TOC of Theorem List - Metamath Proof Explorer

Table of Contents Summary

PART 1  CLASSICAL FIRST-ORDER LOGIC WITH EQUALITY
      1.1  Pre-logic
      1.2  Propositional calculus
      1.3  Other axiomatizations related to classical propositional calculus
      1.4  Predicate calculus with equality: Tarski's system S2 (1 rule, 6 schemes)
      1.5  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes)
      1.6  Uniqueness and unique existence
      1.7  Other axiomatizations related to classical predicate calculus
PART 2  ZF (ZERMELO-FRAENKEL) SET THEORY
      2.1  ZF Set Theory - start with the Axiom of Extensionality
      2.2  ZF Set Theory - add the Axiom of Replacement
      2.3  ZF Set Theory - add the Axiom of Power Sets
      2.4  ZF Set Theory - add the Axiom of Union
      2.5  ZF Set Theory - add the Axiom of Regularity
      2.6  ZF Set Theory - add the Axiom of Infinity
PART 3  ZFC (ZERMELO-FRAENKEL WITH CHOICE) SET THEORY
      3.1  ZFC Set Theory - add Countable Choice and Dependent Choice
      3.2  ZFC Set Theory - add the Axiom of Choice
      3.3  ZFC Axioms with no distinct variable requirements
      3.4  The Generalized Continuum Hypothesis
PART 4  TG (TARSKI-GROTHENDIECK) SET THEORY
      4.1  Inaccessibles
      4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
PART 5  REAL AND COMPLEX NUMBERS
      5.1  Construction and axiomatization of real and complex numbers
      5.2  Derive the basic properties from the field axioms
      5.3  Real and complex numbers - basic operations
      5.4  Integer sets
      5.5  Order sets
      5.6  Elementary integer functions
      5.7  Words over a set
      5.8  Reflexive and transitive closures of relations
      5.9  Elementary real and complex functions
      5.10  Elementary limits and convergence
      5.11  Elementary trigonometry
      5.12  Cardinality of real and complex number subsets
PART 6  ELEMENTARY NUMBER THEORY
      6.1  Elementary properties of divisibility
      6.2  Elementary prime number theory
PART 7  BASIC STRUCTURES
      7.1  Extensible structures
      7.2  Moore spaces
PART 8  BASIC CATEGORY THEORY
      8.1  Categories
      8.2  Arrows (disjointified hom-sets)
      8.3  Examples of categories
      8.4  Categorical constructions
PART 9  BASIC ORDER THEORY
      9.1  Dual of an order structure
      9.2  Preordered sets and directed sets
      9.3  Partially ordered sets (posets)
      9.4  Totally ordered sets (tosets)
      9.5  Lattices
      9.6  Posets, directed sets, and lattices as relations
PART 10  BASIC ALGEBRAIC STRUCTURES
      10.1  Monoids
      10.2  Groups
      10.3  Rings
      10.4  Division rings and fields
      10.5  Left modules
      10.6  Vector spaces
      10.7  Subring algebras and ideals
      10.8  The complex numbers as an algebraic extensible structure
      10.9  Generalized pre-Hilbert and Hilbert spaces
PART 11  BASIC LINEAR ALGEBRA
      11.1  Vectors and free modules
      11.2  Associative algebras
      11.3  Abstract multivariate polynomials
      11.4  Matrices
      11.5  The determinant
      11.6  Polynomial matrices
      11.7  The characteristic polynomial
PART 12  BASIC TOPOLOGY
      12.1  Topology
      12.2  Filters and filter bases
      12.3  Uniform Structures and Spaces
      12.4  Metric spaces
      12.5  Metric subcomplex vector spaces
PART 13  BASIC REAL AND COMPLEX ANALYSIS
      13.1  Continuity
      13.2  Integrals
      13.3  Derivatives
PART 14  BASIC REAL AND COMPLEX FUNCTIONS
      14.1  Polynomials
      14.2  Sequences and series
      14.3  Basic trigonometry
      14.4  Basic number theory
PART 15  SURREAL NUMBERS
      15.1  Sign sequence representation and Alling's axioms
      15.2  Initial consequences of Alling's axioms
      15.3  Conway cut representation
      15.4  Induction and recursion
      15.5  Surreal arithmetic
      15.6  Subsystems of surreals
PART 16  ELEMENTARY GEOMETRY
      16.1  Definition and Tarski's Axioms of Geometry
      16.2  Tarskian Geometry
      16.3  Properties of geometries
      16.4  Geometry in Hilbert spaces
PART 17  GRAPH THEORY
      17.1  Vertices and edges
      17.2  Undirected graphs
      17.3  Walks, paths and cycles
      17.4  Eulerian paths and the Konigsberg Bridge problem
      17.5  The Friendship Theorem
PART 18  GUIDES AND MISCELLANEA
      18.1  Guides (conventions, explanations, and examples)
      18.2  Humor
      18.3  (Future - to be reviewed and classified)
PART 19  COMPLEX TOPOLOGICAL VECTOR SPACES (DEPRECATED)
      19.1  Additional material on group theory (deprecated)
      19.2  Complex vector spaces
      19.3  Normed complex vector spaces
      19.4  Operators on complex vector spaces
      19.5  Inner product (pre-Hilbert) spaces
      19.6  Complex Banach spaces
      19.7  Complex Hilbert spaces
PART 20  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      20.1  Axiomatization of complex pre-Hilbert spaces
      20.2  Inner product and norms
      20.3  Cauchy sequences and completeness axiom
      20.4  Subspaces and projections
      20.5  Properties of Hilbert subspaces
      20.6  Operators on Hilbert spaces
      20.7  States on a Hilbert lattice and Godowski's equation
      20.8  Cover relation, atoms, exchange axiom, and modular symmetry
PART 21  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      ​21.1  Mathboxes for user contributions
      21.2  Mathbox for Stefan Allan
      21.3  Mathbox for Thierry Arnoux
      21.4  Mathbox for Jonathan Ben-Naim
      21.5  Mathbox for BTernaryTau
      21.6  Mathbox for Mario Carneiro
      21.7  Mathbox for Filip Cernatescu
      21.8  Mathbox for Paul Chapman
      21.9  Mathbox for Hongxiu Chen
      21.10  Mathbox for Adrian Ducourtial
      21.11  Mathbox for Scott Fenton
      21.12  Mathbox for Gino Giotto
      21.13  Mathbox for Jeff Hankins
      21.14  Mathbox for Anthony Hart
      21.15  Mathbox for Chen-Pang He
      21.16  Mathbox for Jeff Hoffman
      21.17  Mathbox for Matthew House
      21.18  Mathbox for Asger C. Ipsen
      21.19  Mathbox for BJ
      21.20  Mathbox for Jim Kingdon
      21.21  Mathbox for ML
      21.22  Mathbox for Wolf Lammen
      21.23  Mathbox for Brendan Leahy
      21.24  Mathbox for Jeff Madsen
      21.25  Mathbox for Giovanni Mascellani
      21.26  Mathbox for Peter Mazsa
      21.27  Mathbox for Rodolfo Medina
      21.28  Mathbox for Norm Megill
      21.29  Mathbox for metakunt
      21.30  Mathbox for Steven Nguyen
      21.31  Mathbox for Igor Ieskov
      21.32  Mathbox for OpenAI
      21.33  Mathbox for Stefan O'Rear
      21.34  Mathbox for Noam Pasman
      21.35  Mathbox for Jon Pennant
      21.36  Mathbox for Richard Penner
      21.37  Mathbox for Stanislas Polu
      21.38  Mathbox for Rohan Ridenour
      21.39  Mathbox for Steve Rodriguez
      21.40  Mathbox for Andrew Salmon
      21.41  Mathbox for Alan Sare
      21.42  Mathbox for Eric Schmidt
      21.43  Mathbox for Glauco Siliprandi
      21.44  Mathbox for Saveliy Skresanov
      21.45  Mathbox for Ender Ting
      21.46  Mathbox for Jarvin Udandy
      21.47  Mathbox for Adhemar
      21.48  Mathbox for Alexander van der Vekens
      21.49  Mathbox for Zhi Wang
      21.50  Mathbox for Emmett Weisz
      21.51  Mathbox for David A. Wheeler
      21.52  Mathbox for Kunhao Zheng

Detailed Table of Contents
(* means the section header has a description)

​​*PART 1  CLASSICAL FIRST-ORDER LOGIC WITH EQUALITY
      ​*1.1  Pre-logic
            *1.1.1  Inferences for assisting proof development   idi 1
      ​​*1.2  Propositional calculus
            1.2.1  Recursively define primitive wffs for propositional calculus   wn 3
            ​*1.2.2  The axioms of propositional calculus   ax-mp 5
            ​*1.2.3  Logical implication   mp2 9
            ​*1.2.4  Logical negation   con4 113
            ​*1.2.5  Logical equivalence   wb 206
            ​*1.2.6  Logical conjunction   wa 395
            ​*1.2.7  Logical disjunction   wo 847
            ​*1.2.8  Mixed connectives   jaao 956
            ​*1.2.9  The conditional operator for propositions   wif 1062
            ​*1.2.10  The weak deduction theorem for propositional calculus   elimh 1082
            1.2.11  Abbreviated conjunction and disjunction of three wff's   w3o 1085
            1.2.12  Logical "nand" (Sheffer stroke)   wnan 1491
            1.2.13  Logical "xor"   wxo 1511
            1.2.14  Logical "nor"   wnor 1528
            1.2.15  True and false constants   wal 1538
                  ​*1.2.15.1  Universal quantifier for use by df-tru   wal 1538
                  ​*1.2.15.2  Equality predicate for use by df-tru   cv 1539
                  1.2.15.3  The true constant   wtru 1541
                  1.2.15.4  The false constant   wfal 1552
            ​*1.2.16  Truth tables   truimtru 1563
                  1.2.16.1  Implication   truimtru 1563
                  1.2.16.2  Negation   nottru 1567
                  1.2.16.3  Equivalence   trubitru 1569
                  1.2.16.4  Conjunction   truantru 1573
                  1.2.16.5  Disjunction   truortru 1577
                  1.2.16.6  Alternative denial   trunantru 1581
                  1.2.16.7  Exclusive disjunction   truxortru 1585
                  1.2.16.8  Joint denial   trunortru 1589
            ​*1.2.17  Half adder and full adder in propositional calculus   whad 1593
                  1.2.17.1  Full adder: sum   whad 1593
                  1.2.17.2  Full adder: carry   wcad 1606
      ​1.3  Other axiomatizations related to classical propositional calculus
            ​*1.3.1  Minimal implicational calculus   minimp 1621
            ​*1.3.2  Implicational Calculus   impsingle 1627
            1.3.3  Derive the Lukasiewicz axioms from Meredith's sole axiom   meredith 1641
            1.3.4  Derive the standard axioms from the Lukasiewicz axioms   luklem1 1658
            ​*1.3.5  Derive Nicod's axiom from the standard axioms   nic-dfim 1669
            1.3.6  Derive the Lukasiewicz axioms from Nicod's axiom   nic-imp 1675
            1.3.7  Derive Nicod's Axiom from Lukasiewicz's First Sheffer Stroke Axiom   lukshef-ax1 1694
            1.3.8  Derive the Lukasiewicz Axioms from the Tarski-Bernays-Wajsberg Axioms   tbw-bijust 1698
            1.3.9  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's First CO Axiom   merco1 1713
            1.3.10  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's Second CO Axiom   merco2 1736
            1.3.11  Derive the Lukasiewicz axioms from the Russell-Bernays Axioms   rb-bijust 1749
            ​*1.3.12  Stoic logic non-modal portion (Chrysippus of Soli)   mptnan 1768
      ​​*1.4  Predicate calculus with equality: Tarski's system S2 (1 rule, 6 schemes)
            *1.4.1  Universal quantifier (continued); define "exists" and "not free"   wex 1779
                  1.4.1.1  Existential quantifier   wex 1779
                  1.4.1.2  Nonfreeness predicate   wnf 1783
            1.4.2  Rule scheme ax-gen (Generalization)   ax-gen 1795
            1.4.3  Axiom scheme ax-4 (Quantified Implication)   ax-4 1809
                  ​*1.4.3.1  The empty domain of discourse   empty 1906
            1.4.4  Axiom scheme ax-5 (Distinctness) - first use of $d   ax-5 1910
            ​*1.4.5  Equality predicate (continued)   weq 1962
            1.4.6  Axiom scheme ax-6 (Existence)   ax-6 1967
            1.4.7  Axiom scheme ax-7 (Equality)   ax-7 2008
            1.4.8  Define proper substitution   sbjust 2064
            1.4.9  Membership predicate   wcel 2109
            1.4.10  Axiom scheme ax-8 (Left Equality for Binary Predicate)   ax-8 2111
            1.4.11  Axiom scheme ax-9 (Right Equality for Binary Predicate)   ax-9 2119
            ​*1.4.12  Logical redundancy of ax-10 , ax-11 , ax-12 , ax-13   ax6dgen 2129
      ​​*1.5  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes)
            1.5.1  Axiom scheme ax-10 (Quantified Negation)   ax-10 2142
            1.5.2  Axiom scheme ax-11 (Quantifier Commutation)   ax-11 2158
            1.5.3  Axiom scheme ax-12 (Substitution)   ax-12 2178
            1.5.4  Axiom scheme ax-13 (Quantified Equality)   ax-13 2371
      ​1.6  Uniqueness and unique existence
            1.6.1  Uniqueness: the at-most-one quantifier   wmo 2532
            1.6.2  Unique existence: the unique existential quantifier   weu 2562
      ​1.7  Other axiomatizations related to classical predicate calculus
            ​*1.7.1  Aristotelian logic: Assertic syllogisms   barbara 2657
            ​*1.7.2  Intuitionistic logic   axia1 2687
​​*PART 2  ZF (ZERMELO-FRAENKEL) SET THEORY
      ​2.1  ZF Set Theory - start with the Axiom of Extensionality
            2.1.1  Introduce the Axiom of Extensionality   ax-ext 2702
            2.1.2  Classes   cab 2708
                  2.1.2.1  Class abstractions   cab 2708
                  ​*2.1.2.2  Class equality   df-cleq 2722
                  2.1.2.3  Class membership   df-clel 2804
                  2.1.2.4  Elementary properties of class abstractions   eqabdv 2862
            2.1.3  Class form not-free predicate   wnfc 2877
            2.1.4  Negated equality and membership   wne 2926
                  2.1.4.1  Negated equality   wne 2926
                  2.1.4.2  Negated membership   wnel 3030
            2.1.5  Restricted quantification   wral 3045
                  2.1.5.1  Restricted universal and existential quantification   wral 3045
                  2.1.5.2  Restricted existential uniqueness and at-most-one quantifier   wreu 3354
                  2.1.5.3  Restricted class abstraction   crab 3408
            2.1.6  The universal class   cvv 3450
            ​*2.1.7  Conditional equality (experimental)   wcdeq 3736
            2.1.8  Russell's Paradox   rru 3752
            2.1.9  Proper substitution of classes for sets   wsbc 3755
            2.1.10  Proper substitution of classes for sets into classes   csb 3864
            2.1.11  Define basic set operations and relations   cdif 3913
            2.1.12  Subclasses and subsets   df-ss 3933
            2.1.13  The difference, union, and intersection of two classes   dfdif3 4082
                  2.1.13.1  The difference of two classes   dfdif3 4082
                  2.1.13.2  The union of two classes   elun 4118
                  2.1.13.3  The intersection of two classes   elini 4164
                  2.1.13.4  The symmetric difference of two classes   csymdif 4217
                  2.1.13.5  Combinations of difference, union, and intersection of two classes   unabs 4230
                  2.1.13.6  Class abstractions with difference, union, and intersection of two classes   unabw 4272
                  2.1.13.7  Restricted uniqueness with difference, union, and intersection   reuun2 4290
            2.1.14  The empty set   c0 4298
            ​*2.1.15  The conditional operator for classes   cif 4490
            ​*2.1.16  The weak deduction theorem for set theory   dedth 4549
            2.1.17  Power classes   cpw 4565
            2.1.18  Unordered and ordered pairs   snjust 4590
            2.1.19  The union of a class   cuni 4873
            2.1.20  The intersection of a class   cint 4912
            2.1.21  Indexed union and intersection   ciun 4957
            2.1.22  Disjointness   wdisj 5076
            2.1.23  Binary relations   wbr 5109
            2.1.24  Ordered-pair class abstractions (class builders)   copab 5171
            2.1.25  Functions in maps-to notation   cmpt 5190
            2.1.26  Transitive classes   wtr 5216
      ​2.2  ZF Set Theory - add the Axiom of Replacement
            2.2.1  Introduce the Axiom of Replacement   ax-rep 5236
            2.2.2  Derive the Axiom of Separation   axsepgfromrep 5251
            2.2.3  Derive the Null Set Axiom   axnulALT 5261
            2.2.4  Theorems requiring subset and intersection existence   nalset 5270
            2.2.5  Theorems requiring empty set existence   class2set 5312
      ​2.3  ZF Set Theory - add the Axiom of Power Sets
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 5322
            2.3.2  Derive the Axiom of Pairing   axprlem1 5380
            2.3.3  Ordered pair theorem   opnz 5435
            2.3.4  Ordered-pair class abstractions (cont.)   opabidw 5486
            2.3.5  Power class of union and intersection   pwin 5531
            2.3.6  The identity relation   cid 5534
            2.3.7  The membership relation (or epsilon relation)   cep 5539
            ​*2.3.8  Partial and total orderings   wpo 5546
            2.3.9  Founded and well-ordering relations   wfr 5590
            2.3.10  Relations   cxp 5638
            2.3.11  The Predecessor Class   cpred 6275
            2.3.12  Well-founded induction (variant)   frpomin 6315
            2.3.13  Well-ordered induction   tz6.26 6322
            2.3.14  Ordinals   word 6333
            2.3.15  Definite description binder (inverted iota)   cio 6464
            2.3.16  Functions   wfun 6507
            2.3.17  Cantor's Theorem   canth 7343
            2.3.18  Restricted iota (description binder)   crio 7345
            2.3.19  Operations   co 7389
                  2.3.19.1  Variable-to-class conversion for operations   caovclg 7583
            2.3.20  Maps-to notation   mpondm0 7631
            2.3.21  Function operation   cof 7653
            2.3.22  Proper subset relation   crpss 7700
      ​2.4  ZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 7713
            2.4.2  Ordinals (continued)   epweon 7753
            2.4.3  Transfinite induction   tfi 7831
            2.4.4  The natural numbers (i.e., finite ordinals)   com 7844
            2.4.5  Peano's postulates   peano1 7867
            2.4.6  Finite induction (for finite ordinals)   find 7873
            2.4.7  Relations and functions (cont.)   dmexg 7879
            2.4.8  First and second members of an ordered pair   c1st 7968
            2.4.9  Induction on Cartesian products   frpoins3xpg 8121
            2.4.10  Ordering on Cartesian products   xpord2lem 8123
            2.4.11  Ordering Ordinal Sequences   orderseqlem 8138
            ​*2.4.12  The support of functions   csupp 8141
            ​*2.4.13  Special maps-to operations   opeliunxp2f 8191
            2.4.14  Function transposition   ctpos 8206
            2.4.15  Curry and uncurry   ccur 8246
            2.4.16  Undefined values   cund 8253
            2.4.17  Well-founded recursion   cfrecs 8261
            2.4.18  Well-ordered recursion   cwrecs 8292
            2.4.19  Functions on ordinals; strictly monotone ordinal functions   iunon 8310
            2.4.20  "Strong" transfinite recursion   crecs 8341
            2.4.21  Recursive definition generator   crdg 8379
            2.4.22  Finite recursion   frfnom 8405
            2.4.23  Ordinal arithmetic   c1o 8429
            2.4.24  Natural number arithmetic   nna0 8570
            2.4.25  Natural addition   cnadd 8631
            2.4.26  Equivalence relations and classes   wer 8670
            2.4.27  The mapping operation   cmap 8801
            2.4.28  Infinite Cartesian products   cixp 8872
            2.4.29  Equinumerosity   cen 8917
            2.4.30  Schroeder-Bernstein Theorem   sbthlem1 9056
            2.4.31  Equinumerosity (cont.)   xpf1o 9108
            2.4.32  Finite sets   dif1enlem 9125
            2.4.33  Pigeonhole Principle   phplem1 9173
            2.4.34  Finite sets (cont.)   onomeneq 9183
            2.4.35  Finitely supported functions   cfsupp 9318
            2.4.36  Finite intersections   cfi 9367
            2.4.37  Hall's marriage theorem   marypha1lem 9390
            2.4.38  Supremum and infimum   csup 9397
            2.4.39  Ordinal isomorphism, Hartogs's theorem   coi 9468
            2.4.40  Hartogs function   char 9515
            2.4.41  Weak dominance   cwdom 9523
      ​2.5  ZF Set Theory - add the Axiom of Regularity
            2.5.1  Introduce the Axiom of Regularity   ax-reg 9551
            2.5.2  Axiom of Infinity equivalents   inf0 9580
      ​2.6  ZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-inf 9597
            2.6.2  Existence of omega (the set of natural numbers)   omex 9602
            2.6.3  Cantor normal form   ccnf 9620
            2.6.4  Transitive closure of a relation   cttrcl 9666
            2.6.5  Transitive closure   trcl 9687
            2.6.6  Well-Founded Induction   frmin 9708
            2.6.7  Well-Founded Recursion   frr3g 9715
            2.6.8  Rank   cr1 9721
            2.6.9  Scott's trick; collection principle; Hilbert's epsilon   scottex 9844
            2.6.10  Disjoint union   cdju 9857
            2.6.11  Cardinal numbers   ccrd 9894
            2.6.12  Axiom of Choice equivalents   wac 10074
            ​*2.6.13  Cardinal number arithmetic   undjudom 10127
            2.6.14  The Ackermann bijection   ackbij2lem1 10177
            2.6.15  Cofinality (without Axiom of Choice)   cflem 10204
            2.6.16  Eight inequivalent definitions of finite set   sornom 10236
            2.6.17  Hereditarily size-limited sets without Choice   itunifval 10375
​​*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 10394
            3.1.2  Introduce the Axiom of Dependent Choice   ax-dc 10405
      ​3.2  ZFC Set Theory - add the Axiom of Choice
            3.2.1  Introduce the Axiom of Choice   ax-ac 10418
            3.2.2  AC equivalents: well-ordering, Zorn's lemma   numthcor 10453
            3.2.3  Cardinal number theorems using Axiom of Choice   cardval 10505
            3.2.4  Cardinal number arithmetic using Axiom of Choice   iunctb 10533
            3.2.5  Cofinality using the Axiom of Choice   alephreg 10541
      ​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 10579
            3.4.2  Derivation of the Axiom of Choice   gchaclem 10637
​​*PART 4  TG (TARSKI-GROTHENDIECK) SET THEORY
      ​4.1  Inaccessibles
            4.1.1  Weakly and strongly inaccessible cardinals   cwina 10641
            4.1.2  Weak universes   cwun 10659
            4.1.3  Tarski classes   ctsk 10707
            4.1.4  Grothendieck universes   cgru 10749
      ​4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
            4.2.1  Introduce the Tarski-Grothendieck Axiom   ax-groth 10782
            4.2.2  Derive the Power Set, Infinity and Choice Axioms   grothpw 10785
            4.2.3  Tarski map function   ctskm 10796
​​*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 10803
            5.1.2  Final derivation of real and complex number postulates   axaddf 11104
            5.1.3  Real and complex number postulates restated as axioms   ax-cnex 11130
      ​5.2  Derive the basic properties from the field axioms
            5.2.1  Some deductions from the field axioms for complex numbers   cnex 11155
            5.2.2  Infinity and the extended real number system   cpnf 11211
            5.2.3  Restate the ordering postulates with extended real "less than"   axlttri 11251
            5.2.4  Ordering on reals   lttr 11256
            5.2.5  Initial properties of the complex numbers   mul12 11345
      ​5.3  Real and complex numbers - basic operations
            5.3.1  Addition   add12 11398
            5.3.2  Subtraction   cmin 11411
            5.3.3  Multiplication   kcnktkm1cn 11615
            5.3.4  Ordering on reals (cont.)   gt0ne0 11649
            5.3.5  Reciprocals   ixi 11813
            5.3.6  Division   cdiv 11841
            5.3.7  Ordering on reals (cont.)   elimgt0 12026
            5.3.8  Completeness Axiom and Suprema   fimaxre 12133
            5.3.9  Imaginary and complex number properties   inelr 12177
            5.3.10  Function operation analogue theorems   ofsubeq0 12184
      ​5.4  Integer sets
            5.4.1  Positive integers (as a subset of complex numbers)   cn 12187
            5.4.2  Principle of mathematical induction   nnind 12205
            ​*5.4.3  Decimal representation of numbers   c2 12242
            ​*5.4.4  Some properties of specific numbers   neg1cn 12301
            5.4.5  Simple number properties   halfcl 12414
            5.4.6  The Archimedean property   nnunb 12444
            5.4.7  Nonnegative integers (as a subset of complex numbers)   cn0 12448
            ​*5.4.8  Extended nonnegative integers   cxnn0 12521
            5.4.9  Integers (as a subset of complex numbers)   cz 12535
            5.4.10  Decimal arithmetic   cdc 12655
            5.4.11  Upper sets of integers   cuz 12799
            5.4.12  Well-ordering principle for bounded-below sets of integers   uzwo3 12908
            5.4.13  Rational numbers (as a subset of complex numbers)   cq 12913
            5.4.14  Existence of the set of complex numbers   rpnnen1lem2 12942
      ​5.5  Order sets
            5.5.1  Positive reals (as a subset of complex numbers)   crp 12957
            5.5.2  Infinity and the extended real number system (cont.)   cxne 13075
            5.5.3  Supremum and infimum on the extended reals   xrsupexmnf 13271
            5.5.4  Real number intervals   cioo 13312
            5.5.5  Finite intervals of integers   cfz 13474
            ​*5.5.6  Finite intervals of nonnegative integers   elfz2nn0 13585
            5.5.7  Half-open integer ranges   cfzo 13621
      ​5.6  Elementary integer functions
            5.6.1  The floor and ceiling functions   cfl 13758
            5.6.2  The modulo (remainder) operation   cmo 13837
            5.6.3  Miscellaneous theorems about integers   om2uz0i 13918
            5.6.4  Strong induction over upper sets of integers   uzsinds 13958
            5.6.5  Finitely supported functions over the nonnegative integers   fsuppmapnn0fiublem 13961
            5.6.6  The infinite sequence builder "seq" - extension   cseq 13972
            5.6.7  Integer powers   cexp 14032
            5.6.8  Ordered pair theorem for nonnegative integers   nn0le2msqi 14238
            5.6.9  Factorial function   cfa 14244
            5.6.10  The binomial coefficient operation   cbc 14273
            5.6.11  The ` # ` (set size) function   chash 14301
                  5.6.11.1  Proper unordered pairs and triples (sets of size 2 and 3)   hashprlei 14439
                  5.6.11.2  Functions with a domain containing at least two different elements   fundmge2nop0 14473
                  5.6.11.3  Finite induction on the size of the first component of a binary relation   hashdifsnp1 14477
      ​​*5.7  Words over a set
            5.7.1  Definitions and basic theorems   cword 14484
            5.7.2  Last symbol of a word   clsw 14533
            5.7.3  Concatenations of words   cconcat 14541
            5.7.4  Singleton words   cs1 14566
            5.7.5  Concatenations with singleton words   ccatws1cl 14587
            5.7.6  Subwords/substrings   csubstr 14611
            5.7.7  Prefixes of a word   cpfx 14641
            5.7.8  Subwords of subwords   swrdswrdlem 14675
            5.7.9  Subwords and concatenations   pfxcctswrd 14681
            5.7.10  Subwords of concatenations   swrdccatfn 14695
            5.7.11  Splicing words (substring replacement)   csplice 14720
            5.7.12  Reversing words   creverse 14729
            5.7.13  Repeated symbol words   creps 14739
            ​*5.7.14  Cyclical shifts of words   ccsh 14759
            5.7.15  Mapping words by a function   wrdco 14803
            5.7.16  Longer string literals   cs2 14813
      ​​*5.8  Reflexive and transitive closures of relations
            5.8.1  The reflexive and transitive properties of relations   coss12d 14944
            5.8.2  Basic properties of closures   cleq1lem 14954
            5.8.3  Definitions and basic properties of transitive closures   ctcl 14957
            5.8.4  Exponentiation of relations   crelexp 14991
            5.8.5  Reflexive-transitive closure as an indexed union   crtrcl 15027
            ​*5.8.6  Principle of transitive induction.   relexpindlem 15035
      ​5.9  Elementary real and complex functions
            5.9.1  The "shift" operation   cshi 15038
            5.9.2  Signum (sgn or sign) function   csgn 15058
            5.9.3  Real and imaginary parts; conjugate   ccj 15068
            5.9.4  Square root; absolute value   csqrt 15205
      ​5.10  Elementary limits and convergence
            5.10.1  Superior limit (lim sup)   clsp 15442
            5.10.2  Limits   cli 15456
            5.10.3  Finite and infinite sums   csu 15658
            5.10.4  The binomial theorem   binomlem 15801
            5.10.5  The inclusion/exclusion principle   incexclem 15808
            5.10.6  Infinite sums (cont.)   isumshft 15811
            5.10.7  Miscellaneous converging and diverging sequences   divrcnv 15824
            5.10.8  Arithmetic series   arisum 15832
            5.10.9  Geometric series   expcnv 15836
            5.10.10  Ratio test for infinite series convergence   cvgrat 15855
            5.10.11  Mertens' theorem   mertenslem1 15856
            5.10.12  Finite and infinite products   prodf 15859
                  5.10.12.1  Product sequences   prodf 15859
                  5.10.12.2  Non-trivial convergence   ntrivcvg 15869
                  5.10.12.3  Complex products   cprod 15875
                  5.10.12.4  Finite products   fprod 15913
                  5.10.12.5  Infinite products   iprodclim 15970
            5.10.13  Falling and Rising Factorial   cfallfac 15976
            5.10.14  Bernoulli polynomials and sums of k-th powers   cbp 16018
      ​5.11  Elementary trigonometry
            5.11.1  The exponential, sine, and cosine functions   ce 16033
                  5.11.1.1  The circle constant (tau = 2 pi)   ctau 16176
            5.11.2  _e is irrational   eirrlem 16178
      ​5.12  Cardinality of real and complex number subsets
            5.12.1  Countability of integers and rationals   xpnnen 16185
            5.12.2  The reals are uncountable   rpnnen2lem1 16188
​​*PART 6  ELEMENTARY NUMBER THEORY
      ​6.1  Elementary properties of divisibility
            6.1.1  Irrationality of square root of 2   sqrt2irrlem 16222
            6.1.2  Some Number sets are chains of proper subsets   nthruc 16226
            6.1.3  The divides relation   cdvds 16228
            ​*6.1.4  Even and odd numbers   evenelz 16312
            6.1.5  The division algorithm   divalglem0 16369
            6.1.6  Bit sequences   cbits 16395
            6.1.7  The greatest common divisor operator   cgcd 16470
            6.1.8  Bézout's identity   bezoutlem1 16515
            6.1.9  Algorithms   nn0seqcvgd 16546
            6.1.10  Euclid's Algorithm   eucalgval2 16557
            ​*6.1.11  The least common multiple   clcm 16564
            ​*6.1.12  Coprimality and Euclid's lemma   coprmgcdb 16625
            6.1.13  Cancellability of congruences   congr 16640
      ​6.2  Elementary prime number theory
            ​*6.2.1  Elementary properties   cprime 16647
            ​*6.2.2  Coprimality and Euclid's lemma (cont.)   coprm 16687
            6.2.3  Properties of the canonical representation of a rational   cnumer 16709
            6.2.4  Euler's theorem   codz 16739
            6.2.5  Arithmetic modulo a prime number   modprm1div 16774
            6.2.6  Pythagorean Triples   coprimeprodsq 16785
            6.2.7  The prime count function   cpc 16813
            6.2.8  Pocklington's theorem   prmpwdvds 16881
            6.2.9  Infinite primes theorem   unbenlem 16885
            6.2.10  Sum of prime reciprocals   prmreclem1 16893
            6.2.11  Fundamental theorem of arithmetic   1arithlem1 16900
            6.2.12  Lagrange's four-square theorem   cgz 16906
            6.2.13  Van der Waerden's theorem   cvdwa 16942
            6.2.14  Ramsey's theorem   cram 16976
            ​*6.2.15  Primorial function   cprmo 17008
            ​*6.2.16  Prime gaps   prmgaplem1 17026
            6.2.17  Decimal arithmetic (cont.)   dec2dvds 17040
            6.2.18  Cyclical shifts of words (cont.)   cshwsidrepsw 17070
            6.2.19  Specific prime numbers   prmlem0 17082
            6.2.20  Very large primes   1259lem1 17107
​PART 7  BASIC STRUCTURES
      ​7.1  Extensible structures
            ​*7.1.1  Basic definitions   cstr 17122
                  7.1.1.1  Extensible structures as structures with components   cstr 17122
                  7.1.1.2  Substitution of components   csts 17139
                  7.1.1.3  Slots   cslot 17157
                  ​*7.1.1.4  Structure component indices   cnx 17169
                  7.1.1.5  Base sets   cbs 17185
                  7.1.1.6  Base set restrictions   cress 17206
            7.1.2  Slot definitions   cplusg 17226
            7.1.3  Definition of the structure product   crest 17389
            7.1.4  Definition of the structure quotient   cordt 17468
      ​7.2  Moore spaces
            7.2.1  Moore closures   mrcflem 17573
            7.2.2  Independent sets in a Moore system   mrisval 17597
            7.2.3  Algebraic closure systems   isacs 17618
​PART 8  BASIC CATEGORY THEORY
      ​8.1  Categories
            8.1.1  Categories   ccat 17631
            8.1.2  Opposite category   coppc 17678
            8.1.3  Monomorphisms and epimorphisms   cmon 17696
            8.1.4  Sections, inverses, isomorphisms   csect 17712
            ​*8.1.5  Isomorphic objects   ccic 17763
            8.1.6  Subcategories   cssc 17775
            8.1.7  Functors   cfunc 17822
            8.1.8  Full & faithful functors   cful 17872
            8.1.9  Natural transformations and the functor category   cnat 17912
            8.1.10  Initial, terminal and zero objects of a category   cinito 17949
      ​8.2  Arrows (disjointified hom-sets)
            8.2.1  Identity and composition for arrows   cida 18021
      ​8.3  Examples of categories
            8.3.1  The category of sets   csetc 18043
            8.3.2  The category of categories   ccatc 18066
            ​*8.3.3  The category of extensible structures   fncnvimaeqv 18087
      ​8.4  Categorical constructions
            8.4.1  Product of categories   cxpc 18135
            8.4.2  Functor evaluation   cevlf 18176
            8.4.3  Hom functor   chof 18215
​PART 9  BASIC ORDER THEORY
      ​9.1  Dual of an order structure
      ​9.2  Preordered sets and directed sets
      ​9.3  Partially ordered sets (posets)
      ​9.4  Totally ordered sets (tosets)
      ​9.5  Lattices
            9.5.1  Lattices   clat 18396
            9.5.2  Complete lattices   ccla 18463
            9.5.3  Distributive lattices   cdlat 18485
            9.5.4  Subset order structures   cipo 18492
      ​9.6  Posets, directed sets, and lattices as relations
            ​*9.6.1  Posets and lattices as relations   cps 18529
            9.6.2  Directed sets, nets   cdir 18559
​PART 10  BASIC ALGEBRAIC STRUCTURES
      ​10.1  Monoids
            ​*10.1.1  Magmas   cplusf 18570
            ​*10.1.2  Identity elements   mgmidmo 18593
            ​*10.1.3  Iterated sums in a magma   gsumvalx 18609
            10.1.4  Magma homomorphisms and submagmas   cmgmhm 18623
            ​*10.1.5  Semigroups   csgrp 18651
            ​*10.1.6  Definition and basic properties of monoids   cmnd 18667
            10.1.7  Monoid homomorphisms and submonoids   cmhm 18714
            ​*10.1.8  Iterated sums in a monoid   gsumvallem2 18767
            10.1.9  Free monoids   cfrmd 18780
                  ​*10.1.9.1  Monoid of endofunctions   cefmnd 18801
            10.1.10  Examples and counterexamples for magmas, semigroups and monoids   mgm2nsgrplem1 18851
      ​10.2  Groups
            10.2.1  Definition and basic properties   cgrp 18871
            ​*10.2.2  Group multiple operation   cmg 19005
            10.2.3  Subgroups and Quotient groups   csubg 19058
            ​*10.2.4  Cyclic monoids and groups   cycsubmel 19138
            10.2.5  Elementary theory of group homomorphisms   cghm 19150
            10.2.6  Isomorphisms of groups   cgim 19195
                  10.2.6.1  The first isomorphism theorem of groups   ghmqusnsglem1 19218
            10.2.7  Group actions   cga 19227
            10.2.8  Centralizers and centers   ccntz 19253
            10.2.9  The opposite group   coppg 19283
            10.2.10  Symmetric groups   csymg 19305
                  ​*10.2.10.1  Definition and basic properties   csymg 19305
                  10.2.10.2  Cayley's theorem   cayleylem1 19348
                  10.2.10.3  Permutations fixing one element   symgfix2 19352
                  ​*10.2.10.4  Transpositions in the symmetric group   cpmtr 19377
                  10.2.10.5  The sign of a permutation   cpsgn 19425
            10.2.11  p-Groups and Sylow groups; Sylow's theorems   cod 19460
            10.2.12  Direct products   clsm 19570
                  10.2.12.1  Direct products (extension)   smndlsmidm 19592
            10.2.13  Free groups   cefg 19642
            10.2.14  Abelian groups   ccmn 19716
                  10.2.14.1  Definition and basic properties   ccmn 19716
                  10.2.14.2  Cyclic groups   ccyg 19813
                  10.2.14.3  Group sum operation   gsumval3a 19839
                  10.2.14.4  Group sums over (ranges of) integers   fsfnn0gsumfsffz 19919
                  10.2.14.5  Internal direct products   cdprd 19931
                  10.2.14.6  The Fundamental Theorem of Abelian Groups   ablfacrplem 20003
            10.2.15  Simple groups   csimpg 20028
                  10.2.15.1  Definition and basic properties   csimpg 20028
                  10.2.15.2  Classification of abelian simple groups   ablsimpnosubgd 20042
      ​10.3  Rings
            10.3.1  Multiplicative Group   cmgp 20055
            ​*10.3.2  Non-unital rings ("rngs")   crng 20067
            ​*10.3.3  Ring unity (multiplicative identity)   cur 20096
            10.3.4  Semirings   csrg 20101
                  ​*10.3.4.1  The binomial theorem for semirings   srgbinomlem1 20141
            10.3.5  Unital rings   crg 20148
            10.3.6  Opposite ring   coppr 20251
            10.3.7  Divisibility   cdsr 20269
            10.3.8  Ring primes   crpm 20347
            10.3.9  Homomorphisms of non-unital rings   crnghm 20349
            10.3.10  Ring homomorphisms   crh 20384
            10.3.11  Nonzero rings and zero rings   cnzr 20427
            10.3.12  Local rings   clring 20453
            10.3.13  Subrings   csubrng 20460
                  10.3.13.1  Subrings of non-unital rings   csubrng 20460
                  10.3.13.2  Subrings of unital rings   csubrg 20484
                  10.3.13.3  Subrings generated by a subset   crgspn 20525
            10.3.14  Categories of rings   crngc 20531
                  ​*10.3.14.1  The category of non-unital rings   crngc 20531
                  ​*10.3.14.2  The category of (unital) rings   cringc 20560
                  10.3.14.3  Subcategories of the category of rings   srhmsubclem1 20592
            10.3.15  Left regular elements and domains   crlreg 20606
      ​10.4  Division rings and fields
            10.4.1  Definition and basic properties   cdr 20644
            10.4.2  Sub-division rings   csdrg 20701
            10.4.3  Absolute value (abstract algebra)   cabv 20723
            10.4.4  Star rings   cstf 20752
      ​10.5  Left modules
            10.5.1  Definition and basic properties   clmod 20772
            10.5.2  Subspaces and spans in a left module   clss 20843
            10.5.3  Homomorphisms and isomorphisms of left modules   clmhm 20932
            10.5.4  Subspace sum; bases for a left module   clbs 20987
      ​10.6  Vector spaces
            10.6.1  Definition and basic properties   clvec 21015
      ​10.7  Subring algebras and ideals
            10.7.1  Subring algebras   csra 21084
            ​*10.7.2  Left ideals and spans   clidl 21122
            10.7.3  Two-sided ideals and quotient rings   c2idl 21165
                  ​*10.7.3.1  Condition for a non-unital ring to be unital   rngqiprng1elbas 21202
            10.7.4  Principal ideal rings. Divisibility in the integers   clpidl 21236
            10.7.5  Principal ideal domains   cpid 21252
      ​10.8  The complex numbers as an algebraic extensible structure
            10.8.1  Definition and basic properties   cpsmet 21254
            ​*10.8.2  Ring of integers   czring 21362
                  ​*10.8.2.1  Example for a condition for a non-unital ring to be unital   pzriprnglem1 21397
            10.8.3  Algebraic constructions based on the complex numbers   czrh 21415
            10.8.4  Signs as subgroup of the complex numbers   cnmsgnsubg 21492
            10.8.5  Embedding of permutation signs into a ring   zrhpsgnmhm 21499
            10.8.6  The ordered field of real numbers   crefld 21519
      ​10.9  Generalized pre-Hilbert and Hilbert spaces
            10.9.1  Definition and basic properties   cphl 21539
            10.9.2  Orthocomplements and closed subspaces   cocv 21575
            10.9.3  Orthogonal projection and orthonormal bases   cpj 21615
​​*PART 11  BASIC LINEAR ALGEBRA
      ​11.1  Vectors and free modules
            *11.1.1  Direct sum of left modules   cdsmm 21646
            ​*11.1.2  Free modules   cfrlm 21661
            ​*11.1.3  Standard basis (unit vectors)   cuvc 21697
            ​*11.1.4  Independent sets and families   clindf 21719
            11.1.5  Characterization of free modules   lmimlbs 21751
      ​11.2  Associative algebras
            11.2.1  Definition and basic properties   casa 21765
      ​11.3  Abstract multivariate polynomials
            11.3.1  Definition and basic properties   cmps 21819
            11.3.2  Polynomial evaluation   ces 21985
            11.3.3  Additional definitions for (multivariate) polynomials   cslv 22021
            ​*11.3.4  Univariate polynomials   cps1 22065
            11.3.5  Univariate polynomial evaluation   ces1 22206
                  11.3.5.1  Specialization of polynomial evaluation as a ring homomorphism   evls1scafv 22259
      ​​*11.4  Matrices
            *11.4.1  The matrix multiplication   cmmul 22283
            ​*11.4.2  Square matrices   cmat 22300
            ​*11.4.3  The matrix algebra   matmulr 22331
            ​*11.4.4  Matrices of dimension 0 and 1   mat0dimbas0 22359
            ​*11.4.5  The subalgebras of diagonal and scalar matrices   cdmat 22381
            ​*11.4.6  Multiplication of a matrix with a "column vector"   cmvmul 22433
            11.4.7  Replacement functions for a square matrix   cmarrep 22449
            11.4.8  Submatrices   csubma 22469
      ​11.5  The determinant
            11.5.1  Definition and basic properties   cmdat 22477
            11.5.2  Determinants of 2 x 2 -matrices   m2detleiblem1 22517
            11.5.3  The matrix adjugate/adjunct   cmadu 22525
            ​*11.5.4  Laplace expansion of determinants (special case)   symgmatr01lem 22546
            11.5.5  Inverse matrix   invrvald 22569
            ​*11.5.6  Cramer's rule   slesolvec 22572
      ​​*11.6  Polynomial matrices
            11.6.1  Basic properties   pmatring 22585
            ​*11.6.2  Constant polynomial matrices   ccpmat 22596
            ​*11.6.3  Collecting coefficients of polynomial matrices   cdecpmat 22655
            ​*11.6.4  Ring isomorphism between polynomial matrices and polynomials over matrices   cpm2mp 22685
      ​​*11.7  The characteristic polynomial
            *11.7.1  Definition and basic properties   cchpmat 22719
            ​*11.7.2  The characteristic factor function G   fvmptnn04if 22742
            ​*11.7.3  The Cayley-Hamilton theorem   cpmadurid 22760
​PART 12  BASIC TOPOLOGY
      ​12.1  Topology
            ​*12.1.1  Topological spaces   ctop 22786
                  12.1.1.1  Topologies   ctop 22786
                  12.1.1.2  Topologies on sets   ctopon 22803
                  12.1.1.3  Topological spaces   ctps 22825
            12.1.2  Topological bases   ctb 22838
            12.1.3  Examples of topologies   distop 22888
            12.1.4  Closure and interior   ccld 22909
            12.1.5  Neighborhoods   cnei 22990
            12.1.6  Limit points and perfect sets   clp 23027
            12.1.7  Subspace topologies   restrcl 23050
            12.1.8  Order topology   ordtbaslem 23081
            12.1.9  Limits and continuity in topological spaces   ccn 23117
            12.1.10  Separated spaces: T0, T1, T2 (Hausdorff) ...   ct0 23199
            12.1.11  Compactness   ccmp 23279
            12.1.12  Bolzano-Weierstrass theorem   bwth 23303
            12.1.13  Connectedness   cconn 23304
            12.1.14  First- and second-countability   c1stc 23330
            12.1.15  Local topological properties   clly 23357
            12.1.16  Refinements   cref 23395
            12.1.17  Compactly generated spaces   ckgen 23426
            12.1.18  Product topologies   ctx 23453
            12.1.19  Continuous function-builders   cnmptid 23554
            12.1.20  Quotient maps and quotient topology   ckq 23586
            12.1.21  Homeomorphisms   chmeo 23646
      ​12.2  Filters and filter bases
            12.2.1  Filter bases   elmptrab 23720
            12.2.2  Filters   cfil 23738
            12.2.3  Ultrafilters   cufil 23792
            12.2.4  Filter limits   cfm 23826
            12.2.5  Extension by continuity   ccnext 23952
            12.2.6  Topological groups   ctmd 23963
            12.2.7  Infinite group sum on topological groups   ctsu 24019
            12.2.8  Topological rings, fields, vector spaces   ctrg 24049
      ​12.3  Uniform Structures and Spaces
            12.3.1  Uniform structures   cust 24093
            12.3.2  The topology induced by an uniform structure   cutop 24124
            12.3.3  Uniform Spaces   cuss 24147
            12.3.4  Uniform continuity   cucn 24168
            12.3.5  Cauchy filters in uniform spaces   ccfilu 24179
            12.3.6  Complete uniform spaces   ccusp 24190
      ​12.4  Metric spaces
            12.4.1  Pseudometric spaces   ispsmet 24198
            12.4.2  Basic metric space properties   cxms 24211
            12.4.3  Metric space balls   blfvalps 24277
            12.4.4  Open sets of a metric space   mopnval 24332
            12.4.5  Continuity in metric spaces   metcnp3 24434
            12.4.6  The uniform structure generated by a metric   metuval 24443
            12.4.7  Examples of metric spaces   dscmet 24466
            ​*12.4.8  Normed algebraic structures   cnm 24470
            12.4.9  Normed space homomorphisms (bounded linear operators)   cnmo 24599
            12.4.10  Topology on the reals   qtopbaslem 24652
            12.4.11  Topological definitions using the reals   cii 24774
            12.4.12  Path homotopy   chtpy 24872
            12.4.13  The fundamental group   cpco 24906
      ​12.5  Metric subcomplex vector spaces
            12.5.1  Subcomplex modules   cclm 24968
            ​*12.5.2  Subcomplex vector spaces   ccvs 25029
            ​*12.5.3  Normed subcomplex vector spaces   isncvsngp 25055
            12.5.4  Subcomplex pre-Hilbert spaces   ccph 25072
            12.5.5  Convergence and completeness   ccfil 25158
            12.5.6  Baire's Category Theorem   bcthlem1 25230
            12.5.7  Banach spaces and subcomplex Hilbert spaces   ccms 25238
                  12.5.7.1  The complete ordered field of the real numbers   retopn 25285
            12.5.8  Euclidean spaces   crrx 25289
            12.5.9  Minimizing Vector Theorem   minveclem1 25330
            12.5.10  Projection Theorem   pjthlem1 25343
​PART 13  BASIC REAL AND COMPLEX ANALYSIS
      ​13.1  Continuity
            13.1.1  Intermediate value theorem   pmltpclem1 25355
      ​13.2  Integrals
            13.2.1  Lebesgue measure   covol 25369
            13.2.2  Lebesgue integration   cmbf 25521
                  13.2.2.1  Lesbesgue integral   cmbf 25521
                  13.2.2.2  Lesbesgue directed integral   cdit 25753
      ​13.3  Derivatives
            13.3.1  Real and complex differentiation   climc 25769
                  13.3.1.1  Derivatives of functions of one complex or real variable   climc 25769
                  13.3.1.2  Results on real differentiation   dvferm1lem 25894
​PART 14  BASIC REAL AND COMPLEX FUNCTIONS
      ​14.1  Polynomials
            14.1.1  Polynomial degrees   cmdg 25964
            14.1.2  The division algorithm for univariate polynomials   cmn1 26037
            14.1.3  Elementary properties of complex polynomials   cply 26095
            14.1.4  The division algorithm for polynomials   cquot 26204
            14.1.5  Algebraic numbers   caa 26228
            14.1.6  Liouville's approximation theorem   aalioulem1 26246
      ​14.2  Sequences and series
            14.2.1  Taylor polynomials and Taylor's theorem   ctayl 26266
            14.2.2  Uniform convergence   culm 26291
            14.2.3  Power series   pserval 26325
      ​14.3  Basic trigonometry
            14.3.1  The exponential, sine, and cosine functions (cont.)   efcn 26359
            14.3.2  Properties of pi = 3.14159...   pilem1 26367
            14.3.3  Mapping of the exponential function   efgh 26456
            14.3.4  The natural logarithm on complex numbers   clog 26469
            ​*14.3.5  Logarithms to an arbitrary base   clogb 26680
            14.3.6  Theorems of Pythagoras, isosceles triangles, and intersecting chords   angval 26717
            14.3.7  Solutions of quadratic, cubic, and quartic equations   quad2 26755
            14.3.8  Inverse trigonometric functions   casin 26778
            14.3.9  The Birthday Problem   log2ublem1 26862
            14.3.10  Areas in R^2   carea 26871
            14.3.11  More miscellaneous converging sequences   rlimcnp 26881
            14.3.12  Inequality of arithmetic and geometric means   cvxcl 26901
            14.3.13  Euler-Mascheroni constant   cem 26908
            14.3.14  Zeta function   czeta 26929
            14.3.15  Gamma function   clgam 26932
      ​14.4  Basic number theory
            14.4.1  Wilson's theorem   wilthlem1 26984
            14.4.2  The Fundamental Theorem of Algebra   ftalem1 26989
            14.4.3  The Basel problem (ζ(2) = π²/6)   basellem1 26997
            14.4.4  Number-theoretical functions   ccht 27007
            14.4.5  Perfect Number Theorem   mersenne 27144
            14.4.6  Characters of Z/nZ   cdchr 27149
            14.4.7  Bertrand's postulate   bcctr 27192
            ​*14.4.8  Quadratic residues and the Legendre symbol   clgs 27211
            ​*14.4.9  Gauss' Lemma   gausslemma2dlem0a 27273
            14.4.10  Quadratic reciprocity   lgseisenlem1 27292
            14.4.11  All primes 4n+1 are the sum of two squares   2sqlem1 27334
            14.4.12  Chebyshev's Weak Prime Number Theorem, Dirichlet's Theorem   chebbnd1lem1 27386
            14.4.13  The Prime Number Theorem   mudivsum 27447
            14.4.14  Ostrowski's theorem   abvcxp 27532
​PART 15  SURREAL NUMBERS
      ​​*15.1  Sign sequence representation and Alling's axioms
            15.1.1  Definitions and initial properties   csur 27557
            15.1.2  Ordering   sltsolem1 27593
            15.1.3  Birthday Function   bdayfo 27595
            15.1.4  Density   fvnobday 27596
            ​*15.1.5  Full-Eta Property   bdayimaon 27611
      ​15.2  Initial consequences of Alling's axioms
            15.2.1  Ordering Theorems   csle 27662
            15.2.2  Birthday Theorems   bdayfun 27690
      ​​*15.3  Conway cut representation
            15.3.1  Conway cuts   csslt 27698
            15.3.2  Zero and One   c0s 27740
            15.3.3  Cuts and Options   cmade 27756
            15.3.4  Cofinality and coinitiality   cofsslt 27832
      ​15.4  Induction and recursion
            15.4.1  Induction and recursion on one variable   cnorec 27850
            15.4.2  Induction and recursion on two variables   cnorec2 27861
      ​15.5  Surreal arithmetic
            15.5.1  Addition   cadds 27872
            15.5.2  Negation and Subtraction   cnegs 27931
            15.5.3  Multiplication   cmuls 28015
            15.5.4  Division   cdivs 28096
            15.5.5  Absolute value   cabss 28145
      ​15.6  Subsystems of surreals
            15.6.1  Ordinal numbers   cons 28158
            15.6.2  Surreal recursive sequences   cseqs 28183
            15.6.3  Natural numbers   cnn0s 28212
            15.6.4  Integers   czs 28272
            15.6.5  Dyadic fractions   c2s 28302
            15.6.6  Real numbers   creno 28350
​​*PART 16  ELEMENTARY GEOMETRY
      ​16.1  Definition and Tarski's Axioms of Geometry
            16.1.1  Justification for the congruence notation   tgjustf 28406
      ​16.2  Tarskian Geometry
            16.2.1  Congruence   tgcgrcomimp 28410
            16.2.2  Betweenness   tgbtwntriv2 28420
            16.2.3  Dimension   tglowdim1 28433
            16.2.4  Betweenness and Congruence   tgifscgr 28441
            16.2.5  Congruence of a series of points   ccgrg 28443
            16.2.6  Motions   cismt 28465
            16.2.7  Colinearity   tglng 28479
            16.2.8  Connectivity of betweenness   tgbtwnconn1lem1 28505
            16.2.9  Less-than relation in geometric congruences   cleg 28515
            16.2.10  Rays   chlg 28533
            16.2.11  Lines   btwnlng1 28552
            16.2.12  Point inversions   cmir 28585
            16.2.13  Right angles   crag 28626
            16.2.14  Half-planes   islnopp 28672
            16.2.15  Midpoints and Line Mirroring   cmid 28705
            16.2.16  Congruence of angles   ccgra 28740
            16.2.17  Angle Comparisons   cinag 28768
            16.2.18  Congruence Theorems   tgsas1 28787
            16.2.19  Equilateral triangles   ceqlg 28798
      ​16.3  Properties of geometries
            16.3.1  Isomorphisms between geometries   f1otrgds 28802
      ​16.4  Geometry in Hilbert spaces
            16.4.1  Geometry in the complex plane   cchhllem 28820
            16.4.2  Geometry in Euclidean spaces   cee 28821
                  16.4.2.1  Definition of the Euclidean space   cee 28821
                  16.4.2.2  Tarski's axioms for geometry for the Euclidean space   axdimuniq 28846
                  16.4.2.3  EE^n fulfills Tarski's Axioms   ceeng 28910
​​*PART 17  GRAPH THEORY
      ​*17.1  Vertices and edges
            17.1.1  The edge function extractor for extensible structures   cedgf 28921
            ​*17.1.2  Vertices and indexed edges   cvtx 28929
                  17.1.2.1  Definitions and basic properties   cvtx 28929
                  17.1.2.2  The vertices and edges of a graph represented as ordered pair   opvtxval 28936
                  17.1.2.3  The vertices and edges of a graph represented as extensible structure   funvtxdmge2val 28944
                  17.1.2.4  Representations of graphs without edges   snstrvtxval 28970
                  17.1.2.5  Degenerated cases of representations of graphs   vtxval0 28972
            17.1.3  Edges as range of the edge function   cedg 28980
      ​​*17.2  Undirected graphs
            17.2.1  Undirected hypergraphs   cuhgr 28989
            17.2.2  Undirected pseudographs and multigraphs   cupgr 29013
            ​*17.2.3  Loop-free graphs   umgrislfupgrlem 29055
            17.2.4  Edges as subsets of vertices of graphs   uhgredgiedgb 29059
            ​*17.2.5  Undirected simple graphs   cuspgr 29081
            17.2.6  Examples for graphs   usgr0e 29169
            17.2.7  Subgraphs   csubgr 29200
            17.2.8  Finite undirected simple graphs   cfusgr 29249
            17.2.9  Neighbors, complete graphs and universal vertices   cnbgr 29265
                  17.2.9.1  Neighbors   cnbgr 29265
                  17.2.9.2  Universal vertices   cuvtx 29318
                  17.2.9.3  Complete graphs   ccplgr 29342
            17.2.10  Vertex degree   cvtxdg 29399
            ​*17.2.11  Regular graphs   crgr 29489
      ​​*17.3  Walks, paths and cycles
            *17.3.1  Walks   cewlks 29529
            17.3.2  Walks for loop-free graphs   lfgrwlkprop 29621
            17.3.3  Trails   ctrls 29624
            17.3.4  Paths and simple paths   cpths 29646
            17.3.5  Closed walks   cclwlks 29706
            17.3.6  Circuits and cycles   ccrcts 29720
            ​*17.3.7  Walks as words   cwwlks 29761
            17.3.8  Walks/paths of length 2 (as length 3 strings)   2wlkdlem1 29861
            17.3.9  Walks in regular graphs   rusgrnumwwlkl1 29904
            ​*17.3.10  Closed walks as words   cclwwlk 29916
                  17.3.10.1  Closed walks as words   cclwwlk 29916
                  17.3.10.2  Closed walks of a fixed length as words   cclwwlkn 29959
                  17.3.10.3  Closed walks on a vertex of a fixed length as words   cclwwlknon 30022
            17.3.11  Examples for walks, trails and paths   0ewlk 30049
            17.3.12  Connected graphs   cconngr 30121
      ​17.4  Eulerian paths and the Konigsberg Bridge problem
            ​*17.4.1  Eulerian paths   ceupth 30132
            ​*17.4.2  The Königsberg Bridge problem   konigsbergvtx 30181
      ​17.5  The Friendship Theorem
            17.5.1  Friendship graphs - basics   cfrgr 30193
            17.5.2  The friendship theorem for small graphs   frgr1v 30206
            17.5.3  Theorems according to Mertzios and Unger   2pthfrgrrn 30217
            ​*17.5.4  Huneke's Proof of the Friendship Theorem   frgrncvvdeqlem1 30234
​PART 18  GUIDES AND MISCELLANEA
      ​18.1  Guides (conventions, explanations, and examples)
            ​*18.1.1  Conventions   conventions 30335
            18.1.2  Natural deduction   natded 30338
            ​*18.1.3  Natural deduction examples   ex-natded5.2 30339
            18.1.4  Definitional examples   ex-or 30356
            18.1.5  Other examples   aevdemo 30395
      ​18.2  Humor
            18.2.1  April Fool's theorem   avril1 30398
      ​18.3  (Future - to be reviewed and classified)
            18.3.1  Planar incidence geometry   cplig 30409
            ​*18.3.2  Aliases kept to prevent broken links   dummylink 30422
​​*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 30424
            19.1.2  Abelian groups   cablo 30479
      ​19.2  Complex vector spaces
            19.2.1  Definition and basic properties   cvc 30493
            19.2.2  Examples of complex vector spaces   cnaddabloOLD 30516
      ​19.3  Normed complex vector spaces
            19.3.1  Definition and basic properties   cnv 30519
            19.3.2  Examples of normed complex vector spaces   cnnv 30612
            19.3.3  Induced metric of a normed complex vector space   imsval 30620
            19.3.4  Inner product   cdip 30635
            19.3.5  Subspaces   css 30656
      ​19.4  Operators on complex vector spaces
            19.4.1  Definitions and basic properties   clno 30675
      ​19.5  Inner product (pre-Hilbert) spaces
            19.5.1  Definition and basic properties   ccphlo 30747
            19.5.2  Examples of pre-Hilbert spaces   cncph 30754
            19.5.3  Properties of pre-Hilbert spaces   isph 30757
      ​19.6  Complex Banach spaces
            19.6.1  Definition and basic properties   ccbn 30797
            19.6.2  Examples of complex Banach spaces   cnbn 30804
            19.6.3  Uniform Boundedness Theorem   ubthlem1 30805
            19.6.4  Minimizing Vector Theorem   minvecolem1 30809
      ​19.7  Complex Hilbert spaces
            19.7.1  Definition and basic properties   chlo 30820
            19.7.2  Standard axioms for a complex Hilbert space   hlex 30833
            19.7.3  Examples of complex Hilbert spaces   cnchl 30851
            19.7.4  Hellinger-Toeplitz Theorem   htthlem 30852
​​*PART 20  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      ​20.1  Axiomatization of complex pre-Hilbert spaces
            20.1.1  Basic Hilbert space definitions   chba 30854
            20.1.2  Preliminary ZFC lemmas   df-hnorm 30903
            ​*20.1.3  Derive the Hilbert space axioms from ZFC set theory   axhilex-zf 30916
            ​*20.1.4  Introduce the vector space axioms for a Hilbert space   ax-hilex 30934
            20.1.5  Vector operations   hvmulex 30946
            20.1.6  Inner product postulates for a Hilbert space   ax-hfi 31014
      ​20.2  Inner product and norms
            20.2.1  Inner product   his5 31021
            20.2.2  Norms   dfhnorm2 31057
            20.2.3  Relate Hilbert space to normed complex vector spaces   hilablo 31095
            20.2.4  Bunjakovaskij-Cauchy-Schwarz inequality   bcsiALT 31114
      ​20.3  Cauchy sequences and completeness axiom
            20.3.1  Cauchy sequences and limits   hcau 31119
            20.3.2  Derivation of the completeness axiom from ZF set theory   hilmet 31129
            20.3.3  Completeness postulate for a Hilbert space   ax-hcompl 31137
            20.3.4  Relate Hilbert space to ZFC pre-Hilbert and Hilbert spaces   hhcms 31138
      ​20.4  Subspaces and projections
            20.4.1  Subspaces   df-sh 31142
            20.4.2  Closed subspaces   df-ch 31156
            20.4.3  Orthocomplements   df-oc 31187
            20.4.4  Subspace sum, span, lattice join, lattice supremum   df-shs 31243
            20.4.5  Projection theorem   pjhthlem1 31326
            20.4.6  Projectors   df-pjh 31330
      ​20.5  Properties of Hilbert subspaces
            20.5.1  Orthomodular law   omlsilem 31337
            20.5.2  Projectors (cont.)   pjhtheu2 31351
            20.5.3  Hilbert lattice operations   sh0le 31375
            20.5.4  Span (cont.) and one-dimensional subspaces   spansn0 31476
            20.5.5  Commutes relation for Hilbert lattice elements   df-cm 31518
            20.5.6  Foulis-Holland theorem   fh1 31553
            20.5.7  Quantum Logic Explorer axioms   qlax1i 31562
            20.5.8  Orthogonal subspaces   chscllem1 31572
            20.5.9  Orthoarguesian laws 5OA and 3OA   5oalem1 31589
            20.5.10  Projectors (cont.)   pjorthi 31604
            20.5.11  Mayet's equation E_3   mayete3i 31663
      ​20.6  Operators on Hilbert spaces
            ​*20.6.1  Operator sum, difference, and scalar multiplication   df-hosum 31665
            20.6.2  Zero and identity operators   df-h0op 31683
            20.6.3  Operations on Hilbert space operators   hoaddcl 31693
            20.6.4  Linear, continuous, bounded, Hermitian, unitary operators and norms   df-nmop 31774
            20.6.5  Linear and continuous functionals and norms   df-nmfn 31780
            20.6.6  Adjoint   df-adjh 31784
            20.6.7  Dirac bra-ket notation   df-bra 31785
            20.6.8  Positive operators   df-leop 31787
            20.6.9  Eigenvectors, eigenvalues, spectrum   df-eigvec 31788
            20.6.10  Theorems about operators and functionals   nmopval 31791
            20.6.11  Riesz lemma   riesz3i 31997
            20.6.12  Adjoints (cont.)   cnlnadjlem1 32002
            20.6.13  Quantum computation error bound theorem   unierri 32039
            20.6.14  Dirac bra-ket notation (cont.)   branmfn 32040
            20.6.15  Positive operators (cont.)   leopg 32057
            20.6.16  Projectors as operators   pjhmopi 32081
      ​20.7  States on a Hilbert lattice and Godowski's equation
            20.7.1  States on a Hilbert lattice   df-st 32146
            20.7.2  Godowski's equation   golem1 32206
      ​20.8  Cover relation, atoms, exchange axiom, and modular symmetry
            20.8.1  Covers relation; modular pairs   df-cv 32214
            20.8.2  Atoms   df-at 32273
            20.8.3  Superposition principle   superpos 32289
            20.8.4  Atoms, exchange and covering properties, atomicity   chcv1 32290
            20.8.5  Irreducibility   chirredlem1 32325
            20.8.6  Atoms (cont.)   atcvat3i 32331
            20.8.7  Modular symmetry   mdsymlem1 32338
​PART 21  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      ​21.1  Mathboxes for user contributions
            21.1.1  Mathbox guidelines   mathbox 32377
      ​21.2  Mathbox for Stefan Allan
      ​21.3  Mathbox for Thierry Arnoux
            21.3.1  Propositional Calculus - misc additions   ad11antr 32382
            21.3.2  Predicate Calculus   sbc2iedf 32400
                  21.3.2.1  Predicate Calculus - misc additions   sbc2iedf 32400
                  21.3.2.2  Restricted quantification - misc additions   ralcom4f 32402
                  21.3.2.3  Equality   eqtrb 32409
                  21.3.2.4  Double restricted existential uniqueness quantification   opsbc2ie 32411
                  21.3.2.5  Double restricted existential uniqueness quantification syntax   w2reu 32413
                  21.3.2.6  Substitution (without distinct variables) - misc additions   sbceqbidf 32422
                  21.3.2.7  Existential "at most one" - misc additions   mo5f 32424
                  21.3.2.8  Existential uniqueness - misc additions   reuxfrdf 32426
                  21.3.2.9  Restricted "at most one" - misc additions   rmoxfrd 32428
                  21.3.2.10  Restricted iota (description binder)   riotaeqbidva 32431
            21.3.3  General Set Theory   dmrab 32432
                  21.3.3.1  Class abstractions (a.k.a. class builders)   dmrab 32432
                  21.3.3.2  Image Sets   abrexdomjm 32442
                  21.3.3.3  Set relations and operations - misc additions   nelun 32448
                  21.3.3.4  Unordered pairs   elpreq 32463
                  21.3.3.5  Unordered triples   tpssg 32472
                  21.3.3.6  Conditional operator - misc additions   ifeqeqx 32477
                  21.3.3.7  Set union   uniinn0 32485
                  21.3.3.8  Indexed union - misc additions   cbviunf 32490
                  21.3.3.9  Indexed intersection - misc additions   iinabrex 32504
                  21.3.3.10  Disjointness - misc additions   disjnf 32505
            21.3.4  Relations and Functions   xpdisjres 32533
                  21.3.4.1  Relations - misc additions   xpdisjres 32533
                  21.3.4.2  Functions - misc additions   ac6sf2 32554
                  21.3.4.3  Operations - misc additions   mpomptxf 32607
                  21.3.4.4  The mapping operation   elmaprd 32609
                  21.3.4.5  Support of a function   suppovss 32610
                  21.3.4.6  Explicit Functions with one or two points as a domain   cosnopne 32623
                  21.3.4.7  Isomorphisms - misc. additions   gtiso 32630
                  21.3.4.8  Disjointness (additional proof requiring functions)   disjdsct 32632
                  21.3.4.9  First and second members of an ordered pair - misc additions   df1stres 32633
                  21.3.4.10  Finite Sets   imafi2 32641
                  21.3.4.11  Countable Sets   snct 32643
            21.3.5  Real and Complex Numbers   sgnval2 32664
                  21.3.5.1  Complex operations - misc. additions   creq0 32665
                  21.3.5.2  Ordering on reals - misc additions   lt2addrd 32680
                  21.3.5.3  Extended reals - misc additions   xrlelttric 32681
                  21.3.5.4  Extended nonnegative integers - misc additions   xnn0gt0 32698
                  21.3.5.5  Real number intervals - misc additions   joiniooico 32703
                  21.3.5.6  Finite intervals of integers - misc additions   uzssico 32713
                  21.3.5.7  Half-open integer ranges - misc additions   iundisjfi 32725
                  21.3.5.8  The ` # ` (set size) function - misc additions   hashunif 32737
                  21.3.5.9  The greatest common divisor operator - misc. additions   elq2 32742
                  21.3.5.10  Integers   nn0split01 32748
                  21.3.5.11  Decimal numbers   dfdec100 32761
            21.3.6  Real and complex functions   sgncl 32762
                  21.3.6.1  Signum (sgn or sign) function - misc. additions   sgncl 32762
                  21.3.6.2  Integer powers - misc. additions   nexple 32775
                  21.3.6.3  Indicator Functions   cind 32779
            ​*21.3.7  Decimal expansion   cdp2 32797
                  ​*21.3.7.1  Decimal point   cdp 32814
                  21.3.7.2  Division in the extended real number system   cxdiv 32843
            21.3.8  Words over a set - misc additions   wrdres 32862
                  21.3.8.1  Splicing words (substring replacement)   splfv3 32886
                  21.3.8.2  Cyclic shift of words   1cshid 32887
            21.3.9  Extensible Structures   ressplusf 32891
                  21.3.9.1  Structure restriction operator   ressplusf 32891
                  21.3.9.2  The opposite group   oppgle 32894
                  21.3.9.3  Posets   ressprs 32896
                  21.3.9.4  Complete lattices   clatp0cl 32908
                  21.3.9.5  Order Theory   cmnt 32910
                  21.3.9.6  Chains   cchn 32936
                  21.3.9.7  Extended reals Structure - misc additions   ax-xrssca 32948
                  21.3.9.8  The extended nonnegative real numbers commutative monoid   xrge0base 32958
            21.3.10  Algebra   mndcld 32969
                  21.3.10.1  Monoids   mndcld 32969
                  21.3.10.2  Monoids Homomorphisms   abliso 32983
                  21.3.10.3  Groups - misc additions   grpsubcld 32987
                  21.3.10.4  Finitely supported group sums - misc additions   gsumsubg 32992
                  21.3.10.5  Group or monoid sums over words   gsumwun 33011
                  21.3.10.6  Centralizers and centers - misc additions   cntzun 33014
                  21.3.10.7  Totally ordered monoids and groups   comnd 33017
                  21.3.10.8  The symmetric group   symgfcoeu 33045
                  21.3.10.9  Transpositions   pmtridf1o 33057
                  21.3.10.10  Permutation Signs   psgnid 33060
                  21.3.10.11  Permutation cycles   ctocyc 33069
                  21.3.10.12  The Alternating Group   evpmval 33108
                  21.3.10.13  Signum in an ordered monoid   csgns 33121
                  21.3.10.14  Fixed points   cfxp 33126
                  21.3.10.15  The Archimedean property for generic ordered algebraic structures   cinftm 33136
                  21.3.10.16  Semiring left modules   cslmd 33159
                  21.3.10.17  Simple groups   prmsimpcyc 33187
                  21.3.10.18  Rings - misc additions   ringdi22 33188
                  21.3.10.19  Subrings generated by a set   elrgspnlem1 33199
                  21.3.10.20  The zero ring   irrednzr 33207
                  21.3.10.21  Localization of rings   cerl 33210
                  21.3.10.22  Integral Domains   domnmuln0rd 33231
                  21.3.10.23  Euclidean Domains   ceuf 33244
                  21.3.10.24  Division Rings   ringinveu 33250
                  21.3.10.25  The field of rational numbers   qfld 33253
                  21.3.10.26  Subfields   subsdrg 33254
                  21.3.10.27  Field of fractions   cfrac 33258
                  21.3.10.28  Field extensions generated by a set   cfldgen 33266
                  21.3.10.29  Totally ordered rings and fields   corng 33279
                  21.3.10.30  Ring homomorphisms - misc additions   rhmdvd 33302
                  21.3.10.31  Scalar restriction operation   cresv 33304
                  21.3.10.32  The commutative ring of gaussian integers   gzcrng 33319
                  21.3.10.33  The archimedean ordered field of real numbers   cnfldfld 33320
                  21.3.10.34  The quotient map and quotient modules   qusker 33326
                  21.3.10.35  The ring of integers modulo ` N `   znfermltl 33343
                  21.3.10.36  Independent sets and families   islinds5 33344
                  21.3.10.37  Ring associates, ring units   dvdsruassoi 33361
                  ​*21.3.10.38  Subgroup sum / Sumset / Minkowski sum   elgrplsmsn 33367
                  21.3.10.39  The quotient map   quslsm 33382
                  21.3.10.40  Ideals   lidlmcld 33396
                  21.3.10.41  Prime Ideals   cprmidl 33412
                  21.3.10.42  Maximal Ideals   cmxidl 33436
                  21.3.10.43  The semiring of ideals of a ring   cidlsrg 33477
                  21.3.10.44  Prime Elements   rprmval 33493
                  21.3.10.45  Unique factorization domains   cufd 33515
                  21.3.10.46  The ring of integers   zringidom 33528
                  21.3.10.47  Univariate Polynomials   0ringmon1p 33532
                  21.3.10.48  Polynomial quotient and polynomial remainder   q1pdir 33574
                  21.3.10.49  The subring algebra   sra1r 33583
                  21.3.10.50  Division Ring Extensions   drgext0g 33591
                  21.3.10.51  Vector Spaces   lvecdimfi 33597
                  21.3.10.52  Vector Space Dimension   cldim 33600
            21.3.11  Field Extensions   cfldext 33640
                  21.3.11.1  Algebraic numbers   cirng 33684
                  21.3.11.2  Algebraic extensions   calgext 33693
                  21.3.11.3  Minimal polynomials   cminply 33695
                  21.3.11.4  Quadratic Field Extensions   rtelextdg2lem 33722
                  21.3.11.5  Towers of quadratic extentions   fldext2chn 33724
            ​*21.3.12  Constructible Numbers   cconstr 33725
                  21.3.12.1  Impossible constructions   2sqr3minply 33776
            21.3.13  Matrices   csmat 33789
                  21.3.13.1  Submatrices   csmat 33789
                  21.3.13.2  Matrix literals   clmat 33807
                  21.3.13.3  Laplace expansion of determinants   mdetpmtr1 33819
            21.3.14  Topology   ist0cld 33829
                  21.3.14.1  Open maps   txomap 33830
                  21.3.14.2  Topology of the unit circle   qtopt1 33831
                  21.3.14.3  Refinements   reff 33835
                  21.3.14.4  Open cover refinement property   ccref 33838
                  21.3.14.5  Lindelöf spaces   cldlf 33848
                  21.3.14.6  Paracompact spaces   cpcmp 33851
                  ​*21.3.14.7  Spectrum of a ring   crspec 33858
                  21.3.14.8  Pseudometrics   cmetid 33882
                  21.3.14.9  Continuity - misc additions   hauseqcn 33894
                  21.3.14.10  Topology of the closed unit interval   elunitge0 33895
                  21.3.14.11  Topology of ` ( RR X. RR ) `   unicls 33899
                  21.3.14.12  Order topology - misc. additions   cnvordtrestixx 33909
                  21.3.14.13  Continuity in topological spaces - misc. additions   mndpluscn 33922
                  21.3.14.14  Topology of the extended nonnegative real numbers ordered monoid   xrge0hmph 33928
                  21.3.14.15  Limits - misc additions   lmlim 33943
                  21.3.14.16  Univariate polynomials   pl1cn 33951
            21.3.15  Uniform Stuctures and Spaces   chcmp 33952
                  21.3.15.1  Hausdorff uniform completion   chcmp 33952
            21.3.16  Topology and algebraic structures   zringnm 33954
                  21.3.16.1  The norm on the ring of the integer numbers   zringnm 33954
                  21.3.16.2  Topological ` ZZ ` -modules   zlm0 33956
                  21.3.16.3  Canonical embedding of the field of the rational numbers into a division ring   cqqh 33966
                  21.3.16.4  Canonical embedding of the real numbers into a complete ordered field   crrh 33989
                  21.3.16.5  Embedding from the extended real numbers into a complete lattice   cxrh 34012
                  21.3.16.6  Canonical embeddings into the ordered field of the real numbers   zrhre 34015
                  ​*21.3.16.7  Topological Manifolds   cmntop 34018
                  21.3.16.8  Extended sum   cesum 34023
            21.3.17  Mixed Function/Constant operation   cofc 34091
            21.3.18  Abstract measure   csiga 34104
                  21.3.18.1  Sigma-Algebra   csiga 34104
                  21.3.18.2  Generated sigma-Algebra   csigagen 34134
                  ​*21.3.18.3  lambda and pi-Systems, Rings of Sets   ispisys 34148
                  21.3.18.4  The Borel algebra on the real numbers   cbrsiga 34177
                  21.3.18.5  Product Sigma-Algebra   csx 34184
                  21.3.18.6  Measures   cmeas 34191
                  21.3.18.7  The counting measure   cntmeas 34222
                  21.3.18.8  The Lebesgue measure - misc additions   voliune 34225
                  21.3.18.9  The Dirac delta measure   cdde 34228
                  21.3.18.10  The 'almost everywhere' relation   cae 34233
                  21.3.18.11  Measurable functions   cmbfm 34245
                  21.3.18.12  Borel Algebra on ` ( RR X. RR ) `   br2base 34266
                  ​*21.3.18.13  Caratheodory's extension theorem   coms 34288
            21.3.19  Integration   itgeq12dv 34323
                  21.3.19.1  Lebesgue integral - misc additions   itgeq12dv 34323
                  21.3.19.2  Bochner integral   citgm 34324
            21.3.20  Euler's partition theorem   oddpwdc 34351
            21.3.21  Sequences defined by strong recursion   csseq 34380
            21.3.22  Fibonacci Numbers   cfib 34393
            21.3.23  Probability   cprb 34404
                  21.3.23.1  Probability Theory   cprb 34404
                  21.3.23.2  Conditional Probabilities   ccprob 34428
                  21.3.23.3  Real-valued Random Variables   crrv 34437
                  21.3.23.4  Preimage set mapping operator   corvc 34453
                  21.3.23.5  Distribution Functions   orvcelval 34466
                  21.3.23.6  Cumulative Distribution Functions   orvclteel 34470
                  21.3.23.7  Probabilities - example   coinfliplem 34476
                  21.3.23.8  Bertrand's Ballot Problem   ballotlemoex 34483
            21.3.24  Signum (sgn or sign) function - misc. additions   fzssfzo 34536
                  21.3.24.1  Operations on words   ccatmulgnn0dir 34539
            21.3.25  Polynomials with real coefficients - misc additions   plymul02 34543
            21.3.26  Descartes's rule of signs   signspval 34549
                  21.3.26.1  Sign changes in a word over real numbers   signspval 34549
                  21.3.26.2  Counting sign changes in a word over real numbers   signslema 34559
            21.3.27  Number Theory   iblidicc 34589
                  21.3.27.1  Representations of a number as sums of integers   crepr 34605
                  21.3.27.2  Vinogradov Trigonometric Sums and the Circle Method   cvts 34632
                  21.3.27.3  The Ternary Goldbach Conjecture: Final Statement   ax-hgt749 34641
            21.3.28  Elementary Geometry   cstrkg2d 34661
                  ​*21.3.28.1  Two-dimensional geometry   cstrkg2d 34661
                  21.3.28.2  Outer Five Segment (not used, no need to move to main)   cafs 34666
            ​*21.3.29  LeftPad Project   clpad 34671
      ​​*21.4  Mathbox for Jonathan Ben-Naim
            21.4.1  First-order logic and set theory   bnj170 34694
            21.4.2  Well founded induction and recursion   bnj110 34854
            21.4.3  The existence of a minimal element in certain classes   bnj69 35006
            21.4.4  Well-founded induction   bnj1204 35008
            21.4.5  Well-founded recursion, part 1 of 3   bnj60 35058
            21.4.6  Well-founded recursion, part 2 of 3   bnj1500 35064
            21.4.7  Well-founded recursion, part 3 of 3   bnj1522 35068
      ​21.5  Mathbox for BTernaryTau
            21.5.1  First-order logic   nfan1c 35069
                  21.5.1.1  Auxiliary axiom schemes   nfan1c 35069
            21.5.2  ZF set theory   exdifsn 35075
                  21.5.2.1  Finitism   prcinf 35090
                  21.5.2.2  Global choice   gblacfnacd 35095
            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 35677
      ​21.11  Mathbox for Scott Fenton
            21.11.1  ZFC Axioms in primitive form   axextprim 35683
            21.11.2  Untangled classes   untelirr 35690
            21.11.3  Extra propositional calculus theorems   3jaodd 35697
            21.11.4  Misc. Useful Theorems   nepss 35700
            21.11.5  Properties of real and complex numbers   sqdivzi 35710
            21.11.6  Infinite products   iprodefisumlem 35722
            21.11.7  Factorial limits   faclimlem1 35725
            21.11.8  Greatest common divisor and divisibility   gcd32 35731
            21.11.9  Properties of relationships   dftr6 35733
            21.11.10  Properties of functions and mappings   funpsstri 35748
            21.11.11  Set induction (or epsilon induction)   setinds 35761
            21.11.12  Ordinal numbers   elpotr 35764
            21.11.13  Defined equality axioms   axextdfeq 35780
            21.11.14  Hypothesis builders   hbntg 35788
            21.11.15  Well-founded zero, successor, and limits   cwsuc 35793
            21.11.16  Quantifier-free definitions   ctxp 35813
            21.11.17  Alternate ordered pairs   caltop 35939
            21.11.18  Geometry in the Euclidean space   cofs 35965
                  21.11.18.1  Congruence properties   cofs 35965
                  21.11.18.2  Betweenness properties   btwntriv2 35995
                  21.11.18.3  Segment Transportation   ctransport 36012
                  21.11.18.4  Properties relating betweenness and congruence   cifs 36018
                  21.11.18.5  Connectivity of betweenness   btwnconn1lem1 36070
                  21.11.18.6  Segment less than or equal to   csegle 36089
                  21.11.18.7  Outside-of relationship   coutsideof 36102
                  21.11.18.8  Lines and Rays   cline2 36117
            21.11.19  Forward difference   cfwddif 36141
            21.11.20  Rank theorems   rankung 36149
            21.11.21  Hereditarily Finite Sets   chf 36155
      ​21.12  Mathbox for Gino Giotto
            21.12.1  Equality theorems.   rmoeqi 36170
                  21.12.1.1  Inference versions.   rmoeqi 36170
                  21.12.1.2  Deduction versions.   rmoeqdv 36195
            21.12.2  Change bound variables.   in-ax8 36207
                  21.12.2.1  Change bound variables and domains.   cbvralvw2 36209
                  21.12.2.2  Change bound variables, deduction versions.   cbvmodavw 36233
                  21.12.2.3  Change bound variables and domains, deduction versions.   cbvrmodavw2 36266
            21.12.3  Study of ax-mulf usage.   mpomulnzcnf 36282
      ​21.13  Mathbox for Jeff Hankins
            21.13.1  Miscellany   a1i14 36283
            21.13.2  Basic topological facts   topbnd 36307
            21.13.3  Topology of the real numbers   ivthALT 36318
            21.13.4  Refinements   cfne 36319
            21.13.5  Neighborhood bases determine topologies   neibastop1 36342
            21.13.6  Lattice structure of topologies   topmtcl 36346
            21.13.7  Filter bases   fgmin 36353
            21.13.8  Directed sets, nets   tailfval 36355
      ​21.14  Mathbox for Anthony Hart
            21.14.1  Propositional Calculus   tb-ax1 36366
            21.14.2  Predicate Calculus   nalfal 36386
            21.14.3  Miscellaneous single axioms   meran1 36394
            21.14.4  Connective Symmetry   negsym1 36400
      ​21.15  Mathbox for Chen-Pang He
            21.15.1  Ordinal topology   ontopbas 36411
      ​21.16  Mathbox for Jeff Hoffman
            21.16.1  Inferences for finite induction on generic function values   fveleq 36434
            21.16.2  gdc.mm   nnssi2 36438
      ​21.17  Mathbox for Matthew House
            21.17.1  Relations on well-ordered indexed unions   weiunlem1 36445
      ​21.18  Mathbox for Asger C. Ipsen
            21.18.1  Continuous nowhere differentiable functions   dnival 36454
      ​​*21.19  Mathbox for BJ
            *21.19.1  Propositional calculus   bj-mp2c 36523
                  *21.19.1.1  Derived rules of inference   bj-mp2c 36523
                  ​*21.19.1.2  A syntactic theorem   bj-0 36525
                  21.19.1.3  Minimal implicational calculus   bj-a1k 36527
                  ​*21.19.1.4  Positive calculus   bj-syl66ib 36538
                  21.19.1.5  Implication and negation   bj-con2com 36544
                  ​*21.19.1.6  Disjunction   bj-jaoi1 36554
                  ​*21.19.1.7  Logical equivalence   bj-dfbi4 36556
                  21.19.1.8  The conditional operator for propositions   bj-consensus 36561
                  ​*21.19.1.9  Propositional calculus: miscellaneous   bj-imbi12 36566
            ​*21.19.2  Modal logic   bj-axdd2 36575
            ​*21.19.3  Provability logic   cprvb 36580
            ​*21.19.4  First-order logic   bj-genr 36589
                  21.19.4.1  Adding ax-gen   bj-genr 36589
                  21.19.4.2  Adding ax-4   bj-2alim 36593
                  21.19.4.3  Adding ax-5   bj-ax12wlem 36627
                  21.19.4.4  Equality and substitution   bj-ssbeq 36636
                  21.19.4.5  Adding ax-6   bj-spimvwt 36652
                  21.19.4.6  Adding ax-7   bj-cbvexw 36659
                  21.19.4.7  Membership predicate, ax-8 and ax-9   bj-ax89 36661
                  21.19.4.8  Adding ax-11   bj-alcomexcom 36663
                  21.19.4.9  Adding ax-12   axc11n11 36665
                  21.19.4.10  Nonfreeness   wnnf 36706
                  21.19.4.11  Adding ax-13   bj-axc10 36766
                  ​*21.19.4.12  Removing dependencies on ax-13 (and ax-11)   bj-axc10v 36776
                  ​*21.19.4.13  Distinct var metavariables   bj-hbaeb2 36801
                  ​*21.19.4.14  Around ~ equsal   bj-equsal1t 36805
                  ​*21.19.4.15  Some Principia Mathematica proofs   stdpc5t 36810
                  21.19.4.16  Alternate definition of substitution   bj-sbsb 36820
                  21.19.4.17  Lemmas for substitution   bj-sbf3 36822
                  21.19.4.18  Existential uniqueness   bj-eu3f 36824
                  ​*21.19.4.19  First-order logic: miscellaneous   bj-sblem1 36825
            21.19.5  Set theory   eliminable1 36842
                  ​*21.19.5.1  Eliminability of class terms   eliminable1 36842
                  ​*21.19.5.2  Classes without the axiom of extensionality   bj-denoteslem 36854
                  21.19.5.3  Characterization among sets versus among classes   elelb 36880
                  ​*21.19.5.4  The nonfreeness quantifier for classes   bj-nfcsym 36882
                  ​*21.19.5.5  Lemmas for class substitution   bj-sbeqALT 36883
                  21.19.5.6  Removing some axiom requirements and disjoint variable conditions   bj-exlimvmpi 36894
                  ​*21.19.5.7  Class abstractions   bj-elabd2ALT 36908
                  21.19.5.8  Generalized class abstractions   bj-cgab 36916
                  ​*21.19.5.9  Restricted nonfreeness   wrnf 36924
                  ​*21.19.5.10  Russell's paradox   bj-ru1 36926
                  21.19.5.11  Curry's paradox in set theory   currysetlem 36928
                  ​*21.19.5.12  Some disjointness results   bj-n0i 36934
                  ​*21.19.5.13  Complements on direct products   bj-xpimasn 36938
                  ​*21.19.5.14  "Singletonization" and tagging   bj-snsetex 36946
                  ​*21.19.5.15  Tuples of classes   bj-cproj 36973
                  ​*21.19.5.16  Set theory: elementary operations relative to a universe   bj-rcleqf 37008
                  ​*21.19.5.17  Axioms for finite unions   bj-abex 37013
                  ​*21.19.5.18  Set theory: miscellaneous   eleq2w2ALT 37030
                  ​*21.19.5.19  Evaluation at a class   bj-evaleq 37055
                  21.19.5.20  Elementwise operations   celwise 37062
                  ​*21.19.5.21  Elementwise intersection (families of sets induced on a subset)   bj-rest00 37064
                  21.19.5.22  Moore collections (complements)   bj-raldifsn 37083
                  21.19.5.23  Maps-to notation for functions with three arguments   bj-0nelmpt 37099
                  ​*21.19.5.24  Currying   csethom 37105
                  ​*21.19.5.25  Setting components of extensible structures   cstrset 37117
            ​*21.19.6  Extended real and complex numbers, real and complex projective lines   bj-nfald 37120
                  21.19.6.1  Complements on class abstractions of ordered pairs and binary relations   bj-nfald 37120
                  ​*21.19.6.2  Identity relation (complements)   bj-opabssvv 37133
                  ​*21.19.6.3  Functionalized identity (diagonal in a Cartesian square)   cdiag2 37155
                  ​*21.19.6.4  Direct image and inverse image   cimdir 37161
                  ​*21.19.6.5  Extended numbers and projective lines as sets   cfractemp 37179
                  ​*21.19.6.6  Addition and opposite   caddcc 37220
                  ​*21.19.6.7  Order relation on the extended reals   cltxr 37224
                  ​*21.19.6.8  Argument, multiplication and inverse   carg 37226
                  21.19.6.9  The canonical bijection from the finite ordinals   ciomnn 37232
                  21.19.6.10  Divisibility   cnnbar 37243
            ​*21.19.7  Monoids   bj-smgrpssmgm 37251
                  ​*21.19.7.1  Finite sums in monoids   cfinsum 37266
            ​*21.19.8  Affine, Euclidean, and Cartesian geometry   bj-fvimacnv0 37269
                  *21.19.8.1  Real vector spaces   bj-fvimacnv0 37269
                  ​*21.19.8.2  Complex numbers (supplements)   bj-subcom 37291
                  ​*21.19.8.3  Barycentric coordinates   bj-bary1lem 37293
            21.19.9  Monoid of endomorphisms   cend 37296
      ​21.20  Mathbox for Jim Kingdon
            21.20.1  Circle constant   taupilem3 37302
            21.20.2  Number theory   dfgcd3 37307
            21.20.3  Real numbers   irrdifflemf 37308
      ​21.21  Mathbox for ML
            21.21.1  Miscellaneous   csbrecsg 37311
            21.21.2  Cartesian exponentiation   cfinxp 37366
            21.21.3  Topology   iunctb2 37386
                  ​*21.21.3.1  Pi-base theorems   pibp16 37396
      ​21.22  Mathbox for Wolf Lammen
            21.22.1  1. Bootstrapping   wl-section-boot 37405
            21.22.2  Implication chains   wl-section-impchain 37429
            21.22.3  Theorems around the conditional operator   wl-ifp-ncond1 37447
            21.22.4  Alternative development of hadd, cadd   wl-df-3xor 37451
            21.22.5  An alternative axiom ~ ax-13   ax-wl-13v 37476
            21.22.6  Bootstrapping set theory with classes   wl-cleq-0 37478
            21.22.7  Other stuff   wl-mps 37490
      ​21.23  Mathbox for Brendan Leahy
      ​21.24  Mathbox for Jeff Madsen
            21.24.1  Logic and set theory   unirep 37703
            21.24.2  Real and complex numbers; integers   filbcmb 37729
            21.24.3  Sequences and sums   sdclem2 37731
            21.24.4  Topology   subspopn 37741
            21.24.5  Metric spaces   metf1o 37744
            21.24.6  Continuous maps and homeomorphisms   constcncf 37751
            21.24.7  Boundedness   ctotbnd 37755
            21.24.8  Isometries   cismty 37787
            21.24.9  Heine-Borel Theorem   heibor1lem 37798
            21.24.10  Banach Fixed Point Theorem   bfplem1 37811
            21.24.11  Euclidean space   crrn 37814
            21.24.12  Intervals (continued)   ismrer1 37827
            21.24.13  Operation properties   cass 37831
            21.24.14  Groups and related structures   cmagm 37837
            21.24.15  Group homomorphism and isomorphism   cghomOLD 37872
            21.24.16  Rings   crngo 37883
            21.24.17  Division Rings   cdrng 37937
            21.24.18  Ring homomorphisms   crngohom 37949
            21.24.19  Commutative rings   ccm2 37978
            21.24.20  Ideals   cidl 37996
            21.24.21  Prime rings and integral domains   cprrng 38035
            21.24.22  Ideal generators   cigen 38048
      ​21.25  Mathbox for Giovanni Mascellani
            ​*21.25.1  Tools for automatic proof building   efald2 38067
            ​*21.25.2  Tseitin axioms   fald 38118
            ​*21.25.3  Equality deductions   iuneq2f 38145
            ​*21.25.4  Miscellanea   orcomdd 38156
      ​21.26  Mathbox for Peter Mazsa
            21.26.1  Notations   cxrn 38163
            21.26.2  Preparatory theorems   el2v1 38206
            21.26.3  Range Cartesian product   df-xrn 38348
            21.26.4  Cosets by ` R `   df-coss 38397
            21.26.5  Relations   df-rels 38471
            21.26.6  Subset relations   df-ssr 38484
            21.26.7  Reflexivity   df-refs 38496
            21.26.8  Converse reflexivity   df-cnvrefs 38511
            21.26.9  Symmetry   df-syms 38528
            21.26.10  Reflexivity and symmetry   symrefref2 38549
            21.26.11  Transitivity   df-trs 38558
            21.26.12  Equivalence relations   df-eqvrels 38570
            21.26.13  Redundancy   df-redunds 38609
            21.26.14  Domain quotients   df-dmqss 38624
            21.26.15  Equivalence relations on domain quotients   df-ers 38650
            21.26.16  Functions   df-funss 38667
            21.26.17  Disjoints vs. converse functions   df-disjss 38690
            21.26.18  Antisymmetry   df-antisymrel 38747
            21.26.19  Partitions: disjoints on domain quotients   df-parts 38752
            21.26.20  Partition-Equivalence Theorems   disjim 38768
      ​21.27  Mathbox for Rodolfo Medina
            21.27.1  Partitions   prtlem60 38841
      ​​*21.28  Mathbox for Norm Megill
            *21.28.1  Obsolete schemes ax-c4,c5,c7,c10,c11,c11n,c15,c9,c14,c16   ax-c5 38871
            ​*21.28.2  Rederive new axioms ax-4, ax-10, ax-6, ax-12, ax-13 from old   axc5 38881
            ​*21.28.3  Legacy theorems using obsolete axioms   ax5ALT 38895
            21.28.4  Experiments with weak deduction theorem   elimhyps 38949
            21.28.5  Miscellanea   cnaddcom 38960
            21.28.6  Atoms, hyperplanes, and covering in a left vector space (or module)   clsa 38962
            21.28.7  Functionals and kernels of a left vector space (or module)   clfn 39045
            21.28.8  Opposite rings and dual vector spaces   cld 39111
            21.28.9  Ortholattices and orthomodular lattices   cops 39160
            21.28.10  Atomic lattices with covering property   ccvr 39250
            21.28.11  Hilbert lattices   chlt 39338
            21.28.12  Projective geometries based on Hilbert lattices   clln 39480
            21.28.13  Construction of a vector space from a Hilbert lattice   cdlema1N 39780
            21.28.14  Construction of involution and inner product from a Hilbert lattice   clpoN 41469
      ​21.29  Mathbox for metakunt
            21.29.1  Commutative Semiring   ccsrg 41951
            21.29.2  General helpful statements   rhmzrhval 41954
            21.29.3  Some gcd and lcm results   12gcd5e1 41986
            21.29.4  Least common multiple inequality theorem   3factsumint1 42004
            21.29.5  Logarithm inequalities   3exp7 42036
            21.29.6  Miscellaneous results for AKS formalisation   intlewftc 42044
            21.29.7  Sticks and stones   sticksstones1 42129
            21.29.8  Continuation AKS   aks6d1c6lem1 42153
      ​21.30  Mathbox for Steven Nguyen
            21.30.1  Utility theorems   jarrii 42188
            ​*21.30.2  Arithmetic theorems   c0exALT 42235
            21.30.3  Exponents and divisibility   oexpreposd 42305
            21.30.4  Trigonometry and Calculus   tanhalfpim 42332
            ​*21.30.5  Independence of ax-mulcom   cresub 42348
            21.30.6  Structures   sn-base0 42476
            ​*21.30.7  Projective spaces   cprjsp 42582
            21.30.8  Basic reductions for Fermat's Last Theorem   dffltz 42615
            ​*21.30.9  Exemplar theorems   iddii 42645
                  ​*21.30.9.1  Standard replacements of ax-10 , ax-11 , ax-12   nfa1w 42656
      ​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 42673
            21.33.2  Additional theory of functions   imaiinfv 42674
            21.33.3  Additional topology   elrfi 42675
            21.33.4  Characterization of closure operators. Kuratowski closure axioms   ismrcd1 42679
            21.33.5  Algebraic closure systems   cnacs 42683
            21.33.6  Miscellanea 1. Map utilities   constmap 42694
            21.33.7  Miscellanea for polynomials   mptfcl 42701
            21.33.8  Multivariate polynomials over the integers   cmzpcl 42702
            21.33.9  Miscellanea for Diophantine sets 1   coeq0i 42734
            21.33.10  Diophantine sets 1: definitions   cdioph 42736
            21.33.11  Diophantine sets 2 miscellanea   ellz1 42748
            21.33.12  Diophantine sets 2: union and intersection. Monotone Boolean algebra   diophin 42753
            21.33.13  Diophantine sets 3: construction   diophrex 42756
            21.33.14  Diophantine sets 4 miscellanea   2sbcrex 42765
            21.33.15  Diophantine sets 4: Quantification   rexrabdioph 42775
            21.33.16  Diophantine sets 5: Arithmetic sets   rabdiophlem1 42782
            21.33.17  Diophantine sets 6: reusability. renumbering of variables   eldioph4b 42792
            21.33.18  Pigeonhole Principle and cardinality helpers   fphpd 42797
            21.33.19  A non-closed set of reals is infinite   rencldnfilem 42801
            21.33.20  Lagrange's rational approximation theorem   irrapxlem1 42803
            21.33.21  Pell equations 1: A nontrivial solution always exists   pellexlem1 42810
            21.33.22  Pell equations 2: Algebraic number theory of the solution set   csquarenn 42817
            21.33.23  Pell equations 3: characterizing fundamental solution   infmrgelbi 42859
            ​*21.33.24  Logarithm laws generalized to an arbitrary base   reglogcl 42871
            21.33.25  Pell equations 4: the positive solution group is infinite cyclic   pellfund14 42879
            21.33.26  X and Y sequences 1: Definition and recurrence laws   crmx 42881
            21.33.27  Ordering and induction lemmas for the integers   monotuz 42923
            21.33.28  X and Y sequences 2: Order properties   rmxypos 42929
            21.33.29  Congruential equations   congtr 42947
            21.33.30  Alternating congruential equations   acongid 42957
            21.33.31  Additional theorems on integer divisibility   coprmdvdsb 42967
            21.33.32  X and Y sequences 3: Divisibility properties   jm2.18 42970
            21.33.33  X and Y sequences 4: Diophantine representability of Y   jm2.27a 42987
            21.33.34  X and Y sequences 5: Diophantine representability of X, ^, _C   rmxdiophlem 42997
            21.33.35  Uncategorized stuff not associated with a major project   setindtr 43006
            21.33.36  More equivalents of the Axiom of Choice   axac10 43015
            21.33.37  Finitely generated left modules   clfig 43049
            21.33.38  Noetherian left modules I   clnm 43057
            21.33.39  Addenda for structure powers   pwssplit4 43071
            21.33.40  Every set admits a group structure iff choice   unxpwdom3 43077
            21.33.41  Noetherian rings and left modules II   clnr 43091
            21.33.42  Hilbert's Basis Theorem   cldgis 43103
            21.33.43  Additional material on polynomials [DEPRECATED]   cmnc 43113
            21.33.44  Degree and minimal polynomial of algebraic numbers   cdgraa 43122
            21.33.45  Algebraic integers I   citgo 43139
            21.33.46  Endomorphism algebra   cmend 43153
            21.33.47  Cyclic groups and order   idomodle 43173
            21.33.48  Cyclotomic polynomials   ccytp 43179
            21.33.49  Miscellaneous topology   fgraphopab 43185
      ​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 43199
            21.36.2  Natural addition of Cantor normal forms   oawordex2 43308
            21.36.3  Surreal Contributions   abeqabi 43390
            21.36.4  Short Studies   nlimsuc 43423
                  21.36.4.1  Additional work on conditional logical operator   ifpan123g 43441
                  21.36.4.2  Sophisms   rp-fakeimass 43494
                  ​*21.36.4.3  Finite Sets   rp-isfinite5 43499
                  21.36.4.4  General Observations   intabssd 43501
                  21.36.4.5  Infinite Sets   pwelg 43542
                  ​*21.36.4.6  Finite intersection property   fipjust 43547
                  21.36.4.7  RP ADDTO: Subclasses and subsets   rababg 43556
                  21.36.4.8  RP ADDTO: The intersection of a class   elinintab 43557
                  21.36.4.9  RP ADDTO: Theorems requiring subset and intersection existence   elinintrab 43559
                  21.36.4.10  RP ADDTO: Relations   xpinintabd 43562
                  ​*21.36.4.11  RP ADDTO: Functions   elmapintab 43578
                  ​*21.36.4.12  RP ADDTO: Finite induction (for finite ordinals)   cnvcnvintabd 43582
                  21.36.4.13  RP ADDTO: First and second members of an ordered pair   elcnvlem 43583
                  21.36.4.14  RP ADDTO: The reflexive and transitive properties of relations   undmrnresiss 43586
                  21.36.4.15  RP ADDTO: Basic properties of closures   cleq2lem 43590
                  21.36.4.16  RP REPLACE: Definitions and basic properties of transitive closures   trcleq2lemRP 43612
                  ​*21.36.4.17  Additions for square root; absolute value   sqrtcvallem1 43613
            21.36.5  Additional statements on relations and subclasses   al3im 43629
                  21.36.5.1  Transitive relations (not to be confused with transitive classes).   trrelind 43647
                  21.36.5.2  Reflexive closures   crcl 43654
                  ​*21.36.5.3  Finite relationship composition.   relexp2 43659
                  21.36.5.4  Transitive closure of a relation   dftrcl3 43702
                  ​*21.36.5.5  Adapted from Frege   frege77d 43728
            ​*21.36.6  Propositions from _Begriffsschrift_   dfxor4 43748
                  *21.36.6.1  _Begriffsschrift_ Chapter I   dfxor4 43748
                  ​*21.36.6.2  _Begriffsschrift_ Notation hints   whe 43754
                  21.36.6.3  _Begriffsschrift_ Chapter II Implication   ax-frege1 43772
                  21.36.6.4  _Begriffsschrift_ Chapter II Implication and Negation   axfrege28 43811
                  ​*21.36.6.5  _Begriffsschrift_ Chapter II with logical equivalence   axfrege52a 43838
                  21.36.6.6  _Begriffsschrift_ Chapter II with equivalence of sets   axfrege52c 43869
                  ​*21.36.6.7  _Begriffsschrift_ Chapter II with equivalence of classes   frege53c 43896
                  ​*21.36.6.8  _Begriffsschrift_ Chapter III Properties hereditary in a sequence   dffrege69 43914
                  ​*21.36.6.9  _Begriffsschrift_ Chapter III Following in a sequence   dffrege76 43921
                  ​*21.36.6.10  _Begriffsschrift_ Chapter III Member of sequence   dffrege99 43944
                  ​*21.36.6.11  _Begriffsschrift_ Chapter III Single-valued procedures   dffrege115 43960
            ​*21.36.7  Exploring Topology via Seifert and Threlfall   enrelmap 43979
                  *21.36.7.1  Equinumerosity of sets of relations and maps   enrelmap 43979
                  ​*21.36.7.2  Generic Pseudoclosure Spaces, Pseudointerior Spaces, and Pseudoneighborhoods   or3or 44005
                  ​*21.36.7.3  Generic Neighborhood Spaces   gneispa 44112
            ​*21.36.8  Exploring Higher Homotopy via Kerodon   k0004lem1 44129
                  *21.36.8.1  Simplicial Sets   k0004lem1 44129
      ​21.37  Mathbox for Stanislas Polu
            21.37.1  IMO Problems   wwlemuld 44138
                  21.37.1.1  IMO 1972 B2   wwlemuld 44138
            ​*21.37.2  INT Inequalities Proof Generator   int-addcomd 44155
            ​*21.37.3  N-Digit Addition Proof Generator   unitadd 44177
            21.37.4  AM-GM (for k = 2,3,4)   gsumws3 44178
      ​21.38  Mathbox for Rohan Ridenour
            21.38.1  Misc   spALT 44183
            21.38.2  Monoid rings   cmnring 44193
            21.38.3  Shorter primitive equivalent of ax-groth   gru0eld 44211
                  21.38.3.1  Grothendieck universes are closed under collection   gru0eld 44211
                  21.38.3.2  Minimal universes   ismnu 44243
                  21.38.3.3  Primitive equivalent of ax-groth   expandan 44270
      ​21.39  Mathbox for Steve Rodriguez
            21.39.1  Miscellanea   nanorxor 44287
            21.39.2  Ratio test for infinite series convergence and divergence   dvgrat 44294
            21.39.3  Multiples   reldvds 44297
            21.39.4  Function operations   caofcan 44305
            21.39.5  Calculus   lhe4.4ex1a 44311
            21.39.6  The generalized binomial coefficient operation   cbcc 44318
            21.39.7  Binomial series   uzmptshftfval 44328
      ​21.40  Mathbox for Andrew Salmon
            21.40.1  Principia Mathematica * 10   pm10.12 44340
            21.40.2  Principia Mathematica * 11   2alanimi 44354
            21.40.3  Predicate Calculus   sbeqal1 44380
            21.40.4  Principia Mathematica * 13 and * 14   pm13.13a 44389
            21.40.5  Set Theory   elnev 44420
            21.40.6  Arithmetic   addcomgi 44438
            21.40.7  Geometry   cplusr 44439
      ​​*21.41  Mathbox for Alan Sare
            21.41.1  Auxiliary theorems for the Virtual Deduction tool   idiALT 44461
            21.41.2  Supplementary unification deductions   bi1imp 44465
            21.41.3  Conventional Metamath proofs, some derived from VD proofs   iidn3 44484
            21.41.4  What is Virtual Deduction?   wvd1 44552
            21.41.5  Virtual Deduction Theorems   df-vd1 44553
            21.41.6  Theorems proved using Virtual Deduction   trsspwALT 44800
            21.41.7  Theorems proved using Virtual Deduction with mmj2 assistance   simplbi2VD 44828
            21.41.8  Virtual Deduction transcriptions of textbook proofs   sb5ALTVD 44895
            21.41.9  Theorems proved using conjunction-form Virtual Deduction   elpwgdedVD 44899
            21.41.10  Theorems with a VD proof in conventional notation derived from a VD proof   suctrALT3 44906
            ​*21.41.11  Theorems with a proof in conventional notation derived from a VD proof   notnotrALT2 44909
      ​21.42  Mathbox for Eric Schmidt
            21.42.1  Miscellany   rspesbcd 44920
            21.42.2  Study of dfbi1ALT   dfbi1ALTa 44922
            21.42.3  Relation-preserving functions   wrelp 44925
            21.42.4  Orbits   orbitex 44938
            21.42.5  Well-founded sets   trwf 44942
            21.42.6  Absoluteness in transitive models   ralabso 44951
            21.42.7  Lemmas for showing axioms hold in models   traxext 44960
            21.42.8  The class of well-founded sets is a model for ZFC   wfaxext 44976
            21.42.9  Permutation models   brpermmodel 44986
      ​21.43  Mathbox for Glauco Siliprandi
            21.43.1  Miscellanea   evth2f 45002
            21.43.2  Functions   fnresdmss 45155
            21.43.3  Ordering on real numbers - Real and complex numbers basic operations   sub2times 45264
            21.43.4  Real intervals   gtnelioc 45482
            21.43.5  Finite sums   fsummulc1f 45562
            21.43.6  Finite multiplication of numbers and finite multiplication of functions   fmul01 45571
            21.43.7  Limits   clim1fr1 45592
                  21.43.7.1  Inferior limit (lim inf)   clsi 45742
                  ​*21.43.7.2  Limits for sequences of extended real numbers   clsxlim 45809
            21.43.8  Trigonometry   coseq0 45855
            21.43.9  Continuous Functions   mulcncff 45861
            21.43.10  Derivatives   dvsinexp 45902
            21.43.11  Integrals   itgsin0pilem1 45941
            21.43.12  Stone Weierstrass theorem - real version   stoweidlem1 45992
            21.43.13  Wallis' product for π   wallispilem1 46056
            21.43.14  Stirling's approximation formula for ` n ` factorial   stirlinglem1 46065
            21.43.15  Dirichlet kernel   dirkerval 46082
            21.43.16  Fourier Series   fourierdlem1 46099
            21.43.17  e is transcendental   elaa2lem 46224
            21.43.18  n-dimensional Euclidean space   rrxtopn 46275
            21.43.19  Basic measure theory   csalg 46299
                  ​*21.43.19.1  σ-Algebras   csalg 46299
                  21.43.19.2  Sum of nonnegative extended reals   csumge0 46353
                  ​*21.43.19.3  Measures   cmea 46440
                  ​*21.43.19.4  Outer measures and Caratheodory's construction   come 46480
                  ​*21.43.19.5  Lebesgue measure on n-dimensional Real numbers   covoln 46527
                  ​*21.43.19.6  Measurable functions   csmblfn 46686
      ​21.44  Mathbox for Saveliy Skresanov
            21.44.1  Ceva's theorem   sigarval 46841
            21.44.2  Simple groups   simpcntrab 46861
      ​21.45  Mathbox for Ender Ting
            21.45.1  Increasing sequences and subsequences   et-ltneverrefl 46862
            21.45.2  Scratchpad for number theory   evenwodadd 46879
            21.45.3  Scratchpad for math on real numbers   squeezedltsq 46880
      ​21.46  Mathbox for Jarvin Udandy
      ​21.47  Mathbox for Adhemar
            ​*21.47.1  Minimal implicational calculus   adh-minim 46992
      ​21.48  Mathbox for Alexander van der Vekens
            21.48.1  General auxiliary theorems (1)   n0nsn2el 47016
                  21.48.1.1  Unordered and ordered pairs - extension for singletons   n0nsn2el 47016
                  21.48.1.2  Unordered and ordered pairs - extension for unordered pairs   elprneb 47020
                  21.48.1.3  Unordered and ordered pairs - extension for ordered pairs   oppr 47021
                  21.48.1.4  Relations - extension   eubrv 47026
                  21.48.1.5  Definite description binder (inverted iota) - extension   iota0def 47029
                  21.48.1.6  Functions - extension   fveqvfvv 47031
            21.48.2  Alternative for Russell's definition of a description binder   caiota 47074
            21.48.3  Double restricted existential uniqueness   r19.32 47089
                  21.48.3.1  Restricted quantification (extension)   r19.32 47089
                  21.48.3.2  Restricted uniqueness and "at most one" quantification   reuf1odnf 47098
                  21.48.3.3  Analogs to Existential uniqueness (double quantification)   2reu3 47101
                  21.48.3.4  Additional theorems for double restricted existential uniqueness   2reu8i 47104
            ​*21.48.4  Alternative definitions of function and operation values   wdfat 47107
                  21.48.4.1  Restricted quantification (extension)   ralbinrald 47113
                  21.48.4.2  The universal class (extension)   nvelim 47114
                  21.48.4.3  Introduce the Axiom of Power Sets (extension)   alneu 47115
                  21.48.4.4  Predicate "defined at"   dfateq12d 47117
                  21.48.4.5  Alternative definition of the value of a function   dfafv2 47123
                  21.48.4.6  Alternative definition of the value of an operation   aoveq123d 47169
            ​*21.48.5  Alternative definitions of function values (2)   cafv2 47199
            21.48.6  General auxiliary theorems (2)   an4com24 47259
                  21.48.6.1  Logical conjunction - extension   an4com24 47259
                  21.48.6.2  Abbreviated conjunction and disjunction of three wff's - extension   3an4ancom24 47260
                  21.48.6.3  Negated membership (alternative)   cnelbr 47262
                  21.48.6.4  The empty set - extension   ralralimp 47269
                  21.48.6.5  Indexed union and intersection - extension   otiunsndisjX 47270
                  21.48.6.6  Functions - extension   fvifeq 47271
                  21.48.6.7  Maps-to notation - extension   fvmptrab 47283
                  21.48.6.8  Subtraction - extension   cnambpcma 47285
                  21.48.6.9  Ordering on reals (cont.) - extension   leaddsuble 47288
                  21.48.6.10  Imaginary and complex number properties - extension   readdcnnred 47294
                  21.48.6.11  Nonnegative integers (as a subset of complex numbers) - extension   nn0resubcl 47299
                  21.48.6.12  Integers (as a subset of complex numbers) - extension   zgeltp1eq 47300
                  21.48.6.13  Decimal arithmetic - extension   1t10e1p1e11 47301
                  21.48.6.14  Upper sets of integers - extension   eluzge0nn0 47303
                  21.48.6.15  Infinity and the extended real number system (cont.) - extension   nltle2tri 47304
                  21.48.6.16  Finite intervals of integers - extension   ssfz12 47305
                  21.48.6.17  Half-open integer ranges - extension   fzopred 47313
                  21.48.6.18  The floor and ceiling functions - extension   2ltceilhalf 47319
                  21.48.6.19  The modulo (remainder) operation - extension   fldivmod 47329
                  21.48.6.20  The infinite sequence builder "seq"   smonoord 47362
                  21.48.6.21  Finite and infinite sums - extension   fsummsndifre 47363
                  21.48.6.22  Extensible structures - extension   setsidel 47367
            ​*21.48.7  Preimages of function values   preimafvsnel 47370
            ​*21.48.8  Partitions of real intervals   ciccp 47404
            21.48.9  Shifting functions with an integer range domain   fargshiftfv 47430
            21.48.10  Words over a set (extension)   lswn0 47435
                  21.48.10.1  Last symbol of a word - extension   lswn0 47435
            21.48.11  Unordered pairs   wich 47436
                  21.48.11.1  Interchangeable setvar variables   wich 47436
                  21.48.11.2  Set of unordered pairs   sprid 47465
                  ​*21.48.11.3  Proper (unordered) pairs   prpair 47492
                  21.48.11.4  Set of proper unordered pairs   cprpr 47503
            21.48.12  Number theory (extension)   cfmtno 47518
                  ​*21.48.12.1  Fermat numbers   cfmtno 47518
                  ​*21.48.12.2  Mersenne primes   m2prm 47582
                  21.48.12.3  Proth's theorem   modexp2m1d 47603
                  21.48.12.4  Solutions of quadratic equations   quad1 47611
            ​*21.48.13  Even and odd numbers   ceven 47615
                  21.48.13.1  Definitions and basic properties   ceven 47615
                  21.48.13.2  Alternate definitions using the "divides" relation   dfeven2 47640
                  21.48.13.3  Alternate definitions using the "modulo" operation   dfeven3 47649
                  21.48.13.4  Alternate definitions using the "gcd" operation   iseven5 47655
                  21.48.13.5  Theorems of part 5 revised   zneoALTV 47660
                  21.48.13.6  Theorems of part 6 revised   odd2np1ALTV 47665
                  21.48.13.7  Theorems of AV's mathbox revised   0evenALTV 47679
                  21.48.13.8  Additional theorems   epoo 47694
                  21.48.13.9  Perfect Number Theorem (revised)   perfectALTVlem1 47712
            21.48.14  Number theory (extension 2)   cfppr 47715
                  ​*21.48.14.1  Fermat pseudoprimes   cfppr 47715
                  ​*21.48.14.2  Goldbach's conjectures   cgbe 47736
            21.48.15  Graph theory (extension)   cclnbgr 47809
                  21.48.15.1  Closed neighborhood of a vertex   cclnbgr 47809
                  ​*21.48.15.2  Semiclosed and semiopen neighborhoods (experimental)   dfsclnbgr2 47836
                  21.48.15.3  Induced subgraphs   cisubgr 47850
                  ​*21.48.15.4  Isomorphisms of graphs   cgrisom 47864
                  ​*21.48.15.5  Triangles in graphs   cgrtri 47926
                  ​*21.48.15.6  Star graphs   cstgr 47940
                  ​*21.48.15.7  Local isomorphisms of graphs   cgrlim 47965
                  ​*21.48.15.8  Generalized Petersen graphs   cgpg 48021
                  21.48.15.9  Loop-free graphs - extension   1hegrlfgr 48110
                  21.48.15.10  Walks - extension   cupwlks 48111
                  21.48.15.11  Edges of graphs expressed as sets of unordered pairs   upgredgssspr 48121
            21.48.16  Monoids (extension)   ovn0dmfun 48134
                  21.48.16.1  Auxiliary theorems   ovn0dmfun 48134
                  21.48.16.2  Magmas, Semigroups and Monoids (extension)   plusfreseq 48142
                  21.48.16.3  Examples and counterexamples for magmas, semigroups and monoids (extension)   opmpoismgm 48145
                  21.48.16.4  Group sum operation (extension 1)   gsumsplit2f 48158
            ​*21.48.17  Magmas and internal binary operations (alternate approach)   ccllaw 48161
                  *21.48.17.1  Laws for internal binary operations   ccllaw 48161
                  ​*21.48.17.2  Internal binary operations   cintop 48174
                  21.48.17.3  Alternative definitions for magmas and semigroups   cmgm2 48193
            21.48.18  Rings (extension)   lmod0rng 48207
                  21.48.18.1  Nonzero rings (extension)   lmod0rng 48207
                  21.48.18.2  Ideals as non-unital rings   lidldomn1 48209
                  21.48.18.3  The non-unital ring of even integers   0even 48215
                  21.48.18.4  A constructed not unital ring   cznrnglem 48237
                  ​*21.48.18.5  The category of non-unital rings (alternate definition)   crngcALTV 48241
                  ​*21.48.18.6  The category of (unital) rings (alternate definition)   cringcALTV 48265
            21.48.19  Basic algebraic structures (extension)   eliunxp2 48312
                  21.48.19.1  Auxiliary theorems   eliunxp2 48312
                  21.48.19.2  The binomial coefficient operation (extension)   bcpascm1 48329
                  21.48.19.3  The ` ZZ `-module ` ZZ X. ZZ `   zlmodzxzlmod 48332
                  21.48.19.4  Group sum operation (extension 2)   mgpsumunsn 48339
                  21.48.19.5  Symmetric groups (extension)   exple2lt6 48342
                  21.48.19.6  Divisibility (extension)   invginvrid 48345
                  21.48.19.7  The support of functions (extension)   rmsupp0 48346
                  21.48.19.8  Finitely supported functions (extension)   rmsuppfi 48350
                  21.48.19.9  Left modules (extension)   lmodvsmdi 48357
                  21.48.19.10  Associative algebras (extension)   assaascl0 48359
                  21.48.19.11  Univariate polynomials (extension)   ply1vr1smo 48361
                  21.48.19.12  Univariate polynomials (examples)   linply1 48372
            21.48.20  Linear algebra (extension)   cdmatalt 48375
                  ​*21.48.20.1  The subalgebras of diagonal and scalar matrices (extension)   cdmatalt 48375
                  ​*21.48.20.2  Linear combinations   clinc 48383
                  ​*21.48.20.3  Linear independence   clininds 48419
                  21.48.20.4  Simple left modules and the ` ZZ `-module   lmod1lem1 48466
                  21.48.20.5  Differences between (left) modules and (left) vector spaces   lvecpsslmod 48486
            21.48.21  Complexity theory   suppdm 48489
                  21.48.21.1  Auxiliary theorems   suppdm 48489
                  21.48.21.2  Even and odd integers   nn0onn0ex 48502
                  21.48.21.3  The natural logarithm on complex numbers (extension)   logcxp0 48514
                  21.48.21.4  Division of functions   cfdiv 48516
                  21.48.21.5  Upper bounds   cbigo 48526
                  21.48.21.6  Logarithm to an arbitrary base (extension)   rege1logbrege0 48537
                  ​*21.48.21.7  The binary logarithm   fldivexpfllog2 48544
                  21.48.21.8  Binary length   cblen 48548
                  ​*21.48.21.9  Digits   cdig 48574
                  21.48.21.10  Nonnegative integer as sum of its shifted digits   dignn0flhalflem1 48594
                  21.48.21.11  Algorithms for the multiplication of nonnegative integers   nn0mulfsum 48603
                  ​*21.48.21.12  N-ary functions   cnaryf 48605
                  ​*21.48.21.13  The Ackermann function   citco 48636
            21.48.22  Elementary geometry (extension)   fv1prop 48678
                  21.48.22.1  Auxiliary theorems   fv1prop 48678
                  21.48.22.2  Real euclidean space of dimension 2   rrx2pxel 48690
                  21.48.22.3  Spheres and lines in real Euclidean spaces   cline 48706
      ​21.49  Mathbox for Zhi Wang
            21.49.1  Propositional calculus   pm4.71da 48768
            21.49.2  Predicate calculus with equality   dtrucor3 48777
                  21.49.2.1  Axiom scheme ax-5 (Distinctness)   dtrucor3 48777
            21.49.3  ZF Set Theory - start with the Axiom of Extensionality   ralbidb 48778
                  21.49.3.1  Restricted quantification   ralbidb 48778
                  21.49.3.2  The universal class   reuxfr1dd 48785
                  21.49.3.3  The empty set   ssdisjd 48786
                  21.49.3.4  Unordered and ordered pairs   vsn 48790
                  21.49.3.5  The union of a class   unilbss 48796
                  21.49.3.6  Indexed union and intersection   iuneq0 48797
            21.49.4  ZF Set Theory - add the Axiom of Replacement   inpw 48803
                  21.49.4.1  Theorems requiring subset and intersection existence   inpw 48803
            21.49.5  ZF Set Theory - add the Axiom of Power Sets   opth1neg 48804
                  21.49.5.1  Ordered pair theorem   opth1neg 48804
                  21.49.5.2  Ordered-pair class abstractions (cont.)   brab2dd 48806
                  21.49.5.3  Relations   iinxp 48809
                  21.49.5.4  Functions   mof0 48816
                  21.49.5.5  Operations   ovsng 48834
            21.49.6  ZF Set Theory - add the Axiom of Union   fonex 48843
                  21.49.6.1  Relations and functions (cont.)   fonex 48843
                  21.49.6.2  First and second members of an ordered pair   eloprab1st2nd 48844
                  21.49.6.3  Operations in maps-to notation (continued)   fmpodg 48845
                  21.49.6.4  Function transposition   resinsnlem 48847
                  21.49.6.5  Infinite Cartesian products   ixpv 48866
                  21.49.6.6  Equinumerosity   fvconst0ci 48867
            21.49.7  Order sets   iccin 48872
                  21.49.7.1  Real number intervals   iccin 48872
            21.49.8  Extensible structures   slotresfo 48875
                  21.49.8.1  Basic definitions   slotresfo 48875
            21.49.9  Moore spaces   mreuniss 48876
            ​*21.49.10  Topology   clduni 48877
                  21.49.10.1  Closure and interior   clduni 48877
                  21.49.10.2  Neighborhoods   neircl 48881
                  21.49.10.3  Subspace topologies   restcls2lem 48889
                  21.49.10.4  Limits and continuity in topological spaces   cnneiima 48893
                  21.49.10.5  Topological definitions using the reals   iooii 48894
                  21.49.10.6  Separated sets   sepnsepolem1 48898
                  21.49.10.7  Separated spaces: T0, T1, T2 (Hausdorff) ...   isnrm4 48907
            21.49.11  Preordered sets and directed sets using extensible structures   isprsd 48931
            21.49.12  Posets and lattices using extensible structures   lubeldm2 48932
                  21.49.12.1  Posets   lubeldm2 48932
                  21.49.12.2  Lattices   toslat 48958
                  21.49.12.3  Subset order structures   intubeu 48960
            21.49.13  Rings   elmgpcntrd 48981
                  21.49.13.1  Multiplicative Group   elmgpcntrd 48981
            21.49.14  Associative algebras   asclelbas 48982
                  21.49.14.1  Definition and basic properties   asclelbas 48982
            21.49.15  Categories   homf0 48986
                  21.49.15.1  Categories   homf0 48986
                  21.49.15.2  Opposite category   oppccatb 48993
                  21.49.15.3  Monomorphisms and epimorphisms   idmon 48997
                  21.49.15.4  Sections, inverses, isomorphisms   sectrcl 48999
                  21.49.15.5  Isomorphic objects   cicfn 49019
                  21.49.15.6  Subcategories   dmdm 49030
                  21.49.15.7  Functors   reldmfunc 49052
                  21.49.15.8  Opposite functors   coppf 49099
                  21.49.15.9  Full & faithful functors   imasubc 49127
                  21.49.15.10  Universal property   upciclem1 49139
                  21.49.15.11  Natural transformations and the functor category   isnatd 49194
                  21.49.15.12  Initial, terminal and zero objects of a category   initoo2 49203
                  21.49.15.13  Product of categories   reldmxpc 49217
                  21.49.15.14  Swap functors   cswapf 49230
                  21.49.15.15  Functor evaluation   oppc1stflem 49258
                  21.49.15.16  Transposed curry functors   cofuswapfcl 49264
                  21.49.15.17  Constant functors   diag1 49275
                  21.49.15.18  Functor composition bifunctors   fucofulem1 49281
                  21.49.15.19  Post-composition functors   postcofval 49335
                  21.49.15.20  Pre-composition functors   precofvallem 49337
            21.49.16  Examples of categories   catcrcl 49364
                  21.49.16.1  The category of categories   catcrcl 49364
                  21.49.16.2  Thin categories   cthinc 49386
                  21.49.16.3  Terminal categories   ctermc 49441
                  21.49.16.4  Preordered sets as thin categories   cprstc 49518
                  21.49.16.5  Monoids as categories   cmndtc 49546
                  21.49.16.6  Categories with at most one object and at most two morphisms   2arwcatlem1 49564
            21.49.17  Kan extensions and related concepts   clan 49574
                  21.49.17.1  Kan extensions   clan 49574
                  21.49.17.2  Limits and colimits   clmd 49611
      ​21.50  Mathbox for Emmett Weisz
            ​*21.50.1  Miscellaneous Theorems   nfintd 49639
            21.50.2  Set Recursion   csetrecs 49649
                  ​*21.50.2.1  Basic Properties of Set Recursion   csetrecs 49649
                  21.50.2.2  Examples and properties of set recursion   elsetrecslem 49665
            ​*21.50.3  Construction of Games and Surreal Numbers   cpg 49675
      ​​*21.51  Mathbox for David A. Wheeler
            21.51.1  Natural deduction   sbidd 49684
            ​*21.51.2  Greater than, greater than or equal to.   cge-real 49686
            ​*21.51.3  Hyperbolic trigonometric functions   csinh 49696
            ​*21.51.4  Reciprocal trigonometric functions (sec, csc, cot)   csec 49707
            ​*21.51.5  Identities for "if"   ifnmfalse 49729
            ​*21.51.6  Logarithms generalized to arbitrary base using ` logb `   logb2aval 49730
            ​*21.51.7  Logarithm laws generalized to an arbitrary base - log_   clog- 49731
            ​*21.51.8  Formally define notions such as reflexivity   wreflexive 49733
            ​*21.51.9  Algebra helpers   mvlraddi 49737
            ​*21.51.10  Algebra helper examples   i2linesi 49744
            ​*21.51.11  Formal methods "surprises"   alimp-surprise 49746
            ​*21.51.12  Allsome quantifier   walsi 49752
            ​*21.51.13  Miscellaneous   5m4e1 49763
            21.51.14  Theorems about algebraic numbers   aacllem 49767
      ​21.52  Mathbox for Kunhao Zheng
            21.52.1  Weighted AM-GM inequality   amgmwlem 49768