Conventional Metamath proofs, some derived from VD proofs   iidn3 45030
            21.41.4  What is Virtual Deduction?   wvd1 45098
            21.41.5  Virtual Deduction Theorems   df-vd1 45099
            21.41.6  Theorems proved using Virtual Deduction   trsspwALT 45346
            21.41.7  Theorems proved using Virtual Deduction with mmj2 assistance   simplbi2VD 45374
            21.41.8  Virtual Deduction transcriptions of textbook proofs   sb5ALTVD 45441
            21.41.9  Theorems proved using conjunction-form Virtual Deduction   elpwgdedVD 45445
            21.41.10  Theorems with a VD proof in conventional notation derived from a VD proof   suctrALT3 45452
            ​*21.41.11  Theorems with a proof in conventional notation derived from a VD proof   notnotrALT2 45455
      ​21.42  Mathbox for Eric Schmidt
            21.42.1  Miscellany   rspesbcd 45466
            21.42.2  Study of dfbi1ALT   dfbi1ALTa 45468
            21.42.3  Relation-preserving functions   wrelp 45471
            21.42.4  Orbits   orbitex 45484
            21.42.5  Well-founded sets   trwf 45488
            21.42.6  Absoluteness in transitive models   ralabso 45497
            21.42.7  Lemmas for showing axioms hold in models   traxext 45506
            21.42.8  The class of well-founded sets is a model for ZFC   wfaxext 45522
            21.42.9  Permutation models   brpermmodel 45532
      ​21.43  Mathbox for Glauco Siliprandi
            21.43.1  Miscellanea   evth2f 45548
            21.43.2  Functions   fnresdmss 45699
            21.43.3  Ordering on real numbers - Real and complex numbers basic operations   sub2times 45805
            21.43.4  Real intervals   gtnelioc 46020
            21.43.5  Finite sums   fsummulc1f 46100
            21.43.6  Finite multiplication of numbers and finite multiplication of functions   fmul01 46109
            21.43.7  Limits   clim1fr1 46130
                  21.43.7.1  Inferior limit (lim inf)   clsi 46278
                  ​*21.43.7.2  Limits for sequences of extended real numbers   clsxlim 46345
            21.43.8  Trigonometry   coseq0 46391
            21.43.9  Continuous Functions   mulcncff 46397
            21.43.10  Derivatives   dvsinexp 46438
            21.43.11  Integrals   itgsin0pilem1 46477
            21.43.12  Stone Weierstrass theorem - real version   stoweidlem1 46528
            21.43.13  Wallis' product for π   wallispilem1 46592
            21.43.14  Stirling's approximation formula for ` n ` factorial   stirlinglem1 46601
            21.43.15  Dirichlet kernel   dirkerval 46618
            21.43.16  Fourier Series   fourierdlem1 46635
            21.43.17  e is transcendental   elaa2lem 46760
            21.43.18  n-dimensional Euclidean space   rrxtopn 46811
            21.43.19  Basic measure theory   csalg 46835
                  ​*21.43.19.1  σ-Algebras   csalg 46835
                  21.43.19.2  Sum of nonnegative extended reals   csumge0 46889
                  ​*21.43.19.3  Measures   cmea 46976
                  ​*21.43.19.4  Outer measures and Caratheodory's construction   come 47016
                  ​*21.43.19.5  Lebesgue measure on n-dimensional Real numbers   covoln 47063
                  ​*21.43.19.6  Measurable functions   csmblfn 47222
      ​21.44  Mathbox for Saveliy Skresanov
            21.44.1  Ceva's theorem   sigarval 47377
            21.44.2  Simple groups   simpcntrab 47397
      ​21.45  Mathbox for Ender Ting
            21.45.1  Interesting facts   et-ltneverrefl 47398
            21.45.2  Increasing sequences and subsequences   ormklocald 47403
            21.45.3  Scratchpad for number theory   evenwodadd 47416
            21.45.4  Scratchpad for math on real numbers   squeezedltsq 47417
      ​21.46  Mathbox for Jarvin Udandy
      ​21.47  Mathbox for Adhemar
            ​*21.47.1  Minimal implicational calculus   adh-minim 47548
      ​21.48  Mathbox for Alexander van der Vekens
            21.48.1  General auxiliary theorems (1)   n0nsn2el 47572
                  21.48.1.1  Unordered and ordered pairs - extension for singletons   n0nsn2el 47572
                  21.48.1.2  Unordered and ordered pairs - extension for unordered pairs   elprneb 47576
                  21.48.1.3  Unordered and ordered pairs - extension for ordered pairs   oppr 47577
                  21.48.1.4  Relations - extension   eubrv 47582
                  21.48.1.5  Definite description binder (inverted iota) - extension   iota0def 47585
                  21.48.1.6  Functions - extension   fveqvfvv 47587
            21.48.2  Alternative for Russell's definition of a description binder   caiota 47630
            21.48.3  Double restricted existential uniqueness   r19.32 47645
                  21.48.3.1  Restricted quantification (extension)   r19.32 47645
                  21.48.3.2  Restricted uniqueness and "at most one" quantification   reuf1odnf 47654
                  21.48.3.3  Analogs to Existential uniqueness (double quantification)   2reu3 47657
                  21.48.3.4  Additional theorems for double restricted existential uniqueness   2reu8i 47660
            ​*21.48.4  Alternative definitions of function and operation values   wdfat 47663
                  21.48.4.1  Restricted quantification (extension)   ralbinrald 47669
                  21.48.4.2  The universal class (extension)   nvelim 47670
                  21.48.4.3  Introduce the Axiom of Power Sets (extension)   alneu 47671
                  21.48.4.4  Predicate "defined at"   dfateq12d 47673
                  21.48.4.5  Alternative definition of the value of a function   dfafv2 47679
                  21.48.4.6  Alternative definition of the value of an operation   aoveq123d 47725
            ​*21.48.5  Alternative definitions of function values (2)   cafv2 47755
            21.48.6  General auxiliary theorems (2)   an4com24 47815
                  21.48.6.1  Logical conjunction - extension   an4com24 47815
                  21.48.6.2  Abbreviated conjunction and disjunction of three wff's - extension   3an4ancom24 47816
                  21.48.6.3  Negated membership (alternative)   cnelbr 47818
                  21.48.6.4  The empty set - extension   ralralimp 47825
                  21.48.6.5  Indexed union and intersection - extension   otiunsndisjX 47826
                  21.48.6.6  Functions - extension   fvifeq 47827
                  21.48.6.7  Maps-to notation - extension   fvmptrab 47839
                  21.48.6.8  Subtraction - extension   cnambpcma 47841
                  21.48.6.9  Ordering on reals (cont.) - extension   leaddsuble 47844
                  21.48.6.10  Imaginary and complex number properties - extension   readdcnnred 47850
                  21.48.6.11  Nonnegative integers (as a subset of complex numbers) - extension   nn0resubcl 47855
                  21.48.6.12  Integers (as a subset of complex numbers) - extension   zgeltp1eq 47856
                  21.48.6.13  Decimal arithmetic - extension   1t10e1p1e11 47857
                  21.48.6.14  Upper sets of integers - extension   eluzge0nn0 47859
                  21.48.6.15  Infinity and the extended real number system (cont.) - extension   nltle2tri 47860
                  21.48.6.16  Finite intervals of integers - extension   ssfz12 47861
                  21.48.6.17  Half-open integer ranges - extension   fzopred 47870
                  21.48.6.18  The floor and ceiling functions - extension   2ltceilhalf 47879
                  21.48.6.19  The modulo (remainder) operation - extension   fldivmod 47891
                  21.48.6.20  The infinite sequence builder "seq"   smonoord 47924
                  21.48.6.21  Integer powers - extension   2timesltsq 47925
                  21.48.6.22  Finite and infinite sums - extension   fsummsndifre 47927
                  21.48.6.23  The divides relation - extension   nndivides2 47931
                  21.48.6.24  Extensible structures - extension   setsidel 47935
            ​*21.48.7  Preimages of function values   preimafvsnel 47938
            ​*21.48.8  Partitions of real intervals   ciccp 47972
            21.48.9  Shifting functions with an integer range domain   fargshiftfv 47998
            21.48.10  Words over a set (extension)   lswn0 48003
                  21.48.10.1  Last symbol of a word - extension   lswn0 48003
            21.48.11  Unordered pairs   wich 48004
                  21.48.11.1  Interchangeable setvar variables   wich 48004
                  21.48.11.2  Set of unordered pairs   sprid 48033
                  ​*21.48.11.3  Proper (unordered) pairs   prpair 48060
                  21.48.11.4  Set of proper unordered pairs   cprpr 48071
            21.48.12  Number theory (extension)   nprmmul1 48086
                  21.48.12.1  Properties of non-prime numbers   nprmmul1 48086
                  ​*21.48.12.2  Fermat numbers   cfmtno 48089
                  ​*21.48.12.3  Mersenne primes   m2prm 48153
                  21.48.12.4  Proth's theorem   modexp2m1d 48174
                  21.48.12.5  The prime-counting function according to Ján Mináč   nprmdvdsfacm1lem1 48182
                  21.48.12.6  Solutions of quadratic equations   quad1 48195
            ​*21.48.13  Even and odd numbers   ceven 48199
                  21.48.13.1  Definitions and basic properties   ceven 48199
                  21.48.13.2  Alternate definitions using the "divides" relation   dfeven2 48224
                  21.48.13.3  Alternate definitions using the "modulo" operation   dfeven3 48233
                  21.48.13.4  Alternate definitions using the "gcd" operation   iseven5 48239
                  21.48.13.5  Theorems of part 5 revised   zneoALTV 48244
                  21.48.13.6  Theorems of part 6 revised   odd2np1ALTV 48249
                  21.48.13.7  Theorems of AV's mathbox revised   0evenALTV 48263
                  21.48.13.8  Additional theorems   epoo 48278
                  21.48.13.9  Perfect Number Theorem (revised)   perfectALTVlem1 48296
            21.48.14  Number theory (extension 2)   cfppr 48299
                  ​*21.48.14.1  Fermat pseudoprimes   cfppr 48299
                  ​*21.48.14.2  Goldbach's conjectures   cgbe 48320
            21.48.15  Graph theory (extension)   cclnbgr 48393
                  21.48.15.1  Closed neighborhood of a vertex   cclnbgr 48393
                  ​*21.48.15.2  Semiclosed and semiopen neighborhoods (experimental)   dfsclnbgr2 48421
                  21.48.15.3  Induced subgraphs   cisubgr 48435
                  ​*21.48.15.4  Isomorphisms of graphs   cgrisom 48449
                  ​*21.48.15.5  Triangles in graphs   cgrtri 48512
                  ​*21.48.15.6  Star graphs   cstgr 48526
                  ​*21.48.15.7  Local isomorphisms of graphs   cgrlim 48551
                  ​*21.48.15.8  Generalized Petersen graphs   cgpg 48615
                  21.48.15.9  Loop-free graphs - extension   1hegrlfgr 48707
                  21.48.15.10  Walks - extension   cupwlks 48708
                  21.48.15.11  Edges of graphs expressed as sets of unordered pairs   upgredgssspr 48718
            21.48.16  Monoids (extension)   ovn0dmfun 48731
                  21.48.16.1  Auxiliary theorems   ovn0dmfun 48731
                  21.48.16.2  Magmas, Semigroups and Monoids (extension)   plusfreseq 48739
                  21.48.16.3  Examples and counterexamples for magmas, semigroups and monoids (extension)   opmpoismgm 48742
                  21.48.16.4  Group sum operation (extension 1)   gsumsplit2f 48755
            ​*21.48.17  Magmas and internal binary operations (alternate approach)   ccllaw 48758
                  *21.48.17.1  Laws for internal binary operations   ccllaw 48758
                  ​*21.48.17.2  Internal binary operations   cintop 48771
                  21.48.17.3  Alternative definitions for magmas and semigroups   cmgm2 48790
            21.48.18  Rings (extension)   lmod0rng 48804
                  21.48.18.1  Nonzero rings (extension)   lmod0rng 48804
                  21.48.18.2  Ideals as non-unital rings   lidldomn1 48806
                  21.48.18.3  The non-unital ring of even integers   0even 48812
                  21.48.18.4  A constructed not unital ring   cznrnglem 48834
                  ​*21.48.18.5  The category of non-unital rings (alternate definition)   crngcALTV 48838
                  ​*21.48.18.6  The category of (unital) rings (alternate definition)   cringcALTV 48862
            21.48.19  Basic algebraic structures (extension)   eliunxp2 48909
                  21.48.19.1  Auxiliary theorems   eliunxp2 48909
                  21.48.19.2  The binomial coefficient operation (extension)   bcpascm1 48926
                  21.48.19.3  The ` ZZ `-module ` ZZ X. ZZ `   zlmodzxzlmod 48929
                  21.48.19.4  Group sum operation (extension 2)   mgpsumunsn 48936
                  21.48.19.5  Symmetric groups (extension)   exple2lt6 48939
                  21.48.19.6  Divisibility (extension)   invginvrid 48942
                  21.48.19.7  The support of functions (extension)   rmsupp0 48943
                  21.48.19.8  Finitely supported functions (extension)   rmsuppfi 48947
                  21.48.19.9  Left modules (extension)   lmodvsmdi 48954
                  21.48.19.10  Associative algebras (extension)   assaascl0 48956
                  21.48.19.11  Univariate polynomials (extension)   ply1vr1smo 48958
                  21.48.19.12  Univariate polynomials (examples)   linply1 48968
            21.48.20  Linear algebra (extension)   cdmatalt 48971
                  ​*21.48.20.1  The subalgebras of diagonal and scalar matrices (extension)   cdmatalt 48971
                  ​*21.48.20.2  Linear combinations   clinc 48979
                  ​*21.48.20.3  Linear independence   clininds 49015
                  21.48.20.4  Simple left modules and the ` ZZ `-module   lmod1lem1 49062
                  21.48.20.5  Differences between (left) modules and (left) vector spaces   lvecpsslmod 49082
            21.48.21  Complexity theory   suppdm 49085
                  21.48.21.1  Auxiliary theorems   suppdm 49085
                  21.48.21.2  Even and odd integers   nn0onn0ex 49098
                  21.48.21.3  The natural logarithm on complex numbers (extension)   logcxp0 49110
                  21.48.21.4  Division of functions   cfdiv 49112
                  21.48.21.5  Upper bounds   cbigo 49122
                  21.48.21.6  Logarithm to an arbitrary base (extension)   rege1logbrege0 49133
                  ​*21.48.21.7  The binary logarithm   fldivexpfllog2 49140
                  21.48.21.8  Binary length   cblen 49144
                  ​*21.48.21.9  Digits   cdig 49170
                  21.48.21.10  Nonnegative integer as sum of its shifted digits   dignn0flhalflem1 49190
                  21.48.21.11  Algorithms for the multiplication of nonnegative integers   nn0mulfsum 49199
                  ​*21.48.21.12  N-ary functions   cnaryf 49201
                  ​*21.48.21.13  The Ackermann function   citco 49232
            21.48.22  Elementary geometry (extension)   fv1prop 49274
                  21.48.22.1  Auxiliary theorems   fv1prop 49274
                  21.48.22.2  Real euclidean space of dimension 2   rrx2pxel 49286
                  21.48.22.3  Spheres and lines in real Euclidean spaces   cline 49302
      ​21.49  Mathbox for Zhi Wang
            21.49.1  Propositional calculus   pm4.71da 49364
            21.49.2  Predicate calculus with equality   dtrucor3 49373
                  21.49.2.1  Axiom scheme ax-5 (Distinctness)   dtrucor3 49373
            21.49.3  ZF Set Theory - start with the Axiom of Extensionality   ralbidb 49374
                  21.49.3.1  Restricted quantification   ralbidb 49374
                  21.49.3.2  The universal class   reuxfr1dd 49381
                  21.49.3.3  The empty set   ssdisjd 49382
                  21.49.3.4  Unordered and ordered pairs   vsn 49386
                  21.49.3.5  The union of a class   unilbss 49392
                  21.49.3.6  Indexed union and intersection   iuneq0 49393
            21.49.4  ZF Set Theory - add the Axiom of Replacement   inpw 49399
                  21.49.4.1  Theorems requiring subset and intersection existence   inpw 49399
            21.49.5  ZF Set Theory - add the Axiom of Power Sets   opth1neg 49400
                  21.49.5.1  Ordered pair theorem   opth1neg 49400
                  21.49.5.2  Ordered-pair class abstractions (cont.)   brab2dd 49402
                  21.49.5.3  Relations   iinxp 49405
                  21.49.5.4  Functions   mof0 49412
                  21.49.5.5  Operations   ovsng 49432
            21.49.6  ZF Set Theory - add the Axiom of Union   fonex 49441
                  21.49.6.1  Relations and functions (cont.)   fonex 49441
                  21.49.6.2  First and second members of an ordered pair   eloprab1st2nd 49442
                  21.49.6.3  Operations in maps-to notation (continued)   fmpodg 49443
                  21.49.6.4  Function transposition   resinsnlem 49445
                  21.49.6.5  Infinite Cartesian products   ixpv 49464
                  21.49.6.6  Equinumerosity   fvconst0ci 49465
            21.49.7  Order sets   iccin 49470
                  21.49.7.1  Real number intervals   iccin 49470
            21.49.8  Extensible structures   slotresfo 49473
                  21.49.8.1  Basic definitions   slotresfo 49473
            21.49.9  Moore spaces   mreuniss 49474
            ​*21.49.10  Topology   clduni 49475
                  21.49.10.1  Closure and interior   clduni 49475
                  21.49.10.2  Neighborhoods   neircl 49479
                  21.49.10.3  Subspace topologies   restcls2lem 49487
                  21.49.10.4  Limits and continuity in topological spaces   cnneiima 49491
                  21.49.10.5  Topological definitions using the reals   iooii 49492
                  21.49.10.6  Separated sets   sepnsepolem1 49496
                  21.49.10.7  Separated spaces: T0, T1, T2 (Hausdorff) ...   isnrm4 49505
            21.49.11  Preordered sets and directed sets using extensible structures   isprsd 49529
            21.49.12  Posets and lattices using extensible structures   lubeldm2 49530
                  21.49.12.1  Posets   lubeldm2 49530
                  21.49.12.2  Lattices   toslat 49556
                  21.49.12.3  Subset order structures   intubeu 49558
            21.49.13  Rings   elmgpcntrd 49579
                  21.49.13.1  Multiplicative Group   elmgpcntrd 49579
            21.49.14  Associative algebras   asclelbasALT 49580
                  21.49.14.1  Definition and basic properties   asclelbasALT 49580
            21.49.15  Categories   homf0 49583
                  21.49.15.1  Categories   homf0 49583
                  21.49.15.2  Opposite category   oppccatb 49590
                  21.49.15.3  Monomorphisms and epimorphisms   idmon 49594
                  21.49.15.4  Sections, inverses, isomorphisms   sectrcl 49596
                  21.49.15.5  Isomorphic objects   cicfn 49616
                  21.49.15.6  Subcategories   dmdm 49627
                  21.49.15.7  Functors   reldmfunc 49649
                  21.49.15.8  Opposite functors   coppf 49696
                  21.49.15.9  Full & faithful functors   imasubc 49725
                  21.49.15.10  Universal property   upciclem1 49740
                  21.49.15.11  Natural transformations and the functor category   isnatd 49797
                  21.49.15.12  Initial, terminal and zero objects of a category   initoo2 49806
                  21.49.15.13  Product of categories   reldmxpc 49820
                  21.49.15.14  Swap functors   cswapf 49833
                  21.49.15.15  Functor evaluation   oppc1stflem 49861
                  21.49.15.16  Transposed curry functors   cofuswapfcl 49867
                  21.49.15.17  Constant functors   diag1 49878
                  21.49.15.18  Functor composition bifunctors   fucofulem1 49884
                  21.49.15.19  Post-composition functors   postcofval 49938
                  21.49.15.20  Pre-composition functors   precofvallem 49940
            21.49.16  Examples of categories   catcrcl 49969
                  21.49.16.1  The category of categories   catcrcl 49969
                  21.49.16.2  Thin categories   cthinc 49991
                  21.49.16.3  Terminal categories   ctermc 50046
                  21.49.16.4  Preordered sets as thin categories   cprstc 50123
                  21.49.16.5  Monoids as categories   cmndtc 50151
                  21.49.16.6  Categories with at most one object and at most two morphisms   2arwcatlem1 50169
            21.49.17  Kan extensions and related concepts   clan 50179
                  21.49.17.1  Kan extensions   clan 50179
                  21.49.17.2  Limits and colimits   clmd 50217
      ​21.50  Mathbox for Emmett Weisz
            ​*21.50.1  Miscellaneous Theorems   nfintd 50247
            21.50.2  Set Recursion   csetrecs 50257
                  ​*21.50.2.1  Basic Properties of Set Recursion   csetrecs 50257
                  21.50.2.2  Examples and properties of set recursion   elsetrecslem 50273
            ​*21.50.3  Construction of Games and Surreal Numbers   cpg 50283
      ​​*21.51  Mathbox for David A. Wheeler
            21.51.1  Natural deduction   sbidd 50292
            ​*21.51.2  Greater than, greater than or equal to   cge-real 50294
            ​*21.51.3  Hyperbolic trigonometric functions   csinh 50304
            ​*21.51.4  Reciprocal trigonometric functions (sec, csc, cot)   csec 50315
            ​*21.51.5  Identities for "if"   ifnmfalse 50337
            ​*21.51.6  Logarithms generalized to arbitrary base using ` logb `   logb2aval 50338
            ​*21.51.7  Logarithm laws generalized to an arbitrary base - log_   clog- 50339
            ​*21.51.8  Formally define notions such as reflexivity   wreflexive 50341
            ​*21.51.9  Algebra helpers   mvlraddi 50345
            ​*21.51.10  Algebra helper examples   i2linesi 50352
            ​*21.51.11  Formal methods "surprises"   alimp-surprise 50354
            ​*21.51.12  Allsome quantifier   walsi 50360
            ​*21.51.13  Miscellaneous   5m4e1 50371
            21.51.14  Theorems about algebraic numbers   aacllem 50375
      ​21.52  Mathbox for Kunhao Zheng
            21.52.1  Weighted AM-GM inequality   amgmwlem 50376