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 1492
            1.2.13  Logical "xor"   wxo 1512
            1.2.14  Logical "nor"   wnor 1529
            1.2.15  True and false constants   wal 1539
                  ​*1.2.15.1  Universal quantifier for use by df-tru   wal 1539
                  ​*1.2.15.2  Equality predicate for use by df-tru   cv 1540
                  1.2.15.3  The true constant   wtru 1542
                  1.2.15.4  The false constant   wfal 1553
            ​*1.2.16  Truth tables   truimtru 1564
                  1.2.16.1  Implication   truimtru 1564
                  1.2.16.2  Negation   nottru 1568
                  1.2.16.3  Equivalence   trubitru 1570
                  1.2.16.4  Conjunction   truantru 1574
                  1.2.16.5  Disjunction   truortru 1578
                  1.2.16.6  Alternative denial   trunantru 1582
                  1.2.16.7  Exclusive disjunction   truxortru 1586
                  1.2.16.8  Joint denial   trunortru 1590
            ​*1.2.17  Half adder and full adder in propositional calculus   whad 1594
                  1.2.17.1  Full adder: sum   whad 1594
                  1.2.17.2  Full adder: carry   wcad 1607
      ​1.3  Other axiomatizations related to classical propositional calculus
            ​*1.3.1  Minimal implicational calculus   minimp 1622
            ​*1.3.2  Implicational Calculus   impsingle 1628
            1.3.3  Derive the Lukasiewicz axioms from Meredith's sole axiom   meredith 1642
            1.3.4  Derive the standard axioms from the Lukasiewicz axioms   luklem1 1659
            ​*1.3.5  Derive Nicod's axiom from the standard axioms   nic-dfim 1670
            1.3.6  Derive the Lukasiewicz axioms from Nicod's axiom   nic-imp 1676
            1.3.7  Derive Nicod's Axiom from Lukasiewicz's First Sheffer Stroke Axiom   lukshef-ax1 1695
            1.3.8  Derive the Lukasiewicz Axioms from the Tarski-Bernays-Wajsberg Axioms   tbw-bijust 1699
            1.3.9  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's First CO Axiom   merco1 1714
            1.3.10  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's Second CO Axiom   merco2 1737
            1.3.11  Derive the Lukasiewicz axioms from the Russell-Bernays Axioms   rb-bijust 1750
            ​*1.3.12  Stoic logic non-modal portion (Chrysippus of Soli)   mptnan 1769
      ​​*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 1780
                  1.4.1.1  Existential quantifier   wex 1780
                  1.4.1.2  Nonfreeness predicate   wnf 1784
            1.4.2  Rule scheme ax-gen (Generalization)   ax-gen 1796
            1.4.3  Axiom scheme ax-4 (Quantified Implication)   ax-4 1810
                  ​*1.4.3.1  The empty domain of discourse   empty 1907
            1.4.4  Axiom scheme ax-5 (Distinctness) - first use of $d   ax-5 1911
            ​*1.4.5  Equality predicate (continued)   weq 1963
            1.4.6  Axiom scheme ax-6 (Existence)   ax-6 1968
            1.4.7  Axiom scheme ax-7 (Equality)   ax-7 2009
            1.4.8  Define proper substitution   sbjust 2065
            1.4.9  Membership predicate   wcel 2110
            1.4.10  Axiom scheme ax-8 (Left Equality for Binary Predicate)   ax-8 2112
            1.4.11  Axiom scheme ax-9 (Right Equality for Binary Predicate)   ax-9 2120
            ​*1.4.12  Logical redundancy of ax-10 , ax-11 , ax-12 , ax-13   ax6dgen 2130
      ​​*1.5  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes)
            1.5.1  Axiom scheme ax-10 (Quantified Negation)   ax-10 2143
            1.5.2  Axiom scheme ax-11 (Quantifier Commutation)   ax-11 2159
            1.5.3  Axiom scheme ax-12 (Substitution)   ax-12 2179
            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 3342
                  2.1.5.3  Restricted class abstraction   crab 3393
            2.1.6  The universal class   cvv 3434
            ​*2.1.7  Conditional equality (experimental)   wcdeq 3720
            2.1.8  Russell's Paradox   rru 3736
            2.1.9  Proper substitution of classes for sets   wsbc 3739
            2.1.10  Proper substitution of classes for sets into classes   csb 3848
            2.1.11  Define basic set operations and relations   cdif 3897
            2.1.12  Subclasses and subsets   df-ss 3917
            2.1.13  The difference, union, and intersection of two classes   dfdif3 4065
                  2.1.13.1  The difference of two classes   dfdif3 4065
                  2.1.13.2  The union of two classes   elun 4101
                  2.1.13.3  The intersection of two classes   elini 4147
                  2.1.13.4  The symmetric difference of two classes   csymdif 4200
                  2.1.13.5  Combinations of difference, union, and intersection of two classes   unabs 4213
                  2.1.13.6  Class abstractions with difference, union, and intersection of two classes   unabw 4255
                  2.1.13.7  Restricted uniqueness with difference, union, and intersection   reuun2 4273
            2.1.14  The empty set   c0 4281
            ​*2.1.15  The conditional operator for classes   cif 4473
            ​*2.1.16  The weak deduction theorem for set theory   dedth 4532
            2.1.17  Power classes   cpw 4548
            2.1.18  Unordered and ordered pairs   snjust 4573
            2.1.19  The union of a class   cuni 4857
            2.1.20  The intersection of a class   cint 4895
            2.1.21  Indexed union and intersection   ciun 4939
            2.1.22  Disjointness   wdisj 5056
            2.1.23  Binary relations   wbr 5089
            2.1.24  Ordered-pair class abstractions (class builders)   copab 5151
            2.1.25  Functions in maps-to notation   cmpt 5170
            2.1.26  Transitive classes   wtr 5196
      ​2.2  ZF Set Theory - add the Axiom of Replacement
            2.2.1  Introduce the Axiom of Replacement   ax-rep 5215
            2.2.2  Derive the Axiom of Separation   axsepgfromrep 5230
            2.2.3  Derive the Null Set Axiom   axnulALT 5240
            2.2.4  Theorems requiring subset and intersection existence   nalset 5249
            2.2.5  Theorems requiring empty set existence   class2set 5291
      ​2.3  ZF Set Theory - add the Axiom of Power Sets
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 5301
            2.3.2  Derive the Axiom of Pairing   axprlem1 5359
            2.3.3  Ordered pair theorem   opnz 5411
            2.3.4  Ordered-pair class abstractions (cont.)   opabidw 5462
            2.3.5  Power class of union and intersection   pwin 5505
            2.3.6  The identity relation   cid 5508
            2.3.7  The membership relation (or epsilon relation)   cep 5513
            ​*2.3.8  Partial and total orderings   wpo 5520
            2.3.9  Founded and well-ordering relations   wfr 5564
            2.3.10  Relations   cxp 5612
            2.3.11  The Predecessor Class   cpred 6243
            2.3.12  Well-founded induction (variant)   frpomin 6283
            2.3.13  Well-ordered induction   tz6.26 6290
            2.3.14  Ordinals   word 6301
            2.3.15  Definite description binder (inverted iota)   cio 6431
            2.3.16  Functions   wfun 6471
            2.3.17  Cantor's Theorem   canth 7295
            2.3.18  Restricted iota (description binder)   crio 7297
            2.3.19  Operations   co 7341
                  2.3.19.1  Variable-to-class conversion for operations   caovclg 7533
            2.3.20  Maps-to notation   mpondm0 7581
            2.3.21  Function operation   cof 7603
            2.3.22  Proper subset relation   crpss 7650
      ​2.4  ZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 7663
            2.4.2  Ordinals (continued)   epweon 7703
            2.4.3  Transfinite induction   tfi 7778
            2.4.4  The natural numbers (i.e., finite ordinals)   com 7791
            2.4.5  Peano's postulates   peano1 7814
            2.4.6  Finite induction (for finite ordinals)   find 7820
            2.4.7  Relations and functions (cont.)   dmexg 7826
            2.4.8  First and second members of an ordered pair   c1st 7914
            2.4.9  Induction on Cartesian products   frpoins3xpg 8065
            2.4.10  Ordering on Cartesian products   xpord2lem 8067
            2.4.11  Ordering Ordinal Sequences   orderseqlem 8082
            ​*2.4.12  The support of functions   csupp 8085
            ​*2.4.13  Special maps-to operations   opeliunxp2f 8135
            2.4.14  Function transposition   ctpos 8150
            2.4.15  Curry and uncurry   ccur 8190
            2.4.16  Undefined values   cund 8197
            2.4.17  Well-founded recursion   cfrecs 8205
            2.4.18  Well-ordered recursion   cwrecs 8236
            2.4.19  Functions on ordinals; strictly monotone ordinal functions   iunon 8254
            2.4.20  "Strong" transfinite recursion   crecs 8285
            2.4.21  Recursive definition generator   crdg 8323
            2.4.22  Finite recursion   frfnom 8349
            2.4.23  Ordinal arithmetic   c1o 8373
            2.4.24  Natural number arithmetic   nna0 8514
            2.4.25  Natural addition   cnadd 8575
            2.4.26  Equivalence relations and classes   wer 8614
            2.4.27  The mapping operation   cmap 8745
            2.4.28  Infinite Cartesian products   cixp 8816
            2.4.29  Equinumerosity   cen 8861
            2.4.30  Schroeder-Bernstein Theorem   sbthlem1 8995
            2.4.31  Equinumerosity (cont.)   xpf1o 9047
            2.4.32  Finite sets   dif1enlem 9064
            2.4.33  Pigeonhole Principle   phplem1 9108
            2.4.34  Finite sets (cont.)   onomeneq 9118
            2.4.35  Finitely supported functions   cfsupp 9240
            2.4.36  Finite intersections   cfi 9289
            2.4.37  Hall's marriage theorem   marypha1lem 9312
            2.4.38  Supremum and infimum   csup 9319
            2.4.39  Ordinal isomorphism, Hartogs's theorem   coi 9390
            2.4.40  Hartogs function   char 9437
            2.4.41  Weak dominance   cwdom 9445
      ​2.5  ZF Set Theory - add the Axiom of Regularity
            2.5.1  Introduce the Axiom of Regularity   ax-reg 9473
            2.5.2  Axiom of Infinity equivalents   inf0 9506
      ​2.6  ZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-inf 9523
            2.6.2  Existence of omega (the set of natural numbers)   omex 9528
            2.6.3  Cantor normal form   ccnf 9546
            2.6.4  Transitive closure of a relation   cttrcl 9592
            2.6.5  Transitive closure   trcl 9613
            2.6.6  Well-Founded Induction   frmin 9634
            2.6.7  Well-Founded Recursion   frr3g 9641
            2.6.8  Rank   cr1 9647
            2.6.9  Scott's trick; collection principle; Hilbert's epsilon   scottex 9770
            2.6.10  Disjoint union   cdju 9783
            2.6.11  Cardinal numbers   ccrd 9820
            2.6.12  Axiom of Choice equivalents   wac 9998
            ​*2.6.13  Cardinal number arithmetic   undjudom 10051
            2.6.14  The Ackermann bijection   ackbij2lem1 10101
            2.6.15  Cofinality (without Axiom of Choice)   cflem 10128
            2.6.16  Eight inequivalent definitions of finite set   sornom 10160
            2.6.17  Hereditarily size-limited sets without Choice   itunifval 10299
​​*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 10318
            3.1.2  Introduce the Axiom of Dependent Choice   ax-dc 10329
      ​3.2  ZFC Set Theory - add the Axiom of Choice
            3.2.1  Introduce the Axiom of Choice   ax-ac 10342
            3.2.2  AC equivalents: well-ordering, Zorn's lemma   numthcor 10377
            3.2.3  Cardinal number theorems using Axiom of Choice   cardval 10429
            3.2.4  Cardinal number arithmetic using Axiom of Choice   iunctb 10457
            3.2.5  Cofinality using the Axiom of Choice   alephreg 10465
      ​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 10503
            3.4.2  Derivation of the Axiom of Choice   gchaclem 10561
​​*PART 4  TG (TARSKI-GROTHENDIECK) SET THEORY
      ​4.1  Inaccessibles
            4.1.1  Weakly and strongly inaccessible cardinals   cwina 10565
            4.1.2  Weak universes   cwun 10583
            4.1.3  Tarski classes   ctsk 10631
            4.1.4  Grothendieck universes   cgru 10673
      ​4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
            4.2.1  Introduce the Tarski-Grothendieck Axiom   ax-groth 10706
            4.2.2  Derive the Power Set, Infinity and Choice Axioms   grothpw 10709
            4.2.3  Tarski map function   ctskm 10720
​​*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 10727
            5.1.2  Final derivation of real and complex number postulates   axaddf 11028
            5.1.3  Real and complex number postulates restated as axioms   ax-cnex 11054
      ​5.2  Derive the basic properties from the field axioms
            5.2.1  Some deductions from the field axioms for complex numbers   cnex 11079
            5.2.2  Infinity and the extended real number system   cpnf 11135
            5.2.3  Restate the ordering postulates with extended real "less than"   axlttri 11176
            5.2.4  Ordering on reals   lttr 11181
            5.2.5  Initial properties of the complex numbers   mul12 11270
      ​5.3  Real and complex numbers - basic operations
            5.3.1  Addition   add12 11323
            5.3.2  Subtraction   cmin 11336
            5.3.3  Multiplication   kcnktkm1cn 11540
            5.3.4  Ordering on reals (cont.)   gt0ne0 11574
            5.3.5  Reciprocals   ixi 11738
            5.3.6  Division   cdiv 11766
            5.3.7  Ordering on reals (cont.)   elimgt0 11951
            5.3.8  Completeness Axiom and Suprema   fimaxre 12058
            5.3.9  Imaginary and complex number properties   neg1cn 12102
            5.3.10  Function operation analogue theorems   ofsubeq0 12114
      ​5.4  Integer sets
            5.4.1  Positive integers (as a subset of complex numbers)   cn 12117
            5.4.2  Principle of mathematical induction   nnind 12135
            ​*5.4.3  Decimal representation of numbers   c2 12172
            ​*5.4.4  Some properties of specific numbers   1pneg1e0 12231
            5.4.5  Simple number properties   halfcl 12339
            5.4.6  The Archimedean property   nnunb 12369
            5.4.7  Nonnegative integers (as a subset of complex numbers)   cn0 12373
            ​*5.4.8  Extended nonnegative integers   cxnn0 12446
            5.4.9  Integers (as a subset of complex numbers)   cz 12460
            5.4.10  Decimal arithmetic   cdc 12580
            5.4.11  Upper sets of integers   cuz 12724
            5.4.12  Well-ordering principle for bounded-below sets of integers   uzwo3 12833
            5.4.13  Rational numbers (as a subset of complex numbers)   cq 12838
            5.4.14  Existence of the set of complex numbers   rpnnen1lem2 12867
      ​5.5  Order sets
            5.5.1  Positive reals (as a subset of complex numbers)   crp 12882
            5.5.2  Infinity and the extended real number system (cont.)   cxne 13000
            5.5.3  Supremum and infimum on the extended reals   xrsupexmnf 13196
            5.5.4  Real number intervals   cioo 13237
            5.5.5  Finite intervals of integers   cfz 13399
            ​*5.5.6  Finite intervals of nonnegative integers   elfz2nn0 13510
            5.5.7  Half-open integer ranges   cfzo 13546
      ​5.6  Elementary integer functions
            5.6.1  The floor and ceiling functions   cfl 13686
            5.6.2  The modulo (remainder) operation   cmo 13765
            5.6.3  Miscellaneous theorems about integers   om2uz0i 13846
            5.6.4  Strong induction over upper sets of integers   uzsinds 13886
            5.6.5  Finitely supported functions over the nonnegative integers   fsuppmapnn0fiublem 13889
            5.6.6  The infinite sequence builder "seq" - extension   cseq 13900
            5.6.7  Integer powers   cexp 13960
            5.6.8  Ordered pair theorem for nonnegative integers   nn0le2msqi 14166
            5.6.9  Factorial function   cfa 14172
            5.6.10  The binomial coefficient operation   cbc 14201
            5.6.11  The ` # ` (set size) function   chash 14229
                  5.6.11.1  Proper unordered pairs and triples (sets of size 2 and 3)   hashprlei 14367
                  5.6.11.2  Functions with a domain containing at least two different elements   fundmge2nop0 14401
                  5.6.11.3  Finite induction on the size of the first component of a binary relation   hashdifsnp1 14405
      ​​*5.7  Words over a set
            5.7.1  Definitions and basic theorems   cword 14412
            5.7.2  Last symbol of a word   clsw 14461
            5.7.3  Concatenations of words   cconcat 14469
            5.7.4  Singleton words   cs1 14495
            5.7.5  Concatenations with singleton words   ccatws1cl 14516
            5.7.6  Subwords/substrings   csubstr 14540
            5.7.7  Prefixes of a word   cpfx 14570
            5.7.8  Subwords of subwords   swrdswrdlem 14603
            5.7.9  Subwords and concatenations   pfxcctswrd 14609
            5.7.10  Subwords of concatenations   swrdccatfn 14623
            5.7.11  Splicing words (substring replacement)   csplice 14648
            5.7.12  Reversing words   creverse 14657
            5.7.13  Repeated symbol words   creps 14667
            ​*5.7.14  Cyclical shifts of words   ccsh 14687
            5.7.15  Mapping words by a function   wrdco 14730
            5.7.16  Longer string literals   cs2 14740
      ​​*5.8  Reflexive and transitive closures of relations
            5.8.1  The reflexive and transitive properties of relations   coss12d 14871
            5.8.2  Basic properties of closures   cleq1lem 14881
            5.8.3  Definitions and basic properties of transitive closures   ctcl 14884
            5.8.4  Exponentiation of relations   crelexp 14918
            5.8.5  Reflexive-transitive closure as an indexed union   crtrcl 14954
            ​*5.8.6  Principle of transitive induction.   relexpindlem 14962
      ​5.9  Elementary real and complex functions
            5.9.1  The "shift" operation   cshi 14965
            5.9.2  Signum (sgn or sign) function   csgn 14985
            5.9.3  Real and imaginary parts; conjugate   ccj 14995
            5.9.4  Square root; absolute value   csqrt 15132
      ​5.10  Elementary limits and convergence
            5.10.1  Superior limit (lim sup)   clsp 15369
            5.10.2  Limits   cli 15383
            5.10.3  Finite and infinite sums   csu 15585
            5.10.4  The binomial theorem   binomlem 15728
            5.10.5  The inclusion/exclusion principle   incexclem 15735
            5.10.6  Infinite sums (cont.)   isumshft 15738
            5.10.7  Miscellaneous converging and diverging sequences   divrcnv 15751
            5.10.8  Arithmetic series   arisum 15759
            5.10.9  Geometric series   expcnv 15763
            5.10.10  Ratio test for infinite series convergence   cvgrat 15782
            5.10.11  Mertens' theorem   mertenslem1 15783
            5.10.12  Finite and infinite products   prodf 15786
                  5.10.12.1  Product sequences   prodf 15786
                  5.10.12.2  Non-trivial convergence   ntrivcvg 15796
                  5.10.12.3  Complex products   cprod 15802
                  5.10.12.4  Finite products   fprod 15840
                  5.10.12.5  Infinite products   iprodclim 15897
            5.10.13  Falling and Rising Factorial   cfallfac 15903
            5.10.14  Bernoulli polynomials and sums of k-th powers   cbp 15945
      ​5.11  Elementary trigonometry
            5.11.1  The exponential, sine, and cosine functions   ce 15960
                  5.11.1.1  The circle constant (tau = 2 pi)   ctau 16103
            5.11.2  _e is irrational   eirrlem 16105
      ​5.12  Cardinality of real and complex number subsets
            5.12.1  Countability of integers and rationals   xpnnen 16112
            5.12.2  The reals are uncountable   rpnnen2lem1 16115
​​*PART 6  ELEMENTARY NUMBER THEORY
      ​6.1  Elementary properties of divisibility
            6.1.1  Irrationality of square root of 2   sqrt2irrlem 16149
            6.1.2  Some Number sets are chains of proper subsets   nthruc 16153
            6.1.3  The divides relation   cdvds 16155
            ​*6.1.4  Even and odd numbers   evenelz 16239
            6.1.5  The division algorithm   divalglem0 16296
            6.1.6  Bit sequences   cbits 16322
            6.1.7  The greatest common divisor operator   cgcd 16397
            6.1.8  Bézout's identity   bezoutlem1 16442
            6.1.9  Algorithms   nn0seqcvgd 16473
            6.1.10  Euclid's Algorithm   eucalgval2 16484
            ​*6.1.11  The least common multiple   clcm 16491
            ​*6.1.12  Coprimality and Euclid's lemma   coprmgcdb 16552
            6.1.13  Cancellability of congruences   congr 16567
      ​6.2  Elementary prime number theory
            ​*6.2.1  Elementary properties   cprime 16574
            ​*6.2.2  Coprimality and Euclid's lemma (cont.)   coprm 16614
            6.2.3  Properties of the canonical representation of a rational   cnumer 16636
            6.2.4  Euler's theorem   codz 16666
            6.2.5  Arithmetic modulo a prime number   modprm1div 16701
            6.2.6  Pythagorean Triples   coprimeprodsq 16712
            6.2.7  The prime count function   cpc 16740
            6.2.8  Pocklington's theorem   prmpwdvds 16808
            6.2.9  Infinite primes theorem   unbenlem 16812
            6.2.10  Sum of prime reciprocals   prmreclem1 16820
            6.2.11  Fundamental theorem of arithmetic   1arithlem1 16827
            6.2.12  Lagrange's four-square theorem   cgz 16833
            6.2.13  Van der Waerden's theorem   cvdwa 16869
            6.2.14  Ramsey's theorem   cram 16903
            ​*6.2.15  Primorial function   cprmo 16935
            ​*6.2.16  Prime gaps   prmgaplem1 16953
            6.2.17  Decimal arithmetic (cont.)   dec2dvds 16967
            6.2.18  Cyclical shifts of words (cont.)   cshwsidrepsw 16997
            6.2.19  Specific prime numbers   prmlem0 17009
            6.2.20  Very large primes   1259lem1 17034
​PART 7  BASIC STRUCTURES
      ​7.1  Extensible structures
            ​*7.1.1  Basic definitions   cstr 17049
                  7.1.1.1  Extensible structures as structures with components   cstr 17049
                  7.1.1.2  Substitution of components   csts 17066
                  7.1.1.3  Slots   cslot 17084
                  ​*7.1.1.4  Structure component indices   cnx 17096
                  7.1.1.5  Base sets   cbs 17112
                  7.1.1.6  Base set restrictions   cress 17133
            7.1.2  Slot definitions   cplusg 17153
            7.1.3  Definition of the structure product   crest 17316
            7.1.4  Definition of the structure quotient   cordt 17395
      ​7.2  Moore spaces
            7.2.1  Moore closures   mrcflem 17504
            7.2.2  Independent sets in a Moore system   mrisval 17528
            7.2.3  Algebraic closure systems   isacs 17549
​PART 8  BASIC CATEGORY THEORY
      ​8.1  Categories
            8.1.1  Categories   ccat 17562
            8.1.2  Opposite category   coppc 17609
            8.1.3  Monomorphisms and epimorphisms   cmon 17627
            8.1.4  Sections, inverses, isomorphisms   csect 17643
            ​*8.1.5  Isomorphic objects   ccic 17694
            8.1.6  Subcategories   cssc 17706
            8.1.7  Functors   cfunc 17753
            8.1.8  Full & faithful functors   cful 17803
            8.1.9  Natural transformations and the functor category   cnat 17843
            8.1.10  Initial, terminal and zero objects of a category   cinito 17880
      ​8.2  Arrows (disjointified hom-sets)
            8.2.1  Identity and composition for arrows   cida 17952
      ​8.3  Examples of categories
            8.3.1  The category of sets   csetc 17974
            8.3.2  The category of categories   ccatc 17997
            ​*8.3.3  The category of extensible structures   fncnvimaeqv 18018
      ​8.4  Categorical constructions
            8.4.1  Product of categories   cxpc 18066
            8.4.2  Functor evaluation   cevlf 18107
            8.4.3  Hom functor   chof 18146
​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 18329
            9.5.2  Complete lattices   ccla 18396
            9.5.3  Distributive lattices   cdlat 18418
            9.5.4  Subset order structures   cipo 18425
      ​9.6  Posets, directed sets, and lattices as relations
            ​*9.6.1  Posets and lattices as relations   cps 18462
            9.6.2  Directed sets, nets   cdir 18492
            9.6.3  Chains   cchn 18503
​PART 10  BASIC ALGEBRAIC STRUCTURES
      ​10.1  Monoids
            ​*10.1.1  Magmas   cplusf 18537
            ​*10.1.2  Identity elements   mgmidmo 18560
            ​*10.1.3  Iterated sums in a magma   gsumvalx 18576
            10.1.4  Magma homomorphisms and submagmas   cmgmhm 18590
            ​*10.1.5  Semigroups   csgrp 18618
            ​*10.1.6  Definition and basic properties of monoids   cmnd 18634
            10.1.7  Monoid homomorphisms and submonoids   cmhm 18681
            ​*10.1.8  Iterated sums in a monoid   gsumvallem2 18734
            10.1.9  Free monoids   cfrmd 18747
                  ​*10.1.9.1  Monoid of endofunctions   cefmnd 18768
            10.1.10  Examples and counterexamples for magmas, semigroups and monoids   mgm2nsgrplem1 18818
      ​10.2  Groups
            10.2.1  Definition and basic properties   cgrp 18838
            ​*10.2.2  Group multiple operation   cmg 18972
            10.2.3  Subgroups and Quotient groups   csubg 19025
            ​*10.2.4  Cyclic monoids and groups   cycsubmel 19105
            10.2.5  Elementary theory of group homomorphisms   cghm 19117
            10.2.6  Isomorphisms of groups   cgim 19162
                  10.2.6.1  The first isomorphism theorem of groups   ghmqusnsglem1 19185
            10.2.7  Group actions   cga 19194
            10.2.8  Centralizers and centers   ccntz 19220
            10.2.9  The opposite group   coppg 19250
            10.2.10  Symmetric groups   csymg 19274
                  ​*10.2.10.1  Definition and basic properties   csymg 19274
                  10.2.10.2  Cayley's theorem   cayleylem1 19317
                  10.2.10.3  Permutations fixing one element   symgfix2 19321
                  ​*10.2.10.4  Transpositions in the symmetric group   cpmtr 19346
                  10.2.10.5  The sign of a permutation   cpsgn 19394
            10.2.11  p-Groups and Sylow groups; Sylow's theorems   cod 19429
            10.2.12  Direct products   clsm 19539
                  10.2.12.1  Direct products (extension)   smndlsmidm 19561
            10.2.13  Free groups   cefg 19611
            10.2.14  Abelian groups   ccmn 19685
                  10.2.14.1  Definition and basic properties   ccmn 19685
                  10.2.14.2  Cyclic groups   ccyg 19782
                  10.2.14.3  Group sum operation   gsumval3a 19808
                  10.2.14.4  Group sums over (ranges of) integers   fsfnn0gsumfsffz 19888
                  10.2.14.5  Internal direct products   cdprd 19900
                  10.2.14.6  The Fundamental Theorem of Abelian Groups   ablfacrplem 19972
            10.2.15  Simple groups   csimpg 19997
                  10.2.15.1  Definition and basic properties   csimpg 19997
                  10.2.15.2  Classification of abelian simple groups   ablsimpnosubgd 20011
            10.2.16  Totally ordered monoids and groups   comnd 20024
      ​10.3  Rings
            10.3.1  Multiplicative Group   cmgp 20051
            ​*10.3.2  Non-unital rings ("rngs")   crng 20063
            ​*10.3.3  Ring unity (multiplicative identity)   cur 20092
            10.3.4  Semirings   csrg 20097
                  ​*10.3.4.1  The binomial theorem for semirings   srgbinomlem1 20137
            10.3.5  Unital rings   crg 20144
            10.3.6  Opposite ring   coppr 20247
            10.3.7  Divisibility   cdsr 20265
            10.3.8  Ring primes   crpm 20343
            10.3.9  Homomorphisms of non-unital rings   crnghm 20345
            10.3.10  Ring homomorphisms   crh 20380
            10.3.11  Nonzero rings and zero rings   cnzr 20420
            10.3.12  Local rings   clring 20446
            10.3.13  Subrings   csubrng 20453
                  10.3.13.1  Subrings of non-unital rings   csubrng 20453
                  10.3.13.2  Subrings of unital rings   csubrg 20477
                  10.3.13.3  Subrings generated by a subset   crgspn 20518
            10.3.14  Categories of rings   crngc 20524
                  ​*10.3.14.1  The category of non-unital rings   crngc 20524
                  ​*10.3.14.2  The category of (unital) rings   cringc 20553
                  10.3.14.3  Subcategories of the category of rings   srhmsubclem1 20585
            10.3.15  Left regular elements and domains   crlreg 20599
      ​10.4  Division rings and fields
            10.4.1  Definition and basic properties   cdr 20637
            10.4.2  Sub-division rings   csdrg 20694
            10.4.3  Absolute value (abstract algebra)   cabv 20716
            10.4.4  Star rings   cstf 20745
            10.4.5  Totally ordered rings and fields   corng 20765
      ​10.5  Left modules
            10.5.1  Definition and basic properties   clmod 20786
            10.5.2  Subspaces and spans in a left module   clss 20857
            10.5.3  Homomorphisms and isomorphisms of left modules   clmhm 20946
            10.5.4  Subspace sum; bases for a left module   clbs 21001
      ​10.6  Vector spaces
            10.6.1  Definition and basic properties   clvec 21029
      ​10.7  Subring algebras and ideals
            10.7.1  Subring algebras   csra 21098
            ​*10.7.2  Left ideals and spans   clidl 21136
            10.7.3  Two-sided ideals and quotient rings   c2idl 21179
                  ​*10.7.3.1  Condition for a non-unital ring to be unital   rngqiprng1elbas 21216
            10.7.4  Principal ideal rings. Divisibility in the integers   clpidl 21250
            10.7.5  Principal ideal domains   cpid 21266
      ​10.8  The complex numbers as an algebraic extensible structure
            10.8.1  Definition and basic properties   cpsmet 21268
            ​*10.8.2  Ring of integers   czring 21376
                  ​*10.8.2.1  Example for a condition for a non-unital ring to be unital   pzriprnglem1 21411
            10.8.3  Algebraic constructions based on the complex numbers   czrh 21429
            10.8.4  Signs as subgroup of the complex numbers   cnmsgnsubg 21507
            10.8.5  Embedding of permutation signs into a ring   zrhpsgnmhm 21514
            10.8.6  The ordered field of real numbers   crefld 21534
      ​10.9  Generalized pre-Hilbert and Hilbert spaces
            10.9.1  Definition and basic properties   cphl 21554
            10.9.2  Orthocomplements and closed subspaces   cocv 21590
            10.9.3  Orthogonal projection and orthonormal bases   cpj 21630
​​*PART 11  BASIC LINEAR ALGEBRA
      ​11.1  Vectors and free modules
            *11.1.1  Direct sum of left modules   cdsmm 21661
            ​*11.1.2  Free modules   cfrlm 21676
            ​*11.1.3  Standard basis (unit vectors)   cuvc 21712
            ​*11.1.4  Independent sets and families   clindf 21734
            11.1.5  Characterization of free modules   lmimlbs 21766
      ​11.2  Associative algebras
            11.2.1  Definition and basic properties   casa 21780
      ​11.3  Abstract multivariate polynomials
            11.3.1  Definition and basic properties   cmps 21834
            11.3.2  Polynomial evaluation   ces 22000
            11.3.3  Additional definitions for (multivariate) polynomials   cslv 22036
            ​*11.3.4  Univariate polynomials   cps1 22080
            11.3.5  Univariate polynomial evaluation   ces1 22221
                  11.3.5.1  Specialization of polynomial evaluation as a ring homomorphism   evls1scafv 22274
      ​​*11.4  Matrices
            *11.4.1  The matrix multiplication   cmmul 22298
            ​*11.4.2  Square matrices   cmat 22315
            ​*11.4.3  The matrix algebra   matmulr 22346
            ​*11.4.4  Matrices of dimension 0 and 1   mat0dimbas0 22374
            ​*11.4.5  The subalgebras of diagonal and scalar matrices   cdmat 22396
            ​*11.4.6  Multiplication of a matrix with a "column vector"   cmvmul 22448
            11.4.7  Replacement functions for a square matrix   cmarrep 22464
            11.4.8  Submatrices   csubma 22484
      ​11.5  The determinant
            11.5.1  Definition and basic properties   cmdat 22492
            11.5.2  Determinants of 2 x 2 -matrices   m2detleiblem1 22532
            11.5.3  The matrix adjugate/adjunct   cmadu 22540
            ​*11.5.4  Laplace expansion of determinants (special case)   symgmatr01lem 22561
            11.5.5  Inverse matrix   invrvald 22584
            ​*11.5.6  Cramer's rule   slesolvec 22587
      ​​*11.6  Polynomial matrices
            11.6.1  Basic properties   pmatring 22600
            ​*11.6.2  Constant polynomial matrices   ccpmat 22611
            ​*11.6.3  Collecting coefficients of polynomial matrices   cdecpmat 22670
            ​*11.6.4  Ring isomorphism between polynomial matrices and polynomials over matrices   cpm2mp 22700
      ​​*11.7  The characteristic polynomial
            *11.7.1  Definition and basic properties   cchpmat 22734
            ​*11.7.2  The characteristic factor function G   fvmptnn04if 22757
            ​*11.7.3  The Cayley-Hamilton theorem   cpmadurid 22775
​PART 12  BASIC TOPOLOGY
      ​12.1  Topology
            ​*12.1.1  Topological spaces   ctop 22801
                  12.1.1.1  Topologies   ctop 22801
                  12.1.1.2  Topologies on sets   ctopon 22818
                  12.1.1.3  Topological spaces   ctps 22840
            12.1.2  Topological bases   ctb 22853
            12.1.3  Examples of topologies   distop 22903
            12.1.4  Closure and interior   ccld 22924
            12.1.5  Neighborhoods   cnei 23005
            12.1.6  Limit points and perfect sets   clp 23042
            12.1.7  Subspace topologies   restrcl 23065
            12.1.8  Order topology   ordtbaslem 23096
            12.1.9  Limits and continuity in topological spaces   ccn 23132
            12.1.10  Separated spaces: T0, T1, T2 (Hausdorff) ...   ct0 23214
            12.1.11  Compactness   ccmp 23294
            12.1.12  Bolzano-Weierstrass theorem   bwth 23318
            12.1.13  Connectedness   cconn 23319
            12.1.14  First- and second-countability   c1stc 23345
            12.1.15  Local topological properties   clly 23372
            12.1.16  Refinements   cref 23410
            12.1.17  Compactly generated spaces   ckgen 23441
            12.1.18  Product topologies   ctx 23468
            12.1.19  Continuous function-builders   cnmptid 23569
            12.1.20  Quotient maps and quotient topology   ckq 23601
            12.1.21  Homeomorphisms   chmeo 23661
      ​12.2  Filters and filter bases
            12.2.1  Filter bases   elmptrab 23735
            12.2.2  Filters   cfil 23753
            12.2.3  Ultrafilters   cufil 23807
            12.2.4  Filter limits   cfm 23841
            12.2.5  Extension by continuity   ccnext 23967
            12.2.6  Topological groups   ctmd 23978
            12.2.7  Infinite group sum on topological groups   ctsu 24034
            12.2.8  Topological rings, fields, vector spaces   ctrg 24064
      ​12.3  Uniform Structures and Spaces
            12.3.1  Uniform structures   cust 24108
            12.3.2  The topology induced by an uniform structure   cutop 24138
            12.3.3  Uniform Spaces   cuss 24161
            12.3.4  Uniform continuity   cucn 24182
            12.3.5  Cauchy filters in uniform spaces   ccfilu 24193
            12.3.6  Complete uniform spaces   ccusp 24204
      ​12.4  Metric spaces
            12.4.1  Pseudometric spaces   ispsmet 24212
            12.4.2  Basic metric space properties   cxms 24225
            12.4.3  Metric space balls   blfvalps 24291
            12.4.4  Open sets of a metric space   mopnval 24346
            12.4.5  Continuity in metric spaces   metcnp3 24448
            12.4.6  The uniform structure generated by a metric   metuval 24457
            12.4.7  Examples of metric spaces   dscmet 24480
            ​*12.4.8  Normed algebraic structures   cnm 24484
            12.4.9  Normed space homomorphisms (bounded linear operators)   cnmo 24613
            12.4.10  Topology on the reals   qtopbaslem 24666
            12.4.11  Topological definitions using the reals   cii 24788
            12.4.12  Path homotopy   chtpy 24886
            12.4.13  The fundamental group   cpco 24920
      ​12.5  Metric subcomplex vector spaces
            12.5.1  Subcomplex modules   cclm 24982
            ​*12.5.2  Subcomplex vector spaces   ccvs 25043
            ​*12.5.3  Normed subcomplex vector spaces   isncvsngp 25069
            12.5.4  Subcomplex pre-Hilbert spaces   ccph 25086
            12.5.5  Convergence and completeness   ccfil 25172
            12.5.6  Baire's Category Theorem   bcthlem1 25244
            12.5.7  Banach spaces and subcomplex Hilbert spaces   ccms 25252
                  12.5.7.1  The complete ordered field of the real numbers   retopn 25299
            12.5.8  Euclidean spaces   crrx 25303
            12.5.9  Minimizing Vector Theorem   minveclem1 25344
            12.5.10  Projection Theorem   pjthlem1 25357
​PART 13  BASIC REAL AND COMPLEX ANALYSIS
      ​13.1  Continuity
            13.1.1  Intermediate value theorem   pmltpclem1 25369
      ​13.2  Integrals
            13.2.1  Lebesgue measure   covol 25383
            13.2.2  Lebesgue integration   cmbf 25535
                  13.2.2.1  Lesbesgue integral   cmbf 25535
                  13.2.2.2  Lesbesgue directed integral   cdit 25767
      ​13.3  Derivatives
            13.3.1  Real and complex differentiation   climc 25783
                  13.3.1.1  Derivatives of functions of one complex or real variable   climc 25783
                  13.3.1.2  Results on real differentiation   dvferm1lem 25908
​PART 14  BASIC REAL AND COMPLEX FUNCTIONS
      ​14.1  Polynomials
            14.1.1  Polynomial degrees   cmdg 25978
            14.1.2  The division algorithm for univariate polynomials   cmn1 26051
            14.1.3  Elementary properties of complex polynomials   cply 26109
            14.1.4  The division algorithm for polynomials   cquot 26218
            14.1.5  Algebraic numbers   caa 26242
            14.1.6  Liouville's approximation theorem   aalioulem1 26260
      ​14.2  Sequences and series
            14.2.1  Taylor polynomials and Taylor's theorem   ctayl 26280
            14.2.2  Uniform convergence   culm 26305
            14.2.3  Power series   pserval 26339
      ​14.3  Basic trigonometry
            14.3.1  The exponential, sine, and cosine functions (cont.)   efcn 26373
            14.3.2  Properties of pi = 3.14159...   pilem1 26381
            14.3.3  Mapping of the exponential function   efgh 26470
            14.3.4  The natural logarithm on complex numbers   clog 26483
            ​*14.3.5  Logarithms to an arbitrary base   clogb 26694
            14.3.6  Theorems of Pythagoras, isosceles triangles, and intersecting chords   angval 26731
            14.3.7  Solutions of quadratic, cubic, and quartic equations   quad2 26769
            14.3.8  Inverse trigonometric functions   casin 26792
            14.3.9  The Birthday Problem   log2ublem1 26876
            14.3.10  Areas in R^2   carea 26885
            14.3.11  More miscellaneous converging sequences   rlimcnp 26895
            14.3.12  Inequality of arithmetic and geometric means   cvxcl 26915
            14.3.13  Euler-Mascheroni constant   cem 26922
            14.3.14  Zeta function   czeta 26943
            14.3.15  Gamma function   clgam 26946
      ​14.4  Basic number theory
            14.4.1  Wilson's theorem   wilthlem1 26998
            14.4.2  The Fundamental Theorem of Algebra   ftalem1 27003
            14.4.3  The Basel problem (ζ(2) = π²/6)   basellem1 27011
            14.4.4  Number-theoretical functions   ccht 27021
            14.4.5  Perfect Number Theorem   mersenne 27158
            14.4.6  Characters of Z/nZ   cdchr 27163
            14.4.7  Bertrand's postulate   bcctr 27206
            ​*14.4.8  Quadratic residues and the Legendre symbol   clgs 27225
            ​*14.4.9  Gauss' Lemma   gausslemma2dlem0a 27287
            14.4.10  Quadratic reciprocity   lgseisenlem1 27306
            14.4.11  All primes 4n+1 are the sum of two squares   2sqlem1 27348
            14.4.12  Chebyshev's Weak Prime Number Theorem, Dirichlet's Theorem   chebbnd1lem1 27400
            14.4.13  The Prime Number Theorem   mudivsum 27461
            14.4.14  Ostrowski's theorem   abvcxp 27546
​PART 15  SURREAL NUMBERS
      ​​*15.1  Sign sequence representation and Alling's axioms
            15.1.1  Definitions and initial properties   csur 27571
            15.1.2  Ordering   sltsolem1 27607
            15.1.3  Birthday Function   bdayfo 27609
            15.1.4  Density   fvnobday 27610
            ​*15.1.5  Full-Eta Property   bdayimaon 27625
      ​15.2  Initial consequences of Alling's axioms
            15.2.1  Ordering Theorems   csle 27676
            15.2.2  Birthday Theorems   bdayfun 27704
      ​​*15.3  Conway cut representation
            15.3.1  Conway cuts   csslt 27713
            15.3.2  Zero and One   c0s 27759
            15.3.3  Cuts and Options   cmade 27776
            15.3.4  Cofinality and coinitiality   cofsslt 27855
      ​15.4  Induction and recursion
            15.4.1  Induction and recursion on one variable   cnorec 27873
            15.4.2  Induction and recursion on two variables   cnorec2 27884
      ​15.5  Surreal arithmetic
            15.5.1  Addition   cadds 27895
            15.5.2  Negation and Subtraction   cnegs 27954
            15.5.3  Multiplication   cmuls 28038
            15.5.4  Division   cdivs 28119
            15.5.5  Absolute value   cabss 28168
      ​15.6  Subsystems of surreals
            15.6.1  Ordinal numbers   cons 28181
            15.6.2  Surreal recursive sequences   cseqs 28206
            15.6.3  Natural numbers   cnn0s 28235
            15.6.4  Integers   czs 28295
            15.6.5  Dyadic fractions   c2s 28326
            15.6.6  Real numbers   creno 28388
​​*PART 16  ELEMENTARY GEOMETRY
      ​16.1  Definition and Tarski's Axioms of Geometry
            16.1.1  Justification for the congruence notation   tgjustf 28444
      ​16.2  Tarskian Geometry
            16.2.1  Congruence   tgcgrcomimp 28448
            16.2.2  Betweenness   tgbtwntriv2 28458
            16.2.3  Dimension   tglowdim1 28471
            16.2.4  Betweenness and Congruence   tgifscgr 28479
            16.2.5  Congruence of a series of points   ccgrg 28481
            16.2.6  Motions   cismt 28503
            16.2.7  Colinearity   tglng 28517
            16.2.8  Connectivity of betweenness   tgbtwnconn1lem1 28543
            16.2.9  Less-than relation in geometric congruences   cleg 28553
            16.2.10  Rays   chlg 28571
            16.2.11  Lines   btwnlng1 28590
            16.2.12  Point inversions   cmir 28623
            16.2.13  Right angles   crag 28664
            16.2.14  Half-planes   islnopp 28710
            16.2.15  Midpoints and Line Mirroring   cmid 28743
            16.2.16  Congruence of angles   ccgra 28778
            16.2.17  Angle Comparisons   cinag 28806
            16.2.18  Congruence Theorems   tgsas1 28825
            16.2.19  Equilateral triangles   ceqlg 28836
      ​16.3  Properties of geometries
            16.3.1  Isomorphisms between geometries   f1otrgds 28840
      ​16.4  Geometry in Hilbert spaces
            16.4.1  Geometry in the complex plane   cchhllem 28858
            16.4.2  Geometry in Euclidean spaces   cee 28859
                  16.4.2.1  Definition of the Euclidean space   cee 28859
                  16.4.2.2  Tarski's axioms for geometry for the Euclidean space   axdimuniq 28884
                  16.4.2.3  EE^n fulfills Tarski's Axioms   ceeng 28948
​​*PART 17  GRAPH THEORY
      ​*17.1  Vertices and edges
            17.1.1  The edge function extractor for extensible structures   cedgf 28959
            ​*17.1.2  Vertices and indexed edges   cvtx 28967
                  17.1.2.1  Definitions and basic properties   cvtx 28967
                  17.1.2.2  The vertices and edges of a graph represented as ordered pair   opvtxval 28974
                  17.1.2.3  The vertices and edges of a graph represented as extensible structure   funvtxdmge2val 28982
                  17.1.2.4  Representations of graphs without edges   snstrvtxval 29008
                  17.1.2.5  Degenerated cases of representations of graphs   vtxval0 29010
            17.1.3  Edges as range of the edge function   cedg 29018
      ​​*17.2  Undirected graphs
            17.2.1  Undirected hypergraphs   cuhgr 29027
            17.2.2  Undirected pseudographs and multigraphs   cupgr 29051
            ​*17.2.3  Loop-free graphs   umgrislfupgrlem 29093
            17.2.4  Edges as subsets of vertices of graphs   uhgredgiedgb 29097
            ​*17.2.5  Undirected simple graphs   cuspgr 29119
            17.2.6  Examples for graphs   usgr0e 29207
            17.2.7  Subgraphs   csubgr 29238
            17.2.8  Finite undirected simple graphs   cfusgr 29287
            17.2.9  Neighbors, complete graphs and universal vertices   cnbgr 29303
                  17.2.9.1  Neighbors   cnbgr 29303
                  17.2.9.2  Universal vertices   cuvtx 29356
                  17.2.9.3  Complete graphs   ccplgr 29380
            17.2.10  Vertex degree   cvtxdg 29437
            ​*17.2.11  Regular graphs   crgr 29527
      ​​*17.3  Walks, paths and cycles
            *17.3.1  Walks   cewlks 29567
            17.3.2  Walks for loop-free graphs   lfgrwlkprop 29657
            17.3.3  Trails   ctrls 29660
            17.3.4  Paths and simple paths   cpths 29681
            17.3.5  Closed walks   cclwlks 29741
            17.3.6  Circuits and cycles   ccrcts 29755
            ​*17.3.7  Walks as words   cwwlks 29796
            17.3.8  Walks/paths of length 2 (as length 3 strings)   2wlkdlem1 29896
            17.3.9  Walks in regular graphs   rusgrnumwwlkl1 29939
            ​*17.3.10  Closed walks as words   cclwwlk 29951
                  17.3.10.1  Closed walks as words   cclwwlk 29951
                  17.3.10.2  Closed walks of a fixed length as words   cclwwlkn 29994
                  17.3.10.3  Closed walks on a vertex of a fixed length as words   cclwwlknon 30057
            17.3.11  Examples for walks, trails and paths   0ewlk 30084
            17.3.12  Connected graphs   cconngr 30156
      ​17.4  Eulerian paths and the Konigsberg Bridge problem
            ​*17.4.1  Eulerian paths   ceupth 30167
            ​*17.4.2  The Königsberg Bridge problem   konigsbergvtx 30216
      ​17.5  The Friendship Theorem
            17.5.1  Friendship graphs - basics   cfrgr 30228
            17.5.2  The friendship theorem for small graphs   frgr1v 30241
            17.5.3  Theorems according to Mertzios and Unger   2pthfrgrrn 30252
            ​*17.5.4  Huneke's Proof of the Friendship Theorem   frgrncvvdeqlem1 30269
​PART 18  GUIDES AND MISCELLANEA
      ​18.1  Guides (conventions, explanations, and examples)
            ​*18.1.1  Conventions   conventions 30370
            18.1.2  Natural deduction   natded 30373
            ​*18.1.3  Natural deduction examples   ex-natded5.2 30374
            18.1.4  Definitional examples   ex-or 30391
            18.1.5  Other examples   aevdemo 30430
      ​18.2  Humor
            18.2.1  April Fool's theorem   avril1 30433
      ​18.3  (Future - to be reviewed and classified)
            18.3.1  Planar incidence geometry   cplig 30444
            ​*18.3.2  Aliases kept to prevent broken links   dummylink 30457
​​*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 30459
            19.1.2  Abelian groups   cablo 30514
      ​19.2  Complex vector spaces
            19.2.1  Definition and basic properties   cvc 30528
            19.2.2  Examples of complex vector spaces   cnaddabloOLD 30551
      ​19.3  Normed complex vector spaces
            19.3.1  Definition and basic properties   cnv 30554
            19.3.2  Examples of normed complex vector spaces   cnnv 30647
            19.3.3  Induced metric of a normed complex vector space   imsval 30655
            19.3.4  Inner product   cdip 30670
            19.3.5  Subspaces   css 30691
      ​19.4  Operators on complex vector spaces
            19.4.1  Definitions and basic properties   clno 30710
      ​19.5  Inner product (pre-Hilbert) spaces
            19.5.1  Definition and basic properties   ccphlo 30782
            19.5.2  Examples of pre-Hilbert spaces   cncph 30789
            19.5.3  Properties of pre-Hilbert spaces   isph 30792
      ​19.6  Complex Banach spaces
            19.6.1  Definition and basic properties   ccbn 30832
            19.6.2  Examples of complex Banach spaces   cnbn 30839
            19.6.3  Uniform Boundedness Theorem   ubthlem1 30840
            19.6.4  Minimizing Vector Theorem   minvecolem1 30844
      ​19.7  Complex Hilbert spaces
            19.7.1  Definition and basic properties   chlo 30855
            19.7.2  Standard axioms for a complex Hilbert space   hlex 30868
            19.7.3  Examples of complex Hilbert spaces   cnchl 30886
            19.7.4  Hellinger-Toeplitz Theorem   htthlem 30887
​​*PART 20  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      ​20.1  Axiomatization of complex pre-Hilbert spaces
            20.1.1  Basic Hilbert space definitions   chba 30889
            20.1.2  Preliminary ZFC lemmas   df-hnorm 30938
            ​*20.1.3  Derive the Hilbert space axioms from ZFC set theory   axhilex-zf 30951
            ​*20.1.4  Introduce the vector space axioms for a Hilbert space   ax-hilex 30969
            20.1.5  Vector operations   hvmulex 30981
            20.1.6  Inner product postulates for a Hilbert space   ax-hfi 31049
      ​20.2  Inner product and norms
            20.2.1  Inner product   his5 31056
            20.2.2  Norms   dfhnorm2 31092
            20.2.3  Relate Hilbert space to normed complex vector spaces   hilablo 31130
            20.2.4  Bunjakovaskij-Cauchy-Schwarz inequality   bcsiALT 31149
      ​20.3  Cauchy sequences and completeness axiom
            20.3.1  Cauchy sequences and limits   hcau 31154
            20.3.2  Derivation of the completeness axiom from ZF set theory   hilmet 31164
            20.3.3  Completeness postulate for a Hilbert space   ax-hcompl 31172
            20.3.4  Relate Hilbert space to ZFC pre-Hilbert and Hilbert spaces   hhcms 31173
      ​20.4  Subspaces and projections
            20.4.1  Subspaces   df-sh 31177
            20.4.2  Closed subspaces   df-ch 31191
            20.4.3  Orthocomplements   df-oc 31222
            20.4.4  Subspace sum, span, lattice join, lattice supremum   df-shs 31278
            20.4.5  Projection theorem   pjhthlem1 31361
            20.4.6  Projectors   df-pjh 31365
      ​20.5  Properties of Hilbert subspaces
            20.5.1  Orthomodular law   omlsilem 31372
            20.5.2  Projectors (cont.)   pjhtheu2 31386
            20.5.3  Hilbert lattice operations   sh0le 31410
            20.5.4  Span (cont.) and one-dimensional subspaces   spansn0 31511
            20.5.5  Commutes relation for Hilbert lattice elements   df-cm 31553
            20.5.6  Foulis-Holland theorem   fh1 31588
            20.5.7  Quantum Logic Explorer axioms   qlax1i 31597
            20.5.8  Orthogonal subspaces   chscllem1 31607
            20.5.9  Orthoarguesian laws 5OA and 3OA   5oalem1 31624
            20.5.10  Projectors (cont.)   pjorthi 31639
            20.5.11  Mayet's equation E_3   mayete3i 31698
      ​20.6  Operators on Hilbert spaces
            ​*20.6.1  Operator sum, difference, and scalar multiplication   df-hosum 31700
            20.6.2  Zero and identity operators   df-h0op 31718
            20.6.3  Operations on Hilbert space operators   hoaddcl 31728
            20.6.4  Linear, continuous, bounded, Hermitian, unitary operators and norms   df-nmop 31809
            20.6.5  Linear and continuous functionals and norms   df-nmfn 31815
            20.6.6  Adjoint   df-adjh 31819
            20.6.7  Dirac bra-ket notation   df-bra 31820
            20.6.8  Positive operators   df-leop 31822
            20.6.9  Eigenvectors, eigenvalues, spectrum   df-eigvec 31823
            20.6.10  Theorems about operators and functionals   nmopval 31826
            20.6.11  Riesz lemma   riesz3i 32032
            20.6.12  Adjoints (cont.)   cnlnadjlem1 32037
            20.6.13  Quantum computation error bound theorem   unierri 32074
            20.6.14  Dirac bra-ket notation (cont.)   branmfn 32075
            20.6.15  Positive operators (cont.)   leopg 32092
            20.6.16  Projectors as operators   pjhmopi 32116
      ​20.7  States on a Hilbert lattice and Godowski's equation
            20.7.1  States on a Hilbert lattice   df-st 32181
            20.7.2  Godowski's equation   golem1 32241
      ​20.8  Cover relation, atoms, exchange axiom, and modular symmetry
            20.8.1  Covers relation; modular pairs   df-cv 32249
            20.8.2  Atoms   df-at 32308
            20.8.3  Superposition principle   superpos 32324
            20.8.4  Atoms, exchange and covering properties, atomicity   chcv1 32325
            20.8.5  Irreducibility   chirredlem1 32360
            20.8.6  Atoms (cont.)   atcvat3i 32366
            20.8.7  Modular symmetry   mdsymlem1 32373
​PART 21  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      ​21.1  Mathboxes for user contributions
            21.1.1  Mathbox guidelines   mathbox 32412
      ​21.2  Mathbox for Stefan Allan
      ​21.3  Mathbox for Thierry Arnoux
            21.3.1  Propositional Calculus - misc additions   ad11antr 32417
            21.3.2  Predicate Calculus   sbc2iedf 32434
                  21.3.2.1  Predicate Calculus - misc additions   sbc2iedf 32434
                  21.3.2.2  Restricted quantification - misc additions   ralcom4f 32436
                  21.3.2.3  Equality   eqtrb 32443
                  21.3.2.4  Double restricted existential uniqueness quantification   opsbc2ie 32445
                  21.3.2.5  Double restricted existential uniqueness quantification syntax   w2reu 32447
                  21.3.2.6  Substitution (without distinct variables) - misc additions   sbceqbidf 32456
                  21.3.2.7  Existential "at most one" - misc additions   mo5f 32458
                  21.3.2.8  Existential uniqueness - misc additions   reuxfrdf 32460
                  21.3.2.9  Restricted "at most one" - misc additions   rmoxfrd 32462
                  21.3.2.10  Restricted iota (description binder)   riotaeqbidva 32465
            21.3.3  General Set Theory   dmrab 32466
                  21.3.3.1  Class abstractions (a.k.a. class builders)   dmrab 32466
                  21.3.3.2  Image Sets   abrexdomjm 32477
                  21.3.3.3  Set relations and operations - misc additions   nelun 32483
                  21.3.3.4  Unordered pairs   elpreq 32498
                  21.3.3.5  Unordered triples   tpssg 32507
                  21.3.3.6  Conditional operator - misc additions   ifeqeqx 32512
                  21.3.3.7  Set union   uniinn0 32520
                  21.3.3.8  Indexed union - misc additions   cbviunf 32525
                  21.3.3.9  Indexed intersection - misc additions   iinabrex 32539
                  21.3.3.10  Disjointness - misc additions   disjnf 32540
            21.3.4  Relations and Functions   xpdisjres 32568
                  21.3.4.1  Relations - misc additions   xpdisjres 32568
                  21.3.4.2  Functions - misc additions   fconst7v 32593
                  21.3.4.3  Operations - misc additions   mpomptxf 32649
                  21.3.4.4  The mapping operation   elmaprd 32651
                  21.3.4.5  Support of a function   suppovss 32652
                  21.3.4.6  Explicit Functions with one or two points as a domain   cosnopne 32665
                  21.3.4.7  Isomorphisms - misc. additions   gtiso 32672
                  21.3.4.8  Disjointness (additional proof requiring functions)   disjdsct 32674
                  21.3.4.9  First and second members of an ordered pair - misc additions   df1stres 32675
                  21.3.4.10  Finite Sets   imafi2 32683
                  21.3.4.11  Countable Sets   snct 32685
            21.3.5  Real and Complex Numbers   sgnval2 32708
                  21.3.5.1  Complex operations - misc. additions   creq0 32709
                  21.3.5.2  Ordering on reals - misc additions   lt2addrd 32724
                  21.3.5.3  Extended reals - misc additions   xrlelttric 32725
                  21.3.5.4  Extended nonnegative integers - misc additions   xnn0gt0 32742
                  21.3.5.5  Real number intervals - misc additions   joiniooico 32747
                  21.3.5.6  Finite intervals of integers - misc additions   uzssico 32757
                  21.3.5.7  Half-open integer ranges - misc additions   iundisjfi 32768
                  21.3.5.8  The ` # ` (set size) function - misc additions   hashunif 32778
                  21.3.5.9  The greatest common divisor operator - misc. additions   elq2 32784
                  21.3.5.10  Integers   nn0split01 32790
                  21.3.5.11  Decimal numbers   dfdec100 32803
            21.3.6  Real and complex functions   sgncl 32804
                  21.3.6.1  Signum (sgn or sign) function - misc. additions   sgncl 32804
                  21.3.6.2  Integer powers - misc. additions   nexple 32817
                  21.3.6.3  Indicator Functions   cind 32821
            ​*21.3.7  Decimal expansion   cdp2 32841
                  ​*21.3.7.1  Decimal point   cdp 32858
                  21.3.7.2  Division in the extended real number system   cxdiv 32887
            21.3.8  Words over a set - misc additions   wrdres 32906
                  21.3.8.1  Splicing words (substring replacement)   splfv3 32929
                  21.3.8.2  Cyclic shift of words   1cshid 32930
            21.3.9  Extensible Structures   ressplusf 32934
                  21.3.9.1  Structure restriction operator   ressplusf 32934
                  21.3.9.2  Posets   ressprs 32937
                  21.3.9.3  Complete lattices   clatp0cl 32947
                  21.3.9.4  Order Theory   cmnt 32949
                  21.3.9.5  Extended reals Structure - misc additions   ax-xrssca 32975
                  21.3.9.6  The extended nonnegative real numbers commutative monoid   xrge00 32985
            21.3.10  Algebra   mndcld 32993
                  21.3.10.1  Monoids   mndcld 32993
                  21.3.10.2  Monoids Homomorphisms   abliso 33007
                  21.3.10.3  Groups - misc additions   grpsubcld 33011
                  21.3.10.4  Finitely supported group sums - misc additions   gsumsubg 33016
                  21.3.10.5  Group or monoid sums over words   gsumwun 33035
                  21.3.10.6  Centralizers and centers - misc additions   cntzun 33038
                  21.3.10.7  The symmetric group   symgfcoeu 33041
                  21.3.10.8  Transpositions   pmtridf1o 33053
                  21.3.10.9  Permutation Signs   psgnid 33056
                  21.3.10.10  Permutation cycles   ctocyc 33065
                  21.3.10.11  The Alternating Group   evpmval 33104
                  21.3.10.12  Signum in an ordered monoid   csgns 33117
                  21.3.10.13  Fixed points   cfxp 33122
                  21.3.10.14  The Archimedean property for generic ordered algebraic structures   cinftm 33135
                  21.3.10.15  Semiring left modules   cslmd 33159
                  21.3.10.16  Simple groups   prmsimpcyc 33187
                  21.3.10.17  Rings - misc additions   ringdi22 33188
                  21.3.10.18  Subrings generated by a set   elrgspnlem1 33199
                  21.3.10.19  The zero ring   irrednzr 33207
                  21.3.10.20  Localization of rings   cerl 33210
                  21.3.10.21  Integral Domains   domnmuln0rd 33231
                  21.3.10.22  Euclidean Domains   ceuf 33244
                  21.3.10.23  Division Rings   ringinveu 33250
                  21.3.10.24  The field of rational numbers   qfld 33253
                  21.3.10.25  Subfields   subsdrg 33254
                  21.3.10.26  Field of fractions   cfrac 33258
                  21.3.10.27  Field extensions generated by a set   cfldgen 33266
                  21.3.10.28  Ring homomorphisms - misc additions   rhmdvd 33279
                  21.3.10.29  Scalar restriction operation   cresv 33281
                  21.3.10.30  The commutative ring of gaussian integers   gzcrng 33296
                  21.3.10.31  The archimedean ordered field of real numbers   cnfldfld 33297
                  21.3.10.32  The quotient map and quotient modules   qusker 33304
                  21.3.10.33  The ring of integers modulo ` N `   znfermltl 33321
                  21.3.10.34  Independent sets and families   islinds5 33322
                  21.3.10.35  Ring associates, ring units   dvdsruassoi 33339
                  ​*21.3.10.36  Subgroup sum / Sumset / Minkowski sum   elgrplsmsn 33345
                  21.3.10.37  The quotient map   quslsm 33360
                  21.3.10.38  Ideals   lidlmcld 33374
                  21.3.10.39  Prime Ideals   cprmidl 33390
                  21.3.10.40  Maximal Ideals   cmxidl 33414
                  21.3.10.41  The semiring of ideals of a ring   cidlsrg 33455
                  21.3.10.42  Prime Elements   rprmval 33471
                  21.3.10.43  Unique factorization domains   cufd 33493
                  21.3.10.44  The ring of integers   zringidom 33506
                  21.3.10.45  Univariate Polynomials   0ringmon1p 33510
                  21.3.10.46  Polynomial quotient and polynomial remainder   q1pdir 33553
                  21.3.10.47  Multivariate Polynomials   psrbasfsupp 33562
                  21.3.10.48  The ring of symmetric polynomials   csply 33568
                  21.3.10.49  The subring algebra   sra1r 33583
                  21.3.10.50  Division Ring Extensions   drgext0g 33592
                  21.3.10.51  Vector Spaces   lvecdimfi 33598
                  21.3.10.52  Vector Space Dimension   cldim 33601
            21.3.11  Field Extensions   cfldext 33641
                  21.3.11.1  Algebraic numbers   cirng 33686
                  21.3.11.2  Algebraic extensions   calgext 33698
                  21.3.11.3  Minimal polynomials   cminply 33702
                  21.3.11.4  Quadratic Field Extensions   rtelextdg2lem 33729
                  21.3.11.5  Towers of quadratic extentions   fldext2chn 33731
            ​*21.3.12  Constructible Numbers   cconstr 33732
                  21.3.12.1  Impossible constructions   2sqr3minply 33783
            21.3.13  Matrices   csmat 33796
                  21.3.13.1  Submatrices   csmat 33796
                  21.3.13.2  Matrix literals   clmat 33814
                  21.3.13.3  Laplace expansion of determinants   mdetpmtr1 33826
            21.3.14  Topology   ist0cld 33836
                  21.3.14.1  Open maps   txomap 33837
                  21.3.14.2  Topology of the unit circle   qtopt1 33838
                  21.3.14.3  Refinements   reff 33842
                  21.3.14.4  Open cover refinement property   ccref 33845
                  21.3.14.5  Lindelöf spaces   cldlf 33855
                  21.3.14.6  Paracompact spaces   cpcmp 33858
                  ​*21.3.14.7  Spectrum of a ring   crspec 33865
                  21.3.14.8  Pseudometrics   cmetid 33889
                  21.3.14.9  Continuity - misc additions   hauseqcn 33901
                  21.3.14.10  Topology of the closed unit interval   elunitge0 33902
                  21.3.14.11  Topology of ` ( RR X. RR ) `   unicls 33906
                  21.3.14.12  Order topology - misc. additions   cnvordtrestixx 33916
                  21.3.14.13  Continuity in topological spaces - misc. additions   mndpluscn 33929
                  21.3.14.14  Topology of the extended nonnegative real numbers ordered monoid   xrge0hmph 33935
                  21.3.14.15  Limits - misc additions   lmlim 33950
                  21.3.14.16  Univariate polynomials   pl1cn 33958
            21.3.15  Uniform Stuctures and Spaces   chcmp 33959
                  21.3.15.1  Hausdorff uniform completion   chcmp 33959
            21.3.16  Topology and algebraic structures   zringnm 33961
                  21.3.16.1  The norm on the ring of the integer numbers   zringnm 33961
                  21.3.16.2  Topological ` ZZ ` -modules   zlm0 33963
                  21.3.16.3  Canonical embedding of the field of the rational numbers into a division ring   cqqh 33973
                  21.3.16.4  Canonical embedding of the real numbers into a complete ordered field   crrh 33996
                  21.3.16.5  Embedding from the extended real numbers into a complete lattice   cxrh 34019
                  21.3.16.6  Canonical embeddings into the ordered field of the real numbers   zrhre 34022
                  ​*21.3.16.7  Topological Manifolds   cmntop 34025
                  21.3.16.8  Extended sum   cesum 34030
            21.3.17  Mixed Function/Constant operation   cofc 34098
            21.3.18  Abstract measure   csiga 34111
                  21.3.18.1  Sigma-Algebra   csiga 34111
                  21.3.18.2  Generated sigma-Algebra   csigagen 34141
                  ​*21.3.18.3  lambda and pi-Systems, Rings of Sets   ispisys 34155
                  21.3.18.4  The Borel algebra on the real numbers   cbrsiga 34184
                  21.3.18.5  Product Sigma-Algebra   csx 34191
                  21.3.18.6  Measures   cmeas 34198
                  21.3.18.7  The counting measure   cntmeas 34229
                  21.3.18.8  The Lebesgue measure - misc additions   voliune 34232
                  21.3.18.9  The Dirac delta measure   cdde 34235
                  21.3.18.10  The 'almost everywhere' relation   cae 34240
                  21.3.18.11  Measurable functions   cmbfm 34252
                  21.3.18.12  Borel Algebra on ` ( RR X. RR ) `   br2base 34272
                  ​*21.3.18.13  Caratheodory's extension theorem   coms 34294
            21.3.19  Integration   itgeq12dv 34329
                  21.3.19.1  Lebesgue integral - misc additions   itgeq12dv 34329
                  21.3.19.2  Bochner integral   citgm 34330
            21.3.20  Euler's partition theorem   oddpwdc 34357
            21.3.21  Sequences defined by strong recursion   csseq 34386
            21.3.22  Fibonacci Numbers   cfib 34399
            21.3.23  Probability   cprb 34410
                  21.3.23.1  Probability Theory   cprb 34410
                  21.3.23.2  Conditional Probabilities   ccprob 34434
                  21.3.23.3  Real-valued Random Variables   crrv 34443
                  21.3.23.4  Preimage set mapping operator   corvc 34459
                  21.3.23.5  Distribution Functions   orvcelval 34472
                  21.3.23.6  Cumulative Distribution Functions   orvclteel 34476
                  21.3.23.7  Probabilities - example   coinfliplem 34482
                  21.3.23.8  Bertrand's Ballot Problem   ballotlemoex 34489
            21.3.24  Signum (sgn or sign) function - misc. additions   fzssfzo 34542
                  21.3.24.1  Operations on words   ccatmulgnn0dir 34545
            21.3.25  Polynomials with real coefficients - misc additions   plymul02 34549
            21.3.26  Descartes's rule of signs   signspval 34555
                  21.3.26.1  Sign changes in a word over real numbers   signspval 34555
                  21.3.26.2  Counting sign changes in a word over real numbers   signslema 34565
            21.3.27  Number Theory   iblidicc 34595
                  21.3.27.1  Representations of a number as sums of integers   crepr 34611
                  21.3.27.2  Vinogradov Trigonometric Sums and the Circle Method   cvts 34638
                  21.3.27.3  The Ternary Goldbach Conjecture: Final Statement   ax-hgt749 34647
            21.3.28  Elementary Geometry   cstrkg2d 34667
                  ​*21.3.28.1  Two-dimensional geometry   cstrkg2d 34667
                  21.3.28.2  Outer Five Segment (not used, no need to move to main)   cafs 34672
            ​*21.3.29  LeftPad Project   clpad 34677
      ​​*21.4  Mathbox for Jonathan Ben-Naim
            21.4.1  First-order logic and set theory   bnj170 34700
            21.4.2  Well founded induction and recursion   bnj110 34860
            21.4.3  The existence of a minimal element in certain classes   bnj69 35012
            21.4.4  Well-founded induction   bnj1204 35014
            21.4.5  Well-founded recursion, part 1 of 3   bnj60 35064
            21.4.6  Well-founded recursion, part 2 of 3   bnj1500 35070
            21.4.7  Well-founded recursion, part 3 of 3   bnj1522 35074
      ​21.5  Mathbox for BTernaryTau
            21.5.1  First-order logic   nfan1c 35075
                  21.5.1.1  Auxiliary axiom schemes   nfan1c 35075
            21.5.2  ZF set theory   exdifsn 35081
                  21.5.2.1  Finitism   ax-regs 35096
                  21.5.2.2  Derive ax-regs   axregs 35113
                  21.5.2.3  Global choice   gblacfnacd 35114
            21.5.3  Real and complex numbers   zltp1ne 35122
            21.5.4  Graph theory   lfuhgr 35130
                  21.5.4.1  Acyclic graphs   cacycgr 35154
      ​21.6  Mathbox for Mario Carneiro
            21.6.1  Predicate calculus with all distinct variables   ax-7d 35171
            21.6.2  Miscellaneous stuff   quartfull 35177
            21.6.3  Derangements and the Subfactorial   deranglem 35178
            21.6.4  The Erdős-Szekeres theorem   erdszelem1 35203
            21.6.5  The Kuratowski closure-complement theorem   kur14lem1 35218
            21.6.6  Retracts and sections   cretr 35229
            21.6.7  Path-connected and simply connected spaces   cpconn 35231
            21.6.8  Covering maps   ccvm 35267
            21.6.9  Normal numbers   snmlff 35341
            21.6.10  Godel-sets of formulas - part 1   cgoe 35345
            21.6.11  Godel-sets of formulas - part 2   cgon 35444
            21.6.12  Models of ZF   cgze 35458
            ​*21.6.13  Metamath formal systems   cmcn 35472
            21.6.14  Grammatical formal systems   cm0s 35597
            21.6.15  Models of formal systems   cmuv 35617
            21.6.16  Splitting fields   ccpms 35639
            21.6.17  p-adic number fields   czr 35659
      ​​*21.7  Mathbox for Filip Cernatescu
      ​21.8  Mathbox for Paul Chapman
            21.8.1  Real and complex numbers (cont.)   climuzcnv 35683
            21.8.2  Miscellaneous theorems   elfzm12 35687
      ​21.9  Mathbox for Hongxiu Chen
      ​21.10  Mathbox for Adrian Ducourtial
            21.10.1  Propositional calculus   currybi 35700
            21.10.2  Clone theory   ccloneop 35707
      ​21.11  Mathbox for Scott Fenton
            21.11.1  ZFC Axioms in primitive form   axextprim 35713
            21.11.2  Untangled classes   untelirr 35720
            21.11.3  Extra propositional calculus theorems   3jaodd 35727
            21.11.4  Misc. Useful Theorems   nepss 35730
            21.11.5  Properties of real and complex numbers   sqdivzi 35740
            21.11.6  Infinite products   iprodefisumlem 35752
            21.11.7  Factorial limits   faclimlem1 35755
            21.11.8  Greatest common divisor and divisibility   gcd32 35761
            21.11.9  Properties of relationships   dftr6 35763
            21.11.10  Properties of functions and mappings   funpsstri 35778
            21.11.11  Set induction (or epsilon induction)   setinds 35791
            21.11.12  Ordinal numbers   elpotr 35794
            21.11.13  Defined equality axioms   axextdfeq 35810
            21.11.14  Hypothesis builders   hbntg 35818
            21.11.15  Well-founded zero, successor, and limits   cwsuc 35823
            21.11.16  Quantifier-free definitions   ctxp 35843
            21.11.17  Alternate ordered pairs   caltop 35969
            21.11.18  Geometry in the Euclidean space   cofs 35995
                  21.11.18.1  Congruence properties   cofs 35995
                  21.11.18.2  Betweenness properties   btwntriv2 36025
                  21.11.18.3  Segment Transportation   ctransport 36042
                  21.11.18.4  Properties relating betweenness and congruence   cifs 36048
                  21.11.18.5  Connectivity of betweenness   btwnconn1lem1 36100
                  21.11.18.6  Segment less than or equal to   csegle 36119
                  21.11.18.7  Outside-of relationship   coutsideof 36132
                  21.11.18.8  Lines and Rays   cline2 36147
            21.11.19  Forward difference   cfwddif 36171
            21.11.20  Rank theorems   rankung 36179
            21.11.21  Hereditarily Finite Sets   chf 36185
      ​21.12  Mathbox for Gino Giotto
            21.12.1  Equality theorems.   rmoeqi 36200
                  21.12.1.1  Inference versions.   rmoeqi 36200
                  21.12.1.2  Deduction versions.   rmoeqdv 36225
            21.12.2  Change bound variables.   in-ax8 36237
                  21.12.2.1  Change bound variables and domains.   cbvralvw2 36239
                  21.12.2.2  Change bound variables, deduction versions.   cbvmodavw 36263
                  21.12.2.3  Change bound variables and domains, deduction versions.   cbvrmodavw2 36296
            21.12.3  Study of ax-mulf usage.   mpomulnzcnf 36312
      ​21.13  Mathbox for Jeff Hankins
            21.13.1  Miscellany   a1i14 36313
            21.13.2  Basic topological facts   topbnd 36337
            21.13.3  Topology of the real numbers   ivthALT 36348
            21.13.4  Refinements   cfne 36349
            21.13.5  Neighborhood bases determine topologies   neibastop1 36372
            21.13.6  Lattice structure of topologies   topmtcl 36376
            21.13.7  Filter bases   fgmin 36383
            21.13.8  Directed sets, nets   tailfval 36385
      ​21.14  Mathbox for Anthony Hart
            21.14.1  Propositional Calculus   tb-ax1 36396
            21.14.2  Predicate Calculus   nalfal 36416
            21.14.3  Miscellaneous single axioms   meran1 36424
            21.14.4  Connective Symmetry   negsym1 36430
      ​21.15  Mathbox for Chen-Pang He
            21.15.1  Ordinal topology   ontopbas 36441
      ​21.16  Mathbox for Jeff Hoffman
            21.16.1  Inferences for finite induction on generic function values   fveleq 36464
            21.16.2  gdc.mm   nnssi2 36468
      ​21.17  Mathbox for Matthew House
            21.17.1  Relations on well-ordered indexed unions   weiunlem1 36475
      ​21.18  Mathbox for Asger C. Ipsen
            21.18.1  Continuous nowhere differentiable functions   dnival 36484
      ​​*21.19  Mathbox for BJ
            *21.19.1  Propositional calculus   bj-mp2c 36553
                  *21.19.1.1  Derived rules of inference   bj-mp2c 36553
                  ​*21.19.1.2  A syntactic theorem   bj-0 36555
                  21.19.1.3  Minimal implicational calculus   bj-a1k 36557
                  ​*21.19.1.4  Positive calculus   bj-syl66ib 36568
                  21.19.1.5  Implication and negation   bj-con2com 36574
                  ​*21.19.1.6  Disjunction   bj-jaoi1 36584
                  ​*21.19.1.7  Logical equivalence   bj-dfbi4 36586
                  21.19.1.8  The conditional operator for propositions   bj-consensus 36591
                  ​*21.19.1.9  Propositional calculus: miscellaneous   bj-imbi12 36596
            ​*21.19.2  Modal logic   bj-axdd2 36605
            ​*21.19.3  Provability logic   cprvb 36610
            ​*21.19.4  First-order logic   bj-genr 36619
                  21.19.4.1  Adding ax-gen   bj-genr 36619
                  21.19.4.2  Adding ax-4   bj-2alim 36623
                  21.19.4.3  Adding ax-5   bj-ax12wlem 36657
                  21.19.4.4  Equality and substitution   bj-ssbeq 36666
                  21.19.4.5  Adding ax-6   bj-spimvwt 36682
                  21.19.4.6  Adding ax-7   bj-cbvexw 36689
                  21.19.4.7  Membership predicate, ax-8 and ax-9   bj-ax89 36691
                  21.19.4.8  Adding ax-11   bj-alcomexcom 36693
                  21.19.4.9  Adding ax-12   axc11n11 36695
                  21.19.4.10  Nonfreeness   wnnf 36736
                  21.19.4.11  Adding ax-13   bj-axc10 36796
                  ​*21.19.4.12  Removing dependencies on ax-13 (and ax-11)   bj-axc10v 36806
                  ​*21.19.4.13  Distinct var metavariables   bj-hbaeb2 36831
                  ​*21.19.4.14  Around ~ equsal   bj-equsal1t 36835
                  ​*21.19.4.15  Some Principia Mathematica proofs   stdpc5t 36840
                  21.19.4.16  Alternate definition of substitution   bj-sbsb 36850
                  21.19.4.17  Lemmas for substitution   bj-sbf3 36852
                  21.19.4.18  Existential uniqueness   bj-eu3f 36854
                  ​*21.19.4.19  First-order logic: miscellaneous   bj-sblem1 36855
            21.19.5  Set theory   eliminable1 36872
                  ​*21.19.5.1  Eliminability of class terms   eliminable1 36872
                  ​*21.19.5.2  Classes without the axiom of extensionality   bj-denoteslem 36884
                  21.19.5.3  Characterization among sets versus among classes   elelb 36910
                  ​*21.19.5.4  The nonfreeness quantifier for classes   bj-nfcsym 36912
                  ​*21.19.5.5  Lemmas for class substitution   bj-sbeqALT 36913
                  21.19.5.6  Removing some axiom requirements and disjoint variable conditions   bj-exlimvmpi 36924
                  ​*21.19.5.7  Class abstractions   bj-elabd2ALT 36938
                  21.19.5.8  Generalized class abstractions   bj-cgab 36946
                  ​*21.19.5.9  Restricted nonfreeness   wrnf 36954
                  ​*21.19.5.10  Russell's paradox   bj-ru1 36956
                  21.19.5.11  Curry's paradox in set theory   currysetlem 36958
                  ​*21.19.5.12  Some disjointness results   bj-n0i 36964
                  ​*21.19.5.13  Complements on direct products   bj-xpimasn 36968
                  ​*21.19.5.14  "Singletonization" and tagging   bj-snsetex 36976
                  ​*21.19.5.15  Tuples of classes   bj-cproj 37003
                  ​*21.19.5.16  Set theory: elementary operations relative to a universe   bj-rcleqf 37038
                  ​*21.19.5.17  Axioms for finite unions   bj-abex 37043
                  ​*21.19.5.18  Set theory: miscellaneous   eleq2w2ALT 37060
                  ​*21.19.5.19  Evaluation at a class   bj-evaleq 37085
                  21.19.5.20  Elementwise operations   celwise 37092
                  ​*21.19.5.21  Elementwise intersection (families of sets induced on a subset)   bj-rest00 37094
                  21.19.5.22  Moore collections (complements)   bj-raldifsn 37113
                  21.19.5.23  Maps-to notation for functions with three arguments   bj-0nelmpt 37129
                  ​*21.19.5.24  Currying   csethom 37135
                  ​*21.19.5.25  Setting components of extensible structures   cstrset 37147
            ​*21.19.6  Extended real and complex numbers, real and complex projective lines   bj-nfald 37150
                  21.19.6.1  Complements on class abstractions of ordered pairs and binary relations   bj-nfald 37150
                  ​*21.19.6.2  Identity relation (complements)   bj-opabssvv 37163
                  ​*21.19.6.3  Functionalized identity (diagonal in a Cartesian square)   cdiag2 37185
                  ​*21.19.6.4  Direct image and inverse image   cimdir 37191
                  ​*21.19.6.5  Extended numbers and projective lines as sets   cfractemp 37209
                  ​*21.19.6.6  Addition and opposite   caddcc 37250
                  ​*21.19.6.7  Order relation on the extended reals   cltxr 37254
                  ​*21.19.6.8  Argument, multiplication and inverse   carg 37256
                  21.19.6.9  The canonical bijection from the finite ordinals   ciomnn 37262
                  21.19.6.10  Divisibility   cnnbar 37273
            ​*21.19.7  Monoids   bj-smgrpssmgm 37281
                  ​*21.19.7.1  Finite sums in monoids   cfinsum 37296
            ​*21.19.8  Affine, Euclidean, and Cartesian geometry   bj-fvimacnv0 37299
                  *21.19.8.1  Real vector spaces   bj-fvimacnv0 37299
                  ​*21.19.8.2  Complex numbers (supplements)   bj-subcom 37321
                  ​*21.19.8.3  Barycentric coordinates   bj-bary1lem 37323
            21.19.9  Monoid of endomorphisms   cend 37326
      ​21.20  Mathbox for Jim Kingdon
            21.20.1  Circle constant   taupilem3 37332
            21.20.2  Number theory   dfgcd3 37337
            21.20.3  Real numbers   irrdifflemf 37338
      ​21.21  Mathbox for ML
            21.21.1  Miscellaneous   csbrecsg 37341
            21.21.2  Cartesian exponentiation   cfinxp 37396
            21.21.3  Topology   iunctb2 37416
                  ​*21.21.3.1  Pi-base theorems   pibp16 37426
      ​21.22  Mathbox for Wolf Lammen
            21.22.1  1. Bootstrapping   wl-section-boot 37435
            21.22.2  Implication chains   wl-section-impchain 37459
            21.22.3  Theorems around the conditional operator   wl-ifp-ncond1 37477
            21.22.4  Alternative development of hadd, cadd   wl-df-3xor 37481
            21.22.5  An alternative axiom ~ ax-13   ax-wl-13v 37506
            21.22.6  Bootstrapping set theory with classes   wl-cleq-0 37508
            21.22.7  Other stuff   wl-mps 37520
      ​21.23  Mathbox for Brendan Leahy
      ​21.24  Mathbox for Jeff Madsen
            21.24.1  Logic and set theory   unirep 37733
            21.24.2  Real and complex numbers; integers   filbcmb 37759
            21.24.3  Sequences and sums   sdclem2 37761
            21.24.4  Topology   subspopn 37771
            21.24.5  Metric spaces   metf1o 37774
            21.24.6  Continuous maps and homeomorphisms   constcncf 37781
            21.24.7  Boundedness   ctotbnd 37785
            21.24.8  Isometries   cismty 37817
            21.24.9  Heine-Borel Theorem   heibor1lem 37828
            21.24.10  Banach Fixed Point Theorem   bfplem1 37841
            21.24.11  Euclidean space   crrn 37844
            21.24.12  Intervals (continued)   ismrer1 37857
            21.24.13  Operation properties   cass 37861
            21.24.14  Groups and related structures   cmagm 37867
            21.24.15  Group homomorphism and isomorphism   cghomOLD 37902
            21.24.16  Rings   crngo 37913
            21.24.17  Division Rings   cdrng 37967
            21.24.18  Ring homomorphisms   crngohom 37979
            21.24.19  Commutative rings   ccm2 38008
            21.24.20  Ideals   cidl 38026
            21.24.21  Prime rings and integral domains   cprrng 38065
            21.24.22  Ideal generators   cigen 38078
      ​21.25  Mathbox for Giovanni Mascellani
            ​*21.25.1  Tools for automatic proof building   efald2 38097
            ​*21.25.2  Tseitin axioms   fald 38148
            ​*21.25.3  Equality deductions   iuneq2f 38175
            ​*21.25.4  Miscellanea   orcomdd 38186
      ​21.26  Mathbox for Peter Mazsa
            21.26.1  Notations   cxrn 38193
            21.26.2  Preparatory theorems   el2v1 38236
            21.26.3  Range Cartesian product   df-xrn 38378
            21.26.4  Cosets by ` R `   df-coss 38427
            21.26.5  Relations   df-rels 38501
            21.26.6  Subset relations   df-ssr 38514
            21.26.7  Reflexivity   df-refs 38526
            21.26.8  Converse reflexivity   df-cnvrefs 38541
            21.26.9  Symmetry   df-syms 38558
            21.26.10  Reflexivity and symmetry   symrefref2 38579
            21.26.11  Transitivity   df-trs 38588
            21.26.12  Equivalence relations   df-eqvrels 38600
            21.26.13  Redundancy   df-redunds 38639
            21.26.14  Domain quotients   df-dmqss 38654
            21.26.15  Equivalence relations on domain quotients   df-ers 38680
            21.26.16  Functions   df-funss 38697
            21.26.17  Disjoints vs. converse functions   df-disjss 38720
            21.26.18  Antisymmetry   df-antisymrel 38777
            21.26.19  Partitions: disjoints on domain quotients   df-parts 38782
            21.26.20  Partition-Equivalence Theorems   disjim 38798
      ​21.27  Mathbox for Rodolfo Medina
            21.27.1  Partitions   prtlem60 38871
      ​​*21.28  Mathbox for Norm Megill
            *21.28.1  Obsolete schemes ax-c4,c5,c7,c10,c11,c11n,c15,c9,c14,c16   ax-c5 38901
            ​*21.28.2  Rederive new axioms ax-4, ax-10, ax-6, ax-12, ax-13 from old   axc5 38911
            ​*21.28.3  Legacy theorems using obsolete axioms   ax5ALT 38925
            21.28.4  Experiments with weak deduction theorem   elimhyps 38979
            21.28.5  Miscellanea   cnaddcom 38990
            21.28.6  Atoms, hyperplanes, and covering in a left vector space (or module)   clsa 38992
            21.28.7  Functionals and kernels of a left vector space (or module)   clfn 39075
            21.28.8  Opposite rings and dual vector spaces   cld 39141
            21.28.9  Ortholattices and orthomodular lattices   cops 39190
            21.28.10  Atomic lattices with covering property   ccvr 39280
            21.28.11  Hilbert lattices   chlt 39368
            21.28.12  Projective geometries based on Hilbert lattices   clln 39509
            21.28.13  Construction of a vector space from a Hilbert lattice   cdlema1N 39809
            21.28.14  Construction of involution and inner product from a Hilbert lattice   clpoN 41498
      ​21.29  Mathbox for metakunt
            21.29.1  Commutative Semiring   ccsrg 41980
            21.29.2  General helpful statements   rhmzrhval 41983
            21.29.3  Some gcd and lcm results   12gcd5e1 42015
            21.29.4  Least common multiple inequality theorem   3factsumint1 42033
            21.29.5  Logarithm inequalities   3exp7 42065
            21.29.6  Miscellaneous results for AKS formalisation   intlewftc 42073
            21.29.7  Sticks and stones   sticksstones1 42158
            21.29.8  Continuation AKS   aks6d1c6lem1 42182
      ​21.30  Mathbox for Steven Nguyen
            21.30.1  Utility theorems   jarrii 42217
            ​*21.30.2  Arithmetic theorems   c0exALT 42264
            21.30.3  Exponents and divisibility   oexpreposd 42334
            21.30.4  Trigonometry and Calculus   tanhalfpim 42361
            ​*21.30.5  Independence of ax-mulcom   cresub 42377
            21.30.6  Structures   sn-base0 42507
            ​*21.30.7  Projective spaces   cprjsp 42613
            21.30.8  Basic reductions for Fermat's Last Theorem   dffltz 42646
            ​*21.30.9  Exemplar theorems   iddii 42676
                  ​*21.30.9.1  Standard replacements of ax-10 , ax-11 , ax-12   nfa1w 42687
      ​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 42704
            21.33.2  Additional theory of functions   imaiinfv 42705
            21.33.3  Additional topology   elrfi 42706
            21.33.4  Characterization of closure operators. Kuratowski closure axioms   ismrcd1 42710
            21.33.5  Algebraic closure systems   cnacs 42714
            21.33.6  Miscellanea 1. Map utilities   constmap 42725
            21.33.7  Miscellanea for polynomials   mptfcl 42732
            21.33.8  Multivariate polynomials over the integers   cmzpcl 42733
            21.33.9  Miscellanea for Diophantine sets 1   coeq0i 42765
            21.33.10  Diophantine sets 1: definitions   cdioph 42767
            21.33.11  Diophantine sets 2 miscellanea   ellz1 42779
            21.33.12  Diophantine sets 2: union and intersection. Monotone Boolean algebra   diophin 42784
            21.33.13  Diophantine sets 3: construction   diophrex 42787
            21.33.14  Diophantine sets 4 miscellanea   2sbcrex 42796
            21.33.15  Diophantine sets 4: Quantification   rexrabdioph 42806
            21.33.16  Diophantine sets 5: Arithmetic sets   rabdiophlem1 42813
            21.33.17  Diophantine sets 6: reusability. renumbering of variables   eldioph4b 42823
            21.33.18  Pigeonhole Principle and cardinality helpers   fphpd 42828
            21.33.19  A non-closed set of reals is infinite   rencldnfilem 42832
            21.33.20  Lagrange's rational approximation theorem   irrapxlem1 42834
            21.33.21  Pell equations 1: A nontrivial solution always exists   pellexlem1 42841
            21.33.22  Pell equations 2: Algebraic number theory of the solution set   csquarenn 42848
            21.33.23  Pell equations 3: characterizing fundamental solution   infmrgelbi 42890
            ​*21.33.24  Logarithm laws generalized to an arbitrary base   reglogcl 42902
            21.33.25  Pell equations 4: the positive solution group is infinite cyclic   pellfund14 42910
            21.33.26  X and Y sequences 1: Definition and recurrence laws   crmx 42912
            21.33.27  Ordering and induction lemmas for the integers   monotuz 42953
            21.33.28  X and Y sequences 2: Order properties   rmxypos 42959
            21.33.29  Congruential equations   congtr 42977
            21.33.30  Alternating congruential equations   acongid 42987
            21.33.31  Additional theorems on integer divisibility   coprmdvdsb 42997
            21.33.32  X and Y sequences 3: Divisibility properties   jm2.18 43000
            21.33.33  X and Y sequences 4: Diophantine representability of Y   jm2.27a 43017
            21.33.34  X and Y sequences 5: Diophantine representability of X, ^, _C   rmxdiophlem 43027
            21.33.35  Uncategorized stuff not associated with a major project   setindtr 43036
            21.33.36  More equivalents of the Axiom of Choice   axac10 43045
            21.33.37  Finitely generated left modules   clfig 43079
            21.33.38  Noetherian left modules I   clnm 43087
            21.33.39  Addenda for structure powers   pwssplit4 43101
            21.33.40  Every set admits a group structure iff choice   unxpwdom3 43107
            21.33.41  Noetherian rings and left modules II   clnr 43121
            21.33.42  Hilbert's Basis Theorem   cldgis 43133
            21.33.43  Additional material on polynomials [DEPRECATED]   cmnc 43143
            21.33.44  Degree and minimal polynomial of algebraic numbers   cdgraa 43152
            21.33.45  Algebraic integers I   citgo 43169
            21.33.46  Endomorphism algebra   cmend 43183
            21.33.47  Cyclic groups and order   idomodle 43203
            21.33.48  Cyclotomic polynomials   ccytp 43209
            21.33.49  Miscellaneous topology   fgraphopab 43215
      ​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 43229
            21.36.2  Natural addition of Cantor normal forms   oawordex2 43338
            21.36.3  Surreal Contributions   abeqabi 43420
            21.36.4  Short Studies   nlimsuc 43453
                  21.36.4.1  Additional work on conditional logical operator   ifpan123g 43471
                  21.36.4.2  Sophisms   rp-fakeimass 43524
                  ​*21.36.4.3  Finite Sets   rp-isfinite5 43529
                  21.36.4.4  General Observations   intabssd 43531
                  21.36.4.5  Infinite Sets   pwelg 43572
                  ​*21.36.4.6  Finite intersection property   fipjust 43577
                  21.36.4.7  RP ADDTO: Subclasses and subsets   rababg 43586
                  21.36.4.8  RP ADDTO: The intersection of a class   elinintab 43587
                  21.36.4.9  RP ADDTO: Theorems requiring subset and intersection existence   elinintrab 43589
                  21.36.4.10  RP ADDTO: Relations   xpinintabd 43592
                  ​*21.36.4.11  RP ADDTO: Functions   elmapintab 43608
                  ​*21.36.4.12  RP ADDTO: Finite induction (for finite ordinals)   cnvcnvintabd 43612
                  21.36.4.13  RP ADDTO: First and second members of an ordered pair   elcnvlem 43613
                  21.36.4.14  RP ADDTO: The reflexive and transitive properties of relations   undmrnresiss 43616
                  21.36.4.15  RP ADDTO: Basic properties of closures   cleq2lem 43620
                  21.36.4.16  RP REPLACE: Definitions and basic properties of transitive closures   trcleq2lemRP 43642
                  ​*21.36.4.17  Additions for square root; absolute value   sqrtcvallem1 43643
            21.36.5  Additional statements on relations and subclasses   al3im 43659
                  21.36.5.1  Transitive relations (not to be confused with transitive classes).   trrelind 43677
                  21.36.5.2  Reflexive closures   crcl 43684
                  ​*21.36.5.3  Finite relationship composition.   relexp2 43689
                  21.36.5.4  Transitive closure of a relation   dftrcl3 43732
                  ​*21.36.5.5  Adapted from Frege   frege77d 43758
            ​*21.36.6  Propositions from _Begriffsschrift_   dfxor4 43778
                  *21.36.6.1  _Begriffsschrift_ Chapter I   dfxor4 43778
                  ​*21.36.6.2  _Begriffsschrift_ Notation hints   whe 43784
                  21.36.6.3  _Begriffsschrift_ Chapter II Implication   ax-frege1 43802
                  21.36.6.4  _Begriffsschrift_ Chapter II Implication and Negation   axfrege28 43841
                  ​*21.36.6.5  _Begriffsschrift_ Chapter II with logical equivalence   axfrege52a 43868
                  21.36.6.6  _Begriffsschrift_ Chapter II with equivalence of sets   axfrege52c 43899
                  ​*21.36.6.7  _Begriffsschrift_ Chapter II with equivalence of classes   frege53c 43926
                  ​*21.36.6.8  _Begriffsschrift_ Chapter III Properties hereditary in a sequence   dffrege69 43944
                  ​*21.36.6.9  _Begriffsschrift_ Chapter III Following in a sequence   dffrege76 43951
                  ​*21.36.6.10  _Begriffsschrift_ Chapter III Member of sequence   dffrege99 43974
                  ​*21.36.6.11  _Begriffsschrift_ Chapter III Single-valued procedures   dffrege115 43990
            ​*21.36.7  Exploring Topology via Seifert and Threlfall   enrelmap 44009
                  *21.36.7.1  Equinumerosity of sets of relations and maps   enrelmap 44009
                  ​*21.36.7.2  Generic Pseudoclosure Spaces, Pseudointerior Spaces, and Pseudoneighborhoods   or3or 44035
                  ​*21.36.7.3  Generic Neighborhood Spaces   gneispa 44142
            ​*21.36.8  Exploring Higher Homotopy via Kerodon   k0004lem1 44159
                  *21.36.8.1  Simplicial Sets   k0004lem1 44159
      ​21.37  Mathbox for Stanislas Polu
            21.37.1  IMO Problems   wwlemuld 44168
                  21.37.1.1  IMO 1972 B2   wwlemuld 44168
            ​*21.37.2  INT Inequalities Proof Generator   int-addcomd 44185
            ​*21.37.3  N-Digit Addition Proof Generator   unitadd 44207
            21.37.4  AM-GM (for k = 2,3,4)   gsumws3 44208
      ​21.38  Mathbox for Rohan Ridenour
            21.38.1  Misc   spALT 44213
            21.38.2  Monoid rings   cmnring 44223
            21.38.3  Shorter primitive equivalent of ax-groth   gru0eld 44241
                  21.38.3.1  Grothendieck universes are closed under collection   gru0eld 44241
                  21.38.3.2  Minimal universes   ismnu 44273
                  21.38.3.3  Primitive equivalent of ax-groth   expandan 44300
      ​21.39  Mathbox for Steve Rodriguez
            21.39.1  Miscellanea   nanorxor 44317
            21.39.2  Ratio test for infinite series convergence and divergence   dvgrat 44324
            21.39.3  Multiples   reldvds 44327
            21.39.4  Function operations   caofcan 44335
            21.39.5  Calculus   lhe4.4ex1a 44341
            21.39.6  The generalized binomial coefficient operation   cbcc 44348
            21.39.7  Binomial series   uzmptshftfval 44358
      ​21.40  Mathbox for Andrew Salmon
            21.40.1  Principia Mathematica * 10   pm10.12 44370
            21.40.2  Principia Mathematica * 11   2alanimi 44384
            21.40.3  Predicate Calculus   sbeqal1 44410
            21.40.4  Principia Mathematica * 13 and * 14   pm13.13a 44419
            21.40.5  Set Theory   elnev 44449
            21.40.6  Arithmetic   addcomgi 44467
            21.40.7  Geometry   cplusr 44468
      ​​*21.41  Mathbox for Alan Sare
            21.41.1  Auxiliary theorems for the Virtual Deduction tool   idiALT 44490
            21.41.2  Supplementary unification deductions   bi1imp 44494
            21.41.3  Conventional Metamath proofs, some derived from VD proofs   iidn3 44513
            21.41.4  What is Virtual Deduction?   wvd1 44581
            21.41.5  Virtual Deduction Theorems   df-vd1 44582
            21.41.6  Theorems proved using Virtual Deduction   trsspwALT 44829
            21.41.7  Theorems proved using Virtual Deduction with mmj2 assistance   simplbi2VD 44857
            21.41.8  Virtual Deduction transcriptions of textbook proofs   sb5ALTVD 44924
            21.41.9  Theorems proved using conjunction-form Virtual Deduction   elpwgdedVD 44928
            21.41.10  Theorems with a VD proof in conventional notation derived from a VD proof   suctrALT3 44935
            ​*21.41.11  Theorems with a proof in conventional notation derived from a VD proof   notnotrALT2 44938
      ​21.42  Mathbox for Eric Schmidt
            21.42.1  Miscellany   rspesbcd 44949
            21.42.2  Study of dfbi1ALT   dfbi1ALTa 44951
            21.42.3  Relation-preserving functions   wrelp 44954
            21.42.4  Orbits   orbitex 44967
            21.42.5  Well-founded sets   trwf 44971
            21.42.6  Absoluteness in transitive models   ralabso 44980
            21.42.7  Lemmas for showing axioms hold in models   traxext 44989
            21.42.8  The class of well-founded sets is a model for ZFC   wfaxext 45005
            21.42.9  Permutation models   brpermmodel 45015
      ​21.43  Mathbox for Glauco Siliprandi
            21.43.1  Miscellanea   evth2f 45031
            21.43.2  Functions   fnresdmss 45184
            21.43.3  Ordering on real numbers - Real and complex numbers basic operations   sub2times 45293
            21.43.4  Real intervals   gtnelioc 45510
            21.43.5  Finite sums   fsummulc1f 45590
            21.43.6  Finite multiplication of numbers and finite multiplication of functions   fmul01 45599
            21.43.7  Limits   clim1fr1 45620
                  21.43.7.1  Inferior limit (lim inf)   clsi 45768
                  ​*21.43.7.2  Limits for sequences of extended real numbers   clsxlim 45835
            21.43.8  Trigonometry   coseq0 45881
            21.43.9  Continuous Functions   mulcncff 45887
            21.43.10  Derivatives   dvsinexp 45928
            21.43.11  Integrals   itgsin0pilem1 45967
            21.43.12  Stone Weierstrass theorem - real version   stoweidlem1 46018
            21.43.13  Wallis' product for π   wallispilem1 46082
            21.43.14  Stirling's approximation formula for ` n ` factorial   stirlinglem1 46091
            21.43.15  Dirichlet kernel   dirkerval 46108
            21.43.16  Fourier Series   fourierdlem1 46125
            21.43.17  e is transcendental   elaa2lem 46250
            21.43.18  n-dimensional Euclidean space   rrxtopn 46301
            21.43.19  Basic measure theory   csalg 46325
                  ​*21.43.19.1  σ-Algebras   csalg 46325
                  21.43.19.2  Sum of nonnegative extended reals   csumge0 46379
                  ​*21.43.19.3  Measures   cmea 46466
                  ​*21.43.19.4  Outer measures and Caratheodory's construction   come 46506
                  ​*21.43.19.5  Lebesgue measure on n-dimensional Real numbers   covoln 46553
                  ​*21.43.19.6  Measurable functions   csmblfn 46712
      ​21.44  Mathbox for Saveliy Skresanov
            21.44.1  Ceva's theorem   sigarval 46867
            21.44.2  Simple groups   simpcntrab 46887
      ​21.45  Mathbox for Ender Ting
            21.45.1  Increasing sequences and subsequences   et-ltneverrefl 46888
            21.45.2  Scratchpad for number theory   evenwodadd 46895
            21.45.3  Scratchpad for math on real numbers   squeezedltsq 46896
      ​21.46  Mathbox for Jarvin Udandy
      ​21.47  Mathbox for Adhemar
            ​*21.47.1  Minimal implicational calculus   adh-minim 47011
      ​21.48  Mathbox for Alexander van der Vekens
            21.48.1  General auxiliary theorems (1)   n0nsn2el 47035
                  21.48.1.1  Unordered and ordered pairs - extension for singletons   n0nsn2el 47035
                  21.48.1.2  Unordered and ordered pairs - extension for unordered pairs   elprneb 47039
                  21.48.1.3  Unordered and ordered pairs - extension for ordered pairs   oppr 47040
                  21.48.1.4  Relations - extension   eubrv 47045
                  21.48.1.5  Definite description binder (inverted iota) - extension   iota0def 47048
                  21.48.1.6  Functions - extension   fveqvfvv 47050
            21.48.2  Alternative for Russell's definition of a description binder   caiota 47093
            21.48.3  Double restricted existential uniqueness   r19.32 47108
                  21.48.3.1  Restricted quantification (extension)   r19.32 47108
                  21.48.3.2  Restricted uniqueness and "at most one" quantification   reuf1odnf 47117
                  21.48.3.3  Analogs to Existential uniqueness (double quantification)   2reu3 47120
                  21.48.3.4  Additional theorems for double restricted existential uniqueness   2reu8i 47123
            ​*21.48.4  Alternative definitions of function and operation values   wdfat 47126
                  21.48.4.1  Restricted quantification (extension)   ralbinrald 47132
                  21.48.4.2  The universal class (extension)   nvelim 47133
                  21.48.4.3  Introduce the Axiom of Power Sets (extension)   alneu 47134
                  21.48.4.4  Predicate "defined at"   dfateq12d 47136
                  21.48.4.5  Alternative definition of the value of a function   dfafv2 47142
                  21.48.4.6  Alternative definition of the value of an operation   aoveq123d 47188
            ​*21.48.5  Alternative definitions of function values (2)   cafv2 47218
            21.48.6  General auxiliary theorems (2)   an4com24 47278
                  21.48.6.1  Logical conjunction - extension   an4com24 47278
                  21.48.6.2  Abbreviated conjunction and disjunction of three wff's - extension   3an4ancom24 47279
                  21.48.6.3  Negated membership (alternative)   cnelbr 47281
                  21.48.6.4  The empty set - extension   ralralimp 47288
                  21.48.6.5  Indexed union and intersection - extension   otiunsndisjX 47289
                  21.48.6.6  Functions - extension   fvifeq 47290
                  21.48.6.7  Maps-to notation - extension   fvmptrab 47302
                  21.48.6.8  Subtraction - extension   cnambpcma 47304
                  21.48.6.9  Ordering on reals (cont.) - extension   leaddsuble 47307
                  21.48.6.10  Imaginary and complex number properties - extension   readdcnnred 47313
                  21.48.6.11  Nonnegative integers (as a subset of complex numbers) - extension   nn0resubcl 47318
                  21.48.6.12  Integers (as a subset of complex numbers) - extension   zgeltp1eq 47319
                  21.48.6.13  Decimal arithmetic - extension   1t10e1p1e11 47320
                  21.48.6.14  Upper sets of integers - extension   eluzge0nn0 47322
                  21.48.6.15  Infinity and the extended real number system (cont.) - extension   nltle2tri 47323
                  21.48.6.16  Finite intervals of integers - extension   ssfz12 47324
                  21.48.6.17  Half-open integer ranges - extension   fzopred 47332
                  21.48.6.18  The floor and ceiling functions - extension   2ltceilhalf 47338
                  21.48.6.19  The modulo (remainder) operation - extension   fldivmod 47348
                  21.48.6.20  The infinite sequence builder "seq"   smonoord 47381
                  21.48.6.21  Finite and infinite sums - extension   fsummsndifre 47382
                  21.48.6.22  Extensible structures - extension   setsidel 47386
            ​*21.48.7  Preimages of function values   preimafvsnel 47389
            ​*21.48.8  Partitions of real intervals   ciccp 47423
            21.48.9  Shifting functions with an integer range domain   fargshiftfv 47449
            21.48.10  Words over a set (extension)   lswn0 47454
                  21.48.10.1  Last symbol of a word - extension   lswn0 47454
            21.48.11  Unordered pairs   wich 47455
                  21.48.11.1  Interchangeable setvar variables   wich 47455
                  21.48.11.2  Set of unordered pairs   sprid 47484
                  ​*21.48.11.3  Proper (unordered) pairs   prpair 47511
                  21.48.11.4  Set of proper unordered pairs   cprpr 47522
            21.48.12  Number theory (extension)   cfmtno 47537
                  ​*21.48.12.1  Fermat numbers   cfmtno 47537
                  ​*21.48.12.2  Mersenne primes   m2prm 47601
                  21.48.12.3  Proth's theorem   modexp2m1d 47622
                  21.48.12.4  Solutions of quadratic equations   quad1 47630
            ​*21.48.13  Even and odd numbers   ceven 47634
                  21.48.13.1  Definitions and basic properties   ceven 47634
                  21.48.13.2  Alternate definitions using the "divides" relation   dfeven2 47659
                  21.48.13.3  Alternate definitions using the "modulo" operation   dfeven3 47668
                  21.48.13.4  Alternate definitions using the "gcd" operation   iseven5 47674
                  21.48.13.5  Theorems of part 5 revised   zneoALTV 47679
                  21.48.13.6  Theorems of part 6 revised   odd2np1ALTV 47684
                  21.48.13.7  Theorems of AV's mathbox revised   0evenALTV 47698
                  21.48.13.8  Additional theorems   epoo 47713
                  21.48.13.9  Perfect Number Theorem (revised)   perfectALTVlem1 47731
            21.48.14  Number theory (extension 2)   cfppr 47734
                  ​*21.48.14.1  Fermat pseudoprimes   cfppr 47734
                  ​*21.48.14.2  Goldbach's conjectures   cgbe 47755
            21.48.15  Graph theory (extension)   cclnbgr 47828
                  21.48.15.1  Closed neighborhood of a vertex   cclnbgr 47828
                  ​*21.48.15.2  Semiclosed and semiopen neighborhoods (experimental)   dfsclnbgr2 47856
                  21.48.15.3  Induced subgraphs   cisubgr 47870
                  ​*21.48.15.4  Isomorphisms of graphs   cgrisom 47884
                  ​*21.48.15.5  Triangles in graphs   cgrtri 47947
                  ​*21.48.15.6  Star graphs   cstgr 47961
                  ​*21.48.15.7  Local isomorphisms of graphs   cgrlim 47986
                  ​*21.48.15.8  Generalized Petersen graphs   cgpg 48050
                  21.48.15.9  Loop-free graphs - extension   1hegrlfgr 48142
                  21.48.15.10  Walks - extension   cupwlks 48143
                  21.48.15.11  Edges of graphs expressed as sets of unordered pairs   upgredgssspr 48153
            21.48.16  Monoids (extension)   ovn0dmfun 48166
                  21.48.16.1  Auxiliary theorems   ovn0dmfun 48166
                  21.48.16.2  Magmas, Semigroups and Monoids (extension)   plusfreseq 48174
                  21.48.16.3  Examples and counterexamples for magmas, semigroups and monoids (extension)   opmpoismgm 48177
                  21.48.16.4  Group sum operation (extension 1)   gsumsplit2f 48190
            ​*21.48.17  Magmas and internal binary operations (alternate approach)   ccllaw 48193
                  *21.48.17.1  Laws for internal binary operations   ccllaw 48193
                  ​*21.48.17.2  Internal binary operations   cintop 48206
                  21.48.17.3  Alternative definitions for magmas and semigroups   cmgm2 48225
            21.48.18  Rings (extension)   lmod0rng 48239
                  21.48.18.1  Nonzero rings (extension)   lmod0rng 48239
                  21.48.18.2  Ideals as non-unital rings   lidldomn1 48241
                  21.48.18.3  The non-unital ring of even integers   0even 48247
                  21.48.18.4  A constructed not unital ring   cznrnglem 48269
                  ​*21.48.18.5  The category of non-unital rings (alternate definition)   crngcALTV 48273
                  ​*21.48.18.6  The category of (unital) rings (alternate definition)   cringcALTV 48297
            21.48.19  Basic algebraic structures (extension)   eliunxp2 48344
                  21.48.19.1  Auxiliary theorems   eliunxp2 48344
                  21.48.19.2  The binomial coefficient operation (extension)   bcpascm1 48361
                  21.48.19.3  The ` ZZ `-module ` ZZ X. ZZ `   zlmodzxzlmod 48364
                  21.48.19.4  Group sum operation (extension 2)   mgpsumunsn 48371
                  21.48.19.5  Symmetric groups (extension)   exple2lt6 48374
                  21.48.19.6  Divisibility (extension)   invginvrid 48377
                  21.48.19.7  The support of functions (extension)   rmsupp0 48378
                  21.48.19.8  Finitely supported functions (extension)   rmsuppfi 48382
                  21.48.19.9  Left modules (extension)   lmodvsmdi 48389
                  21.48.19.10  Associative algebras (extension)   assaascl0 48391
                  21.48.19.11  Univariate polynomials (extension)   ply1vr1smo 48393
                  21.48.19.12  Univariate polynomials (examples)   linply1 48404
            21.48.20  Linear algebra (extension)   cdmatalt 48407
                  ​*21.48.20.1  The subalgebras of diagonal and scalar matrices (extension)   cdmatalt 48407
                  ​*21.48.20.2  Linear combinations   clinc 48415
                  ​*21.48.20.3  Linear independence   clininds 48451
                  21.48.20.4  Simple left modules and the ` ZZ `-module   lmod1lem1 48498
                  21.48.20.5  Differences between (left) modules and (left) vector spaces   lvecpsslmod 48518
            21.48.21  Complexity theory   suppdm 48521
                  21.48.21.1  Auxiliary theorems   suppdm 48521
                  21.48.21.2  Even and odd integers   nn0onn0ex 48534
                  21.48.21.3  The natural logarithm on complex numbers (extension)   logcxp0 48546
                  21.48.21.4  Division of functions   cfdiv 48548
                  21.48.21.5  Upper bounds   cbigo 48558
                  21.48.21.6  Logarithm to an arbitrary base (extension)   rege1logbrege0 48569
                  ​*21.48.21.7  The binary logarithm   fldivexpfllog2 48576
                  21.48.21.8  Binary length   cblen 48580
                  ​*21.48.21.9  Digits   cdig 48606
                  21.48.21.10  Nonnegative integer as sum of its shifted digits   dignn0flhalflem1 48626
                  21.48.21.11  Algorithms for the multiplication of nonnegative integers   nn0mulfsum 48635
                  ​*21.48.21.12  N-ary functions   cnaryf 48637
                  ​*21.48.21.13  The Ackermann function   citco 48668
            21.48.22  Elementary geometry (extension)   fv1prop 48710
                  21.48.22.1  Auxiliary theorems   fv1prop 48710
                  21.48.22.2  Real euclidean space of dimension 2   rrx2pxel 48722
                  21.48.22.3  Spheres and lines in real Euclidean spaces   cline 48738
      ​21.49  Mathbox for Zhi Wang
            21.49.1  Propositional calculus   pm4.71da 48800
            21.49.2  Predicate calculus with equality   dtrucor3 48809
                  21.49.2.1  Axiom scheme ax-5 (Distinctness)   dtrucor3 48809
            21.49.3  ZF Set Theory - start with the Axiom of Extensionality   ralbidb 48810
                  21.49.3.1  Restricted quantification   ralbidb 48810
                  21.49.3.2  The universal class   reuxfr1dd 48817
                  21.49.3.3  The empty set   ssdisjd 48818
                  21.49.3.4  Unordered and ordered pairs   vsn 48822
                  21.49.3.5  The union of a class   unilbss 48828
                  21.49.3.6  Indexed union and intersection   iuneq0 48829
            21.49.4  ZF Set Theory - add the Axiom of Replacement   inpw 48835
                  21.49.4.1  Theorems requiring subset and intersection existence   inpw 48835
            21.49.5  ZF Set Theory - add the Axiom of Power Sets   opth1neg 48836
                  21.49.5.1  Ordered pair theorem   opth1neg 48836
                  21.49.5.2  Ordered-pair class abstractions (cont.)   brab2dd 48838
                  21.49.5.3  Relations   iinxp 48841
                  21.49.5.4  Functions   mof0 48848
                  21.49.5.5  Operations   ovsng 48868
            21.49.6  ZF Set Theory - add the Axiom of Union   fonex 48877
                  21.49.6.1  Relations and functions (cont.)   fonex 48877
                  21.49.6.2  First and second members of an ordered pair   eloprab1st2nd 48878
                  21.49.6.3  Operations in maps-to notation (continued)   fmpodg 48879
                  21.49.6.4  Function transposition   resinsnlem 48881
                  21.49.6.5  Infinite Cartesian products   ixpv 48900
                  21.49.6.6  Equinumerosity   fvconst0ci 48901
            21.49.7  Order sets   iccin 48906
                  21.49.7.1  Real number intervals   iccin 48906
            21.49.8  Extensible structures   slotresfo 48909
                  21.49.8.1  Basic definitions   slotresfo 48909
            21.49.9  Moore spaces   mreuniss 48910
            ​*21.49.10  Topology   clduni 48911
                  21.49.10.1  Closure and interior   clduni 48911
                  21.49.10.2  Neighborhoods   neircl 48915
                  21.49.10.3  Subspace topologies   restcls2lem 48923
                  21.49.10.4  Limits and continuity in topological spaces   cnneiima 48927
                  21.49.10.5  Topological definitions using the reals   iooii 48928
                  21.49.10.6  Separated sets   sepnsepolem1 48932
                  21.49.10.7  Separated spaces: T0, T1, T2 (Hausdorff) ...   isnrm4 48941
            21.49.11  Preordered sets and directed sets using extensible structures   isprsd 48965
            21.49.12  Posets and lattices using extensible structures   lubeldm2 48966
                  21.49.12.1  Posets   lubeldm2 48966
                  21.49.12.2  Lattices   toslat 48992
                  21.49.12.3  Subset order structures   intubeu 48994
            21.49.13  Rings   elmgpcntrd 49015
                  21.49.13.1  Multiplicative Group   elmgpcntrd 49015
            21.49.14  Associative algebras   asclelbas 49016
                  21.49.14.1  Definition and basic properties   asclelbas 49016
            21.49.15  Categories   homf0 49020
                  21.49.15.1  Categories   homf0 49020
                  21.49.15.2  Opposite category   oppccatb 49027
                  21.49.15.3  Monomorphisms and epimorphisms   idmon 49031
                  21.49.15.4  Sections, inverses, isomorphisms   sectrcl 49033
                  21.49.15.5  Isomorphic objects   cicfn 49053
                  21.49.15.6  Subcategories   dmdm 49064
                  21.49.15.7  Functors   reldmfunc 49086
                  21.49.15.8  Opposite functors   coppf 49133
                  21.49.15.9  Full & faithful functors   imasubc 49162
                  21.49.15.10  Universal property   upciclem1 49177
                  21.49.15.11  Natural transformations and the functor category   isnatd 49234
                  21.49.15.12  Initial, terminal and zero objects of a category   initoo2 49243
                  21.49.15.13  Product of categories   reldmxpc 49257
                  21.49.15.14  Swap functors   cswapf 49270
                  21.49.15.15  Functor evaluation   oppc1stflem 49298
                  21.49.15.16  Transposed curry functors   cofuswapfcl 49304
                  21.49.15.17  Constant functors   diag1 49315
                  21.49.15.18  Functor composition bifunctors   fucofulem1 49321
                  21.49.15.19  Post-composition functors   postcofval 49375
                  21.49.15.20  Pre-composition functors   precofvallem 49377
            21.49.16  Examples of categories   catcrcl 49406
                  21.49.16.1  The category of categories   catcrcl 49406
                  21.49.16.2  Thin categories   cthinc 49428
                  21.49.16.3  Terminal categories   ctermc 49483
                  21.49.16.4  Preordered sets as thin categories   cprstc 49560
                  21.49.16.5  Monoids as categories   cmndtc 49588
                  21.49.16.6  Categories with at most one object and at most two morphisms   2arwcatlem1 49606
            21.49.17  Kan extensions and related concepts   clan 49616
                  21.49.17.1  Kan extensions   clan 49616
                  21.49.17.2  Limits and colimits   clmd 49654
      ​21.50  Mathbox for Emmett Weisz
            ​*21.50.1  Miscellaneous Theorems   nfintd 49684
            21.50.2  Set Recursion   csetrecs 49694
                  ​*21.50.2.1  Basic Properties of Set Recursion   csetrecs 49694
                  21.50.2.2  Examples and properties of set recursion   elsetrecslem 49710
            ​*21.50.3  Construction of Games and Surreal Numbers   cpg 49720
      ​​*21.51  Mathbox for David A. Wheeler
            21.51.1  Natural deduction   sbidd 49729
            ​*21.51.2  Greater than, greater than or equal to.   cge-real 49731
            ​*21.51.3  Hyperbolic trigonometric functions   csinh 49741
            ​*21.51.4  Reciprocal trigonometric functions (sec, csc, cot)   csec 49752
            ​*21.51.5  Identities for "if"   ifnmfalse 49774
            ​*21.51.6  Logarithms generalized to arbitrary base using ` logb `   logb2aval 49775
            ​*21.51.7  Logarithm laws generalized to an arbitrary base - log_   clog- 49776
            ​*21.51.8  Formally define notions such as reflexivity   wreflexive 49778
            ​*21.51.9  Algebra helpers   mvlraddi 49782
            ​*21.51.10  Algebra helper examples   i2linesi 49789
            ​*21.51.11  Formal methods "surprises"   alimp-surprise 49791
            ​*21.51.12  Allsome quantifier   walsi 49797
            ​*21.51.13  Miscellaneous   5m4e1 49808
            21.51.14  Theorems about algebraic numbers   aacllem 49812
      ​21.52  Mathbox for Kunhao Zheng
            21.52.1  Weighted AM-GM inequality   amgmwlem 49813