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 847
            ​*1.2.8  Mixed connectives   jaao 956
            ​*1.2.9  The conditional operator for propositions   wif 1062
            ​*1.2.10  The weak deduction theorem for propositional calculus   elimh 1082
            1.2.11  Abbreviated conjunction and disjunction of three wff's   w3o 1085
            1.2.12  Logical "nand" (Sheffer stroke)   wnan 1491
            1.2.13  Logical "xor"   wxo 1511
            1.2.14  Logical "nor"   wnor 1528
            1.2.15  True and false constants   wal 1538
                  ​*1.2.15.1  Universal quantifier for use by df-tru   wal 1538
                  ​*1.2.15.2  Equality predicate for use by df-tru   cv 1539
                  1.2.15.3  The true constant   wtru 1541
                  1.2.15.4  The false constant   wfal 1552
            ​*1.2.16  Truth tables   truimtru 1563
                  1.2.16.1  Implication   truimtru 1563
                  1.2.16.2  Negation   nottru 1567
                  1.2.16.3  Equivalence   trubitru 1569
                  1.2.16.4  Conjunction   truantru 1573
                  1.2.16.5  Disjunction   truortru 1577
                  1.2.16.6  Alternative denial   trunantru 1581
                  1.2.16.7  Exclusive disjunction   truxortru 1585
                  1.2.16.8  Joint denial   trunortru 1589
            ​*1.2.17  Half adder and full adder in propositional calculus   whad 1593
                  1.2.17.1  Full adder: sum   whad 1593
                  1.2.17.2  Full adder: carry   wcad 1606
      ​1.3  Other axiomatizations related to classical propositional calculus
            ​*1.3.1  Minimal implicational calculus   minimp 1621
            ​*1.3.2  Implicational Calculus   impsingle 1627
            1.3.3  Derive the Lukasiewicz axioms from Meredith's sole axiom   meredith 1641
            1.3.4  Derive the standard axioms from the Lukasiewicz axioms   luklem1 1658
            ​*1.3.5  Derive Nicod's axiom from the standard axioms   nic-dfim 1669
            1.3.6  Derive the Lukasiewicz axioms from Nicod's axiom   nic-imp 1675
            1.3.7  Derive Nicod's Axiom from Lukasiewicz's First Sheffer Stroke Axiom   lukshef-ax1 1694
            1.3.8  Derive the Lukasiewicz Axioms from the Tarski-Bernays-Wajsberg Axioms   tbw-bijust 1698
            1.3.9  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's First CO Axiom   merco1 1713
            1.3.10  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's Second CO Axiom   merco2 1736
            1.3.11  Derive the Lukasiewicz axioms from the Russell-Bernays Axioms   rb-bijust 1749
            ​*1.3.12  Stoic logic non-modal portion (Chrysippus of Soli)   mptnan 1768
      ​​*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 1779
                  1.4.1.1  Existential quantifier   wex 1779
                  1.4.1.2  Nonfreeness predicate   wnf 1783
            1.4.2  Rule scheme ax-gen (Generalization)   ax-gen 1795
            1.4.3  Axiom scheme ax-4 (Quantified Implication)   ax-4 1809
                  ​*1.4.3.1  The empty domain of discourse   empty 1906
            1.4.4  Axiom scheme ax-5 (Distinctness) - first use of $d   ax-5 1910
            ​*1.4.5  Equality predicate (continued)   weq 1962
            1.4.6  Axiom scheme ax-6 (Existence)   ax-6 1967
            1.4.7  Axiom scheme ax-7 (Equality)   ax-7 2008
            1.4.8  Define proper substitution   sbjust 2064
            1.4.9  Membership predicate   wcel 2109
            1.4.10  Axiom scheme ax-8 (Left Equality for Binary Predicate)   ax-8 2111
            1.4.11  Axiom scheme ax-9 (Right Equality for Binary Predicate)   ax-9 2119
            ​*1.4.12  Logical redundancy of ax-10 , ax-11 , ax-12 , ax-13   ax6dgen 2129
      ​​*1.5  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes)
            1.5.1  Axiom scheme ax-10 (Quantified Negation)   ax-10 2142
            1.5.2  Axiom scheme ax-11 (Quantifier Commutation)   ax-11 2158
            1.5.3  Axiom scheme ax-12 (Substitution)   ax-12 2178
            1.5.4  Axiom scheme ax-13 (Quantified Equality)   ax-13 2371
      ​1.6  Uniqueness and unique existence
            1.6.1  Uniqueness: the at-most-one quantifier   wmo 2532
            1.6.2  Unique existence: the unique existential quantifier   weu 2562
      ​1.7  Other axiomatizations related to classical predicate calculus
            ​*1.7.1  Aristotelian logic: Assertic syllogisms   barbara 2657
            ​*1.7.2  Intuitionistic logic   axia1 2687
​​*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 2702
            2.1.2  Classes   cab 2708
                  2.1.2.1  Class abstractions   cab 2708
                  ​*2.1.2.2  Class equality   df-cleq 2722
                  2.1.2.3  Class membership   df-clel 2804
                  2.1.2.4  Elementary properties of class abstractions   eqabdv 2863
            2.1.3  Class form not-free predicate   wnfc 2878
            2.1.4  Negated equality and membership   wne 2927
                  2.1.4.1  Negated equality   wne 2927
                  2.1.4.2  Negated membership   wnel 3031
            2.1.5  Restricted quantification   wral 3046
                  2.1.5.1  Restricted universal and existential quantification   wral 3046
                  2.1.5.2  Restricted existential uniqueness and at-most-one quantifier   wreu 3355
                  2.1.5.3  Restricted class abstraction   crab 3411
            2.1.6  The universal class   cvv 3455
            ​*2.1.7  Conditional equality (experimental)   wcdeq 3742
            2.1.8  Russell's Paradox   rru 3758
            2.1.9  Proper substitution of classes for sets   wsbc 3761
            2.1.10  Proper substitution of classes for sets into classes   csb 3870
            2.1.11  Define basic set operations and relations   cdif 3919
            2.1.12  Subclasses and subsets   df-ss 3939
            2.1.13  The difference, union, and intersection of two classes   dfdif3 4088
                  2.1.13.1  The difference of two classes   dfdif3 4088
                  2.1.13.2  The union of two classes   elun 4124
                  2.1.13.3  The intersection of two classes   elini 4170
                  2.1.13.4  The symmetric difference of two classes   csymdif 4223
                  2.1.13.5  Combinations of difference, union, and intersection of two classes   unabs 4236
                  2.1.13.6  Class abstractions with difference, union, and intersection of two classes   unabw 4278
                  2.1.13.7  Restricted uniqueness with difference, union, and intersection   reuun2 4296
            2.1.14  The empty set   c0 4304
            ​*2.1.15  The conditional operator for classes   cif 4496
            ​*2.1.16  The weak deduction theorem for set theory   dedth 4555
            2.1.17  Power classes   cpw 4571
            2.1.18  Unordered and ordered pairs   snjust 4596
            2.1.19  The union of a class   cuni 4879
            2.1.20  The intersection of a class   cint 4918
            2.1.21  Indexed union and intersection   ciun 4963
            2.1.22  Disjointness   wdisj 5082
            2.1.23  Binary relations   wbr 5115
            2.1.24  Ordered-pair class abstractions (class builders)   copab 5177
            2.1.25  Functions in maps-to notation   cmpt 5196
            2.1.26  Transitive classes   wtr 5222
      ​2.2  ZF Set Theory - add the Axiom of Replacement
            2.2.1  Introduce the Axiom of Replacement   ax-rep 5242
            2.2.2  Derive the Axiom of Separation   axsepgfromrep 5257
            2.2.3  Derive the Null Set Axiom   axnulALT 5267
            2.2.4  Theorems requiring subset and intersection existence   nalset 5276
            2.2.5  Theorems requiring empty set existence   class2set 5318
      ​2.3  ZF Set Theory - add the Axiom of Power Sets
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 5328
            2.3.2  Derive the Axiom of Pairing   axprlem1 5386
            2.3.3  Ordered pair theorem   opnz 5441
            2.3.4  Ordered-pair class abstractions (cont.)   opabidw 5492
            2.3.5  Power class of union and intersection   pwin 5537
            2.3.6  The identity relation   cid 5540
            2.3.7  The membership relation (or epsilon relation)   cep 5545
            ​*2.3.8  Partial and total orderings   wpo 5552
            2.3.9  Founded and well-ordering relations   wfr 5596
            2.3.10  Relations   cxp 5644
            2.3.11  The Predecessor Class   cpred 6281
            2.3.12  Well-founded induction (variant)   frpomin 6321
            2.3.13  Well-ordered induction   tz6.26 6328
            2.3.14  Ordinals   word 6339
            2.3.15  Definite description binder (inverted iota)   cio 6470
            2.3.16  Functions   wfun 6513
            2.3.17  Cantor's Theorem   canth 7348
            2.3.18  Restricted iota (description binder)   crio 7350
            2.3.19  Operations   co 7394
                  2.3.19.1  Variable-to-class conversion for operations   caovclg 7588
            2.3.20  Maps-to notation   mpondm0 7636
            2.3.21  Function operation   cof 7658
            2.3.22  Proper subset relation   crpss 7705
      ​2.4  ZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 7718
            2.4.2  Ordinals (continued)   epweon 7758
            2.4.3  Transfinite induction   tfi 7837
            2.4.4  The natural numbers (i.e., finite ordinals)   com 7850
            2.4.5  Peano's postulates   peano1 7873
            2.4.6  Finite induction (for finite ordinals)   find 7880
            2.4.7  Relations and functions (cont.)   dmexg 7886
            2.4.8  First and second members of an ordered pair   c1st 7975
            2.4.9  Induction on Cartesian products   frpoins3xpg 8128
            2.4.10  Ordering on Cartesian products   xpord2lem 8130
            2.4.11  Ordering Ordinal Sequences   orderseqlem 8145
            ​*2.4.12  The support of functions   csupp 8148
            ​*2.4.13  Special maps-to operations   opeliunxp2f 8198
            2.4.14  Function transposition   ctpos 8213
            2.4.15  Curry and uncurry   ccur 8253
            2.4.16  Undefined values   cund 8260
            2.4.17  Well-founded recursion   cfrecs 8268
            2.4.18  Well-ordered recursion   cwrecs 8299
            2.4.19  Functions on ordinals; strictly monotone ordinal functions   iunon 8317
            2.4.20  "Strong" transfinite recursion   crecs 8348
            2.4.21  Recursive definition generator   crdg 8386
            2.4.22  Finite recursion   frfnom 8412
            2.4.23  Ordinal arithmetic   c1o 8436
            2.4.24  Natural number arithmetic   nna0 8579
            2.4.25  Natural addition   cnadd 8640
            2.4.26  Equivalence relations and classes   wer 8679
            2.4.27  The mapping operation   cmap 8803
            2.4.28  Infinite Cartesian products   cixp 8874
            2.4.29  Equinumerosity   cen 8919
            2.4.30  Schroeder-Bernstein Theorem   sbthlem1 9060
            2.4.31  Equinumerosity (cont.)   xpf1o 9116
            2.4.32  Finite sets   dif1enlem 9133
            2.4.33  Pigeonhole Principle   phplem1 9181
            2.4.34  Finite sets (cont.)   onomeneq 9194
            2.4.35  Finitely supported functions   cfsupp 9330
            2.4.36  Finite intersections   cfi 9379
            2.4.37  Hall's marriage theorem   marypha1lem 9402
            2.4.38  Supremum and infimum   csup 9409
            2.4.39  Ordinal isomorphism, Hartogs's theorem   coi 9480
            2.4.40  Hartogs function   char 9527
            2.4.41  Weak dominance   cwdom 9535
      ​2.5  ZF Set Theory - add the Axiom of Regularity
            2.5.1  Introduce the Axiom of Regularity   ax-reg 9563
            2.5.2  Axiom of Infinity equivalents   inf0 9592
      ​2.6  ZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-inf 9609
            2.6.2  Existence of omega (the set of natural numbers)   omex 9614
            2.6.3  Cantor normal form   ccnf 9632
            2.6.4  Transitive closure of a relation   cttrcl 9678
            2.6.5  Transitive closure   trcl 9699
            2.6.6  Well-Founded Induction   frmin 9720
            2.6.7  Well-Founded Recursion   frr3g 9727
            2.6.8  Rank   cr1 9733
            2.6.9  Scott's trick; collection principle; Hilbert's epsilon   scottex 9856
            2.6.10  Disjoint union   cdju 9869
            2.6.11  Cardinal numbers   ccrd 9906
            2.6.12  Axiom of Choice equivalents   wac 10086
            ​*2.6.13  Cardinal number arithmetic   undjudom 10139
            2.6.14  The Ackermann bijection   ackbij2lem1 10189
            2.6.15  Cofinality (without Axiom of Choice)   cflem 10216
            2.6.16  Eight inequivalent definitions of finite set   sornom 10248
            2.6.17  Hereditarily size-limited sets without Choice   itunifval 10387
​​*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 10406
            3.1.2  Introduce the Axiom of Dependent Choice   ax-dc 10417
      ​3.2  ZFC Set Theory - add the Axiom of Choice
            3.2.1  Introduce the Axiom of Choice   ax-ac 10430
            3.2.2  AC equivalents: well-ordering, Zorn's lemma   numthcor 10465
            3.2.3  Cardinal number theorems using Axiom of Choice   cardval 10517
            3.2.4  Cardinal number arithmetic using Axiom of Choice   iunctb 10545
            3.2.5  Cofinality using the Axiom of Choice   alephreg 10553
      ​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 10591
            3.4.2  Derivation of the Axiom of Choice   gchaclem 10649
​​*PART 4  TG (TARSKI-GROTHENDIECK) SET THEORY
      ​4.1  Inaccessibles
            4.1.1  Weakly and strongly inaccessible cardinals   cwina 10653
            4.1.2  Weak universes   cwun 10671
            4.1.3  Tarski classes   ctsk 10719
            4.1.4  Grothendieck universes   cgru 10761
      ​4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
            4.2.1  Introduce the Tarski-Grothendieck Axiom   ax-groth 10794
            4.2.2  Derive the Power Set, Infinity and Choice Axioms   grothpw 10797
            4.2.3  Tarski map function   ctskm 10808
​​*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 10815
            5.1.2  Final derivation of real and complex number postulates   axaddf 11116
            5.1.3  Real and complex number postulates restated as axioms   ax-cnex 11142
      ​5.2  Derive the basic properties from the field axioms
            5.2.1  Some deductions from the field axioms for complex numbers   cnex 11167
            5.2.2  Infinity and the extended real number system   cpnf 11223
            5.2.3  Restate the ordering postulates with extended real "less than"   axlttri 11263
            5.2.4  Ordering on reals   lttr 11268
            5.2.5  Initial properties of the complex numbers   mul12 11357
      ​5.3  Real and complex numbers - basic operations
            5.3.1  Addition   add12 11410
            5.3.2  Subtraction   cmin 11423
            5.3.3  Multiplication   kcnktkm1cn 11625
            5.3.4  Ordering on reals (cont.)   gt0ne0 11659
            5.3.5  Reciprocals   ixi 11823
            5.3.6  Division   cdiv 11851
            5.3.7  Ordering on reals (cont.)   elimgt0 12036
            5.3.8  Completeness Axiom and Suprema   fimaxre 12143
            5.3.9  Imaginary and complex number properties   inelr 12187
            5.3.10  Function operation analogue theorems   ofsubeq0 12194
      ​5.4  Integer sets
            5.4.1  Positive integers (as a subset of complex numbers)   cn 12197
            5.4.2  Principle of mathematical induction   nnind 12215
            ​*5.4.3  Decimal representation of numbers   c2 12252
            ​*5.4.4  Some properties of specific numbers   neg1cn 12311
            5.4.5  Simple number properties   halfcl 12424
            5.4.6  The Archimedean property   nnunb 12454
            5.4.7  Nonnegative integers (as a subset of complex numbers)   cn0 12458
            ​*5.4.8  Extended nonnegative integers   cxnn0 12531
            5.4.9  Integers (as a subset of complex numbers)   cz 12545
            5.4.10  Decimal arithmetic   cdc 12665
            5.4.11  Upper sets of integers   cuz 12809
            5.4.12  Well-ordering principle for bounded-below sets of integers   uzwo3 12916
            5.4.13  Rational numbers (as a subset of complex numbers)   cq 12921
            5.4.14  Existence of the set of complex numbers   rpnnen1lem2 12950
      ​5.5  Order sets
            5.5.1  Positive reals (as a subset of complex numbers)   crp 12965
            5.5.2  Infinity and the extended real number system (cont.)   cxne 13082
            5.5.3  Supremum and infimum on the extended reals   xrsupexmnf 13278
            5.5.4  Real number intervals   cioo 13319
            5.5.5  Finite intervals of integers   cfz 13481
            ​*5.5.6  Finite intervals of nonnegative integers   elfz2nn0 13592
            5.5.7  Half-open integer ranges   cfzo 13628
      ​5.6  Elementary integer functions
            5.6.1  The floor and ceiling functions   cfl 13764
            5.6.2  The modulo (remainder) operation   cmo 13843
            5.6.3  Miscellaneous theorems about integers   om2uz0i 13922
            5.6.4  Strong induction over upper sets of integers   uzsinds 13962
            5.6.5  Finitely supported functions over the nonnegative integers   fsuppmapnn0fiublem 13965
            5.6.6  The infinite sequence builder "seq" - extension   cseq 13976
            5.6.7  Integer powers   cexp 14036
            5.6.8  Ordered pair theorem for nonnegative integers   nn0le2msqi 14242
            5.6.9  Factorial function   cfa 14248
            5.6.10  The binomial coefficient operation   cbc 14277
            5.6.11  The ` # ` (set size) function   chash 14305
                  5.6.11.1  Proper unordered pairs and triples (sets of size 2 and 3)   hashprlei 14443
                  5.6.11.2  Functions with a domain containing at least two different elements   fundmge2nop0 14477
                  5.6.11.3  Finite induction on the size of the first component of a binary relation   hashdifsnp1 14481
      ​​*5.7  Words over a set
            5.7.1  Definitions and basic theorems   cword 14488
            5.7.2  Last symbol of a word   clsw 14537
            5.7.3  Concatenations of words   cconcat 14545
            5.7.4  Singleton words   cs1 14570
            5.7.5  Concatenations with singleton words   ccatws1cl 14591
            5.7.6  Subwords/substrings   csubstr 14615
            5.7.7  Prefixes of a word   cpfx 14645
            5.7.8  Subwords of subwords   swrdswrdlem 14679
            5.7.9  Subwords and concatenations   pfxcctswrd 14685
            5.7.10  Subwords of concatenations   swrdccatfn 14699
            5.7.11  Splicing words (substring replacement)   csplice 14724
            5.7.12  Reversing words   creverse 14733
            5.7.13  Repeated symbol words   creps 14743
            ​*5.7.14  Cyclical shifts of words   ccsh 14763
            5.7.15  Mapping words by a function   wrdco 14807
            5.7.16  Longer string literals   cs2 14817
      ​​*5.8  Reflexive and transitive closures of relations
            5.8.1  The reflexive and transitive properties of relations   coss12d 14948
            5.8.2  Basic properties of closures   cleq1lem 14958
            5.8.3  Definitions and basic properties of transitive closures   ctcl 14961
            5.8.4  Exponentiation of relations   crelexp 14995
            5.8.5  Reflexive-transitive closure as an indexed union   crtrcl 15031
            ​*5.8.6  Principle of transitive induction.   relexpindlem 15039
      ​5.9  Elementary real and complex functions
            5.9.1  The "shift" operation   cshi 15042
            5.9.2  Signum (sgn or sign) function   csgn 15062
            5.9.3  Real and imaginary parts; conjugate   ccj 15072
            5.9.4  Square root; absolute value   csqrt 15209
      ​5.10  Elementary limits and convergence
            5.10.1  Superior limit (lim sup)   clsp 15443
            5.10.2  Limits   cli 15457
            5.10.3  Finite and infinite sums   csu 15659
            5.10.4  The binomial theorem   binomlem 15802
            5.10.5  The inclusion/exclusion principle   incexclem 15809
            5.10.6  Infinite sums (cont.)   isumshft 15812
            5.10.7  Miscellaneous converging and diverging sequences   divrcnv 15825
            5.10.8  Arithmetic series   arisum 15833
            5.10.9  Geometric series   expcnv 15837
            5.10.10  Ratio test for infinite series convergence   cvgrat 15856
            5.10.11  Mertens' theorem   mertenslem1 15857
            5.10.12  Finite and infinite products   prodf 15860
                  5.10.12.1  Product sequences   prodf 15860
                  5.10.12.2  Non-trivial convergence   ntrivcvg 15870
                  5.10.12.3  Complex products   cprod 15876
                  5.10.12.4  Finite products   fprod 15914
                  5.10.12.5  Infinite products   iprodclim 15971
            5.10.13  Falling and Rising Factorial   cfallfac 15977
            5.10.14  Bernoulli polynomials and sums of k-th powers   cbp 16019
      ​5.11  Elementary trigonometry
            5.11.1  The exponential, sine, and cosine functions   ce 16034
                  5.11.1.1  The circle constant (tau = 2 pi)   ctau 16177
            5.11.2  _e is irrational   eirrlem 16179
      ​5.12  Cardinality of real and complex number subsets
            5.12.1  Countability of integers and rationals   xpnnen 16186
            5.12.2  The reals are uncountable   rpnnen2lem1 16189
​​*PART 6  ELEMENTARY NUMBER THEORY
      ​6.1  Elementary properties of divisibility
            6.1.1  Irrationality of square root of 2   sqrt2irrlem 16223
            6.1.2  Some Number sets are chains of proper subsets   nthruc 16227
            6.1.3  The divides relation   cdvds 16229
            ​*6.1.4  Even and odd numbers   evenelz 16312
            6.1.5  The division algorithm   divalglem0 16369
            6.1.6  Bit sequences   cbits 16395
            6.1.7  The greatest common divisor operator   cgcd 16470
            6.1.8  Bézout's identity   bezoutlem1 16515
            6.1.9  Algorithms   nn0seqcvgd 16546
            6.1.10  Euclid's Algorithm   eucalgval2 16557
            ​*6.1.11  The least common multiple   clcm 16564
            ​*6.1.12  Coprimality and Euclid's lemma   coprmgcdb 16625
            6.1.13  Cancellability of congruences   congr 16640
      ​6.2  Elementary prime number theory
            ​*6.2.1  Elementary properties   cprime 16647
            ​*6.2.2  Coprimality and Euclid's lemma (cont.)   coprm 16687
            6.2.3  Properties of the canonical representation of a rational   cnumer 16709
            6.2.4  Euler's theorem   codz 16739
            6.2.5  Arithmetic modulo a prime number   modprm1div 16774
            6.2.6  Pythagorean Triples   coprimeprodsq 16785
            6.2.7  The prime count function   cpc 16813
            6.2.8  Pocklington's theorem   prmpwdvds 16881
            6.2.9  Infinite primes theorem   unbenlem 16885
            6.2.10  Sum of prime reciprocals   prmreclem1 16893
            6.2.11  Fundamental theorem of arithmetic   1arithlem1 16900
            6.2.12  Lagrange's four-square theorem   cgz 16906
            6.2.13  Van der Waerden's theorem   cvdwa 16942
            6.2.14  Ramsey's theorem   cram 16976
            ​*6.2.15  Primorial function   cprmo 17008
            ​*6.2.16  Prime gaps   prmgaplem1 17026
            6.2.17  Decimal arithmetic (cont.)   dec2dvds 17040
            6.2.18  Cyclical shifts of words (cont.)   cshwsidrepsw 17070
            6.2.19  Specific prime numbers   prmlem0 17082
            6.2.20  Very large primes   1259lem1 17107
​PART 7  BASIC STRUCTURES
      ​7.1  Extensible structures
            ​*7.1.1  Basic definitions   cstr 17122
                  7.1.1.1  Extensible structures as structures with components   cstr 17122
                  7.1.1.2  Substitution of components   csts 17139
                  7.1.1.3  Slots   cslot 17157
                  ​*7.1.1.4  Structure component indices   cnx 17169
                  7.1.1.5  Base sets   cbs 17185
                  7.1.1.6  Base set restrictions   cress 17206
            7.1.2  Slot definitions   cplusg 17226
            7.1.3  Definition of the structure product   crest 17389
            7.1.4  Definition of the structure quotient   cordt 17468
      ​7.2  Moore spaces
            7.2.1  Moore closures   mrcflem 17573
            7.2.2  Independent sets in a Moore system   mrisval 17597
            7.2.3  Algebraic closure systems   isacs 17618
​PART 8  BASIC CATEGORY THEORY
      ​8.1  Categories
            8.1.1  Categories   ccat 17631
            8.1.2  Opposite category   coppc 17678
            8.1.3  Monomorphisms and epimorphisms   cmon 17696
            8.1.4  Sections, inverses, isomorphisms   csect 17712
            ​*8.1.5  Isomorphic objects   ccic 17763
            8.1.6  Subcategories   cssc 17775
            8.1.7  Functors   cfunc 17822
            8.1.8  Full & faithful functors   cful 17872
            8.1.9  Natural transformations and the functor category   cnat 17912
            8.1.10  Initial, terminal and zero objects of a category   cinito 17949
      ​8.2  Arrows (disjointified hom-sets)
            8.2.1  Identity and composition for arrows   cida 18021
      ​8.3  Examples of categories
            8.3.1  The category of sets   csetc 18043
            8.3.2  The category of categories   ccatc 18066
            ​*8.3.3  The category of extensible structures   fncnvimaeqv 18087
      ​8.4  Categorical constructions
            8.4.1  Product of categories   cxpc 18135
            8.4.2  Functor evaluation   cevlf 18176
            8.4.3  Hom functor   chof 18215
​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 18396
            9.5.2  Complete lattices   ccla 18463
            9.5.3  Distributive lattices   cdlat 18485
            9.5.4  Subset order structures   cipo 18492
      ​9.6  Posets, directed sets, and lattices as relations
            ​*9.6.1  Posets and lattices as relations   cps 18529
            9.6.2  Directed sets, nets   cdir 18559
​PART 10  BASIC ALGEBRAIC STRUCTURES
      ​10.1  Monoids
            ​*10.1.1  Magmas   cplusf 18570
            ​*10.1.2  Identity elements   mgmidmo 18593
            ​*10.1.3  Iterated sums in a magma   gsumvalx 18609
            10.1.4  Magma homomorphisms and submagmas   cmgmhm 18623
            ​*10.1.5  Semigroups   csgrp 18651
            ​*10.1.6  Definition and basic properties of monoids   cmnd 18667
            10.1.7  Monoid homomorphisms and submonoids   cmhm 18714
            ​*10.1.8  Iterated sums in a monoid   gsumvallem2 18767
            10.1.9  Free monoids   cfrmd 18780
                  ​*10.1.9.1  Monoid of endofunctions   cefmnd 18801
            10.1.10  Examples and counterexamples for magmas, semigroups and monoids   mgm2nsgrplem1 18851
      ​10.2  Groups
            10.2.1  Definition and basic properties   cgrp 18871
            ​*10.2.2  Group multiple operation   cmg 19005
            10.2.3  Subgroups and Quotient groups   csubg 19058
            ​*10.2.4  Cyclic monoids and groups   cycsubmel 19138
            10.2.5  Elementary theory of group homomorphisms   cghm 19150
            10.2.6  Isomorphisms of groups   cgim 19195
                  10.2.6.1  The first isomorphism theorem of groups   ghmqusnsglem1 19218
            10.2.7  Group actions   cga 19227
            10.2.8  Centralizers and centers   ccntz 19253
            10.2.9  The opposite group   coppg 19283
            10.2.10  Symmetric groups   csymg 19305
                  ​*10.2.10.1  Definition and basic properties   csymg 19305
                  10.2.10.2  Cayley's theorem   cayleylem1 19348
                  10.2.10.3  Permutations fixing one element   symgfix2 19352
                  ​*10.2.10.4  Transpositions in the symmetric group   cpmtr 19377
                  10.2.10.5  The sign of a permutation   cpsgn 19425
            10.2.11  p-Groups and Sylow groups; Sylow's theorems   cod 19460
            10.2.12  Direct products   clsm 19570
                  10.2.12.1  Direct products (extension)   smndlsmidm 19592
            10.2.13  Free groups   cefg 19642
            10.2.14  Abelian groups   ccmn 19716
                  10.2.14.1  Definition and basic properties   ccmn 19716
                  10.2.14.2  Cyclic groups   ccyg 19813
                  10.2.14.3  Group sum operation   gsumval3a 19839
                  10.2.14.4  Group sums over (ranges of) integers   fsfnn0gsumfsffz 19919
                  10.2.14.5  Internal direct products   cdprd 19931
                  10.2.14.6  The Fundamental Theorem of Abelian Groups   ablfacrplem 20003
            10.2.15  Simple groups   csimpg 20028
                  10.2.15.1  Definition and basic properties   csimpg 20028
                  10.2.15.2  Classification of abelian simple groups   ablsimpnosubgd 20042
      ​10.3  Rings
            10.3.1  Multiplicative Group   cmgp 20055
            ​*10.3.2  Non-unital rings ("rngs")   crng 20067
            ​*10.3.3  Ring unity (multiplicative identity)   cur 20096
            10.3.4  Semirings   csrg 20101
                  ​*10.3.4.1  The binomial theorem for semirings   srgbinomlem1 20141
            10.3.5  Unital rings   crg 20148
            10.3.6  Opposite ring   coppr 20251
            10.3.7  Divisibility   cdsr 20269
            10.3.8  Ring primes   crpm 20347
            10.3.9  Homomorphisms of non-unital rings   crnghm 20349
            10.3.10  Ring homomorphisms   crh 20384
            10.3.11  Nonzero rings and zero rings   cnzr 20427
            10.3.12  Local rings   clring 20453
            10.3.13  Subrings   csubrng 20460
                  10.3.13.1  Subrings of non-unital rings   csubrng 20460
                  10.3.13.2  Subrings of unital rings   csubrg 20484
                  10.3.13.3  Subrings generated by a subset   crgspn 20525
            10.3.14  Categories of rings   crngc 20531
                  ​*10.3.14.1  The category of non-unital rings   crngc 20531
                  ​*10.3.14.2  The category of (unital) rings   cringc 20560
                  10.3.14.3  Subcategories of the category of rings   srhmsubclem1 20592
            10.3.15  Left regular elements and domains   crlreg 20606
      ​10.4  Division rings and fields
            10.4.1  Definition and basic properties   cdr 20644
            10.4.2  Sub-division rings   csdrg 20701
            10.4.3  Absolute value (abstract algebra)   cabv 20723
            10.4.4  Star rings   cstf 20752
      ​10.5  Left modules
            10.5.1  Definition and basic properties   clmod 20772
            10.5.2  Subspaces and spans in a left module   clss 20843
            10.5.3  Homomorphisms and isomorphisms of left modules   clmhm 20932
            10.5.4  Subspace sum; bases for a left module   clbs 20987
      ​10.6  Vector spaces
            10.6.1  Definition and basic properties   clvec 21015
      ​10.7  Subring algebras and ideals
            10.7.1  Subring algebras   csra 21084
            ​*10.7.2  Left ideals and spans   clidl 21122
            10.7.3  Two-sided ideals and quotient rings   c2idl 21165
                  ​*10.7.3.1  Condition for a non-unital ring to be unital   rngqiprng1elbas 21202
            10.7.4  Principal ideal rings. Divisibility in the integers   clpidl 21236
            10.7.5  Principal ideal domains   cpid 21252
      ​10.8  The complex numbers as an algebraic extensible structure
            10.8.1  Definition and basic properties   cpsmet 21254
            ​*10.8.2  Ring of integers   czring 21362
                  ​*10.8.2.1  Example for a condition for a non-unital ring to be unital   pzriprnglem1 21397
            10.8.3  Algebraic constructions based on the complex numbers   czrh 21415
            10.8.4  Signs as subgroup of the complex numbers   cnmsgnsubg 21492
            10.8.5  Embedding of permutation signs into a ring   zrhpsgnmhm 21499
            10.8.6  The ordered field of real numbers   crefld 21519
      ​10.9  Generalized pre-Hilbert and Hilbert spaces
            10.9.1  Definition and basic properties   cphl 21539
            10.9.2  Orthocomplements and closed subspaces   cocv 21575
            10.9.3  Orthogonal projection and orthonormal bases   cpj 21615
​​*PART 11  BASIC LINEAR ALGEBRA
      ​11.1  Vectors and free modules
            *11.1.1  Direct sum of left modules   cdsmm 21646
            ​*11.1.2  Free modules   cfrlm 21661
            ​*11.1.3  Standard basis (unit vectors)   cuvc 21697
            ​*11.1.4  Independent sets and families   clindf 21719
            11.1.5  Characterization of free modules   lmimlbs 21751
      ​11.2  Associative algebras
            11.2.1  Definition and basic properties   casa 21765
      ​11.3  Abstract multivariate polynomials
            11.3.1  Definition and basic properties   cmps 21819
            11.3.2  Polynomial evaluation   ces 21985
            11.3.3  Additional definitions for (multivariate) polynomials   cslv 22021
            ​*11.3.4  Univariate polynomials   cps1 22065
            11.3.5  Univariate polynomial evaluation   ces1 22206
                  11.3.5.1  Specialization of polynomial evaluation as a ring homomorphism   evls1scafv 22259
      ​​*11.4  Matrices
            *11.4.1  The matrix multiplication   cmmul 22283
            ​*11.4.2  Square matrices   cmat 22300
            ​*11.4.3  The matrix algebra   matmulr 22331
            ​*11.4.4  Matrices of dimension 0 and 1   mat0dimbas0 22359
            ​*11.4.5  The subalgebras of diagonal and scalar matrices   cdmat 22381
            ​*11.4.6  Multiplication of a matrix with a "column vector"   cmvmul 22433
            11.4.7  Replacement functions for a square matrix   cmarrep 22449
            11.4.8  Submatrices   csubma 22469
      ​11.5  The determinant
            11.5.1  Definition and basic properties   cmdat 22477
            11.5.2  Determinants of 2 x 2 -matrices   m2detleiblem1 22517
            11.5.3  The matrix adjugate/adjunct   cmadu 22525
            ​*11.5.4  Laplace expansion of determinants (special case)   symgmatr01lem 22546
            11.5.5  Inverse matrix   invrvald 22569
            ​*11.5.6  Cramer's rule   slesolvec 22572
      ​​*11.6  Polynomial matrices
            11.6.1  Basic properties   pmatring 22585
            ​*11.6.2  Constant polynomial matrices   ccpmat 22596
            ​*11.6.3  Collecting coefficients of polynomial matrices   cdecpmat 22655
            ​*11.6.4  Ring isomorphism between polynomial matrices and polynomials over matrices   cpm2mp 22685
      ​​*11.7  The characteristic polynomial
            *11.7.1  Definition and basic properties   cchpmat 22719
            ​*11.7.2  The characteristic factor function G   fvmptnn04if 22742
            ​*11.7.3  The Cayley-Hamilton theorem   cpmadurid 22760
​PART 12  BASIC TOPOLOGY
      ​12.1  Topology
            ​*12.1.1  Topological spaces   ctop 22786
                  12.1.1.1  Topologies   ctop 22786
                  12.1.1.2  Topologies on sets   ctopon 22803
                  12.1.1.3  Topological spaces   ctps 22825
            12.1.2  Topological bases   ctb 22838
            12.1.3  Examples of topologies   distop 22888
            12.1.4  Closure and interior   ccld 22909
            12.1.5  Neighborhoods   cnei 22990
            12.1.6  Limit points and perfect sets   clp 23027
            12.1.7  Subspace topologies   restrcl 23050
            12.1.8  Order topology   ordtbaslem 23081
            12.1.9  Limits and continuity in topological spaces   ccn 23117
            12.1.10  Separated spaces: T0, T1, T2 (Hausdorff) ...   ct0 23199
            12.1.11  Compactness   ccmp 23279
            12.1.12  Bolzano-Weierstrass theorem   bwth 23303
            12.1.13  Connectedness   cconn 23304
            12.1.14  First- and second-countability   c1stc 23330
            12.1.15  Local topological properties   clly 23357
            12.1.16  Refinements   cref 23395
            12.1.17  Compactly generated spaces   ckgen 23426
            12.1.18  Product topologies   ctx 23453
            12.1.19  Continuous function-builders   cnmptid 23554
            12.1.20  Quotient maps and quotient topology   ckq 23586
            12.1.21  Homeomorphisms   chmeo 23646
      ​12.2  Filters and filter bases
            12.2.1  Filter bases   elmptrab 23720
            12.2.2  Filters   cfil 23738
            12.2.3  Ultrafilters   cufil 23792
            12.2.4  Filter limits   cfm 23826
            12.2.5  Extension by continuity   ccnext 23952
            12.2.6  Topological groups   ctmd 23963
            12.2.7  Infinite group sum on topological groups   ctsu 24019
            12.2.8  Topological rings, fields, vector spaces   ctrg 24049
      ​12.3  Uniform Structures and Spaces
            12.3.1  Uniform structures   cust 24093
            12.3.2  The topology induced by an uniform structure   cutop 24124
            12.3.3  Uniform Spaces   cuss 24147
            12.3.4  Uniform continuity   cucn 24168
            12.3.5  Cauchy filters in uniform spaces   ccfilu 24179
            12.3.6  Complete uniform spaces   ccusp 24190
      ​12.4  Metric spaces
            12.4.1  Pseudometric spaces   ispsmet 24198
            12.4.2  Basic metric space properties   cxms 24211
            12.4.3  Metric space balls   blfvalps 24277
            12.4.4  Open sets of a metric space   mopnval 24332
            12.4.5  Continuity in metric spaces   metcnp3 24434
            12.4.6  The uniform structure generated by a metric   metuval 24443
            12.4.7  Examples of metric spaces   dscmet 24466
            ​*12.4.8  Normed algebraic structures   cnm 24470
            12.4.9  Normed space homomorphisms (bounded linear operators)   cnmo 24599
            12.4.10  Topology on the reals   qtopbaslem 24652
            12.4.11  Topological definitions using the reals   cii 24774
            12.4.12  Path homotopy   chtpy 24872
            12.4.13  The fundamental group   cpco 24906
      ​12.5  Metric subcomplex vector spaces
            12.5.1  Subcomplex modules   cclm 24968
            ​*12.5.2  Subcomplex vector spaces   ccvs 25029
            ​*12.5.3  Normed subcomplex vector spaces   isncvsngp 25056
            12.5.4  Subcomplex pre-Hilbert spaces   ccph 25073
            12.5.5  Convergence and completeness   ccfil 25159
            12.5.6  Baire's Category Theorem   bcthlem1 25231
            12.5.7  Banach spaces and subcomplex Hilbert spaces   ccms 25239
                  12.5.7.1  The complete ordered field of the real numbers   retopn 25286
            12.5.8  Euclidean spaces   crrx 25290
            12.5.9  Minimizing Vector Theorem   minveclem1 25331
            12.5.10  Projection Theorem   pjthlem1 25344
​PART 13  BASIC REAL AND COMPLEX ANALYSIS
      ​13.1  Continuity
            13.1.1  Intermediate value theorem   pmltpclem1 25356
      ​13.2  Integrals
            13.2.1  Lebesgue measure   covol 25370
            13.2.2  Lebesgue integration   cmbf 25522
                  13.2.2.1  Lesbesgue integral   cmbf 25522
                  13.2.2.2  Lesbesgue directed integral   cdit 25754
      ​13.3  Derivatives
            13.3.1  Real and complex differentiation   climc 25770
                  13.3.1.1  Derivatives of functions of one complex or real variable   climc 25770
                  13.3.1.2  Results on real differentiation   dvferm1lem 25895
​PART 14  BASIC REAL AND COMPLEX FUNCTIONS
      ​14.1  Polynomials
            14.1.1  Polynomial degrees   cmdg 25965
            14.1.2  The division algorithm for univariate polynomials   cmn1 26038
            14.1.3  Elementary properties of complex polynomials   cply 26096
            14.1.4  The division algorithm for polynomials   cquot 26205
            14.1.5  Algebraic numbers   caa 26229
            14.1.6  Liouville's approximation theorem   aalioulem1 26247
      ​14.2  Sequences and series
            14.2.1  Taylor polynomials and Taylor's theorem   ctayl 26267
            14.2.2  Uniform convergence   culm 26292
            14.2.3  Power series   pserval 26326
      ​14.3  Basic trigonometry
            14.3.1  The exponential, sine, and cosine functions (cont.)   efcn 26360
            14.3.2  Properties of pi = 3.14159...   pilem1 26368
            14.3.3  Mapping of the exponential function   efgh 26457
            14.3.4  The natural logarithm on complex numbers   clog 26470
            ​*14.3.5  Logarithms to an arbitrary base   clogb 26681
            14.3.6  Theorems of Pythagoras, isosceles triangles, and intersecting chords   angval 26718
            14.3.7  Solutions of quadratic, cubic, and quartic equations   quad2 26756
            14.3.8  Inverse trigonometric functions   casin 26779
            14.3.9  The Birthday Problem   log2ublem1 26863
            14.3.10  Areas in R^2   carea 26872
            14.3.11  More miscellaneous converging sequences   rlimcnp 26882
            14.3.12  Inequality of arithmetic and geometric means   cvxcl 26902
            14.3.13  Euler-Mascheroni constant   cem 26909
            14.3.14  Zeta function   czeta 26930
            14.3.15  Gamma function   clgam 26933
      ​14.4  Basic number theory
            14.4.1  Wilson's theorem   wilthlem1 26985
            14.4.2  The Fundamental Theorem of Algebra   ftalem1 26990
            14.4.3  The Basel problem (ζ(2) = π²/6)   basellem1 26998
            14.4.4  Number-theoretical functions   ccht 27008
            14.4.5  Perfect Number Theorem   mersenne 27145
            14.4.6  Characters of Z/nZ   cdchr 27150
            14.4.7  Bertrand's postulate   bcctr 27193
            ​*14.4.8  Quadratic residues and the Legendre symbol   clgs 27212
            ​*14.4.9  Gauss' Lemma   gausslemma2dlem0a 27274
            14.4.10  Quadratic reciprocity   lgseisenlem1 27293
            14.4.11  All primes 4n+1 are the sum of two squares   2sqlem1 27335
            14.4.12  Chebyshev's Weak Prime Number Theorem, Dirichlet's Theorem   chebbnd1lem1 27387
            14.4.13  The Prime Number Theorem   mudivsum 27448
            14.4.14  Ostrowski's theorem   abvcxp 27533
​PART 15  SURREAL NUMBERS
      ​​*15.1  Sign sequence representation and Alling's axioms
            15.1.1  Definitions and initial properties   csur 27558
            15.1.2  Ordering   sltsolem1 27594
            15.1.3  Birthday Function   bdayfo 27596
            15.1.4  Density   fvnobday 27597
            ​*15.1.5  Full-Eta Property   bdayimaon 27612
      ​15.2  Initial consequences of Alling's axioms
            15.2.1  Ordering Theorems   csle 27663
            15.2.2  Birthday Theorems   bdayfun 27691
      ​​*15.3  Conway cut representation
            15.3.1  Conway cuts   csslt 27699
            15.3.2  Zero and One   c0s 27741
            15.3.3  Cuts and Options   cmade 27757
            15.3.4  Cofinality and coinitiality   cofsslt 27833
      ​15.4  Induction and recursion
            15.4.1  Induction and recursion on one variable   cnorec 27851
            15.4.2  Induction and recursion on two variables   cnorec2 27862
      ​15.5  Surreal arithmetic
            15.5.1  Addition   cadds 27873
            15.5.2  Negation and Subtraction   cnegs 27932
            15.5.3  Multiplication   cmuls 28016
            15.5.4  Division   cdivs 28097
            15.5.5  Absolute value   cabss 28146
      ​15.6  Subsystems of surreals
            15.6.1  Ordinal numbers   cons 28159
            15.6.2  Surreal recursive sequences   cseqs 28184
            15.6.3  Natural numbers   cnn0s 28213
            15.6.4  Integers   czs 28273
            15.6.5  Dyadic fractions   c2s 28303
            15.6.6  Real numbers   creno 28351
​​*PART 16  ELEMENTARY GEOMETRY
      ​16.1  Definition and Tarski's Axioms of Geometry
            16.1.1  Justification for the congruence notation   tgjustf 28407
      ​16.2  Tarskian Geometry
            16.2.1  Congruence   tgcgrcomimp 28411
            16.2.2  Betweenness   tgbtwntriv2 28421
            16.2.3  Dimension   tglowdim1 28434
            16.2.4  Betweenness and Congruence   tgifscgr 28442
            16.2.5  Congruence of a series of points   ccgrg 28444
            16.2.6  Motions   cismt 28466
            16.2.7  Colinearity   tglng 28480
            16.2.8  Connectivity of betweenness   tgbtwnconn1lem1 28506
            16.2.9  Less-than relation in geometric congruences   cleg 28516
            16.2.10  Rays   chlg 28534
            16.2.11  Lines   btwnlng1 28553
            16.2.12  Point inversions   cmir 28586
            16.2.13  Right angles   crag 28627
            16.2.14  Half-planes   islnopp 28673
            16.2.15  Midpoints and Line Mirroring   cmid 28706
            16.2.16  Congruence of angles   ccgra 28741
            16.2.17  Angle Comparisons   cinag 28769
            16.2.18  Congruence Theorems   tgsas1 28788
            16.2.19  Equilateral triangles   ceqlg 28799
      ​16.3  Properties of geometries
            16.3.1  Isomorphisms between geometries   f1otrgds 28803
      ​16.4  Geometry in Hilbert spaces
            16.4.1  Geometry in the complex plane   cchhllem 28821
            16.4.2  Geometry in Euclidean spaces   cee 28822
                  16.4.2.1  Definition of the Euclidean space   cee 28822
                  16.4.2.2  Tarski's axioms for geometry for the Euclidean space   axdimuniq 28847
                  16.4.2.3  EE^n fulfills Tarski's Axioms   ceeng 28911
​​*PART 17  GRAPH THEORY
      ​*17.1  Vertices and edges
            17.1.1  The edge function extractor for extensible structures   cedgf 28922
            ​*17.1.2  Vertices and indexed edges   cvtx 28930
                  17.1.2.1  Definitions and basic properties   cvtx 28930
                  17.1.2.2  The vertices and edges of a graph represented as ordered pair   opvtxval 28937
                  17.1.2.3  The vertices and edges of a graph represented as extensible structure   funvtxdmge2val 28945
                  17.1.2.4  Representations of graphs without edges   snstrvtxval 28971
                  17.1.2.5  Degenerated cases of representations of graphs   vtxval0 28973
            17.1.3  Edges as range of the edge function   cedg 28981
      ​​*17.2  Undirected graphs
            17.2.1  Undirected hypergraphs   cuhgr 28990
            17.2.2  Undirected pseudographs and multigraphs   cupgr 29014
            ​*17.2.3  Loop-free graphs   umgrislfupgrlem 29056
            17.2.4  Edges as subsets of vertices of graphs   uhgredgiedgb 29060
            ​*17.2.5  Undirected simple graphs   cuspgr 29082
            17.2.6  Examples for graphs   usgr0e 29170
            17.2.7  Subgraphs   csubgr 29201
            17.2.8  Finite undirected simple graphs   cfusgr 29250
            17.2.9  Neighbors, complete graphs and universal vertices   cnbgr 29266
                  17.2.9.1  Neighbors   cnbgr 29266
                  17.2.9.2  Universal vertices   cuvtx 29319
                  17.2.9.3  Complete graphs   ccplgr 29343
            17.2.10  Vertex degree   cvtxdg 29400
            ​*17.2.11  Regular graphs   crgr 29490
      ​​*17.3  Walks, paths and cycles
            *17.3.1  Walks   cewlks 29530
            17.3.2  Walks for loop-free graphs   lfgrwlkprop 29622
            17.3.3  Trails   ctrls 29625
            17.3.4  Paths and simple paths   cpths 29647
            17.3.5  Closed walks   cclwlks 29707
            17.3.6  Circuits and cycles   ccrcts 29721
            ​*17.3.7  Walks as words   cwwlks 29762
            17.3.8  Walks/paths of length 2 (as length 3 strings)   2wlkdlem1 29862
            17.3.9  Walks in regular graphs   rusgrnumwwlkl1 29905
            ​*17.3.10  Closed walks as words   cclwwlk 29917
                  17.3.10.1  Closed walks as words   cclwwlk 29917
                  17.3.10.2  Closed walks of a fixed length as words   cclwwlkn 29960
                  17.3.10.3  Closed walks on a vertex of a fixed length as words   cclwwlknon 30023
            17.3.11  Examples for walks, trails and paths   0ewlk 30050
            17.3.12  Connected graphs   cconngr 30122
      ​17.4  Eulerian paths and the Konigsberg Bridge problem
            ​*17.4.1  Eulerian paths   ceupth 30133
            ​*17.4.2  The Königsberg Bridge problem   konigsbergvtx 30182
      ​17.5  The Friendship Theorem
            17.5.1  Friendship graphs - basics   cfrgr 30194
            17.5.2  The friendship theorem for small graphs   frgr1v 30207
            17.5.3  Theorems according to Mertzios and Unger   2pthfrgrrn 30218
            ​*17.5.4  Huneke's Proof of the Friendship Theorem   frgrncvvdeqlem1 30235
​PART 18  GUIDES AND MISCELLANEA
      ​18.1  Guides (conventions, explanations, and examples)
            ​*18.1.1  Conventions   conventions 30336
            18.1.2  Natural deduction   natded 30339
            ​*18.1.3  Natural deduction examples   ex-natded5.2 30340
            18.1.4  Definitional examples   ex-or 30357
            18.1.5  Other examples   aevdemo 30396
      ​18.2  Humor
            18.2.1  April Fool's theorem   avril1 30399
      ​18.3  (Future - to be reviewed and classified)
            18.3.1  Planar incidence geometry   cplig 30410
            ​*18.3.2  Aliases kept to prevent broken links   dummylink 30423
​​*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 30425
            19.1.2  Abelian groups   cablo 30480
      ​19.2  Complex vector spaces
            19.2.1  Definition and basic properties   cvc 30494
            19.2.2  Examples of complex vector spaces   cnaddabloOLD 30517
      ​19.3  Normed complex vector spaces
            19.3.1  Definition and basic properties   cnv 30520
            19.3.2  Examples of normed complex vector spaces   cnnv 30613
            19.3.3  Induced metric of a normed complex vector space   imsval 30621
            19.3.4  Inner product   cdip 30636
            19.3.5  Subspaces   css 30657
      ​19.4  Operators on complex vector spaces
            19.4.1  Definitions and basic properties   clno 30676
      ​19.5  Inner product (pre-Hilbert) spaces
            19.5.1  Definition and basic properties   ccphlo 30748
            19.5.2  Examples of pre-Hilbert spaces   cncph 30755
            19.5.3  Properties of pre-Hilbert spaces   isph 30758
      ​19.6  Complex Banach spaces
            19.6.1  Definition and basic properties   ccbn 30798
            19.6.2  Examples of complex Banach spaces   cnbn 30805
            19.6.3  Uniform Boundedness Theorem   ubthlem1 30806
            19.6.4  Minimizing Vector Theorem   minvecolem1 30810
      ​19.7  Complex Hilbert spaces
            19.7.1  Definition and basic properties   chlo 30821
            19.7.2  Standard axioms for a complex Hilbert space   hlex 30834
            19.7.3  Examples of complex Hilbert spaces   cnchl 30852
            19.7.4  Hellinger-Toeplitz Theorem   htthlem 30853
​​*PART 20  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      ​20.1  Axiomatization of complex pre-Hilbert spaces
            20.1.1  Basic Hilbert space definitions   chba 30855
            20.1.2  Preliminary ZFC lemmas   df-hnorm 30904
            ​*20.1.3  Derive the Hilbert space axioms from ZFC set theory   axhilex-zf 30917
            ​*20.1.4  Introduce the vector space axioms for a Hilbert space   ax-hilex 30935
            20.1.5  Vector operations   hvmulex 30947
            20.1.6  Inner product postulates for a Hilbert space   ax-hfi 31015
      ​20.2  Inner product and norms
            20.2.1  Inner product   his5 31022
            20.2.2  Norms   dfhnorm2 31058
            20.2.3  Relate Hilbert space to normed complex vector spaces   hilablo 31096
            20.2.4  Bunjakovaskij-Cauchy-Schwarz inequality   bcsiALT 31115
      ​20.3  Cauchy sequences and completeness axiom
            20.3.1  Cauchy sequences and limits   hcau 31120
            20.3.2  Derivation of the completeness axiom from ZF set theory   hilmet 31130
            20.3.3  Completeness postulate for a Hilbert space   ax-hcompl 31138
            20.3.4  Relate Hilbert space to ZFC pre-Hilbert and Hilbert spaces   hhcms 31139
      ​20.4  Subspaces and projections
            20.4.1  Subspaces   df-sh 31143
            20.4.2  Closed subspaces   df-ch 31157
            20.4.3  Orthocomplements   df-oc 31188
            20.4.4  Subspace sum, span, lattice join, lattice supremum   df-shs 31244
            20.4.5  Projection theorem   pjhthlem1 31327
            20.4.6  Projectors   df-pjh 31331
      ​20.5  Properties of Hilbert subspaces
            20.5.1  Orthomodular law   omlsilem 31338
            20.5.2  Projectors (cont.)   pjhtheu2 31352
            20.5.3  Hilbert lattice operations   sh0le 31376
            20.5.4  Span (cont.) and one-dimensional subspaces   spansn0 31477
            20.5.5  Commutes relation for Hilbert lattice elements   df-cm 31519
            20.5.6  Foulis-Holland theorem   fh1 31554
            20.5.7  Quantum Logic Explorer axioms   qlax1i 31563
            20.5.8  Orthogonal subspaces   chscllem1 31573
            20.5.9  Orthoarguesian laws 5OA and 3OA   5oalem1 31590
            20.5.10  Projectors (cont.)   pjorthi 31605
            20.5.11  Mayet's equation E_3   mayete3i 31664
      ​20.6  Operators on Hilbert spaces
            ​*20.6.1  Operator sum, difference, and scalar multiplication   df-hosum 31666
            20.6.2  Zero and identity operators   df-h0op 31684
            20.6.3  Operations on Hilbert space operators   hoaddcl 31694
            20.6.4  Linear, continuous, bounded, Hermitian, unitary operators and norms   df-nmop 31775
            20.6.5  Linear and continuous functionals and norms   df-nmfn 31781
            20.6.6  Adjoint   df-adjh 31785
            20.6.7  Dirac bra-ket notation   df-bra 31786
            20.6.8  Positive operators   df-leop 31788
            20.6.9  Eigenvectors, eigenvalues, spectrum   df-eigvec 31789
            20.6.10  Theorems about operators and functionals   nmopval 31792
            20.6.11  Riesz lemma   riesz3i 31998
            20.6.12  Adjoints (cont.)   cnlnadjlem1 32003
            20.6.13  Quantum computation error bound theorem   unierri 32040
            20.6.14  Dirac bra-ket notation (cont.)   branmfn 32041
            20.6.15  Positive operators (cont.)   leopg 32058
            20.6.16  Projectors as operators   pjhmopi 32082
      ​20.7  States on a Hilbert lattice and Godowski's equation
            20.7.1  States on a Hilbert lattice   df-st 32147
            20.7.2  Godowski's equation   golem1 32207
      ​20.8  Cover relation, atoms, exchange axiom, and modular symmetry
            20.8.1  Covers relation; modular pairs   df-cv 32215
            20.8.2  Atoms   df-at 32274
            20.8.3  Superposition principle   superpos 32290
            20.8.4  Atoms, exchange and covering properties, atomicity   chcv1 32291
            20.8.5  Irreducibility   chirredlem1 32326
            20.8.6  Atoms (cont.)   atcvat3i 32332
            20.8.7  Modular symmetry   mdsymlem1 32339
​PART 21  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      ​21.1  Mathboxes for user contributions
            21.1.1  Mathbox guidelines   mathbox 32378
      ​21.2  Mathbox for Stefan Allan
      ​21.3  Mathbox for Thierry Arnoux
            21.3.1  Propositional Calculus - misc additions   ad11antr 32383
            21.3.2  Predicate Calculus   sbc2iedf 32401
                  21.3.2.1  Predicate Calculus - misc additions   sbc2iedf 32401
                  21.3.2.2  Restricted quantification - misc additions   ralcom4f 32403
                  21.3.2.3  Equality   eqtrb 32410
                  21.3.2.4  Double restricted existential uniqueness quantification   opsbc2ie 32412
                  21.3.2.5  Double restricted existential uniqueness quantification syntax   w2reu 32414
                  21.3.2.6  Substitution (without distinct variables) - misc additions   sbceqbidf 32423
                  21.3.2.7  Existential "at most one" - misc additions   mo5f 32425
                  21.3.2.8  Existential uniqueness - misc additions   reuxfrdf 32427
                  21.3.2.9  Restricted "at most one" - misc additions   rmoxfrd 32429
                  21.3.2.10  Restricted iota (description binder)   riotaeqbidva 32432
            21.3.3  General Set Theory   dmrab 32433
                  21.3.3.1  Class abstractions (a.k.a. class builders)   dmrab 32433
                  21.3.3.2  Image Sets   abrexdomjm 32443
                  21.3.3.3  Set relations and operations - misc additions   nelun 32449
                  21.3.3.4  Unordered pairs   elpreq 32464
                  21.3.3.5  Unordered triples   tpssg 32473
                  21.3.3.6  Conditional operator - misc additions   ifeqeqx 32478
                  21.3.3.7  Set union   uniinn0 32486
                  21.3.3.8  Indexed union - misc additions   cbviunf 32491
                  21.3.3.9  Indexed intersection - misc additions   iinabrex 32505
                  21.3.3.10  Disjointness - misc additions   disjnf 32506
            21.3.4  Relations and Functions   xpdisjres 32534
                  21.3.4.1  Relations - misc additions   xpdisjres 32534
                  21.3.4.2  Functions - misc additions   feq2dd 32555
                  21.3.4.3  Operations - misc additions   mpomptxf 32609
                  21.3.4.4  The mapping operation   elmaprd 32611
                  21.3.4.5  Support of a function   suppovss 32612
                  21.3.4.6  Explicit Functions with one or two points as a domain   cosnopne 32625
                  21.3.4.7  Isomorphisms - misc. additions   gtiso 32632
                  21.3.4.8  Disjointness (additional proof requiring functions)   disjdsct 32634
                  21.3.4.9  First and second members of an ordered pair - misc additions   df1stres 32635
                  21.3.4.10  Finite Sets   imafi2 32643
                  21.3.4.11  Countable Sets   snct 32645
            21.3.5  Real and Complex Numbers   sgnval2 32666
                  21.3.5.1  Complex operations - misc. additions   creq0 32667
                  21.3.5.2  Ordering on reals - misc additions   lt2addrd 32682
                  21.3.5.3  Extended reals - misc additions   xrlelttric 32683
                  21.3.5.4  Extended nonnegative integers - misc additions   xnn0gt0 32700
                  21.3.5.5  Real number intervals - misc additions   joiniooico 32705
                  21.3.5.6  Finite intervals of integers - misc additions   uzssico 32715
                  21.3.5.7  Half-open integer ranges - misc additions   iundisjfi 32727
                  21.3.5.8  The ` # ` (set size) function - misc additions   hashunif 32739
                  21.3.5.9  The greatest common divisor operator - misc. additions   elq2 32744
                  21.3.5.10  Integers   nn0split01 32750
                  21.3.5.11  Decimal numbers   dfdec100 32763
            21.3.6  Real and complex functions   sgncl 32764
                  21.3.6.1  Signum (sgn or sign) function - misc. additions   sgncl 32764
                  21.3.6.2  Integer powers - misc. additions   nexple 32777
                  21.3.6.3  Indicator Functions   cind 32781
            ​*21.3.7  Decimal expansion   cdp2 32799
                  ​*21.3.7.1  Decimal point   cdp 32816
                  21.3.7.2  Division in the extended real number system   cxdiv 32845
            21.3.8  Words over a set - misc additions   wrdres 32864
                  21.3.8.1  Splicing words (substring replacement)   splfv3 32888
                  21.3.8.2  Cyclic shift of words   1cshid 32889
            21.3.9  Extensible Structures   ressplusf 32893
                  21.3.9.1  Structure restriction operator   ressplusf 32893
                  21.3.9.2  The opposite group   oppgle 32896
                  21.3.9.3  Posets   ressprs 32898
                  21.3.9.4  Complete lattices   clatp0cl 32910
                  21.3.9.5  Order Theory   cmnt 32912
                  21.3.9.6  Chains   cchn 32938
                  21.3.9.7  Extended reals Structure - misc additions   ax-xrssca 32950
                  21.3.9.8  The extended nonnegative real numbers commutative monoid   xrge0base 32960
            21.3.10  Algebra   mndcld 32971
                  21.3.10.1  Monoids   mndcld 32971
                  21.3.10.2  Monoids Homomorphisms   abliso 32985
                  21.3.10.3  Groups - misc additions   grpsubcld 32989
                  21.3.10.4  Finitely supported group sums - misc additions   gsumsubg 32994
                  21.3.10.5  Group or monoid sums over words   gsumwun 33013
                  21.3.10.6  Centralizers and centers - misc additions   cntzun 33016
                  21.3.10.7  Totally ordered monoids and groups   comnd 33019
                  21.3.10.8  The symmetric group   symgfcoeu 33047
                  21.3.10.9  Transpositions   pmtridf1o 33059
                  21.3.10.10  Permutation Signs   psgnid 33062
                  21.3.10.11  Permutation cycles   ctocyc 33071
                  21.3.10.12  The Alternating Group   evpmval 33110
                  21.3.10.13  Signum in an ordered monoid   csgns 33123
                  21.3.10.14  Fixed points   cfxp 33128
                  21.3.10.15  The Archimedean property for generic ordered algebraic structures   cinftm 33138
                  21.3.10.16  Semiring left modules   cslmd 33161
                  21.3.10.17  Simple groups   prmsimpcyc 33189
                  21.3.10.18  Rings - misc additions   ringdi22 33190
                  21.3.10.19  Subrings generated by a set   elrgspnlem1 33201
                  21.3.10.20  The zero ring   irrednzr 33209
                  21.3.10.21  Localization of rings   cerl 33212
                  21.3.10.22  Integral Domains   domnmuln0rd 33233
                  21.3.10.23  Euclidean Domains   ceuf 33246
                  21.3.10.24  Division Rings   ringinveu 33252
                  21.3.10.25  The field of rational numbers   qfld 33255
                  21.3.10.26  Subfields   subsdrg 33256
                  21.3.10.27  Field of fractions   cfrac 33260
                  21.3.10.28  Field extensions generated by a set   cfldgen 33268
                  21.3.10.29  Totally ordered rings and fields   corng 33281
                  21.3.10.30  Ring homomorphisms - misc additions   rhmdvd 33304
                  21.3.10.31  Scalar restriction operation   cresv 33306
                  21.3.10.32  The commutative ring of gaussian integers   gzcrng 33321
                  21.3.10.33  The archimedean ordered field of real numbers   cnfldfld 33322
                  21.3.10.34  The quotient map and quotient modules   qusker 33328
                  21.3.10.35  The ring of integers modulo ` N `   znfermltl 33345
                  21.3.10.36  Independent sets and families   islinds5 33346
                  21.3.10.37  Ring associates, ring units   dvdsruassoi 33363
                  ​*21.3.10.38  Subgroup sum / Sumset / Minkowski sum   elgrplsmsn 33369
                  21.3.10.39  The quotient map   quslsm 33384
                  21.3.10.40  Ideals   lidlmcld 33398
                  21.3.10.41  Prime Ideals   cprmidl 33414
                  21.3.10.42  Maximal Ideals   cmxidl 33438
                  21.3.10.43  The semiring of ideals of a ring   cidlsrg 33479
                  21.3.10.44  Prime Elements   rprmval 33495
                  21.3.10.45  Unique factorization domains   cufd 33517
                  21.3.10.46  The ring of integers   zringidom 33530
                  21.3.10.47  Univariate Polynomials   0ringmon1p 33534
                  21.3.10.48  Polynomial quotient and polynomial remainder   q1pdir 33576
                  21.3.10.49  The subring algebra   sra1r 33585
                  21.3.10.50  Division Ring Extensions   drgext0g 33593
                  21.3.10.51  Vector Spaces   lvecdimfi 33599
                  21.3.10.52  Vector Space Dimension   cldim 33602
            21.3.11  Field Extensions   cfldext 33642
                  21.3.11.1  Algebraic numbers   cirng 33686
                  21.3.11.2  Algebraic extensions   calgext 33695
                  21.3.11.3  Minimal polynomials   cminply 33697
                  21.3.11.4  Quadratic Field Extensions   rtelextdg2lem 33724
                  21.3.11.5  Towers of quadratic extentions   fldext2chn 33726
            ​*21.3.12  Constructible Numbers   cconstr 33727
                  21.3.12.1  Impossible constructions   2sqr3minply 33778
            21.3.13  Matrices   csmat 33791
                  21.3.13.1  Submatrices   csmat 33791
                  21.3.13.2  Matrix literals   clmat 33809
                  21.3.13.3  Laplace expansion of determinants   mdetpmtr1 33821
            21.3.14  Topology   ist0cld 33831
                  21.3.14.1  Open maps   txomap 33832
                  21.3.14.2  Topology of the unit circle   qtopt1 33833
                  21.3.14.3  Refinements   reff 33837
                  21.3.14.4  Open cover refinement property   ccref 33840
                  21.3.14.5  Lindelöf spaces   cldlf 33850
                  21.3.14.6  Paracompact spaces   cpcmp 33853
                  ​*21.3.14.7  Spectrum of a ring   crspec 33860
                  21.3.14.8  Pseudometrics   cmetid 33884
                  21.3.14.9  Continuity - misc additions   hauseqcn 33896
                  21.3.14.10  Topology of the closed unit interval   elunitge0 33897
                  21.3.14.11  Topology of ` ( RR X. RR ) `   unicls 33901
                  21.3.14.12  Order topology - misc. additions   cnvordtrestixx 33911
                  21.3.14.13  Continuity in topological spaces - misc. additions   mndpluscn 33924
                  21.3.14.14  Topology of the extended nonnegative real numbers ordered monoid   xrge0hmph 33930
                  21.3.14.15  Limits - misc additions   lmlim 33945
                  21.3.14.16  Univariate polynomials   pl1cn 33953
            21.3.15  Uniform Stuctures and Spaces   chcmp 33954
                  21.3.15.1  Hausdorff uniform completion   chcmp 33954
            21.3.16  Topology and algebraic structures   zringnm 33956
                  21.3.16.1  The norm on the ring of the integer numbers   zringnm 33956
                  21.3.16.2  Topological ` ZZ ` -modules   zlm0 33958
                  21.3.16.3  Canonical embedding of the field of the rational numbers into a division ring   cqqh 33968
                  21.3.16.4  Canonical embedding of the real numbers into a complete ordered field   crrh 33991
                  21.3.16.5  Embedding from the extended real numbers into a complete lattice   cxrh 34014
                  21.3.16.6  Canonical embeddings into the ordered field of the real numbers   zrhre 34017
                  ​*21.3.16.7  Topological Manifolds   cmntop 34020
                  21.3.16.8  Extended sum   cesum 34025
            21.3.17  Mixed Function/Constant operation   cofc 34093
            21.3.18  Abstract measure   csiga 34106
                  21.3.18.1  Sigma-Algebra   csiga 34106
                  21.3.18.2  Generated sigma-Algebra   csigagen 34136
                  ​*21.3.18.3  lambda and pi-Systems, Rings of Sets   ispisys 34150
                  21.3.18.4  The Borel algebra on the real numbers   cbrsiga 34179
                  21.3.18.5  Product Sigma-Algebra   csx 34186
                  21.3.18.6  Measures   cmeas 34193
                  21.3.18.7  The counting measure   cntmeas 34224
                  21.3.18.8  The Lebesgue measure - misc additions   voliune 34227
                  21.3.18.9  The Dirac delta measure   cdde 34230
                  21.3.18.10  The 'almost everywhere' relation   cae 34235
                  21.3.18.11  Measurable functions   cmbfm 34247
                  21.3.18.12  Borel Algebra on ` ( RR X. RR ) `   br2base 34268
                  ​*21.3.18.13  Caratheodory's extension theorem   coms 34290
            21.3.19  Integration   itgeq12dv 34325
                  21.3.19.1  Lebesgue integral - misc additions   itgeq12dv 34325
                  21.3.19.2  Bochner integral   citgm 34326
            21.3.20  Euler's partition theorem   oddpwdc 34353
            21.3.21  Sequences defined by strong recursion   csseq 34382
            21.3.22  Fibonacci Numbers   cfib 34395
            21.3.23  Probability   cprb 34406
                  21.3.23.1  Probability Theory   cprb 34406
                  21.3.23.2  Conditional Probabilities   ccprob 34430
                  21.3.23.3  Real-valued Random Variables   crrv 34439
                  21.3.23.4  Preimage set mapping operator   corvc 34455
                  21.3.23.5  Distribution Functions   orvcelval 34468
                  21.3.23.6  Cumulative Distribution Functions   orvclteel 34472
                  21.3.23.7  Probabilities - example   coinfliplem 34478
                  21.3.23.8  Bertrand's Ballot Problem   ballotlemoex 34485
            21.3.24  Signum (sgn or sign) function - misc. additions   fzssfzo 34538
                  21.3.24.1  Operations on words   ccatmulgnn0dir 34541
            21.3.25  Polynomials with real coefficients - misc additions   plymul02 34545
            21.3.26  Descartes's rule of signs   signspval 34551
                  21.3.26.1  Sign changes in a word over real numbers   signspval 34551
                  21.3.26.2  Counting sign changes in a word over real numbers   signslema 34561
            21.3.27  Number Theory   iblidicc 34591
                  21.3.27.1  Representations of a number as sums of integers   crepr 34607
                  21.3.27.2  Vinogradov Trigonometric Sums and the Circle Method   cvts 34634
                  21.3.27.3  The Ternary Goldbach Conjecture: Final Statement   ax-hgt749 34643
            21.3.28  Elementary Geometry   cstrkg2d 34663
                  ​*21.3.28.1  Two-dimensional geometry   cstrkg2d 34663
                  21.3.28.2  Outer Five Segment (not used, no need to move to main)   cafs 34668
            ​*21.3.29  LeftPad Project   clpad 34673
      ​​*21.4  Mathbox for Jonathan Ben-Naim
            21.4.1  First-order logic and set theory   bnj170 34696
            21.4.2  Well founded induction and recursion   bnj110 34856
            21.4.3  The existence of a minimal element in certain classes   bnj69 35008
            21.4.4  Well-founded induction   bnj1204 35010
            21.4.5  Well-founded recursion, part 1 of 3   bnj60 35060
            21.4.6  Well-founded recursion, part 2 of 3   bnj1500 35066
            21.4.7  Well-founded recursion, part 3 of 3   bnj1522 35070
      ​21.5  Mathbox for BTernaryTau
            21.5.1  First-order logic   nfan1c 35071
                  21.5.1.1  Auxiliary axiom schemes   nfan1c 35071
            21.5.2  ZF set theory   exdifsn 35077
                  21.5.2.1  Finitism   prcinf 35092
                  21.5.2.2  Global choice   gblacfnacd 35097
            21.5.3  Real and complex numbers   zltp1ne 35099
            21.5.4  Graph theory   lfuhgr 35107
                  21.5.4.1  Acyclic graphs   cacycgr 35131
      ​21.6  Mathbox for Mario Carneiro
            21.6.1  Predicate calculus with all distinct variables   ax-7d 35148
            21.6.2  Miscellaneous stuff   quartfull 35154
            21.6.3  Derangements and the Subfactorial   deranglem 35155
            21.6.4  The Erdős-Szekeres theorem   erdszelem1 35180
            21.6.5  The Kuratowski closure-complement theorem   kur14lem1 35195
            21.6.6  Retracts and sections   cretr 35206
            21.6.7  Path-connected and simply connected spaces   cpconn 35208
            21.6.8  Covering maps   ccvm 35244
            21.6.9  Normal numbers   snmlff 35318
            21.6.10  Godel-sets of formulas - part 1   cgoe 35322
            21.6.11  Godel-sets of formulas - part 2   cgon 35421
            21.6.12  Models of ZF   cgze 35435
            ​*21.6.13  Metamath formal systems   cmcn 35449
            21.6.14  Grammatical formal systems   cm0s 35574
            21.6.15  Models of formal systems   cmuv 35594
            21.6.16  Splitting fields   ccpms 35616
            21.6.17  p-adic number fields   czr 35636
      ​​*21.7  Mathbox for Filip Cernatescu
      ​21.8  Mathbox for Paul Chapman
            21.8.1  Real and complex numbers (cont.)   climuzcnv 35660
            21.8.2  Miscellaneous theorems   elfzm12 35664
      ​21.9  Mathbox for Hongxiu Chen
      ​21.10  Mathbox for Adrian Ducourtial
            21.10.1  Propositional calculus   currybi 35677
            21.10.2  Clone theory   ccloneop 35679
      ​21.11  Mathbox for Scott Fenton
            21.11.1  ZFC Axioms in primitive form   axextprim 35685
            21.11.2  Untangled classes   untelirr 35692
            21.11.3  Extra propositional calculus theorems   3jaodd 35699
            21.11.4  Misc. Useful Theorems   nepss 35702
            21.11.5  Properties of real and complex numbers   sqdivzi 35712
            21.11.6  Infinite products   iprodefisumlem 35724
            21.11.7  Factorial limits   faclimlem1 35727
            21.11.8  Greatest common divisor and divisibility   gcd32 35733
            21.11.9  Properties of relationships   dftr6 35735
            21.11.10  Properties of functions and mappings   funpsstri 35750
            21.11.11  Set induction (or epsilon induction)   setinds 35763
            21.11.12  Ordinal numbers   elpotr 35766
            21.11.13  Defined equality axioms   axextdfeq 35782
            21.11.14  Hypothesis builders   hbntg 35790
            21.11.15  Well-founded zero, successor, and limits   cwsuc 35795
            21.11.16  Quantifier-free definitions   ctxp 35815
            21.11.17  Alternate ordered pairs   caltop 35941
            21.11.18  Geometry in the Euclidean space   cofs 35967
                  21.11.18.1  Congruence properties   cofs 35967
                  21.11.18.2  Betweenness properties   btwntriv2 35997
                  21.11.18.3  Segment Transportation   ctransport 36014
                  21.11.18.4  Properties relating betweenness and congruence   cifs 36020
                  21.11.18.5  Connectivity of betweenness   btwnconn1lem1 36072
                  21.11.18.6  Segment less than or equal to   csegle 36091
                  21.11.18.7  Outside-of relationship   coutsideof 36104
                  21.11.18.8  Lines and Rays   cline2 36119
            21.11.19  Forward difference   cfwddif 36143
            21.11.20  Rank theorems   rankung 36151
            21.11.21  Hereditarily Finite Sets   chf 36157
      ​21.12  Mathbox for Gino Giotto
            21.12.1  Equality theorems.   rmoeqi 36172
                  21.12.1.1  Inference versions.   rmoeqi 36172
                  21.12.1.2  Deduction versions.   rmoeqdv 36197
            21.12.2  Change bound variables.   in-ax8 36209
                  21.12.2.1  Change bound variables and domains.   cbvralvw2 36211
                  21.12.2.2  Change bound variables, deduction versions.   cbvmodavw 36235
                  21.12.2.3  Change bound variables and domains, deduction versions.   cbvrmodavw2 36268
            21.12.3  Study of ax-mulf usage.   mpomulnzcnf 36284
      ​21.13  Mathbox for Jeff Hankins
            21.13.1  Miscellany   a1i14 36285
            21.13.2  Basic topological facts   topbnd 36309
            21.13.3  Topology of the real numbers   ivthALT 36320
            21.13.4  Refinements   cfne 36321
            21.13.5  Neighborhood bases determine topologies   neibastop1 36344
            21.13.6  Lattice structure of topologies   topmtcl 36348
            21.13.7  Filter bases   fgmin 36355
            21.13.8  Directed sets, nets   tailfval 36357
      ​21.14  Mathbox for Anthony Hart
            21.14.1  Propositional Calculus   tb-ax1 36368
            21.14.2  Predicate Calculus   nalfal 36388
            21.14.3  Miscellaneous single axioms   meran1 36396
            21.14.4  Connective Symmetry   negsym1 36402
      ​21.15  Mathbox for Chen-Pang He
            21.15.1  Ordinal topology   ontopbas 36413
      ​21.16  Mathbox for Jeff Hoffman
            21.16.1  Inferences for finite induction on generic function values   fveleq 36436
            21.16.2  gdc.mm   nnssi2 36440
      ​21.17  Mathbox for Matthew House
            21.17.1  Relations on well-ordered indexed unions   weiunlem1 36447
      ​21.18  Mathbox for Asger C. Ipsen
            21.18.1  Continuous nowhere differentiable functions   dnival 36456
      ​​*21.19  Mathbox for BJ
            *21.19.1  Propositional calculus   bj-mp2c 36525
                  *21.19.1.1  Derived rules of inference   bj-mp2c 36525
                  ​*21.19.1.2  A syntactic theorem   bj-0 36527
                  21.19.1.3  Minimal implicational calculus   bj-a1k 36529
                  ​*21.19.1.4  Positive calculus   bj-syl66ib 36540
                  21.19.1.5  Implication and negation   bj-con2com 36546
                  ​*21.19.1.6  Disjunction   bj-jaoi1 36556
                  ​*21.19.1.7  Logical equivalence   bj-dfbi4 36558
                  21.19.1.8  The conditional operator for propositions   bj-consensus 36563
                  ​*21.19.1.9  Propositional calculus: miscellaneous   bj-imbi12 36568
            ​*21.19.2  Modal logic   bj-axdd2 36577
            ​*21.19.3  Provability logic   cprvb 36582
            ​*21.19.4  First-order logic   bj-genr 36591
                  21.19.4.1  Adding ax-gen   bj-genr 36591
                  21.19.4.2  Adding ax-4   bj-2alim 36595
                  21.19.4.3  Adding ax-5   bj-ax12wlem 36629
                  21.19.4.4  Equality and substitution   bj-ssbeq 36638
                  21.19.4.5  Adding ax-6   bj-spimvwt 36654
                  21.19.4.6  Adding ax-7   bj-cbvexw 36661
                  21.19.4.7  Membership predicate, ax-8 and ax-9   bj-ax89 36663
                  21.19.4.8  Adding ax-11   bj-alcomexcom 36665
                  21.19.4.9  Adding ax-12   axc11n11 36667
                  21.19.4.10  Nonfreeness   wnnf 36708
                  21.19.4.11  Adding ax-13   bj-axc10 36768
                  ​*21.19.4.12  Removing dependencies on ax-13 (and ax-11)   bj-axc10v 36778
                  ​*21.19.4.13  Distinct var metavariables   bj-hbaeb2 36803
                  ​*21.19.4.14  Around ~ equsal   bj-equsal1t 36807
                  ​*21.19.4.15  Some Principia Mathematica proofs   stdpc5t 36812
                  21.19.4.16  Alternate definition of substitution   bj-sbsb 36822
                  21.19.4.17  Lemmas for substitution   bj-sbf3 36824
                  21.19.4.18  Existential uniqueness   bj-eu3f 36826
                  ​*21.19.4.19  First-order logic: miscellaneous   bj-sblem1 36827
            21.19.5  Set theory   eliminable1 36844
                  ​*21.19.5.1  Eliminability of class terms   eliminable1 36844
                  ​*21.19.5.2  Classes without the axiom of extensionality   bj-denoteslem 36856
                  21.19.5.3  Characterization among sets versus among classes   elelb 36882
                  ​*21.19.5.4  The nonfreeness quantifier for classes   bj-nfcsym 36884
                  ​*21.19.5.5  Lemmas for class substitution   bj-sbeqALT 36885
                  21.19.5.6  Removing some axiom requirements and disjoint variable conditions   bj-exlimvmpi 36896
                  ​*21.19.5.7  Class abstractions   bj-elabd2ALT 36910
                  21.19.5.8  Generalized class abstractions   bj-cgab 36918
                  ​*21.19.5.9  Restricted nonfreeness   wrnf 36926
                  ​*21.19.5.10  Russell's paradox   bj-ru1 36928
                  21.19.5.11  Curry's paradox in set theory   currysetlem 36930
                  ​*21.19.5.12  Some disjointness results   bj-n0i 36936
                  ​*21.19.5.13  Complements on direct products   bj-xpimasn 36940
                  ​*21.19.5.14  "Singletonization" and tagging   bj-snsetex 36948
                  ​*21.19.5.15  Tuples of classes   bj-cproj 36975
                  ​*21.19.5.16  Set theory: elementary operations relative to a universe   bj-rcleqf 37010
                  ​*21.19.5.17  Axioms for finite unions   bj-abex 37015
                  ​*21.19.5.18  Set theory: miscellaneous   eleq2w2ALT 37032
                  ​*21.19.5.19  Evaluation at a class   bj-evaleq 37057
                  21.19.5.20  Elementwise operations   celwise 37064
                  ​*21.19.5.21  Elementwise intersection (families of sets induced on a subset)   bj-rest00 37066
                  21.19.5.22  Moore collections (complements)   bj-raldifsn 37085
                  21.19.5.23  Maps-to notation for functions with three arguments   bj-0nelmpt 37101
                  ​*21.19.5.24  Currying   csethom 37107
                  ​*21.19.5.25  Setting components of extensible structures   cstrset 37119
            ​*21.19.6  Extended real and complex numbers, real and complex projective lines   bj-nfald 37122
                  21.19.6.1  Complements on class abstractions of ordered pairs and binary relations   bj-nfald 37122
                  ​*21.19.6.2  Identity relation (complements)   bj-opabssvv 37135
                  ​*21.19.6.3  Functionalized identity (diagonal in a Cartesian square)   cdiag2 37157
                  ​*21.19.6.4  Direct image and inverse image   cimdir 37163
                  ​*21.19.6.5  Extended numbers and projective lines as sets   cfractemp 37181
                  ​*21.19.6.6  Addition and opposite   caddcc 37222
                  ​*21.19.6.7  Order relation on the extended reals   cltxr 37226
                  ​*21.19.6.8  Argument, multiplication and inverse   carg 37228
                  21.19.6.9  The canonical bijection from the finite ordinals   ciomnn 37234
                  21.19.6.10  Divisibility   cnnbar 37245
            ​*21.19.7  Monoids   bj-smgrpssmgm 37253
                  ​*21.19.7.1  Finite sums in monoids   cfinsum 37268
            ​*21.19.8  Affine, Euclidean, and Cartesian geometry   bj-fvimacnv0 37271
                  *21.19.8.1  Real vector spaces   bj-fvimacnv0 37271
                  ​*21.19.8.2  Complex numbers (supplements)   bj-subcom 37293
                  ​*21.19.8.3  Barycentric coordinates   bj-bary1lem 37295
            21.19.9  Monoid of endomorphisms   cend 37298
      ​21.20  Mathbox for Jim Kingdon
            21.20.1  Circle constant   taupilem3 37304
            21.20.2  Number theory   dfgcd3 37309
            21.20.3  Real numbers   irrdifflemf 37310
      ​21.21  Mathbox for ML
            21.21.1  Miscellaneous   csbrecsg 37313
            21.21.2  Cartesian exponentiation   cfinxp 37368
            21.21.3  Topology   iunctb2 37388
                  ​*21.21.3.1  Pi-base theorems   pibp16 37398
      ​21.22  Mathbox for Wolf Lammen
            21.22.1  1. Bootstrapping   wl-section-boot 37407
            21.22.2  Implication chains   wl-section-impchain 37431
            21.22.3  Theorems around the conditional operator   wl-ifp-ncond1 37449
            21.22.4  Alternative development of hadd, cadd   wl-df-3xor 37453
            21.22.5  An alternative axiom ~ ax-13   ax-wl-13v 37478
            21.22.6  Bootstrapping set theory with classes   wl-cleq-0 37480
            21.22.7  Other stuff   wl-mps 37492
      ​21.23  Mathbox for Brendan Leahy
      ​21.24  Mathbox for Jeff Madsen
            21.24.1  Logic and set theory   unirep 37705
            21.24.2  Real and complex numbers; integers   filbcmb 37731
            21.24.3  Sequences and sums   sdclem2 37733
            21.24.4  Topology   subspopn 37743
            21.24.5  Metric spaces   metf1o 37746
            21.24.6  Continuous maps and homeomorphisms   constcncf 37753
            21.24.7  Boundedness   ctotbnd 37757
            21.24.8  Isometries   cismty 37789
            21.24.9  Heine-Borel Theorem   heibor1lem 37800
            21.24.10  Banach Fixed Point Theorem   bfplem1 37813
            21.24.11  Euclidean space   crrn 37816
            21.24.12  Intervals (continued)   ismrer1 37829
            21.24.13  Operation properties   cass 37833
            21.24.14  Groups and related structures   cmagm 37839
            21.24.15  Group homomorphism and isomorphism   cghomOLD 37874
            21.24.16  Rings   crngo 37885
            21.24.17  Division Rings   cdrng 37939
            21.24.18  Ring homomorphisms   crngohom 37951
            21.24.19  Commutative rings   ccm2 37980
            21.24.20  Ideals   cidl 37998
            21.24.21  Prime rings and integral domains   cprrng 38037
            21.24.22  Ideal generators   cigen 38050
      ​21.25  Mathbox for Giovanni Mascellani
            ​*21.25.1  Tools for automatic proof building   efald2 38069
            ​*21.25.2  Tseitin axioms   fald 38120
            ​*21.25.3  Equality deductions   iuneq2f 38147
            ​*21.25.4  Miscellanea   orcomdd 38158
      ​21.26  Mathbox for Peter Mazsa
            21.26.1  Notations   cxrn 38165
            21.26.2  Preparatory theorems   el2v1 38208
            21.26.3  Range Cartesian product   df-xrn 38356
            21.26.4  Cosets by ` R `   df-coss 38396
            21.26.5  Relations   df-rels 38470
            21.26.6  Subset relations   df-ssr 38483
            21.26.7  Reflexivity   df-refs 38495
            21.26.8  Converse reflexivity   df-cnvrefs 38510
            21.26.9  Symmetry   df-syms 38527
            21.26.10  Reflexivity and symmetry   symrefref2 38548
            21.26.11  Transitivity   df-trs 38557
            21.26.12  Equivalence relations   df-eqvrels 38569
            21.26.13  Redundancy   df-redunds 38608
            21.26.14  Domain quotients   df-dmqss 38623
            21.26.15  Equivalence relations on domain quotients   df-ers 38648
            21.26.16  Functions   df-funss 38665
            21.26.17  Disjoints vs. converse functions   df-disjss 38688
            21.26.18  Antisymmetry   df-antisymrel 38745
            21.26.19  Partitions: disjoints on domain quotients   df-parts 38750
            21.26.20  Partition-Equivalence Theorems   disjim 38766
      ​21.27  Mathbox for Rodolfo Medina
            21.27.1  Partitions   prtlem60 38838
      ​​*21.28  Mathbox for Norm Megill
            *21.28.1  Obsolete schemes ax-c4,c5,c7,c10,c11,c11n,c15,c9,c14,c16   ax-c5 38868
            ​*21.28.2  Rederive new axioms ax-4, ax-10, ax-6, ax-12, ax-13 from old   axc5 38878
            ​*21.28.3  Legacy theorems using obsolete axioms   ax5ALT 38892
            21.28.4  Experiments with weak deduction theorem   elimhyps 38946
            21.28.5  Miscellanea   cnaddcom 38957
            21.28.6  Atoms, hyperplanes, and covering in a left vector space (or module)   clsa 38959
            21.28.7  Functionals and kernels of a left vector space (or module)   clfn 39042
            21.28.8  Opposite rings and dual vector spaces   cld 39108
            21.28.9  Ortholattices and orthomodular lattices   cops 39157
            21.28.10  Atomic lattices with covering property   ccvr 39247
            21.28.11  Hilbert lattices   chlt 39335
            21.28.12  Projective geometries based on Hilbert lattices   clln 39477
            21.28.13  Construction of a vector space from a Hilbert lattice   cdlema1N 39777
            21.28.14  Construction of involution and inner product from a Hilbert lattice   clpoN 41466
      ​21.29  Mathbox for metakunt
            21.29.1  Commutative Semiring   ccsrg 41948
            21.29.2  General helpful statements   rhmzrhval 41951
            21.29.3  Some gcd and lcm results   12gcd5e1 41983
            21.29.4  Least common multiple inequality theorem   3factsumint1 42001
            21.29.5  Logarithm inequalities   3exp7 42033
            21.29.6  Miscellaneous results for AKS formalisation   intlewftc 42041
            21.29.7  Sticks and stones   sticksstones1 42126
            21.29.8  Continuation AKS   aks6d1c6lem1 42150
      ​21.30  Mathbox for Steven Nguyen
            21.30.1  Utility theorems   jarrii 42185
            ​*21.30.2  Arithmetic theorems   c0exALT 42232
            21.30.3  Exponents and divisibility   oexpreposd 42302
            21.30.4  Trigonometry and Calculus   tanhalfpim 42329
            21.30.5  Real subtraction   cresub 42345
            21.30.6  Structures   sn-base0 42455
            ​*21.30.7  Projective spaces   cprjsp 42561
            21.30.8  Basic reductions for Fermat's Last Theorem   dffltz 42594
            ​*21.30.9  Exemplar theorems   iddii 42624
                  ​*21.30.9.1  Standard replacements of ax-10 , ax-11 , ax-12   nfa1w 42635
      ​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 42652
            21.33.2  Additional theory of functions   imaiinfv 42653
            21.33.3  Additional topology   elrfi 42654
            21.33.4  Characterization of closure operators. Kuratowski closure axioms   ismrcd1 42658
            21.33.5  Algebraic closure systems   cnacs 42662
            21.33.6  Miscellanea 1. Map utilities   constmap 42673
            21.33.7  Miscellanea for polynomials   mptfcl 42680
            21.33.8  Multivariate polynomials over the integers   cmzpcl 42681
            21.33.9  Miscellanea for Diophantine sets 1   coeq0i 42713
            21.33.10  Diophantine sets 1: definitions   cdioph 42715
            21.33.11  Diophantine sets 2 miscellanea   ellz1 42727
            21.33.12  Diophantine sets 2: union and intersection. Monotone Boolean algebra   diophin 42732
            21.33.13  Diophantine sets 3: construction   diophrex 42735
            21.33.14  Diophantine sets 4 miscellanea   2sbcrex 42744
            21.33.15  Diophantine sets 4: Quantification   rexrabdioph 42754
            21.33.16  Diophantine sets 5: Arithmetic sets   rabdiophlem1 42761
            21.33.17  Diophantine sets 6: reusability. renumbering of variables   eldioph4b 42771
            21.33.18  Pigeonhole Principle and cardinality helpers   fphpd 42776
            21.33.19  A non-closed set of reals is infinite   rencldnfilem 42780
            21.33.20  Lagrange's rational approximation theorem   irrapxlem1 42782
            21.33.21  Pell equations 1: A nontrivial solution always exists   pellexlem1 42789
            21.33.22  Pell equations 2: Algebraic number theory of the solution set   csquarenn 42796
            21.33.23  Pell equations 3: characterizing fundamental solution   infmrgelbi 42838
            ​*21.33.24  Logarithm laws generalized to an arbitrary base   reglogcl 42850
            21.33.25  Pell equations 4: the positive solution group is infinite cyclic   pellfund14 42858
            21.33.26  X and Y sequences 1: Definition and recurrence laws   crmx 42860
            21.33.27  Ordering and induction lemmas for the integers   monotuz 42902
            21.33.28  X and Y sequences 2: Order properties   rmxypos 42908
            21.33.29  Congruential equations   congtr 42926
            21.33.30  Alternating congruential equations   acongid 42936
            21.33.31  Additional theorems on integer divisibility   coprmdvdsb 42946
            21.33.32  X and Y sequences 3: Divisibility properties   jm2.18 42949
            21.33.33  X and Y sequences 4: Diophantine representability of Y   jm2.27a 42966
            21.33.34  X and Y sequences 5: Diophantine representability of X, ^, _C   rmxdiophlem 42976
            21.33.35  Uncategorized stuff not associated with a major project   setindtr 42985
            21.33.36  More equivalents of the Axiom of Choice   axac10 42994
            21.33.37  Finitely generated left modules   clfig 43028
            21.33.38  Noetherian left modules I   clnm 43036
            21.33.39  Addenda for structure powers   pwssplit4 43050
            21.33.40  Every set admits a group structure iff choice   unxpwdom3 43056
            21.33.41  Noetherian rings and left modules II   clnr 43070
            21.33.42  Hilbert's Basis Theorem   cldgis 43082
            21.33.43  Additional material on polynomials [DEPRECATED]   cmnc 43092
            21.33.44  Degree and minimal polynomial of algebraic numbers   cdgraa 43101
            21.33.45  Algebraic integers I   citgo 43118
            21.33.46  Endomorphism algebra   cmend 43132
            21.33.47  Cyclic groups and order   idomodle 43152
            21.33.48  Cyclotomic polynomials   ccytp 43158
            21.33.49  Miscellaneous topology   fgraphopab 43164
      ​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 43178
            21.36.2  Natural addition of Cantor normal forms   oawordex2 43287
            21.36.3  Surreal Contributions   abeqabi 43369
            21.36.4  Short Studies   nlimsuc 43402
                  21.36.4.1  Additional work on conditional logical operator   ifpan123g 43420
                  21.36.4.2  Sophisms   rp-fakeimass 43473
                  ​*21.36.4.3  Finite Sets   rp-isfinite5 43478
                  21.36.4.4  General Observations   intabssd 43480
                  21.36.4.5  Infinite Sets   pwelg 43521
                  ​*21.36.4.6  Finite intersection property   fipjust 43526
                  21.36.4.7  RP ADDTO: Subclasses and subsets   rababg 43535
                  21.36.4.8  RP ADDTO: The intersection of a class   elinintab 43536
                  21.36.4.9  RP ADDTO: Theorems requiring subset and intersection existence   elinintrab 43538
                  21.36.4.10  RP ADDTO: Relations   xpinintabd 43541
                  ​*21.36.4.11  RP ADDTO: Functions   elmapintab 43557
                  ​*21.36.4.12  RP ADDTO: Finite induction (for finite ordinals)   cnvcnvintabd 43561
                  21.36.4.13  RP ADDTO: First and second members of an ordered pair   elcnvlem 43562
                  21.36.4.14  RP ADDTO: The reflexive and transitive properties of relations   undmrnresiss 43565
                  21.36.4.15  RP ADDTO: Basic properties of closures   cleq2lem 43569
                  21.36.4.16  RP REPLACE: Definitions and basic properties of transitive closures   trcleq2lemRP 43591
                  ​*21.36.4.17  Additions for square root; absolute value   sqrtcvallem1 43592
            21.36.5  Additional statements on relations and subclasses   al3im 43608
                  21.36.5.1  Transitive relations (not to be confused with transitive classes).   trrelind 43626
                  21.36.5.2  Reflexive closures   crcl 43633
                  ​*21.36.5.3  Finite relationship composition.   relexp2 43638
                  21.36.5.4  Transitive closure of a relation   dftrcl3 43681
                  ​*21.36.5.5  Adapted from Frege   frege77d 43707
            ​*21.36.6  Propositions from _Begriffsschrift_   dfxor4 43727
                  *21.36.6.1  _Begriffsschrift_ Chapter I   dfxor4 43727
                  ​*21.36.6.2  _Begriffsschrift_ Notation hints   whe 43733
                  21.36.6.3  _Begriffsschrift_ Chapter II Implication   ax-frege1 43751
                  21.36.6.4  _Begriffsschrift_ Chapter II Implication and Negation   axfrege28 43790
                  ​*21.36.6.5  _Begriffsschrift_ Chapter II with logical equivalence   axfrege52a 43817
                  21.36.6.6  _Begriffsschrift_ Chapter II with equivalence of sets   axfrege52c 43848
                  ​*21.36.6.7  _Begriffsschrift_ Chapter II with equivalence of classes   frege53c 43875
                  ​*21.36.6.8  _Begriffsschrift_ Chapter III Properties hereditary in a sequence   dffrege69 43893
                  ​*21.36.6.9  _Begriffsschrift_ Chapter III Following in a sequence   dffrege76 43900
                  ​*21.36.6.10  _Begriffsschrift_ Chapter III Member of sequence   dffrege99 43923
                  ​*21.36.6.11  _Begriffsschrift_ Chapter III Single-valued procedures   dffrege115 43939
            ​*21.36.7  Exploring Topology via Seifert and Threlfall   enrelmap 43958
                  *21.36.7.1  Equinumerosity of sets of relations and maps   enrelmap 43958
                  ​*21.36.7.2  Generic Pseudoclosure Spaces, Pseudointerior Spaces, and Pseudoneighborhoods   or3or 43984
                  ​*21.36.7.3  Generic Neighborhood Spaces   gneispa 44091
            ​*21.36.8  Exploring Higher Homotopy via Kerodon   k0004lem1 44108
                  *21.36.8.1  Simplicial Sets   k0004lem1 44108
      ​21.37  Mathbox for Stanislas Polu
            21.37.1  IMO Problems   wwlemuld 44117
                  21.37.1.1  IMO 1972 B2   wwlemuld 44117
            ​*21.37.2  INT Inequalities Proof Generator   int-addcomd 44134
            ​*21.37.3  N-Digit Addition Proof Generator   unitadd 44156
            21.37.4  AM-GM (for k = 2,3,4)   gsumws3 44157
      ​21.38  Mathbox for Rohan Ridenour
            21.38.1  Misc   spALT 44162
            21.38.2  Monoid rings   cmnring 44172
            21.38.3  Shorter primitive equivalent of ax-groth   gru0eld 44190
                  21.38.3.1  Grothendieck universes are closed under collection   gru0eld 44190
                  21.38.3.2  Minimal universes   ismnu 44222
                  21.38.3.3  Primitive equivalent of ax-groth   expandan 44249
      ​21.39  Mathbox for Steve Rodriguez
            21.39.1  Miscellanea   nanorxor 44266
            21.39.2  Ratio test for infinite series convergence and divergence   dvgrat 44273
            21.39.3  Multiples   reldvds 44276
            21.39.4  Function operations   caofcan 44284
            21.39.5  Calculus   lhe4.4ex1a 44290
            21.39.6  The generalized binomial coefficient operation   cbcc 44297
            21.39.7  Binomial series   uzmptshftfval 44307
      ​21.40  Mathbox for Andrew Salmon
            21.40.1  Principia Mathematica * 10   pm10.12 44319
            21.40.2  Principia Mathematica * 11   2alanimi 44333
            21.40.3  Predicate Calculus   sbeqal1 44359
            21.40.4  Principia Mathematica * 13 and * 14   pm13.13a 44368
            21.40.5  Set Theory   elnev 44399
            21.40.6  Arithmetic   addcomgi 44417
            21.40.7  Geometry   cplusr 44418
      ​​*21.41  Mathbox for Alan Sare
            21.41.1  Auxiliary theorems for the Virtual Deduction tool   idiALT 44440
            21.41.2  Supplementary unification deductions   bi1imp 44444
            21.41.3  Conventional Metamath proofs, some derived from VD proofs   iidn3 44463
            21.41.4  What is Virtual Deduction?   wvd1 44531
            21.41.5  Virtual Deduction Theorems   df-vd1 44532
            21.41.6  Theorems proved using Virtual Deduction   trsspwALT 44779
            21.41.7  Theorems proved using Virtual Deduction with mmj2 assistance   simplbi2VD 44807
            21.41.8  Virtual Deduction transcriptions of textbook proofs   sb5ALTVD 44874
            21.41.9  Theorems proved using conjunction-form Virtual Deduction   elpwgdedVD 44878
            21.41.10  Theorems with a VD proof in conventional notation derived from a VD proof   suctrALT3 44885
            ​*21.41.11  Theorems with a proof in conventional notation derived from a VD proof   notnotrALT2 44888
      ​21.42  Mathbox for Eric Schmidt
            21.42.1  Miscellany   rspesbcd 44899
            21.42.2  Study of dfbi1ALT   dfbi1ALTa 44901
            21.42.3  Relation-preserving functions   wrelp 44904
            21.42.4  Orbits   orbitex 44917
            21.42.5  Well-founded sets   trwf 44921
            21.42.6  Absoluteness in transitive models   ralabso 44930
            21.42.7  Lemmas for showing axioms hold in models   traxext 44939
            21.42.8  The class of well-founded sets is a model for ZFC   wfaxext 44955
            21.42.9  Permutation models   brpermmodel 44965
      ​21.43  Mathbox for Glauco Siliprandi
            21.43.1  Miscellanea   evth2f 44981
            21.43.2  Functions   fnresdmss 45134
            21.43.3  Ordering on real numbers - Real and complex numbers basic operations   sub2times 45243
            21.43.4  Real intervals   gtnelioc 45462
            21.43.5  Finite sums   fsummulc1f 45542
            21.43.6  Finite multiplication of numbers and finite multiplication of functions   fmul01 45551
            21.43.7  Limits   clim1fr1 45572
                  21.43.7.1  Inferior limit (lim inf)   clsi 45722
                  ​*21.43.7.2  Limits for sequences of extended real numbers   clsxlim 45789
            21.43.8  Trigonometry   coseq0 45835
            21.43.9  Continuous Functions   mulcncff 45841
            21.43.10  Derivatives   dvsinexp 45882
            21.43.11  Integrals   itgsin0pilem1 45921
            21.43.12  Stone Weierstrass theorem - real version   stoweidlem1 45972
            21.43.13  Wallis' product for π   wallispilem1 46036
            21.43.14  Stirling's approximation formula for ` n ` factorial   stirlinglem1 46045
            21.43.15  Dirichlet kernel   dirkerval 46062
            21.43.16  Fourier Series   fourierdlem1 46079
            21.43.17  e is transcendental   elaa2lem 46204
            21.43.18  n-dimensional Euclidean space   rrxtopn 46255
            21.43.19  Basic measure theory   csalg 46279
                  ​*21.43.19.1  σ-Algebras   csalg 46279
                  21.43.19.2  Sum of nonnegative extended reals   csumge0 46333
                  ​*21.43.19.3  Measures   cmea 46420
                  ​*21.43.19.4  Outer measures and Caratheodory's construction   come 46460
                  ​*21.43.19.5  Lebesgue measure on n-dimensional Real numbers   covoln 46507
                  ​*21.43.19.6  Measurable functions   csmblfn 46666
      ​21.44  Mathbox for Saveliy Skresanov
            21.44.1  Ceva's theorem   sigarval 46821
            21.44.2  Simple groups   simpcntrab 46841
      ​21.45  Mathbox for Ender Ting
            21.45.1  Increasing sequences and subsequences   et-ltneverrefl 46842
            21.45.2  Scratchpad for number theory   evenwodadd 46859
            21.45.3  Scratchpad for math on real numbers   squeezedltsq 46860
      ​21.46  Mathbox for Jarvin Udandy
      ​21.47  Mathbox for Adhemar
            ​*21.47.1  Minimal implicational calculus   adh-minim 46972
      ​21.48  Mathbox for Alexander van der Vekens
            21.48.1  General auxiliary theorems (1)   n0nsn2el 46996
                  21.48.1.1  Unordered and ordered pairs - extension for singletons   n0nsn2el 46996
                  21.48.1.2  Unordered and ordered pairs - extension for unordered pairs   elprneb 47000
                  21.48.1.3  Unordered and ordered pairs - extension for ordered pairs   oppr 47001
                  21.48.1.4  Relations - extension   eubrv 47006
                  21.48.1.5  Definite description binder (inverted iota) - extension   iota0def 47009
                  21.48.1.6  Functions - extension   fveqvfvv 47011
            21.48.2  Alternative for Russell's definition of a description binder   caiota 47054
            21.48.3  Double restricted existential uniqueness   r19.32 47069
                  21.48.3.1  Restricted quantification (extension)   r19.32 47069
                  21.48.3.2  Restricted uniqueness and "at most one" quantification   reuf1odnf 47078
                  21.48.3.3  Analogs to Existential uniqueness (double quantification)   2reu3 47081
                  21.48.3.4  Additional theorems for double restricted existential uniqueness   2reu8i 47084
            ​*21.48.4  Alternative definitions of function and operation values   wdfat 47087
                  21.48.4.1  Restricted quantification (extension)   ralbinrald 47093
                  21.48.4.2  The universal class (extension)   nvelim 47094
                  21.48.4.3  Introduce the Axiom of Power Sets (extension)   alneu 47095
                  21.48.4.4  Predicate "defined at"   dfateq12d 47097
                  21.48.4.5  Alternative definition of the value of a function   dfafv2 47103
                  21.48.4.6  Alternative definition of the value of an operation   aoveq123d 47149
            ​*21.48.5  Alternative definitions of function values (2)   cafv2 47179
            21.48.6  General auxiliary theorems (2)   an4com24 47239
                  21.48.6.1  Logical conjunction - extension   an4com24 47239
                  21.48.6.2  Abbreviated conjunction and disjunction of three wff's - extension   3an4ancom24 47240
                  21.48.6.3  Negated membership (alternative)   cnelbr 47242
                  21.48.6.4  The empty set - extension   ralralimp 47249
                  21.48.6.5  Indexed union and intersection - extension   otiunsndisjX 47250
                  21.48.6.6  Functions - extension   fvifeq 47251
                  21.48.6.7  Maps-to notation - extension   fvmptrab 47263
                  21.48.6.8  Subtraction - extension   cnambpcma 47265
                  21.48.6.9  Ordering on reals (cont.) - extension   leaddsuble 47268
                  21.48.6.10  Imaginary and complex number properties - extension   readdcnnred 47274
                  21.48.6.11  Nonnegative integers (as a subset of complex numbers) - extension   nn0resubcl 47279
                  21.48.6.12  Integers (as a subset of complex numbers) - extension   zgeltp1eq 47280
                  21.48.6.13  Decimal arithmetic - extension   1t10e1p1e11 47281
                  21.48.6.14  Upper sets of integers - extension   eluzge0nn0 47283
                  21.48.6.15  Infinity and the extended real number system (cont.) - extension   nltle2tri 47284
                  21.48.6.16  Finite intervals of integers - extension   ssfz12 47285
                  21.48.6.17  Half-open integer ranges - extension   fzopred 47293
                  21.48.6.18  The floor and ceiling functions - extension   2ltceilhalf 47299
                  21.48.6.19  The modulo (remainder) operation - extension   fldivmod 47309
                  21.48.6.20  The infinite sequence builder "seq"   smonoord 47327
                  21.48.6.21  Finite and infinite sums - extension   fsummsndifre 47328
                  21.48.6.22  Extensible structures - extension   setsidel 47332
            ​*21.48.7  Preimages of function values   preimafvsnel 47335
            ​*21.48.8  Partitions of real intervals   ciccp 47369
            21.48.9  Shifting functions with an integer range domain   fargshiftfv 47395
            21.48.10  Words over a set (extension)   lswn0 47400
                  21.48.10.1  Last symbol of a word - extension   lswn0 47400
            21.48.11  Unordered pairs   wich 47401
                  21.48.11.1  Interchangeable setvar variables   wich 47401
                  21.48.11.2  Set of unordered pairs   sprid 47430
                  ​*21.48.11.3  Proper (unordered) pairs   prpair 47457
                  21.48.11.4  Set of proper unordered pairs   cprpr 47468
            21.48.12  Number theory (extension)   cfmtno 47483
                  ​*21.48.12.1  Fermat numbers   cfmtno 47483
                  ​*21.48.12.2  Mersenne primes   m2prm 47547
                  21.48.12.3  Proth's theorem   modexp2m1d 47568
                  21.48.12.4  Solutions of quadratic equations   quad1 47576
            ​*21.48.13  Even and odd numbers   ceven 47580
                  21.48.13.1  Definitions and basic properties   ceven 47580
                  21.48.13.2  Alternate definitions using the "divides" relation   dfeven2 47605
                  21.48.13.3  Alternate definitions using the "modulo" operation   dfeven3 47614
                  21.48.13.4  Alternate definitions using the "gcd" operation   iseven5 47620
                  21.48.13.5  Theorems of part 5 revised   zneoALTV 47625
                  21.48.13.6  Theorems of part 6 revised   odd2np1ALTV 47630
                  21.48.13.7  Theorems of AV's mathbox revised   0evenALTV 47644
                  21.48.13.8  Additional theorems   epoo 47659
                  21.48.13.9  Perfect Number Theorem (revised)   perfectALTVlem1 47677
            21.48.14  Number theory (extension 2)   cfppr 47680
                  ​*21.48.14.1  Fermat pseudoprimes   cfppr 47680
                  ​*21.48.14.2  Goldbach's conjectures   cgbe 47701
            21.48.15  Graph theory (extension)   cclnbgr 47774
                  21.48.15.1  Closed neighborhood of a vertex   cclnbgr 47774
                  ​*21.48.15.2  Semiclosed and semiopen neighborhoods (experimental)   dfsclnbgr2 47801
                  21.48.15.3  Induced subgraphs   cisubgr 47815
                  ​*21.48.15.4  Isomorphisms of graphs   cgrisom 47829
                  ​*21.48.15.5  Triangles in graphs   cgrtri 47891
                  ​*21.48.15.6  Star graphs   cstgr 47905
                  ​*21.48.15.7  Local isomorphisms of graphs   cgrlim 47930
                  ​*21.48.15.8  Generalized Petersen graphs   cgpg 47986
                  21.48.15.9  Loop-free graphs - extension   1hegrlfgr 48049
                  21.48.15.10  Walks - extension   cupwlks 48050
                  21.48.15.11  Edges of graphs expressed as sets of unordered pairs   upgredgssspr 48060
            21.48.16  Monoids (extension)   ovn0dmfun 48073
                  21.48.16.1  Auxiliary theorems   ovn0dmfun 48073
                  21.48.16.2  Magmas, Semigroups and Monoids (extension)   plusfreseq 48081
                  21.48.16.3  Examples and counterexamples for magmas, semigroups and monoids (extension)   opmpoismgm 48084
                  21.48.16.4  Group sum operation (extension 1)   gsumsplit2f 48097
            ​*21.48.17  Magmas and internal binary operations (alternate approach)   ccllaw 48100
                  *21.48.17.1  Laws for internal binary operations   ccllaw 48100
                  ​*21.48.17.2  Internal binary operations   cintop 48113
                  21.48.17.3  Alternative definitions for magmas and semigroups   cmgm2 48132
            21.48.18  Rings (extension)   lmod0rng 48146
                  21.48.18.1  Nonzero rings (extension)   lmod0rng 48146
                  21.48.18.2  Ideals as non-unital rings   lidldomn1 48148
                  21.48.18.3  The non-unital ring of even integers   0even 48154
                  21.48.18.4  A constructed not unital ring   cznrnglem 48176
                  ​*21.48.18.5  The category of non-unital rings (alternate definition)   crngcALTV 48180
                  ​*21.48.18.6  The category of (unital) rings (alternate definition)   cringcALTV 48204
            21.48.19  Basic algebraic structures (extension)   eliunxp2 48251
                  21.48.19.1  Auxiliary theorems   eliunxp2 48251
                  21.48.19.2  The binomial coefficient operation (extension)   bcpascm1 48268
                  21.48.19.3  The ` ZZ `-module ` ZZ X. ZZ `   zlmodzxzlmod 48271
                  21.48.19.4  Group sum operation (extension 2)   mgpsumunsn 48278
                  21.48.19.5  Symmetric groups (extension)   exple2lt6 48281
                  21.48.19.6  Divisibility (extension)   invginvrid 48284
                  21.48.19.7  The support of functions (extension)   rmsupp0 48285
                  21.48.19.8  Finitely supported functions (extension)   rmsuppfi 48289
                  21.48.19.9  Left modules (extension)   lmodvsmdi 48296
                  21.48.19.10  Associative algebras (extension)   assaascl0 48298
                  21.48.19.11  Univariate polynomials (extension)   ply1vr1smo 48300
                  21.48.19.12  Univariate polynomials (examples)   linply1 48311
            21.48.20  Linear algebra (extension)   cdmatalt 48314
                  ​*21.48.20.1  The subalgebras of diagonal and scalar matrices (extension)   cdmatalt 48314
                  ​*21.48.20.2  Linear combinations   clinc 48322
                  ​*21.48.20.3  Linear independence   clininds 48358
                  21.48.20.4  Simple left modules and the ` ZZ `-module   lmod1lem1 48405
                  21.48.20.5  Differences between (left) modules and (left) vector spaces   lvecpsslmod 48425
            21.48.21  Complexity theory   suppdm 48428
                  21.48.21.1  Auxiliary theorems   suppdm 48428
                  21.48.21.2  The modulo (remainder) operation (extension)   mod0mul 48441
                  21.48.21.3  Even and odd integers   nn0onn0ex 48445
                  21.48.21.4  The natural logarithm on complex numbers (extension)   logcxp0 48457
                  21.48.21.5  Division of functions   cfdiv 48459
                  21.48.21.6  Upper bounds   cbigo 48469
                  21.48.21.7  Logarithm to an arbitrary base (extension)   rege1logbrege0 48480
                  ​*21.48.21.8  The binary logarithm   fldivexpfllog2 48487
                  21.48.21.9  Binary length   cblen 48491
                  ​*21.48.21.10  Digits   cdig 48517
                  21.48.21.11  Nonnegative integer as sum of its shifted digits   dignn0flhalflem1 48537
                  21.48.21.12  Algorithms for the multiplication of nonnegative integers   nn0mulfsum 48546
                  ​*21.48.21.13  N-ary functions   cnaryf 48548
                  ​*21.48.21.14  The Ackermann function   citco 48579
            21.48.22  Elementary geometry (extension)   fv1prop 48621
                  21.48.22.1  Auxiliary theorems   fv1prop 48621
                  21.48.22.2  Real euclidean space of dimension 2   rrx2pxel 48633
                  21.48.22.3  Spheres and lines in real Euclidean spaces   cline 48649
      ​21.49  Mathbox for Zhi Wang
            21.49.1  Propositional calculus   pm4.71da 48711
            21.49.2  Predicate calculus with equality   dtrucor3 48720
                  21.49.2.1  Axiom scheme ax-5 (Distinctness)   dtrucor3 48720
            21.49.3  ZF Set Theory - start with the Axiom of Extensionality   ralbidb 48721
                  21.49.3.1  Restricted quantification   ralbidb 48721
                  21.49.3.2  The universal class   reuxfr1dd 48727
                  21.49.3.3  The empty set   ssdisjd 48728
                  21.49.3.4  Unordered and ordered pairs   vsn 48732
                  21.49.3.5  The union of a class   unilbss 48738
                  21.49.3.6  Indexed union and intersection   iuneq0 48739
            21.49.4  ZF Set Theory - add the Axiom of Replacement   inpw 48745
                  21.49.4.1  Theorems requiring subset and intersection existence   inpw 48745
            21.49.5  ZF Set Theory - add the Axiom of Power Sets   opth1neg 48746
                  21.49.5.1  Ordered pair theorem   opth1neg 48746
                  21.49.5.2  Ordered-pair class abstractions (cont.)   brab2dd 48748
                  21.49.5.3  Relations   iinxp 48751
                  21.49.5.4  Functions   mof0 48758
                  21.49.5.5  Operations   ovsng 48776
            21.49.6  ZF Set Theory - add the Axiom of Union   fonex 48784
                  21.49.6.1  Relations and functions (cont.)   fonex 48784
                  21.49.6.2  First and second members of an ordered pair   eloprab1st2nd 48785
                  21.49.6.3  Operations in maps-to notation (continued)   fmpodg 48786
                  21.49.6.4  Function transposition   resinsnlem 48788
                  21.49.6.5  Infinite Cartesian products   ixpv 48807
                  21.49.6.6  Equinumerosity   fvconst0ci 48808
            21.49.7  Order sets   iccin 48812
                  21.49.7.1  Real number intervals   iccin 48812
            21.49.8  Extensible structures   slotresfo 48815
                  21.49.8.1  Basic definitions   slotresfo 48815
            21.49.9  Moore spaces   mreuniss 48816
            ​*21.49.10  Topology   clduni 48817
                  21.49.10.1  Closure and interior   clduni 48817
                  21.49.10.2  Neighborhoods   neircl 48821
                  21.49.10.3  Subspace topologies   restcls2lem 48829
                  21.49.10.4  Limits and continuity in topological spaces   cnneiima 48833
                  21.49.10.5  Topological definitions using the reals   iooii 48834
                  21.49.10.6  Separated sets   sepnsepolem1 48838
                  21.49.10.7  Separated spaces: T0, T1, T2 (Hausdorff) ...   isnrm4 48847
            21.49.11  Preordered sets and directed sets using extensible structures   isprsd 48871
            21.49.12  Posets and lattices using extensible structures   lubeldm2 48872
                  21.49.12.1  Posets   lubeldm2 48872
                  21.49.12.2  Lattices   toslat 48898
                  21.49.12.3  Subset order structures   intubeu 48900
            21.49.13  Rings   elmgpcntrd 48921
                  21.49.13.1  Multiplicative Group   elmgpcntrd 48921
            21.49.14  Associative algebras   asclelbas 48922
                  21.49.14.1  Definition and basic properties   asclelbas 48922
            21.49.15  Categories   homf0 48926
                  21.49.15.1  Categories   homf0 48926
                  21.49.15.2  Opposite category   oppccatb 48933
                  21.49.15.3  Monomorphisms and epimorphisms   idmon 48937
                  21.49.15.4  Sections, inverses, isomorphisms   sectrcl 48939
                  21.49.15.5  Isomorphic objects   cicfn 48959
                  21.49.15.6  Subcategories   dmdm 48970
                  21.49.15.7  Functors   reldmfunc 48992
                  21.49.15.8  Opposite functors   coppf 49039
                  21.49.15.9  Full & faithful functors   imasubc 49062
                  21.49.15.10  Universal property   upciclem1 49074
                  21.49.15.11  Natural transformations and the functor category   isnatd 49127
                  21.49.15.12  Initial, terminal and zero objects of a category   initoo2 49133
                  21.49.15.13  Product of categories   reldmxpc 49147
                  21.49.15.14  Swap functors   cswapf 49160
                  21.49.15.15  Transposed curry functors   cofuswapfcl 49188
                  21.49.15.16  Constant functors   diag1 49199
                  21.49.15.17  Functor composition bifunctors   fucofulem1 49205
                  21.49.15.18  Post-composition functors   postcofval 49259
                  21.49.15.19  Pre-composition functors   precofvallem 49261
            21.49.16  Examples of categories   catcrcl 49287
                  21.49.16.1  The category of categories   catcrcl 49287
                  21.49.16.2  Thin categories   cthinc 49295
                  21.49.16.3  Terminal categories   ctermc 49350
                  21.49.16.4  Preordered sets as thin categories   cprstc 49427
                  21.49.16.5  Monoids as categories   cmndtc 49455
                  21.49.16.6  Categories with at most one object and at most two morphisms   2arwcatlem1 49473
            21.49.17  Kan extensions and related concepts   clan 49483
                  21.49.17.1  Kan extensions   clan 49483
                  21.49.17.2  Limits and colimits   clmd 49518
      ​21.50  Mathbox for Emmett Weisz
            ​*21.50.1  Miscellaneous Theorems   nfintd 49539
            21.50.2  Set Recursion   csetrecs 49549
                  ​*21.50.2.1  Basic Properties of Set Recursion   csetrecs 49549
                  21.50.2.2  Examples and properties of set recursion   elsetrecslem 49565
            ​*21.50.3  Construction of Games and Surreal Numbers   cpg 49575
      ​​*21.51  Mathbox for David A. Wheeler
            21.51.1  Natural deduction   sbidd 49584
            ​*21.51.2  Greater than, greater than or equal to.   cge-real 49586
            ​*21.51.3  Hyperbolic trigonometric functions   csinh 49596
            ​*21.51.4  Reciprocal trigonometric functions (sec, csc, cot)   csec 49607
            ​*21.51.5  Identities for "if"   ifnmfalse 49629
            ​*21.51.6  Logarithms generalized to arbitrary base using ` logb `   logb2aval 49630
            ​*21.51.7  Logarithm laws generalized to an arbitrary base - log_   clog- 49631
            ​*21.51.8  Formally define notions such as reflexivity   wreflexive 49633
            ​*21.51.9  Algebra helpers   mvlraddi 49637
            ​*21.51.10  Algebra helper examples   i2linesi 49644
            ​*21.51.11  Formal methods "surprises"   alimp-surprise 49646
            ​*21.51.12  Allsome quantifier   walsi 49652
            ​*21.51.13  Miscellaneous   5m4e1 49663
            21.51.14  Theorems about algebraic numbers   aacllem 49667
      ​21.52  Mathbox for Kunhao Zheng
            21.52.1  Weighted AM-GM inequality   amgmwlem 49668