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 Adrian Ducourtial
      21.10  Mathbox for Scott Fenton
      21.11  Mathbox for Gino Giotto
      21.12  Mathbox for Jeff Hankins
      21.13  Mathbox for Anthony Hart
      21.14  Mathbox for Chen-Pang He
      21.15  Mathbox for Jeff Hoffman
      21.16  Mathbox for Asger C. Ipsen
      21.17  Mathbox for BJ
      21.18  Mathbox for Jim Kingdon
      21.19  Mathbox for ML
      21.20  Mathbox for Wolf Lammen
      21.21  Mathbox for Brendan Leahy
      21.22  Mathbox for Jeff Madsen
      21.23  Mathbox for Giovanni Mascellani
      21.24  Mathbox for Peter Mazsa
      21.25  Mathbox for Rodolfo Medina
      21.26  Mathbox for Norm Megill
      21.27  Mathbox for metakunt
      21.28  Mathbox for Steven Nguyen
      21.29  Mathbox for Igor Ieskov
      21.30  Mathbox for OpenAI
      21.31  Mathbox for Stefan O'Rear
      21.32  Mathbox for Noam Pasman
      21.33  Mathbox for Jon Pennant
      21.34  Mathbox for Richard Penner
      21.35  Mathbox for Stanislas Polu
      21.36  Mathbox for Rohan Ridenour
      21.37  Mathbox for Steve Rodriguez
      21.38  Mathbox for Andrew Salmon
      21.39  Mathbox for Alan Sare
      21.40  Mathbox for Glauco Siliprandi
      21.41  Mathbox for Saveliy Skresanov
      21.42  Mathbox for Ender Ting
      21.43  Mathbox for Jarvin Udandy
      21.44  Mathbox for Adhemar
      21.45  Mathbox for Alexander van der Vekens
      21.46  Mathbox for Zhi Wang
      21.47  Mathbox for Emmett Weisz
      21.48  Mathbox for David A. Wheeler
      21.49  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 205
            ​*1.2.6  Logical conjunction   wa 394
            ​*1.2.7  Logical disjunction   wo 845
            ​*1.2.8  Mixed connectives   jaao 952
            ​*1.2.9  The conditional operator for propositions   wif 1060
            ​*1.2.10  The weak deduction theorem for propositional calculus   elimh 1080
            1.2.11  Abbreviated conjunction and disjunction of three wff's   w3o 1083
            1.2.12  Logical "nand" (Sheffer stroke)   wnan 1484
            1.2.13  Logical "xor"   wxo 1504
            1.2.14  Logical "nor"   wnor 1521
            1.2.15  True and false constants   wal 1531
                  ​*1.2.15.1  Universal quantifier for use by df-tru   wal 1531
                  ​*1.2.15.2  Equality predicate for use by df-tru   cv 1532
                  1.2.15.3  The true constant   wtru 1534
                  1.2.15.4  The false constant   wfal 1545
            ​*1.2.16  Truth tables   truimtru 1556
                  1.2.16.1  Implication   truimtru 1556
                  1.2.16.2  Negation   nottru 1560
                  1.2.16.3  Equivalence   trubitru 1562
                  1.2.16.4  Conjunction   truantru 1566
                  1.2.16.5  Disjunction   truortru 1570
                  1.2.16.6  Alternative denial   trunantru 1574
                  1.2.16.7  Exclusive disjunction   truxortru 1578
                  1.2.16.8  Joint denial   trunortru 1582
            ​*1.2.17  Half adder and full adder in propositional calculus   whad 1586
                  1.2.17.1  Full adder: sum   whad 1586
                  1.2.17.2  Full adder: carry   wcad 1599
      ​1.3  Other axiomatizations related to classical propositional calculus
            ​*1.3.1  Minimal implicational calculus   minimp 1615
            ​*1.3.2  Implicational Calculus   impsingle 1621
            1.3.3  Derive the Lukasiewicz axioms from Meredith's sole axiom   meredith 1635
            1.3.4  Derive the standard axioms from the Lukasiewicz axioms   luklem1 1652
            ​*1.3.5  Derive Nicod's axiom from the standard axioms   nic-dfim 1663
            1.3.6  Derive the Lukasiewicz axioms from Nicod's axiom   nic-imp 1669
            1.3.7  Derive Nicod's Axiom from Lukasiewicz's First Sheffer Stroke Axiom   lukshef-ax1 1688
            1.3.8  Derive the Lukasiewicz Axioms from the Tarski-Bernays-Wajsberg Axioms   tbw-bijust 1692
            1.3.9  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's First CO Axiom   merco1 1707
            1.3.10  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's Second CO Axiom   merco2 1730
            1.3.11  Derive the Lukasiewicz axioms from the Russell-Bernays Axioms   rb-bijust 1743
            ​*1.3.12  Stoic logic non-modal portion (Chrysippus of Soli)   mptnan 1762
      ​​*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 1773
                  1.4.1.1  Existential quantifier   wex 1773
                  1.4.1.2  Nonfreeness predicate   wnf 1777
            1.4.2  Rule scheme ax-gen (Generalization)   ax-gen 1789
            1.4.3  Axiom scheme ax-4 (Quantified Implication)   ax-4 1803
                  ​*1.4.3.1  The empty domain of discourse   empty 1901
            1.4.4  Axiom scheme ax-5 (Distinctness) - first use of $d   ax-5 1905
            ​*1.4.5  Equality predicate (continued)   weq 1958
            1.4.6  Axiom scheme ax-6 (Existence)   ax-6 1963
            1.4.7  Axiom scheme ax-7 (Equality)   ax-7 2003
            1.4.8  Define proper substitution   sbjust 2058
            1.4.9  Membership predicate   wcel 2098
            1.4.10  Axiom scheme ax-8 (Left Equality for Binary Predicate)   ax-8 2100
            1.4.11  Axiom scheme ax-9 (Right Equality for Binary Predicate)   ax-9 2108
            ​*1.4.12  Logical redundancy of ax-10 , ax-11 , ax-12 , ax-13   ax6dgen 2116
      ​​*1.5  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes)
            1.5.1  Axiom scheme ax-10 (Quantified Negation)   ax-10 2129
            1.5.2  Axiom scheme ax-11 (Quantifier Commutation)   ax-11 2146
            1.5.3  Axiom scheme ax-12 (Substitution)   ax-12 2166
            1.5.4  Axiom scheme ax-13 (Quantified Equality)   ax-13 2365
      ​1.6  Uniqueness and unique existence
            1.6.1  Uniqueness: the at-most-one quantifier   wmo 2526
            1.6.2  Unique existence: the unique existential quantifier   weu 2556
      ​1.7  Other axiomatizations related to classical predicate calculus
            ​*1.7.1  Aristotelian logic: Assertic syllogisms   barbara 2651
            ​*1.7.2  Intuitionistic logic   axia1 2681
​​*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 2696
            2.1.2  Classes   cab 2702
                  2.1.2.1  Class abstractions   cab 2702
                  ​*2.1.2.2  Class equality   df-cleq 2717
                  2.1.2.3  Class membership   df-clel 2802
                  2.1.2.4  Elementary properties of class abstractions   eqabdv 2859
            2.1.3  Class form not-free predicate   wnfc 2875
            2.1.4  Negated equality and membership   wne 2930
                  2.1.4.1  Negated equality   wne 2930
                  2.1.4.2  Negated membership   wnel 3036
            2.1.5  Restricted quantification   wral 3051
                  2.1.5.1  Restricted universal and existential quantification   wral 3051
                  2.1.5.2  Restricted existential uniqueness and at-most-one quantifier   wreu 3362
                  2.1.5.3  Restricted class abstraction   crab 3419
            2.1.6  The universal class   cvv 3463
            ​*2.1.7  Conditional equality (experimental)   wcdeq 3756
            2.1.8  Russell's Paradox   rru 3772
            2.1.9  Proper substitution of classes for sets   wsbc 3774
            2.1.10  Proper substitution of classes for sets into classes   csb 3890
            2.1.11  Define basic set operations and relations   cdif 3942
            2.1.12  Subclasses and subsets   df-ss 3962
            2.1.13  The difference, union, and intersection of two classes   dfdif3 4111
                  2.1.13.1  The difference of two classes   dfdif3 4111
                  2.1.13.2  The union of two classes   elun 4146
                  2.1.13.3  The intersection of two classes   elini 4192
                  2.1.13.4  The symmetric difference of two classes   csymdif 4241
                  2.1.13.5  Combinations of difference, union, and intersection of two classes   unabs 4254
                  2.1.13.6  Class abstractions with difference, union, and intersection of two classes   unabw 4297
                  2.1.13.7  Restricted uniqueness with difference, union, and intersection   reuun2 4315
            2.1.14  The empty set   c0 4323
            ​*2.1.15  The conditional operator for classes   cif 4529
            ​*2.1.16  The weak deduction theorem for set theory   dedth 4587
            2.1.17  Power classes   cpw 4603
            2.1.18  Unordered and ordered pairs   snjust 4628
            2.1.19  The union of a class   cuni 4908
            2.1.20  The intersection of a class   cint 4949
            2.1.21  Indexed union and intersection   ciun 4996
            2.1.22  Disjointness   wdisj 5113
            2.1.23  Binary relations   wbr 5148
            2.1.24  Ordered-pair class abstractions (class builders)   copab 5210
            2.1.25  Functions in maps-to notation   cmpt 5231
            2.1.26  Transitive classes   wtr 5265
      ​2.2  ZF Set Theory - add the Axiom of Replacement
            2.2.1  Introduce the Axiom of Replacement   ax-rep 5285
            2.2.2  Derive the Axiom of Separation   axsepgfromrep 5297
            2.2.3  Derive the Null Set Axiom   axnulALT 5304
            2.2.4  Theorems requiring subset and intersection existence   nalset 5313
            2.2.5  Theorems requiring empty set existence   class2set 5354
      ​2.3  ZF Set Theory - add the Axiom of Power Sets
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 5364
            2.3.2  Derive the Axiom of Pairing   axprlem1 5422
            2.3.3  Ordered pair theorem   opnz 5474
            2.3.4  Ordered-pair class abstractions (cont.)   opabidw 5525
            2.3.5  Power class of union and intersection   pwin 5571
            2.3.6  The identity relation   cid 5574
            2.3.7  The membership relation (or epsilon relation)   cep 5580
            ​*2.3.8  Partial and total orderings   wpo 5587
            2.3.9  Founded and well-ordering relations   wfr 5629
            2.3.10  Relations   cxp 5675
            2.3.11  The Predecessor Class   cpred 6304
            2.3.12  Well-founded induction (variant)   frpomin 6346
            2.3.13  Well-ordered induction   tz6.26 6353
            2.3.14  Ordinals   word 6368
            2.3.15  Definite description binder (inverted iota)   cio 6497
            2.3.16  Functions   wfun 6541
            2.3.17  Cantor's Theorem   canth 7370
            2.3.18  Restricted iota (description binder)   crio 7372
            2.3.19  Operations   co 7417
                  2.3.19.1  Variable-to-class conversion for operations   caovclg 7611
            2.3.20  Maps-to notation   mpondm0 7659
            2.3.21  Function operation   cof 7681
            2.3.22  Proper subset relation   crpss 7726
      ​2.4  ZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 7739
            2.4.2  Ordinals (continued)   epweon 7776
            2.4.3  Transfinite induction   tfi 7856
            2.4.4  The natural numbers (i.e., finite ordinals)   com 7869
            2.4.5  Peano's postulates   peano1 7893
            2.4.6  Finite induction (for finite ordinals)   find 7901
            2.4.7  Relations and functions (cont.)   dmexg 7907
            2.4.8  First and second members of an ordered pair   c1st 7990
            2.4.9  Induction on Cartesian products   frpoins3xpg 8143
            2.4.10  Ordering on Cartesian products   xpord2lem 8145
            2.4.11  Ordering Ordinal Sequences   orderseqlem 8160
            ​*2.4.12  The support of functions   csupp 8163
            ​*2.4.13  Special maps-to operations   opeliunxp2f 8214
            2.4.14  Function transposition   ctpos 8229
            2.4.15  Curry and uncurry   ccur 8269
            2.4.16  Undefined values   cund 8276
            2.4.17  Well-founded recursion   cfrecs 8284
            2.4.18  Well-ordered recursion   cwrecs 8315
            2.4.19  Functions on ordinals; strictly monotone ordinal functions   iunon 8358
            2.4.20  "Strong" transfinite recursion   crecs 8389
            2.4.21  Recursive definition generator   crdg 8428
            2.4.22  Finite recursion   frfnom 8454
            2.4.23  Ordinal arithmetic   c1o 8478
            2.4.24  Natural number arithmetic   nna0 8623
            2.4.25  Natural addition   cnadd 8684
            2.4.26  Equivalence relations and classes   wer 8720
            2.4.27  The mapping operation   cmap 8843
            2.4.28  Infinite Cartesian products   cixp 8914
            2.4.29  Equinumerosity   cen 8959
            2.4.30  Schroeder-Bernstein Theorem   sbthlem1 9106
            2.4.31  Equinumerosity (cont.)   xpf1o 9162
            2.4.32  Finite sets   dif1enlem 9179
            2.4.33  Pigeonhole Principle   phplem1 9230
            2.4.34  Finite sets (cont.)   onomeneq 9251
            2.4.35  Finitely supported functions   cfsupp 9385
            2.4.36  Finite intersections   cfi 9433
            2.4.37  Hall's marriage theorem   marypha1lem 9456
            2.4.38  Supremum and infimum   csup 9463
            2.4.39  Ordinal isomorphism, Hartogs's theorem   coi 9532
            2.4.40  Hartogs function   char 9579
            2.4.41  Weak dominance   cwdom 9587
      ​2.5  ZF Set Theory - add the Axiom of Regularity
            2.5.1  Introduce the Axiom of Regularity   ax-reg 9615
            2.5.2  Axiom of Infinity equivalents   inf0 9644
      ​2.6  ZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-inf 9661
            2.6.2  Existence of omega (the set of natural numbers)   omex 9666
            2.6.3  Cantor normal form   ccnf 9684
            2.6.4  Transitive closure of a relation   cttrcl 9730
            2.6.5  Transitive closure   trcl 9751
            2.6.6  Well-Founded Induction   frmin 9772
            2.6.7  Well-Founded Recursion   frr3g 9779
            2.6.8  Rank   cr1 9785
            2.6.9  Scott's trick; collection principle; Hilbert's epsilon   scottex 9908
            2.6.10  Disjoint union   cdju 9921
            2.6.11  Cardinal numbers   ccrd 9958
            2.6.12  Axiom of Choice equivalents   wac 10138
            ​*2.6.13  Cardinal number arithmetic   undjudom 10190
            2.6.14  The Ackermann bijection   ackbij2lem1 10242
            2.6.15  Cofinality (without Axiom of Choice)   cflem 10269
            2.6.16  Eight inequivalent definitions of finite set   sornom 10300
            2.6.17  Hereditarily size-limited sets without Choice   itunifval 10439
​​*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 10458
            3.1.2  Introduce the Axiom of Dependent Choice   ax-dc 10469
      ​3.2  ZFC Set Theory - add the Axiom of Choice
            3.2.1  Introduce the Axiom of Choice   ax-ac 10482
            3.2.2  AC equivalents: well-ordering, Zorn's lemma   numthcor 10517
            3.2.3  Cardinal number theorems using Axiom of Choice   cardval 10569
            3.2.4  Cardinal number arithmetic using Axiom of Choice   iunctb 10597
            3.2.5  Cofinality using the Axiom of Choice   alephreg 10605
      ​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 10643
            3.4.2  Derivation of the Axiom of Choice   gchaclem 10701
​​*PART 4  TG (TARSKI-GROTHENDIECK) SET THEORY
      ​4.1  Inaccessibles
            4.1.1  Weakly and strongly inaccessible cardinals   cwina 10705
            4.1.2  Weak universes   cwun 10723
            4.1.3  Tarski classes   ctsk 10771
            4.1.4  Grothendieck universes   cgru 10813
      ​4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
            4.2.1  Introduce the Tarski-Grothendieck Axiom   ax-groth 10846
            4.2.2  Derive the Power Set, Infinity and Choice Axioms   grothpw 10849
            4.2.3  Tarski map function   ctskm 10860
​​*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 10867
            5.1.2  Final derivation of real and complex number postulates   axaddf 11168
            5.1.3  Real and complex number postulates restated as axioms   ax-cnex 11194
      ​5.2  Derive the basic properties from the field axioms
            5.2.1  Some deductions from the field axioms for complex numbers   cnex 11219
            5.2.2  Infinity and the extended real number system   cpnf 11275
            5.2.3  Restate the ordering postulates with extended real "less than"   axlttri 11315
            5.2.4  Ordering on reals   lttr 11320
            5.2.5  Initial properties of the complex numbers   mul12 11409
      ​5.3  Real and complex numbers - basic operations
            5.3.1  Addition   add12 11461
            5.3.2  Subtraction   cmin 11474
            5.3.3  Multiplication   kcnktkm1cn 11675
            5.3.4  Ordering on reals (cont.)   gt0ne0 11709
            5.3.5  Reciprocals   ixi 11873
            5.3.6  Division   cdiv 11901
            5.3.7  Ordering on reals (cont.)   elimgt0 12082
            5.3.8  Completeness Axiom and Suprema   fimaxre 12188
            5.3.9  Imaginary and complex number properties   inelr 12232
            5.3.10  Function operation analogue theorems   ofsubeq0 12239
      ​5.4  Integer sets
            5.4.1  Positive integers (as a subset of complex numbers)   cn 12242
            5.4.2  Principle of mathematical induction   nnind 12260
            ​*5.4.3  Decimal representation of numbers   c2 12297
            ​*5.4.4  Some properties of specific numbers   neg1cn 12356
            5.4.5  Simple number properties   halfcl 12467
            5.4.6  The Archimedean property   nnunb 12498
            5.4.7  Nonnegative integers (as a subset of complex numbers)   cn0 12502
            ​*5.4.8  Extended nonnegative integers   cxnn0 12574
            5.4.9  Integers (as a subset of complex numbers)   cz 12588
            5.4.10  Decimal arithmetic   cdc 12707
            5.4.11  Upper sets of integers   cuz 12852
            5.4.12  Well-ordering principle for bounded-below sets of integers   uzwo3 12957
            5.4.13  Rational numbers (as a subset of complex numbers)   cq 12962
            5.4.14  Existence of the set of complex numbers   rpnnen1lem2 12991
      ​5.5  Order sets
            5.5.1  Positive reals (as a subset of complex numbers)   crp 13006
            5.5.2  Infinity and the extended real number system (cont.)   cxne 13121
            5.5.3  Supremum and infimum on the extended reals   xrsupexmnf 13316
            5.5.4  Real number intervals   cioo 13356
            5.5.5  Finite intervals of integers   cfz 13516
            ​*5.5.6  Finite intervals of nonnegative integers   elfz2nn0 13624
            5.5.7  Half-open integer ranges   cfzo 13659
      ​5.6  Elementary integer functions
            5.6.1  The floor and ceiling functions   cfl 13787
            5.6.2  The modulo (remainder) operation   cmo 13866
            5.6.3  Miscellaneous theorems about integers   om2uz0i 13944
            5.6.4  Strong induction over upper sets of integers   uzsinds 13984
            5.6.5  Finitely supported functions over the nonnegative integers   fsuppmapnn0fiublem 13987
            5.6.6  The infinite sequence builder "seq" - extension   cseq 13998
            5.6.7  Integer powers   cexp 14058
            5.6.8  Ordered pair theorem for nonnegative integers   nn0le2msqi 14258
            5.6.9  Factorial function   cfa 14264
            5.6.10  The binomial coefficient operation   cbc 14293
            5.6.11  The ` # ` (set size) function   chash 14321
                  5.6.11.1  Proper unordered pairs and triples (sets of size 2 and 3)   hashprlei 14461
                  5.6.11.2  Functions with a domain containing at least two different elements   fundmge2nop0 14485
                  5.6.11.3  Finite induction on the size of the first component of a binary relation   hashdifsnp1 14489
      ​​*5.7  Words over a set
            5.7.1  Definitions and basic theorems   cword 14496
            5.7.2  Last symbol of a word   clsw 14544
            5.7.3  Concatenations of words   cconcat 14552
            5.7.4  Singleton words   cs1 14577
            5.7.5  Concatenations with singleton words   ccatws1cl 14598
            5.7.6  Subwords/substrings   csubstr 14622
            5.7.7  Prefixes of a word   cpfx 14652
            5.7.8  Subwords of subwords   swrdswrdlem 14686
            5.7.9  Subwords and concatenations   pfxcctswrd 14692
            5.7.10  Subwords of concatenations   swrdccatfn 14706
            5.7.11  Splicing words (substring replacement)   csplice 14731
            5.7.12  Reversing words   creverse 14740
            5.7.13  Repeated symbol words   creps 14750
            ​*5.7.14  Cyclical shifts of words   ccsh 14770
            5.7.15  Mapping words by a function   wrdco 14814
            5.7.16  Longer string literals   cs2 14824
      ​​*5.8  Reflexive and transitive closures of relations
            5.8.1  The reflexive and transitive properties of relations   coss12d 14951
            5.8.2  Basic properties of closures   cleq1lem 14961
            5.8.3  Definitions and basic properties of transitive closures   ctcl 14964
            5.8.4  Exponentiation of relations   crelexp 14998
            5.8.5  Reflexive-transitive closure as an indexed union   crtrcl 15034
            ​*5.8.6  Principle of transitive induction.   relexpindlem 15042
      ​5.9  Elementary real and complex functions
            5.9.1  The "shift" operation   cshi 15045
            5.9.2  Signum (sgn or sign) function   csgn 15065
            5.9.3  Real and imaginary parts; conjugate   ccj 15075
            5.9.4  Square root; absolute value   csqrt 15212
      ​5.10  Elementary limits and convergence
            5.10.1  Superior limit (lim sup)   clsp 15446
            5.10.2  Limits   cli 15460
            5.10.3  Finite and infinite sums   csu 15664
            5.10.4  The binomial theorem   binomlem 15807
            5.10.5  The inclusion/exclusion principle   incexclem 15814
            5.10.6  Infinite sums (cont.)   isumshft 15817
            5.10.7  Miscellaneous converging and diverging sequences   divrcnv 15830
            5.10.8  Arithmetic series   arisum 15838
            5.10.9  Geometric series   expcnv 15842
            5.10.10  Ratio test for infinite series convergence   cvgrat 15861
            5.10.11  Mertens' theorem   mertenslem1 15862
            5.10.12  Finite and infinite products   prodf 15865
                  5.10.12.1  Product sequences   prodf 15865
                  5.10.12.2  Non-trivial convergence   ntrivcvg 15875
                  5.10.12.3  Complex products   cprod 15881
                  5.10.12.4  Finite products   fprod 15917
                  5.10.12.5  Infinite products   iprodclim 15974
            5.10.13  Falling and Rising Factorial   cfallfac 15980
            5.10.14  Bernoulli polynomials and sums of k-th powers   cbp 16022
      ​5.11  Elementary trigonometry
            5.11.1  The exponential, sine, and cosine functions   ce 16037
                  5.11.1.1  The circle constant (tau = 2 pi)   ctau 16178
            5.11.2  _e is irrational   eirrlem 16180
      ​5.12  Cardinality of real and complex number subsets
            5.12.1  Countability of integers and rationals   xpnnen 16187
            5.12.2  The reals are uncountable   rpnnen2lem1 16190
​​*PART 6  ELEMENTARY NUMBER THEORY
      ​6.1  Elementary properties of divisibility
            6.1.1  Irrationality of square root of 2   sqrt2irrlem 16224
            6.1.2  Some Number sets are chains of proper subsets   nthruc 16228
            6.1.3  The divides relation   cdvds 16230
            ​*6.1.4  Even and odd numbers   evenelz 16312
            6.1.5  The division algorithm   divalglem0 16369
            6.1.6  Bit sequences   cbits 16393
            6.1.7  The greatest common divisor operator   cgcd 16468
            6.1.8  Bézout's identity   bezoutlem1 16514
            6.1.9  Algorithms   nn0seqcvgd 16540
            6.1.10  Euclid's Algorithm   eucalgval2 16551
            ​*6.1.11  The least common multiple   clcm 16558
            ​*6.1.12  Coprimality and Euclid's lemma   coprmgcdb 16619
            6.1.13  Cancellability of congruences   congr 16634
      ​6.2  Elementary prime number theory
            ​*6.2.1  Elementary properties   cprime 16641
            ​*6.2.2  Coprimality and Euclid's lemma (cont.)   coprm 16681
            6.2.3  Properties of the canonical representation of a rational   cnumer 16704
            6.2.4  Euler's theorem   codz 16731
            6.2.5  Arithmetic modulo a prime number   modprm1div 16765
            6.2.6  Pythagorean Triples   coprimeprodsq 16776
            6.2.7  The prime count function   cpc 16804
            6.2.8  Pocklington's theorem   prmpwdvds 16872
            6.2.9  Infinite primes theorem   unbenlem 16876
            6.2.10  Sum of prime reciprocals   prmreclem1 16884
            6.2.11  Fundamental theorem of arithmetic   1arithlem1 16891
            6.2.12  Lagrange's four-square theorem   cgz 16897
            6.2.13  Van der Waerden's theorem   cvdwa 16933
            6.2.14  Ramsey's theorem   cram 16967
            ​*6.2.15  Primorial function   cprmo 16999
            ​*6.2.16  Prime gaps   prmgaplem1 17017
            6.2.17  Decimal arithmetic (cont.)   dec2dvds 17031
            6.2.18  Cyclical shifts of words (cont.)   cshwsidrepsw 17062
            6.2.19  Specific prime numbers   prmlem0 17074
            6.2.20  Very large primes   1259lem1 17099
​PART 7  BASIC STRUCTURES
      ​7.1  Extensible structures
            ​*7.1.1  Basic definitions   cstr 17114
                  7.1.1.1  Extensible structures as structures with components   cstr 17114
                  7.1.1.2  Substitution of components   csts 17131
                  7.1.1.3  Slots   cslot 17149
                  ​*7.1.1.4  Structure component indices   cnx 17161
                  7.1.1.5  Base sets   cbs 17179
                  7.1.1.6  Base set restrictions   cress 17208
            7.1.2  Slot definitions   cplusg 17232
            7.1.3  Definition of the structure product   crest 17401
            7.1.4  Definition of the structure quotient   cordt 17480
      ​7.2  Moore spaces
            7.2.1  Moore closures   mrcflem 17585
            7.2.2  Independent sets in a Moore system   mrisval 17609
            7.2.3  Algebraic closure systems   isacs 17630
​PART 8  BASIC CATEGORY THEORY
      ​8.1  Categories
            8.1.1  Categories   ccat 17643
            8.1.2  Opposite category   coppc 17690
            8.1.3  Monomorphisms and epimorphisms   cmon 17710
            8.1.4  Sections, inverses, isomorphisms   csect 17726
            ​*8.1.5  Isomorphic objects   ccic 17777
            8.1.6  Subcategories   cssc 17789
            8.1.7  Functors   cfunc 17839
            8.1.8  Full & faithful functors   cful 17890
            8.1.9  Natural transformations and the functor category   cnat 17930
            8.1.10  Initial, terminal and zero objects of a category   cinito 17969
      ​8.2  Arrows (disjointified hom-sets)
            8.2.1  Identity and composition for arrows   cida 18041
      ​8.3  Examples of categories
            8.3.1  The category of sets   csetc 18063
            8.3.2  The category of categories   ccatc 18086
            ​*8.3.3  The category of extensible structures   fncnvimaeqv 18109
      ​8.4  Categorical constructions
            8.4.1  Product of categories   cxpc 18158
            8.4.2  Functor evaluation   cevlf 18200
            8.4.3  Hom functor   chof 18239
​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 18422
            9.5.2  Complete lattices   ccla 18489
            9.5.3  Distributive lattices   cdlat 18511
            9.5.4  Subset order structures   cipo 18518
      ​9.6  Posets, directed sets, and lattices as relations
            ​*9.6.1  Posets and lattices as relations   cps 18555
            9.6.2  Directed sets, nets   cdir 18585
​PART 10  BASIC ALGEBRAIC STRUCTURES
      ​10.1  Monoids
            ​*10.1.1  Magmas   cplusf 18596
            ​*10.1.2  Identity elements   mgmidmo 18619
            ​*10.1.3  Iterated sums in a magma   gsumvalx 18635
            10.1.4  Magma homomorphisms and submagmas   cmgmhm 18649
            ​*10.1.5  Semigroups   csgrp 18677
            ​*10.1.6  Definition and basic properties of monoids   cmnd 18693
            10.1.7  Monoid homomorphisms and submonoids   cmhm 18737
            ​*10.1.8  Iterated sums in a monoid   gsumvallem2 18790
            10.1.9  Free monoids   cfrmd 18803
                  ​*10.1.9.1  Monoid of endofunctions   cefmnd 18824
            10.1.10  Examples and counterexamples for magmas, semigroups and monoids   mgm2nsgrplem1 18874
      ​10.2  Groups
            10.2.1  Definition and basic properties   cgrp 18894
            ​*10.2.2  Group multiple operation   cmg 19027
            10.2.3  Subgroups and Quotient groups   csubg 19079
            ​*10.2.4  Cyclic monoids and groups   cycsubmel 19159
            10.2.5  Elementary theory of group homomorphisms   cghm 19171
            10.2.6  Isomorphisms of groups   cgim 19215
                  10.2.6.1  The first isomorphism theorem of groups   ghmquskerlem1 19238
            10.2.7  Group actions   cga 19244
            10.2.8  Centralizers and centers   ccntz 19270
            10.2.9  The opposite group   coppg 19300
            10.2.10  Symmetric groups   csymg 19325
                  ​*10.2.10.1  Definition and basic properties   csymg 19325
                  10.2.10.2  Cayley's theorem   cayleylem1 19371
                  10.2.10.3  Permutations fixing one element   symgfix2 19375
                  ​*10.2.10.4  Transpositions in the symmetric group   cpmtr 19400
                  10.2.10.5  The sign of a permutation   cpsgn 19448
            10.2.11  p-Groups and Sylow groups; Sylow's theorems   cod 19483
            10.2.12  Direct products   clsm 19593
                  10.2.12.1  Direct products (extension)   smndlsmidm 19615
            10.2.13  Free groups   cefg 19665
            10.2.14  Abelian groups   ccmn 19739
                  10.2.14.1  Definition and basic properties   ccmn 19739
                  10.2.14.2  Cyclic groups   ccyg 19836
                  10.2.14.3  Group sum operation   gsumval3a 19862
                  10.2.14.4  Group sums over (ranges of) integers   fsfnn0gsumfsffz 19942
                  10.2.14.5  Internal direct products   cdprd 19954
                  10.2.14.6  The Fundamental Theorem of Abelian Groups   ablfacrplem 20026
            10.2.15  Simple groups   csimpg 20051
                  10.2.15.1  Definition and basic properties   csimpg 20051
                  10.2.15.2  Classification of abelian simple groups   ablsimpnosubgd 20065
      ​10.3  Rings
            10.3.1  Multiplicative Group   cmgp 20078
            ​*10.3.2  Non-unital rings ("rngs")   crng 20096
            ​*10.3.3  Ring unity (multiplicative identity)   cur 20125
            10.3.4  Semirings   csrg 20130
                  ​*10.3.4.1  The binomial theorem for semirings   srgbinomlem1 20170
            10.3.5  Unital rings   crg 20177
            10.3.6  Opposite ring   coppr 20276
            10.3.7  Divisibility   cdsr 20297
            10.3.8  Ring primes   crpm 20375
            10.3.9  Homomorphisms of non-unital rings   crnghm 20377
            10.3.10  Ring homomorphisms   crh 20412
            10.3.11  Nonzero rings and zero rings   cnzr 20455
            10.3.12  Local rings   clring 20479
            10.3.13  Subrings   csubrng 20486
                  10.3.13.1  Subrings of non-unital rings   csubrng 20486
                  10.3.13.2  Subrings of unital rings   csubrg 20510
            10.3.14  Categories of rings   crngc 20553
                  ​*10.3.14.1  The category of non-unital rings   crngc 20553
                  ​*10.3.14.2  The category of (unital) rings   cringc 20582
                  10.3.14.3  Subcategories of the category of rings   srhmsubclem1 20614
      ​10.4  Division rings and fields
            10.4.1  Definition and basic properties   cdr 20628
            10.4.2  Sub-division rings   csdrg 20678
            10.4.3  Absolute value (abstract algebra)   cabv 20700
            10.4.4  Star rings   cstf 20727
      ​10.5  Left modules
            10.5.1  Definition and basic properties   clmod 20747
            10.5.2  Subspaces and spans in a left module   clss 20819
            10.5.3  Homomorphisms and isomorphisms of left modules   clmhm 20908
            10.5.4  Subspace sum; bases for a left module   clbs 20963
      ​10.6  Vector spaces
            10.6.1  Definition and basic properties   clvec 20991
      ​10.7  Subring algebras and ideals
            10.7.1  Subring algebras   csra 21060
            ​*10.7.2  Left ideals and spans   clidl 21106
            10.7.3  Two-sided ideals and quotient rings   c2idl 21147
                  ​*10.7.3.1  Condition for a non-unital ring to be unital   rngqiprng1elbas 21180
            10.7.4  Principal ideal rings. Divisibility in the integers   clpidl 21214
            10.7.5  Left regular elements. More kinds of rings   crlreg 21230
      ​10.8  The complex numbers as an algebraic extensible structure
            10.8.1  Definition and basic properties   cpsmet 21267
            ​*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 21513
            10.8.5  Embedding of permutation signs into a ring   zrhpsgnmhm 21520
            10.8.6  The ordered field of real numbers   crefld 21540
      ​10.9  Generalized pre-Hilbert and Hilbert spaces
            10.9.1  Definition and basic properties   cphl 21560
            10.9.2  Orthocomplements and closed subspaces   cocv 21596
            10.9.3  Orthogonal projection and orthonormal bases   cpj 21638
​​*PART 11  BASIC LINEAR ALGEBRA
      ​11.1  Vectors and free modules
            *11.1.1  Direct sum of left modules   cdsmm 21669
            ​*11.1.2  Free modules   cfrlm 21684
            ​*11.1.3  Standard basis (unit vectors)   cuvc 21720
            ​*11.1.4  Independent sets and families   clindf 21742
            11.1.5  Characterization of free modules   lmimlbs 21774
      ​11.2  Associative algebras
            11.2.1  Definition and basic properties   casa 21788
      ​11.3  Abstract multivariate polynomials
            11.3.1  Definition and basic properties   cmps 21841
            11.3.2  Polynomial evaluation   ces 22023
            11.3.3  Additional definitions for (multivariate) polynomials   cslv 22061
            ​*11.3.4  Univariate polynomials   cps1 22102
            11.3.5  Univariate polynomial evaluation   ces1 22241
                  11.3.5.1  Specialization of polynomial evaluation as a ring homomorphism   evls1scafv 22294
      ​​*11.4  Matrices
            *11.4.1  The matrix multiplication   cmmul 22320
            ​*11.4.2  Square matrices   cmat 22337
            ​*11.4.3  The matrix algebra   matmulr 22370
            ​*11.4.4  Matrices of dimension 0 and 1   mat0dimbas0 22398
            ​*11.4.5  The subalgebras of diagonal and scalar matrices   cdmat 22420
            ​*11.4.6  Multiplication of a matrix with a "column vector"   cmvmul 22472
            11.4.7  Replacement functions for a square matrix   cmarrep 22488
            11.4.8  Submatrices   csubma 22508
      ​11.5  The determinant
            11.5.1  Definition and basic properties   cmdat 22516
            11.5.2  Determinants of 2 x 2 -matrices   m2detleiblem1 22556
            11.5.3  The matrix adjugate/adjunct   cmadu 22564
            ​*11.5.4  Laplace expansion of determinants (special case)   symgmatr01lem 22585
            11.5.5  Inverse matrix   invrvald 22608
            ​*11.5.6  Cramer's rule   slesolvec 22611
      ​​*11.6  Polynomial matrices
            11.6.1  Basic properties   pmatring 22624
            ​*11.6.2  Constant polynomial matrices   ccpmat 22635
            ​*11.6.3  Collecting coefficients of polynomial matrices   cdecpmat 22694
            ​*11.6.4  Ring isomorphism between polynomial matrices and polynomials over matrices   cpm2mp 22724
      ​​*11.7  The characteristic polynomial
            *11.7.1  Definition and basic properties   cchpmat 22758
            ​*11.7.2  The characteristic factor function G   fvmptnn04if 22781
            ​*11.7.3  The Cayley-Hamilton theorem   cpmadurid 22799
​PART 12  BASIC TOPOLOGY
      ​12.1  Topology
            ​*12.1.1  Topological spaces   ctop 22825
                  12.1.1.1  Topologies   ctop 22825
                  12.1.1.2  Topologies on sets   ctopon 22842
                  12.1.1.3  Topological spaces   ctps 22864
            12.1.2  Topological bases   ctb 22878
            12.1.3  Examples of topologies   distop 22928
            12.1.4  Closure and interior   ccld 22950
            12.1.5  Neighborhoods   cnei 23031
            12.1.6  Limit points and perfect sets   clp 23068
            12.1.7  Subspace topologies   restrcl 23091
            12.1.8  Order topology   ordtbaslem 23122
            12.1.9  Limits and continuity in topological spaces   ccn 23158
            12.1.10  Separated spaces: T0, T1, T2 (Hausdorff) ...   ct0 23240
            12.1.11  Compactness   ccmp 23320
            12.1.12  Bolzano-Weierstrass theorem   bwth 23344
            12.1.13  Connectedness   cconn 23345
            12.1.14  First- and second-countability   c1stc 23371
            12.1.15  Local topological properties   clly 23398
            12.1.16  Refinements   cref 23436
            12.1.17  Compactly generated spaces   ckgen 23467
            12.1.18  Product topologies   ctx 23494
            12.1.19  Continuous function-builders   cnmptid 23595
            12.1.20  Quotient maps and quotient topology   ckq 23627
            12.1.21  Homeomorphisms   chmeo 23687
      ​12.2  Filters and filter bases
            12.2.1  Filter bases   elmptrab 23761
            12.2.2  Filters   cfil 23779
            12.2.3  Ultrafilters   cufil 23833
            12.2.4  Filter limits   cfm 23867
            12.2.5  Extension by continuity   ccnext 23993
            12.2.6  Topological groups   ctmd 24004
            12.2.7  Infinite group sum on topological groups   ctsu 24060
            12.2.8  Topological rings, fields, vector spaces   ctrg 24090
      ​12.3  Uniform Structures and Spaces
            12.3.1  Uniform structures   cust 24134
            12.3.2  The topology induced by an uniform structure   cutop 24165
            12.3.3  Uniform Spaces   cuss 24188
            12.3.4  Uniform continuity   cucn 24210
            12.3.5  Cauchy filters in uniform spaces   ccfilu 24221
            12.3.6  Complete uniform spaces   ccusp 24232
      ​12.4  Metric spaces
            12.4.1  Pseudometric spaces   ispsmet 24240
            12.4.2  Basic metric space properties   cxms 24253
            12.4.3  Metric space balls   blfvalps 24319
            12.4.4  Open sets of a metric space   mopnval 24374
            12.4.5  Continuity in metric spaces   metcnp3 24479
            12.4.6  The uniform structure generated by a metric   metuval 24488
            12.4.7  Examples of metric spaces   dscmet 24511
            ​*12.4.8  Normed algebraic structures   cnm 24515
            12.4.9  Normed space homomorphisms (bounded linear operators)   cnmo 24652
            12.4.10  Topology on the reals   qtopbaslem 24705
            12.4.11  Topological definitions using the reals   cii 24825
            12.4.12  Path homotopy   chtpy 24923
            12.4.13  The fundamental group   cpco 24957
      ​12.5  Metric subcomplex vector spaces
            12.5.1  Subcomplex modules   cclm 25019
            ​*12.5.2  Subcomplex vector spaces   ccvs 25080
            ​*12.5.3  Normed subcomplex vector spaces   isncvsngp 25107
            12.5.4  Subcomplex pre-Hilbert spaces   ccph 25124
            12.5.5  Convergence and completeness   ccfil 25210
            12.5.6  Baire's Category Theorem   bcthlem1 25282
            12.5.7  Banach spaces and subcomplex Hilbert spaces   ccms 25290
                  12.5.7.1  The complete ordered field of the real numbers   retopn 25337
            12.5.8  Euclidean spaces   crrx 25341
            12.5.9  Minimizing Vector Theorem   minveclem1 25382
            12.5.10  Projection Theorem   pjthlem1 25395
​PART 13  BASIC REAL AND COMPLEX ANALYSIS
      ​13.1  Continuity
            13.1.1  Intermediate value theorem   pmltpclem1 25407
      ​13.2  Integrals
            13.2.1  Lebesgue measure   covol 25421
            13.2.2  Lebesgue integration   cmbf 25573
                  13.2.2.1  Lesbesgue integral   cmbf 25573
                  13.2.2.2  Lesbesgue directed integral   cdit 25805
      ​13.3  Derivatives
            13.3.1  Real and complex differentiation   climc 25821
                  13.3.1.1  Derivatives of functions of one complex or real variable   climc 25821
                  13.3.1.2  Results on real differentiation   dvferm1lem 25946
​PART 14  BASIC REAL AND COMPLEX FUNCTIONS
      ​14.1  Polynomials
            14.1.1  Polynomial degrees   cmdg 26016
            14.1.2  The division algorithm for univariate polynomials   cmn1 26091
            14.1.3  Elementary properties of complex polynomials   cply 26148
            14.1.4  The division algorithm for polynomials   cquot 26255
            14.1.5  Algebraic numbers   caa 26279
            14.1.6  Liouville's approximation theorem   aalioulem1 26297
      ​14.2  Sequences and series
            14.2.1  Taylor polynomials and Taylor's theorem   ctayl 26317
            14.2.2  Uniform convergence   culm 26342
            14.2.3  Power series   pserval 26376
      ​14.3  Basic trigonometry
            14.3.1  The exponential, sine, and cosine functions (cont.)   efcn 26410
            14.3.2  Properties of pi = 3.14159...   pilem1 26418
            14.3.3  Mapping of the exponential function   efgh 26505
            14.3.4  The natural logarithm on complex numbers   clog 26518
            ​*14.3.5  Logarithms to an arbitrary base   clogb 26726
            14.3.6  Theorems of Pythagoras, isosceles triangles, and intersecting chords   angval 26763
            14.3.7  Solutions of quadratic, cubic, and quartic equations   quad2 26801
            14.3.8  Inverse trigonometric functions   casin 26824
            14.3.9  The Birthday Problem   log2ublem1 26908
            14.3.10  Areas in R^2   carea 26917
            14.3.11  More miscellaneous converging sequences   rlimcnp 26927
            14.3.12  Inequality of arithmetic and geometric means   cvxcl 26947
            14.3.13  Euler-Mascheroni constant   cem 26954
            14.3.14  Zeta function   czeta 26975
            14.3.15  Gamma function   clgam 26978
      ​14.4  Basic number theory
            14.4.1  Wilson's theorem   wilthlem1 27030
            14.4.2  The Fundamental Theorem of Algebra   ftalem1 27035
            14.4.3  The Basel problem (ζ(2) = π²/6)   basellem1 27043
            14.4.4  Number-theoretical functions   ccht 27053
            14.4.5  Perfect Number Theorem   mersenne 27190
            14.4.6  Characters of Z/nZ   cdchr 27195
            14.4.7  Bertrand's postulate   bcctr 27238
            ​*14.4.8  Quadratic residues and the Legendre symbol   clgs 27257
            ​*14.4.9  Gauss' Lemma   gausslemma2dlem0a 27319
            14.4.10  Quadratic reciprocity   lgseisenlem1 27338
            14.4.11  All primes 4n+1 are the sum of two squares   2sqlem1 27380
            14.4.12  Chebyshev's Weak Prime Number Theorem, Dirichlet's Theorem   chebbnd1lem1 27432
            14.4.13  The Prime Number Theorem   mudivsum 27493
            14.4.14  Ostrowski's theorem   abvcxp 27578
​PART 15  SURREAL NUMBERS
      ​​*15.1  Sign sequence representation and Alling's axioms
            15.1.1  Definitions and initial properties   csur 27603
            15.1.2  Ordering   sltsolem1 27638
            15.1.3  Birthday Function   bdayfo 27640
            15.1.4  Density   fvnobday 27641
            ​*15.1.5  Full-Eta Property   bdayimaon 27656
      ​15.2  Initial consequences of Alling's axioms
            15.2.1  Ordering Theorems   csle 27707
            15.2.2  Birthday Theorems   bdayfun 27735
      ​​*15.3  Conway cut representation
            15.3.1  Conway cuts   csslt 27743
            15.3.2  Zero and One   c0s 27785
            15.3.3  Cuts and Options   cmade 27799
            15.3.4  Cofinality and coinitiality   cofsslt 27868
      ​15.4  Induction and recursion
            15.4.1  Induction and recursion on one variable   cnorec 27884
            15.4.2  Induction and recursion on two variables   cnorec2 27895
      ​15.5  Surreal arithmetic
            15.5.1  Addition   cadds 27906
            15.5.2  Negation and Subtraction   cnegs 27962
            15.5.3  Multiplication   cmuls 28040
            15.5.4  Division   cdivs 28121
            15.5.5  Absolute value   cabss 28165
      ​15.6  Subsystems of surreals
            15.6.1  Ordinal numbers   cons 28178
            15.6.2  Surreal recursive sequences   cseqs 28190
            15.6.3  Natural numbers   cnn0s 28219
            15.6.4  Integers   czs 28261
            15.6.5  Real numbers   creno 28277
​​*PART 16  ELEMENTARY GEOMETRY
      ​16.1  Definition and Tarski's Axioms of Geometry
            16.1.1  Justification for the congruence notation   tgjustf 28333
      ​16.2  Tarskian Geometry
            16.2.1  Congruence   tgcgrcomimp 28337
            16.2.2  Betweenness   tgbtwntriv2 28347
            16.2.3  Dimension   tglowdim1 28360
            16.2.4  Betweenness and Congruence   tgifscgr 28368
            16.2.5  Congruence of a series of points   ccgrg 28370
            16.2.6  Motions   cismt 28392
            16.2.7  Colinearity   tglng 28406
            16.2.8  Connectivity of betweenness   tgbtwnconn1lem1 28432
            16.2.9  Less-than relation in geometric congruences   cleg 28442
            16.2.10  Rays   chlg 28460
            16.2.11  Lines   btwnlng1 28479
            16.2.12  Point inversions   cmir 28512
            16.2.13  Right angles   crag 28553
            16.2.14  Half-planes   islnopp 28599
            16.2.15  Midpoints and Line Mirroring   cmid 28632
            16.2.16  Congruence of angles   ccgra 28667
            16.2.17  Angle Comparisons   cinag 28695
            16.2.18  Congruence Theorems   tgsas1 28714
            16.2.19  Equilateral triangles   ceqlg 28725
      ​16.3  Properties of geometries
            16.3.1  Isomorphisms between geometries   f1otrgds 28729
      ​16.4  Geometry in Hilbert spaces
            16.4.1  Geometry in the complex plane   cchhllem 28753
            16.4.2  Geometry in Euclidean spaces   cee 28755
                  16.4.2.1  Definition of the Euclidean space   cee 28755
                  16.4.2.2  Tarski's axioms for geometry for the Euclidean space   axdimuniq 28780
                  16.4.2.3  EE^n fulfills Tarski's Axioms   ceeng 28844
​​*PART 17  GRAPH THEORY
      ​*17.1  Vertices and edges
            17.1.1  The edge function extractor for extensible structures   cedgf 28855
            ​*17.1.2  Vertices and indexed edges   cvtx 28865
                  17.1.2.1  Definitions and basic properties   cvtx 28865
                  17.1.2.2  The vertices and edges of a graph represented as ordered pair   opvtxval 28872
                  17.1.2.3  The vertices and edges of a graph represented as extensible structure   funvtxdmge2val 28880
                  17.1.2.4  Representations of graphs without edges   snstrvtxval 28906
                  17.1.2.5  Degenerated cases of representations of graphs   vtxval0 28908
            17.1.3  Edges as range of the edge function   cedg 28916
      ​​*17.2  Undirected graphs
            17.2.1  Undirected hypergraphs   cuhgr 28925
            17.2.2  Undirected pseudographs and multigraphs   cupgr 28949
            ​*17.2.3  Loop-free graphs   umgrislfupgrlem 28991
            17.2.4  Edges as subsets of vertices of graphs   uhgredgiedgb 28995
            ​*17.2.5  Undirected simple graphs   cuspgr 29017
            17.2.6  Examples for graphs   usgr0e 29105
            17.2.7  Subgraphs   csubgr 29136
            17.2.8  Finite undirected simple graphs   cfusgr 29185
            17.2.9  Neighbors, complete graphs and universal vertices   cnbgr 29201
                  17.2.9.1  Neighbors   cnbgr 29201
                  17.2.9.2  Universal vertices   cuvtx 29254
                  17.2.9.3  Complete graphs   ccplgr 29278
            17.2.10  Vertex degree   cvtxdg 29335
            ​*17.2.11  Regular graphs   crgr 29425
      ​​*17.3  Walks, paths and cycles
            *17.3.1  Walks   cewlks 29465
            17.3.2  Walks for loop-free graphs   lfgrwlkprop 29557
            17.3.3  Trails   ctrls 29560
            17.3.4  Paths and simple paths   cpths 29582
            17.3.5  Closed walks   cclwlks 29640
            17.3.6  Circuits and cycles   ccrcts 29654
            ​*17.3.7  Walks as words   cwwlks 29692
            17.3.8  Walks/paths of length 2 (as length 3 strings)   2wlkdlem1 29792
            17.3.9  Walks in regular graphs   rusgrnumwwlkl1 29835
            ​*17.3.10  Closed walks as words   cclwwlk 29847
                  17.3.10.1  Closed walks as words   cclwwlk 29847
                  17.3.10.2  Closed walks of a fixed length as words   cclwwlkn 29890
                  17.3.10.3  Closed walks on a vertex of a fixed length as words   cclwwlknon 29953
            17.3.11  Examples for walks, trails and paths   0ewlk 29980
            17.3.12  Connected graphs   cconngr 30052
      ​17.4  Eulerian paths and the Konigsberg Bridge problem
            ​*17.4.1  Eulerian paths   ceupth 30063
            ​*17.4.2  The Königsberg Bridge problem   konigsbergvtx 30112
      ​17.5  The Friendship Theorem
            17.5.1  Friendship graphs - basics   cfrgr 30124
            17.5.2  The friendship theorem for small graphs   frgr1v 30137
            17.5.3  Theorems according to Mertzios and Unger   2pthfrgrrn 30148
            ​*17.5.4  Huneke's Proof of the Friendship Theorem   frgrncvvdeqlem1 30165
​PART 18  GUIDES AND MISCELLANEA
      ​18.1  Guides (conventions, explanations, and examples)
            ​*18.1.1  Conventions   conventions 30266
            18.1.2  Natural deduction   natded 30269
            ​*18.1.3  Natural deduction examples   ex-natded5.2 30270
            18.1.4  Definitional examples   ex-or 30287
            18.1.5  Other examples   aevdemo 30326
      ​18.2  Humor
            18.2.1  April Fool's theorem   avril1 30329
      ​18.3  (Future - to be reviewed and classified)
            18.3.1  Planar incidence geometry   cplig 30340
            ​*18.3.2  Aliases kept to prevent broken links   dummylink 30353
​​*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 30355
            19.1.2  Abelian groups   cablo 30410
      ​19.2  Complex vector spaces
            19.2.1  Definition and basic properties   cvc 30424
            19.2.2  Examples of complex vector spaces   cnaddabloOLD 30447
      ​19.3  Normed complex vector spaces
            19.3.1  Definition and basic properties   cnv 30450
            19.3.2  Examples of normed complex vector spaces   cnnv 30543
            19.3.3  Induced metric of a normed complex vector space   imsval 30551
            19.3.4  Inner product   cdip 30566
            19.3.5  Subspaces   css 30587
      ​19.4  Operators on complex vector spaces
            19.4.1  Definitions and basic properties   clno 30606
      ​19.5  Inner product (pre-Hilbert) spaces
            19.5.1  Definition and basic properties   ccphlo 30678
            19.5.2  Examples of pre-Hilbert spaces   cncph 30685
            19.5.3  Properties of pre-Hilbert spaces   isph 30688
      ​19.6  Complex Banach spaces
            19.6.1  Definition and basic properties   ccbn 30728
            19.6.2  Examples of complex Banach spaces   cnbn 30735
            19.6.3  Uniform Boundedness Theorem   ubthlem1 30736
            19.6.4  Minimizing Vector Theorem   minvecolem1 30740
      ​19.7  Complex Hilbert spaces
            19.7.1  Definition and basic properties   chlo 30751
            19.7.2  Standard axioms for a complex Hilbert space   hlex 30764
            19.7.3  Examples of complex Hilbert spaces   cnchl 30782
            19.7.4  Hellinger-Toeplitz Theorem   htthlem 30783
​​*PART 20  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      ​20.1  Axiomatization of complex pre-Hilbert spaces
            20.1.1  Basic Hilbert space definitions   chba 30785
            20.1.2  Preliminary ZFC lemmas   df-hnorm 30834
            ​*20.1.3  Derive the Hilbert space axioms from ZFC set theory   axhilex-zf 30847
            ​*20.1.4  Introduce the vector space axioms for a Hilbert space   ax-hilex 30865
            20.1.5  Vector operations   hvmulex 30877
            20.1.6  Inner product postulates for a Hilbert space   ax-hfi 30945
      ​20.2  Inner product and norms
            20.2.1  Inner product   his5 30952
            20.2.2  Norms   dfhnorm2 30988
            20.2.3  Relate Hilbert space to normed complex vector spaces   hilablo 31026
            20.2.4  Bunjakovaskij-Cauchy-Schwarz inequality   bcsiALT 31045
      ​20.3  Cauchy sequences and completeness axiom
            20.3.1  Cauchy sequences and limits   hcau 31050
            20.3.2  Derivation of the completeness axiom from ZF set theory   hilmet 31060
            20.3.3  Completeness postulate for a Hilbert space   ax-hcompl 31068
            20.3.4  Relate Hilbert space to ZFC pre-Hilbert and Hilbert spaces   hhcms 31069
      ​20.4  Subspaces and projections
            20.4.1  Subspaces   df-sh 31073
            20.4.2  Closed subspaces   df-ch 31087
            20.4.3  Orthocomplements   df-oc 31118
            20.4.4  Subspace sum, span, lattice join, lattice supremum   df-shs 31174
            20.4.5  Projection theorem   pjhthlem1 31257
            20.4.6  Projectors   df-pjh 31261
      ​20.5  Properties of Hilbert subspaces
            20.5.1  Orthomodular law   omlsilem 31268
            20.5.2  Projectors (cont.)   pjhtheu2 31282
            20.5.3  Hilbert lattice operations   sh0le 31306
            20.5.4  Span (cont.) and one-dimensional subspaces   spansn0 31407
            20.5.5  Commutes relation for Hilbert lattice elements   df-cm 31449
            20.5.6  Foulis-Holland theorem   fh1 31484
            20.5.7  Quantum Logic Explorer axioms   qlax1i 31493
            20.5.8  Orthogonal subspaces   chscllem1 31503
            20.5.9  Orthoarguesian laws 5OA and 3OA   5oalem1 31520
            20.5.10  Projectors (cont.)   pjorthi 31535
            20.5.11  Mayet's equation E_3   mayete3i 31594
      ​20.6  Operators on Hilbert spaces
            ​*20.6.1  Operator sum, difference, and scalar multiplication   df-hosum 31596
            20.6.2  Zero and identity operators   df-h0op 31614
            20.6.3  Operations on Hilbert space operators   hoaddcl 31624
            20.6.4  Linear, continuous, bounded, Hermitian, unitary operators and norms   df-nmop 31705
            20.6.5  Linear and continuous functionals and norms   df-nmfn 31711
            20.6.6  Adjoint   df-adjh 31715
            20.6.7  Dirac bra-ket notation   df-bra 31716
            20.6.8  Positive operators   df-leop 31718
            20.6.9  Eigenvectors, eigenvalues, spectrum   df-eigvec 31719
            20.6.10  Theorems about operators and functionals   nmopval 31722
            20.6.11  Riesz lemma   riesz3i 31928
            20.6.12  Adjoints (cont.)   cnlnadjlem1 31933
            20.6.13  Quantum computation error bound theorem   unierri 31970
            20.6.14  Dirac bra-ket notation (cont.)   branmfn 31971
            20.6.15  Positive operators (cont.)   leopg 31988
            20.6.16  Projectors as operators   pjhmopi 32012
      ​20.7  States on a Hilbert lattice and Godowski's equation
            20.7.1  States on a Hilbert lattice   df-st 32077
            20.7.2  Godowski's equation   golem1 32137
      ​20.8  Cover relation, atoms, exchange axiom, and modular symmetry
            20.8.1  Covers relation; modular pairs   df-cv 32145
            20.8.2  Atoms   df-at 32204
            20.8.3  Superposition principle   superpos 32220
            20.8.4  Atoms, exchange and covering properties, atomicity   chcv1 32221
            20.8.5  Irreducibility   chirredlem1 32256
            20.8.6  Atoms (cont.)   atcvat3i 32262
            20.8.7  Modular symmetry   mdsymlem1 32269
​PART 21  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      ​21.1  Mathboxes for user contributions
            21.1.1  Mathbox guidelines   mathbox 32308
      ​21.2  Mathbox for Stefan Allan
      ​21.3  Mathbox for Thierry Arnoux
            21.3.1  Propositional Calculus - misc additions   bian1d 32313
            21.3.2  Predicate Calculus   sbc2iedf 32322
                  21.3.2.1  Predicate Calculus - misc additions   sbc2iedf 32322
                  21.3.2.2  Restricted quantification - misc additions   ralcom4f 32324
                  21.3.2.3  Equality   eqtrb 32329
                  21.3.2.4  Double restricted existential uniqueness quantification   opsbc2ie 32331
                  21.3.2.5  Double restricted existential uniqueness quantification syntax   w2reu 32333
                  21.3.2.6  Substitution (without distinct variables) - misc additions   sbceqbidf 32342
                  21.3.2.7  Existential "at most one" - misc additions   mo5f 32344
                  21.3.2.8  Existential uniqueness - misc additions   reuxfrdf 32346
                  21.3.2.9  Restricted "at most one" - misc additions   rmoxfrd 32348
                  21.3.2.10  Restricted iota (description binder)   riotaeqbidva 32351
            21.3.3  General Set Theory   dmrab 32352
                  21.3.3.1  Class abstractions (a.k.a. class builders)   dmrab 32352
                  21.3.3.2  Image Sets   abrexdomjm 32359
                  21.3.3.3  Set relations and operations - misc additions   elunsn 32365
                  21.3.3.4  Unordered pairs   eqsnd 32382
                  21.3.3.5  Conditional operator - misc additions   ifeqeqx 32390
                  21.3.3.6  Set union   uniinn0 32398
                  21.3.3.7  Indexed union - misc additions   cbviunf 32403
                  21.3.3.8  Indexed intersection - misc additions   iinabrex 32416
                  21.3.3.9  Disjointness - misc additions   disjnf 32417
            21.3.4  Relations and Functions   xpdisjres 32445
                  21.3.4.1  Relations - misc additions   xpdisjres 32445
                  21.3.4.2  Functions - misc additions   ac6sf2 32467
                  21.3.4.3  Operations - misc additions   mpomptxf 32521
                  21.3.4.4  Support of a function   suppovss 32522
                  21.3.4.5  Explicit Functions with one or two points as a domain   cosnopne 32531
                  21.3.4.6  Isomorphisms - misc. additions   gtiso 32537
                  21.3.4.7  Disjointness (additional proof requiring functions)   disjdsct 32539
                  21.3.4.8  First and second members of an ordered pair - misc additions   df1stres 32540
                  21.3.4.9  Supremum - misc additions   supssd 32548
                  21.3.4.10  Finite Sets   imafi2 32550
                  21.3.4.11  Countable Sets   snct 32552
            21.3.5  Real and Complex Numbers   creq0 32574
                  21.3.5.1  Complex operations - misc. additions   creq0 32574
                  21.3.5.2  Ordering on reals - misc additions   lt2addrd 32578
                  21.3.5.3  Extended reals - misc additions   xrlelttric 32579
                  21.3.5.4  Extended nonnegative integers - misc additions   xnn0gt0 32596
                  21.3.5.5  Real number intervals - misc additions   joiniooico 32599
                  21.3.5.6  Finite intervals of integers - misc additions   uzssico 32609
                  21.3.5.7  Half-open integer ranges - misc additions   iundisjfi 32621
                  21.3.5.8  The ` # ` (set size) function - misc additions   hashunif 32632
                  21.3.5.9  The greatest common divisor operator - misc. additions   znumd 32635
                  21.3.5.10  Integers   nnindf 32639
                  21.3.5.11  Decimal numbers   dfdec100 32650
            ​*21.3.6  Decimal expansion   cdp2 32651
                  ​*21.3.6.1  Decimal point   cdp 32668
                  21.3.6.2  Division in the extended real number system   cxdiv 32697
            21.3.7  Words over a set - misc additions   wrdfd 32716
                  21.3.7.1  Splicing words (substring replacement)   splfv3 32736
                  21.3.7.2  Cyclic shift of words   1cshid 32737
            21.3.8  Extensible Structures   ressplusf 32741
                  21.3.8.1  Structure restriction operator   ressplusf 32741
                  21.3.8.2  The opposite group   oppgle 32744
                  21.3.8.3  Posets   ressprs 32747
                  21.3.8.4  Complete lattices   clatp0cl 32760
                  21.3.8.5  Order Theory   cmnt 32762
                  21.3.8.6  Extended reals Structure - misc additions   ax-xrssca 32788
                  21.3.8.7  The extended nonnegative real numbers commutative monoid   xrge0base 32798
            21.3.9  Algebra   cmn4d 32809
                  21.3.9.1  Monoids   cmn4d 32809
                  21.3.9.2  Monoids Homomorphisms   abliso 32813
                  21.3.9.3  Finitely supported group sums - misc additions   gsumsubg 32817
                  21.3.9.4  Centralizers and centers - misc additions   cntzun 32831
                  21.3.9.5  Totally ordered monoids and groups   comnd 32834
                  21.3.9.6  The symmetric group   symgfcoeu 32862
                  21.3.9.7  Transpositions   pmtridf1o 32872
                  21.3.9.8  Permutation Signs   psgnid 32875
                  21.3.9.9  Permutation cycles   ctocyc 32884
                  21.3.9.10  The Alternating Group   evpmval 32923
                  21.3.9.11  Signum in an ordered monoid   csgns 32936
                  21.3.9.12  The Archimedean property for generic ordered algebraic structures   cinftm 32941
                  21.3.9.13  Semiring left modules   cslmd 32964
                  21.3.9.14  Simple groups   prmsimpcyc 32992
                  21.3.9.15  Rings - misc additions   cringmul32d 32993
                  21.3.9.16  The zero ring   irrednzr 33004
                  21.3.9.17  Localization of rings   cerl 33007
                  21.3.9.18  Integral Domains   domnlcan 33028
                  21.3.9.19  Euclidean Domains   ceuf 33037
                  21.3.9.20  Division Rings   ringinveu 33043
                  21.3.9.21  Subfields   sdrgdvcl 33046
                  21.3.9.22  Field of fractions   cfrac 33049
                  21.3.9.23  Field extensions generated by a set   cfldgen 33057
                  21.3.9.24  Totally ordered rings and fields   corng 33070
                  21.3.9.25  Ring homomorphisms - misc additions   rhmdvd 33093
                  21.3.9.26  Scalar restriction operation   cresv 33095
                  21.3.9.27  The commutative ring of gaussian integers   gzcrng 33115
                  21.3.9.28  The archimedean ordered field of real numbers   reofld 33116
                  21.3.9.29  The quotient map and quotient modules   qusker 33121
                  21.3.9.30  The ring of integers modulo ` N `   znfermltl 33138
                  21.3.9.31  Independent sets and families   islinds5 33139
                  ​*21.3.9.32  Subgroup sum / Sumset / Minkowski sum   elgrplsmsn 33161
                  21.3.9.33  The quotient map   qusmul 33176
                  21.3.9.34  Ideals   intlidl 33194
                  21.3.9.35  Prime Ideals   cprmidl 33213
                  21.3.9.36  Maximal Ideals   cmxidl 33234
                  21.3.9.37  The semiring of ideals of a ring   cidlsrg 33273
                  21.3.9.38  Prime Elements   rprmval 33289
                  21.3.9.39  Unique factorization domains   cufd 33305
                  21.3.9.40  The ring of integers   zringidom 33311
                  21.3.9.41  Univariate Polynomials   0ringmon1p 33315
                  21.3.9.42  Polynomial quotient and polynomial remainder   q1pdir 33343
                  21.3.9.43  The subring algebra   sra1r 33352
                  21.3.9.44  Division Ring Extensions   drgext0g 33359
                  21.3.9.45  Vector Spaces   lvecdimfi 33365
                  21.3.9.46  Vector Space Dimension   cldim 33366
            21.3.10  Field Extensions   cfldext 33400
                  21.3.10.1  Algebraic numbers   cirng 33431
                  21.3.10.2  Minimal polynomials   cminply 33440
            21.3.11  Matrices   csmat 33464
                  21.3.11.1  Submatrices   csmat 33464
                  21.3.11.2  Matrix literals   clmat 33482
                  21.3.11.3  Laplace expansion of determinants   mdetpmtr1 33494
            21.3.12  Topology   ist0cld 33504
                  21.3.12.1  Open maps   txomap 33505
                  21.3.12.2  Topology of the unit circle   qtopt1 33506
                  21.3.12.3  Refinements   reff 33510
                  21.3.12.4  Open cover refinement property   ccref 33513
                  21.3.12.5  Lindelöf spaces   cldlf 33523
                  21.3.12.6  Paracompact spaces   cpcmp 33526
                  ​*21.3.12.7  Spectrum of a ring   crspec 33533
                  21.3.12.8  Pseudometrics   cmetid 33557
                  21.3.12.9  Continuity - misc additions   hauseqcn 33569
                  21.3.12.10  Topology of the closed unit interval   elunitge0 33570
                  21.3.12.11  Topology of ` ( RR X. RR ) `   unicls 33574
                  21.3.12.12  Order topology - misc. additions   cnvordtrestixx 33584
                  21.3.12.13  Continuity in topological spaces - misc. additions   mndpluscn 33597
                  21.3.12.14  Topology of the extended nonnegative real numbers ordered monoid   xrge0hmph 33603
                  21.3.12.15  Limits - misc additions   lmlim 33618
                  21.3.12.16  Univariate polynomials   pl1cn 33626
            21.3.13  Uniform Stuctures and Spaces   chcmp 33627
                  21.3.13.1  Hausdorff uniform completion   chcmp 33627
            21.3.14  Topology and algebraic structures   zringnm 33629
                  21.3.14.1  The norm on the ring of the integer numbers   zringnm 33629
                  21.3.14.2  Topological ` ZZ ` -modules   zlm0 33631
                  21.3.14.3  Canonical embedding of the field of the rational numbers into a division ring   cqqh 33643
                  21.3.14.4  Canonical embedding of the real numbers into a complete ordered field   crrh 33664
                  21.3.14.5  Embedding from the extended real numbers into a complete lattice   cxrh 33687
                  21.3.14.6  Canonical embeddings into the ordered field of the real numbers   zrhre 33690
                  ​*21.3.14.7  Topological Manifolds   cmntop 33693
            21.3.15  Real and complex functions   nexple 33698
                  21.3.15.1  Integer powers - misc. additions   nexple 33698
                  21.3.15.2  Indicator Functions   cind 33699
                  21.3.15.3  Extended sum   cesum 33716
            21.3.16  Mixed Function/Constant operation   cofc 33784
            21.3.17  Abstract measure   csiga 33797
                  21.3.17.1  Sigma-Algebra   csiga 33797
                  21.3.17.2  Generated sigma-Algebra   csigagen 33827
                  ​*21.3.17.3  lambda and pi-Systems, Rings of Sets   ispisys 33841
                  21.3.17.4  The Borel algebra on the real numbers   cbrsiga 33870
                  21.3.17.5  Product Sigma-Algebra   csx 33877
                  21.3.17.6  Measures   cmeas 33884
                  21.3.17.7  The counting measure   cntmeas 33915
                  21.3.17.8  The Lebesgue measure - misc additions   voliune 33918
                  21.3.17.9  The Dirac delta measure   cdde 33921
                  21.3.17.10  The 'almost everywhere' relation   cae 33926
                  21.3.17.11  Measurable functions   cmbfm 33938
                  21.3.17.12  Borel Algebra on ` ( RR X. RR ) `   br2base 33959
                  ​*21.3.17.13  Caratheodory's extension theorem   coms 33981
            21.3.18  Integration   itgeq12dv 34016
                  21.3.18.1  Lebesgue integral - misc additions   itgeq12dv 34016
                  21.3.18.2  Bochner integral   citgm 34017
            21.3.19  Euler's partition theorem   oddpwdc 34044
            21.3.20  Sequences defined by strong recursion   csseq 34073
            21.3.21  Fibonacci Numbers   cfib 34086
            21.3.22  Probability   cprb 34097
                  21.3.22.1  Probability Theory   cprb 34097
                  21.3.22.2  Conditional Probabilities   ccprob 34121
                  21.3.22.3  Real-valued Random Variables   crrv 34130
                  21.3.22.4  Preimage set mapping operator   corvc 34145
                  21.3.22.5  Distribution Functions   orvcelval 34158
                  21.3.22.6  Cumulative Distribution Functions   orvclteel 34162
                  21.3.22.7  Probabilities - example   coinfliplem 34168
                  21.3.22.8  Bertrand's Ballot Problem   ballotlemoex 34175
            21.3.23  Signum (sgn or sign) function - misc. additions   sgncl 34228
                  21.3.23.1  Operations on words   ccatmulgnn0dir 34244
            21.3.24  Polynomials with real coefficients - misc additions   plymul02 34248
            21.3.25  Descartes's rule of signs   signspval 34254
                  21.3.25.1  Sign changes in a word over real numbers   signspval 34254
                  21.3.25.2  Counting sign changes in a word over real numbers   signslema 34264
            21.3.26  Number Theory   iblidicc 34294
                  21.3.26.1  Representations of a number as sums of integers   crepr 34310
                  21.3.26.2  Vinogradov Trigonometric Sums and the Circle Method   cvts 34337
                  21.3.26.3  The Ternary Goldbach Conjecture: Final Statement   ax-hgt749 34346
            21.3.27  Elementary Geometry   cstrkg2d 34366
                  ​*21.3.27.1  Two-dimensional geometry   cstrkg2d 34366
                  21.3.27.2  Outer Five Segment (not used, no need to move to main)   cafs 34371
            ​*21.3.28  LeftPad Project   clpad 34376
      ​​*21.4  Mathbox for Jonathan Ben-Naim
            21.4.1  First-order logic and set theory   bnj170 34399
            21.4.2  Well founded induction and recursion   bnj110 34559
            21.4.3  The existence of a minimal element in certain classes   bnj69 34711
            21.4.4  Well-founded induction   bnj1204 34713
            21.4.5  Well-founded recursion, part 1 of 3   bnj60 34763
            21.4.6  Well-founded recursion, part 2 of 3   bnj1500 34769
            21.4.7  Well-founded recursion, part 3 of 3   bnj1522 34773
      ​21.5  Mathbox for BTernaryTau
            21.5.1  ZF set theory   exdifsn 34774
                  21.5.1.1  Finitism   fineqvrep 34785
            21.5.2  Real and complex numbers   zltp1ne 34789
            21.5.3  Graph theory   lfuhgr 34797
                  21.5.3.1  Acyclic graphs   cacycgr 34822
      ​21.6  Mathbox for Mario Carneiro
            21.6.1  Predicate calculus with all distinct variables   ax-7d 34839
            21.6.2  Miscellaneous stuff   quartfull 34845
            21.6.3  Derangements and the Subfactorial   deranglem 34846
            21.6.4  The Erdős-Szekeres theorem   erdszelem1 34871
            21.6.5  The Kuratowski closure-complement theorem   kur14lem1 34886
            21.6.6  Retracts and sections   cretr 34897
            21.6.7  Path-connected and simply connected spaces   cpconn 34899
            21.6.8  Covering maps   ccvm 34935
            21.6.9  Normal numbers   snmlff 35009
            21.6.10  Godel-sets of formulas - part 1   cgoe 35013
            21.6.11  Godel-sets of formulas - part 2   cgon 35112
            21.6.12  Models of ZF   cgze 35126
            ​*21.6.13  Metamath formal systems   cmcn 35140
            21.6.14  Grammatical formal systems   cm0s 35265
            21.6.15  Models of formal systems   cmuv 35285
            21.6.16  Splitting fields   ccpms 35307
            21.6.17  p-adic number fields   czr 35321
      ​​*21.7  Mathbox for Filip Cernatescu
      ​21.8  Mathbox for Paul Chapman
            21.8.1  Real and complex numbers (cont.)   climuzcnv 35345
            21.8.2  Miscellaneous theorems   elfzm12 35349
      ​21.9  Mathbox for Adrian Ducourtial
            21.9.1  Propositional calculus   currybi 35358
            21.9.2  Clone theory   ccloneop 35359
      ​21.10  Mathbox for Scott Fenton
            21.10.1  ZFC Axioms in primitive form   axextprim 35365
            21.10.2  Untangled classes   untelirr 35372
            21.10.3  Extra propositional calculus theorems   3jaodd 35379
            21.10.4  Misc. Useful Theorems   nepss 35382
            21.10.5  Properties of real and complex numbers   sqdivzi 35392
            21.10.6  Infinite products   iprodefisumlem 35404
            21.10.7  Factorial limits   faclimlem1 35407
            21.10.8  Greatest common divisor and divisibility   gcd32 35413
            21.10.9  Properties of relationships   dftr6 35415
            21.10.10  Properties of functions and mappings   funpsstri 35431
            21.10.11  Set induction (or epsilon induction)   setinds 35444
            21.10.12  Ordinal numbers   elpotr 35447
            21.10.13  Defined equality axioms   axextdfeq 35463
            21.10.14  Hypothesis builders   hbntg 35471
            21.10.15  Well-founded zero, successor, and limits   cwsuc 35476
            21.10.16  Quantifier-free definitions   ctxp 35496
            21.10.17  Alternate ordered pairs   caltop 35622
            21.10.18  Geometry in the Euclidean space   cofs 35648
                  21.10.18.1  Congruence properties   cofs 35648
                  21.10.18.2  Betweenness properties   btwntriv2 35678
                  21.10.18.3  Segment Transportation   ctransport 35695
                  21.10.18.4  Properties relating betweenness and congruence   cifs 35701
                  21.10.18.5  Connectivity of betweenness   btwnconn1lem1 35753
                  21.10.18.6  Segment less than or equal to   csegle 35772
                  21.10.18.7  Outside-of relationship   coutsideof 35785
                  21.10.18.8  Lines and Rays   cline2 35800
            21.10.19  Forward difference   cfwddif 35824
            21.10.20  Rank theorems   rankung 35832
            21.10.21  Hereditarily Finite Sets   chf 35838
      ​21.11  Mathbox for Gino Giotto
            21.11.1  Study of ax-mulf usage.   mpomulnzcnf 35853
      ​21.12  Mathbox for Jeff Hankins
            21.12.1  Miscellany   a1i14 35854
            21.12.2  Basic topological facts   topbnd 35878
            21.12.3  Topology of the real numbers   ivthALT 35889
            21.12.4  Refinements   cfne 35890
            21.12.5  Neighborhood bases determine topologies   neibastop1 35913
            21.12.6  Lattice structure of topologies   topmtcl 35917
            21.12.7  Filter bases   fgmin 35924
            21.12.8  Directed sets, nets   tailfval 35926
      ​21.13  Mathbox for Anthony Hart
            21.13.1  Propositional Calculus   tb-ax1 35937
            21.13.2  Predicate Calculus   nalfal 35957
            21.13.3  Miscellaneous single axioms   meran1 35965
            21.13.4  Connective Symmetry   negsym1 35971
      ​21.14  Mathbox for Chen-Pang He
            21.14.1  Ordinal topology   ontopbas 35982
      ​21.15  Mathbox for Jeff Hoffman
            21.15.1  Inferences for finite induction on generic function values   fveleq 36005
            21.15.2  gdc.mm   nnssi2 36009
      ​21.16  Mathbox for Asger C. Ipsen
            21.16.1  Continuous nowhere differentiable functions   dnival 36016
      ​​*21.17  Mathbox for BJ
            *21.17.1  Propositional calculus   bj-mp2c 36085
                  *21.17.1.1  Derived rules of inference   bj-mp2c 36085
                  ​*21.17.1.2  A syntactic theorem   bj-0 36087
                  21.17.1.3  Minimal implicational calculus   bj-a1k 36089
                  ​*21.17.1.4  Positive calculus   bj-syl66ib 36100
                  21.17.1.5  Implication and negation   bj-con2com 36106
                  ​*21.17.1.6  Disjunction   bj-jaoi1 36117
                  ​*21.17.1.7  Logical equivalence   bj-dfbi4 36119
                  21.17.1.8  The conditional operator for propositions   bj-consensus 36124
                  ​*21.17.1.9  Propositional calculus: miscellaneous   bj-imbi12 36129
            ​*21.17.2  Modal logic   bj-axdd2 36139
            ​*21.17.3  Provability logic   cprvb 36144
            ​*21.17.4  First-order logic   bj-genr 36153
                  21.17.4.1  Adding ax-gen   bj-genr 36153
                  21.17.4.2  Adding ax-4   bj-2alim 36157
                  21.17.4.3  Adding ax-5   bj-ax12wlem 36190
                  21.17.4.4  Equality and substitution   bj-ssbeq 36199
                  21.17.4.5  Adding ax-6   bj-spimvwt 36215
                  21.17.4.6  Adding ax-7   bj-cbvexw 36222
                  21.17.4.7  Membership predicate, ax-8 and ax-9   bj-ax89 36224
                  21.17.4.8  Adding ax-11   bj-alcomexcom 36227
                  21.17.4.9  Adding ax-12   axc11n11 36229
                  21.17.4.10  Nonfreeness   wnnf 36270
                  21.17.4.11  Adding ax-13   bj-axc10 36330
                  ​*21.17.4.12  Removing dependencies on ax-13 (and ax-11)   bj-axc10v 36340
                  ​*21.17.4.13  Distinct var metavariables   bj-hbaeb2 36365
                  ​*21.17.4.14  Around ~ equsal   bj-equsal1t 36369
                  ​*21.17.4.15  Some Principia Mathematica proofs   stdpc5t 36374
                  21.17.4.16  Alternate definition of substitution   bj-sbsb 36384
                  21.17.4.17  Lemmas for substitution   bj-sbf3 36386
                  21.17.4.18  Existential uniqueness   bj-eu3f 36388
                  ​*21.17.4.19  First-order logic: miscellaneous   bj-sblem1 36389
            21.17.5  Set theory   eliminable1 36406
                  ​*21.17.5.1  Eliminability of class terms   eliminable1 36406
                  ​*21.17.5.2  Classes without the axiom of extensionality   bj-denoteslem 36418
                  21.17.5.3  Characterization among sets versus among classes   elelb 36445
                  ​*21.17.5.4  The nonfreeness quantifier for classes   bj-nfcsym 36447
                  ​*21.17.5.5  Lemmas for class substitution   bj-sbeqALT 36448
                  21.17.5.6  Removing some axiom requirements and disjoint variable conditions   bj-exlimvmpi 36459
                  ​*21.17.5.7  Class abstractions   bj-elabd2ALT 36473
                  21.17.5.8  Generalized class abstractions   bj-cgab 36481
                  ​*21.17.5.9  Restricted nonfreeness   wrnf 36489
                  ​*21.17.5.10  Russell's paradox   bj-ru0 36491
                  21.17.5.11  Curry's paradox in set theory   currysetlem 36494
                  ​*21.17.5.12  Some disjointness results   bj-n0i 36500
                  ​*21.17.5.13  Complements on direct products   bj-xpimasn 36504
                  ​*21.17.5.14  "Singletonization" and tagging   bj-snsetex 36512
                  ​*21.17.5.15  Tuples of classes   bj-cproj 36539
                  ​*21.17.5.16  Set theory: elementary operations relative to a universe   bj-rcleqf 36574
                  ​*21.17.5.17  Axioms for finite unions   bj-abex 36579
                  ​*21.17.5.18  Set theory: miscellaneous   eleq2w2ALT 36596
                  ​*21.17.5.19  Evaluation at a class   bj-evaleq 36621
                  21.17.5.20  Elementwise operations   celwise 36628
                  ​*21.17.5.21  Elementwise intersection (families of sets induced on a subset)   bj-rest00 36630
                  21.17.5.22  Moore collections (complements)   bj-raldifsn 36649
                  21.17.5.23  Maps-to notation for functions with three arguments   bj-0nelmpt 36665
                  ​*21.17.5.24  Currying   csethom 36671
                  ​*21.17.5.25  Setting components of extensible structures   cstrset 36683
            ​*21.17.6  Extended real and complex numbers, real and complex projective lines   bj-nfald 36686
                  21.17.6.1  Complements on class abstractions of ordered pairs and binary relations   bj-nfald 36686
                  ​*21.17.6.2  Identity relation (complements)   bj-opabssvv 36699
                  ​*21.17.6.3  Functionalized identity (diagonal in a Cartesian square)   cdiag2 36721
                  ​*21.17.6.4  Direct image and inverse image   cimdir 36727
                  ​*21.17.6.5  Extended numbers and projective lines as sets   cfractemp 36745
                  ​*21.17.6.6  Addition and opposite   caddcc 36786
                  ​*21.17.6.7  Order relation on the extended reals   cltxr 36790
                  ​*21.17.6.8  Argument, multiplication and inverse   carg 36792
                  21.17.6.9  The canonical bijection from the finite ordinals   ciomnn 36798
                  21.17.6.10  Divisibility   cnnbar 36809
            ​*21.17.7  Monoids   bj-smgrpssmgm 36817
                  ​*21.17.7.1  Finite sums in monoids   cfinsum 36832
            ​*21.17.8  Affine, Euclidean, and Cartesian geometry   bj-fvimacnv0 36835
                  *21.17.8.1  Real vector spaces   bj-fvimacnv0 36835
                  ​*21.17.8.2  Complex numbers (supplements)   bj-subcom 36857
                  ​*21.17.8.3  Barycentric coordinates   bj-bary1lem 36859
            21.17.9  Monoid of endomorphisms   cend 36862
      ​21.18  Mathbox for Jim Kingdon
                  21.18.0.1  Circle constant   taupilem3 36868
                  21.18.0.2  Number theory   dfgcd3 36873
                  21.18.0.3  Real numbers   irrdifflemf 36874
      ​21.19  Mathbox for ML
            21.19.1  Miscellaneous   csbrecsg 36877
            21.19.2  Cartesian exponentiation   cfinxp 36932
            21.19.3  Topology   iunctb2 36952
                  ​*21.19.3.1  Pi-base theorems   pibp16 36962
      ​21.20  Mathbox for Wolf Lammen
            21.20.1  1. Bootstrapping   wl-section-boot 36971
            21.20.2  Implication chains   wl-section-impchain 36995
            21.20.3  Theorems around the conditional operator   wl-ifp-ncond1 37013
            21.20.4  Alternative development of hadd, cadd   wl-df-3xor 37017
            21.20.5  An alternative axiom ~ ax-13   ax-wl-13v 37042
            21.20.6  Other stuff   wl-mps 37044
      ​21.21  Mathbox for Brendan Leahy
      ​21.22  Mathbox for Jeff Madsen
            21.22.1  Logic and set theory   unirep 37257
            21.22.2  Real and complex numbers; integers   filbcmb 37283
            21.22.3  Sequences and sums   sdclem2 37285
            21.22.4  Topology   subspopn 37295
            21.22.5  Metric spaces   metf1o 37298
            21.22.6  Continuous maps and homeomorphisms   constcncf 37305
            21.22.7  Boundedness   ctotbnd 37309
            21.22.8  Isometries   cismty 37341
            21.22.9  Heine-Borel Theorem   heibor1lem 37352
            21.22.10  Banach Fixed Point Theorem   bfplem1 37365
            21.22.11  Euclidean space   crrn 37368
            21.22.12  Intervals (continued)   ismrer1 37381
            21.22.13  Operation properties   cass 37385
            21.22.14  Groups and related structures   cmagm 37391
            21.22.15  Group homomorphism and isomorphism   cghomOLD 37426
            21.22.16  Rings   crngo 37437
            21.22.17  Division Rings   cdrng 37491
            21.22.18  Ring homomorphisms   crngohom 37503
            21.22.19  Commutative rings   ccm2 37532
            21.22.20  Ideals   cidl 37550
            21.22.21  Prime rings and integral domains   cprrng 37589
            21.22.22  Ideal generators   cigen 37602
      ​21.23  Mathbox for Giovanni Mascellani
            ​*21.23.1  Tools for automatic proof building   efald2 37621
            ​*21.23.2  Tseitin axioms   fald 37672
            ​*21.23.3  Equality deductions   iuneq2f 37699
            ​*21.23.4  Miscellanea   orcomdd 37710
      ​21.24  Mathbox for Peter Mazsa
            21.24.1  Notations   cxrn 37717
            21.24.2  Preparatory theorems   el2v1 37760
            21.24.3  Range Cartesian product   df-xrn 37912
            21.24.4  Cosets by ` R `   df-coss 37952
            21.24.5  Relations   df-rels 38026
            21.24.6  Subset relations   df-ssr 38039
            21.24.7  Reflexivity   df-refs 38051
            21.24.8  Converse reflexivity   df-cnvrefs 38066
            21.24.9  Symmetry   df-syms 38083
            21.24.10  Reflexivity and symmetry   symrefref2 38104
            21.24.11  Transitivity   df-trs 38113
            21.24.12  Equivalence relations   df-eqvrels 38125
            21.24.13  Redundancy   df-redunds 38164
            21.24.14  Domain quotients   df-dmqss 38179
            21.24.15  Equivalence relations on domain quotients   df-ers 38204
            21.24.16  Functions   df-funss 38221
            21.24.17  Disjoints vs. converse functions   df-disjss 38244
            21.24.18  Antisymmetry   df-antisymrel 38301
            21.24.19  Partitions: disjoints on domain quotients   df-parts 38306
            21.24.20  Partition-Equivalence Theorems   disjim 38322
      ​21.25  Mathbox for Rodolfo Medina
            21.25.1  Partitions   prtlem60 38394
      ​​*21.26  Mathbox for Norm Megill
            *21.26.1  Obsolete schemes ax-c4,c5,c7,c10,c11,c11n,c15,c9,c14,c16   ax-c5 38424
            ​*21.26.2  Rederive new axioms ax-4, ax-10, ax-6, ax-12, ax-13 from old   axc5 38434
            ​*21.26.3  Legacy theorems using obsolete axioms   ax5ALT 38448
            21.26.4  Experiments with weak deduction theorem   elimhyps 38502
            21.26.5  Miscellanea   cnaddcom 38513
            21.26.6  Atoms, hyperplanes, and covering in a left vector space (or module)   clsa 38515
            21.26.7  Functionals and kernels of a left vector space (or module)   clfn 38598
            21.26.8  Opposite rings and dual vector spaces   cld 38664
            21.26.9  Ortholattices and orthomodular lattices   cops 38713
            21.26.10  Atomic lattices with covering property   ccvr 38803
            21.26.11  Hilbert lattices   chlt 38891
            21.26.12  Projective geometries based on Hilbert lattices   clln 39033
            21.26.13  Construction of a vector space from a Hilbert lattice   cdlema1N 39333
            21.26.14  Construction of involution and inner product from a Hilbert lattice   clpoN 41022
      ​21.27  Mathbox for metakunt
            21.27.1  Commutative Semiring   ccsrg 41508
            21.27.2  General helpful statements   leexp1ad 41511
            21.27.3  Some gcd and lcm results   12gcd5e1 41543
            21.27.4  Least common multiple inequality theorem   3factsumint1 41561
            21.27.5  Logarithm inequalities   3exp7 41593
            21.27.6  Miscellaneous results for AKS formalisation   intlewftc 41601
            21.27.7  Sticks and stones   sticksstones1 41687
            21.27.8  Continuation AKS   aks6d1c6lem1 41711
            21.27.9  Permutation results   metakunt1 41726
            21.27.10  Unused lemmas scheduled for deletion   fac2xp3 41760
      ​21.28  Mathbox for Steven Nguyen
            21.28.1  Utility theorems   intnanrt 41764
            21.28.2  Structures   nelsubginvcld 41804
            ​*21.28.3  Arithmetic theorems   c0exALT 41899
            21.28.4  Exponents and divisibility   oexpreposd 41946
            21.28.5  Real subtraction   cresub 41985
            ​*21.28.6  Projective spaces   cprjsp 42090
            21.28.7  Basic reductions for Fermat's Last Theorem   dffltz 42123
            ​*21.28.8  Exemplar theorems   iddii 42153
      ​21.29  Mathbox for Igor Ieskov
      ​21.30  Mathbox for OpenAI
      ​21.31  Mathbox for Stefan O'Rear
            21.31.1  Additional elementary logic and set theory   moxfr 42177
            21.31.2  Additional theory of functions   imaiinfv 42178
            21.31.3  Additional topology   elrfi 42179
            21.31.4  Characterization of closure operators. Kuratowski closure axioms   ismrcd1 42183
            21.31.5  Algebraic closure systems   cnacs 42187
            21.31.6  Miscellanea 1. Map utilities   constmap 42198
            21.31.7  Miscellanea for polynomials   mptfcl 42205
            21.31.8  Multivariate polynomials over the integers   cmzpcl 42206
            21.31.9  Miscellanea for Diophantine sets 1   coeq0i 42238
            21.31.10  Diophantine sets 1: definitions   cdioph 42240
            21.31.11  Diophantine sets 2 miscellanea   ellz1 42252
            21.31.12  Diophantine sets 2: union and intersection. Monotone Boolean algebra   diophin 42257
            21.31.13  Diophantine sets 3: construction   diophrex 42260
            21.31.14  Diophantine sets 4 miscellanea   2sbcrex 42269
            21.31.15  Diophantine sets 4: Quantification   rexrabdioph 42279
            21.31.16  Diophantine sets 5: Arithmetic sets   rabdiophlem1 42286
            21.31.17  Diophantine sets 6: reusability. renumbering of variables   eldioph4b 42296
            21.31.18  Pigeonhole Principle and cardinality helpers   fphpd 42301
            21.31.19  A non-closed set of reals is infinite   rencldnfilem 42305
            21.31.20  Lagrange's rational approximation theorem   irrapxlem1 42307
            21.31.21  Pell equations 1: A nontrivial solution always exists   pellexlem1 42314
            21.31.22  Pell equations 2: Algebraic number theory of the solution set   csquarenn 42321
            21.31.23  Pell equations 3: characterizing fundamental solution   infmrgelbi 42363
            ​*21.31.24  Logarithm laws generalized to an arbitrary base   reglogcl 42375
            21.31.25  Pell equations 4: the positive solution group is infinite cyclic   pellfund14 42383
            21.31.26  X and Y sequences 1: Definition and recurrence laws   crmx 42385
            21.31.27  Ordering and induction lemmas for the integers   monotuz 42427
            21.31.28  X and Y sequences 2: Order properties   rmxypos 42433
            21.31.29  Congruential equations   congtr 42451
            21.31.30  Alternating congruential equations   acongid 42461
            21.31.31  Additional theorems on integer divisibility   coprmdvdsb 42471
            21.31.32  X and Y sequences 3: Divisibility properties   jm2.18 42474
            21.31.33  X and Y sequences 4: Diophantine representability of Y   jm2.27a 42491
            21.31.34  X and Y sequences 5: Diophantine representability of X, ^, _C   rmxdiophlem 42501
            21.31.35  Uncategorized stuff not associated with a major project   setindtr 42510
            21.31.36  More equivalents of the Axiom of Choice   axac10 42519
            21.31.37  Finitely generated left modules   clfig 42556
            21.31.38  Noetherian left modules I   clnm 42564
            21.31.39  Addenda for structure powers   pwssplit4 42578
            21.31.40  Every set admits a group structure iff choice   unxpwdom3 42584
            21.31.41  Noetherian rings and left modules II   clnr 42598
            21.31.42  Hilbert's Basis Theorem   cldgis 42610
            21.31.43  Additional material on polynomials [DEPRECATED]   cmnc 42620
            21.31.44  Degree and minimal polynomial of algebraic numbers   cdgraa 42629
            21.31.45  Algebraic integers I   citgo 42646
            21.31.46  Endomorphism algebra   cmend 42664
            21.31.47  Cyclic groups and order   idomodle 42684
            21.31.48  Cyclotomic polynomials   ccytp 42690
            21.31.49  Miscellaneous topology   fgraphopab 42696
      ​21.32  Mathbox for Noam Pasman
      ​21.33  Mathbox for Jon Pennant
      ​21.34  Mathbox for Richard Penner
            21.34.1  Set Theory and Ordinal Numbers   uniel 42710
            21.34.2  Natural addition of Cantor normal forms   oawordex2 42820
            21.34.3  Surreal Contributions   abeqabi 42903
            21.34.4  Short Studies   nlimsuc 42936
                  21.34.4.1  Additional work on conditional logical operator   ifpan123g 42954
                  21.34.4.2  Sophisms   rp-fakeimass 43007
                  ​*21.34.4.3  Finite Sets   rp-isfinite5 43012
                  21.34.4.4  General Observations   intabssd 43014
                  21.34.4.5  Infinite Sets   pwelg 43055
                  ​*21.34.4.6  Finite intersection property   fipjust 43060
                  21.34.4.7  RP ADDTO: Subclasses and subsets   rababg 43069
                  21.34.4.8  RP ADDTO: The intersection of a class   elinintab 43070
                  21.34.4.9  RP ADDTO: Theorems requiring subset and intersection existence   elinintrab 43072
                  21.34.4.10  RP ADDTO: Relations   xpinintabd 43075
                  ​*21.34.4.11  RP ADDTO: Functions   elmapintab 43091
                  ​*21.34.4.12  RP ADDTO: Finite induction (for finite ordinals)   cnvcnvintabd 43095
                  21.34.4.13  RP ADDTO: First and second members of an ordered pair   elcnvlem 43096
                  21.34.4.14  RP ADDTO: The reflexive and transitive properties of relations   undmrnresiss 43099
                  21.34.4.15  RP ADDTO: Basic properties of closures   cleq2lem 43103
                  21.34.4.16  RP REPLACE: Definitions and basic properties of transitive closures   trcleq2lemRP 43125
                  ​*21.34.4.17  Additions for square root; absolute value   sqrtcvallem1 43126
            21.34.5  Additional statements on relations and subclasses   al3im 43142
                  21.34.5.1  Transitive relations (not to be confused with transitive classes).   trrelind 43160
                  21.34.5.2  Reflexive closures   crcl 43167
                  ​*21.34.5.3  Finite relationship composition.   relexp2 43172
                  21.34.5.4  Transitive closure of a relation   dftrcl3 43215
                  ​*21.34.5.5  Adapted from Frege   frege77d 43241
            ​*21.34.6  Propositions from _Begriffsschrift_   dfxor4 43261
                  *21.34.6.1  _Begriffsschrift_ Chapter I   dfxor4 43261
                  ​*21.34.6.2  _Begriffsschrift_ Notation hints   whe 43267
                  21.34.6.3  _Begriffsschrift_ Chapter II Implication   ax-frege1 43285
                  21.34.6.4  _Begriffsschrift_ Chapter II Implication and Negation   axfrege28 43324
                  ​*21.34.6.5  _Begriffsschrift_ Chapter II with logical equivalence   axfrege52a 43351
                  21.34.6.6  _Begriffsschrift_ Chapter II with equivalence of sets   axfrege52c 43382
                  ​*21.34.6.7  _Begriffsschrift_ Chapter II with equivalence of classes   frege53c 43409
                  ​*21.34.6.8  _Begriffsschrift_ Chapter III Properties hereditary in a sequence   dffrege69 43427
                  ​*21.34.6.9  _Begriffsschrift_ Chapter III Following in a sequence   dffrege76 43434
                  ​*21.34.6.10  _Begriffsschrift_ Chapter III Member of sequence   dffrege99 43457
                  ​*21.34.6.11  _Begriffsschrift_ Chapter III Single-valued procedures   dffrege115 43473
            ​*21.34.7  Exploring Topology via Seifert and Threlfall   enrelmap 43492
                  *21.34.7.1  Equinumerosity of sets of relations and maps   enrelmap 43492
                  ​*21.34.7.2  Generic Pseudoclosure Spaces, Pseudointerior Spaces, and Pseudoneighborhoods   or3or 43518
                  ​*21.34.7.3  Generic Neighborhood Spaces   gneispa 43625
            ​*21.34.8  Exploring Higher Homotopy via Kerodon   k0004lem1 43642
                  *21.34.8.1  Simplicial Sets   k0004lem1 43642
      ​21.35  Mathbox for Stanislas Polu
            21.35.1  IMO Problems   wwlemuld 43651
                  21.35.1.1  IMO 1972 B2   wwlemuld 43651
            ​*21.35.2  INT Inequalities Proof Generator   int-addcomd 43668
            ​*21.35.3  N-Digit Addition Proof Generator   unitadd 43690
            21.35.4  AM-GM (for k = 2,3,4)   gsumws3 43691
      ​21.36  Mathbox for Rohan Ridenour
            21.36.1  Misc   spALT 43696
            21.36.2  Monoid rings   cmnring 43708
            21.36.3  Shorter primitive equivalent of ax-groth   gru0eld 43731
                  21.36.3.1  Grothendieck universes are closed under collection   gru0eld 43731
                  21.36.3.2  Minimal universes   ismnu 43763
                  21.36.3.3  Primitive equivalent of ax-groth   expandan 43790
      ​21.37  Mathbox for Steve Rodriguez
            21.37.1  Miscellanea   nanorxor 43807
            21.37.2  Ratio test for infinite series convergence and divergence   dvgrat 43814
            21.37.3  Multiples   reldvds 43817
            21.37.4  Function operations   caofcan 43825
            21.37.5  Calculus   lhe4.4ex1a 43831
            21.37.6  The generalized binomial coefficient operation   cbcc 43838
            21.37.7  Binomial series   uzmptshftfval 43848
      ​21.38  Mathbox for Andrew Salmon
            21.38.1  Principia Mathematica * 10   pm10.12 43860
            21.38.2  Principia Mathematica * 11   2alanimi 43874
            21.38.3  Predicate Calculus   sbeqal1 43900
            21.38.4  Principia Mathematica * 13 and * 14   pm13.13a 43909
            21.38.5  Set Theory   elnev 43940
            21.38.6  Arithmetic   addcomgi 43958
            21.38.7  Geometry   cplusr 43959
      ​​*21.39  Mathbox for Alan Sare
            21.39.1  Auxiliary theorems for the Virtual Deduction tool   idiALT 43981
            21.39.2  Supplementary unification deductions   bi1imp 43985
            21.39.3  Conventional Metamath proofs, some derived from VD proofs   iidn3 44005
            21.39.4  What is Virtual Deduction?   wvd1 44073
            21.39.5  Virtual Deduction Theorems   df-vd1 44074
            21.39.6  Theorems proved using Virtual Deduction   trsspwALT 44322
            21.39.7  Theorems proved using Virtual Deduction with mmj2 assistance   simplbi2VD 44350
            21.39.8  Virtual Deduction transcriptions of textbook proofs   sb5ALTVD 44417
            21.39.9  Theorems proved using conjunction-form Virtual Deduction   elpwgdedVD 44421
            21.39.10  Theorems with a VD proof in conventional notation derived from a VD proof   suctrALT3 44428
            ​*21.39.11  Theorems with a proof in conventional notation derived from a VD proof   notnotrALT2 44431
      ​21.40  Mathbox for Glauco Siliprandi
            21.40.1  Miscellanea   evth2f 44442
            21.40.2  Functions   feq1dd 44604
            21.40.3  Ordering on real numbers - Real and complex numbers basic operations   sub2times 44717
            21.40.4  Real intervals   gtnelioc 44939
            21.40.5  Finite sums   fsummulc1f 45022
            21.40.6  Finite multiplication of numbers and finite multiplication of functions   fmul01 45031
            21.40.7  Limits   clim1fr1 45052
                  21.40.7.1  Inferior limit (lim inf)   clsi 45202
                  ​*21.40.7.2  Limits for sequences of extended real numbers   clsxlim 45269
            21.40.8  Trigonometry   coseq0 45315
            21.40.9  Continuous Functions   mulcncff 45321
            21.40.10  Derivatives   dvsinexp 45362
            21.40.11  Integrals   itgsin0pilem1 45401
            21.40.12  Stone Weierstrass theorem - real version   stoweidlem1 45452
            21.40.13  Wallis' product for π   wallispilem1 45516
            21.40.14  Stirling's approximation formula for ` n ` factorial   stirlinglem1 45525
            21.40.15  Dirichlet kernel   dirkerval 45542
            21.40.16  Fourier Series   fourierdlem1 45559
            21.40.17  e is transcendental   elaa2lem 45684
            21.40.18  n-dimensional Euclidean space   rrxtopn 45735
            21.40.19  Basic measure theory   csalg 45759
                  ​*21.40.19.1  σ-Algebras   csalg 45759
                  21.40.19.2  Sum of nonnegative extended reals   csumge0 45813
                  ​*21.40.19.3  Measures   cmea 45900
                  ​*21.40.19.4  Outer measures and Caratheodory's construction   come 45940
                  ​*21.40.19.5  Lebesgue measure on n-dimensional Real numbers   covoln 45987
                  ​*21.40.19.6  Measurable functions   csmblfn 46146
      ​21.41  Mathbox for Saveliy Skresanov
            21.41.1  Ceva's theorem   sigarval 46301
            21.41.2  Simple groups   simpcntrab 46321
      ​21.42  Mathbox for Ender Ting
            21.42.1  Increasing sequences and subsequences   et-ltneverrefl 46322
      ​21.43  Mathbox for Jarvin Udandy
      ​21.44  Mathbox for Adhemar
            ​*21.44.1  Minimal implicational calculus   adh-minim 46446
      ​21.45  Mathbox for Alexander van der Vekens
            21.45.1  General auxiliary theorems (1)   n0nsn2el 46470
                  21.45.1.1  Unordered and ordered pairs - extension for singletons   n0nsn2el 46470
                  21.45.1.2  Unordered and ordered pairs - extension for unordered pairs   elprneb 46474
                  21.45.1.3  Unordered and ordered pairs - extension for ordered pairs   oppr 46475
                  21.45.1.4  Relations - extension   eubrv 46480
                  21.45.1.5  Definite description binder (inverted iota) - extension   iota0def 46483
                  21.45.1.6  Functions - extension   fveqvfvv 46485
            21.45.2  Alternative for Russell's definition of a description binder   caiota 46526
            21.45.3  Double restricted existential uniqueness   r19.32 46541
                  21.45.3.1  Restricted quantification (extension)   r19.32 46541
                  21.45.3.2  Restricted uniqueness and "at most one" quantification   reuf1odnf 46550
                  21.45.3.3  Analogs to Existential uniqueness (double quantification)   2reu3 46553
                  21.45.3.4  Additional theorems for double restricted existential uniqueness   2reu8i 46556
            ​*21.45.4  Alternative definitions of function and operation values   wdfat 46559
                  21.45.4.1  Restricted quantification (extension)   ralbinrald 46565
                  21.45.4.2  The universal class (extension)   nvelim 46566
                  21.45.4.3  Introduce the Axiom of Power Sets (extension)   alneu 46567
                  21.45.4.4  Predicate "defined at"   dfateq12d 46569
                  21.45.4.5  Alternative definition of the value of a function   dfafv2 46575
                  21.45.4.6  Alternative definition of the value of an operation   aoveq123d 46621
            ​*21.45.5  Alternative definitions of function values (2)   cafv2 46651
            21.45.6  General auxiliary theorems (2)   an4com24 46711
                  21.45.6.1  Logical conjunction - extension   an4com24 46711
                  21.45.6.2  Abbreviated conjunction and disjunction of three wff's - extension   3an4ancom24 46712
                  21.45.6.3  Negated membership (alternative)   cnelbr 46714
                  21.45.6.4  The empty set - extension   ralralimp 46721
                  21.45.6.5  Indexed union and intersection - extension   otiunsndisjX 46722
                  21.45.6.6  Functions - extension   fvifeq 46723
                  21.45.6.7  Maps-to notation - extension   fvmptrab 46735
                  21.45.6.8  Subtraction - extension   cnambpcma 46737
                  21.45.6.9  Ordering on reals (cont.) - extension   leaddsuble 46740
                  21.45.6.10  Imaginary and complex number properties - extension   readdcnnred 46746
                  21.45.6.11  Nonnegative integers (as a subset of complex numbers) - extension   nn0resubcl 46751
                  21.45.6.12  Integers (as a subset of complex numbers) - extension   zgeltp1eq 46752
                  21.45.6.13  Decimal arithmetic - extension   1t10e1p1e11 46753
                  21.45.6.14  Upper sets of integers - extension   eluzge0nn0 46755
                  21.45.6.15  Infinity and the extended real number system (cont.) - extension   nltle2tri 46756
                  21.45.6.16  Finite intervals of integers - extension   ssfz12 46757
                  21.45.6.17  Half-open integer ranges - extension   fzopred 46765
                  21.45.6.18  The modulo (remainder) operation - extension   m1mod0mod1 46772
                  21.45.6.19  The infinite sequence builder "seq"   smonoord 46774
                  21.45.6.20  Finite and infinite sums - extension   fsummsndifre 46775
                  21.45.6.21  Extensible structures - extension   setsidel 46779
            ​*21.45.7  Preimages of function values   preimafvsnel 46782
            ​*21.45.8  Partitions of real intervals   ciccp 46816
            21.45.9  Shifting functions with an integer range domain   fargshiftfv 46842
            21.45.10  Words over a set (extension)   lswn0 46847
                  21.45.10.1  Last symbol of a word - extension   lswn0 46847
            21.45.11  Unordered pairs   wich 46848
                  21.45.11.1  Interchangeable setvar variables   wich 46848
                  21.45.11.2  Set of unordered pairs   sprid 46877
                  ​*21.45.11.3  Proper (unordered) pairs   prpair 46904
                  21.45.11.4  Set of proper unordered pairs   cprpr 46915
            21.45.12  Number theory (extension)   cfmtno 46930
                  ​*21.45.12.1  Fermat numbers   cfmtno 46930
                  ​*21.45.12.2  Mersenne primes   m2prm 46994
                  21.45.12.3  Proth's theorem   modexp2m1d 47015
                  21.45.12.4  Solutions of quadratic equations   quad1 47023
            ​*21.45.13  Even and odd numbers   ceven 47027
                  21.45.13.1  Definitions and basic properties   ceven 47027
                  21.45.13.2  Alternate definitions using the "divides" relation   dfeven2 47052
                  21.45.13.3  Alternate definitions using the "modulo" operation   dfeven3 47061
                  21.45.13.4  Alternate definitions using the "gcd" operation   iseven5 47067
                  21.45.13.5  Theorems of part 5 revised   zneoALTV 47072
                  21.45.13.6  Theorems of part 6 revised   odd2np1ALTV 47077
                  21.45.13.7  Theorems of AV's mathbox revised   0evenALTV 47091
                  21.45.13.8  Additional theorems   epoo 47106
                  21.45.13.9  Perfect Number Theorem (revised)   perfectALTVlem1 47124
            21.45.14  Number theory (extension 2)   cfppr 47127
                  ​*21.45.14.1  Fermat pseudoprimes   cfppr 47127
                  ​*21.45.14.2  Goldbach's conjectures   cgbe 47148
            21.45.15  Graph theory (extension)   cclnbgr 47221
                  21.45.15.1  Closed neighborhood of a vertex   cclnbgr 47221
                  ​*21.45.15.2  Semiclosed and semiopen neighborhoods (experimental)   dfsclnbgr2 47244
                  21.45.15.3  Induced subgraphs   cisubgr 47258
                  ​*21.45.15.4  Isomorphisms of graphs   cgrisom 47270
                  21.45.15.5  Loop-free graphs - extension   1hegrlfgr 47306
                  21.45.15.6  Walks - extension   cupwlks 47307
                  21.45.15.7  Edges of graphs expressed as sets of unordered pairs   upgredgssspr 47317
            21.45.16  Monoids (extension)   ovn0dmfun 47330
                  21.45.16.1  Auxiliary theorems   ovn0dmfun 47330
                  21.45.16.2  Magmas, Semigroups and Monoids (extension)   plusfreseq 47338
                  21.45.16.3  Examples and counterexamples for magmas, semigroups and monoids (extension)   opmpoismgm 47341
                  21.45.16.4  Group sum operation (extension 1)   gsumsplit2f 47354
            ​*21.45.17  Magmas and internal binary operations (alternate approach)   ccllaw 47357
                  *21.45.17.1  Laws for internal binary operations   ccllaw 47357
                  ​*21.45.17.2  Internal binary operations   cintop 47370
                  21.45.17.3  Alternative definitions for magmas and semigroups   cmgm2 47389
            21.45.18  Rings (extension)   lmod0rng 47403
                  21.45.18.1  Nonzero rings (extension)   lmod0rng 47403
                  21.45.18.2  Ideals as non-unital rings   lidldomn1 47405
                  21.45.18.3  The non-unital ring of even integers   0even 47411
                  21.45.18.4  A constructed not unital ring   cznrnglem 47433
                  ​*21.45.18.5  The category of non-unital rings (alternate definition)   crngcALTV 47437
                  ​*21.45.18.6  The category of (unital) rings (alternate definition)   cringcALTV 47461
            21.45.19  Basic algebraic structures (extension)   opeliun2xp 47508
                  21.45.19.1  Auxiliary theorems   opeliun2xp 47508
                  21.45.19.2  The binomial coefficient operation (extension)   bcpascm1 47527
                  21.45.19.3  The ` ZZ `-module ` ZZ X. ZZ `   zlmodzxzlmod 47530
                  21.45.19.4  Group sum operation (extension 2)   mgpsumunsn 47537
                  21.45.19.5  Symmetric groups (extension)   exple2lt6 47540
                  21.45.19.6  Divisibility (extension)   invginvrid 47543
                  21.45.19.7  The support of functions (extension)   rmsupp0 47544
                  21.45.19.8  Finitely supported functions (extension)   rmsuppfi 47549
                  21.45.19.9  Left modules (extension)   lmodvsmdi 47558
                  21.45.19.10  Associative algebras (extension)   assaascl0 47560
                  21.45.19.11  Univariate polynomials (extension)   ply1vr1smo 47562
                  21.45.19.12  Univariate polynomials (examples)   linply1 47573
            21.45.20  Linear algebra (extension)   cdmatalt 47576
                  ​*21.45.20.1  The subalgebras of diagonal and scalar matrices (extension)   cdmatalt 47576
                  ​*21.45.20.2  Linear combinations   clinc 47584
                  ​*21.45.20.3  Linear independence   clininds 47620
                  21.45.20.4  Simple left modules and the ` ZZ `-module   lmod1lem1 47667
                  21.45.20.5  Differences between (left) modules and (left) vector spaces   lvecpsslmod 47687
            21.45.21  Complexity theory   suppdm 47690
                  21.45.21.1  Auxiliary theorems   suppdm 47690
                  21.45.21.2  The modulo (remainder) operation (extension)   fldivmod 47703
                  21.45.21.3  Even and odd integers   nn0onn0ex 47708
                  21.45.21.4  The natural logarithm on complex numbers (extension)   logcxp0 47720
                  21.45.21.5  Division of functions   cfdiv 47722
                  21.45.21.6  Upper bounds   cbigo 47732
                  21.45.21.7  Logarithm to an arbitrary base (extension)   rege1logbrege0 47743
                  ​*21.45.21.8  The binary logarithm   fldivexpfllog2 47750
                  21.45.21.9  Binary length   cblen 47754
                  ​*21.45.21.10  Digits   cdig 47780
                  21.45.21.11  Nonnegative integer as sum of its shifted digits   dignn0flhalflem1 47800
                  21.45.21.12  Algorithms for the multiplication of nonnegative integers   nn0mulfsum 47809
                  ​*21.45.21.13  N-ary functions   cnaryf 47811
                  ​*21.45.21.14  The Ackermann function   citco 47842
            21.45.22  Elementary geometry (extension)   fv1prop 47884
                  21.45.22.1  Auxiliary theorems   fv1prop 47884
                  21.45.22.2  Real euclidean space of dimension 2   rrx2pxel 47896
                  21.45.22.3  Spheres and lines in real Euclidean spaces   cline 47912
      ​21.46  Mathbox for Zhi Wang
            21.46.1  Propositional calculus   pm4.71da 47974
            21.46.2  Predicate calculus with equality   dtrucor3 47983
                  21.46.2.1  Axiom scheme ax-5 (Distinctness)   dtrucor3 47983
            21.46.3  ZF Set Theory - start with the Axiom of Extensionality   ralbidb 47984
                  21.46.3.1  Restricted quantification   ralbidb 47984
                  21.46.3.2  The empty set   ssdisjd 47990
                  21.46.3.3  Unordered and ordered pairs   vsn 47994
                  21.46.3.4  The union of a class   unilbss 48000
            21.46.4  ZF Set Theory - add the Axiom of Replacement   inpw 48001
                  21.46.4.1  Theorems requiring subset and intersection existence   inpw 48001
            21.46.5  ZF Set Theory - add the Axiom of Power Sets   mof0 48002
                  21.46.5.1  Functions   mof0 48002
                  21.46.5.2  Operations   fvconstr 48020
            21.46.6  ZF Set Theory - add the Axiom of Union   fvconst0ci 48023
                  21.46.6.1  Equinumerosity   fvconst0ci 48023
            21.46.7  Order sets   iccin 48027
                  21.46.7.1  Real number intervals   iccin 48027
            21.46.8  Moore spaces   mreuniss 48030
            ​*21.46.9  Topology   clduni 48031
                  21.46.9.1  Closure and interior   clduni 48031
                  21.46.9.2  Neighborhoods   neircl 48035
                  21.46.9.3  Subspace topologies   restcls2lem 48043
                  21.46.9.4  Limits and continuity in topological spaces   cnneiima 48047
                  21.46.9.5  Topological definitions using the reals   iooii 48048
                  21.46.9.6  Separated sets   sepnsepolem1 48052
                  21.46.9.7  Separated spaces: T0, T1, T2 (Hausdorff) ...   isnrm4 48061
            21.46.10  Preordered sets and directed sets using extensible structures   isprsd 48086
            21.46.11  Posets and lattices using extensible structures   lubeldm2 48087
                  21.46.11.1  Posets   lubeldm2 48087
                  21.46.11.2  Lattices   toslat 48105
                  21.46.11.3  Subset order structures   intubeu 48107
            21.46.12  Categories   catprslem 48128
                  21.46.12.1  Categories   catprslem 48128
                  21.46.12.2  Monomorphisms and epimorphisms   idmon 48134
                  21.46.12.3  Functors   funcf2lem 48136
            21.46.13  Examples of categories   cthinc 48137
                  21.46.13.1  Thin categories   cthinc 48137
                  21.46.13.2  Preordered sets as thin categories   cprstc 48180
                  21.46.13.3  Monoids as categories   cmndtc 48201
      ​21.47  Mathbox for Emmett Weisz
            ​*21.47.1  Miscellaneous Theorems   nfintd 48216
            21.47.2  Set Recursion   csetrecs 48226
                  ​*21.47.2.1  Basic Properties of Set Recursion   csetrecs 48226
                  21.47.2.2  Examples and properties of set recursion   elsetrecslem 48242
            ​*21.47.3  Construction of Games and Surreal Numbers   cpg 48252
      ​​*21.48  Mathbox for David A. Wheeler
            21.48.1  Natural deduction   sbidd 48261
            ​*21.48.2  Greater than, greater than or equal to.   cge-real 48263
            ​*21.48.3  Hyperbolic trigonometric functions   csinh 48273
            ​*21.48.4  Reciprocal trigonometric functions (sec, csc, cot)   csec 48284
            ​*21.48.5  Identities for "if"   ifnmfalse 48306
            ​*21.48.6  Logarithms generalized to arbitrary base using ` logb `   logb2aval 48307
            ​*21.48.7  Logarithm laws generalized to an arbitrary base - log_   clog- 48308
            ​*21.48.8  Formally define notions such as reflexivity   wreflexive 48310
            ​*21.48.9  Algebra helpers   comraddi 48314
            ​*21.48.10  Algebra helper examples   i2linesi 48323
            ​*21.48.11  Formal methods "surprises"   alimp-surprise 48325
            ​*21.48.12  Allsome quantifier   walsi 48331
            ​*21.48.13  Miscellaneous   5m4e1 48342
            21.48.14  Theorems about algebraic numbers   aacllem 48346
      ​21.49  Mathbox for Kunhao Zheng
            21.49.1  Weighted AM-GM inequality   amgmwlem 48347