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
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 206
            ​*1.2.6  Logical conjunction   wa 395
            ​*1.2.7  Logical disjunction   wo 848
            ​*1.2.8  Mixed connectives   jaao 957
            ​*1.2.9  The conditional operator for propositions   wif 1063
            ​*1.2.10  The weak deduction theorem for propositional calculus   elimh 1083
            1.2.11  Abbreviated conjunction and disjunction of three wff's   w3o 1086
            1.2.12  Logical "nand" (Sheffer stroke)   wnan 1493
            1.2.13  Logical "xor"   wxo 1513
            1.2.14  Logical "nor"   wnor 1530
            1.2.15  True and false constants   wal 1540
                  ​*1.2.15.1  Universal quantifier for use by df-tru   wal 1540
                  ​*1.2.15.2  Equality predicate for use by df-tru   cv 1541
                  1.2.15.3  The true constant   wtru 1543
                  1.2.15.4  The false constant   wfal 1554
            ​*1.2.16  Truth tables   truimtru 1565
                  1.2.16.1  Implication   truimtru 1565
                  1.2.16.2  Negation   nottru 1569
                  1.2.16.3  Equivalence   trubitru 1571
                  1.2.16.4  Conjunction   truantru 1575
                  1.2.16.5  Disjunction   truortru 1579
                  1.2.16.6  Alternative denial   trunantru 1583
                  1.2.16.7  Exclusive disjunction   truxortru 1587
                  1.2.16.8  Joint denial   trunortru 1591
            ​*1.2.17  Half adder and full adder in propositional calculus   whad 1595
                  1.2.17.1  Full adder: sum   whad 1595
                  1.2.17.2  Full adder: carry   wcad 1608
      ​1.3  Other axiomatizations related to classical propositional calculus
            ​*1.3.1  Minimal implicational calculus   minimp 1623
            ​*1.3.2  Implicational Calculus   impsingle 1629
            1.3.3  Derive the Lukasiewicz axioms from Meredith's sole axiom   meredith 1643
            1.3.4  Derive the standard axioms from the Lukasiewicz axioms   luklem1 1660
            ​*1.3.5  Derive Nicod's axiom from the standard axioms   nic-dfim 1671
            1.3.6  Derive the Lukasiewicz axioms from Nicod's axiom   nic-imp 1677
            1.3.7  Derive Nicod's Axiom from Lukasiewicz's First Sheffer Stroke Axiom   lukshef-ax1 1696
            1.3.8  Derive the Lukasiewicz Axioms from the Tarski-Bernays-Wajsberg Axioms   tbw-bijust 1700
            1.3.9  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's First CO Axiom   merco1 1715
            1.3.10  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's Second CO Axiom   merco2 1738
            1.3.11  Derive the Lukasiewicz axioms from the Russell-Bernays Axioms   rb-bijust 1751
            ​*1.3.12  Stoic logic non-modal portion (Chrysippus of Soli)   mptnan 1770
      ​​*1.4  Predicate calculus with equality: Tarski's system S2 (1 rule, 6 schemes)
            *1.4.1  Universal quantifier (continued); define "exists" and "not free"   wex 1781
                  1.4.1.1  Existential quantifier   wex 1781
                  1.4.1.2  Nonfreeness predicate   wnf 1785
            1.4.2  Rule scheme ax-gen (Generalization)   ax-gen 1797
            1.4.3  Axiom scheme ax-4 (Quantified Implication)   ax-4 1811
                  ​*1.4.3.1  The empty domain of discourse   empty 1908
            1.4.4  Axiom scheme ax-5 (Distinctness) - first use of $d   ax-5 1912
            ​*1.4.5  Equality predicate (continued)   weq 1964
            1.4.6  Axiom scheme ax-6 (Existence)   ax-6 1969
            1.4.7  Axiom scheme ax-7 (Equality)   ax-7 2010
            1.4.8  Define proper substitution   sbjust 2067
            1.4.9  Membership predicate   wcel 2114
            1.4.10  Axiom scheme ax-8 (Left Equality for Binary Predicate)   ax-8 2116
            1.4.11  Axiom scheme ax-9 (Right Equality for Binary Predicate)   ax-9 2124
            ​*1.4.12  Logical redundancy of ax-10 , ax-11 , ax-12 , ax-13   ax6dgen 2134
      ​​*1.5  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes)
            1.5.1  Axiom scheme ax-10 (Quantified Negation)   ax-10 2147
            1.5.2  Axiom scheme ax-11 (Quantifier Commutation)   ax-11 2163
            1.5.3  Axiom scheme ax-12 (Substitution)   ax-12 2185
            1.5.4  Axiom scheme ax-13 (Quantified Equality)   ax-13 2377
      ​1.6  Uniqueness and unique existence
            1.6.1  Uniqueness: the at-most-one quantifier   wmo 2538
            1.6.2  Unique existence: the unique existential quantifier   weu 2569
      ​1.7  Other axiomatizations related to classical predicate calculus
            ​*1.7.1  Aristotelian logic: Assertic syllogisms   barbara 2664
            ​*1.7.2  Intuitionistic logic   axia1 2694
​​*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 2709
            2.1.2  Classes   cab 2715
                  2.1.2.1  Class abstractions   cab 2715
                  ​*2.1.2.2  Class equality   df-cleq 2729
                  2.1.2.3  Class membership   df-clel 2812
                  2.1.2.4  Elementary properties of class abstractions   eqabdv 2870
            2.1.3  Class form not-free predicate   wnfc 2884
            2.1.4  Negated equality and membership   wne 2933
                  2.1.4.1  Negated equality   wne 2933
                  2.1.4.2  Negated membership   wnel 3037
            2.1.5  Restricted quantification   wral 3052
                  2.1.5.1  Restricted universal and existential quantification   wral 3052
                  2.1.5.2  Restricted existential uniqueness and at-most-one quantifier   wreu 3341
                  2.1.5.3  Restricted class abstraction   crab 3390
            2.1.6  The universal class   cvv 3430
            ​*2.1.7  Conditional equality (experimental)   wcdeq 3710
            2.1.8  Russell's Paradox   rru 3726
            2.1.9  Proper substitution of classes for sets   wsbc 3729
            2.1.10  Proper substitution of classes for sets into classes   csb 3838
            2.1.11  Define basic set operations and relations   cdif 3887
            2.1.12  Subclasses and subsets   df-ss 3907
            2.1.13  The difference, union, and intersection of two classes   dfdif3 4058
                  2.1.13.1  The difference of two classes   dfdif3 4058
                  2.1.13.2  The union of two classes   elun 4094
                  2.1.13.3  The intersection of two classes   elini 4140
                  2.1.13.4  The symmetric difference of two classes   csymdif 4193
                  2.1.13.5  Combinations of difference, union, and intersection of two classes   unabs 4206
                  2.1.13.6  Class abstractions with difference, union, and intersection of two classes   unabw 4248
                  2.1.13.7  Restricted uniqueness with difference, union, and intersection   reuun2 4266
            2.1.14  The empty set   c0 4274
            ​*2.1.15  The conditional operator for classes   cif 4467
            ​*2.1.16  The weak deduction theorem for set theory   dedth 4526
            2.1.17  Power classes   cpw 4542
            2.1.18  Unordered and ordered pairs   snjust 4567
            2.1.19  The union of a class   cuni 4851
            2.1.20  The intersection of a class   cint 4890
            2.1.21  Indexed union and intersection   ciun 4934
            2.1.22  Disjointness   wdisj 5053
            2.1.23  Binary relations   wbr 5086
            2.1.24  Ordered-pair class abstractions (class builders)   copab 5148
            2.1.25  Functions in maps-to notation   cmpt 5167
            2.1.26  Transitive classes   wtr 5193
      ​2.2  ZF Set Theory - add the Axiom of Replacement
            2.2.1  Introduce the Axiom of Replacement   ax-rep 5212
            2.2.2  Derive the Axiom of Separation   axsepgfromrep 5229
            2.2.3  Derive the Null Set Axiom   axnulALT 5239
            2.2.4  Theorems requiring subset and intersection existence   nalset 5248
            2.2.5  Theorems requiring empty set existence   class2set 5290
      ​2.3  ZF Set Theory - add the Axiom of Power Sets
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 5300
            2.3.2  Derive the Axiom of Pairing   axprlem1 5358
            2.3.3  Ordered pair theorem   opnz 5419
            2.3.4  Ordered-pair class abstractions (cont.)   opabidw 5470
            2.3.5  Power class of union and intersection   pwin 5513
            2.3.6  The identity relation   cid 5516
            2.3.7  The membership relation (or epsilon relation)   cep 5521
            ​*2.3.8  Partial and total orderings   wpo 5528
            2.3.9  Founded and well-ordering relations   wfr 5572
            2.3.10  Relations   cxp 5620
            2.3.11  The Predecessor Class   cpred 6256
            2.3.12  Well-founded induction (variant)   frpomin 6296
            2.3.13  Well-ordered induction   tz6.26 6303
            2.3.14  Ordinals   word 6314
            2.3.15  Definite description binder (inverted iota)   cio 6444
            2.3.16  Functions   wfun 6484
            2.3.17  Cantor's Theorem   canth 7312
            2.3.18  Restricted iota (description binder)   crio 7314
            2.3.19  Operations   co 7358
                  2.3.19.1  Variable-to-class conversion for operations   caovclg 7550
            2.3.20  Maps-to notation   mpondm0 7598
            2.3.21  Function operation   cof 7620
            2.3.22  Proper subset relation   crpss 7667
      ​2.4  ZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 7680
            2.4.2  Ordinals (continued)   epweon 7720
            2.4.3  Transfinite induction   tfi 7795
            2.4.4  The natural numbers (i.e., finite ordinals)   com 7808
            2.4.5  Peano's postulates   peano1 7831
            2.4.6  Finite induction (for finite ordinals)   find 7837
            2.4.7  Relations and functions (cont.)   dmexg 7843
            2.4.8  First and second members of an ordered pair   c1st 7931
            2.4.9  Induction on Cartesian products   frpoins3xpg 8081
            2.4.10  Ordering on Cartesian products   xpord2lem 8083
            2.4.11  Ordering Ordinal Sequences   orderseqlem 8098
            ​*2.4.12  The support of functions   csupp 8101
            ​*2.4.13  Special maps-to operations   opeliunxp2f 8151
            2.4.14  Function transposition   ctpos 8166
            2.4.15  Curry and uncurry   ccur 8206
            2.4.16  Undefined values   cund 8213
            2.4.17  Well-founded recursion   cfrecs 8221
            2.4.18  Well-ordered recursion   cwrecs 8252
            2.4.19  Functions on ordinals; strictly monotone ordinal functions   iunon 8270
            2.4.20  "Strong" transfinite recursion   crecs 8301
            2.4.21  Recursive definition generator   crdg 8339
            2.4.22  Finite recursion   frfnom 8365
            2.4.23  Ordinal arithmetic   c1o 8389
            2.4.24  Natural number arithmetic   nna0 8531
            2.4.25  Natural addition   cnadd 8592
            2.4.26  Equivalence relations and classes   wer 8631
            2.4.27  The mapping operation   cmap 8764
            2.4.28  Infinite Cartesian products   cixp 8836
            2.4.29  Equinumerosity   cen 8881
            2.4.30  Schroeder-Bernstein Theorem   sbthlem1 9016
            2.4.31  Equinumerosity (cont.)   xpf1o 9068
            2.4.32  Finite sets   dif1enlem 9085
            2.4.33  Pigeonhole Principle   phplem1 9129
            2.4.34  Finite sets (cont.)   onomeneq 9139
            2.4.35  Finitely supported functions   cfsupp 9265
            2.4.36  Finite intersections   cfi 9314
            2.4.37  Hall's marriage theorem   marypha1lem 9337
            2.4.38  Supremum and infimum   csup 9344
            2.4.39  Ordinal isomorphism, Hartogs's theorem   coi 9415
            2.4.40  Hartogs function   char 9462
            2.4.41  Weak dominance   cwdom 9470
      ​2.5  ZF Set Theory - add the Axiom of Regularity
            2.5.1  Introduce the Axiom of Regularity   ax-reg 9498
            2.5.2  Axiom of Infinity equivalents   inf0 9531
      ​2.6  ZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-inf 9548
            2.6.2  Existence of omega (the set of natural numbers)   omex 9553
            2.6.3  Cantor normal form   ccnf 9571
            2.6.4  Transitive closure of a relation   cttrcl 9617
            2.6.5  Transitive closure   trcl 9638
            2.6.6  Set induction (or epsilon induction)   setind 9657
            2.6.7  Well-Founded Induction   frmin 9662
            2.6.8  Well-Founded Recursion   frr3g 9669
            2.6.9  Rank   cr1 9675
            2.6.10  Scott's trick; collection principle; Hilbert's epsilon   scottex 9798
            2.6.11  Disjoint union   cdju 9811
            2.6.12  Cardinal numbers   ccrd 9848
            2.6.13  Axiom of Choice equivalents   wac 10026
            ​*2.6.14  Cardinal number arithmetic   undjudom 10079
            2.6.15  The Ackermann bijection   ackbij2lem1 10129
            2.6.16  Cofinality (without Axiom of Choice)   cflem 10156
            2.6.17  Eight inequivalent definitions of finite set   sornom 10188
            2.6.18  Hereditarily size-limited sets without Choice   itunifval 10327
​​*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 10346
            3.1.2  Introduce the Axiom of Dependent Choice   ax-dc 10357
      ​3.2  ZFC Set Theory - add the Axiom of Choice
            3.2.1  Introduce the Axiom of Choice   ax-ac 10370
            3.2.2  AC equivalents: well-ordering, Zorn's lemma   numthcor 10405
            3.2.3  Cardinal number theorems using Axiom of Choice   cardval 10457
            3.2.4  Cardinal number arithmetic using Axiom of Choice   iunctb 10486
            3.2.5  Cofinality using the Axiom of Choice   alephreg 10494
      ​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 10532
            3.4.2  Derivation of the Axiom of Choice   gchaclem 10590
​​*PART 4  TG (TARSKI-GROTHENDIECK) SET THEORY
      ​4.1  Inaccessibles
            4.1.1  Weakly and strongly inaccessible cardinals   cwina 10594
            4.1.2  Weak universes   cwun 10612
            4.1.3  Tarski classes   ctsk 10660
            4.1.4  Grothendieck universes   cgru 10702
      ​4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
            4.2.1  Introduce the Tarski-Grothendieck Axiom   ax-groth 10735
            4.2.2  Derive the Power Set, Infinity and Choice Axioms   grothpw 10738
            4.2.3  Tarski map function   ctskm 10749
​​*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 10756
            5.1.2  Final derivation of real and complex number postulates   axaddf 11057
            5.1.3  Real and complex number postulates restated as axioms   ax-cnex 11083
      ​5.2  Derive the basic properties from the field axioms
            5.2.1  Some deductions from the field axioms for complex numbers   cnex 11108
            5.2.2  Infinity and the extended real number system   cpnf 11165
            5.2.3  Restate the ordering postulates with extended real "less than"   axlttri 11206
            5.2.4  Ordering on reals   lttr 11211
            5.2.5  Initial properties of the complex numbers   mul12 11300
      ​5.3  Real and complex numbers - basic operations
            5.3.1  Addition   add12 11353
            5.3.2  Subtraction   cmin 11366
            5.3.3  Multiplication   kcnktkm1cn 11570
            5.3.4  Ordering on reals (cont.)   gt0ne0 11604
            5.3.5  Reciprocals   ixi 11768
            5.3.6  Division   cdiv 11796
            5.3.7  Ordering on reals (cont.)   elimgt0 11982
            5.3.8  Completeness Axiom and Suprema   fimaxre 12089
            5.3.9  Imaginary and complex number properties   neg1cn 12133
            5.3.10  Function operation analogue theorems   ofsubeq0 12145
            ​*5.3.11  Indicator Functions   cind 12148
      ​5.4  Integer sets
            5.4.1  Positive integers (as a subset of complex numbers)   cn 12163
            5.4.2  Principle of mathematical induction   nnind 12181
            ​*5.4.3  Decimal representation of numbers   c2 12225
            ​*5.4.4  Some properties of specific numbers   1pneg1e0 12284
            5.4.5  Simple number properties   halfcl 12392
            5.4.6  The Archimedean property   nnunb 12422
            5.4.7  Nonnegative integers (as a subset of complex numbers)   cn0 12426
            ​*5.4.8  Extended nonnegative integers   cxnn0 12499
            5.4.9  Integers (as a subset of complex numbers)   cz 12513
            5.4.10  Decimal arithmetic   cdc 12633
            5.4.11  Upper sets of integers   cuz 12777
            5.4.12  Well-ordering principle for bounded-below sets of integers   uzwo3 12882
            5.4.13  Rational numbers (as a subset of complex numbers)   cq 12887
            5.4.14  Existence of the set of complex numbers   rpnnen1lem2 12916
      ​5.5  Order sets
            5.5.1  Positive reals (as a subset of complex numbers)   crp 12931
            5.5.2  Infinity and the extended real number system (cont.)   cxne 13049
            5.5.3  Supremum and infimum on the extended reals   xrsupexmnf 13246
            5.5.4  Real number intervals   cioo 13287
            5.5.5  Finite intervals of integers   cfz 13450
            ​*5.5.6  Finite intervals of nonnegative integers   elfz2nn0 13561
            5.5.7  Half-open integer ranges   cfzo 13597
      ​5.6  Elementary integer functions
            5.6.1  The floor and ceiling functions   cfl 13738
            5.6.2  The modulo (remainder) operation   cmo 13817
            5.6.3  Miscellaneous theorems about integers   om2uz0i 13898
            5.6.4  Strong induction over upper sets of integers   uzsinds 13938
            5.6.5  Finitely supported functions over the nonnegative integers   fsuppmapnn0fiublem 13941
            5.6.6  The infinite sequence builder "seq" - extension   cseq 13952
            5.6.7  Integer powers   cexp 14012
            5.6.8  Ordered pair theorem for nonnegative integers   nn0le2msqi 14218
            5.6.9  Factorial function   cfa 14224
            5.6.10  The binomial coefficient operation   cbc 14253
            5.6.11  The ` # ` (set size) function   chash 14281
                  5.6.11.1  Proper unordered pairs and triples (sets of size 2 and 3)   hashprlei 14419
                  5.6.11.2  Functions with a domain containing at least two different elements   fundmge2nop0 14453
                  5.6.11.3  Finite induction on the size of the first component of a binary relation   hashdifsnp1 14457
      ​​*5.7  Words over a set
            5.7.1  Definitions and basic theorems   cword 14464
            5.7.2  Last symbol of a word   clsw 14513
            5.7.3  Concatenations of words   cconcat 14521
            5.7.4  Singleton words   cs1 14547
            5.7.5  Concatenations with singleton words   ccatws1cl 14568
            5.7.6  Subwords/substrings   csubstr 14592
            5.7.7  Prefixes of a word   cpfx 14622
            5.7.8  Subwords of subwords   swrdswrdlem 14655
            5.7.9  Subwords and concatenations   pfxcctswrd 14661
            5.7.10  Subwords of concatenations   swrdccatfn 14675
            5.7.11  Splicing words (substring replacement)   csplice 14700
            5.7.12  Reversing words   creverse 14709
            5.7.13  Repeated symbol words   creps 14719
            ​*5.7.14  Cyclical shifts of words   ccsh 14739
            5.7.15  Mapping words by a function   wrdco 14782
            5.7.16  Longer string literals   cs2 14792
      ​​*5.8  Reflexive and transitive closures of relations
            5.8.1  The reflexive and transitive properties of relations   coss12d 14923
            5.8.2  Basic properties of closures   cleq1lem 14933
            5.8.3  Definitions and basic properties of transitive closures   ctcl 14936
            5.8.4  Exponentiation of relations   crelexp 14970
            5.8.5  Reflexive-transitive closure as an indexed union   crtrcl 15006
            ​*5.8.6  Principle of transitive induction   relexpindlem 15014
      ​5.9  Elementary real and complex functions
            5.9.1  The "shift" operation   cshi 15017
            5.9.2  Signum (sgn or sign) function   csgn 15037
            5.9.3  Real and imaginary parts; conjugate   ccj 15047
            5.9.4  Square root; absolute value   csqrt 15184
      ​5.10  Elementary limits and convergence
            5.10.1  Superior limit (lim sup)   clsp 15421
            5.10.2  Limits   cli 15435
            5.10.3  Finite and infinite sums   csu 15637
            5.10.4  The binomial theorem   binomlem 15783
            5.10.5  The inclusion/exclusion principle   incexclem 15790
            5.10.6  Infinite sums (cont.)   isumshft 15793
            5.10.7  Miscellaneous converging and diverging sequences   divrcnv 15806
            5.10.8  Arithmetic series   arisum 15814
            5.10.9  Geometric series   expcnv 15818
            5.10.10  Ratio test for infinite series convergence   cvgrat 15837
            5.10.11  Mertens' theorem   mertenslem1 15838
            5.10.12  Finite and infinite products   prodf 15841
                  5.10.12.1  Product sequences   prodf 15841
                  5.10.12.2  Non-trivial convergence   ntrivcvg 15851
                  5.10.12.3  Complex products   cprod 15857
                  5.10.12.4  Finite products   fprod 15895
                  5.10.12.5  Infinite products   iprodclim 15952
            5.10.13  Falling and Rising Factorial   cfallfac 15958
            5.10.14  Bernoulli polynomials and sums of k-th powers   cbp 16000
      ​5.11  Elementary trigonometry
            5.11.1  The exponential, sine, and cosine functions   ce 16015
                  5.11.1.1  The circle constant (tau = 2 pi)   ctau 16158
            5.11.2  _e is irrational   eirrlem 16160
      ​5.12  Cardinality of real and complex number subsets
            5.12.1  Countability of integers and rationals   xpnnen 16167
            5.12.2  The reals are uncountable   rpnnen2lem1 16170
​​*PART 6  ELEMENTARY NUMBER THEORY
      ​6.1  Elementary properties of divisibility
            6.1.1  Irrationality of square root of 2   sqrt2irrlem 16204
            6.1.2  Some Number sets are chains of proper subsets   nthruc 16208
            6.1.3  The divides relation   cdvds 16210
            ​*6.1.4  Even and odd numbers   evenelz 16294
            6.1.5  The division algorithm   divalglem0 16351
            6.1.6  Bit sequences   cbits 16377
            6.1.7  The greatest common divisor operator   cgcd 16452
            6.1.8  Bézout's identity   bezoutlem1 16497
            6.1.9  Algorithms   nn0seqcvgd 16528
            6.1.10  Euclid's Algorithm   eucalgval2 16539
            ​*6.1.11  The least common multiple   clcm 16546
            ​*6.1.12  Coprimality and Euclid's lemma   coprmgcdb 16607
            6.1.13  Cancellability of congruences   congr 16622
      ​6.2  Elementary prime number theory
            ​*6.2.1  Elementary properties   cprime 16629
            ​*6.2.2  Coprimality and Euclid's lemma (cont.)   coprm 16670
            6.2.3  Properties of the canonical representation of a rational   cnumer 16692
            6.2.4  Euler's theorem   codz 16722
            6.2.5  Arithmetic modulo a prime number   modprm1div 16757
            6.2.6  Pythagorean Triples   coprimeprodsq 16768
            6.2.7  The prime count function   cpc 16796
            6.2.8  Pocklington's theorem   prmpwdvds 16864
            6.2.9  Infinite primes theorem   unbenlem 16868
            6.2.10  Sum of prime reciprocals   prmreclem1 16876
            6.2.11  Fundamental theorem of arithmetic   1arithlem1 16883
            6.2.12  Lagrange's four-square theorem   cgz 16889
            6.2.13  Van der Waerden's theorem   cvdwa 16925
            6.2.14  Ramsey's theorem   cram 16959
            ​*6.2.15  Primorial function   cprmo 16991
            ​*6.2.16  Prime gaps   prmgaplem1 17009
            6.2.17  Decimal arithmetic (cont.)   dec2dvds 17023
            6.2.18  Cyclical shifts of words (cont.)   cshwsidrepsw 17053
            6.2.19  Specific prime numbers   prmlem0 17065
            6.2.20  Very large primes   1259lem1 17090
​PART 7  BASIC STRUCTURES
      ​7.1  Extensible structures
            ​*7.1.1  Basic definitions   cstr 17105
                  7.1.1.1  Extensible structures as structures with components   cstr 17105
                  7.1.1.2  Substitution of components   csts 17122
                  7.1.1.3  Slots   cslot 17140
                  ​*7.1.1.4  Structure component indices   cnx 17152
                  7.1.1.5  Base sets   cbs 17168
                  7.1.1.6  Base set restrictions   cress 17189
            7.1.2  Slot definitions   cplusg 17209
            7.1.3  Definition of the structure product   crest 17372
            7.1.4  Definition of the structure quotient   cordt 17452
      ​7.2  Moore spaces
            7.2.1  Moore closures   mrcflem 17561
            7.2.2  Independent sets in a Moore system   mrisval 17585
            7.2.3  Algebraic closure systems   isacs 17606
​PART 8  BASIC CATEGORY THEORY
      ​8.1  Categories
            8.1.1  Categories   ccat 17619
            8.1.2  Opposite category   coppc 17666
            8.1.3  Monomorphisms and epimorphisms   cmon 17684
            8.1.4  Sections, inverses, isomorphisms   csect 17700
            ​*8.1.5  Isomorphic objects   ccic 17751
            8.1.6  Subcategories   cssc 17763
            8.1.7  Functors   cfunc 17810
            8.1.8  Full & faithful functors   cful 17860
            8.1.9  Natural transformations and the functor category   cnat 17900
            8.1.10  Initial, terminal and zero objects of a category   cinito 17937
      ​8.2  Arrows (disjointified hom-sets)
            8.2.1  Identity and composition for arrows   cida 18009
      ​8.3  Examples of categories
            8.3.1  The category of sets   csetc 18031
            8.3.2  The category of categories   ccatc 18054
            ​*8.3.3  The category of extensible structures   fncnvimaeqv 18075
      ​8.4  Categorical constructions
            8.4.1  Product of categories   cxpc 18123
            8.4.2  Functor evaluation   cevlf 18164
            8.4.3  Hom functor   chof 18203
​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 18386
            9.5.2  Complete lattices   ccla 18453
            9.5.3  Distributive lattices   cdlat 18475
            9.5.4  Subset order structures   cipo 18482
      ​9.6  Posets, directed sets, and lattices as relations
            ​*9.6.1  Posets and lattices as relations   cps 18519
            9.6.2  Directed sets, nets   cdir 18549
            9.6.3  Chains   cchn 18560
​PART 10  BASIC ALGEBRAIC STRUCTURES
      ​10.1  Monoids
            ​*10.1.1  Magmas   cplusf 18594
            ​*10.1.2  Identity elements   mgmidmo 18617
            ​*10.1.3  Iterated sums in a magma   gsumvalx 18633
            10.1.4  Magma homomorphisms and submagmas   cmgmhm 18647
            ​*10.1.5  Semigroups   csgrp 18675
            ​*10.1.6  Definition and basic properties of monoids   cmnd 18691
            10.1.7  Monoid homomorphisms and submonoids   cmhm 18738
            ​*10.1.8  Iterated sums in a monoid   gsumvallem2 18791
            10.1.9  Free monoids   cfrmd 18804
                  ​*10.1.9.1  Monoid of endofunctions   cefmnd 18825
            10.1.10  Examples and counterexamples for magmas, semigroups and monoids   mgm2nsgrplem1 18878
      ​10.2  Groups
            10.2.1  Definition and basic properties   cgrp 18898
            ​*10.2.2  Group multiple operation   cmg 19032
            10.2.3  Subgroups and Quotient groups   csubg 19085
            ​*10.2.4  Cyclic monoids and groups   cycsubmel 19164
            10.2.5  Elementary theory of group homomorphisms   cghm 19176
            10.2.6  Isomorphisms of groups   cgim 19221
                  10.2.6.1  The first isomorphism theorem of groups   ghmqusnsglem1 19244
            10.2.7  Group actions   cga 19253
            10.2.8  Centralizers and centers   ccntz 19279
            10.2.9  The opposite group   coppg 19309
            10.2.10  Symmetric groups   csymg 19333
                  ​*10.2.10.1  Definition and basic properties   csymg 19333
                  10.2.10.2  Cayley's theorem   cayleylem1 19376
                  10.2.10.3  Permutations fixing one element   symgfix2 19380
                  ​*10.2.10.4  Transpositions in the symmetric group   cpmtr 19405
                  10.2.10.5  The sign of a permutation   cpsgn 19453
            10.2.11  p-Groups and Sylow groups; Sylow's theorems   cod 19488
            10.2.12  Direct products   clsm 19598
                  10.2.12.1  Direct products (extension)   smndlsmidm 19620
            10.2.13  Free groups   cefg 19670
            10.2.14  Abelian groups   ccmn 19744
                  10.2.14.1  Definition and basic properties   ccmn 19744
                  10.2.14.2  Cyclic groups   ccyg 19841
                  10.2.14.3  Group sum operation   gsumval3a 19867
                  10.2.14.4  Group sums over (ranges of) integers   fsfnn0gsumfsffz 19947
                  10.2.14.5  Internal direct products   cdprd 19959
                  10.2.14.6  The Fundamental Theorem of Abelian Groups   ablfacrplem 20031
            10.2.15  Simple groups   csimpg 20056
                  10.2.15.1  Definition and basic properties   csimpg 20056
                  10.2.15.2  Classification of abelian simple groups   ablsimpnosubgd 20070
            10.2.16  Totally ordered monoids and groups   comnd 20083
      ​10.3  Rings
            10.3.1  Multiplicative Group   cmgp 20110
            ​*10.3.2  Non-unital rings ("rngs")   crng 20122
            ​*10.3.3  Ring unity (multiplicative identity)   cur 20151
            10.3.4  Semirings   csrg 20156
                  ​*10.3.4.1  The binomial theorem for semirings   srgbinomlem1 20196
            10.3.5  Unital rings   crg 20203
            10.3.6  Opposite ring   coppr 20305
            10.3.7  Divisibility   cdsr 20323
            10.3.8  Ring primes   crpm 20401
            10.3.9  Homomorphisms of non-unital rings   crnghm 20403
            10.3.10  Ring homomorphisms   crh 20438
            10.3.11  Nonzero rings and zero rings   cnzr 20478
            10.3.12  Local rings   clring 20504
            10.3.13  Subrings   csubrng 20511
                  10.3.13.1  Subrings of non-unital rings   csubrng 20511
                  10.3.13.2  Subrings of unital rings   csubrg 20535
                  10.3.13.3  Subrings generated by a subset   crgspn 20576
            10.3.14  Categories of rings   crngc 20582
                  ​*10.3.14.1  The category of non-unital rings   crngc 20582
                  ​*10.3.14.2  The category of (unital) rings   cringc 20611
                  10.3.14.3  Subcategories of the category of rings   srhmsubclem1 20643
            10.3.15  Left regular elements and domains   crlreg 20657
      ​10.4  Division rings and fields
            10.4.1  Definition and basic properties   cdr 20695
            10.4.2  Sub-division rings   csdrg 20752
            10.4.3  Absolute value (abstract algebra)   cabv 20774
            10.4.4  Star rings   cstf 20803
            10.4.5  Totally ordered rings and fields   corng 20823
      ​10.5  Left modules
            10.5.1  Definition and basic properties   clmod 20844
            10.5.2  Subspaces and spans in a left module   clss 20915
            10.5.3  Homomorphisms and isomorphisms of left modules   clmhm 21004
            10.5.4  Subspace sum; bases for a left module   clbs 21059
      ​10.6  Vector spaces
            10.6.1  Definition and basic properties   clvec 21087
      ​10.7  Subring algebras and ideals
            10.7.1  Subring algebras   csra 21156
            ​*10.7.2  Left ideals and spans   clidl 21194
            10.7.3  Two-sided ideals and quotient rings   c2idl 21237
                  ​*10.7.3.1  Condition for a non-unital ring to be unital   rngqiprng1elbas 21274
            10.7.4  Principal ideal rings. Divisibility in the integers   clpidl 21308
            10.7.5  Principal ideal domains   cpid 21324
      ​10.8  The complex numbers as an algebraic extensible structure
            10.8.1  Definition and basic properties   cpsmet 21326
            ​*10.8.2  Ring of integers   czring 21434
                  ​*10.8.2.1  Example for a condition for a non-unital ring to be unital   pzriprnglem1 21469
            10.8.3  Algebraic constructions based on the complex numbers   czrh 21487
            10.8.4  Signs as subgroup of the complex numbers   cnmsgnsubg 21565
            10.8.5  Embedding of permutation signs into a ring   zrhpsgnmhm 21572
            10.8.6  The ordered field of real numbers   crefld 21592
      ​10.9  Generalized pre-Hilbert and Hilbert spaces
            10.9.1  Definition and basic properties   cphl 21612
            10.9.2  Orthocomplements and closed subspaces   cocv 21648
            10.9.3  Orthogonal projection and orthonormal bases   cpj 21688
​​*PART 11  BASIC LINEAR ALGEBRA
      ​11.1  Vectors and free modules
            *11.1.1  Direct sum of left modules   cdsmm 21719
            ​*11.1.2  Free modules   cfrlm 21734
            ​*11.1.3  Standard basis (unit vectors)   cuvc 21770
            ​*11.1.4  Independent sets and families   clindf 21792
            11.1.5  Characterization of free modules   lmimlbs 21824
      ​11.2  Associative algebras
            11.2.1  Definition and basic properties   casa 21838
      ​11.3  Abstract multivariate polynomials
            11.3.1  Definition and basic properties   cmps 21892
            11.3.2  Polynomial evaluation   ces 22059
            11.3.3  Additional definitions for (multivariate) polynomials   cslv 22103
            ​*11.3.4  Univariate polynomials   cps1 22147
            11.3.5  Univariate polynomial evaluation   ces1 22287
                  11.3.5.1  Specialization of polynomial evaluation as a ring homomorphism   evls1scafv 22340
      ​​*11.4  Matrices
            *11.4.1  The matrix multiplication   cmmul 22364
            ​*11.4.2  Square matrices   cmat 22381
            ​*11.4.3  The matrix algebra   matmulr 22412
            ​*11.4.4  Matrices of dimension 0 and 1   mat0dimbas0 22440
            ​*11.4.5  The subalgebras of diagonal and scalar matrices   cdmat 22462
            ​*11.4.6  Multiplication of a matrix with a "column vector"   cmvmul 22514
            11.4.7  Replacement functions for a square matrix   cmarrep 22530
            11.4.8  Submatrices   csubma 22550
      ​11.5  The determinant
            11.5.1  Definition and basic properties   cmdat 22558
            11.5.2  Determinants of 2 x 2 -matrices   m2detleiblem1 22598
            11.5.3  The matrix adjugate/adjunct   cmadu 22606
            ​*11.5.4  Laplace expansion of determinants (special case)   symgmatr01lem 22627
            11.5.5  Inverse matrix   invrvald 22650
            ​*11.5.6  Cramer's rule   slesolvec 22653
      ​​*11.6  Polynomial matrices
            11.6.1  Basic properties   pmatring 22666
            ​*11.6.2  Constant polynomial matrices   ccpmat 22677
            ​*11.6.3  Collecting coefficients of polynomial matrices   cdecpmat 22736
            ​*11.6.4  Ring isomorphism between polynomial matrices and polynomials over matrices   cpm2mp 22766
      ​​*11.7  The characteristic polynomial
            *11.7.1  Definition and basic properties   cchpmat 22800
            ​*11.7.2  The characteristic factor function G   fvmptnn04if 22823
            ​*11.7.3  The Cayley-Hamilton theorem   cpmadurid 22841
​PART 12  BASIC TOPOLOGY
      ​12.1  Topology
            ​*12.1.1  Topological spaces   ctop 22867
                  12.1.1.1  Topologies   ctop 22867
                  12.1.1.2  Topologies on sets   ctopon 22884
                  12.1.1.3  Topological spaces   ctps 22906
            12.1.2  Topological bases   ctb 22919
            12.1.3  Examples of topologies   distop 22969
            12.1.4  Closure and interior   ccld 22990
            12.1.5  Neighborhoods   cnei 23071
            12.1.6  Limit points and perfect sets   clp 23108
            12.1.7  Subspace topologies   restrcl 23131
            12.1.8  Order topology   ordtbaslem 23162
            12.1.9  Limits and continuity in topological spaces   ccn 23198
            12.1.10  Separated spaces: T0, T1, T2 (Hausdorff) ...   ct0 23280
            12.1.11  Compactness   ccmp 23360
            12.1.12  Bolzano-Weierstrass theorem   bwth 23384
            12.1.13  Connectedness   cconn 23385
            12.1.14  First- and second-countability   c1stc 23411
            12.1.15  Local topological properties   clly 23438
            12.1.16  Refinements   cref 23476
            12.1.17  Compactly generated spaces   ckgen 23507
            12.1.18  Product topologies   ctx 23534
            12.1.19  Continuous function-builders   cnmptid 23635
            12.1.20  Quotient maps and quotient topology   ckq 23667
            12.1.21  Homeomorphisms   chmeo 23727
      ​12.2  Filters and filter bases
            12.2.1  Filter bases   elmptrab 23801
            12.2.2  Filters   cfil 23819
            12.2.3  Ultrafilters   cufil 23873
            12.2.4  Filter limits   cfm 23907
            12.2.5  Extension by continuity   ccnext 24033
            12.2.6  Topological groups   ctmd 24044
            12.2.7  Infinite group sum on topological groups   ctsu 24100
            12.2.8  Topological rings, fields, vector spaces   ctrg 24130
      ​12.3  Uniform Structures and Spaces
            12.3.1  Uniform structures   cust 24174
            12.3.2  The topology induced by an uniform structure   cutop 24204
            12.3.3  Uniform Spaces   cuss 24227
            12.3.4  Uniform continuity   cucn 24248
            12.3.5  Cauchy filters in uniform spaces   ccfilu 24259
            12.3.6  Complete uniform spaces   ccusp 24270
      ​12.4  Metric spaces
            12.4.1  Pseudometric spaces   ispsmet 24278
            12.4.2  Basic metric space properties   cxms 24291
            12.4.3  Metric space balls   blfvalps 24357
            12.4.4  Open sets of a metric space   mopnval 24412
            12.4.5  Continuity in metric spaces   metcnp3 24514
            12.4.6  The uniform structure generated by a metric   metuval 24523
            12.4.7  Examples of metric spaces   dscmet 24546
            ​*12.4.8  Normed algebraic structures   cnm 24550
            12.4.9  Normed space homomorphisms (bounded linear operators)   cnmo 24679
            12.4.10  Topology on the reals   qtopbaslem 24732
            12.4.11  Topological definitions using the reals   cii 24851
            12.4.12  Path homotopy   chtpy 24943
            12.4.13  The fundamental group   cpco 24976
      ​12.5  Metric subcomplex vector spaces
            12.5.1  Subcomplex modules   cclm 25038
            ​*12.5.2  Subcomplex vector spaces   ccvs 25099
            ​*12.5.3  Normed subcomplex vector spaces   isncvsngp 25125
            12.5.4  Subcomplex pre-Hilbert spaces   ccph 25142
            12.5.5  Convergence and completeness   ccfil 25228
            12.5.6  Baire's Category Theorem   bcthlem1 25300
            12.5.7  Banach spaces and subcomplex Hilbert spaces   ccms 25308
                  12.5.7.1  The complete ordered field of the real numbers   retopn 25355
            12.5.8  Euclidean spaces   crrx 25359
            12.5.9  Minimizing Vector Theorem   minveclem1 25400
            12.5.10  Projection Theorem   pjthlem1 25413
​PART 13  BASIC REAL AND COMPLEX ANALYSIS
      ​13.1  Continuity
            13.1.1  Intermediate value theorem   pmltpclem1 25424
      ​13.2  Integrals
            13.2.1  Lebesgue measure   covol 25438
            13.2.2  Lebesgue integration   cmbf 25590
                  13.2.2.1  Lesbesgue integral   cmbf 25590
                  13.2.2.2  Lesbesgue directed integral   cdit 25822
      ​13.3  Derivatives
            13.3.1  Real and complex differentiation   climc 25838
                  13.3.1.1  Derivatives of functions of one complex or real variable   climc 25838
                  13.3.1.2  Results on real differentiation   dvferm1lem 25960
​PART 14  BASIC REAL AND COMPLEX FUNCTIONS
      ​14.1  Polynomials
            14.1.1  Polynomial degrees   cmdg 26030
            14.1.2  The division algorithm for univariate polynomials   cmn1 26103
            14.1.3  Elementary properties of complex polynomials   cply 26161
            14.1.4  The division algorithm for polynomials   cquot 26269
            14.1.5  Algebraic numbers   caa 26293
            14.1.6  Liouville's approximation theorem   aalioulem1 26311
      ​14.2  Sequences and series
            14.2.1  Taylor polynomials and Taylor's theorem   ctayl 26331
            14.2.2  Uniform convergence   culm 26356
            14.2.3  Power series   pserval 26390
      ​14.3  Basic trigonometry
            14.3.1  The exponential, sine, and cosine functions (cont.)   efcn 26424
            14.3.2  Properties of pi = 3.14159...   pilem1 26432
            14.3.3  Mapping of the exponential function   efgh 26521
            14.3.4  The natural logarithm on complex numbers   clog 26534
            ​*14.3.5  Logarithms to an arbitrary base   clogb 26745
            14.3.6  Theorems of Pythagoras, isosceles triangles, and intersecting chords   angval 26782
            14.3.7  Solutions of quadratic, cubic, and quartic equations   quad2 26820
            14.3.8  Inverse trigonometric functions   casin 26843
            14.3.9  The Birthday Problem   log2ublem1 26927
            14.3.10  Areas in R^2   carea 26936
            14.3.11  More miscellaneous converging sequences   rlimcnp 26946
            14.3.12  Inequality of arithmetic and geometric means   cvxcl 26966
            14.3.13  Euler-Mascheroni constant   cem 26973
            14.3.14  Zeta function   czeta 26994
            14.3.15  Gamma function   clgam 26997
      ​14.4  Basic number theory
            14.4.1  Wilson's theorem   wilthlem1 27049
            14.4.2  The Fundamental Theorem of Algebra   ftalem1 27054
            14.4.3  The Basel problem (ζ(2) = π²/6)   basellem1 27062
            14.4.4  Number-theoretical functions   ccht 27072
            14.4.5  Perfect Number Theorem   mersenne 27209
            14.4.6  Characters of Z/nZ   cdchr 27214
            14.4.7  Bertrand's postulate   bcctr 27257
            ​*14.4.8  Quadratic residues and the Legendre symbol   clgs 27276
            ​*14.4.9  Gauss' Lemma   gausslemma2dlem0a 27338
            14.4.10  Quadratic reciprocity   lgseisenlem1 27357
            14.4.11  All primes 4n+1 are the sum of two squares   2sqlem1 27399
            14.4.12  Chebyshev's Weak Prime Number Theorem, Dirichlet's Theorem   chebbnd1lem1 27451
            14.4.13  The Prime Number Theorem   mudivsum 27512
            14.4.14  Ostrowski's theorem   abvcxp 27597
​PART 15  SURREAL NUMBERS
      ​​*15.1  Sign sequence representation and Alling's axioms
            15.1.1  Definitions and initial properties   csur 27622
            15.1.2  Ordering   ltssolem1 27658
            15.1.3  Birthday Function   bdayfo 27660
            15.1.4  Density   fvnobday 27661
            ​*15.1.5  Full-Eta Property   bdayimaon 27676
      ​15.2  Initial consequences of Alling's axioms
            15.2.1  Ordering Theorems   cles 27727
            15.2.2  Birthday Theorems   bdayfun 27759
      ​​*15.3  Conway cut representation
            15.3.1  Conway cuts   cslts 27768
            15.3.2  Zero and One   c0s 27816
            15.3.3  Cuts and Options   cmade 27833
            15.3.4  Cofinality and coinitiality   cofslts 27929
      ​15.4  Induction and recursion
            15.4.1  Induction and recursion on one variable   cnorec 27948
            15.4.2  Induction and recursion on two variables   cnorec2 27959
      ​15.5  Surreal arithmetic
            15.5.1  Addition   cadds 27970
            15.5.2  Negation and Subtraction   cnegs 28030
            15.5.3  Multiplication   cmuls 28117
            15.5.4  Division   cdivs 28198
            15.5.5  Absolute value   cabss 28248
      ​15.6  Subsystems of surreals
            15.6.1  Ordinal numbers   cons 28262
            15.6.2  Surreal recursive sequences   cseqs 28294
            15.6.3  Natural numbers   cn0s 28323
            15.6.4  Integers   czs 28389
            15.6.5  Dyadic fractions   c2s 28421
            15.6.6  Real numbers   creno 28500
​​*PART 16  ELEMENTARY GEOMETRY
      ​16.1  Definition and Tarski's Axioms of Geometry
            16.1.1  Justification for the congruence notation   tgjustf 28560
      ​16.2  Tarskian Geometry
            16.2.1  Congruence   tgcgrcomimp 28564
            16.2.2  Betweenness   tgbtwntriv2 28574
            16.2.3  Dimension   tglowdim1 28587
            16.2.4  Betweenness and Congruence   tgifscgr 28595
            16.2.5  Congruence of a series of points   ccgrg 28597
            16.2.6  Motions   cismt 28619
            16.2.7  Colinearity   tglng 28633
            16.2.8  Connectivity of betweenness   tgbtwnconn1lem1 28659
            16.2.9  Less-than relation in geometric congruences   cleg 28669
            16.2.10  Rays   chlg 28687
            16.2.11  Lines   btwnlng1 28706
            16.2.12  Point inversions   cmir 28739
            16.2.13  Right angles   crag 28780
            16.2.14  Half-planes   islnopp 28826
            16.2.15  Midpoints and Line Mirroring   cmid 28859
            16.2.16  Congruence of angles   ccgra 28894
            16.2.17  Angle Comparisons   cinag 28922
            16.2.18  Congruence Theorems   tgsas1 28941
            16.2.19  Equilateral triangles   ceqlg 28952
      ​16.3  Properties of geometries
            16.3.1  Isomorphisms between geometries   f1otrgds 28956
      ​16.4  Geometry in Hilbert spaces
            16.4.1  Geometry in the complex plane   cchhllem 28974
            16.4.2  Geometry in Euclidean spaces   cee 28975
                  16.4.2.1  Definition of the Euclidean space   cee 28975
                  16.4.2.2  Tarski's axioms for geometry for the Euclidean space   axdimuniq 29001
                  16.4.2.3  EE^n fulfills Tarski's Axioms   ceeng 29065
​​*PART 17  GRAPH THEORY
      ​*17.1  Vertices and edges
            17.1.1  The edge function extractor for extensible structures   cedgf 29076
            ​*17.1.2  Vertices and indexed edges   cvtx 29084
                  17.1.2.1  Definitions and basic properties   cvtx 29084
                  17.1.2.2  The vertices and edges of a graph represented as ordered pair   opvtxval 29091
                  17.1.2.3  The vertices and edges of a graph represented as extensible structure   funvtxdmge2val 29099
                  17.1.2.4  Representations of graphs without edges   snstrvtxval 29125
                  17.1.2.5  Degenerated cases of representations of graphs   vtxval0 29127
            17.1.3  Edges as range of the edge function   cedg 29135
      ​​*17.2  Undirected graphs
            17.2.1  Undirected hypergraphs   cuhgr 29144
            17.2.2  Undirected pseudographs and multigraphs   cupgr 29168
            ​*17.2.3  Loop-free graphs   umgrislfupgrlem 29210
            17.2.4  Edges as subsets of vertices of graphs   uhgredgiedgb 29214
            ​*17.2.5  Undirected simple graphs   cuspgr 29236
            17.2.6  Examples for graphs   usgr0e 29324
            17.2.7  Subgraphs   csubgr 29355
            17.2.8  Finite undirected simple graphs   cfusgr 29404
            17.2.9  Neighbors, complete graphs and universal vertices   cnbgr 29420
                  17.2.9.1  Neighbors   cnbgr 29420
                  17.2.9.2  Universal vertices   cuvtx 29473
                  17.2.9.3  Complete graphs   ccplgr 29497
            17.2.10  Vertex degree   cvtxdg 29554
            ​*17.2.11  Regular graphs   crgr 29644
      ​​*17.3  Walks, paths and cycles
            *17.3.1  Walks   cewlks 29684
            17.3.2  Walks for loop-free graphs   lfgrwlkprop 29774
            17.3.3  Trails   ctrls 29777
            17.3.4  Paths and simple paths   cpths 29798
            17.3.5  Closed walks   cclwlks 29858
            17.3.6  Circuits and cycles   ccrcts 29872
            ​*17.3.7  Walks as words   cwwlks 29913
            17.3.8  Walks/paths of length 2 (as length 3 strings)   2wlkdlem1 30013
            17.3.9  Walks in regular graphs   rusgrnumwwlkl1 30059
            ​*17.3.10  Closed walks as words   cclwwlk 30071
                  17.3.10.1  Closed walks as words   cclwwlk 30071
                  17.3.10.2  Closed walks of a fixed length as words   cclwwlkn 30114
                  17.3.10.3  Closed walks on a vertex of a fixed length as words   cclwwlknon 30177
            17.3.11  Examples for walks, trails and paths   0ewlk 30204
            17.3.12  Connected graphs   cconngr 30276
      ​17.4  Eulerian paths and the Konigsberg Bridge problem
            ​*17.4.1  Eulerian paths   ceupth 30287
            ​*17.4.2  The Königsberg Bridge problem   konigsbergvtx 30336
      ​17.5  The Friendship Theorem
            17.5.1  Friendship graphs - basics   cfrgr 30348
            17.5.2  The friendship theorem for small graphs   frgr1v 30361
            17.5.3  Theorems according to Mertzios and Unger   2pthfrgrrn 30372
            ​*17.5.4  Huneke's Proof of the Friendship Theorem   frgrncvvdeqlem1 30389
​PART 18  GUIDES AND MISCELLANEA
      ​18.1  Guides (conventions, explanations, and examples)
            ​*18.1.1  Conventions   conventions 30490
            18.1.2  Natural deduction   natded 30493
            ​*18.1.3  Natural deduction examples   ex-natded5.2 30494
            18.1.4  Definitional examples   ex-or 30511
            18.1.5  Other examples   aevdemo 30550
      ​18.2  Humor
            18.2.1  April Fool's theorem   avril1 30553
      ​18.3  (Future - to be reviewed and classified)
            18.3.1  Planar incidence geometry   cplig 30565
            ​*18.3.2  Aliases kept to prevent broken links   dummylink 30578
​​*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 30580
            19.1.2  Abelian groups   cablo 30635
      ​19.2  Complex vector spaces
            19.2.1  Definition and basic properties   cvc 30649
            19.2.2  Examples of complex vector spaces   cnaddabloOLD 30672
      ​19.3  Normed complex vector spaces
            19.3.1  Definition and basic properties   cnv 30675
            19.3.2  Examples of normed complex vector spaces   cnnv 30768
            19.3.3  Induced metric of a normed complex vector space   imsval 30776
            19.3.4  Inner product   cdip 30791
            19.3.5  Subspaces   css 30812
      ​19.4  Operators on complex vector spaces
            19.4.1  Definitions and basic properties   clno 30831
      ​19.5  Inner product (pre-Hilbert) spaces
            19.5.1  Definition and basic properties   ccphlo 30903
            19.5.2  Examples of pre-Hilbert spaces   cncph 30910
            19.5.3  Properties of pre-Hilbert spaces   isph 30913
      ​19.6  Complex Banach spaces
            19.6.1  Definition and basic properties   ccbn 30953
            19.6.2  Examples of complex Banach spaces   cnbn 30960
            19.6.3  Uniform Boundedness Theorem   ubthlem1 30961
            19.6.4  Minimizing Vector Theorem   minvecolem1 30965
      ​19.7  Complex Hilbert spaces
            19.7.1  Definition and basic properties   chlo 30976
            19.7.2  Standard axioms for a complex Hilbert space   hlex 30989
            19.7.3  Examples of complex Hilbert spaces   cnchl 31007
            19.7.4  Hellinger-Toeplitz Theorem   htthlem 31008
​​*PART 20  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      ​20.1  Axiomatization of complex pre-Hilbert spaces
            20.1.1  Basic Hilbert space definitions   chba 31010
            20.1.2  Preliminary ZFC lemmas   df-hnorm 31059
            ​*20.1.3  Derive the Hilbert space axioms from ZFC set theory   axhilex-zf 31072
            ​*20.1.4  Introduce the vector space axioms for a Hilbert space   ax-hilex 31090
            20.1.5  Vector operations   hvmulex 31102
            20.1.6  Inner product postulates for a Hilbert space   ax-hfi 31170
      ​20.2  Inner product and norms
            20.2.1  Inner product   his5 31177
            20.2.2  Norms   dfhnorm2 31213
            20.2.3  Relate Hilbert space to normed complex vector spaces   hilablo 31251
            20.2.4  Bunjakovaskij-Cauchy-Schwarz inequality   bcsiALT 31270
      ​20.3  Cauchy sequences and completeness axiom
            20.3.1  Cauchy sequences and limits   hcau 31275
            20.3.2  Derivation of the completeness axiom from ZF set theory   hilmet 31285
            20.3.3  Completeness postulate for a Hilbert space   ax-hcompl 31293
            20.3.4  Relate Hilbert space to ZFC pre-Hilbert and Hilbert spaces   hhcms 31294
      ​20.4  Subspaces and projections
            20.4.1  Subspaces   df-sh 31298
            20.4.2  Closed subspaces   df-ch 31312
            20.4.3  Orthocomplements   df-oc 31343
            20.4.4  Subspace sum, span, lattice join, lattice supremum   df-shs 31399
            20.4.5  Projection theorem   pjhthlem1 31482
            20.4.6  Projectors   df-pjh 31486
      ​20.5  Properties of Hilbert subspaces
            20.5.1  Orthomodular law   omlsilem 31493
            20.5.2  Projectors (cont.)   pjhtheu2 31507
            20.5.3  Hilbert lattice operations   sh0le 31531
            20.5.4  Span (cont.) and one-dimensional subspaces   spansn0 31632
            20.5.5  Commutes relation for Hilbert lattice elements   df-cm 31674
            20.5.6  Foulis-Holland theorem   fh1 31709
            20.5.7  Quantum Logic Explorer axioms   qlax1i 31718
            20.5.8  Orthogonal subspaces   chscllem1 31728
            20.5.9  Orthoarguesian laws 5OA and 3OA   5oalem1 31745
            20.5.10  Projectors (cont.)   pjorthi 31760
            20.5.11  Mayet's equation E_3   mayete3i 31819
      ​20.6  Operators on Hilbert spaces
            ​*20.6.1  Operator sum, difference, and scalar multiplication   df-hosum 31821
            20.6.2  Zero and identity operators   df-h0op 31839
            20.6.3  Operations on Hilbert space operators   hoaddcl 31849
            20.6.4  Linear, continuous, bounded, Hermitian, unitary operators and norms   df-nmop 31930
            20.6.5  Linear and continuous functionals and norms   df-nmfn 31936
            20.6.6  Adjoint   df-adjh 31940
            20.6.7  Dirac bra-ket notation   df-bra 31941
            20.6.8  Positive operators   df-leop 31943
            20.6.9  Eigenvectors, eigenvalues, spectrum   df-eigvec 31944
            20.6.10  Theorems about operators and functionals   nmopval 31947
            20.6.11  Riesz lemma   riesz3i 32153
            20.6.12  Adjoints (cont.)   cnlnadjlem1 32158
            20.6.13  Quantum computation error bound theorem   unierri 32195
            20.6.14  Dirac bra-ket notation (cont.)   branmfn 32196
            20.6.15  Positive operators (cont.)   leopg 32213
            20.6.16  Projectors as operators   pjhmopi 32237
      ​20.7  States on a Hilbert lattice and Godowski's equation
            20.7.1  States on a Hilbert lattice   df-st 32302
            20.7.2  Godowski's equation   golem1 32362
      ​20.8  Cover relation, atoms, exchange axiom, and modular symmetry
            20.8.1  Covers relation; modular pairs   df-cv 32370
            20.8.2  Atoms   df-at 32429
            20.8.3  Superposition principle   superpos 32445
            20.8.4  Atoms, exchange and covering properties, atomicity   chcv1 32446
            20.8.5  Irreducibility   chirredlem1 32481
            20.8.6  Atoms (cont.)   atcvat3i 32487
            20.8.7  Modular symmetry   mdsymlem1 32494
​PART 21  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      ​21.1  Mathboxes for user contributions
            21.1.1  Mathbox guidelines   mathbox 32533
      ​21.2  Mathbox for Stefan Allan
      ​21.3  Mathbox for Thierry Arnoux
            21.3.1  Propositional Calculus - misc additions   ad11antr 32538
            21.3.2  Predicate Calculus   sbc2iedf 32554
                  21.3.2.1  Predicate Calculus - misc additions   sbc2iedf 32554
                  21.3.2.2  Restricted quantification - misc additions   ralcom4f 32556
                  21.3.2.3  Equality   eqtrb 32563
                  21.3.2.4  Double restricted existential uniqueness quantification   opsbc2ie 32565
                  21.3.2.5  Double restricted existential uniqueness quantification syntax   w2reu 32567
                  21.3.2.6  Substitution (without distinct variables) - misc additions   sbceqbidf 32576
                  21.3.2.7  Existential "at most one" - misc additions   mo5f 32578
                  21.3.2.8  Existential uniqueness - misc additions   reuxfrdf 32580
                  21.3.2.9  Restricted "at most one" - misc additions   rmoxfrd 32582
                  21.3.2.10  Restricted iota (description binder)   riotaeqbidva 32585
            21.3.3  General Set Theory   dmrab 32586
                  21.3.3.1  Class abstractions (a.k.a. class builders)   dmrab 32586
                  21.3.3.2  Image Sets   abrexdomjm 32597
                  21.3.3.3  Set relations and operations - misc additions   nelun 32603
                  21.3.3.4  Unordered pairs   elpreq 32618
                  21.3.3.5  Unordered triples   tpssg 32627
                  21.3.3.6  Conditional operator - misc additions   ifeqeqx 32632
                  21.3.3.7  Set union   uniinn0 32640
                  21.3.3.8  Indexed union - misc additions   cbviunf 32645
                  21.3.3.9  Indexed intersection - misc additions   iinabrex 32659
                  21.3.3.10  Disjointness - misc additions   disjnf 32660
            21.3.4  Relations and Functions   xpdisjres 32688
                  21.3.4.1  Relations - misc additions   xpdisjres 32688
                  21.3.4.2  Functions - misc additions   fconst7v 32713
                  21.3.4.3  Operations - misc additions   mpomptxf 32771
                  21.3.4.4  The mapping operation   elmaprd 32773
                  21.3.4.5  Support of a function   suppovss 32774
                  21.3.4.6  Explicit Functions with one or two points as a domain   cosnopne 32787
                  21.3.4.7  Isomorphisms - misc. additions   gtiso 32794
                  21.3.4.8  Disjointness (additional proof requiring functions)   disjdsct 32796
                  21.3.4.9  First and second members of an ordered pair - misc additions   df1stres 32797
                  21.3.4.10  Countable Sets   snct 32805
            21.3.5  Real and Complex Numbers   sgnval2 32828
                  21.3.5.1  Complex operations - misc. additions   creq0 32829
                  21.3.5.2  Ordering on reals - misc additions   lt2addrd 32843
                  21.3.5.3  Extended reals - misc additions   nn0mnfxrd 32844
                  21.3.5.4  Extended nonnegative integers - misc additions   xnn0gt0 32862
                  21.3.5.5  Real number intervals - misc additions   joiniooico 32867
                  21.3.5.6  Finite intervals of integers - misc additions   uzssico 32877
                  21.3.5.7  Half-open integer ranges - misc additions   iundisjfi 32889
                  21.3.5.8  The ` # ` (set size) function - misc additions   hashunif 32899
                  21.3.5.9  The greatest common divisor operator - misc. additions   elq2 32905
                  21.3.5.10  Integers   nn0split01 32911
                  21.3.5.11  Decimal numbers   dfdec100 32923
            21.3.6  Real and complex functions   sgncl 32924
                  21.3.6.1  Signum (sgn or sign) function - misc. additions   sgncl 32924
                  21.3.6.2  Integer powers - misc. additions   nexple 32937
                  21.3.6.3  Indicator Functions (continued)   indsumin 32941
            ​*21.3.7  Decimal expansion   cdp2 32950
                  ​*21.3.7.1  Decimal point   cdp 32967
                  21.3.7.2  Division in the extended real number system   cxdiv 32996
            21.3.8  Words over a set - misc additions   wrdres 33015
                  21.3.8.1  Splicing words (substring replacement)   splfv3 33038
                  21.3.8.2  Cyclic shift of words   1cshid 33039
            21.3.9  Extensible Structures   ressplusf 33043
                  21.3.9.1  Structure restriction operator   ressplusf 33043
                  21.3.9.2  Posets   ressprs 33046
                  21.3.9.3  Complete lattices   clatp0cl 33056
                  21.3.9.4  Order Theory   cmnt 33058
                  21.3.9.5  Extended reals Structure - misc additions   ax-xrssca 33084
                  21.3.9.6  The extended nonnegative real numbers commutative monoid   xrge00 33094
            21.3.10  Algebra   mndcld 33102
                  21.3.10.1  Monoids   mndcld 33102
                  21.3.10.2  Monoids Homomorphisms   abliso 33116
                  21.3.10.3  Groups - misc additions   grpinvinvd 33120
                  21.3.10.4  Abelian Groups - misc additions   ablcomd 33126
                  21.3.10.5  Finitely supported group sums - misc additions   gsumsubg 33127
                  21.3.10.6  Group or monoid sums over words   gsumwun 33157
                  21.3.10.7  Centralizers and centers - misc additions   cntzun 33160
                  21.3.10.8  The symmetric group   symgfcoeu 33163
                  21.3.10.9  Transpositions   pmtridf1o 33175
                  21.3.10.10  Permutation Signs   psgnid 33178
                  21.3.10.11  Permutation cycles   ctocyc 33187
                  21.3.10.12  The Alternating Group   evpmval 33226
                  21.3.10.13  Signum in an ordered monoid   csgns 33239
                  21.3.10.14  Fixed points   cfxp 33244
                  21.3.10.15  The Archimedean property for generic ordered algebraic structures   cinftm 33257
                  21.3.10.16  Semiring left modules   cslmd 33281
                  21.3.10.17  Simple groups   prmsimpcyc 33309
                  21.3.10.18  Rings - misc additions   ringrngd 33310
                  21.3.10.19  Subrings generated by a set   elrgspnlem1 33323
                  21.3.10.20  The zero ring   irrednzr 33331
                  21.3.10.21  Localization of rings   cerl 33334
                  21.3.10.22  Integral Domains   domnmuln0rd 33355
                  21.3.10.23  Euclidean Domains   ceuf 33369
                  21.3.10.24  Division Rings   ringinveu 33375
                  21.3.10.25  The field of rational numbers   qfld 33378
                  21.3.10.26  Subfields   subsdrg 33379
                  21.3.10.27  Field of fractions   cfrac 33383
                  21.3.10.28  Field extensions generated by a set   cfldgen 33391
                  21.3.10.29  Ring homomorphisms - misc additions   rhmdvd 33404
                  21.3.10.30  Scalar restriction operation   cresv 33406
                  21.3.10.31  The commutative ring of gaussian integers   gzcrng 33421
                  21.3.10.32  The archimedean ordered field of real numbers   cnfldfld 33422
                  21.3.10.33  The quotient map and quotient modules   qusker 33429
                  21.3.10.34  The ring of integers modulo ` N `   znfermltl 33446
                  21.3.10.35  Independent sets and families   islinds5 33447
                  21.3.10.36  Ring associates, ring units   dvdsruassoi 33464
                  ​*21.3.10.37  Subgroup sum / Sumset / Minkowski sum   elgrplsmsn 33470
                  21.3.10.38  The quotient map   quslsm 33485
                  21.3.10.39  Ideals   lidlmcld 33499
                  21.3.10.40  Prime Ideals   cprmidl 33515
                  21.3.10.41  Maximal Ideals   cmxidl 33539
                  21.3.10.42  The semiring of ideals of a ring   cidlsrg 33580
                  21.3.10.43  Prime Elements   rprmval 33596
                  21.3.10.44  Unique factorization domains   cufd 33618
                  21.3.10.45  The ring of integers   zringidom 33631
                  21.3.10.46  Associative Algebra   assaassd 33635
                  21.3.10.47  Univariate Polynomials   0ringmon1p 33637
                  21.3.10.48  Polynomial quotient and polynomial remainder   q1pdir 33683
                  21.3.10.49  Multivariate Polynomials   psrbasfsupp 33692
                  21.3.10.50  The ring of symmetric polynomials   csply 33719
                  21.3.10.51  The subring algebra   sra1r 33745
                  21.3.10.52  Division Ring Extensions   drgext0g 33754
                  21.3.10.53  Vector Spaces   lvecdimfi 33760
                  21.3.10.54  Vector Space Dimension   cldim 33763
            21.3.11  Field Extensions   cfldext 33803
                  21.3.11.1  Algebraic numbers   cirng 33848
                  21.3.11.2  Algebraic extensions   calgext 33860
                  21.3.11.3  Minimal polynomials   cminply 33864
                  21.3.11.4  Quadratic Field Extensions   rtelextdg2lem 33891
                  21.3.11.5  Towers of quadratic extentions   fldext2chn 33893
            ​*21.3.12  Constructible Numbers   cconstr 33894
                  21.3.12.1  Impossible constructions   2sqr3minply 33945
            21.3.13  Matrices   csmat 33958
                  21.3.13.1  Submatrices   csmat 33958
                  21.3.13.2  Matrix literals   clmat 33976
                  21.3.13.3  Laplace expansion of determinants   mdetpmtr1 33988
            21.3.14  Topology   ist0cld 33998
                  21.3.14.1  Open maps   txomap 33999
                  21.3.14.2  Topology of the unit circle   qtopt1 34000
                  21.3.14.3  Refinements   reff 34004
                  21.3.14.4  Open cover refinement property   ccref 34007
                  21.3.14.5  Lindelöf spaces   cldlf 34017
                  21.3.14.6  Paracompact spaces   cpcmp 34020
                  ​*21.3.14.7  Spectrum of a ring   crspec 34027
                  21.3.14.8  Pseudometrics   cmetid 34051
                  21.3.14.9  Continuity - misc additions   hauseqcn 34063
                  21.3.14.10  Topology of the closed unit interval   elunitge0 34064
                  21.3.14.11  Topology of ` ( RR X. RR ) `   unicls 34068
                  21.3.14.12  Order topology - misc. additions   cnvordtrestixx 34078
                  21.3.14.13  Continuity in topological spaces - misc. additions   mndpluscn 34091
                  21.3.14.14  Topology of the extended nonnegative real numbers ordered monoid   xrge0hmph 34097
                  21.3.14.15  Limits - misc additions   lmlim 34112
                  21.3.14.16  Univariate polynomials   pl1cn 34120
            21.3.15  Uniform Stuctures and Spaces   chcmp 34121
                  21.3.15.1  Hausdorff uniform completion   chcmp 34121
            21.3.16  Topology and algebraic structures   zringnm 34123
                  21.3.16.1  The norm on the ring of the integer numbers   zringnm 34123
                  21.3.16.2  Topological ` ZZ ` -modules   zlm0 34125
                  21.3.16.3  Canonical embedding of the field of the rational numbers into a division ring   cqqh 34135
                  21.3.16.4  Canonical embedding of the real numbers into a complete ordered field   crrh 34158
                  21.3.16.5  Embedding from the extended real numbers into a complete lattice   cxrh 34181
                  21.3.16.6  Canonical embeddings into the ordered field of the real numbers   zrhre 34184
                  ​*21.3.16.7  Topological Manifolds   cmntop 34187
                  21.3.16.8  Extended sum   cesum 34192
            21.3.17  Mixed Function/Constant operation   cofc 34260
            21.3.18  Abstract measure   csiga 34273
                  21.3.18.1  Sigma-Algebra   csiga 34273
                  21.3.18.2  Generated sigma-Algebra   csigagen 34303
                  ​*21.3.18.3  lambda and pi-Systems, Rings of Sets   ispisys 34317
                  21.3.18.4  The Borel algebra on the real numbers   cbrsiga 34346
                  21.3.18.5  Product Sigma-Algebra   csx 34353
                  21.3.18.6  Measures   cmeas 34360
                  21.3.18.7  The counting measure   cntmeas 34391
                  21.3.18.8  The Lebesgue measure - misc additions   voliune 34394
                  21.3.18.9  The Dirac delta measure   cdde 34397
                  21.3.18.10  The 'almost everywhere' relation   cae 34402
                  21.3.18.11  Measurable functions   cmbfm 34414
                  21.3.18.12  Borel Algebra on ` ( RR X. RR ) `   br2base 34434
                  ​*21.3.18.13  Caratheodory's extension theorem   coms 34456
            21.3.19  Integration   itgeq12dv 34491
                  21.3.19.1  Lebesgue integral - misc additions   itgeq12dv 34491
                  21.3.19.2  Bochner integral   citgm 34492
            21.3.20  Euler's partition theorem   oddpwdc 34519
            21.3.21  Sequences defined by strong recursion   csseq 34548
            21.3.22  Fibonacci Numbers   cfib 34561
            21.3.23  Probability   cprb 34572
                  21.3.23.1  Probability Theory   cprb 34572
                  21.3.23.2  Conditional Probabilities   ccprob 34596
                  21.3.23.3  Real-valued Random Variables   crrv 34605
                  21.3.23.4  Preimage set mapping operator   corvc 34621
                  21.3.23.5  Distribution Functions   orvcelval 34634
                  21.3.23.6  Cumulative Distribution Functions   orvclteel 34638
                  21.3.23.7  Probabilities - example   coinfliplem 34644
                  21.3.23.8  Bertrand's Ballot Problem   ballotlemoex 34651
            21.3.24  Signum (sgn or sign) function - misc. additions   fzssfzo 34704
                  21.3.24.1  Operations on words   ccatmulgnn0dir 34707
            21.3.25  Polynomials with real coefficients - misc additions   plymul02 34711
            21.3.26  Descartes's rule of signs   signspval 34717
                  21.3.26.1  Sign changes in a word over real numbers   signspval 34717
                  21.3.26.2  Counting sign changes in a word over real numbers   signslema 34727
            21.3.27  Number Theory   iblidicc 34757
                  21.3.27.1  Representations of a number as sums of integers   crepr 34773
                  21.3.27.2  Vinogradov Trigonometric Sums and the Circle Method   cvts 34800
                  21.3.27.3  The Ternary Goldbach Conjecture: Final Statement   ax-hgt749 34809
            21.3.28  Elementary Geometry   cstrkg2d 34829
                  ​*21.3.28.1  Two-dimensional geometry   cstrkg2d 34829
                  21.3.28.2  Outer Five Segment (not used, no need to move to main)   cafs 34834
            ​*21.3.29  LeftPad Project   clpad 34839
      ​​*21.4  Mathbox for Jonathan Ben-Naim
            21.4.1  First-order logic and set theory   bnj170 34862
            21.4.2  Well founded induction and recursion   bnj110 35021
            21.4.3  The existence of a minimal element in certain classes   bnj69 35173
            21.4.4  Well-founded induction   bnj1204 35175
            21.4.5  Well-founded recursion, part 1 of 3   bnj60 35225
            21.4.6  Well-founded recursion, part 2 of 3   bnj1500 35231
            21.4.7  Well-founded recursion, part 3 of 3   bnj1522 35235
      ​21.5  Mathbox for BTernaryTau
            21.5.1  First-order logic   nfan1c 35236
                  21.5.1.1  Auxiliary axiom schemes   nfan1c 35236
            21.5.2  ZF set theory   exdifsn 35242
                  21.5.2.1  Finitism   prcinf 35278
                  21.5.2.2  Introduce ax-regs   ax-regs 35291
                  21.5.2.3  Derive ax-regs   axregs 35304
                  21.5.2.4  Global choice   gblacfnacd 35305
            21.5.3  Real and complex numbers   zltp1ne 35313
            21.5.4  Graph theory   lfuhgr 35321
                  21.5.4.1  Acyclic graphs   cacycgr 35345
      ​21.6  Mathbox for Mario Carneiro
            21.6.1  Predicate calculus with all distinct variables   ax-7d 35362
            21.6.2  Miscellaneous stuff   quartfull 35368
            21.6.3  Derangements and the Subfactorial   deranglem 35369
            21.6.4  The Erdős-Szekeres theorem   erdszelem1 35394
            21.6.5  The Kuratowski closure-complement theorem   kur14lem1 35409
            21.6.6  Retracts and sections   cretr 35420
            21.6.7  Path-connected and simply connected spaces   cpconn 35422
            21.6.8  Covering maps   ccvm 35458
            21.6.9  Normal numbers   snmlff 35532
            21.6.10  Godel-sets of formulas - part 1   cgoe 35536
            21.6.11  Godel-sets of formulas - part 2   cgon 35635
            21.6.12  Models of ZF   cgze 35649
            ​*21.6.13  Metamath formal systems   cmcn 35663
            21.6.14  Grammatical formal systems   cm0s 35788
            21.6.15  Models of formal systems   cmuv 35808
            21.6.16  Splitting fields   ccpms 35830
            21.6.17  p-adic number fields   czr 35850
      ​​*21.7  Mathbox for Filip Cernatescu
      ​21.8  Mathbox for Paul Chapman
            21.8.1  Real and complex numbers (cont.)   climuzcnv 35874
            21.8.2  Miscellaneous theorems   elfzm12 35878
      ​21.9  Mathbox for Hongxiu Chen
      ​21.10  Mathbox for Adrian Ducourtial
            21.10.1  Propositional calculus   currybi 35891
            21.10.2  Clone theory   ccloneop 35898
      ​21.11  Mathbox for Scott Fenton
            21.11.1  ZFC Axioms in primitive form   axextprim 35904
            21.11.2  Untangled classes   untelirr 35911
            21.11.3  Extra propositional calculus theorems   3jaodd 35918
            21.11.4  Misc. Useful Theorems   nepss 35921
            21.11.5  Properties of real and complex numbers   sqdivzi 35931
            21.11.6  Infinite products   iprodefisumlem 35943
            21.11.7  Factorial limits   faclimlem1 35946
            21.11.8  Greatest common divisor and divisibility   gcd32 35952
            21.11.9  Properties of relationships   dftr6 35954
            21.11.10  Properties of functions and mappings   funpsstri 35969
            21.11.11  Ordinal numbers   elpotr 35982
            21.11.12  Defined equality axioms   axextdfeq 35998
            21.11.13  Hypothesis builders   hbntg 36006
            21.11.14  Well-founded zero, successor, and limits   cwsuc 36011
            21.11.15  Quantifier-free definitions   ctxp 36031
            21.11.16  Alternate ordered pairs   caltop 36159
            21.11.17  Geometry in the Euclidean space   cofs 36185
                  21.11.17.1  Congruence properties   cofs 36185
                  21.11.17.2  Betweenness properties   btwntriv2 36215
                  21.11.17.3  Segment Transportation   ctransport 36232
                  21.11.17.4  Properties relating betweenness and congruence   cifs 36238
                  21.11.17.5  Connectivity of betweenness   btwnconn1lem1 36290
                  21.11.17.6  Segment less than or equal to   csegle 36309
                  21.11.17.7  Outside-of relationship   coutsideof 36322
                  21.11.17.8  Lines and Rays   cline2 36337
            21.11.18  Forward difference   cfwddif 36361
            21.11.19  Rank theorems   rankung 36369
            21.11.20  Hereditarily Finite Sets   chf 36375
      ​21.12  Mathbox for Gino Giotto
            21.12.1  Equality theorems   rmoeqi 36390
                  21.12.1.1  Inference versions   rmoeqi 36390
                  21.12.1.2  Deduction versions   rmoeqdv 36415
            21.12.2  Change bound variables   in-ax8 36427
                  21.12.2.1  Change bound variables and domains   cbvralvw2 36429
                  21.12.2.2  Change bound variables, deduction versions   cbvmodavw 36453
                  21.12.2.3  Change bound variables and domains, deduction versions   cbvrmodavw2 36486
            21.12.3  Study of ax-mulf usage   mpomulnzcnf 36502
      ​21.13  Mathbox for Jeff Hankins
            21.13.1  Miscellany   a1i14 36503
            21.13.2  Basic topological facts   topbnd 36527
            21.13.3  Topology of the real numbers   ivthALT 36538
            21.13.4  Refinements   cfne 36539
            21.13.5  Neighborhood bases determine topologies   neibastop1 36562
            21.13.6  Lattice structure of topologies   topmtcl 36566
            21.13.7  Filter bases   fgmin 36573
            21.13.8  Directed sets, nets   tailfval 36575
      ​21.14  Mathbox for Anthony Hart
            21.14.1  Propositional Calculus   tb-ax1 36586
            21.14.2  Predicate Calculus   nalfal 36606
            21.14.3  Miscellaneous single axioms   meran1 36614
            21.14.4  Connective Symmetry   negsym1 36620
      ​21.15  Mathbox for Chen-Pang He
            21.15.1  Ordinal topology   ontopbas 36631
      ​21.16  Mathbox for Jeff Hoffman
            21.16.1  Inferences for finite induction on generic function values   fveleq 36654
            21.16.2  gdc.mm   nnssi2 36658
      ​21.17  Mathbox for Matthew House
            21.17.1  Relations on well-ordered indexed unions   weiunval 36665
            21.17.2  Axiom of Transitive Containment   axtco 36674
            21.17.3  Transitive closure of a class   tr0elw 36687
            ​*21.17.4  Stronger axioms of regularity   mh-setind 36739
      ​21.18  Mathbox for Asger C. Ipsen
            21.18.1  Continuous nowhere differentiable functions   dnival 36744
      ​​*21.19  Mathbox for BJ
            *21.19.1  Propositional calculus   bj-mp2c 36813
                  *21.19.1.1  Derived rules of inference   bj-mp2c 36813
                  ​*21.19.1.2  A syntactic theorem   bj-0 36815
                  ​*21.19.1.3  Minimal implicational calculus   bj-a1k 36817
                  ​*21.19.1.4  Positive calculus   bj-bisimpl 36830
                  ​*21.19.1.5  Implication and negation   bj-con2com 36838
                  ​*21.19.1.6  Disjunction   bj-jaoi1 36849
                  ​*21.19.1.7  Logical equivalence   bj-dfbi4 36851
                  21.19.1.8  The conditional operator for propositions   bj-consensus 36856
                  ​*21.19.1.9  Propositional calculus: miscellaneous   bj-imbi12 36861
            ​*21.19.2  Modal logic   bj-axdd2 36870
            ​*21.19.3  Provability logic   cprvb 36875
            ​*21.19.4  First-order logic   bj-exexalal 36884
                  21.19.4.1  Universal and existential quantifiers, nonfreeness predicate   bj-exexalal 36884
                  21.19.4.2  Adding ax-gen   bj-genr 36885
                  21.19.4.3  Adding ax-4   bj-almp 36889
                  21.19.4.4  Adding ax-5   bj-spvw 36942
                  21.19.4.5  Equality and substitution   bj-df-sb 36957
                  21.19.4.6  Adding ax-6   bj-spim0 36976
                  21.19.4.7  Adding ax-7   bj-cbvexw 36984
                  21.19.4.8  Membership predicate, ax-8 and ax-9   bj-ax89 36986
                  21.19.4.9  Adding ax-11   bj-alcomexcom 36988
                  21.19.4.10  Adding ax-12   axc11n11 36992
                  ​*21.19.4.11  Really adding ax-12   bj-substax12 37034
                  21.19.4.12  Nonfreeness   wnnf 37036
                  21.19.4.13  Adding ax-13   bj-axc10 37103
                  ​*21.19.4.14  Removing dependencies on ax-13 (and ax-11)   bj-axc10v 37113
                  ​*21.19.4.15  Distinct var metavariables   bj-hbaeb2 37138
                  ​*21.19.4.16  Around ~ equsal   bj-equsal1t 37142
                  ​*21.19.4.17  Some Principia Mathematica proofs   stdpc5t 37147
                  21.19.4.18  Alternate definition of substitution   bj-sbsb 37157
                  21.19.4.19  Lemmas for substitution   bj-sbf3 37159
                  21.19.4.20  Existential uniqueness   bj-eu3f 37161
                  ​*21.19.4.21  First-order logic: miscellaneous   bj-sblem1 37162
            21.19.5  Set theory   eliminable1 37179
                  ​*21.19.5.1  Eliminability of class terms   eliminable1 37179
                  ​*21.19.5.2  Classes without the axiom of extensionality   bj-denoteslem 37191
                  21.19.5.3  Characterization among sets versus among classes   elelb 37217
                  ​*21.19.5.4  The nonfreeness quantifier for classes   bj-nfcsym 37219
                  ​*21.19.5.5  Lemmas for class substitution   bj-sbeqALT 37220
                  21.19.5.6  Removing some axiom requirements and disjoint variable conditions   bj-exlimvmpi 37231
                  ​*21.19.5.7  Class abstractions   bj-elabd2ALT 37245
                  21.19.5.8  Generalized class abstractions   bj-cgab 37253
                  ​*21.19.5.9  Restricted nonfreeness   wrnf 37261
                  ​*21.19.5.10  Russell's paradox   bj-ru1 37263
                  21.19.5.11  Curry's paradox in set theory   currysetlem 37265
                  ​*21.19.5.12  Some disjointness results   bj-n0i 37271
                  ​*21.19.5.13  Complements on direct products   bj-xpimasn 37275
                  ​*21.19.5.14  "Singletonization" and tagging   bj-snsetex 37283
                  ​*21.19.5.15  Tuples of classes   bj-cproj 37310
                  ​*21.19.5.16  Set theory: elementary operations relative to a universe   bj-rcleqf 37345
                  ​*21.19.5.17  Axioms for finite unions   bj-abex 37350
                  ​*21.19.5.18  Set theory: miscellaneous   eleq2w2ALT 37367
                  ​*21.19.5.19  Axioms of separation and replacement   bj-axnul 37392
                  ​*21.19.5.20  Evaluation at a class   bj-evaleq 37396
                  21.19.5.21  Elementwise operations   celwise 37404
                  ​*21.19.5.22  Elementwise intersection (families of sets induced on a subset)   bj-rest00 37406
                  21.19.5.23  Moore collections (complements)   bj-raldifsn 37425
                  21.19.5.24  Maps-to notation for functions with three arguments   bj-0nelmpt 37441
                  ​*21.19.5.25  Currying   csethom 37447
                  ​*21.19.5.26  Setting components of extensible structures   cstrset 37459
            ​*21.19.6  Extended real and complex numbers, real and complex projective lines   bj-nfald 37462
                  21.19.6.1  Complements on class abstractions of ordered pairs and binary relations   bj-nfald 37462
                  ​*21.19.6.2  Identity relation (complements)   bj-opabssvv 37477
                  ​*21.19.6.3  Functionalized identity (diagonal in a Cartesian square)   cdiag2 37499
                  ​*21.19.6.4  Direct image and inverse image   cimdir 37505
                  ​*21.19.6.5  Extended numbers and projective lines as sets   cfractemp 37523
                  ​*21.19.6.6  Addition and opposite   caddcc 37564
                  ​*21.19.6.7  Order relation on the extended reals   cltxr 37568
                  ​*21.19.6.8  Argument, multiplication and inverse   carg 37570
                  21.19.6.9  The canonical bijection from the finite ordinals   ciomnn 37576
                  21.19.6.10  Divisibility   cnnbar 37587
            ​*21.19.7  Monoids   bj-smgrpssmgm 37595
                  ​*21.19.7.1  Finite sums in monoids   cfinsum 37610
            ​*21.19.8  Affine, Euclidean, and Cartesian geometry   bj-fvimacnv0 37613
                  *21.19.8.1  Real vector spaces   bj-fvimacnv0 37613
                  ​*21.19.8.2  Complex numbers (supplements)   bj-subcom 37635
                  ​*21.19.8.3  Barycentric coordinates   bj-bary1lem 37637
            21.19.9  Monoid of endomorphisms   cend 37640
      ​21.20  Mathbox for Jim Kingdon
            21.20.1  Circle constant   taupilem3 37646
            21.20.2  Number theory   dfgcd3 37651
            21.20.3  Real numbers   irrdifflemf 37652
      ​21.21  Mathbox for ML
            21.21.1  Miscellaneous   csbrecsg 37655
            21.21.2  Cartesian exponentiation   cfinxp 37710
            21.21.3  Topology   iunctb2 37730
                  ​*21.21.3.1  Pi-base theorems   pibp16 37740
      ​21.22  Mathbox for Wolf Lammen
            21.22.1  1. Bootstrapping   wl-section-boot 37749
            21.22.2  Implication chains   wl-section-impchain 37773
            21.22.3  Theorems around the conditional operator   wl-ifp-ncond1 37791
            21.22.4  Alternative development of hadd, cadd   wl-df-3xor 37795
            21.22.5  An alternative axiom ~ ax-13   ax-wl-13v 37820
            21.22.6  Bootstrapping set theory with classes   wl-cleq-0 37822
            21.22.7  Other stuff   wl-mps 37843
      ​21.23  Mathbox for Brendan Leahy
      ​21.24  Mathbox for Jeff Madsen
            21.24.1  Logic and set theory   unirep 38046
            21.24.2  Real and complex numbers; integers   filbcmb 38072
            21.24.3  Sequences and sums   sdclem2 38074
            21.24.4  Topology   subspopn 38084
            21.24.5  Metric spaces   metf1o 38087
            21.24.6  Continuous maps and homeomorphisms   constcncf 38094
            21.24.7  Boundedness   ctotbnd 38098
            21.24.8  Isometries   cismty 38130
            21.24.9  Heine-Borel Theorem   heibor1lem 38141
            21.24.10  Banach Fixed Point Theorem   bfplem1 38154
            21.24.11  Euclidean space   crrn 38157
            21.24.12  Intervals (continued)   ismrer1 38170
            21.24.13  Operation properties   cass 38174
            21.24.14  Groups and related structures   cmagm 38180
            21.24.15  Group homomorphism and isomorphism   cghomOLD 38215
            21.24.16  Rings   crngo 38226
            21.24.17  Division Rings   cdrng 38280
            21.24.18  Ring homomorphisms   crngohom 38292
            21.24.19  Commutative rings   ccm2 38321
            21.24.20  Ideals   cidl 38339
            21.24.21  Prime rings and integral domains   cprrng 38378
            21.24.22  Ideal generators   cigen 38391
      ​21.25  Mathbox for Giovanni Mascellani
            ​*21.25.1  Tools for automatic proof building   efald2 38410
            ​*21.25.2  Tseitin axioms   fald 38461
            ​*21.25.3  Equality deductions   iuneq2f 38488
            ​*21.25.4  Miscellanea   orcomdd 38499
      ​21.26  Mathbox for Peter Mazsa
            21.26.1  Notations   cxrn 38506
            21.26.2  Preparatory theorems   el2v1 38561
            21.26.3  Range Cartesian product   df-xrn 38712
            21.26.4  Relations   df-rels 38772
            21.26.5  Quotient map (coset map)   df-qmap 38778
            21.26.6  Lifts, shifts, successor, and predecessor   df-adjliftmap 38787
            21.26.7  Cosets by ` R `   df-coss 38833
            21.26.8  Subset relations   df-ssr 38910
            21.26.9  Reflexivity   df-refs 38922
            21.26.10  Converse reflexivity   df-cnvrefs 38937
            21.26.11  Symmetry   df-syms 38954
            21.26.12  Reflexivity and symmetry   symrefref2 38979
            21.26.13  Transitivity   df-trs 38988
            21.26.14  Equivalence relations   df-eqvrels 39000
            21.26.15  Redundancy   df-redunds 39039
            21.26.16  Domain quotients   df-dmqss 39054
            21.26.17  Equivalence relations on domain quotients   df-ers 39080
            21.26.18  Functions   df-funss 39097
            21.26.19  Disjoints vs. converse functions   df-disjss 39120
            21.26.20  Antisymmetry   df-antisymrel 39195
            21.26.21  Partitions: disjoints on domain quotients   df-parts 39200
            21.26.22  Partition-Equivalence Theorems   disjim 39216
            21.26.23  Type-safe Partition-Equivalence: PetParts, PetErs, Pet2Parts, Pet2Ers   df-petparts 39300
      ​21.27  Mathbox for Rodolfo Medina
            21.27.1  Partitions   prtlem60 39310
      ​​*21.28  Mathbox for Norm Megill
            *21.28.1  Obsolete schemes ax-c4,c5,c7,c10,c11,c11n,c15,c9,c14,c16   ax-c5 39340
            ​*21.28.2  Rederive new axioms ax-4, ax-10, ax-6, ax-12, ax-13 from old   axc5 39350
            ​*21.28.3  Legacy theorems using obsolete axioms   ax5ALT 39364
            21.28.4  Experiments with weak deduction theorem   elimhyps 39418
            21.28.5  Miscellanea   cnaddcom 39429
            21.28.6  Atoms, hyperplanes, and covering in a left vector space (or module)   clsa 39431
            21.28.7  Functionals and kernels of a left vector space (or module)   clfn 39514
            21.28.8  Opposite rings and dual vector spaces   cld 39580
            21.28.9  Ortholattices and orthomodular lattices   cops 39629
            21.28.10  Atomic lattices with covering property   ccvr 39719
            21.28.11  Hilbert lattices   chlt 39807
            21.28.12  Projective geometries based on Hilbert lattices   clln 39948
            21.28.13  Construction of a vector space from a Hilbert lattice   cdlema1N 40248
            21.28.14  Construction of involution and inner product from a Hilbert lattice   clpoN 41937
      ​21.29  Mathbox for metakunt
            21.29.1  Commutative Semiring   ccsrg 42419
            21.29.2  General helpful statements   rhmzrhval 42422
            21.29.3  Some gcd and lcm results   12gcd5e1 42453
            21.29.4  Least common multiple inequality theorem   3factsumint1 42471
            21.29.5  Logarithm inequalities   3exp7 42503
            21.29.6  Miscellaneous results for AKS formalisation   intlewftc 42511
            21.29.7  Sticks and stones   sticksstones1 42596
            21.29.8  Continuation AKS   aks6d1c6lem1 42620
      ​21.30  Mathbox for Steven Nguyen
            21.30.1  Utility theorems   jarrii 42655
            ​*21.30.2  Arithmetic theorems   c0exALT 42702
            21.30.3  Exponents and divisibility   oexpreposd 42765
            21.30.4  Trigonometry and Calculus   tanhalfpim 42792
            ​*21.30.5  Independence of ax-mulcom   cresub 42808
            21.30.6  Structures   sn-base0 42951
            ​*21.30.7  Projective spaces   cprjsp 43045
            21.30.8  Basic reductions for Fermat's Last Theorem   dffltz 43078
            ​*21.30.9  Exemplar theorems   iddii 43108
                  ​*21.30.9.1  Standard replacements of ax-10 , ax-11 , ax-12   nfa1w 43119
      ​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 43135
            21.33.2  Additional theory of functions   imaiinfv 43136
            21.33.3  Additional topology   elrfi 43137
            21.33.4  Characterization of closure operators. Kuratowski closure axioms   ismrcd1 43141
            21.33.5  Algebraic closure systems   cnacs 43145
            21.33.6  Miscellanea 1. Map utilities   constmap 43156
            21.33.7  Miscellanea for polynomials   mptfcl 43163
            21.33.8  Multivariate polynomials over the integers   cmzpcl 43164
            21.33.9  Miscellanea for Diophantine sets 1   coeq0i 43196
            21.33.10  Diophantine sets 1: definitions   cdioph 43198
            21.33.11  Diophantine sets 2 miscellanea   ellz1 43210
            21.33.12  Diophantine sets 2: union and intersection. Monotone Boolean algebra   diophin 43215
            21.33.13  Diophantine sets 3: construction   diophrex 43218
            21.33.14  Diophantine sets 4 miscellanea   2sbcrex 43227
            21.33.15  Diophantine sets 4: Quantification   rexrabdioph 43237
            21.33.16  Diophantine sets 5: Arithmetic sets   rabdiophlem1 43244
            21.33.17  Diophantine sets 6: reusability. renumbering of variables   eldioph4b 43254
            21.33.18  Pigeonhole Principle and cardinality helpers   fphpd 43259
            21.33.19  A non-closed set of reals is infinite   rencldnfilem 43263
            21.33.20  Lagrange's rational approximation theorem   irrapxlem1 43265
            21.33.21  Pell equations 1: A nontrivial solution always exists   pellexlem1 43272
            21.33.22  Pell equations 2: Algebraic number theory of the solution set   csquarenn 43279
            21.33.23  Pell equations 3: characterizing fundamental solution   infmrgelbi 43321
            ​*21.33.24  Logarithm laws generalized to an arbitrary base   reglogcl 43333
            21.33.25  Pell equations 4: the positive solution group is infinite cyclic   pellfund14 43341
            21.33.26  X and Y sequences 1: Definition and recurrence laws   crmx 43343
            21.33.27  Ordering and induction lemmas for the integers   monotuz 43384
            21.33.28  X and Y sequences 2: Order properties   rmxypos 43390
            21.33.29  Congruential equations   congtr 43408
            21.33.30  Alternating congruential equations   acongid 43418
            21.33.31  Additional theorems on integer divisibility   coprmdvdsb 43428
            21.33.32  X and Y sequences 3: Divisibility properties   jm2.18 43431
            21.33.33  X and Y sequences 4: Diophantine representability of Y   jm2.27a 43448
            21.33.34  X and Y sequences 5: Diophantine representability of X, ^, _C   rmxdiophlem 43458
            21.33.35  Uncategorized stuff not associated with a major project   setindtr 43467
            21.33.36  More equivalents of the Axiom of Choice   axac10 43476
            21.33.37  Finitely generated left modules   clfig 43510
            21.33.38  Noetherian left modules I   clnm 43518
            21.33.39  Addenda for structure powers   pwssplit4 43532
            21.33.40  Every set admits a group structure iff choice   unxpwdom3 43538
            21.33.41  Noetherian rings and left modules II   clnr 43552
            21.33.42  Hilbert's Basis Theorem   cldgis 43564
            21.33.43  Additional material on polynomials [DEPRECATED]   cmnc 43574
            21.33.44  Degree and minimal polynomial of algebraic numbers   cdgraa 43583
            21.33.45  Algebraic integers I   citgo 43600
            21.33.46  Endomorphism algebra   cmend 43614
            21.33.47  Cyclic groups and order   idomodle 43634
            21.33.48  Cyclotomic polynomials   ccytp 43640
            21.33.49  Miscellaneous topology   fgraphopab 43646
      ​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 43660
            21.36.2  Natural addition of Cantor normal forms   oawordex2 43769
            21.36.3  Surreal Contributions   abeqabi 43850
            21.36.4  Short Studies   nlimsuc 43883
                  21.36.4.1  Additional work on conditional logical operator   ifpan123g 43901
                  21.36.4.2  Sophisms   rp-fakeimass 43954
                  ​*21.36.4.3  Finite Sets   rp-isfinite5 43959
                  21.36.4.4  General Observations   intabssd 43961
                  21.36.4.5  Infinite Sets   pwelg 44002
                  ​*21.36.4.6  Finite intersection property   fipjust 44007
                  21.36.4.7  RP ADDTO: Subclasses and subsets   rababg 44016
                  21.36.4.8  RP ADDTO: The intersection of a class   elinintab 44017
                  21.36.4.9  RP ADDTO: Theorems requiring subset and intersection existence   elinintrab 44019
                  21.36.4.10  RP ADDTO: Relations   xpinintabd 44022
                  ​*21.36.4.11  RP ADDTO: Functions   elmapintab 44038
                  ​*21.36.4.12  RP ADDTO: Finite induction (for finite ordinals)   cnvcnvintabd 44042
                  21.36.4.13  RP ADDTO: First and second members of an ordered pair   elcnvlem 44043
                  21.36.4.14  RP ADDTO: The reflexive and transitive properties of relations   undmrnresiss 44046
                  21.36.4.15  RP ADDTO: Basic properties of closures   cleq2lem 44050
                  21.36.4.16  RP REPLACE: Definitions and basic properties of transitive closures   trcleq2lemRP 44072
                  ​*21.36.4.17  Additions for square root; absolute value   sqrtcvallem1 44073
            21.36.5  Additional statements on relations and subclasses   al3im 44089
                  21.36.5.1  Transitive relations (not to be confused with transitive classes)   trrelind 44107
                  21.36.5.2  Reflexive closures   crcl 44114
                  ​*21.36.5.3  Finite relationship composition   relexp2 44119
                  21.36.5.4  Transitive closure of a relation   dftrcl3 44162
                  ​*21.36.5.5  Adapted from Frege   frege77d 44188
            ​*21.36.6  Propositions from _Begriffsschrift_   dfxor4 44208
                  *21.36.6.1  _Begriffsschrift_ Chapter I   dfxor4 44208
                  ​*21.36.6.2  _Begriffsschrift_ Notation hints   whe 44214
                  21.36.6.3  _Begriffsschrift_ Chapter II Implication   ax-frege1 44232
                  21.36.6.4  _Begriffsschrift_ Chapter II Implication and Negation   axfrege28 44271
                  ​*21.36.6.5  _Begriffsschrift_ Chapter II with logical equivalence   axfrege52a 44298
                  21.36.6.6  _Begriffsschrift_ Chapter II with equivalence of sets   axfrege52c 44329
                  ​*21.36.6.7  _Begriffsschrift_ Chapter II with equivalence of classes   frege53c 44356
                  ​*21.36.6.8  _Begriffsschrift_ Chapter III Properties hereditary in a sequence   dffrege69 44374
                  ​*21.36.6.9  _Begriffsschrift_ Chapter III Following in a sequence   dffrege76 44381
                  ​*21.36.6.10  _Begriffsschrift_ Chapter III Member of sequence   dffrege99 44404
                  ​*21.36.6.11  _Begriffsschrift_ Chapter III Single-valued procedures   dffrege115 44420
            ​*21.36.7  Exploring Topology via Seifert and Threlfall   enrelmap 44439
                  *21.36.7.1  Equinumerosity of sets of relations and maps   enrelmap 44439
                  ​*21.36.7.2  Generic Pseudoclosure Spaces, Pseudointerior Spaces, and Pseudoneighborhoods   or3or 44465
                  ​*21.36.7.3  Generic Neighborhood Spaces   gneispa 44572
            ​*21.36.8  Exploring Higher Homotopy via Kerodon   k0004lem1 44589
                  *21.36.8.1  Simplicial Sets   k0004lem1 44589
      ​21.37  Mathbox for Stanislas Polu
            21.37.1  IMO Problems   wwlemuld 44598
                  21.37.1.1  IMO 1972 B2   wwlemuld 44598
            ​*21.37.2  INT Inequalities Proof Generator   int-addcomd 44615
            ​*21.37.3  N-Digit Addition Proof Generator   unitadd 44637
            21.37.4  AM-GM (for k = 2,3,4)   gsumws3 44638
      ​21.38  Mathbox for Rohan Ridenour
            21.38.1  Misc   spALT 44643
            21.38.2  Monoid rings   cmnring 44653
            21.38.3  Shorter primitive equivalent of ax-groth   gru0eld 44671
                  21.38.3.1  Grothendieck universes are closed under collection   gru0eld 44671
                  21.38.3.2  Minimal universes   ismnu 44703
                  21.38.3.3  Primitive equivalent of ax-groth   expandan 44730
      ​21.39  Mathbox for Steve Rodriguez
            21.39.1  Miscellanea   nanorxor 44747
            21.39.2  Ratio test for infinite series convergence and divergence   dvgrat 44754
            21.39.3  Multiples   reldvds 44757
            21.39.4  Function operations   caofcan 44765
            21.39.5  Calculus   lhe4.4ex1a 44771
            21.39.6  The generalized binomial coefficient operation   cbcc 44778
            21.39.7  Binomial series   uzmptshftfval 44788
      ​21.40  Mathbox for Andrew Salmon
            21.40.1  Principia Mathematica * 10   pm10.12 44800
            21.40.2  Principia Mathematica * 11   2alanimi 44814
            21.40.3  Predicate Calculus   sbeqal1 44840
            21.40.4  Principia Mathematica * 13 and * 14   pm13.13a 44849
            21.40.5  Set Theory   elnev 44879
            21.40.6  Arithmetic   addcomgi 44897
            21.40.7  Geometry   cplusr 44898
      ​​*21.41  Mathbox for Alan Sare
            21.41.1  Auxiliary theorems for the Virtual Deduction tool   idiALT 44920
            21.41.2  Supplementary unification deductions   bi1imp 44924
            21.41.3  Conventional Metamath proofs, some derived from VD proofs   iidn3 44943
            21.41.4  What is Virtual Deduction?   wvd1 45011
            21.41.5  Virtual Deduction Theorems   df-vd1 45012
            21.41.6  Theorems proved using Virtual Deduction   trsspwALT 45259
            21.41.7  Theorems proved using Virtual Deduction with mmj2 assistance   simplbi2VD 45287
            21.41.8  Virtual Deduction transcriptions of textbook proofs   sb5ALTVD 45354
            21.41.9  Theorems proved using conjunction-form Virtual Deduction   elpwgdedVD 45358
            21.41.10  Theorems with a VD proof in conventional notation derived from a VD proof   suctrALT3 45365
            ​*21.41.11  Theorems with a proof in conventional notation derived from a VD proof   notnotrALT2 45368
      ​21.42  Mathbox for Eric Schmidt
            21.42.1  Miscellany   rspesbcd 45379
            21.42.2  Study of dfbi1ALT   dfbi1ALTa 45381
            21.42.3  Relation-preserving functions   wrelp 45384
            21.42.4  Orbits   orbitex 45397
            21.42.5  Well-founded sets   trwf 45401
            21.42.6  Absoluteness in transitive models   ralabso 45410
            21.42.7  Lemmas for showing axioms hold in models   traxext 45419
            21.42.8  The class of well-founded sets is a model for ZFC   wfaxext 45435
            21.42.9  Permutation models   brpermmodel 45445
      ​21.43  Mathbox for Glauco Siliprandi
            21.43.1  Miscellanea   evth2f 45461
            21.43.2  Functions   fnresdmss 45613
            21.43.3  Ordering on real numbers - Real and complex numbers basic operations   sub2times 45721
            21.43.4  Real intervals   gtnelioc 45936
            21.43.5  Finite sums   fsummulc1f 46016
            21.43.6  Finite multiplication of numbers and finite multiplication of functions   fmul01 46025
            21.43.7  Limits   clim1fr1 46046
                  21.43.7.1  Inferior limit (lim inf)   clsi 46194
                  ​*21.43.7.2  Limits for sequences of extended real numbers   clsxlim 46261
            21.43.8  Trigonometry   coseq0 46307
            21.43.9  Continuous Functions   mulcncff 46313
            21.43.10  Derivatives   dvsinexp 46354
            21.43.11  Integrals   itgsin0pilem1 46393
            21.43.12  Stone Weierstrass theorem - real version   stoweidlem1 46444
            21.43.13  Wallis' product for π   wallispilem1 46508
            21.43.14  Stirling's approximation formula for ` n ` factorial   stirlinglem1 46517
            21.43.15  Dirichlet kernel   dirkerval 46534
            21.43.16  Fourier Series   fourierdlem1 46551
            21.43.17  e is transcendental   elaa2lem 46676
            21.43.18  n-dimensional Euclidean space   rrxtopn 46727
            21.43.19  Basic measure theory   csalg 46751
                  ​*21.43.19.1  σ-Algebras   csalg 46751
                  21.43.19.2  Sum of nonnegative extended reals   csumge0 46805
                  ​*21.43.19.3  Measures   cmea 46892
                  ​*21.43.19.4  Outer measures and Caratheodory's construction   come 46932
                  ​*21.43.19.5  Lebesgue measure on n-dimensional Real numbers   covoln 46979
                  ​*21.43.19.6  Measurable functions   csmblfn 47138
      ​21.44  Mathbox for Saveliy Skresanov
            21.44.1  Ceva's theorem   sigarval 47293
            21.44.2  Simple groups   simpcntrab 47313
      ​21.45  Mathbox for Ender Ting
            21.45.1  Increasing sequences and subsequences   et-ltneverrefl 47314
            21.45.2  Scratchpad for number theory   evenwodadd 47330
            21.45.3  Scratchpad for math on real numbers   squeezedltsq 47331
      ​21.46  Mathbox for Jarvin Udandy
      ​21.47  Mathbox for Adhemar
            ​*21.47.1  Minimal implicational calculus   adh-minim 47446
      ​21.48  Mathbox for Alexander van der Vekens
            21.48.1  General auxiliary theorems (1)   n0nsn2el 47470
                  21.48.1.1  Unordered and ordered pairs - extension for singletons   n0nsn2el 47470
                  21.48.1.2  Unordered and ordered pairs - extension for unordered pairs   elprneb 47474
                  21.48.1.3  Unordered and ordered pairs - extension for ordered pairs   oppr 47475
                  21.48.1.4  Relations - extension   eubrv 47480
                  21.48.1.5  Definite description binder (inverted iota) - extension   iota0def 47483
                  21.48.1.6  Functions - extension   fveqvfvv 47485
            21.48.2  Alternative for Russell's definition of a description binder   caiota 47528
            21.48.3  Double restricted existential uniqueness   r19.32 47543
                  21.48.3.1  Restricted quantification (extension)   r19.32 47543
                  21.48.3.2  Restricted uniqueness and "at most one" quantification   reuf1odnf 47552
                  21.48.3.3  Analogs to Existential uniqueness (double quantification)   2reu3 47555
                  21.48.3.4  Additional theorems for double restricted existential uniqueness   2reu8i 47558
            ​*21.48.4  Alternative definitions of function and operation values   wdfat 47561
                  21.48.4.1  Restricted quantification (extension)   ralbinrald 47567
                  21.48.4.2  The universal class (extension)   nvelim 47568
                  21.48.4.3  Introduce the Axiom of Power Sets (extension)   alneu 47569
                  21.48.4.4  Predicate "defined at"   dfateq12d 47571
                  21.48.4.5  Alternative definition of the value of a function   dfafv2 47577
                  21.48.4.6  Alternative definition of the value of an operation   aoveq123d 47623
            ​*21.48.5  Alternative definitions of function values (2)   cafv2 47653
            21.48.6  General auxiliary theorems (2)   an4com24 47713
                  21.48.6.1  Logical conjunction - extension   an4com24 47713
                  21.48.6.2  Abbreviated conjunction and disjunction of three wff's - extension   3an4ancom24 47714
                  21.48.6.3  Negated membership (alternative)   cnelbr 47716
                  21.48.6.4  The empty set - extension   ralralimp 47723
                  21.48.6.5  Indexed union and intersection - extension   otiunsndisjX 47724
                  21.48.6.6  Functions - extension   fvifeq 47725
                  21.48.6.7  Maps-to notation - extension   fvmptrab 47737
                  21.48.6.8  Subtraction - extension   cnambpcma 47739
                  21.48.6.9  Ordering on reals (cont.) - extension   leaddsuble 47742
                  21.48.6.10  Imaginary and complex number properties - extension   readdcnnred 47748
                  21.48.6.11  Nonnegative integers (as a subset of complex numbers) - extension   nn0resubcl 47753
                  21.48.6.12  Integers (as a subset of complex numbers) - extension   zgeltp1eq 47754
                  21.48.6.13  Decimal arithmetic - extension   1t10e1p1e11 47755
                  21.48.6.14  Upper sets of integers - extension   eluzge0nn0 47757
                  21.48.6.15  Infinity and the extended real number system (cont.) - extension   nltle2tri 47758
                  21.48.6.16  Finite intervals of integers - extension   ssfz12 47759
                  21.48.6.17  Half-open integer ranges - extension   fzopred 47768
                  21.48.6.18  The floor and ceiling functions - extension   2ltceilhalf 47777
                  21.48.6.19  The modulo (remainder) operation - extension   fldivmod 47789
                  21.48.6.20  The infinite sequence builder "seq"   smonoord 47822
                  21.48.6.21  Integer powers - extension   2timesltsq 47823
                  21.48.6.22  Finite and infinite sums - extension   fsummsndifre 47825
                  21.48.6.23  The divides relation - extension   nndivides2 47829
                  21.48.6.24  Extensible structures - extension   setsidel 47833
            ​*21.48.7  Preimages of function values   preimafvsnel 47836
            ​*21.48.8  Partitions of real intervals   ciccp 47870
            21.48.9  Shifting functions with an integer range domain   fargshiftfv 47896
            21.48.10  Words over a set (extension)   lswn0 47901
                  21.48.10.1  Last symbol of a word - extension   lswn0 47901
            21.48.11  Unordered pairs   wich 47902
                  21.48.11.1  Interchangeable setvar variables   wich 47902
                  21.48.11.2  Set of unordered pairs   sprid 47931
                  ​*21.48.11.3  Proper (unordered) pairs   prpair 47958
                  21.48.11.4  Set of proper unordered pairs   cprpr 47969
            21.48.12  Number theory (extension)   nprmmul1 47984
                  21.48.12.1  Properties of non-prime numbers   nprmmul1 47984
                  ​*21.48.12.2  Fermat numbers   cfmtno 47987
                  ​*21.48.12.3  Mersenne primes   m2prm 48051
                  21.48.12.4  Proth's theorem   modexp2m1d 48072
                  21.48.12.5  The prime-counting function according to Ján Mináč   nprmdvdsfacm1lem1 48080
                  21.48.12.6  Solutions of quadratic equations   quad1 48093
            ​*21.48.13  Even and odd numbers   ceven 48097
                  21.48.13.1  Definitions and basic properties   ceven 48097
                  21.48.13.2  Alternate definitions using the "divides" relation   dfeven2 48122
                  21.48.13.3  Alternate definitions using the "modulo" operation   dfeven3 48131
                  21.48.13.4  Alternate definitions using the "gcd" operation   iseven5 48137
                  21.48.13.5  Theorems of part 5 revised   zneoALTV 48142
                  21.48.13.6  Theorems of part 6 revised   odd2np1ALTV 48147
                  21.48.13.7  Theorems of AV's mathbox revised   0evenALTV 48161
                  21.48.13.8  Additional theorems   epoo 48176
                  21.48.13.9  Perfect Number Theorem (revised)   perfectALTVlem1 48194
            21.48.14  Number theory (extension 2)   cfppr 48197
                  ​*21.48.14.1  Fermat pseudoprimes   cfppr 48197
                  ​*21.48.14.2  Goldbach's conjectures   cgbe 48218
            21.48.15  Graph theory (extension)   cclnbgr 48291
                  21.48.15.1  Closed neighborhood of a vertex   cclnbgr 48291
                  ​*21.48.15.2  Semiclosed and semiopen neighborhoods (experimental)   dfsclnbgr2 48319
                  21.48.15.3  Induced subgraphs   cisubgr 48333
                  ​*21.48.15.4  Isomorphisms of graphs   cgrisom 48347
                  ​*21.48.15.5  Triangles in graphs   cgrtri 48410
                  ​*21.48.15.6  Star graphs   cstgr 48424
                  ​*21.48.15.7  Local isomorphisms of graphs   cgrlim 48449
                  ​*21.48.15.8  Generalized Petersen graphs   cgpg 48513
                  21.48.15.9  Loop-free graphs - extension   1hegrlfgr 48605
                  21.48.15.10  Walks - extension   cupwlks 48606
                  21.48.15.11  Edges of graphs expressed as sets of unordered pairs   upgredgssspr 48616
            21.48.16  Monoids (extension)   ovn0dmfun 48629
                  21.48.16.1  Auxiliary theorems   ovn0dmfun 48629
                  21.48.16.2  Magmas, Semigroups and Monoids (extension)   plusfreseq 48637
                  21.48.16.3  Examples and counterexamples for magmas, semigroups and monoids (extension)   opmpoismgm 48640
                  21.48.16.4  Group sum operation (extension 1)   gsumsplit2f 48653
            ​*21.48.17  Magmas and internal binary operations (alternate approach)   ccllaw 48656
                  *21.48.17.1  Laws for internal binary operations   ccllaw 48656
                  ​*21.48.17.2  Internal binary operations   cintop 48669
                  21.48.17.3  Alternative definitions for magmas and semigroups   cmgm2 48688
            21.48.18  Rings (extension)   lmod0rng 48702
                  21.48.18.1  Nonzero rings (extension)   lmod0rng 48702
                  21.48.18.2  Ideals as non-unital rings   lidldomn1 48704
                  21.48.18.3  The non-unital ring of even integers   0even 48710
                  21.48.18.4  A constructed not unital ring   cznrnglem 48732
                  ​*21.48.18.5  The category of non-unital rings (alternate definition)   crngcALTV 48736
                  ​*21.48.18.6  The category of (unital) rings (alternate definition)   cringcALTV 48760
            21.48.19  Basic algebraic structures (extension)   eliunxp2 48807
                  21.48.19.1  Auxiliary theorems   eliunxp2 48807
                  21.48.19.2  The binomial coefficient operation (extension)   bcpascm1 48824
                  21.48.19.3  The ` ZZ `-module ` ZZ X. ZZ `   zlmodzxzlmod 48827
                  21.48.19.4  Group sum operation (extension 2)   mgpsumunsn 48834
                  21.48.19.5  Symmetric groups (extension)   exple2lt6 48837
                  21.48.19.6  Divisibility (extension)   invginvrid 48840
                  21.48.19.7  The support of functions (extension)   rmsupp0 48841
                  21.48.19.8  Finitely supported functions (extension)   rmsuppfi 48845
                  21.48.19.9  Left modules (extension)   lmodvsmdi 48852
                  21.48.19.10  Associative algebras (extension)   assaascl0 48854
                  21.48.19.11  Univariate polynomials (extension)   ply1vr1smo 48856
                  21.48.19.12  Univariate polynomials (examples)   linply1 48866
            21.48.20  Linear algebra (extension)   cdmatalt 48869
                  ​*21.48.20.1  The subalgebras of diagonal and scalar matrices (extension)   cdmatalt 48869
                  ​*21.48.20.2  Linear combinations   clinc 48877
                  ​*21.48.20.3  Linear independence   clininds 48913
                  21.48.20.4  Simple left modules and the ` ZZ `-module   lmod1lem1 48960
                  21.48.20.5  Differences between (left) modules and (left) vector spaces   lvecpsslmod 48980
            21.48.21  Complexity theory   suppdm 48983
                  21.48.21.1  Auxiliary theorems   suppdm 48983
                  21.48.21.2  Even and odd integers   nn0onn0ex 48996
                  21.48.21.3  The natural logarithm on complex numbers (extension)   logcxp0 49008
                  21.48.21.4  Division of functions   cfdiv 49010
                  21.48.21.5  Upper bounds   cbigo 49020
                  21.48.21.6  Logarithm to an arbitrary base (extension)   rege1logbrege0 49031
                  ​*21.48.21.7  The binary logarithm   fldivexpfllog2 49038
                  21.48.21.8  Binary length   cblen 49042
                  ​*21.48.21.9  Digits   cdig 49068
                  21.48.21.10  Nonnegative integer as sum of its shifted digits   dignn0flhalflem1 49088
                  21.48.21.11  Algorithms for the multiplication of nonnegative integers   nn0mulfsum 49097
                  ​*21.48.21.12  N-ary functions   cnaryf 49099
                  ​*21.48.21.13  The Ackermann function   citco 49130
            21.48.22  Elementary geometry (extension)   fv1prop 49172
                  21.48.22.1  Auxiliary theorems   fv1prop 49172
                  21.48.22.2  Real euclidean space of dimension 2   rrx2pxel 49184
                  21.48.22.3  Spheres and lines in real Euclidean spaces   cline 49200
      ​21.49  Mathbox for Zhi Wang
            21.49.1  Propositional calculus   pm4.71da 49262
            21.49.2  Predicate calculus with equality   dtrucor3 49271
                  21.49.2.1  Axiom scheme ax-5 (Distinctness)   dtrucor3 49271
            21.49.3  ZF Set Theory - start with the Axiom of Extensionality   ralbidb 49272
                  21.49.3.1  Restricted quantification   ralbidb 49272
                  21.49.3.2  The universal class   reuxfr1dd 49279
                  21.49.3.3  The empty set   ssdisjd 49280
                  21.49.3.4  Unordered and ordered pairs   vsn 49284
                  21.49.3.5  The union of a class   unilbss 49290
                  21.49.3.6  Indexed union and intersection   iuneq0 49291
            21.49.4  ZF Set Theory - add the Axiom of Replacement   inpw 49297
                  21.49.4.1  Theorems requiring subset and intersection existence   inpw 49297
            21.49.5  ZF Set Theory - add the Axiom of Power Sets   opth1neg 49298
                  21.49.5.1  Ordered pair theorem   opth1neg 49298
                  21.49.5.2  Ordered-pair class abstractions (cont.)   brab2dd 49300
                  21.49.5.3  Relations   iinxp 49303
                  21.49.5.4  Functions   mof0 49310
                  21.49.5.5  Operations   ovsng 49330
            21.49.6  ZF Set Theory - add the Axiom of Union   fonex 49339
                  21.49.6.1  Relations and functions (cont.)   fonex 49339
                  21.49.6.2  First and second members of an ordered pair   eloprab1st2nd 49340
                  21.49.6.3  Operations in maps-to notation (continued)   fmpodg 49341
                  21.49.6.4  Function transposition   resinsnlem 49343
                  21.49.6.5  Infinite Cartesian products   ixpv 49362
                  21.49.6.6  Equinumerosity   fvconst0ci 49363
            21.49.7  Order sets   iccin 49368
                  21.49.7.1  Real number intervals   iccin 49368
            21.49.8  Extensible structures   slotresfo 49371
                  21.49.8.1  Basic definitions   slotresfo 49371
            21.49.9  Moore spaces   mreuniss 49372
            ​*21.49.10  Topology   clduni 49373
                  21.49.10.1  Closure and interior   clduni 49373
                  21.49.10.2  Neighborhoods   neircl 49377
                  21.49.10.3  Subspace topologies   restcls2lem 49385
                  21.49.10.4  Limits and continuity in topological spaces   cnneiima 49389
                  21.49.10.5  Topological definitions using the reals   iooii 49390
                  21.49.10.6  Separated sets   sepnsepolem1 49394
                  21.49.10.7  Separated spaces: T0, T1, T2 (Hausdorff) ...   isnrm4 49403
            21.49.11  Preordered sets and directed sets using extensible structures   isprsd 49427
            21.49.12  Posets and lattices using extensible structures   lubeldm2 49428
                  21.49.12.1  Posets   lubeldm2 49428
                  21.49.12.2  Lattices   toslat 49454
                  21.49.12.3  Subset order structures   intubeu 49456
            21.49.13  Rings   elmgpcntrd 49477
                  21.49.13.1  Multiplicative Group   elmgpcntrd 49477
            21.49.14  Associative algebras   asclelbasALT 49478
                  21.49.14.1  Definition and basic properties   asclelbasALT 49478
            21.49.15  Categories   homf0 49481
                  21.49.15.1  Categories   homf0 49481
                  21.49.15.2  Opposite category   oppccatb 49488
                  21.49.15.3  Monomorphisms and epimorphisms   idmon 49492
                  21.49.15.4  Sections, inverses, isomorphisms   sectrcl 49494
                  21.49.15.5  Isomorphic objects   cicfn 49514
                  21.49.15.6  Subcategories   dmdm 49525
                  21.49.15.7  Functors   reldmfunc 49547
                  21.49.15.8  Opposite functors   coppf 49594
                  21.49.15.9  Full & faithful functors   imasubc 49623
                  21.49.15.10  Universal property   upciclem1 49638
                  21.49.15.11  Natural transformations and the functor category   isnatd 49695
                  21.49.15.12  Initial, terminal and zero objects of a category   initoo2 49704
                  21.49.15.13  Product of categories   reldmxpc 49718
                  21.49.15.14  Swap functors   cswapf 49731
                  21.49.15.15  Functor evaluation   oppc1stflem 49759
                  21.49.15.16  Transposed curry functors   cofuswapfcl 49765
                  21.49.15.17  Constant functors   diag1 49776
                  21.49.15.18  Functor composition bifunctors   fucofulem1 49782
                  21.49.15.19  Post-composition functors   postcofval 49836
                  21.49.15.20  Pre-composition functors   precofvallem 49838
            21.49.16  Examples of categories   catcrcl 49867
                  21.49.16.1  The category of categories   catcrcl 49867
                  21.49.16.2  Thin categories   cthinc 49889
                  21.49.16.3  Terminal categories   ctermc 49944
                  21.49.16.4  Preordered sets as thin categories   cprstc 50021
                  21.49.16.5  Monoids as categories   cmndtc 50049
                  21.49.16.6  Categories with at most one object and at most two morphisms   2arwcatlem1 50067
            21.49.17  Kan extensions and related concepts   clan 50077
                  21.49.17.1  Kan extensions   clan 50077
                  21.49.17.2  Limits and colimits   clmd 50115
      ​21.50  Mathbox for Emmett Weisz
            ​*21.50.1  Miscellaneous Theorems   nfintd 50145
            21.50.2  Set Recursion   csetrecs 50155
                  ​*21.50.2.1  Basic Properties of Set Recursion   csetrecs 50155
                  21.50.2.2  Examples and properties of set recursion   elsetrecslem 50171
            ​*21.50.3  Construction of Games and Surreal Numbers   cpg 50181
      ​​*21.51  Mathbox for David A. Wheeler
            21.51.1  Natural deduction   sbidd 50190
            ​*21.51.2  Greater than, greater than or equal to   cge-real 50192
            ​*21.51.3  Hyperbolic trigonometric functions   csinh 50202
            ​*21.51.4  Reciprocal trigonometric functions (sec, csc, cot)   csec 50213
            ​*21.51.5  Identities for "if"   ifnmfalse 50235
            ​*21.51.6  Logarithms generalized to arbitrary base using ` logb `   logb2aval 50236
            ​*21.51.7  Logarithm laws generalized to an arbitrary base - log_   clog- 50237
            ​*21.51.8  Formally define notions such as reflexivity   wreflexive 50239
            ​*21.51.9  Algebra helpers   mvlraddi 50243
            ​*21.51.10  Algebra helper examples   i2linesi 50250
            ​*21.51.11  Formal methods "surprises"   alimp-surprise 50252
            ​*21.51.12  Allsome quantifier   walsi 50258
            ​*21.51.13  Miscellaneous   5m4e1 50269
            21.51.14  Theorems about algebraic numbers   aacllem 50273
      ​21.52  Mathbox for Kunhao Zheng
            21.52.1  Weighted AM-GM inequality   amgmwlem 50274