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 3352
                  2.1.5.3  Restricted class abstraction   crab 3405
            2.1.6  The universal class   cvv 3447
            ​*2.1.7  Conditional equality (experimental)   wcdeq 3734
            2.1.8  Russell's Paradox   rru 3750
            2.1.9  Proper substitution of classes for sets   wsbc 3753
            2.1.10  Proper substitution of classes for sets into classes   csb 3862
            2.1.11  Define basic set operations and relations   cdif 3911
            2.1.12  Subclasses and subsets   df-ss 3931
            2.1.13  The difference, union, and intersection of two classes   dfdif3 4080
                  2.1.13.1  The difference of two classes   dfdif3 4080
                  2.1.13.2  The union of two classes   elun 4116
                  2.1.13.3  The intersection of two classes   elini 4162
                  2.1.13.4  The symmetric difference of two classes   csymdif 4215
                  2.1.13.5  Combinations of difference, union, and intersection of two classes   unabs 4228
                  2.1.13.6  Class abstractions with difference, union, and intersection of two classes   unabw 4270
                  2.1.13.7  Restricted uniqueness with difference, union, and intersection   reuun2 4288
            2.1.14  The empty set   c0 4296
            ​*2.1.15  The conditional operator for classes   cif 4488
            ​*2.1.16  The weak deduction theorem for set theory   dedth 4547
            2.1.17  Power classes   cpw 4563
            2.1.18  Unordered and ordered pairs   snjust 4588
            2.1.19  The union of a class   cuni 4871
            2.1.20  The intersection of a class   cint 4910
            2.1.21  Indexed union and intersection   ciun 4955
            2.1.22  Disjointness   wdisj 5074
            2.1.23  Binary relations   wbr 5107
            2.1.24  Ordered-pair class abstractions (class builders)   copab 5169
            2.1.25  Functions in maps-to notation   cmpt 5188
            2.1.26  Transitive classes   wtr 5214
      ​2.2  ZF Set Theory - add the Axiom of Replacement
            2.2.1  Introduce the Axiom of Replacement   ax-rep 5234
            2.2.2  Derive the Axiom of Separation   axsepgfromrep 5249
            2.2.3  Derive the Null Set Axiom   axnulALT 5259
            2.2.4  Theorems requiring subset and intersection existence   nalset 5268
            2.2.5  Theorems requiring empty set existence   class2set 5310
      ​2.3  ZF Set Theory - add the Axiom of Power Sets
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 5320
            2.3.2  Derive the Axiom of Pairing   axprlem1 5378
            2.3.3  Ordered pair theorem   opnz 5433
            2.3.4  Ordered-pair class abstractions (cont.)   opabidw 5484
            2.3.5  Power class of union and intersection   pwin 5529
            2.3.6  The identity relation   cid 5532
            2.3.7  The membership relation (or epsilon relation)   cep 5537
            ​*2.3.8  Partial and total orderings   wpo 5544
            2.3.9  Founded and well-ordering relations   wfr 5588
            2.3.10  Relations   cxp 5636
            2.3.11  The Predecessor Class   cpred 6273
            2.3.12  Well-founded induction (variant)   frpomin 6313
            2.3.13  Well-ordered induction   tz6.26 6320
            2.3.14  Ordinals   word 6331
            2.3.15  Definite description binder (inverted iota)   cio 6462
            2.3.16  Functions   wfun 6505
            2.3.17  Cantor's Theorem   canth 7341
            2.3.18  Restricted iota (description binder)   crio 7343
            2.3.19  Operations   co 7387
                  2.3.19.1  Variable-to-class conversion for operations   caovclg 7581
            2.3.20  Maps-to notation   mpondm0 7629
            2.3.21  Function operation   cof 7651
            2.3.22  Proper subset relation   crpss 7698
      ​2.4  ZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 7711
            2.4.2  Ordinals (continued)   epweon 7751
            2.4.3  Transfinite induction   tfi 7829
            2.4.4  The natural numbers (i.e., finite ordinals)   com 7842
            2.4.5  Peano's postulates   peano1 7865
            2.4.6  Finite induction (for finite ordinals)   find 7871
            2.4.7  Relations and functions (cont.)   dmexg 7877
            2.4.8  First and second members of an ordered pair   c1st 7966
            2.4.9  Induction on Cartesian products   frpoins3xpg 8119
            2.4.10  Ordering on Cartesian products   xpord2lem 8121
            2.4.11  Ordering Ordinal Sequences   orderseqlem 8136
            ​*2.4.12  The support of functions   csupp 8139
            ​*2.4.13  Special maps-to operations   opeliunxp2f 8189
            2.4.14  Function transposition   ctpos 8204
            2.4.15  Curry and uncurry   ccur 8244
            2.4.16  Undefined values   cund 8251
            2.4.17  Well-founded recursion   cfrecs 8259
            2.4.18  Well-ordered recursion   cwrecs 8290
            2.4.19  Functions on ordinals; strictly monotone ordinal functions   iunon 8308
            2.4.20  "Strong" transfinite recursion   crecs 8339
            2.4.21  Recursive definition generator   crdg 8377
            2.4.22  Finite recursion   frfnom 8403
            2.4.23  Ordinal arithmetic   c1o 8427
            2.4.24  Natural number arithmetic   nna0 8568
            2.4.25  Natural addition   cnadd 8629
            2.4.26  Equivalence relations and classes   wer 8668
            2.4.27  The mapping operation   cmap 8799
            2.4.28  Infinite Cartesian products   cixp 8870
            2.4.29  Equinumerosity   cen 8915
            2.4.30  Schroeder-Bernstein Theorem   sbthlem1 9051
            2.4.31  Equinumerosity (cont.)   xpf1o 9103
            2.4.32  Finite sets   dif1enlem 9120
            2.4.33  Pigeonhole Principle   phplem1 9168
            2.4.34  Finite sets (cont.)   onomeneq 9178
            2.4.35  Finitely supported functions   cfsupp 9312
            2.4.36  Finite intersections   cfi 9361
            2.4.37  Hall's marriage theorem   marypha1lem 9384
            2.4.38  Supremum and infimum   csup 9391
            2.4.39  Ordinal isomorphism, Hartogs's theorem   coi 9462
            2.4.40  Hartogs function   char 9509
            2.4.41  Weak dominance   cwdom 9517
      ​2.5  ZF Set Theory - add the Axiom of Regularity
            2.5.1  Introduce the Axiom of Regularity   ax-reg 9545
            2.5.2  Axiom of Infinity equivalents   inf0 9574
      ​2.6  ZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-inf 9591
            2.6.2  Existence of omega (the set of natural numbers)   omex 9596
            2.6.3  Cantor normal form   ccnf 9614
            2.6.4  Transitive closure of a relation   cttrcl 9660
            2.6.5  Transitive closure   trcl 9681
            2.6.6  Well-Founded Induction   frmin 9702
            2.6.7  Well-Founded Recursion   frr3g 9709
            2.6.8  Rank   cr1 9715
            2.6.9  Scott's trick; collection principle; Hilbert's epsilon   scottex 9838
            2.6.10  Disjoint union   cdju 9851
            2.6.11  Cardinal numbers   ccrd 9888
            2.6.12  Axiom of Choice equivalents   wac 10068
            ​*2.6.13  Cardinal number arithmetic   undjudom 10121
            2.6.14  The Ackermann bijection   ackbij2lem1 10171
            2.6.15  Cofinality (without Axiom of Choice)   cflem 10198
            2.6.16  Eight inequivalent definitions of finite set   sornom 10230
            2.6.17  Hereditarily size-limited sets without Choice   itunifval 10369
​​*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 10388
            3.1.2  Introduce the Axiom of Dependent Choice   ax-dc 10399
      ​3.2  ZFC Set Theory - add the Axiom of Choice
            3.2.1  Introduce the Axiom of Choice   ax-ac 10412
            3.2.2  AC equivalents: well-ordering, Zorn's lemma   numthcor 10447
            3.2.3  Cardinal number theorems using Axiom of Choice   cardval 10499
            3.2.4  Cardinal number arithmetic using Axiom of Choice   iunctb 10527
            3.2.5  Cofinality using the Axiom of Choice   alephreg 10535
      ​3.3  ZFC Axioms with no distinct variable requirements
      ​3.4  The Generalized Continuum Hypothesis
            3.4.1  Sets satisfying the Generalized Continuum Hypothesis   cgch 10573
            3.4.2  Derivation of the Axiom of Choice   gchaclem 10631
​​*PART 4  TG (TARSKI-GROTHENDIECK) SET THEORY
      ​4.1  Inaccessibles
            4.1.1  Weakly and strongly inaccessible cardinals   cwina 10635
            4.1.2  Weak universes   cwun 10653
            4.1.3  Tarski classes   ctsk 10701
            4.1.4  Grothendieck universes   cgru 10743
      ​4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
            4.2.1  Introduce the Tarski-Grothendieck Axiom   ax-groth 10776
            4.2.2  Derive the Power Set, Infinity and Choice Axioms   grothpw 10779
            4.2.3  Tarski map function   ctskm 10790
​​*PART 5  REAL AND COMPLEX NUMBERS
      ​5.1  Construction and axiomatization of real and complex numbers
            5.1.1  Dedekind-cut construction of real and complex numbers   cnpi 10797
            5.1.2  Final derivation of real and complex number postulates   axaddf 11098
            5.1.3  Real and complex number postulates restated as axioms   ax-cnex 11124
      ​5.2  Derive the basic properties from the field axioms
            5.2.1  Some deductions from the field axioms for complex numbers   cnex 11149
            5.2.2  Infinity and the extended real number system   cpnf 11205
            5.2.3  Restate the ordering postulates with extended real "less than"   axlttri 11245
            5.2.4  Ordering on reals   lttr 11250
            5.2.5  Initial properties of the complex numbers   mul12 11339
      ​5.3  Real and complex numbers - basic operations
            5.3.1  Addition   add12 11392
            5.3.2  Subtraction   cmin 11405
            5.3.3  Multiplication   kcnktkm1cn 11609
            5.3.4  Ordering on reals (cont.)   gt0ne0 11643
            5.3.5  Reciprocals   ixi 11807
            5.3.6  Division   cdiv 11835
            5.3.7  Ordering on reals (cont.)   elimgt0 12020
            5.3.8  Completeness Axiom and Suprema   fimaxre 12127
            5.3.9  Imaginary and complex number properties   neg1cn 12171
            5.3.10  Function operation analogue theorems   ofsubeq0 12183
      ​5.4  Integer sets
            5.4.1  Positive integers (as a subset of complex numbers)   cn 12186
            5.4.2  Principle of mathematical induction   nnind 12204
            ​*5.4.3  Decimal representation of numbers   c2 12241
            ​*5.4.4  Some properties of specific numbers   1pneg1e0 12300
            5.4.5  Simple number properties   halfcl 12408
            5.4.6  The Archimedean property   nnunb 12438
            5.4.7  Nonnegative integers (as a subset of complex numbers)   cn0 12442
            ​*5.4.8  Extended nonnegative integers   cxnn0 12515
            5.4.9  Integers (as a subset of complex numbers)   cz 12529
            5.4.10  Decimal arithmetic   cdc 12649
            5.4.11  Upper sets of integers   cuz 12793
            5.4.12  Well-ordering principle for bounded-below sets of integers   uzwo3 12902
            5.4.13  Rational numbers (as a subset of complex numbers)   cq 12907
            5.4.14  Existence of the set of complex numbers   rpnnen1lem2 12936
      ​5.5  Order sets
            5.5.1  Positive reals (as a subset of complex numbers)   crp 12951
            5.5.2  Infinity and the extended real number system (cont.)   cxne 13069
            5.5.3  Supremum and infimum on the extended reals   xrsupexmnf 13265
            5.5.4  Real number intervals   cioo 13306
            5.5.5  Finite intervals of integers   cfz 13468
            ​*5.5.6  Finite intervals of nonnegative integers   elfz2nn0 13579
            5.5.7  Half-open integer ranges   cfzo 13615
      ​5.6  Elementary integer functions
            5.6.1  The floor and ceiling functions   cfl 13752
            5.6.2  The modulo (remainder) operation   cmo 13831
            5.6.3  Miscellaneous theorems about integers   om2uz0i 13912
            5.6.4  Strong induction over upper sets of integers   uzsinds 13952
            5.6.5  Finitely supported functions over the nonnegative integers   fsuppmapnn0fiublem 13955
            5.6.6  The infinite sequence builder "seq" - extension   cseq 13966
            5.6.7  Integer powers   cexp 14026
            5.6.8  Ordered pair theorem for nonnegative integers   nn0le2msqi 14232
            5.6.9  Factorial function   cfa 14238
            5.6.10  The binomial coefficient operation   cbc 14267
            5.6.11  The ` # ` (set size) function   chash 14295
                  5.6.11.1  Proper unordered pairs and triples (sets of size 2 and 3)   hashprlei 14433
                  5.6.11.2  Functions with a domain containing at least two different elements   fundmge2nop0 14467
                  5.6.11.3  Finite induction on the size of the first component of a binary relation   hashdifsnp1 14471
      ​​*5.7  Words over a set
            5.7.1  Definitions and basic theorems   cword 14478
            5.7.2  Last symbol of a word   clsw 14527
            5.7.3  Concatenations of words   cconcat 14535
            5.7.4  Singleton words   cs1 14560
            5.7.5  Concatenations with singleton words   ccatws1cl 14581
            5.7.6  Subwords/substrings   csubstr 14605
            5.7.7  Prefixes of a word   cpfx 14635
            5.7.8  Subwords of subwords   swrdswrdlem 14669
            5.7.9  Subwords and concatenations   pfxcctswrd 14675
            5.7.10  Subwords of concatenations   swrdccatfn 14689
            5.7.11  Splicing words (substring replacement)   csplice 14714
            5.7.12  Reversing words   creverse 14723
            5.7.13  Repeated symbol words   creps 14733
            ​*5.7.14  Cyclical shifts of words   ccsh 14753
            5.7.15  Mapping words by a function   wrdco 14797
            5.7.16  Longer string literals   cs2 14807
      ​​*5.8  Reflexive and transitive closures of relations
            5.8.1  The reflexive and transitive properties of relations   coss12d 14938
            5.8.2  Basic properties of closures   cleq1lem 14948
            5.8.3  Definitions and basic properties of transitive closures   ctcl 14951
            5.8.4  Exponentiation of relations   crelexp 14985
            5.8.5  Reflexive-transitive closure as an indexed union   crtrcl 15021
            ​*5.8.6  Principle of transitive induction.   relexpindlem 15029
      ​5.9  Elementary real and complex functions
            5.9.1  The "shift" operation   cshi 15032
            5.9.2  Signum (sgn or sign) function   csgn 15052
            5.9.3  Real and imaginary parts; conjugate   ccj 15062
            5.9.4  Square root; absolute value   csqrt 15199
      ​5.10  Elementary limits and convergence
            5.10.1  Superior limit (lim sup)   clsp 15436
            5.10.2  Limits   cli 15450
            5.10.3  Finite and infinite sums   csu 15652
            5.10.4  The binomial theorem   binomlem 15795
            5.10.5  The inclusion/exclusion principle   incexclem 15802
            5.10.6  Infinite sums (cont.)   isumshft 15805
            5.10.7  Miscellaneous converging and diverging sequences   divrcnv 15818
            5.10.8  Arithmetic series   arisum 15826
            5.10.9  Geometric series   expcnv 15830
            5.10.10  Ratio test for infinite series convergence   cvgrat 15849
            5.10.11  Mertens' theorem   mertenslem1 15850
            5.10.12  Finite and infinite products   prodf 15853
                  5.10.12.1  Product sequences   prodf 15853
                  5.10.12.2  Non-trivial convergence   ntrivcvg 15863
                  5.10.12.3  Complex products   cprod 15869
                  5.10.12.4  Finite products   fprod 15907
                  5.10.12.5  Infinite products   iprodclim 15964
            5.10.13  Falling and Rising Factorial   cfallfac 15970
            5.10.14  Bernoulli polynomials and sums of k-th powers   cbp 16012
      ​5.11  Elementary trigonometry
            5.11.1  The exponential, sine, and cosine functions   ce 16027
                  5.11.1.1  The circle constant (tau = 2 pi)   ctau 16170
            5.11.2  _e is irrational   eirrlem 16172
      ​5.12  Cardinality of real and complex number subsets
            5.12.1  Countability of integers and rationals   xpnnen 16179
            5.12.2  The reals are uncountable   rpnnen2lem1 16182
​​*PART 6  ELEMENTARY NUMBER THEORY
      ​6.1  Elementary properties of divisibility
            6.1.1  Irrationality of square root of 2   sqrt2irrlem 16216
            6.1.2  Some Number sets are chains of proper subsets   nthruc 16220
            6.1.3  The divides relation   cdvds 16222
            ​*6.1.4  Even and odd numbers   evenelz 16306
            6.1.5  The division algorithm   divalglem0 16363
            6.1.6  Bit sequences   cbits 16389
            6.1.7  The greatest common divisor operator   cgcd 16464
            6.1.8  Bézout's identity   bezoutlem1 16509
            6.1.9  Algorithms   nn0seqcvgd 16540
            6.1.10  Euclid's Algorithm   eucalgval2 16551
            ​*6.1.11  The least common multiple   clcm 16558
            ​*6.1.12  Coprimality and Euclid's lemma   coprmgcdb 16619
            6.1.13  Cancellability of congruences   congr 16634
      ​6.2  Elementary prime number theory
            ​*6.2.1  Elementary properties   cprime 16641
            ​*6.2.2  Coprimality and Euclid's lemma (cont.)   coprm 16681
            6.2.3  Properties of the canonical representation of a rational   cnumer 16703
            6.2.4  Euler's theorem   codz 16733
            6.2.5  Arithmetic modulo a prime number   modprm1div 16768
            6.2.6  Pythagorean Triples   coprimeprodsq 16779
            6.2.7  The prime count function   cpc 16807
            6.2.8  Pocklington's theorem   prmpwdvds 16875
            6.2.9  Infinite primes theorem   unbenlem 16879
            6.2.10  Sum of prime reciprocals   prmreclem1 16887
            6.2.11  Fundamental theorem of arithmetic   1arithlem1 16894
            6.2.12  Lagrange's four-square theorem   cgz 16900
            6.2.13  Van der Waerden's theorem   cvdwa 16936
            6.2.14  Ramsey's theorem   cram 16970
            ​*6.2.15  Primorial function   cprmo 17002
            ​*6.2.16  Prime gaps   prmgaplem1 17020
            6.2.17  Decimal arithmetic (cont.)   dec2dvds 17034
            6.2.18  Cyclical shifts of words (cont.)   cshwsidrepsw 17064
            6.2.19  Specific prime numbers   prmlem0 17076
            6.2.20  Very large primes   1259lem1 17101
​PART 7  BASIC STRUCTURES
      ​7.1  Extensible structures
            ​*7.1.1  Basic definitions   cstr 17116
                  7.1.1.1  Extensible structures as structures with components   cstr 17116
                  7.1.1.2  Substitution of components   csts 17133
                  7.1.1.3  Slots   cslot 17151
                  ​*7.1.1.4  Structure component indices   cnx 17163
                  7.1.1.5  Base sets   cbs 17179
                  7.1.1.6  Base set restrictions   cress 17200
            7.1.2  Slot definitions   cplusg 17220
            7.1.3  Definition of the structure product   crest 17383
            7.1.4  Definition of the structure quotient   cordt 17462
      ​7.2  Moore spaces
            7.2.1  Moore closures   mrcflem 17567
            7.2.2  Independent sets in a Moore system   mrisval 17591
            7.2.3  Algebraic closure systems   isacs 17612
​PART 8  BASIC CATEGORY THEORY
      ​8.1  Categories
            8.1.1  Categories   ccat 17625
            8.1.2  Opposite category   coppc 17672
            8.1.3  Monomorphisms and epimorphisms   cmon 17690
            8.1.4  Sections, inverses, isomorphisms   csect 17706
            ​*8.1.5  Isomorphic objects   ccic 17757
            8.1.6  Subcategories   cssc 17769
            8.1.7  Functors   cfunc 17816
            8.1.8  Full & faithful functors   cful 17866
            8.1.9  Natural transformations and the functor category   cnat 17906
            8.1.10  Initial, terminal and zero objects of a category   cinito 17943
      ​8.2  Arrows (disjointified hom-sets)
            8.2.1  Identity and composition for arrows   cida 18015
      ​8.3  Examples of categories
            8.3.1  The category of sets   csetc 18037
            8.3.2  The category of categories   ccatc 18060
            ​*8.3.3  The category of extensible structures   fncnvimaeqv 18081
      ​8.4  Categorical constructions
            8.4.1  Product of categories   cxpc 18129
            8.4.2  Functor evaluation   cevlf 18170
            8.4.3  Hom functor   chof 18209
​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 18390
            9.5.2  Complete lattices   ccla 18457
            9.5.3  Distributive lattices   cdlat 18479
            9.5.4  Subset order structures   cipo 18486
      ​9.6  Posets, directed sets, and lattices as relations
            ​*9.6.1  Posets and lattices as relations   cps 18523
            9.6.2  Directed sets, nets   cdir 18553
​PART 10  BASIC ALGEBRAIC STRUCTURES
      ​10.1  Monoids
            ​*10.1.1  Magmas   cplusf 18564
            ​*10.1.2  Identity elements   mgmidmo 18587
            ​*10.1.3  Iterated sums in a magma   gsumvalx 18603
            10.1.4  Magma homomorphisms and submagmas   cmgmhm 18617
            ​*10.1.5  Semigroups   csgrp 18645
            ​*10.1.6  Definition and basic properties of monoids   cmnd 18661
            10.1.7  Monoid homomorphisms and submonoids   cmhm 18708
            ​*10.1.8  Iterated sums in a monoid   gsumvallem2 18761
            10.1.9  Free monoids   cfrmd 18774
                  ​*10.1.9.1  Monoid of endofunctions   cefmnd 18795
            10.1.10  Examples and counterexamples for magmas, semigroups and monoids   mgm2nsgrplem1 18845
      ​10.2  Groups
            10.2.1  Definition and basic properties   cgrp 18865
            ​*10.2.2  Group multiple operation   cmg 18999
            10.2.3  Subgroups and Quotient groups   csubg 19052
            ​*10.2.4  Cyclic monoids and groups   cycsubmel 19132
            10.2.5  Elementary theory of group homomorphisms   cghm 19144
            10.2.6  Isomorphisms of groups   cgim 19189
                  10.2.6.1  The first isomorphism theorem of groups   ghmqusnsglem1 19212
            10.2.7  Group actions   cga 19221
            10.2.8  Centralizers and centers   ccntz 19247
            10.2.9  The opposite group   coppg 19277
            10.2.10  Symmetric groups   csymg 19299
                  ​*10.2.10.1  Definition and basic properties   csymg 19299
                  10.2.10.2  Cayley's theorem   cayleylem1 19342
                  10.2.10.3  Permutations fixing one element   symgfix2 19346
                  ​*10.2.10.4  Transpositions in the symmetric group   cpmtr 19371
                  10.2.10.5  The sign of a permutation   cpsgn 19419
            10.2.11  p-Groups and Sylow groups; Sylow's theorems   cod 19454
            10.2.12  Direct products   clsm 19564
                  10.2.12.1  Direct products (extension)   smndlsmidm 19586
            10.2.13  Free groups   cefg 19636
            10.2.14  Abelian groups   ccmn 19710
                  10.2.14.1  Definition and basic properties   ccmn 19710
                  10.2.14.2  Cyclic groups   ccyg 19807
                  10.2.14.3  Group sum operation   gsumval3a 19833
                  10.2.14.4  Group sums over (ranges of) integers   fsfnn0gsumfsffz 19913
                  10.2.14.5  Internal direct products   cdprd 19925
                  10.2.14.6  The Fundamental Theorem of Abelian Groups   ablfacrplem 19997
            10.2.15  Simple groups   csimpg 20022
                  10.2.15.1  Definition and basic properties   csimpg 20022
                  10.2.15.2  Classification of abelian simple groups   ablsimpnosubgd 20036
      ​10.3  Rings
            10.3.1  Multiplicative Group   cmgp 20049
            ​*10.3.2  Non-unital rings ("rngs")   crng 20061
            ​*10.3.3  Ring unity (multiplicative identity)   cur 20090
            10.3.4  Semirings   csrg 20095
                  ​*10.3.4.1  The binomial theorem for semirings   srgbinomlem1 20135
            10.3.5  Unital rings   crg 20142
            10.3.6  Opposite ring   coppr 20245
            10.3.7  Divisibility   cdsr 20263
            10.3.8  Ring primes   crpm 20341
            10.3.9  Homomorphisms of non-unital rings   crnghm 20343
            10.3.10  Ring homomorphisms   crh 20378
            10.3.11  Nonzero rings and zero rings   cnzr 20421
            10.3.12  Local rings   clring 20447
            10.3.13  Subrings   csubrng 20454
                  10.3.13.1  Subrings of non-unital rings   csubrng 20454
                  10.3.13.2  Subrings of unital rings   csubrg 20478
                  10.3.13.3  Subrings generated by a subset   crgspn 20519
            10.3.14  Categories of rings   crngc 20525
                  ​*10.3.14.1  The category of non-unital rings   crngc 20525
                  ​*10.3.14.2  The category of (unital) rings   cringc 20554
                  10.3.14.3  Subcategories of the category of rings   srhmsubclem1 20586
            10.3.15  Left regular elements and domains   crlreg 20600
      ​10.4  Division rings and fields
            10.4.1  Definition and basic properties   cdr 20638
            10.4.2  Sub-division rings   csdrg 20695
            10.4.3  Absolute value (abstract algebra)   cabv 20717
            10.4.4  Star rings   cstf 20746
      ​10.5  Left modules
            10.5.1  Definition and basic properties   clmod 20766
            10.5.2  Subspaces and spans in a left module   clss 20837
            10.5.3  Homomorphisms and isomorphisms of left modules   clmhm 20926
            10.5.4  Subspace sum; bases for a left module   clbs 20981
      ​10.6  Vector spaces
            10.6.1  Definition and basic properties   clvec 21009
      ​10.7  Subring algebras and ideals
            10.7.1  Subring algebras   csra 21078
            ​*10.7.2  Left ideals and spans   clidl 21116
            10.7.3  Two-sided ideals and quotient rings   c2idl 21159
                  ​*10.7.3.1  Condition for a non-unital ring to be unital   rngqiprng1elbas 21196
            10.7.4  Principal ideal rings. Divisibility in the integers   clpidl 21230
            10.7.5  Principal ideal domains   cpid 21246
      ​10.8  The complex numbers as an algebraic extensible structure
            10.8.1  Definition and basic properties   cpsmet 21248
            ​*10.8.2  Ring of integers   czring 21356
                  ​*10.8.2.1  Example for a condition for a non-unital ring to be unital   pzriprnglem1 21391
            10.8.3  Algebraic constructions based on the complex numbers   czrh 21409
            10.8.4  Signs as subgroup of the complex numbers   cnmsgnsubg 21486
            10.8.5  Embedding of permutation signs into a ring   zrhpsgnmhm 21493
            10.8.6  The ordered field of real numbers   crefld 21513
      ​10.9  Generalized pre-Hilbert and Hilbert spaces
            10.9.1  Definition and basic properties   cphl 21533
            10.9.2  Orthocomplements and closed subspaces   cocv 21569
            10.9.3  Orthogonal projection and orthonormal bases   cpj 21609
​​*PART 11  BASIC LINEAR ALGEBRA
      ​11.1  Vectors and free modules
            *11.1.1  Direct sum of left modules   cdsmm 21640
            ​*11.1.2  Free modules   cfrlm 21655
            ​*11.1.3  Standard basis (unit vectors)   cuvc 21691
            ​*11.1.4  Independent sets and families   clindf 21713
            11.1.5  Characterization of free modules   lmimlbs 21745
      ​11.2  Associative algebras
            11.2.1  Definition and basic properties   casa 21759
      ​11.3  Abstract multivariate polynomials
            11.3.1  Definition and basic properties   cmps 21813
            11.3.2  Polynomial evaluation   ces 21979
            11.3.3  Additional definitions for (multivariate) polynomials   cslv 22015
            ​*11.3.4  Univariate polynomials   cps1 22059
            11.3.5  Univariate polynomial evaluation   ces1 22200
                  11.3.5.1  Specialization of polynomial evaluation as a ring homomorphism   evls1scafv 22253
      ​​*11.4  Matrices
            *11.4.1  The matrix multiplication   cmmul 22277
            ​*11.4.2  Square matrices   cmat 22294
            ​*11.4.3  The matrix algebra   matmulr 22325
            ​*11.4.4  Matrices of dimension 0 and 1   mat0dimbas0 22353
            ​*11.4.5  The subalgebras of diagonal and scalar matrices   cdmat 22375
            ​*11.4.6  Multiplication of a matrix with a "column vector"   cmvmul 22427
            11.4.7  Replacement functions for a square matrix   cmarrep 22443
            11.4.8  Submatrices   csubma 22463
      ​11.5  The determinant
            11.5.1  Definition and basic properties   cmdat 22471
            11.5.2  Determinants of 2 x 2 -matrices   m2detleiblem1 22511
            11.5.3  The matrix adjugate/adjunct   cmadu 22519
            ​*11.5.4  Laplace expansion of determinants (special case)   symgmatr01lem 22540
            11.5.5  Inverse matrix   invrvald 22563
            ​*11.5.6  Cramer's rule   slesolvec 22566
      ​​*11.6  Polynomial matrices
            11.6.1  Basic properties   pmatring 22579
            ​*11.6.2  Constant polynomial matrices   ccpmat 22590
            ​*11.6.3  Collecting coefficients of polynomial matrices   cdecpmat 22649
            ​*11.6.4  Ring isomorphism between polynomial matrices and polynomials over matrices   cpm2mp 22679
      ​​*11.7  The characteristic polynomial
            *11.7.1  Definition and basic properties   cchpmat 22713
            ​*11.7.2  The characteristic factor function G   fvmptnn04if 22736
            ​*11.7.3  The Cayley-Hamilton theorem   cpmadurid 22754
​PART 12  BASIC TOPOLOGY
      ​12.1  Topology
            ​*12.1.1  Topological spaces   ctop 22780
                  12.1.1.1  Topologies   ctop 22780
                  12.1.1.2  Topologies on sets   ctopon 22797
                  12.1.1.3  Topological spaces   ctps 22819
            12.1.2  Topological bases   ctb 22832
            12.1.3  Examples of topologies   distop 22882
            12.1.4  Closure and interior   ccld 22903
            12.1.5  Neighborhoods   cnei 22984
            12.1.6  Limit points and perfect sets   clp 23021
            12.1.7  Subspace topologies   restrcl 23044
            12.1.8  Order topology   ordtbaslem 23075
            12.1.9  Limits and continuity in topological spaces   ccn 23111
            12.1.10  Separated spaces: T0, T1, T2 (Hausdorff) ...   ct0 23193
            12.1.11  Compactness   ccmp 23273
            12.1.12  Bolzano-Weierstrass theorem   bwth 23297
            12.1.13  Connectedness   cconn 23298
            12.1.14  First- and second-countability   c1stc 23324
            12.1.15  Local topological properties   clly 23351
            12.1.16  Refinements   cref 23389
            12.1.17  Compactly generated spaces   ckgen 23420
            12.1.18  Product topologies   ctx 23447
            12.1.19  Continuous function-builders   cnmptid 23548
            12.1.20  Quotient maps and quotient topology   ckq 23580
            12.1.21  Homeomorphisms   chmeo 23640
      ​12.2  Filters and filter bases
            12.2.1  Filter bases   elmptrab 23714
            12.2.2  Filters   cfil 23732
            12.2.3  Ultrafilters   cufil 23786
            12.2.4  Filter limits   cfm 23820
            12.2.5  Extension by continuity   ccnext 23946
            12.2.6  Topological groups   ctmd 23957
            12.2.7  Infinite group sum on topological groups   ctsu 24013
            12.2.8  Topological rings, fields, vector spaces   ctrg 24043
      ​12.3  Uniform Structures and Spaces
            12.3.1  Uniform structures   cust 24087
            12.3.2  The topology induced by an uniform structure   cutop 24118
            12.3.3  Uniform Spaces   cuss 24141
            12.3.4  Uniform continuity   cucn 24162
            12.3.5  Cauchy filters in uniform spaces   ccfilu 24173
            12.3.6  Complete uniform spaces   ccusp 24184
      ​12.4  Metric spaces
            12.4.1  Pseudometric spaces   ispsmet 24192
            12.4.2  Basic metric space properties   cxms 24205
            12.4.3  Metric space balls   blfvalps 24271
            12.4.4  Open sets of a metric space   mopnval 24326
            12.4.5  Continuity in metric spaces   metcnp3 24428
            12.4.6  The uniform structure generated by a metric   metuval 24437
            12.4.7  Examples of metric spaces   dscmet 24460
            ​*12.4.8  Normed algebraic structures   cnm 24464
            12.4.9  Normed space homomorphisms (bounded linear operators)   cnmo 24593
            12.4.10  Topology on the reals   qtopbaslem 24646
            12.4.11  Topological definitions using the reals   cii 24768
            12.4.12  Path homotopy   chtpy 24866
            12.4.13  The fundamental group   cpco 24900
      ​12.5  Metric subcomplex vector spaces
            12.5.1  Subcomplex modules   cclm 24962
            ​*12.5.2  Subcomplex vector spaces   ccvs 25023
            ​*12.5.3  Normed subcomplex vector spaces   isncvsngp 25049
            12.5.4  Subcomplex pre-Hilbert spaces   ccph 25066
            12.5.5  Convergence and completeness   ccfil 25152
            12.5.6  Baire's Category Theorem   bcthlem1 25224
            12.5.7  Banach spaces and subcomplex Hilbert spaces   ccms 25232
                  12.5.7.1  The complete ordered field of the real numbers   retopn 25279
            12.5.8  Euclidean spaces   crrx 25283
            12.5.9  Minimizing Vector Theorem   minveclem1 25324
            12.5.10  Projection Theorem   pjthlem1 25337
​PART 13  BASIC REAL AND COMPLEX ANALYSIS
      ​13.1  Continuity
            13.1.1  Intermediate value theorem   pmltpclem1 25349
      ​13.2  Integrals
            13.2.1  Lebesgue measure   covol 25363
            13.2.2  Lebesgue integration   cmbf 25515
                  13.2.2.1  Lesbesgue integral   cmbf 25515
                  13.2.2.2  Lesbesgue directed integral   cdit 25747
      ​13.3  Derivatives
            13.3.1  Real and complex differentiation   climc 25763
                  13.3.1.1  Derivatives of functions of one complex or real variable   climc 25763
                  13.3.1.2  Results on real differentiation   dvferm1lem 25888
​PART 14  BASIC REAL AND COMPLEX FUNCTIONS
      ​14.1  Polynomials
            14.1.1  Polynomial degrees   cmdg 25958
            14.1.2  The division algorithm for univariate polynomials   cmn1 26031
            14.1.3  Elementary properties of complex polynomials   cply 26089
            14.1.4  The division algorithm for polynomials   cquot 26198
            14.1.5  Algebraic numbers   caa 26222
            14.1.6  Liouville's approximation theorem   aalioulem1 26240
      ​14.2  Sequences and series
            14.2.1  Taylor polynomials and Taylor's theorem   ctayl 26260
            14.2.2  Uniform convergence   culm 26285
            14.2.3  Power series   pserval 26319
      ​14.3  Basic trigonometry
            14.3.1  The exponential, sine, and cosine functions (cont.)   efcn 26353
            14.3.2  Properties of pi = 3.14159...   pilem1 26361
            14.3.3  Mapping of the exponential function   efgh 26450
            14.3.4  The natural logarithm on complex numbers   clog 26463
            ​*14.3.5  Logarithms to an arbitrary base   clogb 26674
            14.3.6  Theorems of Pythagoras, isosceles triangles, and intersecting chords   angval 26711
            14.3.7  Solutions of quadratic, cubic, and quartic equations   quad2 26749
            14.3.8  Inverse trigonometric functions   casin 26772
            14.3.9  The Birthday Problem   log2ublem1 26856
            14.3.10  Areas in R^2   carea 26865
            14.3.11  More miscellaneous converging sequences   rlimcnp 26875
            14.3.12  Inequality of arithmetic and geometric means   cvxcl 26895
            14.3.13  Euler-Mascheroni constant   cem 26902
            14.3.14  Zeta function   czeta 26923
            14.3.15  Gamma function   clgam 26926
      ​14.4  Basic number theory
            14.4.1  Wilson's theorem   wilthlem1 26978
            14.4.2  The Fundamental Theorem of Algebra   ftalem1 26983
            14.4.3  The Basel problem (ζ(2) = π²/6)   basellem1 26991
            14.4.4  Number-theoretical functions   ccht 27001
            14.4.5  Perfect Number Theorem   mersenne 27138
            14.4.6  Characters of Z/nZ   cdchr 27143
            14.4.7  Bertrand's postulate   bcctr 27186
            ​*14.4.8  Quadratic residues and the Legendre symbol   clgs 27205
            ​*14.4.9  Gauss' Lemma   gausslemma2dlem0a 27267
            14.4.10  Quadratic reciprocity   lgseisenlem1 27286
            14.4.11  All primes 4n+1 are the sum of two squares   2sqlem1 27328
            14.4.12  Chebyshev's Weak Prime Number Theorem, Dirichlet's Theorem   chebbnd1lem1 27380
            14.4.13  The Prime Number Theorem   mudivsum 27441
            14.4.14  Ostrowski's theorem   abvcxp 27526
​PART 15  SURREAL NUMBERS
      ​​*15.1  Sign sequence representation and Alling's axioms
            15.1.1  Definitions and initial properties   csur 27551
            15.1.2  Ordering   sltsolem1 27587
            15.1.3  Birthday Function   bdayfo 27589
            15.1.4  Density   fvnobday 27590
            ​*15.1.5  Full-Eta Property   bdayimaon 27605
      ​15.2  Initial consequences of Alling's axioms
            15.2.1  Ordering Theorems   csle 27656
            15.2.2  Birthday Theorems   bdayfun 27684
      ​​*15.3  Conway cut representation
            15.3.1  Conway cuts   csslt 27692
            15.3.2  Zero and One   c0s 27734
            15.3.3  Cuts and Options   cmade 27750
            15.3.4  Cofinality and coinitiality   cofsslt 27826
      ​15.4  Induction and recursion
            15.4.1  Induction and recursion on one variable   cnorec 27844
            15.4.2  Induction and recursion on two variables   cnorec2 27855
      ​15.5  Surreal arithmetic
            15.5.1  Addition   cadds 27866
            15.5.2  Negation and Subtraction   cnegs 27925
            15.5.3  Multiplication   cmuls 28009
            15.5.4  Division   cdivs 28090
            15.5.5  Absolute value   cabss 28139
      ​15.6  Subsystems of surreals
            15.6.1  Ordinal numbers   cons 28152
            15.6.2  Surreal recursive sequences   cseqs 28177
            15.6.3  Natural numbers   cnn0s 28206
            15.6.4  Integers   czs 28266
            15.6.5  Dyadic fractions   c2s 28296
            15.6.6  Real numbers   creno 28344
​​*PART 16  ELEMENTARY GEOMETRY
      ​16.1  Definition and Tarski's Axioms of Geometry
            16.1.1  Justification for the congruence notation   tgjustf 28400
      ​16.2  Tarskian Geometry
            16.2.1  Congruence   tgcgrcomimp 28404
            16.2.2  Betweenness   tgbtwntriv2 28414
            16.2.3  Dimension   tglowdim1 28427
            16.2.4  Betweenness and Congruence   tgifscgr 28435
            16.2.5  Congruence of a series of points   ccgrg 28437
            16.2.6  Motions   cismt 28459
            16.2.7  Colinearity   tglng 28473
            16.2.8  Connectivity of betweenness   tgbtwnconn1lem1 28499
            16.2.9  Less-than relation in geometric congruences   cleg 28509
            16.2.10  Rays   chlg 28527
            16.2.11  Lines   btwnlng1 28546
            16.2.12  Point inversions   cmir 28579
            16.2.13  Right angles   crag 28620
            16.2.14  Half-planes   islnopp 28666
            16.2.15  Midpoints and Line Mirroring   cmid 28699
            16.2.16  Congruence of angles   ccgra 28734
            16.2.17  Angle Comparisons   cinag 28762
            16.2.18  Congruence Theorems   tgsas1 28781
            16.2.19  Equilateral triangles   ceqlg 28792
      ​16.3  Properties of geometries
            16.3.1  Isomorphisms between geometries   f1otrgds 28796
      ​16.4  Geometry in Hilbert spaces
            16.4.1  Geometry in the complex plane   cchhllem 28814
            16.4.2  Geometry in Euclidean spaces   cee 28815
                  16.4.2.1  Definition of the Euclidean space   cee 28815
                  16.4.2.2  Tarski's axioms for geometry for the Euclidean space   axdimuniq 28840
                  16.4.2.3  EE^n fulfills Tarski's Axioms   ceeng 28904
​​*PART 17  GRAPH THEORY
      ​*17.1  Vertices and edges
            17.1.1  The edge function extractor for extensible structures   cedgf 28915
            ​*17.1.2  Vertices and indexed edges   cvtx 28923
                  17.1.2.1  Definitions and basic properties   cvtx 28923
                  17.1.2.2  The vertices and edges of a graph represented as ordered pair   opvtxval 28930
                  17.1.2.3  The vertices and edges of a graph represented as extensible structure   funvtxdmge2val 28938
                  17.1.2.4  Representations of graphs without edges   snstrvtxval 28964
                  17.1.2.5  Degenerated cases of representations of graphs   vtxval0 28966
            17.1.3  Edges as range of the edge function   cedg 28974
      ​​*17.2  Undirected graphs
            17.2.1  Undirected hypergraphs   cuhgr 28983
            17.2.2  Undirected pseudographs and multigraphs   cupgr 29007
            ​*17.2.3  Loop-free graphs   umgrislfupgrlem 29049
            17.2.4  Edges as subsets of vertices of graphs   uhgredgiedgb 29053
            ​*17.2.5  Undirected simple graphs   cuspgr 29075
            17.2.6  Examples for graphs   usgr0e 29163
            17.2.7  Subgraphs   csubgr 29194
            17.2.8  Finite undirected simple graphs   cfusgr 29243
            17.2.9  Neighbors, complete graphs and universal vertices   cnbgr 29259
                  17.2.9.1  Neighbors   cnbgr 29259
                  17.2.9.2  Universal vertices   cuvtx 29312
                  17.2.9.3  Complete graphs   ccplgr 29336
            17.2.10  Vertex degree   cvtxdg 29393
            ​*17.2.11  Regular graphs   crgr 29483
      ​​*17.3  Walks, paths and cycles
            *17.3.1  Walks   cewlks 29523
            17.3.2  Walks for loop-free graphs   lfgrwlkprop 29615
            17.3.3  Trails   ctrls 29618
            17.3.4  Paths and simple paths   cpths 29640
            17.3.5  Closed walks   cclwlks 29700
            17.3.6  Circuits and cycles   ccrcts 29714
            ​*17.3.7  Walks as words   cwwlks 29755
            17.3.8  Walks/paths of length 2 (as length 3 strings)   2wlkdlem1 29855
            17.3.9  Walks in regular graphs   rusgrnumwwlkl1 29898
            ​*17.3.10  Closed walks as words   cclwwlk 29910
                  17.3.10.1  Closed walks as words   cclwwlk 29910
                  17.3.10.2  Closed walks of a fixed length as words   cclwwlkn 29953
                  17.3.10.3  Closed walks on a vertex of a fixed length as words   cclwwlknon 30016
            17.3.11  Examples for walks, trails and paths   0ewlk 30043
            17.3.12  Connected graphs   cconngr 30115
      ​17.4  Eulerian paths and the Konigsberg Bridge problem
            ​*17.4.1  Eulerian paths   ceupth 30126
            ​*17.4.2  The Königsberg Bridge problem   konigsbergvtx 30175
      ​17.5  The Friendship Theorem
            17.5.1  Friendship graphs - basics   cfrgr 30187
            17.5.2  The friendship theorem for small graphs   frgr1v 30200
            17.5.3  Theorems according to Mertzios and Unger   2pthfrgrrn 30211
            ​*17.5.4  Huneke's Proof of the Friendship Theorem   frgrncvvdeqlem1 30228
​PART 18  GUIDES AND MISCELLANEA
      ​18.1  Guides (conventions, explanations, and examples)
            ​*18.1.1  Conventions   conventions 30329
            18.1.2  Natural deduction   natded 30332
            ​*18.1.3  Natural deduction examples   ex-natded5.2 30333
            18.1.4  Definitional examples   ex-or 30350
            18.1.5  Other examples   aevdemo 30389
      ​18.2  Humor
            18.2.1  April Fool's theorem   avril1 30392
      ​18.3  (Future - to be reviewed and classified)
            18.3.1  Planar incidence geometry   cplig 30403
            ​*18.3.2  Aliases kept to prevent broken links   dummylink 30416
​​*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 30418
            19.1.2  Abelian groups   cablo 30473
      ​19.2  Complex vector spaces
            19.2.1  Definition and basic properties   cvc 30487
            19.2.2  Examples of complex vector spaces   cnaddabloOLD 30510
      ​19.3  Normed complex vector spaces
            19.3.1  Definition and basic properties   cnv 30513
            19.3.2  Examples of normed complex vector spaces   cnnv 30606
            19.3.3  Induced metric of a normed complex vector space   imsval 30614
            19.3.4  Inner product   cdip 30629
            19.3.5  Subspaces   css 30650
      ​19.4  Operators on complex vector spaces
            19.4.1  Definitions and basic properties   clno 30669
      ​19.5  Inner product (pre-Hilbert) spaces
            19.5.1  Definition and basic properties   ccphlo 30741
            19.5.2  Examples of pre-Hilbert spaces   cncph 30748
            19.5.3  Properties of pre-Hilbert spaces   isph 30751
      ​19.6  Complex Banach spaces
            19.6.1  Definition and basic properties   ccbn 30791
            19.6.2  Examples of complex Banach spaces   cnbn 30798
            19.6.3  Uniform Boundedness Theorem   ubthlem1 30799
            19.6.4  Minimizing Vector Theorem   minvecolem1 30803
      ​19.7  Complex Hilbert spaces
            19.7.1  Definition and basic properties   chlo 30814
            19.7.2  Standard axioms for a complex Hilbert space   hlex 30827
            19.7.3  Examples of complex Hilbert spaces   cnchl 30845
            19.7.4  Hellinger-Toeplitz Theorem   htthlem 30846
​​*PART 20  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      ​20.1  Axiomatization of complex pre-Hilbert spaces
            20.1.1  Basic Hilbert space definitions   chba 30848
            20.1.2  Preliminary ZFC lemmas   df-hnorm 30897
            ​*20.1.3  Derive the Hilbert space axioms from ZFC set theory   axhilex-zf 30910
            ​*20.1.4  Introduce the vector space axioms for a Hilbert space   ax-hilex 30928
            20.1.5  Vector operations   hvmulex 30940
            20.1.6  Inner product postulates for a Hilbert space   ax-hfi 31008
      ​20.2  Inner product and norms
            20.2.1  Inner product   his5 31015
            20.2.2  Norms   dfhnorm2 31051
            20.2.3  Relate Hilbert space to normed complex vector spaces   hilablo 31089
            20.2.4  Bunjakovaskij-Cauchy-Schwarz inequality   bcsiALT 31108
      ​20.3  Cauchy sequences and completeness axiom
            20.3.1  Cauchy sequences and limits   hcau 31113
            20.3.2  Derivation of the completeness axiom from ZF set theory   hilmet 31123
            20.3.3  Completeness postulate for a Hilbert space   ax-hcompl 31131
            20.3.4  Relate Hilbert space to ZFC pre-Hilbert and Hilbert spaces   hhcms 31132
      ​20.4  Subspaces and projections
            20.4.1  Subspaces   df-sh 31136
            20.4.2  Closed subspaces   df-ch 31150
            20.4.3  Orthocomplements   df-oc 31181
            20.4.4  Subspace sum, span, lattice join, lattice supremum   df-shs 31237
            20.4.5  Projection theorem   pjhthlem1 31320
            20.4.6  Projectors   df-pjh 31324
      ​20.5  Properties of Hilbert subspaces
            20.5.1  Orthomodular law   omlsilem 31331
            20.5.2  Projectors (cont.)   pjhtheu2 31345
            20.5.3  Hilbert lattice operations   sh0le 31369
            20.5.4  Span (cont.) and one-dimensional subspaces   spansn0 31470
            20.5.5  Commutes relation for Hilbert lattice elements   df-cm 31512
            20.5.6  Foulis-Holland theorem   fh1 31547
            20.5.7  Quantum Logic Explorer axioms   qlax1i 31556
            20.5.8  Orthogonal subspaces   chscllem1 31566
            20.5.9  Orthoarguesian laws 5OA and 3OA   5oalem1 31583
            20.5.10  Projectors (cont.)   pjorthi 31598
            20.5.11  Mayet's equation E_3   mayete3i 31657
      ​20.6  Operators on Hilbert spaces
            ​*20.6.1  Operator sum, difference, and scalar multiplication   df-hosum 31659
            20.6.2  Zero and identity operators   df-h0op 31677
            20.6.3  Operations on Hilbert space operators   hoaddcl 31687
            20.6.4  Linear, continuous, bounded, Hermitian, unitary operators and norms   df-nmop 31768
            20.6.5  Linear and continuous functionals and norms   df-nmfn 31774
            20.6.6  Adjoint   df-adjh 31778
            20.6.7  Dirac bra-ket notation   df-bra 31779
            20.6.8  Positive operators   df-leop 31781
            20.6.9  Eigenvectors, eigenvalues, spectrum   df-eigvec 31782
            20.6.10  Theorems about operators and functionals   nmopval 31785
            20.6.11  Riesz lemma   riesz3i 31991
            20.6.12  Adjoints (cont.)   cnlnadjlem1 31996
            20.6.13  Quantum computation error bound theorem   unierri 32033
            20.6.14  Dirac bra-ket notation (cont.)   branmfn 32034
            20.6.15  Positive operators (cont.)   leopg 32051
            20.6.16  Projectors as operators   pjhmopi 32075
      ​20.7  States on a Hilbert lattice and Godowski's equation
            20.7.1  States on a Hilbert lattice   df-st 32140
            20.7.2  Godowski's equation   golem1 32200
      ​20.8  Cover relation, atoms, exchange axiom, and modular symmetry
            20.8.1  Covers relation; modular pairs   df-cv 32208
            20.8.2  Atoms   df-at 32267
            20.8.3  Superposition principle   superpos 32283
            20.8.4  Atoms, exchange and covering properties, atomicity   chcv1 32284
            20.8.5  Irreducibility   chirredlem1 32319
            20.8.6  Atoms (cont.)   atcvat3i 32325
            20.8.7  Modular symmetry   mdsymlem1 32332
​PART 21  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      ​21.1  Mathboxes for user contributions
            21.1.1  Mathbox guidelines   mathbox 32371
      ​21.2  Mathbox for Stefan Allan
      ​21.3  Mathbox for Thierry Arnoux
            21.3.1  Propositional Calculus - misc additions   ad11antr 32376
            21.3.2  Predicate Calculus   sbc2iedf 32394
                  21.3.2.1  Predicate Calculus - misc additions   sbc2iedf 32394
                  21.3.2.2  Restricted quantification - misc additions   ralcom4f 32396
                  21.3.2.3  Equality   eqtrb 32403
                  21.3.2.4  Double restricted existential uniqueness quantification   opsbc2ie 32405
                  21.3.2.5  Double restricted existential uniqueness quantification syntax   w2reu 32407
                  21.3.2.6  Substitution (without distinct variables) - misc additions   sbceqbidf 32416
                  21.3.2.7  Existential "at most one" - misc additions   mo5f 32418
                  21.3.2.8  Existential uniqueness - misc additions   reuxfrdf 32420
                  21.3.2.9  Restricted "at most one" - misc additions   rmoxfrd 32422
                  21.3.2.10  Restricted iota (description binder)   riotaeqbidva 32425
            21.3.3  General Set Theory   dmrab 32426
                  21.3.3.1  Class abstractions (a.k.a. class builders)   dmrab 32426
                  21.3.3.2  Image Sets   abrexdomjm 32436
                  21.3.3.3  Set relations and operations - misc additions   nelun 32442
                  21.3.3.4  Unordered pairs   elpreq 32457
                  21.3.3.5  Unordered triples   tpssg 32466
                  21.3.3.6  Conditional operator - misc additions   ifeqeqx 32471
                  21.3.3.7  Set union   uniinn0 32479
                  21.3.3.8  Indexed union - misc additions   cbviunf 32484
                  21.3.3.9  Indexed intersection - misc additions   iinabrex 32498
                  21.3.3.10  Disjointness - misc additions   disjnf 32499
            21.3.4  Relations and Functions   xpdisjres 32527
                  21.3.4.1  Relations - misc additions   xpdisjres 32527
                  21.3.4.2  Functions - misc additions   ac6sf2 32548
                  21.3.4.3  Operations - misc additions   mpomptxf 32601
                  21.3.4.4  The mapping operation   elmaprd 32603
                  21.3.4.5  Support of a function   suppovss 32604
                  21.3.4.6  Explicit Functions with one or two points as a domain   cosnopne 32617
                  21.3.4.7  Isomorphisms - misc. additions   gtiso 32624
                  21.3.4.8  Disjointness (additional proof requiring functions)   disjdsct 32626
                  21.3.4.9  First and second members of an ordered pair - misc additions   df1stres 32627
                  21.3.4.10  Finite Sets   imafi2 32635
                  21.3.4.11  Countable Sets   snct 32637
            21.3.5  Real and Complex Numbers   sgnval2 32658
                  21.3.5.1  Complex operations - misc. additions   creq0 32659
                  21.3.5.2  Ordering on reals - misc additions   lt2addrd 32674
                  21.3.5.3  Extended reals - misc additions   xrlelttric 32675
                  21.3.5.4  Extended nonnegative integers - misc additions   xnn0gt0 32692
                  21.3.5.5  Real number intervals - misc additions   joiniooico 32697
                  21.3.5.6  Finite intervals of integers - misc additions   uzssico 32707
                  21.3.5.7  Half-open integer ranges - misc additions   iundisjfi 32719
                  21.3.5.8  The ` # ` (set size) function - misc additions   hashunif 32731
                  21.3.5.9  The greatest common divisor operator - misc. additions   elq2 32736
                  21.3.5.10  Integers   nn0split01 32742
                  21.3.5.11  Decimal numbers   dfdec100 32755
            21.3.6  Real and complex functions   sgncl 32756
                  21.3.6.1  Signum (sgn or sign) function - misc. additions   sgncl 32756
                  21.3.6.2  Integer powers - misc. additions   nexple 32769
                  21.3.6.3  Indicator Functions   cind 32773
            ​*21.3.7  Decimal expansion   cdp2 32791
                  ​*21.3.7.1  Decimal point   cdp 32808
                  21.3.7.2  Division in the extended real number system   cxdiv 32837
            21.3.8  Words over a set - misc additions   wrdres 32856
                  21.3.8.1  Splicing words (substring replacement)   splfv3 32880
                  21.3.8.2  Cyclic shift of words   1cshid 32881
            21.3.9  Extensible Structures   ressplusf 32885
                  21.3.9.1  Structure restriction operator   ressplusf 32885
                  21.3.9.2  The opposite group   oppgle 32888
                  21.3.9.3  Posets   ressprs 32890
                  21.3.9.4  Complete lattices   clatp0cl 32902
                  21.3.9.5  Order Theory   cmnt 32904
                  21.3.9.6  Chains   cchn 32930
                  21.3.9.7  Extended reals Structure - misc additions   ax-xrssca 32942
                  21.3.9.8  The extended nonnegative real numbers commutative monoid   xrge0base 32952
            21.3.10  Algebra   mndcld 32963
                  21.3.10.1  Monoids   mndcld 32963
                  21.3.10.2  Monoids Homomorphisms   abliso 32977
                  21.3.10.3  Groups - misc additions   grpsubcld 32981
                  21.3.10.4  Finitely supported group sums - misc additions   gsumsubg 32986
                  21.3.10.5  Group or monoid sums over words   gsumwun 33005
                  21.3.10.6  Centralizers and centers - misc additions   cntzun 33008
                  21.3.10.7  Totally ordered monoids and groups   comnd 33011
                  21.3.10.8  The symmetric group   symgfcoeu 33039
                  21.3.10.9  Transpositions   pmtridf1o 33051
                  21.3.10.10  Permutation Signs   psgnid 33054
                  21.3.10.11  Permutation cycles   ctocyc 33063
                  21.3.10.12  The Alternating Group   evpmval 33102
                  21.3.10.13  Signum in an ordered monoid   csgns 33115
                  21.3.10.14  Fixed points   cfxp 33120
                  21.3.10.15  The Archimedean property for generic ordered algebraic structures   cinftm 33130
                  21.3.10.16  Semiring left modules   cslmd 33153
                  21.3.10.17  Simple groups   prmsimpcyc 33181
                  21.3.10.18  Rings - misc additions   ringdi22 33182
                  21.3.10.19  Subrings generated by a set   elrgspnlem1 33193
                  21.3.10.20  The zero ring   irrednzr 33201
                  21.3.10.21  Localization of rings   cerl 33204
                  21.3.10.22  Integral Domains   domnmuln0rd 33225
                  21.3.10.23  Euclidean Domains   ceuf 33238
                  21.3.10.24  Division Rings   ringinveu 33244
                  21.3.10.25  The field of rational numbers   qfld 33247
                  21.3.10.26  Subfields   subsdrg 33248
                  21.3.10.27  Field of fractions   cfrac 33252
                  21.3.10.28  Field extensions generated by a set   cfldgen 33260
                  21.3.10.29  Totally ordered rings and fields   corng 33273
                  21.3.10.30  Ring homomorphisms - misc additions   rhmdvd 33296
                  21.3.10.31  Scalar restriction operation   cresv 33298
                  21.3.10.32  The commutative ring of gaussian integers   gzcrng 33313
                  21.3.10.33  The archimedean ordered field of real numbers   cnfldfld 33314
                  21.3.10.34  The quotient map and quotient modules   qusker 33320
                  21.3.10.35  The ring of integers modulo ` N `   znfermltl 33337
                  21.3.10.36  Independent sets and families   islinds5 33338
                  21.3.10.37  Ring associates, ring units   dvdsruassoi 33355
                  ​*21.3.10.38  Subgroup sum / Sumset / Minkowski sum   elgrplsmsn 33361
                  21.3.10.39  The quotient map   quslsm 33376
                  21.3.10.40  Ideals   lidlmcld 33390
                  21.3.10.41  Prime Ideals   cprmidl 33406
                  21.3.10.42  Maximal Ideals   cmxidl 33430
                  21.3.10.43  The semiring of ideals of a ring   cidlsrg 33471
                  21.3.10.44  Prime Elements   rprmval 33487
                  21.3.10.45  Unique factorization domains   cufd 33509
                  21.3.10.46  The ring of integers   zringidom 33522
                  21.3.10.47  Univariate Polynomials   0ringmon1p 33526
                  21.3.10.48  Polynomial quotient and polynomial remainder   q1pdir 33568
                  21.3.10.49  The subring algebra   sra1r 33577
                  21.3.10.50  Division Ring Extensions   drgext0g 33585
                  21.3.10.51  Vector Spaces   lvecdimfi 33591
                  21.3.10.52  Vector Space Dimension   cldim 33594
            21.3.11  Field Extensions   cfldext 33634
                  21.3.11.1  Algebraic numbers   cirng 33678
                  21.3.11.2  Algebraic extensions   calgext 33687
                  21.3.11.3  Minimal polynomials   cminply 33689
                  21.3.11.4  Quadratic Field Extensions   rtelextdg2lem 33716
                  21.3.11.5  Towers of quadratic extentions   fldext2chn 33718
            ​*21.3.12  Constructible Numbers   cconstr 33719
                  21.3.12.1  Impossible constructions   2sqr3minply 33770
            21.3.13  Matrices   csmat 33783
                  21.3.13.1  Submatrices   csmat 33783
                  21.3.13.2  Matrix literals   clmat 33801
                  21.3.13.3  Laplace expansion of determinants   mdetpmtr1 33813
            21.3.14  Topology   ist0cld 33823
                  21.3.14.1  Open maps   txomap 33824
                  21.3.14.2  Topology of the unit circle   qtopt1 33825
                  21.3.14.3  Refinements   reff 33829
                  21.3.14.4  Open cover refinement property   ccref 33832
                  21.3.14.5  Lindelöf spaces   cldlf 33842
                  21.3.14.6  Paracompact spaces   cpcmp 33845
                  ​*21.3.14.7  Spectrum of a ring   crspec 33852
                  21.3.14.8  Pseudometrics   cmetid 33876
                  21.3.14.9  Continuity - misc additions   hauseqcn 33888
                  21.3.14.10  Topology of the closed unit interval   elunitge0 33889
                  21.3.14.11  Topology of ` ( RR X. RR ) `   unicls 33893
                  21.3.14.12  Order topology - misc. additions   cnvordtrestixx 33903
                  21.3.14.13  Continuity in topological spaces - misc. additions   mndpluscn 33916
                  21.3.14.14  Topology of the extended nonnegative real numbers ordered monoid   xrge0hmph 33922
                  21.3.14.15  Limits - misc additions   lmlim 33937
                  21.3.14.16  Univariate polynomials   pl1cn 33945
            21.3.15  Uniform Stuctures and Spaces   chcmp 33946
                  21.3.15.1  Hausdorff uniform completion   chcmp 33946
            21.3.16  Topology and algebraic structures   zringnm 33948
                  21.3.16.1  The norm on the ring of the integer numbers   zringnm 33948
                  21.3.16.2  Topological ` ZZ ` -modules   zlm0 33950
                  21.3.16.3  Canonical embedding of the field of the rational numbers into a division ring   cqqh 33960
                  21.3.16.4  Canonical embedding of the real numbers into a complete ordered field   crrh 33983
                  21.3.16.5  Embedding from the extended real numbers into a complete lattice   cxrh 34006
                  21.3.16.6  Canonical embeddings into the ordered field of the real numbers   zrhre 34009
                  ​*21.3.16.7  Topological Manifolds   cmntop 34012
                  21.3.16.8  Extended sum   cesum 34017
            21.3.17  Mixed Function/Constant operation   cofc 34085
            21.3.18  Abstract measure   csiga 34098
                  21.3.18.1  Sigma-Algebra   csiga 34098
                  21.3.18.2  Generated sigma-Algebra   csigagen 34128
                  ​*21.3.18.3  lambda and pi-Systems, Rings of Sets   ispisys 34142
                  21.3.18.4  The Borel algebra on the real numbers   cbrsiga 34171
                  21.3.18.5  Product Sigma-Algebra   csx 34178
                  21.3.18.6  Measures   cmeas 34185
                  21.3.18.7  The counting measure   cntmeas 34216
                  21.3.18.8  The Lebesgue measure - misc additions   voliune 34219
                  21.3.18.9  The Dirac delta measure   cdde 34222
                  21.3.18.10  The 'almost everywhere' relation   cae 34227
                  21.3.18.11  Measurable functions   cmbfm 34239
                  21.3.18.12  Borel Algebra on ` ( RR X. RR ) `   br2base 34260
                  ​*21.3.18.13  Caratheodory's extension theorem   coms 34282
            21.3.19  Integration   itgeq12dv 34317
                  21.3.19.1  Lebesgue integral - misc additions   itgeq12dv 34317
                  21.3.19.2  Bochner integral   citgm 34318
            21.3.20  Euler's partition theorem   oddpwdc 34345
            21.3.21  Sequences defined by strong recursion   csseq 34374
            21.3.22  Fibonacci Numbers   cfib 34387
            21.3.23  Probability   cprb 34398
                  21.3.23.1  Probability Theory   cprb 34398
                  21.3.23.2  Conditional Probabilities   ccprob 34422
                  21.3.23.3  Real-valued Random Variables   crrv 34431
                  21.3.23.4  Preimage set mapping operator   corvc 34447
                  21.3.23.5  Distribution Functions   orvcelval 34460
                  21.3.23.6  Cumulative Distribution Functions   orvclteel 34464
                  21.3.23.7  Probabilities - example   coinfliplem 34470
                  21.3.23.8  Bertrand's Ballot Problem   ballotlemoex 34477
            21.3.24  Signum (sgn or sign) function - misc. additions   fzssfzo 34530
                  21.3.24.1  Operations on words   ccatmulgnn0dir 34533
            21.3.25  Polynomials with real coefficients - misc additions   plymul02 34537
            21.3.26  Descartes's rule of signs   signspval 34543
                  21.3.26.1  Sign changes in a word over real numbers   signspval 34543
                  21.3.26.2  Counting sign changes in a word over real numbers   signslema 34553
            21.3.27  Number Theory   iblidicc 34583
                  21.3.27.1  Representations of a number as sums of integers   crepr 34599
                  21.3.27.2  Vinogradov Trigonometric Sums and the Circle Method   cvts 34626
                  21.3.27.3  The Ternary Goldbach Conjecture: Final Statement   ax-hgt749 34635
            21.3.28  Elementary Geometry   cstrkg2d 34655
                  ​*21.3.28.1  Two-dimensional geometry   cstrkg2d 34655
                  21.3.28.2  Outer Five Segment (not used, no need to move to main)   cafs 34660
            ​*21.3.29  LeftPad Project   clpad 34665
      ​​*21.4  Mathbox for Jonathan Ben-Naim
            21.4.1  First-order logic and set theory   bnj170 34688
            21.4.2  Well founded induction and recursion   bnj110 34848
            21.4.3  The existence of a minimal element in certain classes   bnj69 35000
            21.4.4  Well-founded induction   bnj1204 35002
            21.4.5  Well-founded recursion, part 1 of 3   bnj60 35052
            21.4.6  Well-founded recursion, part 2 of 3   bnj1500 35058
            21.4.7  Well-founded recursion, part 3 of 3   bnj1522 35062
      ​21.5  Mathbox for BTernaryTau
            21.5.1  First-order logic   nfan1c 35063
                  21.5.1.1  Auxiliary axiom schemes   nfan1c 35063
            21.5.2  ZF set theory   exdifsn 35069
                  21.5.2.1  Finitism   prcinf 35084
                  21.5.2.2  Global choice   gblacfnacd 35089
            21.5.3  Real and complex numbers   zltp1ne 35097
            21.5.4  Graph theory   lfuhgr 35105
                  21.5.4.1  Acyclic graphs   cacycgr 35129
      ​21.6  Mathbox for Mario Carneiro
            21.6.1  Predicate calculus with all distinct variables   ax-7d 35146
            21.6.2  Miscellaneous stuff   quartfull 35152
            21.6.3  Derangements and the Subfactorial   deranglem 35153
            21.6.4  The Erdős-Szekeres theorem   erdszelem1 35178
            21.6.5  The Kuratowski closure-complement theorem   kur14lem1 35193
            21.6.6  Retracts and sections   cretr 35204
            21.6.7  Path-connected and simply connected spaces   cpconn 35206
            21.6.8  Covering maps   ccvm 35242
            21.6.9  Normal numbers   snmlff 35316
            21.6.10  Godel-sets of formulas - part 1   cgoe 35320
            21.6.11  Godel-sets of formulas - part 2   cgon 35419
            21.6.12  Models of ZF   cgze 35433
            ​*21.6.13  Metamath formal systems   cmcn 35447
            21.6.14  Grammatical formal systems   cm0s 35572
            21.6.15  Models of formal systems   cmuv 35592
            21.6.16  Splitting fields   ccpms 35614
            21.6.17  p-adic number fields   czr 35634
      ​​*21.7  Mathbox for Filip Cernatescu
      ​21.8  Mathbox for Paul Chapman
            21.8.1  Real and complex numbers (cont.)   climuzcnv 35658
            21.8.2  Miscellaneous theorems   elfzm12 35662
      ​21.9  Mathbox for Hongxiu Chen
      ​21.10  Mathbox for Adrian Ducourtial
            21.10.1  Propositional calculus   currybi 35675
            21.10.2  Clone theory   ccloneop 35682
      ​21.11  Mathbox for Scott Fenton
            21.11.1  ZFC Axioms in primitive form   axextprim 35688
            21.11.2  Untangled classes   untelirr 35695
            21.11.3  Extra propositional calculus theorems   3jaodd 35702
            21.11.4  Misc. Useful Theorems   nepss 35705
            21.11.5  Properties of real and complex numbers   sqdivzi 35715
            21.11.6  Infinite products   iprodefisumlem 35727
            21.11.7  Factorial limits   faclimlem1 35730
            21.11.8  Greatest common divisor and divisibility   gcd32 35736
            21.11.9  Properties of relationships   dftr6 35738
            21.11.10  Properties of functions and mappings   funpsstri 35753
            21.11.11  Set induction (or epsilon induction)   setinds 35766
            21.11.12  Ordinal numbers   elpotr 35769
            21.11.13  Defined equality axioms   axextdfeq 35785
            21.11.14  Hypothesis builders   hbntg 35793
            21.11.15  Well-founded zero, successor, and limits   cwsuc 35798
            21.11.16  Quantifier-free definitions   ctxp 35818
            21.11.17  Alternate ordered pairs   caltop 35944
            21.11.18  Geometry in the Euclidean space   cofs 35970
                  21.11.18.1  Congruence properties   cofs 35970
                  21.11.18.2  Betweenness properties   btwntriv2 36000
                  21.11.18.3  Segment Transportation   ctransport 36017
                  21.11.18.4  Properties relating betweenness and congruence   cifs 36023
                  21.11.18.5  Connectivity of betweenness   btwnconn1lem1 36075
                  21.11.18.6  Segment less than or equal to   csegle 36094
                  21.11.18.7  Outside-of relationship   coutsideof 36107
                  21.11.18.8  Lines and Rays   cline2 36122
            21.11.19  Forward difference   cfwddif 36146
            21.11.20  Rank theorems   rankung 36154
            21.11.21  Hereditarily Finite Sets   chf 36160
      ​21.12  Mathbox for Gino Giotto
            21.12.1  Equality theorems.   rmoeqi 36175
                  21.12.1.1  Inference versions.   rmoeqi 36175
                  21.12.1.2  Deduction versions.   rmoeqdv 36200
            21.12.2  Change bound variables.   in-ax8 36212
                  21.12.2.1  Change bound variables and domains.   cbvralvw2 36214
                  21.12.2.2  Change bound variables, deduction versions.   cbvmodavw 36238
                  21.12.2.3  Change bound variables and domains, deduction versions.   cbvrmodavw2 36271
            21.12.3  Study of ax-mulf usage.   mpomulnzcnf 36287
      ​21.13  Mathbox for Jeff Hankins
            21.13.1  Miscellany   a1i14 36288
            21.13.2  Basic topological facts   topbnd 36312
            21.13.3  Topology of the real numbers   ivthALT 36323
            21.13.4  Refinements   cfne 36324
            21.13.5  Neighborhood bases determine topologies   neibastop1 36347
            21.13.6  Lattice structure of topologies   topmtcl 36351
            21.13.7  Filter bases   fgmin 36358
            21.13.8  Directed sets, nets   tailfval 36360
      ​21.14  Mathbox for Anthony Hart
            21.14.1  Propositional Calculus   tb-ax1 36371
            21.14.2  Predicate Calculus   nalfal 36391
            21.14.3  Miscellaneous single axioms   meran1 36399
            21.14.4  Connective Symmetry   negsym1 36405
      ​21.15  Mathbox for Chen-Pang He
            21.15.1  Ordinal topology   ontopbas 36416
      ​21.16  Mathbox for Jeff Hoffman
            21.16.1  Inferences for finite induction on generic function values   fveleq 36439
            21.16.2  gdc.mm   nnssi2 36443
      ​21.17  Mathbox for Matthew House
            21.17.1  Relations on well-ordered indexed unions   weiunlem1 36450
      ​21.18  Mathbox for Asger C. Ipsen
            21.18.1  Continuous nowhere differentiable functions   dnival 36459
      ​​*21.19  Mathbox for BJ
            *21.19.1  Propositional calculus   bj-mp2c 36528
                  *21.19.1.1  Derived rules of inference   bj-mp2c 36528
                  ​*21.19.1.2  A syntactic theorem   bj-0 36530
                  21.19.1.3  Minimal implicational calculus   bj-a1k 36532
                  ​*21.19.1.4  Positive calculus   bj-syl66ib 36543
                  21.19.1.5  Implication and negation   bj-con2com 36549
                  ​*21.19.1.6  Disjunction   bj-jaoi1 36559
                  ​*21.19.1.7  Logical equivalence   bj-dfbi4 36561
                  21.19.1.8  The conditional operator for propositions   bj-consensus 36566
                  ​*21.19.1.9  Propositional calculus: miscellaneous   bj-imbi12 36571
            ​*21.19.2  Modal logic   bj-axdd2 36580
            ​*21.19.3  Provability logic   cprvb 36585
            ​*21.19.4  First-order logic   bj-genr 36594
                  21.19.4.1  Adding ax-gen   bj-genr 36594
                  21.19.4.2  Adding ax-4   bj-2alim 36598
                  21.19.4.3  Adding ax-5   bj-ax12wlem 36632
                  21.19.4.4  Equality and substitution   bj-ssbeq 36641
                  21.19.4.5  Adding ax-6   bj-spimvwt 36657
                  21.19.4.6  Adding ax-7   bj-cbvexw 36664
                  21.19.4.7  Membership predicate, ax-8 and ax-9   bj-ax89 36666
                  21.19.4.8  Adding ax-11   bj-alcomexcom 36668
                  21.19.4.9  Adding ax-12   axc11n11 36670
                  21.19.4.10  Nonfreeness   wnnf 36711
                  21.19.4.11  Adding ax-13   bj-axc10 36771
                  ​*21.19.4.12  Removing dependencies on ax-13 (and ax-11)   bj-axc10v 36781
                  ​*21.19.4.13  Distinct var metavariables   bj-hbaeb2 36806
                  ​*21.19.4.14  Around ~ equsal   bj-equsal1t 36810
                  ​*21.19.4.15  Some Principia Mathematica proofs   stdpc5t 36815
                  21.19.4.16  Alternate definition of substitution   bj-sbsb 36825
                  21.19.4.17  Lemmas for substitution   bj-sbf3 36827
                  21.19.4.18  Existential uniqueness   bj-eu3f 36829
                  ​*21.19.4.19  First-order logic: miscellaneous   bj-sblem1 36830
            21.19.5  Set theory   eliminable1 36847
                  ​*21.19.5.1  Eliminability of class terms   eliminable1 36847
                  ​*21.19.5.2  Classes without the axiom of extensionality   bj-denoteslem 36859
                  21.19.5.3  Characterization among sets versus among classes   elelb 36885
                  ​*21.19.5.4  The nonfreeness quantifier for classes   bj-nfcsym 36887
                  ​*21.19.5.5  Lemmas for class substitution   bj-sbeqALT 36888
                  21.19.5.6  Removing some axiom requirements and disjoint variable conditions   bj-exlimvmpi 36899
                  ​*21.19.5.7  Class abstractions   bj-elabd2ALT 36913
                  21.19.5.8  Generalized class abstractions   bj-cgab 36921
                  ​*21.19.5.9  Restricted nonfreeness   wrnf 36929
                  ​*21.19.5.10  Russell's paradox   bj-ru1 36931
                  21.19.5.11  Curry's paradox in set theory   currysetlem 36933
                  ​*21.19.5.12  Some disjointness results   bj-n0i 36939
                  ​*21.19.5.13  Complements on direct products   bj-xpimasn 36943
                  ​*21.19.5.14  "Singletonization" and tagging   bj-snsetex 36951
                  ​*21.19.5.15  Tuples of classes   bj-cproj 36978
                  ​*21.19.5.16  Set theory: elementary operations relative to a universe   bj-rcleqf 37013
                  ​*21.19.5.17  Axioms for finite unions   bj-abex 37018
                  ​*21.19.5.18  Set theory: miscellaneous   eleq2w2ALT 37035
                  ​*21.19.5.19  Evaluation at a class   bj-evaleq 37060
                  21.19.5.20  Elementwise operations   celwise 37067
                  ​*21.19.5.21  Elementwise intersection (families of sets induced on a subset)   bj-rest00 37069
                  21.19.5.22  Moore collections (complements)   bj-raldifsn 37088
                  21.19.5.23  Maps-to notation for functions with three arguments   bj-0nelmpt 37104
                  ​*21.19.5.24  Currying   csethom 37110
                  ​*21.19.5.25  Setting components of extensible structures   cstrset 37122
            ​*21.19.6  Extended real and complex numbers, real and complex projective lines   bj-nfald 37125
                  21.19.6.1  Complements on class abstractions of ordered pairs and binary relations   bj-nfald 37125
                  ​*21.19.6.2  Identity relation (complements)   bj-opabssvv 37138
                  ​*21.19.6.3  Functionalized identity (diagonal in a Cartesian square)   cdiag2 37160
                  ​*21.19.6.4  Direct image and inverse image   cimdir 37166
                  ​*21.19.6.5  Extended numbers and projective lines as sets   cfractemp 37184
                  ​*21.19.6.6  Addition and opposite   caddcc 37225
                  ​*21.19.6.7  Order relation on the extended reals   cltxr 37229
                  ​*21.19.6.8  Argument, multiplication and inverse   carg 37231
                  21.19.6.9  The canonical bijection from the finite ordinals   ciomnn 37237
                  21.19.6.10  Divisibility   cnnbar 37248
            ​*21.19.7  Monoids   bj-smgrpssmgm 37256
                  ​*21.19.7.1  Finite sums in monoids   cfinsum 37271
            ​*21.19.8  Affine, Euclidean, and Cartesian geometry   bj-fvimacnv0 37274
                  *21.19.8.1  Real vector spaces   bj-fvimacnv0 37274
                  ​*21.19.8.2  Complex numbers (supplements)   bj-subcom 37296
                  ​*21.19.8.3  Barycentric coordinates   bj-bary1lem 37298
            21.19.9  Monoid of endomorphisms   cend 37301
      ​21.20  Mathbox for Jim Kingdon
            21.20.1  Circle constant   taupilem3 37307
            21.20.2  Number theory   dfgcd3 37312
            21.20.3  Real numbers   irrdifflemf 37313
      ​21.21  Mathbox for ML
            21.21.1  Miscellaneous   csbrecsg 37316
            21.21.2  Cartesian exponentiation   cfinxp 37371
            21.21.3  Topology   iunctb2 37391
                  ​*21.21.3.1  Pi-base theorems   pibp16 37401
      ​21.22  Mathbox for Wolf Lammen
            21.22.1  1. Bootstrapping   wl-section-boot 37410
            21.22.2  Implication chains   wl-section-impchain 37434
            21.22.3  Theorems around the conditional operator   wl-ifp-ncond1 37452
            21.22.4  Alternative development of hadd, cadd   wl-df-3xor 37456
            21.22.5  An alternative axiom ~ ax-13   ax-wl-13v 37481
            21.22.6  Bootstrapping set theory with classes   wl-cleq-0 37483
            21.22.7  Other stuff   wl-mps 37495
      ​21.23  Mathbox for Brendan Leahy
      ​21.24  Mathbox for Jeff Madsen
            21.24.1  Logic and set theory   unirep 37708
            21.24.2  Real and complex numbers; integers   filbcmb 37734
            21.24.3  Sequences and sums   sdclem2 37736
            21.24.4  Topology   subspopn 37746
            21.24.5  Metric spaces   metf1o 37749
            21.24.6  Continuous maps and homeomorphisms   constcncf 37756
            21.24.7  Boundedness   ctotbnd 37760
            21.24.8  Isometries   cismty 37792
            21.24.9  Heine-Borel Theorem   heibor1lem 37803
            21.24.10  Banach Fixed Point Theorem   bfplem1 37816
            21.24.11  Euclidean space   crrn 37819
            21.24.12  Intervals (continued)   ismrer1 37832
            21.24.13  Operation properties   cass 37836
            21.24.14  Groups and related structures   cmagm 37842
            21.24.15  Group homomorphism and isomorphism   cghomOLD 37877
            21.24.16  Rings   crngo 37888
            21.24.17  Division Rings   cdrng 37942
            21.24.18  Ring homomorphisms   crngohom 37954
            21.24.19  Commutative rings   ccm2 37983
            21.24.20  Ideals   cidl 38001
            21.24.21  Prime rings and integral domains   cprrng 38040
            21.24.22  Ideal generators   cigen 38053
      ​21.25  Mathbox for Giovanni Mascellani
            ​*21.25.1  Tools for automatic proof building   efald2 38072
            ​*21.25.2  Tseitin axioms   fald 38123
            ​*21.25.3  Equality deductions   iuneq2f 38150
            ​*21.25.4  Miscellanea   orcomdd 38161
      ​21.26  Mathbox for Peter Mazsa
            21.26.1  Notations   cxrn 38168
            21.26.2  Preparatory theorems   el2v1 38211
            21.26.3  Range Cartesian product   df-xrn 38353
            21.26.4  Cosets by ` R `   df-coss 38402
            21.26.5  Relations   df-rels 38476
            21.26.6  Subset relations   df-ssr 38489
            21.26.7  Reflexivity   df-refs 38501
            21.26.8  Converse reflexivity   df-cnvrefs 38516
            21.26.9  Symmetry   df-syms 38533
            21.26.10  Reflexivity and symmetry   symrefref2 38554
            21.26.11  Transitivity   df-trs 38563
            21.26.12  Equivalence relations   df-eqvrels 38575
            21.26.13  Redundancy   df-redunds 38614
            21.26.14  Domain quotients   df-dmqss 38629
            21.26.15  Equivalence relations on domain quotients   df-ers 38655
            21.26.16  Functions   df-funss 38672
            21.26.17  Disjoints vs. converse functions   df-disjss 38695
            21.26.18  Antisymmetry   df-antisymrel 38752
            21.26.19  Partitions: disjoints on domain quotients   df-parts 38757
            21.26.20  Partition-Equivalence Theorems   disjim 38773
      ​21.27  Mathbox for Rodolfo Medina
            21.27.1  Partitions   prtlem60 38846
      ​​*21.28  Mathbox for Norm Megill
            *21.28.1  Obsolete schemes ax-c4,c5,c7,c10,c11,c11n,c15,c9,c14,c16   ax-c5 38876
            ​*21.28.2  Rederive new axioms ax-4, ax-10, ax-6, ax-12, ax-13 from old   axc5 38886
            ​*21.28.3  Legacy theorems using obsolete axioms   ax5ALT 38900
            21.28.4  Experiments with weak deduction theorem   elimhyps 38954
            21.28.5  Miscellanea   cnaddcom 38965
            21.28.6  Atoms, hyperplanes, and covering in a left vector space (or module)   clsa 38967
            21.28.7  Functionals and kernels of a left vector space (or module)   clfn 39050
            21.28.8  Opposite rings and dual vector spaces   cld 39116
            21.28.9  Ortholattices and orthomodular lattices   cops 39165
            21.28.10  Atomic lattices with covering property   ccvr 39255
            21.28.11  Hilbert lattices   chlt 39343
            21.28.12  Projective geometries based on Hilbert lattices   clln 39485
            21.28.13  Construction of a vector space from a Hilbert lattice   cdlema1N 39785
            21.28.14  Construction of involution and inner product from a Hilbert lattice   clpoN 41474
      ​21.29  Mathbox for metakunt
            21.29.1  Commutative Semiring   ccsrg 41956
            21.29.2  General helpful statements   rhmzrhval 41959
            21.29.3  Some gcd and lcm results   12gcd5e1 41991
            21.29.4  Least common multiple inequality theorem   3factsumint1 42009
            21.29.5  Logarithm inequalities   3exp7 42041
            21.29.6  Miscellaneous results for AKS formalisation   intlewftc 42049
            21.29.7  Sticks and stones   sticksstones1 42134
            21.29.8  Continuation AKS   aks6d1c6lem1 42158
      ​21.30  Mathbox for Steven Nguyen
            21.30.1  Utility theorems   jarrii 42193
            ​*21.30.2  Arithmetic theorems   c0exALT 42240
            21.30.3  Exponents and divisibility   oexpreposd 42310
            21.30.4  Trigonometry and Calculus   tanhalfpim 42337
            ​*21.30.5  Independence of ax-mulcom   cresub 42353
            21.30.6  Structures   sn-base0 42483
            ​*21.30.7  Projective spaces   cprjsp 42589
            21.30.8  Basic reductions for Fermat's Last Theorem   dffltz 42622
            ​*21.30.9  Exemplar theorems   iddii 42652
                  ​*21.30.9.1  Standard replacements of ax-10 , ax-11 , ax-12   nfa1w 42663
      ​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 42680
            21.33.2  Additional theory of functions   imaiinfv 42681
            21.33.3  Additional topology   elrfi 42682
            21.33.4  Characterization of closure operators. Kuratowski closure axioms   ismrcd1 42686
            21.33.5  Algebraic closure systems   cnacs 42690
            21.33.6  Miscellanea 1. Map utilities   constmap 42701
            21.33.7  Miscellanea for polynomials   mptfcl 42708
            21.33.8  Multivariate polynomials over the integers   cmzpcl 42709
            21.33.9  Miscellanea for Diophantine sets 1   coeq0i 42741
            21.33.10  Diophantine sets 1: definitions   cdioph 42743
            21.33.11  Diophantine sets 2 miscellanea   ellz1 42755
            21.33.12  Diophantine sets 2: union and intersection. Monotone Boolean algebra   diophin 42760
            21.33.13  Diophantine sets 3: construction   diophrex 42763
            21.33.14  Diophantine sets 4 miscellanea   2sbcrex 42772
            21.33.15  Diophantine sets 4: Quantification   rexrabdioph 42782
            21.33.16  Diophantine sets 5: Arithmetic sets   rabdiophlem1 42789
            21.33.17  Diophantine sets 6: reusability. renumbering of variables   eldioph4b 42799
            21.33.18  Pigeonhole Principle and cardinality helpers   fphpd 42804
            21.33.19  A non-closed set of reals is infinite   rencldnfilem 42808
            21.33.20  Lagrange's rational approximation theorem   irrapxlem1 42810
            21.33.21  Pell equations 1: A nontrivial solution always exists   pellexlem1 42817
            21.33.22  Pell equations 2: Algebraic number theory of the solution set   csquarenn 42824
            21.33.23  Pell equations 3: characterizing fundamental solution   infmrgelbi 42866
            ​*21.33.24  Logarithm laws generalized to an arbitrary base   reglogcl 42878
            21.33.25  Pell equations 4: the positive solution group is infinite cyclic   pellfund14 42886
            21.33.26  X and Y sequences 1: Definition and recurrence laws   crmx 42888
            21.33.27  Ordering and induction lemmas for the integers   monotuz 42930
            21.33.28  X and Y sequences 2: Order properties   rmxypos 42936
            21.33.29  Congruential equations   congtr 42954
            21.33.30  Alternating congruential equations   acongid 42964
            21.33.31  Additional theorems on integer divisibility   coprmdvdsb 42974
            21.33.32  X and Y sequences 3: Divisibility properties   jm2.18 42977
            21.33.33  X and Y sequences 4: Diophantine representability of Y   jm2.27a 42994
            21.33.34  X and Y sequences 5: Diophantine representability of X, ^, _C   rmxdiophlem 43004
            21.33.35  Uncategorized stuff not associated with a major project   setindtr 43013
            21.33.36  More equivalents of the Axiom of Choice   axac10 43022
            21.33.37  Finitely generated left modules   clfig 43056
            21.33.38  Noetherian left modules I   clnm 43064
            21.33.39  Addenda for structure powers   pwssplit4 43078
            21.33.40  Every set admits a group structure iff choice   unxpwdom3 43084
            21.33.41  Noetherian rings and left modules II   clnr 43098
            21.33.42  Hilbert's Basis Theorem   cldgis 43110
            21.33.43  Additional material on polynomials [DEPRECATED]   cmnc 43120
            21.33.44  Degree and minimal polynomial of algebraic numbers   cdgraa 43129
            21.33.45  Algebraic integers I   citgo 43146
            21.33.46  Endomorphism algebra   cmend 43160
            21.33.47  Cyclic groups and order   idomodle 43180
            21.33.48  Cyclotomic polynomials   ccytp 43186
            21.33.49  Miscellaneous topology   fgraphopab 43192
      ​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 43206
            21.36.2  Natural addition of Cantor normal forms   oawordex2 43315
            21.36.3  Surreal Contributions   abeqabi 43397
            21.36.4  Short Studies   nlimsuc 43430
                  21.36.4.1  Additional work on conditional logical operator   ifpan123g 43448
                  21.36.4.2  Sophisms   rp-fakeimass 43501
                  ​*21.36.4.3  Finite Sets   rp-isfinite5 43506
                  21.36.4.4  General Observations   intabssd 43508
                  21.36.4.5  Infinite Sets   pwelg 43549
                  ​*21.36.4.6  Finite intersection property   fipjust 43554
                  21.36.4.7  RP ADDTO: Subclasses and subsets   rababg 43563
                  21.36.4.8  RP ADDTO: The intersection of a class   elinintab 43564
                  21.36.4.9  RP ADDTO: Theorems requiring subset and intersection existence   elinintrab 43566
                  21.36.4.10  RP ADDTO: Relations   xpinintabd 43569
                  ​*21.36.4.11  RP ADDTO: Functions   elmapintab 43585
                  ​*21.36.4.12  RP ADDTO: Finite induction (for finite ordinals)   cnvcnvintabd 43589
                  21.36.4.13  RP ADDTO: First and second members of an ordered pair   elcnvlem 43590
                  21.36.4.14  RP ADDTO: The reflexive and transitive properties of relations   undmrnresiss 43593
                  21.36.4.15  RP ADDTO: Basic properties of closures   cleq2lem 43597
                  21.36.4.16  RP REPLACE: Definitions and basic properties of transitive closures   trcleq2lemRP 43619
                  ​*21.36.4.17  Additions for square root; absolute value   sqrtcvallem1 43620
            21.36.5  Additional statements on relations and subclasses   al3im 43636
                  21.36.5.1  Transitive relations (not to be confused with transitive classes).   trrelind 43654
                  21.36.5.2  Reflexive closures   crcl 43661
                  ​*21.36.5.3  Finite relationship composition.   relexp2 43666
                  21.36.5.4  Transitive closure of a relation   dftrcl3 43709
                  ​*21.36.5.5  Adapted from Frege   frege77d 43735
            ​*21.36.6  Propositions from _Begriffsschrift_   dfxor4 43755
                  *21.36.6.1  _Begriffsschrift_ Chapter I   dfxor4 43755
                  ​*21.36.6.2  _Begriffsschrift_ Notation hints   whe 43761
                  21.36.6.3  _Begriffsschrift_ Chapter II Implication   ax-frege1 43779
                  21.36.6.4  _Begriffsschrift_ Chapter II Implication and Negation   axfrege28 43818
                  ​*21.36.6.5  _Begriffsschrift_ Chapter II with logical equivalence   axfrege52a 43845
                  21.36.6.6  _Begriffsschrift_ Chapter II with equivalence of sets   axfrege52c 43876
                  ​*21.36.6.7  _Begriffsschrift_ Chapter II with equivalence of classes   frege53c 43903
                  ​*21.36.6.8  _Begriffsschrift_ Chapter III Properties hereditary in a sequence   dffrege69 43921
                  ​*21.36.6.9  _Begriffsschrift_ Chapter III Following in a sequence   dffrege76 43928
                  ​*21.36.6.10  _Begriffsschrift_ Chapter III Member of sequence   dffrege99 43951
                  ​*21.36.6.11  _Begriffsschrift_ Chapter III Single-valued procedures   dffrege115 43967
            ​*21.36.7  Exploring Topology via Seifert and Threlfall   enrelmap 43986
                  *21.36.7.1  Equinumerosity of sets of relations and maps   enrelmap 43986
                  ​*21.36.7.2  Generic Pseudoclosure Spaces, Pseudointerior Spaces, and Pseudoneighborhoods   or3or 44012
                  ​*21.36.7.3  Generic Neighborhood Spaces   gneispa 44119
            ​*21.36.8  Exploring Higher Homotopy via Kerodon   k0004lem1 44136
                  *21.36.8.1  Simplicial Sets   k0004lem1 44136
      ​21.37  Mathbox for Stanislas Polu
            21.37.1  IMO Problems   wwlemuld 44145
                  21.37.1.1  IMO 1972 B2   wwlemuld 44145
            ​*21.37.2  INT Inequalities Proof Generator   int-addcomd 44162
            ​*21.37.3  N-Digit Addition Proof Generator   unitadd 44184
            21.37.4  AM-GM (for k = 2,3,4)   gsumws3 44185
      ​21.38  Mathbox for Rohan Ridenour
            21.38.1  Misc   spALT 44190
            21.38.2  Monoid rings   cmnring 44200
            21.38.3  Shorter primitive equivalent of ax-groth   gru0eld 44218
                  21.38.3.1  Grothendieck universes are closed under collection   gru0eld 44218
                  21.38.3.2  Minimal universes   ismnu 44250
                  21.38.3.3  Primitive equivalent of ax-groth   expandan 44277
      ​21.39  Mathbox for Steve Rodriguez
            21.39.1  Miscellanea   nanorxor 44294
            21.39.2  Ratio test for infinite series convergence and divergence   dvgrat 44301
            21.39.3  Multiples   reldvds 44304
            21.39.4  Function operations   caofcan 44312
            21.39.5  Calculus   lhe4.4ex1a 44318
            21.39.6  The generalized binomial coefficient operation   cbcc 44325
            21.39.7  Binomial series   uzmptshftfval 44335
      ​21.40  Mathbox for Andrew Salmon
            21.40.1  Principia Mathematica * 10   pm10.12 44347
            21.40.2  Principia Mathematica * 11   2alanimi 44361
            21.40.3  Predicate Calculus   sbeqal1 44387
            21.40.4  Principia Mathematica * 13 and * 14   pm13.13a 44396
            21.40.5  Set Theory   elnev 44427
            21.40.6  Arithmetic   addcomgi 44445
            21.40.7  Geometry   cplusr 44446
      ​​*21.41  Mathbox for Alan Sare
            21.41.1  Auxiliary theorems for the Virtual Deduction tool   idiALT 44468
            21.41.2  Supplementary unification deductions   bi1imp 44472
            21.41.3  Conventional Metamath proofs, some derived from VD proofs   iidn3 44491
            21.41.4  What is Virtual Deduction?   wvd1 44559
            21.41.5  Virtual Deduction Theorems   df-vd1 44560
            21.41.6  Theorems proved using Virtual Deduction   trsspwALT 44807
            21.41.7  Theorems proved using Virtual Deduction with mmj2 assistance   simplbi2VD 44835
            21.41.8  Virtual Deduction transcriptions of textbook proofs   sb5ALTVD 44902
            21.41.9  Theorems proved using conjunction-form Virtual Deduction   elpwgdedVD 44906
            21.41.10  Theorems with a VD proof in conventional notation derived from a VD proof   suctrALT3 44913
            ​*21.41.11  Theorems with a proof in conventional notation derived from a VD proof   notnotrALT2 44916
      ​21.42  Mathbox for Eric Schmidt
            21.42.1  Miscellany   rspesbcd 44927
            21.42.2  Study of dfbi1ALT   dfbi1ALTa 44929
            21.42.3  Relation-preserving functions   wrelp 44932
            21.42.4  Orbits   orbitex 44945
            21.42.5  Well-founded sets   trwf 44949
            21.42.6  Absoluteness in transitive models   ralabso 44958
            21.42.7  Lemmas for showing axioms hold in models   traxext 44967
            21.42.8  The class of well-founded sets is a model for ZFC   wfaxext 44983
            21.42.9  Permutation models   brpermmodel 44993
      ​21.43  Mathbox for Glauco Siliprandi
            21.43.1  Miscellanea   evth2f 45009
            21.43.2  Functions   fnresdmss 45162
            21.43.3  Ordering on real numbers - Real and complex numbers basic operations   sub2times 45271
            21.43.4  Real intervals   gtnelioc 45489
            21.43.5  Finite sums   fsummulc1f 45569
            21.43.6  Finite multiplication of numbers and finite multiplication of functions   fmul01 45578
            21.43.7  Limits   clim1fr1 45599
                  21.43.7.1  Inferior limit (lim inf)   clsi 45749
                  ​*21.43.7.2  Limits for sequences of extended real numbers   clsxlim 45816
            21.43.8  Trigonometry   coseq0 45862
            21.43.9  Continuous Functions   mulcncff 45868
            21.43.10  Derivatives   dvsinexp 45909
            21.43.11  Integrals   itgsin0pilem1 45948
            21.43.12  Stone Weierstrass theorem - real version   stoweidlem1 45999
            21.43.13  Wallis' product for π   wallispilem1 46063
            21.43.14  Stirling's approximation formula for ` n ` factorial   stirlinglem1 46072
            21.43.15  Dirichlet kernel   dirkerval 46089
            21.43.16  Fourier Series   fourierdlem1 46106
            21.43.17  e is transcendental   elaa2lem 46231
            21.43.18  n-dimensional Euclidean space   rrxtopn 46282
            21.43.19  Basic measure theory   csalg 46306
                  ​*21.43.19.1  σ-Algebras   csalg 46306
                  21.43.19.2  Sum of nonnegative extended reals   csumge0 46360
                  ​*21.43.19.3  Measures   cmea 46447
                  ​*21.43.19.4  Outer measures and Caratheodory's construction   come 46487
                  ​*21.43.19.5  Lebesgue measure on n-dimensional Real numbers   covoln 46534
                  ​*21.43.19.6  Measurable functions   csmblfn 46693
      ​21.44  Mathbox for Saveliy Skresanov
            21.44.1  Ceva's theorem   sigarval 46848
            21.44.2  Simple groups   simpcntrab 46868
      ​21.45  Mathbox for Ender Ting
            21.45.1  Increasing sequences and subsequences   et-ltneverrefl 46869
            21.45.2  Scratchpad for number theory   evenwodadd 46886
            21.45.3  Scratchpad for math on real numbers   squeezedltsq 46887
      ​21.46  Mathbox for Jarvin Udandy
      ​21.47  Mathbox for Adhemar
            ​*21.47.1  Minimal implicational calculus   adh-minim 47002
      ​21.48  Mathbox for Alexander van der Vekens
            21.48.1  General auxiliary theorems (1)   n0nsn2el 47026
                  21.48.1.1  Unordered and ordered pairs - extension for singletons   n0nsn2el 47026
                  21.48.1.2  Unordered and ordered pairs - extension for unordered pairs   elprneb 47030
                  21.48.1.3  Unordered and ordered pairs - extension for ordered pairs   oppr 47031
                  21.48.1.4  Relations - extension   eubrv 47036
                  21.48.1.5  Definite description binder (inverted iota) - extension   iota0def 47039
                  21.48.1.6  Functions - extension   fveqvfvv 47041
            21.48.2  Alternative for Russell's definition of a description binder   caiota 47084
            21.48.3  Double restricted existential uniqueness   r19.32 47099
                  21.48.3.1  Restricted quantification (extension)   r19.32 47099
                  21.48.3.2  Restricted uniqueness and "at most one" quantification   reuf1odnf 47108
                  21.48.3.3  Analogs to Existential uniqueness (double quantification)   2reu3 47111
                  21.48.3.4  Additional theorems for double restricted existential uniqueness   2reu8i 47114
            ​*21.48.4  Alternative definitions of function and operation values   wdfat 47117
                  21.48.4.1  Restricted quantification (extension)   ralbinrald 47123
                  21.48.4.2  The universal class (extension)   nvelim 47124
                  21.48.4.3  Introduce the Axiom of Power Sets (extension)   alneu 47125
                  21.48.4.4  Predicate "defined at"   dfateq12d 47127
                  21.48.4.5  Alternative definition of the value of a function   dfafv2 47133
                  21.48.4.6  Alternative definition of the value of an operation   aoveq123d 47179
            ​*21.48.5  Alternative definitions of function values (2)   cafv2 47209
            21.48.6  General auxiliary theorems (2)   an4com24 47269
                  21.48.6.1  Logical conjunction - extension   an4com24 47269
                  21.48.6.2  Abbreviated conjunction and disjunction of three wff's - extension   3an4ancom24 47270
                  21.48.6.3  Negated membership (alternative)   cnelbr 47272
                  21.48.6.4  The empty set - extension   ralralimp 47279
                  21.48.6.5  Indexed union and intersection - extension   otiunsndisjX 47280
                  21.48.6.6  Functions - extension   fvifeq 47281
                  21.48.6.7  Maps-to notation - extension   fvmptrab 47293
                  21.48.6.8  Subtraction - extension   cnambpcma 47295
                  21.48.6.9  Ordering on reals (cont.) - extension   leaddsuble 47298
                  21.48.6.10  Imaginary and complex number properties - extension   readdcnnred 47304
                  21.48.6.11  Nonnegative integers (as a subset of complex numbers) - extension   nn0resubcl 47309
                  21.48.6.12  Integers (as a subset of complex numbers) - extension   zgeltp1eq 47310
                  21.48.6.13  Decimal arithmetic - extension   1t10e1p1e11 47311
                  21.48.6.14  Upper sets of integers - extension   eluzge0nn0 47313
                  21.48.6.15  Infinity and the extended real number system (cont.) - extension   nltle2tri 47314
                  21.48.6.16  Finite intervals of integers - extension   ssfz12 47315
                  21.48.6.17  Half-open integer ranges - extension   fzopred 47323
                  21.48.6.18  The floor and ceiling functions - extension   2ltceilhalf 47329
                  21.48.6.19  The modulo (remainder) operation - extension   fldivmod 47339
                  21.48.6.20  The infinite sequence builder "seq"   smonoord 47372
                  21.48.6.21  Finite and infinite sums - extension   fsummsndifre 47373
                  21.48.6.22  Extensible structures - extension   setsidel 47377
            ​*21.48.7  Preimages of function values   preimafvsnel 47380
            ​*21.48.8  Partitions of real intervals   ciccp 47414
            21.48.9  Shifting functions with an integer range domain   fargshiftfv 47440
            21.48.10  Words over a set (extension)   lswn0 47445
                  21.48.10.1  Last symbol of a word - extension   lswn0 47445
            21.48.11  Unordered pairs   wich 47446
                  21.48.11.1  Interchangeable setvar variables   wich 47446
                  21.48.11.2  Set of unordered pairs   sprid 47475
                  ​*21.48.11.3  Proper (unordered) pairs   prpair 47502
                  21.48.11.4  Set of proper unordered pairs   cprpr 47513
            21.48.12  Number theory (extension)   cfmtno 47528
                  ​*21.48.12.1  Fermat numbers   cfmtno 47528
                  ​*21.48.12.2  Mersenne primes   m2prm 47592
                  21.48.12.3  Proth's theorem   modexp2m1d 47613
                  21.48.12.4  Solutions of quadratic equations   quad1 47621
            ​*21.48.13  Even and odd numbers   ceven 47625
                  21.48.13.1  Definitions and basic properties   ceven 47625
                  21.48.13.2  Alternate definitions using the "divides" relation   dfeven2 47650
                  21.48.13.3  Alternate definitions using the "modulo" operation   dfeven3 47659
                  21.48.13.4  Alternate definitions using the "gcd" operation   iseven5 47665
                  21.48.13.5  Theorems of part 5 revised   zneoALTV 47670
                  21.48.13.6  Theorems of part 6 revised   odd2np1ALTV 47675
                  21.48.13.7  Theorems of AV's mathbox revised   0evenALTV 47689
                  21.48.13.8  Additional theorems   epoo 47704
                  21.48.13.9  Perfect Number Theorem (revised)   perfectALTVlem1 47722
            21.48.14  Number theory (extension 2)   cfppr 47725
                  ​*21.48.14.1  Fermat pseudoprimes   cfppr 47725
                  ​*21.48.14.2  Goldbach's conjectures   cgbe 47746
            21.48.15  Graph theory (extension)   cclnbgr 47819
                  21.48.15.1  Closed neighborhood of a vertex   cclnbgr 47819
                  ​*21.48.15.2  Semiclosed and semiopen neighborhoods (experimental)   dfsclnbgr2 47846
                  21.48.15.3  Induced subgraphs   cisubgr 47860
                  ​*21.48.15.4  Isomorphisms of graphs   cgrisom 47874
                  ​*21.48.15.5  Triangles in graphs   cgrtri 47936
                  ​*21.48.15.6  Star graphs   cstgr 47950
                  ​*21.48.15.7  Local isomorphisms of graphs   cgrlim 47975
                  ​*21.48.15.8  Generalized Petersen graphs   cgpg 48031
                  21.48.15.9  Loop-free graphs - extension   1hegrlfgr 48120
                  21.48.15.10  Walks - extension   cupwlks 48121
                  21.48.15.11  Edges of graphs expressed as sets of unordered pairs   upgredgssspr 48131
            21.48.16  Monoids (extension)   ovn0dmfun 48144
                  21.48.16.1  Auxiliary theorems   ovn0dmfun 48144
                  21.48.16.2  Magmas, Semigroups and Monoids (extension)   plusfreseq 48152
                  21.48.16.3  Examples and counterexamples for magmas, semigroups and monoids (extension)   opmpoismgm 48155
                  21.48.16.4  Group sum operation (extension 1)   gsumsplit2f 48168
            ​*21.48.17  Magmas and internal binary operations (alternate approach)   ccllaw 48171
                  *21.48.17.1  Laws for internal binary operations   ccllaw 48171
                  ​*21.48.17.2  Internal binary operations   cintop 48184
                  21.48.17.3  Alternative definitions for magmas and semigroups   cmgm2 48203
            21.48.18  Rings (extension)   lmod0rng 48217
                  21.48.18.1  Nonzero rings (extension)   lmod0rng 48217
                  21.48.18.2  Ideals as non-unital rings   lidldomn1 48219
                  21.48.18.3  The non-unital ring of even integers   0even 48225
                  21.48.18.4  A constructed not unital ring   cznrnglem 48247
                  ​*21.48.18.5  The category of non-unital rings (alternate definition)   crngcALTV 48251
                  ​*21.48.18.6  The category of (unital) rings (alternate definition)   cringcALTV 48275
            21.48.19  Basic algebraic structures (extension)   eliunxp2 48322
                  21.48.19.1  Auxiliary theorems   eliunxp2 48322
                  21.48.19.2  The binomial coefficient operation (extension)   bcpascm1 48339
                  21.48.19.3  The ` ZZ `-module ` ZZ X. ZZ `   zlmodzxzlmod 48342
                  21.48.19.4  Group sum operation (extension 2)   mgpsumunsn 48349
                  21.48.19.5  Symmetric groups (extension)   exple2lt6 48352
                  21.48.19.6  Divisibility (extension)   invginvrid 48355
                  21.48.19.7  The support of functions (extension)   rmsupp0 48356
                  21.48.19.8  Finitely supported functions (extension)   rmsuppfi 48360
                  21.48.19.9  Left modules (extension)   lmodvsmdi 48367
                  21.48.19.10  Associative algebras (extension)   assaascl0 48369
                  21.48.19.11  Univariate polynomials (extension)   ply1vr1smo 48371
                  21.48.19.12  Univariate polynomials (examples)   linply1 48382
            21.48.20  Linear algebra (extension)   cdmatalt 48385
                  ​*21.48.20.1  The subalgebras of diagonal and scalar matrices (extension)   cdmatalt 48385
                  ​*21.48.20.2  Linear combinations   clinc 48393
                  ​*21.48.20.3  Linear independence   clininds 48429
                  21.48.20.4  Simple left modules and the ` ZZ `-module   lmod1lem1 48476
                  21.48.20.5  Differences between (left) modules and (left) vector spaces   lvecpsslmod 48496
            21.48.21  Complexity theory   suppdm 48499
                  21.48.21.1  Auxiliary theorems   suppdm 48499
                  21.48.21.2  Even and odd integers   nn0onn0ex 48512
                  21.48.21.3  The natural logarithm on complex numbers (extension)   logcxp0 48524
                  21.48.21.4  Division of functions   cfdiv 48526
                  21.48.21.5  Upper bounds   cbigo 48536
                  21.48.21.6  Logarithm to an arbitrary base (extension)   rege1logbrege0 48547
                  ​*21.48.21.7  The binary logarithm   fldivexpfllog2 48554
                  21.48.21.8  Binary length   cblen 48558
                  ​*21.48.21.9  Digits   cdig 48584
                  21.48.21.10  Nonnegative integer as sum of its shifted digits   dignn0flhalflem1 48604
                  21.48.21.11  Algorithms for the multiplication of nonnegative integers   nn0mulfsum 48613
                  ​*21.48.21.12  N-ary functions   cnaryf 48615
                  ​*21.48.21.13  The Ackermann function   citco 48646
            21.48.22  Elementary geometry (extension)   fv1prop 48688
                  21.48.22.1  Auxiliary theorems   fv1prop 48688
                  21.48.22.2  Real euclidean space of dimension 2   rrx2pxel 48700
                  21.48.22.3  Spheres and lines in real Euclidean spaces   cline 48716
      ​21.49  Mathbox for Zhi Wang
            21.49.1  Propositional calculus   pm4.71da 48778
            21.49.2  Predicate calculus with equality   dtrucor3 48787
                  21.49.2.1  Axiom scheme ax-5 (Distinctness)   dtrucor3 48787
            21.49.3  ZF Set Theory - start with the Axiom of Extensionality   ralbidb 48788
                  21.49.3.1  Restricted quantification   ralbidb 48788
                  21.49.3.2  The universal class   reuxfr1dd 48795
                  21.49.3.3  The empty set   ssdisjd 48796
                  21.49.3.4  Unordered and ordered pairs   vsn 48800
                  21.49.3.5  The union of a class   unilbss 48806
                  21.49.3.6  Indexed union and intersection   iuneq0 48807
            21.49.4  ZF Set Theory - add the Axiom of Replacement   inpw 48813
                  21.49.4.1  Theorems requiring subset and intersection existence   inpw 48813
            21.49.5  ZF Set Theory - add the Axiom of Power Sets   opth1neg 48814
                  21.49.5.1  Ordered pair theorem   opth1neg 48814
                  21.49.5.2  Ordered-pair class abstractions (cont.)   brab2dd 48816
                  21.49.5.3  Relations   iinxp 48819
                  21.49.5.4  Functions   mof0 48826
                  21.49.5.5  Operations   ovsng 48846
            21.49.6  ZF Set Theory - add the Axiom of Union   fonex 48855
                  21.49.6.1  Relations and functions (cont.)   fonex 48855
                  21.49.6.2  First and second members of an ordered pair   eloprab1st2nd 48856
                  21.49.6.3  Operations in maps-to notation (continued)   fmpodg 48857
                  21.49.6.4  Function transposition   resinsnlem 48859
                  21.49.6.5  Infinite Cartesian products   ixpv 48878
                  21.49.6.6  Equinumerosity   fvconst0ci 48879
            21.49.7  Order sets   iccin 48884
                  21.49.7.1  Real number intervals   iccin 48884
            21.49.8  Extensible structures   slotresfo 48887
                  21.49.8.1  Basic definitions   slotresfo 48887
            21.49.9  Moore spaces   mreuniss 48888
            ​*21.49.10  Topology   clduni 48889
                  21.49.10.1  Closure and interior   clduni 48889
                  21.49.10.2  Neighborhoods   neircl 48893
                  21.49.10.3  Subspace topologies   restcls2lem 48901
                  21.49.10.4  Limits and continuity in topological spaces   cnneiima 48905
                  21.49.10.5  Topological definitions using the reals   iooii 48906
                  21.49.10.6  Separated sets   sepnsepolem1 48910
                  21.49.10.7  Separated spaces: T0, T1, T2 (Hausdorff) ...   isnrm4 48919
            21.49.11  Preordered sets and directed sets using extensible structures   isprsd 48943
            21.49.12  Posets and lattices using extensible structures   lubeldm2 48944
                  21.49.12.1  Posets   lubeldm2 48944
                  21.49.12.2  Lattices   toslat 48970
                  21.49.12.3  Subset order structures   intubeu 48972
            21.49.13  Rings   elmgpcntrd 48993
                  21.49.13.1  Multiplicative Group   elmgpcntrd 48993
            21.49.14  Associative algebras   asclelbas 48994
                  21.49.14.1  Definition and basic properties   asclelbas 48994
            21.49.15  Categories   homf0 48998
                  21.49.15.1  Categories   homf0 48998
                  21.49.15.2  Opposite category   oppccatb 49005
                  21.49.15.3  Monomorphisms and epimorphisms   idmon 49009
                  21.49.15.4  Sections, inverses, isomorphisms   sectrcl 49011
                  21.49.15.5  Isomorphic objects   cicfn 49031
                  21.49.15.6  Subcategories   dmdm 49042
                  21.49.15.7  Functors   reldmfunc 49064
                  21.49.15.8  Opposite functors   coppf 49111
                  21.49.15.9  Full & faithful functors   imasubc 49140
                  21.49.15.10  Universal property   upciclem1 49155
                  21.49.15.11  Natural transformations and the functor category   isnatd 49212
                  21.49.15.12  Initial, terminal and zero objects of a category   initoo2 49221
                  21.49.15.13  Product of categories   reldmxpc 49235
                  21.49.15.14  Swap functors   cswapf 49248
                  21.49.15.15  Functor evaluation   oppc1stflem 49276
                  21.49.15.16  Transposed curry functors   cofuswapfcl 49282
                  21.49.15.17  Constant functors   diag1 49293
                  21.49.15.18  Functor composition bifunctors   fucofulem1 49299
                  21.49.15.19  Post-composition functors   postcofval 49353
                  21.49.15.20  Pre-composition functors   precofvallem 49355
            21.49.16  Examples of categories   catcrcl 49384
                  21.49.16.1  The category of categories   catcrcl 49384
                  21.49.16.2  Thin categories   cthinc 49406
                  21.49.16.3  Terminal categories   ctermc 49461
                  21.49.16.4  Preordered sets as thin categories   cprstc 49538
                  21.49.16.5  Monoids as categories   cmndtc 49566
                  21.49.16.6  Categories with at most one object and at most two morphisms   2arwcatlem1 49584
            21.49.17  Kan extensions and related concepts   clan 49594
                  21.49.17.1  Kan extensions   clan 49594
                  21.49.17.2  Limits and colimits   clmd 49632
      ​21.50  Mathbox for Emmett Weisz
            ​*21.50.1  Miscellaneous Theorems   nfintd 49662
            21.50.2  Set Recursion   csetrecs 49672
                  ​*21.50.2.1  Basic Properties of Set Recursion   csetrecs 49672
                  21.50.2.2  Examples and properties of set recursion   elsetrecslem 49688
            ​*21.50.3  Construction of Games and Surreal Numbers   cpg 49698
      ​​*21.51  Mathbox for David A. Wheeler
            21.51.1  Natural deduction   sbidd 49707
            ​*21.51.2  Greater than, greater than or equal to.   cge-real 49709
            ​*21.51.3  Hyperbolic trigonometric functions   csinh 49719
            ​*21.51.4  Reciprocal trigonometric functions (sec, csc, cot)   csec 49730
            ​*21.51.5  Identities for "if"   ifnmfalse 49752
            ​*21.51.6  Logarithms generalized to arbitrary base using ` logb `   logb2aval 49753
            ​*21.51.7  Logarithm laws generalized to an arbitrary base - log_   clog- 49754
            ​*21.51.8  Formally define notions such as reflexivity   wreflexive 49756
            ​*21.51.9  Algebra helpers   mvlraddi 49760
            ​*21.51.10  Algebra helper examples   i2linesi 49767
            ​*21.51.11  Formal methods "surprises"   alimp-surprise 49769
            ​*21.51.12  Allsome quantifier   walsi 49775
            ​*21.51.13  Miscellaneous   5m4e1 49786
            21.51.14  Theorems about algebraic numbers   aacllem 49790
      ​21.52  Mathbox for Kunhao Zheng
            21.52.1  Weighted AM-GM inequality   amgmwlem 49791