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 2370
      ​1.6  Uniqueness and unique existence
            1.6.1  Uniqueness: the at-most-one quantifier   wmo 2531
            1.6.2  Unique existence: the unique existential quantifier   weu 2561
      ​1.7  Other axiomatizations related to classical predicate calculus
            ​*1.7.1  Aristotelian logic: Assertic syllogisms   barbara 2656
            ​*1.7.2  Intuitionistic logic   axia1 2686
​​*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 2701
            2.1.2  Classes   cab 2707
                  2.1.2.1  Class abstractions   cab 2707
                  ​*2.1.2.2  Class equality   df-cleq 2721
                  2.1.2.3  Class membership   df-clel 2803
                  2.1.2.4  Elementary properties of class abstractions   eqabdv 2861
            2.1.3  Class form not-free predicate   wnfc 2876
            2.1.4  Negated equality and membership   wne 2925
                  2.1.4.1  Negated equality   wne 2925
                  2.1.4.2  Negated membership   wnel 3029
            2.1.5  Restricted quantification   wral 3044
                  2.1.5.1  Restricted universal and existential quantification   wral 3044
                  2.1.5.2  Restricted existential uniqueness and at-most-one quantifier   wreu 3343
                  2.1.5.3  Restricted class abstraction   crab 3396
            2.1.6  The universal class   cvv 3438
            ​*2.1.7  Conditional equality (experimental)   wcdeq 3725
            2.1.8  Russell's Paradox   rru 3741
            2.1.9  Proper substitution of classes for sets   wsbc 3744
            2.1.10  Proper substitution of classes for sets into classes   csb 3853
            2.1.11  Define basic set operations and relations   cdif 3902
            2.1.12  Subclasses and subsets   df-ss 3922
            2.1.13  The difference, union, and intersection of two classes   dfdif3 4070
                  2.1.13.1  The difference of two classes   dfdif3 4070
                  2.1.13.2  The union of two classes   elun 4106
                  2.1.13.3  The intersection of two classes   elini 4152
                  2.1.13.4  The symmetric difference of two classes   csymdif 4205
                  2.1.13.5  Combinations of difference, union, and intersection of two classes   unabs 4218
                  2.1.13.6  Class abstractions with difference, union, and intersection of two classes   unabw 4260
                  2.1.13.7  Restricted uniqueness with difference, union, and intersection   reuun2 4278
            2.1.14  The empty set   c0 4286
            ​*2.1.15  The conditional operator for classes   cif 4478
            ​*2.1.16  The weak deduction theorem for set theory   dedth 4537
            2.1.17  Power classes   cpw 4553
            2.1.18  Unordered and ordered pairs   snjust 4578
            2.1.19  The union of a class   cuni 4861
            2.1.20  The intersection of a class   cint 4899
            2.1.21  Indexed union and intersection   ciun 4944
            2.1.22  Disjointness   wdisj 5062
            2.1.23  Binary relations   wbr 5095
            2.1.24  Ordered-pair class abstractions (class builders)   copab 5157
            2.1.25  Functions in maps-to notation   cmpt 5176
            2.1.26  Transitive classes   wtr 5202
      ​2.2  ZF Set Theory - add the Axiom of Replacement
            2.2.1  Introduce the Axiom of Replacement   ax-rep 5221
            2.2.2  Derive the Axiom of Separation   axsepgfromrep 5236
            2.2.3  Derive the Null Set Axiom   axnulALT 5246
            2.2.4  Theorems requiring subset and intersection existence   nalset 5255
            2.2.5  Theorems requiring empty set existence   class2set 5297
      ​2.3  ZF Set Theory - add the Axiom of Power Sets
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 5307
            2.3.2  Derive the Axiom of Pairing   axprlem1 5365
            2.3.3  Ordered pair theorem   opnz 5420
            2.3.4  Ordered-pair class abstractions (cont.)   opabidw 5471
            2.3.5  Power class of union and intersection   pwin 5514
            2.3.6  The identity relation   cid 5517
            2.3.7  The membership relation (or epsilon relation)   cep 5522
            ​*2.3.8  Partial and total orderings   wpo 5529
            2.3.9  Founded and well-ordering relations   wfr 5573
            2.3.10  Relations   cxp 5621
            2.3.11  The Predecessor Class   cpred 6252
            2.3.12  Well-founded induction (variant)   frpomin 6292
            2.3.13  Well-ordered induction   tz6.26 6299
            2.3.14  Ordinals   word 6310
            2.3.15  Definite description binder (inverted iota)   cio 6440
            2.3.16  Functions   wfun 6480
            2.3.17  Cantor's Theorem   canth 7307
            2.3.18  Restricted iota (description binder)   crio 7309
            2.3.19  Operations   co 7353
                  2.3.19.1  Variable-to-class conversion for operations   caovclg 7545
            2.3.20  Maps-to notation   mpondm0 7593
            2.3.21  Function operation   cof 7615
            2.3.22  Proper subset relation   crpss 7662
      ​2.4  ZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 7675
            2.4.2  Ordinals (continued)   epweon 7715
            2.4.3  Transfinite induction   tfi 7793
            2.4.4  The natural numbers (i.e., finite ordinals)   com 7806
            2.4.5  Peano's postulates   peano1 7829
            2.4.6  Finite induction (for finite ordinals)   find 7835
            2.4.7  Relations and functions (cont.)   dmexg 7841
            2.4.8  First and second members of an ordered pair   c1st 7929
            2.4.9  Induction on Cartesian products   frpoins3xpg 8080
            2.4.10  Ordering on Cartesian products   xpord2lem 8082
            2.4.11  Ordering Ordinal Sequences   orderseqlem 8097
            ​*2.4.12  The support of functions   csupp 8100
            ​*2.4.13  Special maps-to operations   opeliunxp2f 8150
            2.4.14  Function transposition   ctpos 8165
            2.4.15  Curry and uncurry   ccur 8205
            2.4.16  Undefined values   cund 8212
            2.4.17  Well-founded recursion   cfrecs 8220
            2.4.18  Well-ordered recursion   cwrecs 8251
            2.4.19  Functions on ordinals; strictly monotone ordinal functions   iunon 8269
            2.4.20  "Strong" transfinite recursion   crecs 8300
            2.4.21  Recursive definition generator   crdg 8338
            2.4.22  Finite recursion   frfnom 8364
            2.4.23  Ordinal arithmetic   c1o 8388
            2.4.24  Natural number arithmetic   nna0 8529
            2.4.25  Natural addition   cnadd 8590
            2.4.26  Equivalence relations and classes   wer 8629
            2.4.27  The mapping operation   cmap 8760
            2.4.28  Infinite Cartesian products   cixp 8831
            2.4.29  Equinumerosity   cen 8876
            2.4.30  Schroeder-Bernstein Theorem   sbthlem1 9011
            2.4.31  Equinumerosity (cont.)   xpf1o 9063
            2.4.32  Finite sets   dif1enlem 9080
            2.4.33  Pigeonhole Principle   phplem1 9128
            2.4.34  Finite sets (cont.)   onomeneq 9138
            2.4.35  Finitely supported functions   cfsupp 9270
            2.4.36  Finite intersections   cfi 9319
            2.4.37  Hall's marriage theorem   marypha1lem 9342
            2.4.38  Supremum and infimum   csup 9349
            2.4.39  Ordinal isomorphism, Hartogs's theorem   coi 9420
            2.4.40  Hartogs function   char 9467
            2.4.41  Weak dominance   cwdom 9475
      ​2.5  ZF Set Theory - add the Axiom of Regularity
            2.5.1  Introduce the Axiom of Regularity   ax-reg 9503
            2.5.2  Axiom of Infinity equivalents   inf0 9536
      ​2.6  ZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-inf 9553
            2.6.2  Existence of omega (the set of natural numbers)   omex 9558
            2.6.3  Cantor normal form   ccnf 9576
            2.6.4  Transitive closure of a relation   cttrcl 9622
            2.6.5  Transitive closure   trcl 9643
            2.6.6  Well-Founded Induction   frmin 9664
            2.6.7  Well-Founded Recursion   frr3g 9671
            2.6.8  Rank   cr1 9677
            2.6.9  Scott's trick; collection principle; Hilbert's epsilon   scottex 9800
            2.6.10  Disjoint union   cdju 9813
            2.6.11  Cardinal numbers   ccrd 9850
            2.6.12  Axiom of Choice equivalents   wac 10028
            ​*2.6.13  Cardinal number arithmetic   undjudom 10081
            2.6.14  The Ackermann bijection   ackbij2lem1 10131
            2.6.15  Cofinality (without Axiom of Choice)   cflem 10158
            2.6.16  Eight inequivalent definitions of finite set   sornom 10190
            2.6.17  Hereditarily size-limited sets without Choice   itunifval 10329
​​*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 10348
            3.1.2  Introduce the Axiom of Dependent Choice   ax-dc 10359
      ​3.2  ZFC Set Theory - add the Axiom of Choice
            3.2.1  Introduce the Axiom of Choice   ax-ac 10372
            3.2.2  AC equivalents: well-ordering, Zorn's lemma   numthcor 10407
            3.2.3  Cardinal number theorems using Axiom of Choice   cardval 10459
            3.2.4  Cardinal number arithmetic using Axiom of Choice   iunctb 10487
            3.2.5  Cofinality using the Axiom of Choice   alephreg 10495
      ​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 10533
            3.4.2  Derivation of the Axiom of Choice   gchaclem 10591
​​*PART 4  TG (TARSKI-GROTHENDIECK) SET THEORY
      ​4.1  Inaccessibles
            4.1.1  Weakly and strongly inaccessible cardinals   cwina 10595
            4.1.2  Weak universes   cwun 10613
            4.1.3  Tarski classes   ctsk 10661
            4.1.4  Grothendieck universes   cgru 10703
      ​4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
            4.2.1  Introduce the Tarski-Grothendieck Axiom   ax-groth 10736
            4.2.2  Derive the Power Set, Infinity and Choice Axioms   grothpw 10739
            4.2.3  Tarski map function   ctskm 10750
​​*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 10757
            5.1.2  Final derivation of real and complex number postulates   axaddf 11058
            5.1.3  Real and complex number postulates restated as axioms   ax-cnex 11084
      ​5.2  Derive the basic properties from the field axioms
            5.2.1  Some deductions from the field axioms for complex numbers   cnex 11109
            5.2.2  Infinity and the extended real number system   cpnf 11165
            5.2.3  Restate the ordering postulates with extended real "less than"   axlttri 11205
            5.2.4  Ordering on reals   lttr 11210
            5.2.5  Initial properties of the complex numbers   mul12 11299
      ​5.3  Real and complex numbers - basic operations
            5.3.1  Addition   add12 11352
            5.3.2  Subtraction   cmin 11365
            5.3.3  Multiplication   kcnktkm1cn 11569
            5.3.4  Ordering on reals (cont.)   gt0ne0 11603
            5.3.5  Reciprocals   ixi 11767
            5.3.6  Division   cdiv 11795
            5.3.7  Ordering on reals (cont.)   elimgt0 11980
            5.3.8  Completeness Axiom and Suprema   fimaxre 12087
            5.3.9  Imaginary and complex number properties   neg1cn 12131
            5.3.10  Function operation analogue theorems   ofsubeq0 12143
      ​5.4  Integer sets
            5.4.1  Positive integers (as a subset of complex numbers)   cn 12146
            5.4.2  Principle of mathematical induction   nnind 12164
            ​*5.4.3  Decimal representation of numbers   c2 12201
            ​*5.4.4  Some properties of specific numbers   1pneg1e0 12260
            5.4.5  Simple number properties   halfcl 12368
            5.4.6  The Archimedean property   nnunb 12398
            5.4.7  Nonnegative integers (as a subset of complex numbers)   cn0 12402
            ​*5.4.8  Extended nonnegative integers   cxnn0 12475
            5.4.9  Integers (as a subset of complex numbers)   cz 12489
            5.4.10  Decimal arithmetic   cdc 12609
            5.4.11  Upper sets of integers   cuz 12753
            5.4.12  Well-ordering principle for bounded-below sets of integers   uzwo3 12862
            5.4.13  Rational numbers (as a subset of complex numbers)   cq 12867
            5.4.14  Existence of the set of complex numbers   rpnnen1lem2 12896
      ​5.5  Order sets
            5.5.1  Positive reals (as a subset of complex numbers)   crp 12911
            5.5.2  Infinity and the extended real number system (cont.)   cxne 13029
            5.5.3  Supremum and infimum on the extended reals   xrsupexmnf 13225
            5.5.4  Real number intervals   cioo 13266
            5.5.5  Finite intervals of integers   cfz 13428
            ​*5.5.6  Finite intervals of nonnegative integers   elfz2nn0 13539
            5.5.7  Half-open integer ranges   cfzo 13575
      ​5.6  Elementary integer functions
            5.6.1  The floor and ceiling functions   cfl 13712
            5.6.2  The modulo (remainder) operation   cmo 13791
            5.6.3  Miscellaneous theorems about integers   om2uz0i 13872
            5.6.4  Strong induction over upper sets of integers   uzsinds 13912
            5.6.5  Finitely supported functions over the nonnegative integers   fsuppmapnn0fiublem 13915
            5.6.6  The infinite sequence builder "seq" - extension   cseq 13926
            5.6.7  Integer powers   cexp 13986
            5.6.8  Ordered pair theorem for nonnegative integers   nn0le2msqi 14192
            5.6.9  Factorial function   cfa 14198
            5.6.10  The binomial coefficient operation   cbc 14227
            5.6.11  The ` # ` (set size) function   chash 14255
                  5.6.11.1  Proper unordered pairs and triples (sets of size 2 and 3)   hashprlei 14393
                  5.6.11.2  Functions with a domain containing at least two different elements   fundmge2nop0 14427
                  5.6.11.3  Finite induction on the size of the first component of a binary relation   hashdifsnp1 14431
      ​​*5.7  Words over a set
            5.7.1  Definitions and basic theorems   cword 14438
            5.7.2  Last symbol of a word   clsw 14487
            5.7.3  Concatenations of words   cconcat 14495
            5.7.4  Singleton words   cs1 14520
            5.7.5  Concatenations with singleton words   ccatws1cl 14541
            5.7.6  Subwords/substrings   csubstr 14565
            5.7.7  Prefixes of a word   cpfx 14595
            5.7.8  Subwords of subwords   swrdswrdlem 14628
            5.7.9  Subwords and concatenations   pfxcctswrd 14634
            5.7.10  Subwords of concatenations   swrdccatfn 14648
            5.7.11  Splicing words (substring replacement)   csplice 14673
            5.7.12  Reversing words   creverse 14682
            5.7.13  Repeated symbol words   creps 14692
            ​*5.7.14  Cyclical shifts of words   ccsh 14712
            5.7.15  Mapping words by a function   wrdco 14756
            5.7.16  Longer string literals   cs2 14766
      ​​*5.8  Reflexive and transitive closures of relations
            5.8.1  The reflexive and transitive properties of relations   coss12d 14897
            5.8.2  Basic properties of closures   cleq1lem 14907
            5.8.3  Definitions and basic properties of transitive closures   ctcl 14910
            5.8.4  Exponentiation of relations   crelexp 14944
            5.8.5  Reflexive-transitive closure as an indexed union   crtrcl 14980
            ​*5.8.6  Principle of transitive induction.   relexpindlem 14988
      ​5.9  Elementary real and complex functions
            5.9.1  The "shift" operation   cshi 14991
            5.9.2  Signum (sgn or sign) function   csgn 15011
            5.9.3  Real and imaginary parts; conjugate   ccj 15021
            5.9.4  Square root; absolute value   csqrt 15158
      ​5.10  Elementary limits and convergence
            5.10.1  Superior limit (lim sup)   clsp 15395
            5.10.2  Limits   cli 15409
            5.10.3  Finite and infinite sums   csu 15611
            5.10.4  The binomial theorem   binomlem 15754
            5.10.5  The inclusion/exclusion principle   incexclem 15761
            5.10.6  Infinite sums (cont.)   isumshft 15764
            5.10.7  Miscellaneous converging and diverging sequences   divrcnv 15777
            5.10.8  Arithmetic series   arisum 15785
            5.10.9  Geometric series   expcnv 15789
            5.10.10  Ratio test for infinite series convergence   cvgrat 15808
            5.10.11  Mertens' theorem   mertenslem1 15809
            5.10.12  Finite and infinite products   prodf 15812
                  5.10.12.1  Product sequences   prodf 15812
                  5.10.12.2  Non-trivial convergence   ntrivcvg 15822
                  5.10.12.3  Complex products   cprod 15828
                  5.10.12.4  Finite products   fprod 15866
                  5.10.12.5  Infinite products   iprodclim 15923
            5.10.13  Falling and Rising Factorial   cfallfac 15929
            5.10.14  Bernoulli polynomials and sums of k-th powers   cbp 15971
      ​5.11  Elementary trigonometry
            5.11.1  The exponential, sine, and cosine functions   ce 15986
                  5.11.1.1  The circle constant (tau = 2 pi)   ctau 16129
            5.11.2  _e is irrational   eirrlem 16131
      ​5.12  Cardinality of real and complex number subsets
            5.12.1  Countability of integers and rationals   xpnnen 16138
            5.12.2  The reals are uncountable   rpnnen2lem1 16141
​​*PART 6  ELEMENTARY NUMBER THEORY
      ​6.1  Elementary properties of divisibility
            6.1.1  Irrationality of square root of 2   sqrt2irrlem 16175
            6.1.2  Some Number sets are chains of proper subsets   nthruc 16179
            6.1.3  The divides relation   cdvds 16181
            ​*6.1.4  Even and odd numbers   evenelz 16265
            6.1.5  The division algorithm   divalglem0 16322
            6.1.6  Bit sequences   cbits 16348
            6.1.7  The greatest common divisor operator   cgcd 16423
            6.1.8  Bézout's identity   bezoutlem1 16468
            6.1.9  Algorithms   nn0seqcvgd 16499
            6.1.10  Euclid's Algorithm   eucalgval2 16510
            ​*6.1.11  The least common multiple   clcm 16517
            ​*6.1.12  Coprimality and Euclid's lemma   coprmgcdb 16578
            6.1.13  Cancellability of congruences   congr 16593
      ​6.2  Elementary prime number theory
            ​*6.2.1  Elementary properties   cprime 16600
            ​*6.2.2  Coprimality and Euclid's lemma (cont.)   coprm 16640
            6.2.3  Properties of the canonical representation of a rational   cnumer 16662
            6.2.4  Euler's theorem   codz 16692
            6.2.5  Arithmetic modulo a prime number   modprm1div 16727
            6.2.6  Pythagorean Triples   coprimeprodsq 16738
            6.2.7  The prime count function   cpc 16766
            6.2.8  Pocklington's theorem   prmpwdvds 16834
            6.2.9  Infinite primes theorem   unbenlem 16838
            6.2.10  Sum of prime reciprocals   prmreclem1 16846
            6.2.11  Fundamental theorem of arithmetic   1arithlem1 16853
            6.2.12  Lagrange's four-square theorem   cgz 16859
            6.2.13  Van der Waerden's theorem   cvdwa 16895
            6.2.14  Ramsey's theorem   cram 16929
            ​*6.2.15  Primorial function   cprmo 16961
            ​*6.2.16  Prime gaps   prmgaplem1 16979
            6.2.17  Decimal arithmetic (cont.)   dec2dvds 16993
            6.2.18  Cyclical shifts of words (cont.)   cshwsidrepsw 17023
            6.2.19  Specific prime numbers   prmlem0 17035
            6.2.20  Very large primes   1259lem1 17060
​PART 7  BASIC STRUCTURES
      ​7.1  Extensible structures
            ​*7.1.1  Basic definitions   cstr 17075
                  7.1.1.1  Extensible structures as structures with components   cstr 17075
                  7.1.1.2  Substitution of components   csts 17092
                  7.1.1.3  Slots   cslot 17110
                  ​*7.1.1.4  Structure component indices   cnx 17122
                  7.1.1.5  Base sets   cbs 17138
                  7.1.1.6  Base set restrictions   cress 17159
            7.1.2  Slot definitions   cplusg 17179
            7.1.3  Definition of the structure product   crest 17342
            7.1.4  Definition of the structure quotient   cordt 17421
      ​7.2  Moore spaces
            7.2.1  Moore closures   mrcflem 17530
            7.2.2  Independent sets in a Moore system   mrisval 17554
            7.2.3  Algebraic closure systems   isacs 17575
​PART 8  BASIC CATEGORY THEORY
      ​8.1  Categories
            8.1.1  Categories   ccat 17588
            8.1.2  Opposite category   coppc 17635
            8.1.3  Monomorphisms and epimorphisms   cmon 17653
            8.1.4  Sections, inverses, isomorphisms   csect 17669
            ​*8.1.5  Isomorphic objects   ccic 17720
            8.1.6  Subcategories   cssc 17732
            8.1.7  Functors   cfunc 17779
            8.1.8  Full & faithful functors   cful 17829
            8.1.9  Natural transformations and the functor category   cnat 17869
            8.1.10  Initial, terminal and zero objects of a category   cinito 17906
      ​8.2  Arrows (disjointified hom-sets)
            8.2.1  Identity and composition for arrows   cida 17978
      ​8.3  Examples of categories
            8.3.1  The category of sets   csetc 18000
            8.3.2  The category of categories   ccatc 18023
            ​*8.3.3  The category of extensible structures   fncnvimaeqv 18044
      ​8.4  Categorical constructions
            8.4.1  Product of categories   cxpc 18092
            8.4.2  Functor evaluation   cevlf 18133
            8.4.3  Hom functor   chof 18172
​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 18355
            9.5.2  Complete lattices   ccla 18422
            9.5.3  Distributive lattices   cdlat 18444
            9.5.4  Subset order structures   cipo 18451
      ​9.6  Posets, directed sets, and lattices as relations
            ​*9.6.1  Posets and lattices as relations   cps 18488
            9.6.2  Directed sets, nets   cdir 18518
​PART 10  BASIC ALGEBRAIC STRUCTURES
      ​10.1  Monoids
            ​*10.1.1  Magmas   cplusf 18529
            ​*10.1.2  Identity elements   mgmidmo 18552
            ​*10.1.3  Iterated sums in a magma   gsumvalx 18568
            10.1.4  Magma homomorphisms and submagmas   cmgmhm 18582
            ​*10.1.5  Semigroups   csgrp 18610
            ​*10.1.6  Definition and basic properties of monoids   cmnd 18626
            10.1.7  Monoid homomorphisms and submonoids   cmhm 18673
            ​*10.1.8  Iterated sums in a monoid   gsumvallem2 18726
            10.1.9  Free monoids   cfrmd 18739
                  ​*10.1.9.1  Monoid of endofunctions   cefmnd 18760
            10.1.10  Examples and counterexamples for magmas, semigroups and monoids   mgm2nsgrplem1 18810
      ​10.2  Groups
            10.2.1  Definition and basic properties   cgrp 18830
            ​*10.2.2  Group multiple operation   cmg 18964
            10.2.3  Subgroups and Quotient groups   csubg 19017
            ​*10.2.4  Cyclic monoids and groups   cycsubmel 19097
            10.2.5  Elementary theory of group homomorphisms   cghm 19109
            10.2.6  Isomorphisms of groups   cgim 19154
                  10.2.6.1  The first isomorphism theorem of groups   ghmqusnsglem1 19177
            10.2.7  Group actions   cga 19186
            10.2.8  Centralizers and centers   ccntz 19212
            10.2.9  The opposite group   coppg 19242
            10.2.10  Symmetric groups   csymg 19266
                  ​*10.2.10.1  Definition and basic properties   csymg 19266
                  10.2.10.2  Cayley's theorem   cayleylem1 19309
                  10.2.10.3  Permutations fixing one element   symgfix2 19313
                  ​*10.2.10.4  Transpositions in the symmetric group   cpmtr 19338
                  10.2.10.5  The sign of a permutation   cpsgn 19386
            10.2.11  p-Groups and Sylow groups; Sylow's theorems   cod 19421
            10.2.12  Direct products   clsm 19531
                  10.2.12.1  Direct products (extension)   smndlsmidm 19553
            10.2.13  Free groups   cefg 19603
            10.2.14  Abelian groups   ccmn 19677
                  10.2.14.1  Definition and basic properties   ccmn 19677
                  10.2.14.2  Cyclic groups   ccyg 19774
                  10.2.14.3  Group sum operation   gsumval3a 19800
                  10.2.14.4  Group sums over (ranges of) integers   fsfnn0gsumfsffz 19880
                  10.2.14.5  Internal direct products   cdprd 19892
                  10.2.14.6  The Fundamental Theorem of Abelian Groups   ablfacrplem 19964
            10.2.15  Simple groups   csimpg 19989
                  10.2.15.1  Definition and basic properties   csimpg 19989
                  10.2.15.2  Classification of abelian simple groups   ablsimpnosubgd 20003
            10.2.16  Totally ordered monoids and groups   comnd 20016
      ​10.3  Rings
            10.3.1  Multiplicative Group   cmgp 20043
            ​*10.3.2  Non-unital rings ("rngs")   crng 20055
            ​*10.3.3  Ring unity (multiplicative identity)   cur 20084
            10.3.4  Semirings   csrg 20089
                  ​*10.3.4.1  The binomial theorem for semirings   srgbinomlem1 20129
            10.3.5  Unital rings   crg 20136
            10.3.6  Opposite ring   coppr 20239
            10.3.7  Divisibility   cdsr 20257
            10.3.8  Ring primes   crpm 20335
            10.3.9  Homomorphisms of non-unital rings   crnghm 20337
            10.3.10  Ring homomorphisms   crh 20372
            10.3.11  Nonzero rings and zero rings   cnzr 20415
            10.3.12  Local rings   clring 20441
            10.3.13  Subrings   csubrng 20448
                  10.3.13.1  Subrings of non-unital rings   csubrng 20448
                  10.3.13.2  Subrings of unital rings   csubrg 20472
                  10.3.13.3  Subrings generated by a subset   crgspn 20513
            10.3.14  Categories of rings   crngc 20519
                  ​*10.3.14.1  The category of non-unital rings   crngc 20519
                  ​*10.3.14.2  The category of (unital) rings   cringc 20548
                  10.3.14.3  Subcategories of the category of rings   srhmsubclem1 20580
            10.3.15  Left regular elements and domains   crlreg 20594
      ​10.4  Division rings and fields
            10.4.1  Definition and basic properties   cdr 20632
            10.4.2  Sub-division rings   csdrg 20689
            10.4.3  Absolute value (abstract algebra)   cabv 20711
            10.4.4  Star rings   cstf 20740
            10.4.5  Totally ordered rings and fields   corng 20760
      ​10.5  Left modules
            10.5.1  Definition and basic properties   clmod 20781
            10.5.2  Subspaces and spans in a left module   clss 20852
            10.5.3  Homomorphisms and isomorphisms of left modules   clmhm 20941
            10.5.4  Subspace sum; bases for a left module   clbs 20996
      ​10.6  Vector spaces
            10.6.1  Definition and basic properties   clvec 21024
      ​10.7  Subring algebras and ideals
            10.7.1  Subring algebras   csra 21093
            ​*10.7.2  Left ideals and spans   clidl 21131
            10.7.3  Two-sided ideals and quotient rings   c2idl 21174
                  ​*10.7.3.1  Condition for a non-unital ring to be unital   rngqiprng1elbas 21211
            10.7.4  Principal ideal rings. Divisibility in the integers   clpidl 21245
            10.7.5  Principal ideal domains   cpid 21261
      ​10.8  The complex numbers as an algebraic extensible structure
            10.8.1  Definition and basic properties   cpsmet 21263
            ​*10.8.2  Ring of integers   czring 21371
                  ​*10.8.2.1  Example for a condition for a non-unital ring to be unital   pzriprnglem1 21406
            10.8.3  Algebraic constructions based on the complex numbers   czrh 21424
            10.8.4  Signs as subgroup of the complex numbers   cnmsgnsubg 21502
            10.8.5  Embedding of permutation signs into a ring   zrhpsgnmhm 21509
            10.8.6  The ordered field of real numbers   crefld 21529
      ​10.9  Generalized pre-Hilbert and Hilbert spaces
            10.9.1  Definition and basic properties   cphl 21549
            10.9.2  Orthocomplements and closed subspaces   cocv 21585
            10.9.3  Orthogonal projection and orthonormal bases   cpj 21625
​​*PART 11  BASIC LINEAR ALGEBRA
      ​11.1  Vectors and free modules
            *11.1.1  Direct sum of left modules   cdsmm 21656
            ​*11.1.2  Free modules   cfrlm 21671
            ​*11.1.3  Standard basis (unit vectors)   cuvc 21707
            ​*11.1.4  Independent sets and families   clindf 21729
            11.1.5  Characterization of free modules   lmimlbs 21761
      ​11.2  Associative algebras
            11.2.1  Definition and basic properties   casa 21775
      ​11.3  Abstract multivariate polynomials
            11.3.1  Definition and basic properties   cmps 21829
            11.3.2  Polynomial evaluation   ces 21995
            11.3.3  Additional definitions for (multivariate) polynomials   cslv 22031
            ​*11.3.4  Univariate polynomials   cps1 22075
            11.3.5  Univariate polynomial evaluation   ces1 22216
                  11.3.5.1  Specialization of polynomial evaluation as a ring homomorphism   evls1scafv 22269
      ​​*11.4  Matrices
            *11.4.1  The matrix multiplication   cmmul 22293
            ​*11.4.2  Square matrices   cmat 22310
            ​*11.4.3  The matrix algebra   matmulr 22341
            ​*11.4.4  Matrices of dimension 0 and 1   mat0dimbas0 22369
            ​*11.4.5  The subalgebras of diagonal and scalar matrices   cdmat 22391
            ​*11.4.6  Multiplication of a matrix with a "column vector"   cmvmul 22443
            11.4.7  Replacement functions for a square matrix   cmarrep 22459
            11.4.8  Submatrices   csubma 22479
      ​11.5  The determinant
            11.5.1  Definition and basic properties   cmdat 22487
            11.5.2  Determinants of 2 x 2 -matrices   m2detleiblem1 22527
            11.5.3  The matrix adjugate/adjunct   cmadu 22535
            ​*11.5.4  Laplace expansion of determinants (special case)   symgmatr01lem 22556
            11.5.5  Inverse matrix   invrvald 22579
            ​*11.5.6  Cramer's rule   slesolvec 22582
      ​​*11.6  Polynomial matrices
            11.6.1  Basic properties   pmatring 22595
            ​*11.6.2  Constant polynomial matrices   ccpmat 22606
            ​*11.6.3  Collecting coefficients of polynomial matrices   cdecpmat 22665
            ​*11.6.4  Ring isomorphism between polynomial matrices and polynomials over matrices   cpm2mp 22695
      ​​*11.7  The characteristic polynomial
            *11.7.1  Definition and basic properties   cchpmat 22729
            ​*11.7.2  The characteristic factor function G   fvmptnn04if 22752
            ​*11.7.3  The Cayley-Hamilton theorem   cpmadurid 22770
​PART 12  BASIC TOPOLOGY
      ​12.1  Topology
            ​*12.1.1  Topological spaces   ctop 22796
                  12.1.1.1  Topologies   ctop 22796
                  12.1.1.2  Topologies on sets   ctopon 22813
                  12.1.1.3  Topological spaces   ctps 22835
            12.1.2  Topological bases   ctb 22848
            12.1.3  Examples of topologies   distop 22898
            12.1.4  Closure and interior   ccld 22919
            12.1.5  Neighborhoods   cnei 23000
            12.1.6  Limit points and perfect sets   clp 23037
            12.1.7  Subspace topologies   restrcl 23060
            12.1.8  Order topology   ordtbaslem 23091
            12.1.9  Limits and continuity in topological spaces   ccn 23127
            12.1.10  Separated spaces: T0, T1, T2 (Hausdorff) ...   ct0 23209
            12.1.11  Compactness   ccmp 23289
            12.1.12  Bolzano-Weierstrass theorem   bwth 23313
            12.1.13  Connectedness   cconn 23314
            12.1.14  First- and second-countability   c1stc 23340
            12.1.15  Local topological properties   clly 23367
            12.1.16  Refinements   cref 23405
            12.1.17  Compactly generated spaces   ckgen 23436
            12.1.18  Product topologies   ctx 23463
            12.1.19  Continuous function-builders   cnmptid 23564
            12.1.20  Quotient maps and quotient topology   ckq 23596
            12.1.21  Homeomorphisms   chmeo 23656
      ​12.2  Filters and filter bases
            12.2.1  Filter bases   elmptrab 23730
            12.2.2  Filters   cfil 23748
            12.2.3  Ultrafilters   cufil 23802
            12.2.4  Filter limits   cfm 23836
            12.2.5  Extension by continuity   ccnext 23962
            12.2.6  Topological groups   ctmd 23973
            12.2.7  Infinite group sum on topological groups   ctsu 24029
            12.2.8  Topological rings, fields, vector spaces   ctrg 24059
      ​12.3  Uniform Structures and Spaces
            12.3.1  Uniform structures   cust 24103
            12.3.2  The topology induced by an uniform structure   cutop 24134
            12.3.3  Uniform Spaces   cuss 24157
            12.3.4  Uniform continuity   cucn 24178
            12.3.5  Cauchy filters in uniform spaces   ccfilu 24189
            12.3.6  Complete uniform spaces   ccusp 24200
      ​12.4  Metric spaces
            12.4.1  Pseudometric spaces   ispsmet 24208
            12.4.2  Basic metric space properties   cxms 24221
            12.4.3  Metric space balls   blfvalps 24287
            12.4.4  Open sets of a metric space   mopnval 24342
            12.4.5  Continuity in metric spaces   metcnp3 24444
            12.4.6  The uniform structure generated by a metric   metuval 24453
            12.4.7  Examples of metric spaces   dscmet 24476
            ​*12.4.8  Normed algebraic structures   cnm 24480
            12.4.9  Normed space homomorphisms (bounded linear operators)   cnmo 24609
            12.4.10  Topology on the reals   qtopbaslem 24662
            12.4.11  Topological definitions using the reals   cii 24784
            12.4.12  Path homotopy   chtpy 24882
            12.4.13  The fundamental group   cpco 24916
      ​12.5  Metric subcomplex vector spaces
            12.5.1  Subcomplex modules   cclm 24978
            ​*12.5.2  Subcomplex vector spaces   ccvs 25039
            ​*12.5.3  Normed subcomplex vector spaces   isncvsngp 25065
            12.5.4  Subcomplex pre-Hilbert spaces   ccph 25082
            12.5.5  Convergence and completeness   ccfil 25168
            12.5.6  Baire's Category Theorem   bcthlem1 25240
            12.5.7  Banach spaces and subcomplex Hilbert spaces   ccms 25248
                  12.5.7.1  The complete ordered field of the real numbers   retopn 25295
            12.5.8  Euclidean spaces   crrx 25299
            12.5.9  Minimizing Vector Theorem   minveclem1 25340
            12.5.10  Projection Theorem   pjthlem1 25353
​PART 13  BASIC REAL AND COMPLEX ANALYSIS
      ​13.1  Continuity
            13.1.1  Intermediate value theorem   pmltpclem1 25365
      ​13.2  Integrals
            13.2.1  Lebesgue measure   covol 25379
            13.2.2  Lebesgue integration   cmbf 25531
                  13.2.2.1  Lesbesgue integral   cmbf 25531
                  13.2.2.2  Lesbesgue directed integral   cdit 25763
      ​13.3  Derivatives
            13.3.1  Real and complex differentiation   climc 25779
                  13.3.1.1  Derivatives of functions of one complex or real variable   climc 25779
                  13.3.1.2  Results on real differentiation   dvferm1lem 25904
​PART 14  BASIC REAL AND COMPLEX FUNCTIONS
      ​14.1  Polynomials
            14.1.1  Polynomial degrees   cmdg 25974
            14.1.2  The division algorithm for univariate polynomials   cmn1 26047
            14.1.3  Elementary properties of complex polynomials   cply 26105
            14.1.4  The division algorithm for polynomials   cquot 26214
            14.1.5  Algebraic numbers   caa 26238
            14.1.6  Liouville's approximation theorem   aalioulem1 26256
      ​14.2  Sequences and series
            14.2.1  Taylor polynomials and Taylor's theorem   ctayl 26276
            14.2.2  Uniform convergence   culm 26301
            14.2.3  Power series   pserval 26335
      ​14.3  Basic trigonometry
            14.3.1  The exponential, sine, and cosine functions (cont.)   efcn 26369
            14.3.2  Properties of pi = 3.14159...   pilem1 26377
            14.3.3  Mapping of the exponential function   efgh 26466
            14.3.4  The natural logarithm on complex numbers   clog 26479
            ​*14.3.5  Logarithms to an arbitrary base   clogb 26690
            14.3.6  Theorems of Pythagoras, isosceles triangles, and intersecting chords   angval 26727
            14.3.7  Solutions of quadratic, cubic, and quartic equations   quad2 26765
            14.3.8  Inverse trigonometric functions   casin 26788
            14.3.9  The Birthday Problem   log2ublem1 26872
            14.3.10  Areas in R^2   carea 26881
            14.3.11  More miscellaneous converging sequences   rlimcnp 26891
            14.3.12  Inequality of arithmetic and geometric means   cvxcl 26911
            14.3.13  Euler-Mascheroni constant   cem 26918
            14.3.14  Zeta function   czeta 26939
            14.3.15  Gamma function   clgam 26942
      ​14.4  Basic number theory
            14.4.1  Wilson's theorem   wilthlem1 26994
            14.4.2  The Fundamental Theorem of Algebra   ftalem1 26999
            14.4.3  The Basel problem (ζ(2) = π²/6)   basellem1 27007
            14.4.4  Number-theoretical functions   ccht 27017
            14.4.5  Perfect Number Theorem   mersenne 27154
            14.4.6  Characters of Z/nZ   cdchr 27159
            14.4.7  Bertrand's postulate   bcctr 27202
            ​*14.4.8  Quadratic residues and the Legendre symbol   clgs 27221
            ​*14.4.9  Gauss' Lemma   gausslemma2dlem0a 27283
            14.4.10  Quadratic reciprocity   lgseisenlem1 27302
            14.4.11  All primes 4n+1 are the sum of two squares   2sqlem1 27344
            14.4.12  Chebyshev's Weak Prime Number Theorem, Dirichlet's Theorem   chebbnd1lem1 27396
            14.4.13  The Prime Number Theorem   mudivsum 27457
            14.4.14  Ostrowski's theorem   abvcxp 27542
​PART 15  SURREAL NUMBERS
      ​​*15.1  Sign sequence representation and Alling's axioms
            15.1.1  Definitions and initial properties   csur 27567
            15.1.2  Ordering   sltsolem1 27603
            15.1.3  Birthday Function   bdayfo 27605
            15.1.4  Density   fvnobday 27606
            ​*15.1.5  Full-Eta Property   bdayimaon 27621
      ​15.2  Initial consequences of Alling's axioms
            15.2.1  Ordering Theorems   csle 27672
            15.2.2  Birthday Theorems   bdayfun 27700
      ​​*15.3  Conway cut representation
            15.3.1  Conway cuts   csslt 27709
            15.3.2  Zero and One   c0s 27754
            15.3.3  Cuts and Options   cmade 27770
            15.3.4  Cofinality and coinitiality   cofsslt 27849
      ​15.4  Induction and recursion
            15.4.1  Induction and recursion on one variable   cnorec 27867
            15.4.2  Induction and recursion on two variables   cnorec2 27878
      ​15.5  Surreal arithmetic
            15.5.1  Addition   cadds 27889
            15.5.2  Negation and Subtraction   cnegs 27948
            15.5.3  Multiplication   cmuls 28032
            15.5.4  Division   cdivs 28113
            15.5.5  Absolute value   cabss 28162
      ​15.6  Subsystems of surreals
            15.6.1  Ordinal numbers   cons 28175
            15.6.2  Surreal recursive sequences   cseqs 28200
            15.6.3  Natural numbers   cnn0s 28229
            15.6.4  Integers   czs 28289
            15.6.5  Dyadic fractions   c2s 28320
            15.6.6  Real numbers   creno 28380
​​*PART 16  ELEMENTARY GEOMETRY
      ​16.1  Definition and Tarski's Axioms of Geometry
            16.1.1  Justification for the congruence notation   tgjustf 28436
      ​16.2  Tarskian Geometry
            16.2.1  Congruence   tgcgrcomimp 28440
            16.2.2  Betweenness   tgbtwntriv2 28450
            16.2.3  Dimension   tglowdim1 28463
            16.2.4  Betweenness and Congruence   tgifscgr 28471
            16.2.5  Congruence of a series of points   ccgrg 28473
            16.2.6  Motions   cismt 28495
            16.2.7  Colinearity   tglng 28509
            16.2.8  Connectivity of betweenness   tgbtwnconn1lem1 28535
            16.2.9  Less-than relation in geometric congruences   cleg 28545
            16.2.10  Rays   chlg 28563
            16.2.11  Lines   btwnlng1 28582
            16.2.12  Point inversions   cmir 28615
            16.2.13  Right angles   crag 28656
            16.2.14  Half-planes   islnopp 28702
            16.2.15  Midpoints and Line Mirroring   cmid 28735
            16.2.16  Congruence of angles   ccgra 28770
            16.2.17  Angle Comparisons   cinag 28798
            16.2.18  Congruence Theorems   tgsas1 28817
            16.2.19  Equilateral triangles   ceqlg 28828
      ​16.3  Properties of geometries
            16.3.1  Isomorphisms between geometries   f1otrgds 28832
      ​16.4  Geometry in Hilbert spaces
            16.4.1  Geometry in the complex plane   cchhllem 28850
            16.4.2  Geometry in Euclidean spaces   cee 28851
                  16.4.2.1  Definition of the Euclidean space   cee 28851
                  16.4.2.2  Tarski's axioms for geometry for the Euclidean space   axdimuniq 28876
                  16.4.2.3  EE^n fulfills Tarski's Axioms   ceeng 28940
​​*PART 17  GRAPH THEORY
      ​*17.1  Vertices and edges
            17.1.1  The edge function extractor for extensible structures   cedgf 28951
            ​*17.1.2  Vertices and indexed edges   cvtx 28959
                  17.1.2.1  Definitions and basic properties   cvtx 28959
                  17.1.2.2  The vertices and edges of a graph represented as ordered pair   opvtxval 28966
                  17.1.2.3  The vertices and edges of a graph represented as extensible structure   funvtxdmge2val 28974
                  17.1.2.4  Representations of graphs without edges   snstrvtxval 29000
                  17.1.2.5  Degenerated cases of representations of graphs   vtxval0 29002
            17.1.3  Edges as range of the edge function   cedg 29010
      ​​*17.2  Undirected graphs
            17.2.1  Undirected hypergraphs   cuhgr 29019
            17.2.2  Undirected pseudographs and multigraphs   cupgr 29043
            ​*17.2.3  Loop-free graphs   umgrislfupgrlem 29085
            17.2.4  Edges as subsets of vertices of graphs   uhgredgiedgb 29089
            ​*17.2.5  Undirected simple graphs   cuspgr 29111
            17.2.6  Examples for graphs   usgr0e 29199
            17.2.7  Subgraphs   csubgr 29230
            17.2.8  Finite undirected simple graphs   cfusgr 29279
            17.2.9  Neighbors, complete graphs and universal vertices   cnbgr 29295
                  17.2.9.1  Neighbors   cnbgr 29295
                  17.2.9.2  Universal vertices   cuvtx 29348
                  17.2.9.3  Complete graphs   ccplgr 29372
            17.2.10  Vertex degree   cvtxdg 29429
            ​*17.2.11  Regular graphs   crgr 29519
      ​​*17.3  Walks, paths and cycles
            *17.3.1  Walks   cewlks 29559
            17.3.2  Walks for loop-free graphs   lfgrwlkprop 29649
            17.3.3  Trails   ctrls 29652
            17.3.4  Paths and simple paths   cpths 29673
            17.3.5  Closed walks   cclwlks 29733
            17.3.6  Circuits and cycles   ccrcts 29747
            ​*17.3.7  Walks as words   cwwlks 29788
            17.3.8  Walks/paths of length 2 (as length 3 strings)   2wlkdlem1 29888
            17.3.9  Walks in regular graphs   rusgrnumwwlkl1 29931
            ​*17.3.10  Closed walks as words   cclwwlk 29943
                  17.3.10.1  Closed walks as words   cclwwlk 29943
                  17.3.10.2  Closed walks of a fixed length as words   cclwwlkn 29986
                  17.3.10.3  Closed walks on a vertex of a fixed length as words   cclwwlknon 30049
            17.3.11  Examples for walks, trails and paths   0ewlk 30076
            17.3.12  Connected graphs   cconngr 30148
      ​17.4  Eulerian paths and the Konigsberg Bridge problem
            ​*17.4.1  Eulerian paths   ceupth 30159
            ​*17.4.2  The Königsberg Bridge problem   konigsbergvtx 30208
      ​17.5  The Friendship Theorem
            17.5.1  Friendship graphs - basics   cfrgr 30220
            17.5.2  The friendship theorem for small graphs   frgr1v 30233
            17.5.3  Theorems according to Mertzios and Unger   2pthfrgrrn 30244
            ​*17.5.4  Huneke's Proof of the Friendship Theorem   frgrncvvdeqlem1 30261
​PART 18  GUIDES AND MISCELLANEA
      ​18.1  Guides (conventions, explanations, and examples)
            ​*18.1.1  Conventions   conventions 30362
            18.1.2  Natural deduction   natded 30365
            ​*18.1.3  Natural deduction examples   ex-natded5.2 30366
            18.1.4  Definitional examples   ex-or 30383
            18.1.5  Other examples   aevdemo 30422
      ​18.2  Humor
            18.2.1  April Fool's theorem   avril1 30425
      ​18.3  (Future - to be reviewed and classified)
            18.3.1  Planar incidence geometry   cplig 30436
            ​*18.3.2  Aliases kept to prevent broken links   dummylink 30449
​​*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 30451
            19.1.2  Abelian groups   cablo 30506
      ​19.2  Complex vector spaces
            19.2.1  Definition and basic properties   cvc 30520
            19.2.2  Examples of complex vector spaces   cnaddabloOLD 30543
      ​19.3  Normed complex vector spaces
            19.3.1  Definition and basic properties   cnv 30546
            19.3.2  Examples of normed complex vector spaces   cnnv 30639
            19.3.3  Induced metric of a normed complex vector space   imsval 30647
            19.3.4  Inner product   cdip 30662
            19.3.5  Subspaces   css 30683
      ​19.4  Operators on complex vector spaces
            19.4.1  Definitions and basic properties   clno 30702
      ​19.5  Inner product (pre-Hilbert) spaces
            19.5.1  Definition and basic properties   ccphlo 30774
            19.5.2  Examples of pre-Hilbert spaces   cncph 30781
            19.5.3  Properties of pre-Hilbert spaces   isph 30784
      ​19.6  Complex Banach spaces
            19.6.1  Definition and basic properties   ccbn 30824
            19.6.2  Examples of complex Banach spaces   cnbn 30831
            19.6.3  Uniform Boundedness Theorem   ubthlem1 30832
            19.6.4  Minimizing Vector Theorem   minvecolem1 30836
      ​19.7  Complex Hilbert spaces
            19.7.1  Definition and basic properties   chlo 30847
            19.7.2  Standard axioms for a complex Hilbert space   hlex 30860
            19.7.3  Examples of complex Hilbert spaces   cnchl 30878
            19.7.4  Hellinger-Toeplitz Theorem   htthlem 30879
​​*PART 20  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      ​20.1  Axiomatization of complex pre-Hilbert spaces
            20.1.1  Basic Hilbert space definitions   chba 30881
            20.1.2  Preliminary ZFC lemmas   df-hnorm 30930
            ​*20.1.3  Derive the Hilbert space axioms from ZFC set theory   axhilex-zf 30943
            ​*20.1.4  Introduce the vector space axioms for a Hilbert space   ax-hilex 30961
            20.1.5  Vector operations   hvmulex 30973
            20.1.6  Inner product postulates for a Hilbert space   ax-hfi 31041
      ​20.2  Inner product and norms
            20.2.1  Inner product   his5 31048
            20.2.2  Norms   dfhnorm2 31084
            20.2.3  Relate Hilbert space to normed complex vector spaces   hilablo 31122
            20.2.4  Bunjakovaskij-Cauchy-Schwarz inequality   bcsiALT 31141
      ​20.3  Cauchy sequences and completeness axiom
            20.3.1  Cauchy sequences and limits   hcau 31146
            20.3.2  Derivation of the completeness axiom from ZF set theory   hilmet 31156
            20.3.3  Completeness postulate for a Hilbert space   ax-hcompl 31164
            20.3.4  Relate Hilbert space to ZFC pre-Hilbert and Hilbert spaces   hhcms 31165
      ​20.4  Subspaces and projections
            20.4.1  Subspaces   df-sh 31169
            20.4.2  Closed subspaces   df-ch 31183
            20.4.3  Orthocomplements   df-oc 31214
            20.4.4  Subspace sum, span, lattice join, lattice supremum   df-shs 31270
            20.4.5  Projection theorem   pjhthlem1 31353
            20.4.6  Projectors   df-pjh 31357
      ​20.5  Properties of Hilbert subspaces
            20.5.1  Orthomodular law   omlsilem 31364
            20.5.2  Projectors (cont.)   pjhtheu2 31378
            20.5.3  Hilbert lattice operations   sh0le 31402
            20.5.4  Span (cont.) and one-dimensional subspaces   spansn0 31503
            20.5.5  Commutes relation for Hilbert lattice elements   df-cm 31545
            20.5.6  Foulis-Holland theorem   fh1 31580
            20.5.7  Quantum Logic Explorer axioms   qlax1i 31589
            20.5.8  Orthogonal subspaces   chscllem1 31599
            20.5.9  Orthoarguesian laws 5OA and 3OA   5oalem1 31616
            20.5.10  Projectors (cont.)   pjorthi 31631
            20.5.11  Mayet's equation E_3   mayete3i 31690
      ​20.6  Operators on Hilbert spaces
            ​*20.6.1  Operator sum, difference, and scalar multiplication   df-hosum 31692
            20.6.2  Zero and identity operators   df-h0op 31710
            20.6.3  Operations on Hilbert space operators   hoaddcl 31720
            20.6.4  Linear, continuous, bounded, Hermitian, unitary operators and norms   df-nmop 31801
            20.6.5  Linear and continuous functionals and norms   df-nmfn 31807
            20.6.6  Adjoint   df-adjh 31811
            20.6.7  Dirac bra-ket notation   df-bra 31812
            20.6.8  Positive operators   df-leop 31814
            20.6.9  Eigenvectors, eigenvalues, spectrum   df-eigvec 31815
            20.6.10  Theorems about operators and functionals   nmopval 31818
            20.6.11  Riesz lemma   riesz3i 32024
            20.6.12  Adjoints (cont.)   cnlnadjlem1 32029
            20.6.13  Quantum computation error bound theorem   unierri 32066
            20.6.14  Dirac bra-ket notation (cont.)   branmfn 32067
            20.6.15  Positive operators (cont.)   leopg 32084
            20.6.16  Projectors as operators   pjhmopi 32108
      ​20.7  States on a Hilbert lattice and Godowski's equation
            20.7.1  States on a Hilbert lattice   df-st 32173
            20.7.2  Godowski's equation   golem1 32233
      ​20.8  Cover relation, atoms, exchange axiom, and modular symmetry
            20.8.1  Covers relation; modular pairs   df-cv 32241
            20.8.2  Atoms   df-at 32300
            20.8.3  Superposition principle   superpos 32316
            20.8.4  Atoms, exchange and covering properties, atomicity   chcv1 32317
            20.8.5  Irreducibility   chirredlem1 32352
            20.8.6  Atoms (cont.)   atcvat3i 32358
            20.8.7  Modular symmetry   mdsymlem1 32365
​PART 21  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      ​21.1  Mathboxes for user contributions
            21.1.1  Mathbox guidelines   mathbox 32404
      ​21.2  Mathbox for Stefan Allan
      ​21.3  Mathbox for Thierry Arnoux
            21.3.1  Propositional Calculus - misc additions   ad11antr 32409
            21.3.2  Predicate Calculus   sbc2iedf 32427
                  21.3.2.1  Predicate Calculus - misc additions   sbc2iedf 32427
                  21.3.2.2  Restricted quantification - misc additions   ralcom4f 32429
                  21.3.2.3  Equality   eqtrb 32436
                  21.3.2.4  Double restricted existential uniqueness quantification   opsbc2ie 32438
                  21.3.2.5  Double restricted existential uniqueness quantification syntax   w2reu 32440
                  21.3.2.6  Substitution (without distinct variables) - misc additions   sbceqbidf 32449
                  21.3.2.7  Existential "at most one" - misc additions   mo5f 32451
                  21.3.2.8  Existential uniqueness - misc additions   reuxfrdf 32453
                  21.3.2.9  Restricted "at most one" - misc additions   rmoxfrd 32455
                  21.3.2.10  Restricted iota (description binder)   riotaeqbidva 32458
            21.3.3  General Set Theory   dmrab 32459
                  21.3.3.1  Class abstractions (a.k.a. class builders)   dmrab 32459
                  21.3.3.2  Image Sets   abrexdomjm 32469
                  21.3.3.3  Set relations and operations - misc additions   nelun 32475
                  21.3.3.4  Unordered pairs   elpreq 32490
                  21.3.3.5  Unordered triples   tpssg 32499
                  21.3.3.6  Conditional operator - misc additions   ifeqeqx 32504
                  21.3.3.7  Set union   uniinn0 32512
                  21.3.3.8  Indexed union - misc additions   cbviunf 32517
                  21.3.3.9  Indexed intersection - misc additions   iinabrex 32531
                  21.3.3.10  Disjointness - misc additions   disjnf 32532
            21.3.4  Relations and Functions   xpdisjres 32560
                  21.3.4.1  Relations - misc additions   xpdisjres 32560
                  21.3.4.2  Functions - misc additions   ac6sf2 32581
                  21.3.4.3  Operations - misc additions   mpomptxf 32634
                  21.3.4.4  The mapping operation   elmaprd 32636
                  21.3.4.5  Support of a function   suppovss 32637
                  21.3.4.6  Explicit Functions with one or two points as a domain   cosnopne 32650
                  21.3.4.7  Isomorphisms - misc. additions   gtiso 32657
                  21.3.4.8  Disjointness (additional proof requiring functions)   disjdsct 32659
                  21.3.4.9  First and second members of an ordered pair - misc additions   df1stres 32660
                  21.3.4.10  Finite Sets   imafi2 32668
                  21.3.4.11  Countable Sets   snct 32670
            21.3.5  Real and Complex Numbers   sgnval2 32691
                  21.3.5.1  Complex operations - misc. additions   creq0 32692
                  21.3.5.2  Ordering on reals - misc additions   lt2addrd 32707
                  21.3.5.3  Extended reals - misc additions   xrlelttric 32708
                  21.3.5.4  Extended nonnegative integers - misc additions   xnn0gt0 32725
                  21.3.5.5  Real number intervals - misc additions   joiniooico 32730
                  21.3.5.6  Finite intervals of integers - misc additions   uzssico 32740
                  21.3.5.7  Half-open integer ranges - misc additions   iundisjfi 32752
                  21.3.5.8  The ` # ` (set size) function - misc additions   hashunif 32764
                  21.3.5.9  The greatest common divisor operator - misc. additions   elq2 32769
                  21.3.5.10  Integers   nn0split01 32775
                  21.3.5.11  Decimal numbers   dfdec100 32788
            21.3.6  Real and complex functions   sgncl 32789
                  21.3.6.1  Signum (sgn or sign) function - misc. additions   sgncl 32789
                  21.3.6.2  Integer powers - misc. additions   nexple 32802
                  21.3.6.3  Indicator Functions   cind 32806
            ​*21.3.7  Decimal expansion   cdp2 32824
                  ​*21.3.7.1  Decimal point   cdp 32841
                  21.3.7.2  Division in the extended real number system   cxdiv 32870
            21.3.8  Words over a set - misc additions   wrdres 32889
                  21.3.8.1  Splicing words (substring replacement)   splfv3 32913
                  21.3.8.2  Cyclic shift of words   1cshid 32914
            21.3.9  Extensible Structures   ressplusf 32918
                  21.3.9.1  Structure restriction operator   ressplusf 32918
                  21.3.9.2  Posets   ressprs 32921
                  21.3.9.3  Complete lattices   clatp0cl 32931
                  21.3.9.4  Order Theory   cmnt 32933
                  21.3.9.5  Chains   cchn 32959
                  21.3.9.6  Extended reals Structure - misc additions   ax-xrssca 32971
                  21.3.9.7  The extended nonnegative real numbers commutative monoid   xrge00 32981
            21.3.10  Algebra   mndcld 32989
                  21.3.10.1  Monoids   mndcld 32989
                  21.3.10.2  Monoids Homomorphisms   abliso 33003
                  21.3.10.3  Groups - misc additions   grpsubcld 33007
                  21.3.10.4  Finitely supported group sums - misc additions   gsumsubg 33012
                  21.3.10.5  Group or monoid sums over words   gsumwun 33031
                  21.3.10.6  Centralizers and centers - misc additions   cntzun 33034
                  21.3.10.7  The symmetric group   symgfcoeu 33037
                  21.3.10.8  Transpositions   pmtridf1o 33049
                  21.3.10.9  Permutation Signs   psgnid 33052
                  21.3.10.10  Permutation cycles   ctocyc 33061
                  21.3.10.11  The Alternating Group   evpmval 33100
                  21.3.10.12  Signum in an ordered monoid   csgns 33113
                  21.3.10.13  Fixed points   cfxp 33118
                  21.3.10.14  The Archimedean property for generic ordered algebraic structures   cinftm 33128
                  21.3.10.15  Semiring left modules   cslmd 33152
                  21.3.10.16  Simple groups   prmsimpcyc 33180
                  21.3.10.17  Rings - misc additions   ringdi22 33181
                  21.3.10.18  Subrings generated by a set   elrgspnlem1 33192
                  21.3.10.19  The zero ring   irrednzr 33200
                  21.3.10.20  Localization of rings   cerl 33203
                  21.3.10.21  Integral Domains   domnmuln0rd 33224
                  21.3.10.22  Euclidean Domains   ceuf 33237
                  21.3.10.23  Division Rings   ringinveu 33243
                  21.3.10.24  The field of rational numbers   qfld 33246
                  21.3.10.25  Subfields   subsdrg 33247
                  21.3.10.26  Field of fractions   cfrac 33251
                  21.3.10.27  Field extensions generated by a set   cfldgen 33259
                  21.3.10.28  Ring homomorphisms - misc additions   rhmdvd 33272
                  21.3.10.29  Scalar restriction operation   cresv 33274
                  21.3.10.30  The commutative ring of gaussian integers   gzcrng 33289
                  21.3.10.31  The archimedean ordered field of real numbers   cnfldfld 33290
                  21.3.10.32  The quotient map and quotient modules   qusker 33296
                  21.3.10.33  The ring of integers modulo ` N `   znfermltl 33313
                  21.3.10.34  Independent sets and families   islinds5 33314
                  21.3.10.35  Ring associates, ring units   dvdsruassoi 33331
                  ​*21.3.10.36  Subgroup sum / Sumset / Minkowski sum   elgrplsmsn 33337
                  21.3.10.37  The quotient map   quslsm 33352
                  21.3.10.38  Ideals   lidlmcld 33366
                  21.3.10.39  Prime Ideals   cprmidl 33382
                  21.3.10.40  Maximal Ideals   cmxidl 33406
                  21.3.10.41  The semiring of ideals of a ring   cidlsrg 33447
                  21.3.10.42  Prime Elements   rprmval 33463
                  21.3.10.43  Unique factorization domains   cufd 33485
                  21.3.10.44  The ring of integers   zringidom 33498
                  21.3.10.45  Univariate Polynomials   0ringmon1p 33502
                  21.3.10.46  Polynomial quotient and polynomial remainder   q1pdir 33544
                  21.3.10.47  The subring algebra   sra1r 33553
                  21.3.10.48  Division Ring Extensions   drgext0g 33561
                  21.3.10.49  Vector Spaces   lvecdimfi 33567
                  21.3.10.50  Vector Space Dimension   cldim 33570
            21.3.11  Field Extensions   cfldext 33610
                  21.3.11.1  Algebraic numbers   cirng 33654
                  21.3.11.2  Algebraic extensions   calgext 33663
                  21.3.11.3  Minimal polynomials   cminply 33665
                  21.3.11.4  Quadratic Field Extensions   rtelextdg2lem 33692
                  21.3.11.5  Towers of quadratic extentions   fldext2chn 33694
            ​*21.3.12  Constructible Numbers   cconstr 33695
                  21.3.12.1  Impossible constructions   2sqr3minply 33746
            21.3.13  Matrices   csmat 33759
                  21.3.13.1  Submatrices   csmat 33759
                  21.3.13.2  Matrix literals   clmat 33777
                  21.3.13.3  Laplace expansion of determinants   mdetpmtr1 33789
            21.3.14  Topology   ist0cld 33799
                  21.3.14.1  Open maps   txomap 33800
                  21.3.14.2  Topology of the unit circle   qtopt1 33801
                  21.3.14.3  Refinements   reff 33805
                  21.3.14.4  Open cover refinement property   ccref 33808
                  21.3.14.5  Lindelöf spaces   cldlf 33818
                  21.3.14.6  Paracompact spaces   cpcmp 33821
                  ​*21.3.14.7  Spectrum of a ring   crspec 33828
                  21.3.14.8  Pseudometrics   cmetid 33852
                  21.3.14.9  Continuity - misc additions   hauseqcn 33864
                  21.3.14.10  Topology of the closed unit interval   elunitge0 33865
                  21.3.14.11  Topology of ` ( RR X. RR ) `   unicls 33869
                  21.3.14.12  Order topology - misc. additions   cnvordtrestixx 33879
                  21.3.14.13  Continuity in topological spaces - misc. additions   mndpluscn 33892
                  21.3.14.14  Topology of the extended nonnegative real numbers ordered monoid   xrge0hmph 33898
                  21.3.14.15  Limits - misc additions   lmlim 33913
                  21.3.14.16  Univariate polynomials   pl1cn 33921
            21.3.15  Uniform Stuctures and Spaces   chcmp 33922
                  21.3.15.1  Hausdorff uniform completion   chcmp 33922
            21.3.16  Topology and algebraic structures   zringnm 33924
                  21.3.16.1  The norm on the ring of the integer numbers   zringnm 33924
                  21.3.16.2  Topological ` ZZ ` -modules   zlm0 33926
                  21.3.16.3  Canonical embedding of the field of the rational numbers into a division ring   cqqh 33936
                  21.3.16.4  Canonical embedding of the real numbers into a complete ordered field   crrh 33959
                  21.3.16.5  Embedding from the extended real numbers into a complete lattice   cxrh 33982
                  21.3.16.6  Canonical embeddings into the ordered field of the real numbers   zrhre 33985
                  ​*21.3.16.7  Topological Manifolds   cmntop 33988
                  21.3.16.8  Extended sum   cesum 33993
            21.3.17  Mixed Function/Constant operation   cofc 34061
            21.3.18  Abstract measure   csiga 34074
                  21.3.18.1  Sigma-Algebra   csiga 34074
                  21.3.18.2  Generated sigma-Algebra   csigagen 34104
                  ​*21.3.18.3  lambda and pi-Systems, Rings of Sets   ispisys 34118
                  21.3.18.4  The Borel algebra on the real numbers   cbrsiga 34147
                  21.3.18.5  Product Sigma-Algebra   csx 34154
                  21.3.18.6  Measures   cmeas 34161
                  21.3.18.7  The counting measure   cntmeas 34192
                  21.3.18.8  The Lebesgue measure - misc additions   voliune 34195
                  21.3.18.9  The Dirac delta measure   cdde 34198
                  21.3.18.10  The 'almost everywhere' relation   cae 34203
                  21.3.18.11  Measurable functions   cmbfm 34215
                  21.3.18.12  Borel Algebra on ` ( RR X. RR ) `   br2base 34236
                  ​*21.3.18.13  Caratheodory's extension theorem   coms 34258
            21.3.19  Integration   itgeq12dv 34293
                  21.3.19.1  Lebesgue integral - misc additions   itgeq12dv 34293
                  21.3.19.2  Bochner integral   citgm 34294
            21.3.20  Euler's partition theorem   oddpwdc 34321
            21.3.21  Sequences defined by strong recursion   csseq 34350
            21.3.22  Fibonacci Numbers   cfib 34363
            21.3.23  Probability   cprb 34374
                  21.3.23.1  Probability Theory   cprb 34374
                  21.3.23.2  Conditional Probabilities   ccprob 34398
                  21.3.23.3  Real-valued Random Variables   crrv 34407
                  21.3.23.4  Preimage set mapping operator   corvc 34423
                  21.3.23.5  Distribution Functions   orvcelval 34436
                  21.3.23.6  Cumulative Distribution Functions   orvclteel 34440
                  21.3.23.7  Probabilities - example   coinfliplem 34446
                  21.3.23.8  Bertrand's Ballot Problem   ballotlemoex 34453
            21.3.24  Signum (sgn or sign) function - misc. additions   fzssfzo 34506
                  21.3.24.1  Operations on words   ccatmulgnn0dir 34509
            21.3.25  Polynomials with real coefficients - misc additions   plymul02 34513
            21.3.26  Descartes's rule of signs   signspval 34519
                  21.3.26.1  Sign changes in a word over real numbers   signspval 34519
                  21.3.26.2  Counting sign changes in a word over real numbers   signslema 34529
            21.3.27  Number Theory   iblidicc 34559
                  21.3.27.1  Representations of a number as sums of integers   crepr 34575
                  21.3.27.2  Vinogradov Trigonometric Sums and the Circle Method   cvts 34602
                  21.3.27.3  The Ternary Goldbach Conjecture: Final Statement   ax-hgt749 34611
            21.3.28  Elementary Geometry   cstrkg2d 34631
                  ​*21.3.28.1  Two-dimensional geometry   cstrkg2d 34631
                  21.3.28.2  Outer Five Segment (not used, no need to move to main)   cafs 34636
            ​*21.3.29  LeftPad Project   clpad 34641
      ​​*21.4  Mathbox for Jonathan Ben-Naim
            21.4.1  First-order logic and set theory   bnj170 34664
            21.4.2  Well founded induction and recursion   bnj110 34824
            21.4.3  The existence of a minimal element in certain classes   bnj69 34976
            21.4.4  Well-founded induction   bnj1204 34978
            21.4.5  Well-founded recursion, part 1 of 3   bnj60 35028
            21.4.6  Well-founded recursion, part 2 of 3   bnj1500 35034
            21.4.7  Well-founded recursion, part 3 of 3   bnj1522 35038
      ​21.5  Mathbox for BTernaryTau
            21.5.1  First-order logic   nfan1c 35039
                  21.5.1.1  Auxiliary axiom schemes   nfan1c 35039
            21.5.2  ZF set theory   exdifsn 35045
                  21.5.2.1  Finitism   ax-regs 35060
                  21.5.2.2  Derive ax-regs   axregs 35073
                  21.5.2.3  Global choice   gblacfnacd 35074
            21.5.3  Real and complex numbers   zltp1ne 35082
            21.5.4  Graph theory   lfuhgr 35090
                  21.5.4.1  Acyclic graphs   cacycgr 35114
      ​21.6  Mathbox for Mario Carneiro
            21.6.1  Predicate calculus with all distinct variables   ax-7d 35131
            21.6.2  Miscellaneous stuff   quartfull 35137
            21.6.3  Derangements and the Subfactorial   deranglem 35138
            21.6.4  The Erdős-Szekeres theorem   erdszelem1 35163
            21.6.5  The Kuratowski closure-complement theorem   kur14lem1 35178
            21.6.6  Retracts and sections   cretr 35189
            21.6.7  Path-connected and simply connected spaces   cpconn 35191
            21.6.8  Covering maps   ccvm 35227
            21.6.9  Normal numbers   snmlff 35301
            21.6.10  Godel-sets of formulas - part 1   cgoe 35305
            21.6.11  Godel-sets of formulas - part 2   cgon 35404
            21.6.12  Models of ZF   cgze 35418
            ​*21.6.13  Metamath formal systems   cmcn 35432
            21.6.14  Grammatical formal systems   cm0s 35557
            21.6.15  Models of formal systems   cmuv 35577
            21.6.16  Splitting fields   ccpms 35599
            21.6.17  p-adic number fields   czr 35619
      ​​*21.7  Mathbox for Filip Cernatescu
      ​21.8  Mathbox for Paul Chapman
            21.8.1  Real and complex numbers (cont.)   climuzcnv 35643
            21.8.2  Miscellaneous theorems   elfzm12 35647
      ​21.9  Mathbox for Hongxiu Chen
      ​21.10  Mathbox for Adrian Ducourtial
            21.10.1  Propositional calculus   currybi 35660
            21.10.2  Clone theory   ccloneop 35667
      ​21.11  Mathbox for Scott Fenton
            21.11.1  ZFC Axioms in primitive form   axextprim 35673
            21.11.2  Untangled classes   untelirr 35680
            21.11.3  Extra propositional calculus theorems   3jaodd 35687
            21.11.4  Misc. Useful Theorems   nepss 35690
            21.11.5  Properties of real and complex numbers   sqdivzi 35700
            21.11.6  Infinite products   iprodefisumlem 35712
            21.11.7  Factorial limits   faclimlem1 35715
            21.11.8  Greatest common divisor and divisibility   gcd32 35721
            21.11.9  Properties of relationships   dftr6 35723
            21.11.10  Properties of functions and mappings   funpsstri 35738
            21.11.11  Set induction (or epsilon induction)   setinds 35751
            21.11.12  Ordinal numbers   elpotr 35754
            21.11.13  Defined equality axioms   axextdfeq 35770
            21.11.14  Hypothesis builders   hbntg 35778
            21.11.15  Well-founded zero, successor, and limits   cwsuc 35783
            21.11.16  Quantifier-free definitions   ctxp 35803
            21.11.17  Alternate ordered pairs   caltop 35929
            21.11.18  Geometry in the Euclidean space   cofs 35955
                  21.11.18.1  Congruence properties   cofs 35955
                  21.11.18.2  Betweenness properties   btwntriv2 35985
                  21.11.18.3  Segment Transportation   ctransport 36002
                  21.11.18.4  Properties relating betweenness and congruence   cifs 36008
                  21.11.18.5  Connectivity of betweenness   btwnconn1lem1 36060
                  21.11.18.6  Segment less than or equal to   csegle 36079
                  21.11.18.7  Outside-of relationship   coutsideof 36092
                  21.11.18.8  Lines and Rays   cline2 36107
            21.11.19  Forward difference   cfwddif 36131
            21.11.20  Rank theorems   rankung 36139
            21.11.21  Hereditarily Finite Sets   chf 36145
      ​21.12  Mathbox for Gino Giotto
            21.12.1  Equality theorems.   rmoeqi 36160
                  21.12.1.1  Inference versions.   rmoeqi 36160
                  21.12.1.2  Deduction versions.   rmoeqdv 36185
            21.12.2  Change bound variables.   in-ax8 36197
                  21.12.2.1  Change bound variables and domains.   cbvralvw2 36199
                  21.12.2.2  Change bound variables, deduction versions.   cbvmodavw 36223
                  21.12.2.3  Change bound variables and domains, deduction versions.   cbvrmodavw2 36256
            21.12.3  Study of ax-mulf usage.   mpomulnzcnf 36272
      ​21.13  Mathbox for Jeff Hankins
            21.13.1  Miscellany   a1i14 36273
            21.13.2  Basic topological facts   topbnd 36297
            21.13.3  Topology of the real numbers   ivthALT 36308
            21.13.4  Refinements   cfne 36309
            21.13.5  Neighborhood bases determine topologies   neibastop1 36332
            21.13.6  Lattice structure of topologies   topmtcl 36336
            21.13.7  Filter bases   fgmin 36343
            21.13.8  Directed sets, nets   tailfval 36345
      ​21.14  Mathbox for Anthony Hart
            21.14.1  Propositional Calculus   tb-ax1 36356
            21.14.2  Predicate Calculus   nalfal 36376
            21.14.3  Miscellaneous single axioms   meran1 36384
            21.14.4  Connective Symmetry   negsym1 36390
      ​21.15  Mathbox for Chen-Pang He
            21.15.1  Ordinal topology   ontopbas 36401
      ​21.16  Mathbox for Jeff Hoffman
            21.16.1  Inferences for finite induction on generic function values   fveleq 36424
            21.16.2  gdc.mm   nnssi2 36428
      ​21.17  Mathbox for Matthew House
            21.17.1  Relations on well-ordered indexed unions   weiunlem1 36435
      ​21.18  Mathbox for Asger C. Ipsen
            21.18.1  Continuous nowhere differentiable functions   dnival 36444
      ​​*21.19  Mathbox for BJ
            *21.19.1  Propositional calculus   bj-mp2c 36513
                  *21.19.1.1  Derived rules of inference   bj-mp2c 36513
                  ​*21.19.1.2  A syntactic theorem   bj-0 36515
                  21.19.1.3  Minimal implicational calculus   bj-a1k 36517
                  ​*21.19.1.4  Positive calculus   bj-syl66ib 36528
                  21.19.1.5  Implication and negation   bj-con2com 36534
                  ​*21.19.1.6  Disjunction   bj-jaoi1 36544
                  ​*21.19.1.7  Logical equivalence   bj-dfbi4 36546
                  21.19.1.8  The conditional operator for propositions   bj-consensus 36551
                  ​*21.19.1.9  Propositional calculus: miscellaneous   bj-imbi12 36556
            ​*21.19.2  Modal logic   bj-axdd2 36565
            ​*21.19.3  Provability logic   cprvb 36570
            ​*21.19.4  First-order logic   bj-genr 36579
                  21.19.4.1  Adding ax-gen   bj-genr 36579
                  21.19.4.2  Adding ax-4   bj-2alim 36583
                  21.19.4.3  Adding ax-5   bj-ax12wlem 36617
                  21.19.4.4  Equality and substitution   bj-ssbeq 36626
                  21.19.4.5  Adding ax-6   bj-spimvwt 36642
                  21.19.4.6  Adding ax-7   bj-cbvexw 36649
                  21.19.4.7  Membership predicate, ax-8 and ax-9   bj-ax89 36651
                  21.19.4.8  Adding ax-11   bj-alcomexcom 36653
                  21.19.4.9  Adding ax-12   axc11n11 36655
                  21.19.4.10  Nonfreeness   wnnf 36696
                  21.19.4.11  Adding ax-13   bj-axc10 36756
                  ​*21.19.4.12  Removing dependencies on ax-13 (and ax-11)   bj-axc10v 36766
                  ​*21.19.4.13  Distinct var metavariables   bj-hbaeb2 36791
                  ​*21.19.4.14  Around ~ equsal   bj-equsal1t 36795
                  ​*21.19.4.15  Some Principia Mathematica proofs   stdpc5t 36800
                  21.19.4.16  Alternate definition of substitution   bj-sbsb 36810
                  21.19.4.17  Lemmas for substitution   bj-sbf3 36812
                  21.19.4.18  Existential uniqueness   bj-eu3f 36814
                  ​*21.19.4.19  First-order logic: miscellaneous   bj-sblem1 36815
            21.19.5  Set theory   eliminable1 36832
                  ​*21.19.5.1  Eliminability of class terms   eliminable1 36832
                  ​*21.19.5.2  Classes without the axiom of extensionality   bj-denoteslem 36844
                  21.19.5.3  Characterization among sets versus among classes   elelb 36870
                  ​*21.19.5.4  The nonfreeness quantifier for classes   bj-nfcsym 36872
                  ​*21.19.5.5  Lemmas for class substitution   bj-sbeqALT 36873
                  21.19.5.6  Removing some axiom requirements and disjoint variable conditions   bj-exlimvmpi 36884
                  ​*21.19.5.7  Class abstractions   bj-elabd2ALT 36898
                  21.19.5.8  Generalized class abstractions   bj-cgab 36906
                  ​*21.19.5.9  Restricted nonfreeness   wrnf 36914
                  ​*21.19.5.10  Russell's paradox   bj-ru1 36916
                  21.19.5.11  Curry's paradox in set theory   currysetlem 36918
                  ​*21.19.5.12  Some disjointness results   bj-n0i 36924
                  ​*21.19.5.13  Complements on direct products   bj-xpimasn 36928
                  ​*21.19.5.14  "Singletonization" and tagging   bj-snsetex 36936
                  ​*21.19.5.15  Tuples of classes   bj-cproj 36963
                  ​*21.19.5.16  Set theory: elementary operations relative to a universe   bj-rcleqf 36998
                  ​*21.19.5.17  Axioms for finite unions   bj-abex 37003
                  ​*21.19.5.18  Set theory: miscellaneous   eleq2w2ALT 37020
                  ​*21.19.5.19  Evaluation at a class   bj-evaleq 37045
                  21.19.5.20  Elementwise operations   celwise 37052
                  ​*21.19.5.21  Elementwise intersection (families of sets induced on a subset)   bj-rest00 37054
                  21.19.5.22  Moore collections (complements)   bj-raldifsn 37073
                  21.19.5.23  Maps-to notation for functions with three arguments   bj-0nelmpt 37089
                  ​*21.19.5.24  Currying   csethom 37095
                  ​*21.19.5.25  Setting components of extensible structures   cstrset 37107
            ​*21.19.6  Extended real and complex numbers, real and complex projective lines   bj-nfald 37110
                  21.19.6.1  Complements on class abstractions of ordered pairs and binary relations   bj-nfald 37110
                  ​*21.19.6.2  Identity relation (complements)   bj-opabssvv 37123
                  ​*21.19.6.3  Functionalized identity (diagonal in a Cartesian square)   cdiag2 37145
                  ​*21.19.6.4  Direct image and inverse image   cimdir 37151
                  ​*21.19.6.5  Extended numbers and projective lines as sets   cfractemp 37169
                  ​*21.19.6.6  Addition and opposite   caddcc 37210
                  ​*21.19.6.7  Order relation on the extended reals   cltxr 37214
                  ​*21.19.6.8  Argument, multiplication and inverse   carg 37216
                  21.19.6.9  The canonical bijection from the finite ordinals   ciomnn 37222
                  21.19.6.10  Divisibility   cnnbar 37233
            ​*21.19.7  Monoids   bj-smgrpssmgm 37241
                  ​*21.19.7.1  Finite sums in monoids   cfinsum 37256
            ​*21.19.8  Affine, Euclidean, and Cartesian geometry   bj-fvimacnv0 37259
                  *21.19.8.1  Real vector spaces   bj-fvimacnv0 37259
                  ​*21.19.8.2  Complex numbers (supplements)   bj-subcom 37281
                  ​*21.19.8.3  Barycentric coordinates   bj-bary1lem 37283
            21.19.9  Monoid of endomorphisms   cend 37286
      ​21.20  Mathbox for Jim Kingdon
            21.20.1  Circle constant   taupilem3 37292
            21.20.2  Number theory   dfgcd3 37297
            21.20.3  Real numbers   irrdifflemf 37298
      ​21.21  Mathbox for ML
            21.21.1  Miscellaneous   csbrecsg 37301
            21.21.2  Cartesian exponentiation   cfinxp 37356
            21.21.3  Topology   iunctb2 37376
                  ​*21.21.3.1  Pi-base theorems   pibp16 37386
      ​21.22  Mathbox for Wolf Lammen
            21.22.1  1. Bootstrapping   wl-section-boot 37395
            21.22.2  Implication chains   wl-section-impchain 37419
            21.22.3  Theorems around the conditional operator   wl-ifp-ncond1 37437
            21.22.4  Alternative development of hadd, cadd   wl-df-3xor 37441
            21.22.5  An alternative axiom ~ ax-13   ax-wl-13v 37466
            21.22.6  Bootstrapping set theory with classes   wl-cleq-0 37468
            21.22.7  Other stuff   wl-mps 37480
      ​21.23  Mathbox for Brendan Leahy
      ​21.24  Mathbox for Jeff Madsen
            21.24.1  Logic and set theory   unirep 37693
            21.24.2  Real and complex numbers; integers   filbcmb 37719
            21.24.3  Sequences and sums   sdclem2 37721
            21.24.4  Topology   subspopn 37731
            21.24.5  Metric spaces   metf1o 37734
            21.24.6  Continuous maps and homeomorphisms   constcncf 37741
            21.24.7  Boundedness   ctotbnd 37745
            21.24.8  Isometries   cismty 37777
            21.24.9  Heine-Borel Theorem   heibor1lem 37788
            21.24.10  Banach Fixed Point Theorem   bfplem1 37801
            21.24.11  Euclidean space   crrn 37804
            21.24.12  Intervals (continued)   ismrer1 37817
            21.24.13  Operation properties   cass 37821
            21.24.14  Groups and related structures   cmagm 37827
            21.24.15  Group homomorphism and isomorphism   cghomOLD 37862
            21.24.16  Rings   crngo 37873
            21.24.17  Division Rings   cdrng 37927
            21.24.18  Ring homomorphisms   crngohom 37939
            21.24.19  Commutative rings   ccm2 37968
            21.24.20  Ideals   cidl 37986
            21.24.21  Prime rings and integral domains   cprrng 38025
            21.24.22  Ideal generators   cigen 38038
      ​21.25  Mathbox for Giovanni Mascellani
            ​*21.25.1  Tools for automatic proof building   efald2 38057
            ​*21.25.2  Tseitin axioms   fald 38108
            ​*21.25.3  Equality deductions   iuneq2f 38135
            ​*21.25.4  Miscellanea   orcomdd 38146
      ​21.26  Mathbox for Peter Mazsa
            21.26.1  Notations   cxrn 38153
            21.26.2  Preparatory theorems   el2v1 38196
            21.26.3  Range Cartesian product   df-xrn 38338
            21.26.4  Cosets by ` R `   df-coss 38387
            21.26.5  Relations   df-rels 38461
            21.26.6  Subset relations   df-ssr 38474
            21.26.7  Reflexivity   df-refs 38486
            21.26.8  Converse reflexivity   df-cnvrefs 38501
            21.26.9  Symmetry   df-syms 38518
            21.26.10  Reflexivity and symmetry   symrefref2 38539
            21.26.11  Transitivity   df-trs 38548
            21.26.12  Equivalence relations   df-eqvrels 38560
            21.26.13  Redundancy   df-redunds 38599
            21.26.14  Domain quotients   df-dmqss 38614
            21.26.15  Equivalence relations on domain quotients   df-ers 38640
            21.26.16  Functions   df-funss 38657
            21.26.17  Disjoints vs. converse functions   df-disjss 38680
            21.26.18  Antisymmetry   df-antisymrel 38737
            21.26.19  Partitions: disjoints on domain quotients   df-parts 38742
            21.26.20  Partition-Equivalence Theorems   disjim 38758
      ​21.27  Mathbox for Rodolfo Medina
            21.27.1  Partitions   prtlem60 38831
      ​​*21.28  Mathbox for Norm Megill
            *21.28.1  Obsolete schemes ax-c4,c5,c7,c10,c11,c11n,c15,c9,c14,c16   ax-c5 38861
            ​*21.28.2  Rederive new axioms ax-4, ax-10, ax-6, ax-12, ax-13 from old   axc5 38871
            ​*21.28.3  Legacy theorems using obsolete axioms   ax5ALT 38885
            21.28.4  Experiments with weak deduction theorem   elimhyps 38939
            21.28.5  Miscellanea   cnaddcom 38950
            21.28.6  Atoms, hyperplanes, and covering in a left vector space (or module)   clsa 38952
            21.28.7  Functionals and kernels of a left vector space (or module)   clfn 39035
            21.28.8  Opposite rings and dual vector spaces   cld 39101
            21.28.9  Ortholattices and orthomodular lattices   cops 39150
            21.28.10  Atomic lattices with covering property   ccvr 39240
            21.28.11  Hilbert lattices   chlt 39328
            21.28.12  Projective geometries based on Hilbert lattices   clln 39470
            21.28.13  Construction of a vector space from a Hilbert lattice   cdlema1N 39770
            21.28.14  Construction of involution and inner product from a Hilbert lattice   clpoN 41459
      ​21.29  Mathbox for metakunt
            21.29.1  Commutative Semiring   ccsrg 41941
            21.29.2  General helpful statements   rhmzrhval 41944
            21.29.3  Some gcd and lcm results   12gcd5e1 41976
            21.29.4  Least common multiple inequality theorem   3factsumint1 41994
            21.29.5  Logarithm inequalities   3exp7 42026
            21.29.6  Miscellaneous results for AKS formalisation   intlewftc 42034
            21.29.7  Sticks and stones   sticksstones1 42119
            21.29.8  Continuation AKS   aks6d1c6lem1 42143
      ​21.30  Mathbox for Steven Nguyen
            21.30.1  Utility theorems   jarrii 42178
            ​*21.30.2  Arithmetic theorems   c0exALT 42225
            21.30.3  Exponents and divisibility   oexpreposd 42295
            21.30.4  Trigonometry and Calculus   tanhalfpim 42322
            ​*21.30.5  Independence of ax-mulcom   cresub 42338
            21.30.6  Structures   sn-base0 42468
            ​*21.30.7  Projective spaces   cprjsp 42574
            21.30.8  Basic reductions for Fermat's Last Theorem   dffltz 42607
            ​*21.30.9  Exemplar theorems   iddii 42637
                  ​*21.30.9.1  Standard replacements of ax-10 , ax-11 , ax-12   nfa1w 42648
      ​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 42665
            21.33.2  Additional theory of functions   imaiinfv 42666
            21.33.3  Additional topology   elrfi 42667
            21.33.4  Characterization of closure operators. Kuratowski closure axioms   ismrcd1 42671
            21.33.5  Algebraic closure systems   cnacs 42675
            21.33.6  Miscellanea 1. Map utilities   constmap 42686
            21.33.7  Miscellanea for polynomials   mptfcl 42693
            21.33.8  Multivariate polynomials over the integers   cmzpcl 42694
            21.33.9  Miscellanea for Diophantine sets 1   coeq0i 42726
            21.33.10  Diophantine sets 1: definitions   cdioph 42728
            21.33.11  Diophantine sets 2 miscellanea   ellz1 42740
            21.33.12  Diophantine sets 2: union and intersection. Monotone Boolean algebra   diophin 42745
            21.33.13  Diophantine sets 3: construction   diophrex 42748
            21.33.14  Diophantine sets 4 miscellanea   2sbcrex 42757
            21.33.15  Diophantine sets 4: Quantification   rexrabdioph 42767
            21.33.16  Diophantine sets 5: Arithmetic sets   rabdiophlem1 42774
            21.33.17  Diophantine sets 6: reusability. renumbering of variables   eldioph4b 42784
            21.33.18  Pigeonhole Principle and cardinality helpers   fphpd 42789
            21.33.19  A non-closed set of reals is infinite   rencldnfilem 42793
            21.33.20  Lagrange's rational approximation theorem   irrapxlem1 42795
            21.33.21  Pell equations 1: A nontrivial solution always exists   pellexlem1 42802
            21.33.22  Pell equations 2: Algebraic number theory of the solution set   csquarenn 42809
            21.33.23  Pell equations 3: characterizing fundamental solution   infmrgelbi 42851
            ​*21.33.24  Logarithm laws generalized to an arbitrary base   reglogcl 42863
            21.33.25  Pell equations 4: the positive solution group is infinite cyclic   pellfund14 42871
            21.33.26  X and Y sequences 1: Definition and recurrence laws   crmx 42873
            21.33.27  Ordering and induction lemmas for the integers   monotuz 42914
            21.33.28  X and Y sequences 2: Order properties   rmxypos 42920
            21.33.29  Congruential equations   congtr 42938
            21.33.30  Alternating congruential equations   acongid 42948
            21.33.31  Additional theorems on integer divisibility   coprmdvdsb 42958
            21.33.32  X and Y sequences 3: Divisibility properties   jm2.18 42961
            21.33.33  X and Y sequences 4: Diophantine representability of Y   jm2.27a 42978
            21.33.34  X and Y sequences 5: Diophantine representability of X, ^, _C   rmxdiophlem 42988
            21.33.35  Uncategorized stuff not associated with a major project   setindtr 42997
            21.33.36  More equivalents of the Axiom of Choice   axac10 43006
            21.33.37  Finitely generated left modules   clfig 43040
            21.33.38  Noetherian left modules I   clnm 43048
            21.33.39  Addenda for structure powers   pwssplit4 43062
            21.33.40  Every set admits a group structure iff choice   unxpwdom3 43068
            21.33.41  Noetherian rings and left modules II   clnr 43082
            21.33.42  Hilbert's Basis Theorem   cldgis 43094
            21.33.43  Additional material on polynomials [DEPRECATED]   cmnc 43104
            21.33.44  Degree and minimal polynomial of algebraic numbers   cdgraa 43113
            21.33.45  Algebraic integers I   citgo 43130
            21.33.46  Endomorphism algebra   cmend 43144
            21.33.47  Cyclic groups and order   idomodle 43164
            21.33.48  Cyclotomic polynomials   ccytp 43170
            21.33.49  Miscellaneous topology   fgraphopab 43176
      ​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 43190
            21.36.2  Natural addition of Cantor normal forms   oawordex2 43299
            21.36.3  Surreal Contributions   abeqabi 43381
            21.36.4  Short Studies   nlimsuc 43414
                  21.36.4.1  Additional work on conditional logical operator   ifpan123g 43432
                  21.36.4.2  Sophisms   rp-fakeimass 43485
                  ​*21.36.4.3  Finite Sets   rp-isfinite5 43490
                  21.36.4.4  General Observations   intabssd 43492
                  21.36.4.5  Infinite Sets   pwelg 43533
                  ​*21.36.4.6  Finite intersection property   fipjust 43538
                  21.36.4.7  RP ADDTO: Subclasses and subsets   rababg 43547
                  21.36.4.8  RP ADDTO: The intersection of a class   elinintab 43548
                  21.36.4.9  RP ADDTO: Theorems requiring subset and intersection existence   elinintrab 43550
                  21.36.4.10  RP ADDTO: Relations   xpinintabd 43553
                  ​*21.36.4.11  RP ADDTO: Functions   elmapintab 43569
                  ​*21.36.4.12  RP ADDTO: Finite induction (for finite ordinals)   cnvcnvintabd 43573
                  21.36.4.13  RP ADDTO: First and second members of an ordered pair   elcnvlem 43574
                  21.36.4.14  RP ADDTO: The reflexive and transitive properties of relations   undmrnresiss 43577
                  21.36.4.15  RP ADDTO: Basic properties of closures   cleq2lem 43581
                  21.36.4.16  RP REPLACE: Definitions and basic properties of transitive closures   trcleq2lemRP 43603
                  ​*21.36.4.17  Additions for square root; absolute value   sqrtcvallem1 43604
            21.36.5  Additional statements on relations and subclasses   al3im 43620
                  21.36.5.1  Transitive relations (not to be confused with transitive classes).   trrelind 43638
                  21.36.5.2  Reflexive closures   crcl 43645
                  ​*21.36.5.3  Finite relationship composition.   relexp2 43650
                  21.36.5.4  Transitive closure of a relation   dftrcl3 43693
                  ​*21.36.5.5  Adapted from Frege   frege77d 43719
            ​*21.36.6  Propositions from _Begriffsschrift_   dfxor4 43739
                  *21.36.6.1  _Begriffsschrift_ Chapter I   dfxor4 43739
                  ​*21.36.6.2  _Begriffsschrift_ Notation hints   whe 43745
                  21.36.6.3  _Begriffsschrift_ Chapter II Implication   ax-frege1 43763
                  21.36.6.4  _Begriffsschrift_ Chapter II Implication and Negation   axfrege28 43802
                  ​*21.36.6.5  _Begriffsschrift_ Chapter II with logical equivalence   axfrege52a 43829
                  21.36.6.6  _Begriffsschrift_ Chapter II with equivalence of sets   axfrege52c 43860
                  ​*21.36.6.7  _Begriffsschrift_ Chapter II with equivalence of classes   frege53c 43887
                  ​*21.36.6.8  _Begriffsschrift_ Chapter III Properties hereditary in a sequence   dffrege69 43905
                  ​*21.36.6.9  _Begriffsschrift_ Chapter III Following in a sequence   dffrege76 43912
                  ​*21.36.6.10  _Begriffsschrift_ Chapter III Member of sequence   dffrege99 43935
                  ​*21.36.6.11  _Begriffsschrift_ Chapter III Single-valued procedures   dffrege115 43951
            ​*21.36.7  Exploring Topology via Seifert and Threlfall   enrelmap 43970
                  *21.36.7.1  Equinumerosity of sets of relations and maps   enrelmap 43970
                  ​*21.36.7.2  Generic Pseudoclosure Spaces, Pseudointerior Spaces, and Pseudoneighborhoods   or3or 43996
                  ​*21.36.7.3  Generic Neighborhood Spaces   gneispa 44103
            ​*21.36.8  Exploring Higher Homotopy via Kerodon   k0004lem1 44120
                  *21.36.8.1  Simplicial Sets   k0004lem1 44120
      ​21.37  Mathbox for Stanislas Polu
            21.37.1  IMO Problems   wwlemuld 44129
                  21.37.1.1  IMO 1972 B2   wwlemuld 44129
            ​*21.37.2  INT Inequalities Proof Generator   int-addcomd 44146
            ​*21.37.3  N-Digit Addition Proof Generator   unitadd 44168
            21.37.4  AM-GM (for k = 2,3,4)   gsumws3 44169
      ​21.38  Mathbox for Rohan Ridenour
            21.38.1  Misc   spALT 44174
            21.38.2  Monoid rings   cmnring 44184
            21.38.3  Shorter primitive equivalent of ax-groth   gru0eld 44202
                  21.38.3.1  Grothendieck universes are closed under collection   gru0eld 44202
                  21.38.3.2  Minimal universes   ismnu 44234
                  21.38.3.3  Primitive equivalent of ax-groth   expandan 44261
      ​21.39  Mathbox for Steve Rodriguez
            21.39.1  Miscellanea   nanorxor 44278
            21.39.2  Ratio test for infinite series convergence and divergence   dvgrat 44285
            21.39.3  Multiples   reldvds 44288
            21.39.4  Function operations   caofcan 44296
            21.39.5  Calculus   lhe4.4ex1a 44302
            21.39.6  The generalized binomial coefficient operation   cbcc 44309
            21.39.7  Binomial series   uzmptshftfval 44319
      ​21.40  Mathbox for Andrew Salmon
            21.40.1  Principia Mathematica * 10   pm10.12 44331
            21.40.2  Principia Mathematica * 11   2alanimi 44345
            21.40.3  Predicate Calculus   sbeqal1 44371
            21.40.4  Principia Mathematica * 13 and * 14   pm13.13a 44380
            21.40.5  Set Theory   elnev 44411
            21.40.6  Arithmetic   addcomgi 44429
            21.40.7  Geometry   cplusr 44430
      ​​*21.41  Mathbox for Alan Sare
            21.41.1  Auxiliary theorems for the Virtual Deduction tool   idiALT 44452
            21.41.2  Supplementary unification deductions   bi1imp 44456
            21.41.3  Conventional Metamath proofs, some derived from VD proofs   iidn3 44475
            21.41.4  What is Virtual Deduction?   wvd1 44543
            21.41.5  Virtual Deduction Theorems   df-vd1 44544
            21.41.6  Theorems proved using Virtual Deduction   trsspwALT 44791
            21.41.7  Theorems proved using Virtual Deduction with mmj2 assistance   simplbi2VD 44819
            21.41.8  Virtual Deduction transcriptions of textbook proofs   sb5ALTVD 44886
            21.41.9  Theorems proved using conjunction-form Virtual Deduction   elpwgdedVD 44890
            21.41.10  Theorems with a VD proof in conventional notation derived from a VD proof   suctrALT3 44897
            ​*21.41.11  Theorems with a proof in conventional notation derived from a VD proof   notnotrALT2 44900
      ​21.42  Mathbox for Eric Schmidt
            21.42.1  Miscellany   rspesbcd 44911
            21.42.2  Study of dfbi1ALT   dfbi1ALTa 44913
            21.42.3  Relation-preserving functions   wrelp 44916
            21.42.4  Orbits   orbitex 44929
            21.42.5  Well-founded sets   trwf 44933
            21.42.6  Absoluteness in transitive models   ralabso 44942
            21.42.7  Lemmas for showing axioms hold in models   traxext 44951
            21.42.8  The class of well-founded sets is a model for ZFC   wfaxext 44967
            21.42.9  Permutation models   brpermmodel 44977
      ​21.43  Mathbox for Glauco Siliprandi
            21.43.1  Miscellanea   evth2f 44993
            21.43.2  Functions   fnresdmss 45146
            21.43.3  Ordering on real numbers - Real and complex numbers basic operations   sub2times 45255
            21.43.4  Real intervals   gtnelioc 45473
            21.43.5  Finite sums   fsummulc1f 45553
            21.43.6  Finite multiplication of numbers and finite multiplication of functions   fmul01 45562
            21.43.7  Limits   clim1fr1 45583
                  21.43.7.1  Inferior limit (lim inf)   clsi 45733
                  ​*21.43.7.2  Limits for sequences of extended real numbers   clsxlim 45800
            21.43.8  Trigonometry   coseq0 45846
            21.43.9  Continuous Functions   mulcncff 45852
            21.43.10  Derivatives   dvsinexp 45893
            21.43.11  Integrals   itgsin0pilem1 45932
            21.43.12  Stone Weierstrass theorem - real version   stoweidlem1 45983
            21.43.13  Wallis' product for π   wallispilem1 46047
            21.43.14  Stirling's approximation formula for ` n ` factorial   stirlinglem1 46056
            21.43.15  Dirichlet kernel   dirkerval 46073
            21.43.16  Fourier Series   fourierdlem1 46090
            21.43.17  e is transcendental   elaa2lem 46215
            21.43.18  n-dimensional Euclidean space   rrxtopn 46266
            21.43.19  Basic measure theory   csalg 46290
                  ​*21.43.19.1  σ-Algebras   csalg 46290
                  21.43.19.2  Sum of nonnegative extended reals   csumge0 46344
                  ​*21.43.19.3  Measures   cmea 46431
                  ​*21.43.19.4  Outer measures and Caratheodory's construction   come 46471
                  ​*21.43.19.5  Lebesgue measure on n-dimensional Real numbers   covoln 46518
                  ​*21.43.19.6  Measurable functions   csmblfn 46677
      ​21.44  Mathbox for Saveliy Skresanov
            21.44.1  Ceva's theorem   sigarval 46832
            21.44.2  Simple groups   simpcntrab 46852
      ​21.45  Mathbox for Ender Ting
            21.45.1  Increasing sequences and subsequences   et-ltneverrefl 46853
            21.45.2  Scratchpad for number theory   evenwodadd 46870
            21.45.3  Scratchpad for math on real numbers   squeezedltsq 46871
      ​21.46  Mathbox for Jarvin Udandy
      ​21.47  Mathbox for Adhemar
            ​*21.47.1  Minimal implicational calculus   adh-minim 46986
      ​21.48  Mathbox for Alexander van der Vekens
            21.48.1  General auxiliary theorems (1)   n0nsn2el 47010
                  21.48.1.1  Unordered and ordered pairs - extension for singletons   n0nsn2el 47010
                  21.48.1.2  Unordered and ordered pairs - extension for unordered pairs   elprneb 47014
                  21.48.1.3  Unordered and ordered pairs - extension for ordered pairs   oppr 47015
                  21.48.1.4  Relations - extension   eubrv 47020
                  21.48.1.5  Definite description binder (inverted iota) - extension   iota0def 47023
                  21.48.1.6  Functions - extension   fveqvfvv 47025
            21.48.2  Alternative for Russell's definition of a description binder   caiota 47068
            21.48.3  Double restricted existential uniqueness   r19.32 47083
                  21.48.3.1  Restricted quantification (extension)   r19.32 47083
                  21.48.3.2  Restricted uniqueness and "at most one" quantification   reuf1odnf 47092
                  21.48.3.3  Analogs to Existential uniqueness (double quantification)   2reu3 47095
                  21.48.3.4  Additional theorems for double restricted existential uniqueness   2reu8i 47098
            ​*21.48.4  Alternative definitions of function and operation values   wdfat 47101
                  21.48.4.1  Restricted quantification (extension)   ralbinrald 47107
                  21.48.4.2  The universal class (extension)   nvelim 47108
                  21.48.4.3  Introduce the Axiom of Power Sets (extension)   alneu 47109
                  21.48.4.4  Predicate "defined at"   dfateq12d 47111
                  21.48.4.5  Alternative definition of the value of a function   dfafv2 47117
                  21.48.4.6  Alternative definition of the value of an operation   aoveq123d 47163
            ​*21.48.5  Alternative definitions of function values (2)   cafv2 47193
            21.48.6  General auxiliary theorems (2)   an4com24 47253
                  21.48.6.1  Logical conjunction - extension   an4com24 47253
                  21.48.6.2  Abbreviated conjunction and disjunction of three wff's - extension   3an4ancom24 47254
                  21.48.6.3  Negated membership (alternative)   cnelbr 47256
                  21.48.6.4  The empty set - extension   ralralimp 47263
                  21.48.6.5  Indexed union and intersection - extension   otiunsndisjX 47264
                  21.48.6.6  Functions - extension   fvifeq 47265
                  21.48.6.7  Maps-to notation - extension   fvmptrab 47277
                  21.48.6.8  Subtraction - extension   cnambpcma 47279
                  21.48.6.9  Ordering on reals (cont.) - extension   leaddsuble 47282
                  21.48.6.10  Imaginary and complex number properties - extension   readdcnnred 47288
                  21.48.6.11  Nonnegative integers (as a subset of complex numbers) - extension   nn0resubcl 47293
                  21.48.6.12  Integers (as a subset of complex numbers) - extension   zgeltp1eq 47294
                  21.48.6.13  Decimal arithmetic - extension   1t10e1p1e11 47295
                  21.48.6.14  Upper sets of integers - extension   eluzge0nn0 47297
                  21.48.6.15  Infinity and the extended real number system (cont.) - extension   nltle2tri 47298
                  21.48.6.16  Finite intervals of integers - extension   ssfz12 47299
                  21.48.6.17  Half-open integer ranges - extension   fzopred 47307
                  21.48.6.18  The floor and ceiling functions - extension   2ltceilhalf 47313
                  21.48.6.19  The modulo (remainder) operation - extension   fldivmod 47323
                  21.48.6.20  The infinite sequence builder "seq"   smonoord 47356
                  21.48.6.21  Finite and infinite sums - extension   fsummsndifre 47357
                  21.48.6.22  Extensible structures - extension   setsidel 47361
            ​*21.48.7  Preimages of function values   preimafvsnel 47364
            ​*21.48.8  Partitions of real intervals   ciccp 47398
            21.48.9  Shifting functions with an integer range domain   fargshiftfv 47424
            21.48.10  Words over a set (extension)   lswn0 47429
                  21.48.10.1  Last symbol of a word - extension   lswn0 47429
            21.48.11  Unordered pairs   wich 47430
                  21.48.11.1  Interchangeable setvar variables   wich 47430
                  21.48.11.2  Set of unordered pairs   sprid 47459
                  ​*21.48.11.3  Proper (unordered) pairs   prpair 47486
                  21.48.11.4  Set of proper unordered pairs   cprpr 47497
            21.48.12  Number theory (extension)   cfmtno 47512
                  ​*21.48.12.1  Fermat numbers   cfmtno 47512
                  ​*21.48.12.2  Mersenne primes   m2prm 47576
                  21.48.12.3  Proth's theorem   modexp2m1d 47597
                  21.48.12.4  Solutions of quadratic equations   quad1 47605
            ​*21.48.13  Even and odd numbers   ceven 47609
                  21.48.13.1  Definitions and basic properties   ceven 47609
                  21.48.13.2  Alternate definitions using the "divides" relation   dfeven2 47634
                  21.48.13.3  Alternate definitions using the "modulo" operation   dfeven3 47643
                  21.48.13.4  Alternate definitions using the "gcd" operation   iseven5 47649
                  21.48.13.5  Theorems of part 5 revised   zneoALTV 47654
                  21.48.13.6  Theorems of part 6 revised   odd2np1ALTV 47659
                  21.48.13.7  Theorems of AV's mathbox revised   0evenALTV 47673
                  21.48.13.8  Additional theorems   epoo 47688
                  21.48.13.9  Perfect Number Theorem (revised)   perfectALTVlem1 47706
            21.48.14  Number theory (extension 2)   cfppr 47709
                  ​*21.48.14.1  Fermat pseudoprimes   cfppr 47709
                  ​*21.48.14.2  Goldbach's conjectures   cgbe 47730
            21.48.15  Graph theory (extension)   cclnbgr 47803
                  21.48.15.1  Closed neighborhood of a vertex   cclnbgr 47803
                  ​*21.48.15.2  Semiclosed and semiopen neighborhoods (experimental)   dfsclnbgr2 47830
                  21.48.15.3  Induced subgraphs   cisubgr 47844
                  ​*21.48.15.4  Isomorphisms of graphs   cgrisom 47858
                  ​*21.48.15.5  Triangles in graphs   cgrtri 47920
                  ​*21.48.15.6  Star graphs   cstgr 47934
                  ​*21.48.15.7  Local isomorphisms of graphs   cgrlim 47959
                  ​*21.48.15.8  Generalized Petersen graphs   cgpg 48015
                  21.48.15.9  Loop-free graphs - extension   1hegrlfgr 48104
                  21.48.15.10  Walks - extension   cupwlks 48105
                  21.48.15.11  Edges of graphs expressed as sets of unordered pairs   upgredgssspr 48115
            21.48.16  Monoids (extension)   ovn0dmfun 48128
                  21.48.16.1  Auxiliary theorems   ovn0dmfun 48128
                  21.48.16.2  Magmas, Semigroups and Monoids (extension)   plusfreseq 48136
                  21.48.16.3  Examples and counterexamples for magmas, semigroups and monoids (extension)   opmpoismgm 48139
                  21.48.16.4  Group sum operation (extension 1)   gsumsplit2f 48152
            ​*21.48.17  Magmas and internal binary operations (alternate approach)   ccllaw 48155
                  *21.48.17.1  Laws for internal binary operations   ccllaw 48155
                  ​*21.48.17.2  Internal binary operations   cintop 48168
                  21.48.17.3  Alternative definitions for magmas and semigroups   cmgm2 48187
            21.48.18  Rings (extension)   lmod0rng 48201
                  21.48.18.1  Nonzero rings (extension)   lmod0rng 48201
                  21.48.18.2  Ideals as non-unital rings   lidldomn1 48203
                  21.48.18.3  The non-unital ring of even integers   0even 48209
                  21.48.18.4  A constructed not unital ring   cznrnglem 48231
                  ​*21.48.18.5  The category of non-unital rings (alternate definition)   crngcALTV 48235
                  ​*21.48.18.6  The category of (unital) rings (alternate definition)   cringcALTV 48259
            21.48.19  Basic algebraic structures (extension)   eliunxp2 48306
                  21.48.19.1  Auxiliary theorems   eliunxp2 48306
                  21.48.19.2  The binomial coefficient operation (extension)   bcpascm1 48323
                  21.48.19.3  The ` ZZ `-module ` ZZ X. ZZ `   zlmodzxzlmod 48326
                  21.48.19.4  Group sum operation (extension 2)   mgpsumunsn 48333
                  21.48.19.5  Symmetric groups (extension)   exple2lt6 48336
                  21.48.19.6  Divisibility (extension)   invginvrid 48339
                  21.48.19.7  The support of functions (extension)   rmsupp0 48340
                  21.48.19.8  Finitely supported functions (extension)   rmsuppfi 48344
                  21.48.19.9  Left modules (extension)   lmodvsmdi 48351
                  21.48.19.10  Associative algebras (extension)   assaascl0 48353
                  21.48.19.11  Univariate polynomials (extension)   ply1vr1smo 48355
                  21.48.19.12  Univariate polynomials (examples)   linply1 48366
            21.48.20  Linear algebra (extension)   cdmatalt 48369
                  ​*21.48.20.1  The subalgebras of diagonal and scalar matrices (extension)   cdmatalt 48369
                  ​*21.48.20.2  Linear combinations   clinc 48377
                  ​*21.48.20.3  Linear independence   clininds 48413
                  21.48.20.4  Simple left modules and the ` ZZ `-module   lmod1lem1 48460
                  21.48.20.5  Differences between (left) modules and (left) vector spaces   lvecpsslmod 48480
            21.48.21  Complexity theory   suppdm 48483
                  21.48.21.1  Auxiliary theorems   suppdm 48483
                  21.48.21.2  Even and odd integers   nn0onn0ex 48496
                  21.48.21.3  The natural logarithm on complex numbers (extension)   logcxp0 48508
                  21.48.21.4  Division of functions   cfdiv 48510
                  21.48.21.5  Upper bounds   cbigo 48520
                  21.48.21.6  Logarithm to an arbitrary base (extension)   rege1logbrege0 48531
                  ​*21.48.21.7  The binary logarithm   fldivexpfllog2 48538
                  21.48.21.8  Binary length   cblen 48542
                  ​*21.48.21.9  Digits   cdig 48568
                  21.48.21.10  Nonnegative integer as sum of its shifted digits   dignn0flhalflem1 48588
                  21.48.21.11  Algorithms for the multiplication of nonnegative integers   nn0mulfsum 48597
                  ​*21.48.21.12  N-ary functions   cnaryf 48599
                  ​*21.48.21.13  The Ackermann function   citco 48630
            21.48.22  Elementary geometry (extension)   fv1prop 48672
                  21.48.22.1  Auxiliary theorems   fv1prop 48672
                  21.48.22.2  Real euclidean space of dimension 2   rrx2pxel 48684
                  21.48.22.3  Spheres and lines in real Euclidean spaces   cline 48700
      ​21.49  Mathbox for Zhi Wang
            21.49.1  Propositional calculus   pm4.71da 48762
            21.49.2  Predicate calculus with equality   dtrucor3 48771
                  21.49.2.1  Axiom scheme ax-5 (Distinctness)   dtrucor3 48771
            21.49.3  ZF Set Theory - start with the Axiom of Extensionality   ralbidb 48772
                  21.49.3.1  Restricted quantification   ralbidb 48772
                  21.49.3.2  The universal class   reuxfr1dd 48779
                  21.49.3.3  The empty set   ssdisjd 48780
                  21.49.3.4  Unordered and ordered pairs   vsn 48784
                  21.49.3.5  The union of a class   unilbss 48790
                  21.49.3.6  Indexed union and intersection   iuneq0 48791
            21.49.4  ZF Set Theory - add the Axiom of Replacement   inpw 48797
                  21.49.4.1  Theorems requiring subset and intersection existence   inpw 48797
            21.49.5  ZF Set Theory - add the Axiom of Power Sets   opth1neg 48798
                  21.49.5.1  Ordered pair theorem   opth1neg 48798
                  21.49.5.2  Ordered-pair class abstractions (cont.)   brab2dd 48800
                  21.49.5.3  Relations   iinxp 48803
                  21.49.5.4  Functions   mof0 48810
                  21.49.5.5  Operations   ovsng 48830
            21.49.6  ZF Set Theory - add the Axiom of Union   fonex 48839
                  21.49.6.1  Relations and functions (cont.)   fonex 48839
                  21.49.6.2  First and second members of an ordered pair   eloprab1st2nd 48840
                  21.49.6.3  Operations in maps-to notation (continued)   fmpodg 48841
                  21.49.6.4  Function transposition   resinsnlem 48843
                  21.49.6.5  Infinite Cartesian products   ixpv 48862
                  21.49.6.6  Equinumerosity   fvconst0ci 48863
            21.49.7  Order sets   iccin 48868
                  21.49.7.1  Real number intervals   iccin 48868
            21.49.8  Extensible structures   slotresfo 48871
                  21.49.8.1  Basic definitions   slotresfo 48871
            21.49.9  Moore spaces   mreuniss 48872
            ​*21.49.10  Topology   clduni 48873
                  21.49.10.1  Closure and interior   clduni 48873
                  21.49.10.2  Neighborhoods   neircl 48877
                  21.49.10.3  Subspace topologies   restcls2lem 48885
                  21.49.10.4  Limits and continuity in topological spaces   cnneiima 48889
                  21.49.10.5  Topological definitions using the reals   iooii 48890
                  21.49.10.6  Separated sets   sepnsepolem1 48894
                  21.49.10.7  Separated spaces: T0, T1, T2 (Hausdorff) ...   isnrm4 48903
            21.49.11  Preordered sets and directed sets using extensible structures   isprsd 48927
            21.49.12  Posets and lattices using extensible structures   lubeldm2 48928
                  21.49.12.1  Posets   lubeldm2 48928
                  21.49.12.2  Lattices   toslat 48954
                  21.49.12.3  Subset order structures   intubeu 48956
            21.49.13  Rings   elmgpcntrd 48977
                  21.49.13.1  Multiplicative Group   elmgpcntrd 48977
            21.49.14  Associative algebras   asclelbas 48978
                  21.49.14.1  Definition and basic properties   asclelbas 48978
            21.49.15  Categories   homf0 48982
                  21.49.15.1  Categories   homf0 48982
                  21.49.15.2  Opposite category   oppccatb 48989
                  21.49.15.3  Monomorphisms and epimorphisms   idmon 48993
                  21.49.15.4  Sections, inverses, isomorphisms   sectrcl 48995
                  21.49.15.5  Isomorphic objects   cicfn 49015
                  21.49.15.6  Subcategories   dmdm 49026
                  21.49.15.7  Functors   reldmfunc 49048
                  21.49.15.8  Opposite functors   coppf 49095
                  21.49.15.9  Full & faithful functors   imasubc 49124
                  21.49.15.10  Universal property   upciclem1 49139
                  21.49.15.11  Natural transformations and the functor category   isnatd 49196
                  21.49.15.12  Initial, terminal and zero objects of a category   initoo2 49205
                  21.49.15.13  Product of categories   reldmxpc 49219
                  21.49.15.14  Swap functors   cswapf 49232
                  21.49.15.15  Functor evaluation   oppc1stflem 49260
                  21.49.15.16  Transposed curry functors   cofuswapfcl 49266
                  21.49.15.17  Constant functors   diag1 49277
                  21.49.15.18  Functor composition bifunctors   fucofulem1 49283
                  21.49.15.19  Post-composition functors   postcofval 49337
                  21.49.15.20  Pre-composition functors   precofvallem 49339
            21.49.16  Examples of categories   catcrcl 49368
                  21.49.16.1  The category of categories   catcrcl 49368
                  21.49.16.2  Thin categories   cthinc 49390
                  21.49.16.3  Terminal categories   ctermc 49445
                  21.49.16.4  Preordered sets as thin categories   cprstc 49522
                  21.49.16.5  Monoids as categories   cmndtc 49550
                  21.49.16.6  Categories with at most one object and at most two morphisms   2arwcatlem1 49568
            21.49.17  Kan extensions and related concepts   clan 49578
                  21.49.17.1  Kan extensions   clan 49578
                  21.49.17.2  Limits and colimits   clmd 49616
      ​21.50  Mathbox for Emmett Weisz
            ​*21.50.1  Miscellaneous Theorems   nfintd 49646
            21.50.2  Set Recursion   csetrecs 49656
                  ​*21.50.2.1  Basic Properties of Set Recursion   csetrecs 49656
                  21.50.2.2  Examples and properties of set recursion   elsetrecslem 49672
            ​*21.50.3  Construction of Games and Surreal Numbers   cpg 49682
      ​​*21.51  Mathbox for David A. Wheeler
            21.51.1  Natural deduction   sbidd 49691
            ​*21.51.2  Greater than, greater than or equal to.   cge-real 49693
            ​*21.51.3  Hyperbolic trigonometric functions   csinh 49703
            ​*21.51.4  Reciprocal trigonometric functions (sec, csc, cot)   csec 49714
            ​*21.51.5  Identities for "if"   ifnmfalse 49736
            ​*21.51.6  Logarithms generalized to arbitrary base using ` logb `   logb2aval 49737
            ​*21.51.7  Logarithm laws generalized to an arbitrary base - log_   clog- 49738
            ​*21.51.8  Formally define notions such as reflexivity   wreflexive 49740
            ​*21.51.9  Algebra helpers   mvlraddi 49744
            ​*21.51.10  Algebra helper examples   i2linesi 49751
            ​*21.51.11  Formal methods "surprises"   alimp-surprise 49753
            ​*21.51.12  Allsome quantifier   walsi 49759
            ​*21.51.13  Miscellaneous   5m4e1 49770
            21.51.14  Theorems about algebraic numbers   aacllem 49774
      ​21.52  Mathbox for Kunhao Zheng
            21.52.1  Weighted AM-GM inequality   amgmwlem 49775