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
      9.7  Chains
PART 10  BASIC ALGEBRAIC STRUCTURES
      10.1  Monoids
      10.2  Groups
      10.3  Rings
      10.4  Division rings and fields
      10.5  Left modules
      10.6  Vector spaces
      10.7  Subring algebras and ideals
      10.8  The complex numbers as an algebraic extensible structure
      10.9  Generalized pre-Hilbert and Hilbert spaces
PART 11  BASIC LINEAR ALGEBRA
      11.1  Vectors and free modules
      11.2  Associative algebras
      11.3  Abstract multivariate polynomials
      11.4  Matrices
      11.5  The determinant
      11.6  Polynomial matrices
      11.7  The characteristic polynomial
PART 12  BASIC TOPOLOGY
      12.1  Topology
      12.2  Filters and filter bases
      12.3  Uniform Structures and Spaces
      12.4  Metric spaces
      12.5  Metric subcomplex vector spaces
PART 13  BASIC REAL AND COMPLEX ANALYSIS
      13.1  Continuity
      13.2  Integrals
      13.3  Derivatives
PART 14  BASIC REAL AND COMPLEX FUNCTIONS
      14.1  Polynomials
      14.2  Sequences and series
      14.3  Basic trigonometry
      14.4  Basic number theory
PART 15  SURREAL NUMBERS
      15.1  Sign sequence representation and Alling's axioms
      15.2  Initial consequences of Alling's axioms
      15.3  Conway cut representation
      15.4  Induction and recursion
      15.5  Surreal arithmetic
      15.6  Subsystems of surreals
PART 16  ELEMENTARY GEOMETRY
      16.1  Definition and Tarski's Axioms of Geometry
      16.2  Tarskian Geometry
      16.3  Properties of geometries
      16.4  Geometry in Hilbert spaces
PART 17  GRAPH THEORY
      17.1  Vertices and edges
      17.2  Undirected graphs
      17.3  Walks, paths and cycles
      17.4  Eulerian paths and the Konigsberg Bridge problem
      17.5  The Friendship Theorem
PART 18  GUIDES AND MISCELLANEA
      18.1  Guides (conventions, explanations, and examples)
      18.2  Humor
      18.3  (Future - to be reviewed and classified)
PART 19  COMPLEX TOPOLOGICAL VECTOR SPACES (DEPRECATED)
      19.1  Additional material on group theory (deprecated)
      19.2  Complex vector spaces
      19.3  Normed complex vector spaces
      19.4  Operators on complex vector spaces
      19.5  Inner product (pre-Hilbert) spaces
      19.6  Complex Banach spaces
      19.7  Complex Hilbert spaces
PART 20  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      20.1  Axiomatization of complex pre-Hilbert spaces
      20.2  Inner product and norms
      20.3  Cauchy sequences and completeness axiom
      20.4  Subspaces and projections
      20.5  Properties of Hilbert subspaces
      20.6  Operators on Hilbert spaces
      20.7  States on a Hilbert lattice and Godowski's equation
      20.8  Cover relation, atoms, exchange axiom, and modular symmetry
PART 21  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      ​21.1  Mathboxes for user contributions
      21.2  Mathbox for Stefan Allan
      21.3  Mathbox for Thierry Arnoux
      21.4  Mathbox for Jonathan Ben-Naim
      21.5  Mathbox for BTernaryTau
      21.6  Mathbox for Mario Carneiro
      21.7  Mathbox for Filip Cernatescu
      21.8  Mathbox for Paul Chapman
      21.9  Mathbox for Hongxiu Chen
      21.10  Mathbox for Adrian Ducourtial
      21.11  Mathbox for Scott Fenton
      21.12  Mathbox for Gino Giotto
      21.13  Mathbox for Jeff Hankins
      21.14  Mathbox for Anthony Hart
      21.15  Mathbox for Chen-Pang He
      21.16  Mathbox for Jeff Hoffman
      21.17  Mathbox for Matthew House
      21.18  Mathbox for Asger C. Ipsen
      21.19  Mathbox for BJ
      21.20  Mathbox for Jim Kingdon
      21.21  Mathbox for ML
      21.22  Mathbox for Wolf Lammen
      21.23  Mathbox for Brendan Leahy
      21.24  Mathbox for Jeff Madsen
      21.25  Mathbox for Giovanni Mascellani
      21.26  Mathbox for Peter Mazsa
      21.27  Mathbox for Rodolfo Medina
      21.28  Mathbox for Norm Megill
      21.29  Mathbox for metakunt
      21.30  Mathbox for Luke Murphy
      21.31  Mathbox for Steven Nguyen
      21.32  Mathbox for Igor Ieskov
      21.33  Mathbox for OpenAI
      21.34  Mathbox for Stefan O'Rear
      21.35  Mathbox for Noam Pasman
      21.36  Mathbox for Jon Pennant
      21.37  Mathbox for Richard Penner
      21.38  Mathbox for Stanislas Polu
      21.39  Mathbox for Rohan Ridenour
      21.40  Mathbox for Steve Rodriguez
      21.41  Mathbox for Andrew Salmon
      21.42  Mathbox for Alan Sare
      21.43  Mathbox for Eric Schmidt
      21.44  Mathbox for Glauco Siliprandi
      21.45  Mathbox for Saveliy Skresanov
      21.46  Mathbox for Ender Ting
      21.47  Mathbox for Jarvin Udandy
      21.48  Mathbox for Adhemar
      21.49  Mathbox for Alexander van der Vekens
      21.50  Mathbox for Zhi Wang
      21.51  Mathbox for Emmett Weisz
      21.52  Mathbox for David A. Wheeler
      21.53  Mathbox for Mingli Yuan
      21.54  Mathbox for Jiamin Zhao
      21.55  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 114
            ​*1.2.5  Logical equivalence   wb 209
            ​*1.2.6  Logical conjunction   wa 401
            ​*1.2.7  Logical disjunction   wo 861
            ​*1.2.8  Mixed connectives   jaao 969
            ​*1.2.9  The conditional operator for propositions   wif 1078
            ​*1.2.10  The weak deduction theorem for propositional calculus   elimh 1099
            1.2.11  Abbreviated conjunction and disjunction of three wff's   w3o 1102
            1.2.12  Logical "nand" (Sheffer stroke)   wnan 1521
            1.2.13  Logical "xor"   wxo 1541
            1.2.14  Logical "nor"   wnor 1558
            1.2.15  True and false constants   wal 1568
                  ​*1.2.15.1  Universal quantifier for use by df-tru   wal 1568
                  ​*1.2.15.2  Equality predicate for use by df-tru   cv 1569
                  1.2.15.3  The true constant   wtru 1571
                  1.2.15.4  The false constant   wfal 1582
            ​*1.2.16  Truth tables   truimtru 1593
                  1.2.16.1  Implication   truimtru 1593
                  1.2.16.2  Negation   nottru 1597
                  1.2.16.3  Equivalence   trubitru 1599
                  1.2.16.4  Conjunction   truantru 1603
                  1.2.16.5  Disjunction   truortru 1607
                  1.2.16.6  Alternative denial   trunantru 1611
                  1.2.16.7  Exclusive disjunction   truxortru 1615
                  1.2.16.8  Joint denial   trunortru 1619
            ​*1.2.17  Half adder and full adder in propositional calculus   whad 1623
                  1.2.17.1  Full adder: sum   whad 1623
                  1.2.17.2  Full adder: carry   wcad 1639
      ​1.3  Other axiomatizations related to classical propositional calculus
            ​*1.3.1  Minimal implicational calculus   minimp 1654
            ​*1.3.2  Implicational Calculus   impsingle 1660
            1.3.3  Derive the Lukasiewicz axioms from Meredith's sole axiom   meredith 1674
            1.3.4  Derive the standard axioms from the Lukasiewicz axioms   luklem1 1691
            ​*1.3.5  Derive Nicod's axiom from the standard axioms   nic-dfim 1702
            1.3.6  Derive the Lukasiewicz axioms from Nicod's axiom   nic-imp 1708
            1.3.7  Derive Nicod's Axiom from Lukasiewicz's First Sheffer Stroke Axiom   lukshef-ax1 1727
            1.3.8  Derive the Lukasiewicz Axioms from the Tarski-Bernays-Wajsberg Axioms   tbw-bijust 1731
            1.3.9  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's First CO Axiom   merco1 1746
            1.3.10  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's Second CO Axiom   merco2 1769
            1.3.11  Derive the Lukasiewicz axioms from the Russell-Bernays Axioms   rb-bijust 1782
            ​*1.3.12  Stoic logic non-modal portion (Chrysippus of Soli)   mptnan 1801
      ​​*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 1812
                  1.4.1.1  Existential quantifier   wex 1812
                  1.4.1.2  Nonfreeness predicate   wnf 1816
            1.4.2  Rule scheme ax-gen (Generalization)   ax-gen 1828
            1.4.3  Axiom scheme ax-4 (Quantified Implication)   ax-4 1842
                  ​*1.4.3.1  The empty domain of discourse   empty 1939
            1.4.4  Axiom scheme ax-5 (Distinctness) - first use of $d   ax-5 1943
            ​*1.4.5  Equality predicate (continued)   weq 1995
            1.4.6  Axiom scheme ax-6 (Existence)   ax-6 2000
            1.4.7  Axiom scheme ax-7 (Equality)   ax-7 2041
            1.4.8  Define proper substitution   sbjust 2098
            1.4.9  Membership predicate   wcel 2146
            1.4.10  Axiom scheme ax-8 (Left Equality for Binary Predicate)   ax-8 2148
            1.4.11  Axiom scheme ax-9 (Right Equality for Binary Predicate)   ax-9 2156
            ​*1.4.12  Logical redundancy of ax-10 , ax-11 , ax-12 , ax-13   ax6dgen 2166
      ​​*1.5  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes)
            1.5.1  Axiom scheme ax-10 (Quantified Negation)   ax-10 2179
            1.5.2  Axiom scheme ax-11 (Quantifier Commutation)   ax-11 2195
            1.5.3  Axiom scheme ax-12 (Substitution)   ax-12 2216
            1.5.4  Axiom scheme ax-13 (Quantified Equality)   ax-13 2407
      ​1.6  Uniqueness and unique existence
            1.6.1  Uniqueness: the at-most-one quantifier   wmo 2568
            1.6.2  Unique existence: the unique existential quantifier   weu 2599
      ​1.7  Other axiomatizations related to classical predicate calculus
            ​*1.7.1  Aristotelian logic: Assertic syllogisms   barbara 2693
            ​*1.7.2  Intuitionistic logic   axia1 2723
​​*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 2738
            2.1.2  Classes   cab 2744
                  2.1.2.1  Class abstractions   cab 2744
                  ​*2.1.2.2  Class equality   df-cleq 2758
                  2.1.2.3  Class membership   df-clel 2841
                  2.1.2.4  Elementary properties of class abstractions   eqabdv 2899
            2.1.3  Class form not-free predicate   wnfc 2913
            2.1.4  Negated equality and membership   wne 2961
                  2.1.4.1  Negated equality   wne 2961
                  2.1.4.2  Negated membership   wnel 3067
            2.1.5  Restricted quantification   wral 3082
                  2.1.5.1  Restricted universal and existential quantification   wral 3082
                  2.1.5.2  Restricted existential uniqueness and at-most-one quantifier   wreu 3370
                  2.1.5.3  Restricted class abstraction   crab 3419
            2.1.6  The universal class   cvv 3458
            ​*2.1.7  Conditional equality (experimental)   wcdeq 3729
            2.1.8  Russell's Paradox   rru 3745
            2.1.9  Proper substitution of classes for sets   wsbc 3747
            2.1.10  Proper substitution of classes for sets into classes   csb 3856
            2.1.11  Define basic set operations and relations   cdif 3905
            2.1.12  Subclasses and subsets   df-ss 3925
            2.1.13  The difference, union, and intersection of two classes   dfdif3 4075
                  2.1.13.1  The difference of two classes   dfdif3 4075
                  2.1.13.2  The union of two classes   elun 4110
                  2.1.13.3  The intersection of two classes   elini 4155
                  2.1.13.4  The symmetric difference of two classes   csymdif 4208
                  2.1.13.5  Combinations of difference, union, and intersection of two classes   unabs 4221
                  2.1.13.6  Class abstractions with difference, union, and intersection of two classes   unabw 4263
                  2.1.13.7  Restricted uniqueness with difference, union, and intersection   reuun2 4281
            2.1.14  The empty set   c0 4289
            ​*2.1.15  The conditional operator for classes   cif 4492
            ​*2.1.16  The weak deduction theorem for set theory   dedth 4551
            2.1.17  Power classes   cpw 4567
            2.1.18  Unordered and ordered pairs   snjust 4593
            2.1.19  The union of a class   cuni 4877
            2.1.20  The intersection of a class   cint 4917
            2.1.21  Indexed union and intersection   ciun 4961
            2.1.22  Disjointness   wdisj 5081
            2.1.23  Binary relations   wbr 5114
            2.1.24  Ordered-pair class abstractions (class builders)   copab 5178
            2.1.25  Functions in maps-to notation   cmpt 5197
            2.1.26  Transitive classes   wtr 5223
      ​2.2  ZF Set Theory - add the Axiom of Replacement
            2.2.1  Introduce the Axiom of Replacement   ax-rep 5243
            2.2.2  Derive the Axiom of Separation   axsepgfromrep 5260
            2.2.3  Derive the Null Set Axiom   axnulALT 5272
            2.2.4  Theorems requiring subset and intersection existence   exnelv 5281
            2.2.5  Theorems requiring empty set existence   class2set 5330
      ​2.3  ZF Set Theory - add the Axiom of Power Sets
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 5341
            2.3.2  Derive the Axiom of Pairing   axprlem1 5399
            2.3.3  Ordered pair theorem   opnz 5460
            2.3.4  Ordered-pair class abstractions (cont.)   opabidw 5513
            2.3.5  Power class of union and intersection   pwin 5557
            2.3.6  The identity relation   cid 5560
            2.3.7  The membership relation (or epsilon relation)   cep 5565
            ​*2.3.8  Partial and total orderings   wpo 5572
            2.3.9  Founded and well-ordering relations   wfr 5616
            2.3.10  Relations   cxp 5664
            2.3.11  The Predecessor Class   cpred 6308
            2.3.12  Well-founded induction (variant)   frpomin 6348
            2.3.13  Well-ordered induction   tz6.26 6355
            2.3.14  Ordinals   word 6366
            2.3.15  Definite description binder (inverted iota)   cio 6497
            2.3.16  Functions   wfun 6537
            2.3.17  Cantor's Theorem   canth 7377
            2.3.18  Restricted iota (description binder)   crio 7379
            2.3.19  Operations   co 7423
                  2.3.19.1  Variable-to-class conversion for operations   caovclg 7615
            2.3.20  Maps-to notation   mpondm0 7663
            2.3.21  Function operation   cof 7685
            2.3.22  Proper subset relation   crpss 7732
      ​2.4  ZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 7745
            2.4.2  Ordinals (continued)   epweon 7783
            2.4.3  Transfinite induction   tfi 7858
            2.4.4  The natural numbers (i.e., finite ordinals)   com 7871
            2.4.5  Peano's postulates   peano1 7894
            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 7993
            2.4.9  Induction on Cartesian products   frpoins3xpg 8145
            2.4.10  Ordering on Cartesian products   xpord2lem 8147
            2.4.11  Ordering Ordinal Sequences   orderseqlem 8162
            ​*2.4.12  The support of functions   csupp 8165
            ​*2.4.13  Special maps-to operations   opeliunxp2f 8215
            2.4.14  Function transposition   ctpos 8230
            2.4.15  Curry and uncurry   ccur 8270
            2.4.16  Undefined values   cund 8277
            2.4.17  Well-founded recursion   cfrecs 8286
            2.4.18  Well-ordered recursion   cwrecs 8317
            2.4.19  Functions on ordinals; strictly monotone ordinal functions   iunon 8335
            2.4.20  "Strong" transfinite recursion   crecs 8366
            2.4.21  Recursive definition generator   crdg 8405
            2.4.22  Finite recursion   frfnom 8431
            2.4.23  Ordinal arithmetic   c1o 8455
            2.4.24  Natural number arithmetic   nna0 8599
            2.4.25  Natural addition   cnadd 8660
            2.4.26  Equivalence relations and classes   wer 8700
            2.4.27  The mapping operation   cmap 8833
            2.4.28  Infinite Cartesian products   cixp 8904
            2.4.29  Equinumerosity   cen 8949
            2.4.30  Schroeder-Bernstein Theorem   sbthlem1 9085
            2.4.31  Equinumerosity (cont.)   xpf1o 9137
            2.4.32  Finite sets   dif1enlem 9154
            2.4.33  Pigeonhole Principle   phplem1 9198
            2.4.34  Finite sets (cont.)   onomeneq 9208
            2.4.35  Finitely supported functions   cfsupp 9331
            2.4.36  Finite intersections   cfi 9380
            2.4.37  Hall's marriage theorem   marypha1lem 9403
            2.4.38  Supremum and infimum   csup 9410
            2.4.39  Ordinal isomorphism, Hartogs's theorem   coi 9481
            2.4.40  Hartogs function   char 9528
            2.4.41  Weak dominance   cwdom 9536
      ​2.5  ZF Set Theory - add the Axiom of Regularity
            2.5.1  Introduce the Axiom of Regularity   ax-reg 9564
            2.5.2  Axiom of Infinity equivalents   inf0 9600
      ​2.6  ZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-inf 9617
            2.6.2  Existence of omega (the set of natural numbers)   omex 9622
            2.6.3  Cantor normal form   ccnf 9640
            2.6.4  Transitive closure of a relation   cttrcl 9686
            2.6.5  Transitive closure   trcl 9707
            2.6.6  Set induction (or epsilon induction)   setind 9726
            2.6.7  Well-Founded Induction   frmin 9731
            2.6.8  Well-Founded Recursion   frr3g 9738
            2.6.9  Rank   cr1 9744
            2.6.10  Scott's trick; collection principle; Hilbert's epsilon   cscott 9867
            2.6.11  Disjoint union   cdju 9903
            2.6.12  Cardinal numbers   ccrd 9940
            2.6.13  Axiom of Choice equivalents   wac 10118
            ​*2.6.14  Cardinal number arithmetic   undjudom 10170
            2.6.15  The Ackermann bijection   ackbij2lem1 10220
            2.6.16  Cofinality (without Axiom of Choice)   cflem 10247
            2.6.17  Eight inequivalent definitions of finite set   sornom 10279
            2.6.18  Hereditarily size-limited sets without Choice   itunifval 10418
​​*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 10437
            3.1.2  Introduce the Axiom of Dependent Choice   ax-dc 10448
      ​3.2  ZFC Set Theory - add the Axiom of Choice
            3.2.1  Introduce the Axiom of Choice   ax-ac 10461
            3.2.2  AC equivalents: well-ordering, Zorn's lemma   numthcor 10496
            3.2.3  Cardinal number theorems using Axiom of Choice   cardval 10548
            3.2.4  Cardinal number arithmetic using Axiom of Choice   iunctb 10577
            3.2.5  Cofinality using the Axiom of Choice   alephreg 10585
      ​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 10623
            3.4.2  Derivation of the Axiom of Choice   gchaclem 10681
​​*PART 4  TG (TARSKI-GROTHENDIECK) SET THEORY
      ​4.1  Inaccessibles
            4.1.1  Weakly and strongly inaccessible cardinals   cwina 10685
            4.1.2  Weak universes   cwun 10703
            4.1.3  Tarski classes   ctsk 10751
            4.1.4  Grothendieck universes   cgru 10793
      ​4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
            4.2.1  Introduce the Tarski-Grothendieck Axiom   ax-groth 10826
            4.2.2  Derive the Power Set, Infinity and Choice Axioms   grothpw 10829
            4.2.3  Tarski map function   ctskm 10840
​​*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 10847
            5.1.2  Final derivation of real and complex number postulates   axaddf 11148
            5.1.3  Real and complex number postulates restated as axioms   ax-cnex 11174
      ​5.2  Derive the basic properties from the field axioms
            5.2.1  Some deductions from the field axioms for complex numbers   cnex 11199
            5.2.2  Infinity and the extended real number system   cpnf 11258
            5.2.3  Restate the ordering postulates with extended real "less than"   axlttri 11299
            5.2.4  Ordering on reals   lttr 11304
            5.2.5  Initial properties of the complex numbers   mul12 11393
      ​5.3  Real and complex numbers - basic operations
            5.3.1  Addition   add12 11446
            5.3.2  Subtraction   cmin 11459
            5.3.3  Multiplication   kcnktkm1cn 11663
            5.3.4  Ordering on reals (cont.)   gt0ne0 11697
            5.3.5  Reciprocals   ixi 11861
            5.3.6  Division   cdiv 11889
            5.3.7  Ordering on reals (cont.)   elimgt0 12071
            5.3.8  Completeness Axiom and Suprema   fimaxre 12177
            5.3.9  Imaginary and complex number properties   neg1cn 12221
            5.3.10  Function operation analogue theorems   ofsubeq0 12233
            ​*5.3.11  Indicator Functions   cind 12236
      ​5.4  Integer sets
            5.4.1  Positive integers (as a subset of complex numbers)   cn 12251
            5.4.2  Principle of mathematical induction   nnind 12269
            ​*5.4.3  Decimal representation of numbers   c2 12313
            ​*5.4.4  Some properties of specific numbers   1pneg1e0 12376
            5.4.5  Simple number properties   halfcl 12488
            5.4.6  The Archimedean property   nnunb 12518
            5.4.7  Nonnegative integers (as a subset of complex numbers)   cn0 12522
            ​*5.4.8  Extended nonnegative integers   cxnn0 12595
            5.4.9  Integers (as a subset of complex numbers)   cz 12609
            5.4.10  Decimal arithmetic   cdc 12729
            5.4.11  Upper sets of integers   cuz 12880
            5.4.12  Well-ordering principle for bounded-below sets of integers   uzwo3 12985
            5.4.13  Rational numbers (as a subset of complex numbers)   cq 12990
            5.4.14  Existence of the set of complex numbers   rpnnen1lem2 13019
      ​5.5  Order sets
            5.5.1  Positive reals (as a subset of complex numbers)   crp 13034
            5.5.2  Infinity and the extended real number system (cont.)   cxne 13152
            5.5.3  Supremum and infimum on the extended reals   xrsupexmnf 13349
            5.5.4  Real number intervals   cioo 13390
            5.5.5  Finite intervals of integers   cfz 13553
            ​*5.5.6  Finite intervals of nonnegative integers   elfz2nn0 13665
            5.5.7  Half-open integer ranges   cfzo 13701
      ​5.6  Elementary integer functions
            5.6.1  The floor and ceiling functions   cfl 13843
            5.6.2  The modulo (remainder) operation   cmo 13922
            5.6.3  Miscellaneous theorems about integers   om2uz0i 14003
            5.6.4  Strong induction over upper sets of integers   uzsinds 14043
            5.6.5  Finitely supported functions over the nonnegative integers   fsuppmapnn0fiublem 14046
            5.6.6  The infinite sequence builder "seq" - extension   cseq 14057
            5.6.7  Integer powers   cexp 14117
            5.6.8  Ordered pair theorem for nonnegative integers   nn0le2msqi 14323
            5.6.9  Factorial function   cfa 14329
            5.6.10  The binomial coefficient operation   cbc 14358
            5.6.11  The ` # ` (set size) function   chash 14386
                  5.6.11.1  Proper unordered pairs and triples (sets of size 2 and 3)   hashprlei 14525
                  5.6.11.2  Functions with a domain containing at least two different elements   fundmge2nop0 14559
                  5.6.11.3  Finite induction on the size of the first component of a binary relation   hashdifsnp1 14563
      ​​*5.7  Words over a set
            5.7.1  Definitions and basic theorems   cword 14570
            5.7.2  Last symbol of a word   clsw 14619
            5.7.3  Concatenations of words   cconcat 14627
            5.7.4  Singleton words   cs1 14654
            5.7.5  Concatenations with singleton words   ccatws1cl 14676
            5.7.6  Subwords/substrings   csubstr 14700
            5.7.7  Prefixes of a word   cpfx 14732
            5.7.8  Subwords of subwords   swrdswrdlem 14765
            5.7.9  Subwords and concatenations   pfxcctswrd 14771
            5.7.10  Subwords of concatenations   swrdccatfn 14785
            5.7.11  Splicing words (substring replacement)   csplice 14810
            5.7.12  Reversing words   creverse 14819
            5.7.13  Repeated symbol words   creps 14831
            ​*5.7.14  Cyclical shifts of words   ccsh 14851
            5.7.15  Mapping words by a function   wrdco 14894
            5.7.16  Longer string literals   cs2 14904
      ​​*5.8  Reflexive and transitive closures of relations
            5.8.1  The reflexive and transitive properties of relations   coss12d 15035
            5.8.2  Basic properties of closures   cleq1lem 15045
            5.8.3  Definitions and basic properties of transitive closures   ctcl 15048
            5.8.4  Exponentiation of relations   crelexp 15082
            5.8.5  Reflexive-transitive closure as an indexed union   crtrcl 15118
            ​*5.8.6  Principle of transitive induction   relexpindlem 15126
      ​5.9  Elementary real and complex functions
            5.9.1  The "shift" operation   cshi 15129
            5.9.2  Signum (sgn or sign) function   csgn 15149
            5.9.3  Real and imaginary parts; conjugate   ccj 15173
            5.9.4  Square root; absolute value   csqrt 15310
      ​5.10  Elementary limits and convergence
            5.10.1  Superior limit (lim sup)   clsp 15547
            5.10.2  Limits   cli 15561
            5.10.3  Finite and infinite sums   csu 15763
            5.10.4  The binomial theorem   binomlem 15909
            5.10.5  The inclusion/exclusion principle   incexclem 15916
            5.10.6  Infinite sums (cont.)   isumshft 15919
            5.10.7  Miscellaneous converging and diverging sequences   divrcnv 15932
            5.10.8  Arithmetic series   arisum 15940
            5.10.9  Geometric series   expcnv 15944
            5.10.10  Ratio test for infinite series convergence   cvgrat 15963
            5.10.11  Mertens' theorem   mertenslem1 15964
            5.10.12  Finite and infinite products   prodf 15967
                  5.10.12.1  Product sequences   prodf 15967
                  5.10.12.2  Non-trivial convergence   ntrivcvg 15977
                  5.10.12.3  Complex products   cprod 15983
                  5.10.12.4  Finite products   fprod 16021
                  5.10.12.5  Infinite products   iprodclim 16078
            5.10.13  Falling and Rising Factorial   cfallfac 16084
            5.10.14  Bernoulli polynomials and sums of k-th powers   cbp 16125
      ​5.11  Elementary trigonometry
            5.11.1  The exponential, sine, and cosine functions   ce 16140
                  5.11.1.1  The circle constant (tau = 2 pi)   ctau 16283
            5.11.2  _e is irrational   eirrlem 16285
      ​5.12  Cardinality of real and complex number subsets
            5.12.1  Countability of integers and rationals   xpnnen 16292
            5.12.2  The reals are uncountable   rpnnen2lem1 16295
​​*PART 6  ELEMENTARY NUMBER THEORY
      ​6.1  Elementary properties of divisibility
            6.1.1  Irrationality of square root of 2   sqrt2irrlem 16329
            6.1.2  Some Number sets are chains of proper subsets   nthruc 16333
            6.1.3  The divides relation   cdvds 16335
            ​*6.1.4  Even and odd numbers   evenelz 16419
            6.1.5  The division algorithm   divalglem0 16476
            6.1.6  Bit sequences   cbits 16502
            6.1.7  The greatest common divisor operator   cgcd 16577
            6.1.8  Bézout's identity   bezoutlem1 16622
            6.1.9  Algorithms   nn0seqcvgd 16653
            6.1.10  Euclid's Algorithm   eucalgval2 16664
            ​*6.1.11  The least common multiple   clcm 16671
            ​*6.1.12  Coprimality and Euclid's lemma   coprmgcdb 16732
            6.1.13  Cancellability of congruences   congr 16747
      ​6.2  Elementary prime number theory
            ​*6.2.1  Elementary properties   cprime 16754
            ​*6.2.2  Coprimality and Euclid's lemma (cont.)   coprm 16795
            6.2.3  Properties of the canonical representation of a rational   cnumer 16817
            6.2.4  Euler's theorem   codz 16847
            6.2.5  Arithmetic modulo a prime number   modprm1div 16882
            6.2.6  Pythagorean Triples   coprimeprodsq 16893
            6.2.7  The prime count function   cpc 16921
            6.2.8  Pocklington's theorem   prmpwdvds 16989
            6.2.9  Infinite primes theorem   unbenlem 16993
            6.2.10  Sum of prime reciprocals   prmreclem1 17001
            6.2.11  Fundamental theorem of arithmetic   1arithlem1 17008
            6.2.12  Lagrange's four-square theorem   cgz 17014
            6.2.13  Van der Waerden's theorem   cvdwa 17050
            6.2.14  Ramsey's theorem   cram 17084
            ​*6.2.15  Primorial function   cprmo 17116
            ​*6.2.16  Prime gaps   prmgaplem1 17134
            6.2.17  Decimal arithmetic (cont.)   dec2dvds 17148
            6.2.18  Cyclical shifts of words (cont.)   cshwsidrepsw 17178
            6.2.19  Specific prime numbers   prmlem0 17190
            6.2.20  Very large primes   1259lem1 17216
​PART 7  BASIC STRUCTURES
      ​7.1  Extensible structures
            ​*7.1.1  Basic definitions   cstr 17231
                  7.1.1.1  Extensible structures as structures with components   cstr 17231
                  7.1.1.2  Substitution of components   csts 17248
                  7.1.1.3  Slots   cslot 17266
                  ​*7.1.1.4  Structure component indices   cnx 17278
                  7.1.1.5  Base sets   cbs 17294
                  7.1.1.6  Base set restrictions   cress 17315
            7.1.2  Slot definitions   cplusg 17335
            7.1.3  Definition of the structure product   crest 17498
            7.1.4  Definition of the structure quotient   cordt 17578
      ​7.2  Moore spaces
            7.2.1  Moore closures   mrcflem 17687
            7.2.2  Independent sets in a Moore system   mrisval 17711
            7.2.3  Algebraic closure systems   isacs 17732
​PART 8  BASIC CATEGORY THEORY
      ​8.1  Categories
            8.1.1  Categories   ccat 17745
            8.1.2  Opposite category   coppc 17792
            8.1.3  Monomorphisms and epimorphisms   cmon 17810
            8.1.4  Sections, inverses, isomorphisms   csect 17826
            ​*8.1.5  Isomorphic objects   ccic 17877
            8.1.6  Subcategories   cssc 17889
            8.1.7  Functors   cfunc 17936
            8.1.8  Full & faithful functors   cful 17986
            8.1.9  Natural transformations and the functor category   cnat 18026
            8.1.10  Initial, terminal and zero objects of a category   cinito 18063
      ​8.2  Arrows (disjointified hom-sets)
            8.2.1  Identity and composition for arrows   cida 18135
      ​8.3  Examples of categories
            8.3.1  The category of sets   csetc 18157
            8.3.2  The category of categories   ccatc 18180
            ​*8.3.3  The category of extensible structures   fncnvimaeqv 18201
      ​8.4  Categorical constructions
            8.4.1  Product of categories   cxpc 18249
            8.4.2  Functor evaluation   cevlf 18290
            8.4.3  Hom functor   chof 18329
​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 18512
            9.5.2  Complete lattices   ccla 18579
            9.5.3  Distributive lattices   cdlat 18601
            9.5.4  Subset order structures   cipo 18608
      ​9.6  Posets, directed sets, and lattices as relations
            ​*9.6.1  Posets and lattices as relations   cps 18645
            9.6.2  Directed sets, nets   cdir 18675
      ​9.7  Chains
​PART 10  BASIC ALGEBRAIC STRUCTURES
      ​10.1  Monoids
            ​*10.1.1  Magmas   cplusf 18720
            ​*10.1.2  Identity elements   mgmidmo 18743
            ​*10.1.3  Iterated sums in a magma   gsumvalx 18763
            10.1.4  Magma homomorphisms and submagmas   cmgmhm 18777
            ​*10.1.5  Semigroups   csgrp 18805
            ​*10.1.6  Definition and basic properties of monoids   cmnd 18821
            10.1.7  Monoid homomorphisms and submonoids   cmhm 18870
            ​*10.1.8  Iterated sums in a monoid   gsumvallem2 18924
            10.1.9  Free monoids   cfrmd 18937
                  ​*10.1.9.1  Monoid of endofunctions   cefmnd 18958
            10.1.10  Examples and counterexamples for magmas, semigroups and monoids   mgm2nsgrplem1 19011
      ​10.2  Groups
            10.2.1  Definition and basic properties   cgrp 19031
            ​*10.2.2  Group multiple operation   cmg 19164
            10.2.3  Subgroups and Quotient groups   csubg 19217
            ​*10.2.4  Cyclic monoids and groups   cycsubmel 19302
            10.2.5  Elementary theory of group homomorphisms   cghm 19314
            10.2.6  Isomorphisms of groups   cgim 19358
                  10.2.6.1  The first isomorphism theorem of groups   ghmqusnsglem1 19381
            10.2.7  Group actions   cga 19390
            10.2.8  Centralizers and centers   ccntz 19416
            10.2.9  The opposite group   coppg 19446
            10.2.10  Symmetric groups   csymg 19470
                  ​*10.2.10.1  Definition and basic properties   csymg 19470
                  10.2.10.2  Cayley's theorem   cayleylem1 19513
                  10.2.10.3  Permutations fixing one element   symgfix2 19517
                  ​*10.2.10.4  Transpositions in the symmetric group   cpmtr 19542
                  10.2.10.5  The sign of a permutation   cpsgn 19590
            10.2.11  p-Groups and Sylow groups; Sylow's theorems   cod 19625
            10.2.12  Direct products   clsm 19735
                  10.2.12.1  Direct products (extension)   smndlsmidm 19757
            10.2.13  Free groups   cefg 19807
            10.2.14  Abelian groups   ccmn 19881
                  10.2.14.1  Definition and basic properties   ccmn 19881
                  10.2.14.2  Cyclic groups   ccyg 19978
                  10.2.14.3  Group sum operation   gsumval3a 20004
                  10.2.14.4  Group sums over (ranges of) integers   fsfnn0gsumfsffz 20084
                  10.2.14.5  Internal direct products   cdprd 20096
                  10.2.14.6  The Fundamental Theorem of Abelian Groups   ablfacrplem 20168
            10.2.15  Simple groups   csimpg 20193
                  10.2.15.1  Definition and basic properties   csimpg 20193
                  10.2.15.2  Classification of abelian simple groups   ablsimpnosubgd 20207
            10.2.16  Totally ordered monoids and groups   comnd 20220
      ​10.3  Rings
            10.3.1  Multiplicative Group   cmgp 20247
            ​*10.3.2  Non-unital rings ("rngs")   crng 20261
            ​*10.3.3  Ring unity (multiplicative identity)   cur 20294
            10.3.4  Semirings   csrg 20299
                  ​*10.3.4.1  The binomial theorem for semirings   srgbinomlem1 20339
            10.3.5  Unital rings   crg 20346
            10.3.6  Opposite ring   coppr 20451
            10.3.7  Divisibility   cdsr 20469
            10.3.8  Ring primes   crpm 20547
            10.3.9  Homomorphisms of non-unital rings   crnghm 20549
            10.3.10  Ring homomorphisms   crh 20584
            10.3.11  Nonzero rings and zero rings   cnzr 20646
            10.3.12  Local rings   clring 20674
            10.3.13  Subrings   csubrng 20681
                  10.3.13.1  Subrings of non-unital rings   csubrng 20681
                  10.3.13.2  Subrings of unital rings   csubrg 20705
                  10.3.13.3  Subrings generated by a subset   crgspn 20746
            10.3.14  Categories of rings   crngc 20752
                  ​*10.3.14.1  The category of non-unital rings   crngc 20752
                  ​*10.3.14.2  The category of (unital) rings   cringc 20781
                  10.3.14.3  Subcategories of the category of rings   srhmsubclem1 20813
            10.3.15  Left regular elements and domains   crlreg 20827
      ​10.4  Division rings and fields
            10.4.1  Definition and basic properties   cdr 20864
            10.4.2  Sub-division rings   csdrg 20926
            10.4.3  Absolute value (abstract algebra)   cabv 20948
            10.4.4  Star rings   cstf 20977
            10.4.5  Totally ordered rings and fields   corng 20997
      ​10.5  Left modules
            10.5.1  Definition and basic properties   clmod 21018
            10.5.2  Subspaces and spans in a left module   clss 21089
            10.5.3  Homomorphisms and isomorphisms of left modules   clmhm 21177
            10.5.4  Subspace sum; bases for a left module   clbs 21232
      ​10.6  Vector spaces
            10.6.1  Definition and basic properties   clvec 21260
      ​10.7  Subring algebras and ideals
            10.7.1  Subring algebras   csra 21329
            ​*10.7.2  Left ideals and spans   clidl 21367
            10.7.3  Two-sided ideals and quotient rings   c2idl 21425
                  ​*10.7.3.1  Condition for a non-unital ring to be unital   rngqiprng1elbas 21463
                  10.7.3.2  Prime Ideals   cprmidl 21497
            10.7.4  Principal ideal rings. Divisibility in the integers   clpidl 21525
            10.7.5  Principal ideal domains   cpid 21541
      ​10.8  The complex numbers as an algebraic extensible structure
            10.8.1  Definition and basic properties   cpsmet 21543
            ​*10.8.2  Ring of integers   czring 21633
                  ​*10.8.2.1  Example for a condition for a non-unital ring to be unital   pzriprnglem1 21668
            10.8.3  Algebraic constructions based on the complex numbers   czrh 21686
            10.8.4  Signs as subgroup of the complex numbers   cnmsgnsubg 21764
            10.8.5  Embedding of permutation signs into a ring   zrhpsgnmhm 21771
            10.8.6  The ordered field of real numbers   crefld 21791
      ​10.9  Generalized pre-Hilbert and Hilbert spaces
            10.9.1  Definition and basic properties   cphl 21811
            10.9.2  Orthocomplements and closed subspaces   cocv 21847
            10.9.3  Orthogonal projection and orthonormal bases   cpj 21887
​​*PART 11  BASIC LINEAR ALGEBRA
      ​11.1  Vectors and free modules
            *11.1.1  Direct sum of left modules   cdsmm 21918
            ​*11.1.2  Free modules   cfrlm 21933
            ​*11.1.3  Standard basis (unit vectors)   cuvc 21969
            ​*11.1.4  Independent sets and families   clindf 21991
            11.1.5  Characterization of free modules   lmimlbs 22023
      ​11.2  Associative algebras
            11.2.1  Definition and basic properties   casa 22037
      ​11.3  Abstract multivariate polynomials
            11.3.1  Definition and basic properties   cmps 22091
            11.3.2  Polynomial evaluation   ces 22260
            11.3.3  The "variable selection" function   cslv 22304
            11.3.4  Additional definitions for (multivariate) polynomials   cmhp 22333
            ​*11.3.5  Univariate polynomials   cps1 22372
            11.3.6  Univariate polynomial evaluation   ces1 22510
                  11.3.6.1  Specialization of polynomial evaluation as a ring homomorphism   evls1scafv 22563
      ​​*11.4  Matrices
            *11.4.1  The matrix multiplication   cmmul 22584
            ​*11.4.2  Square matrices   cmat 22601
            ​*11.4.3  The matrix algebra   matmulr 22632
            ​*11.4.4  Matrices of dimension 0 and 1   mat0dimbas0 22660
            ​*11.4.5  The subalgebras of diagonal and scalar matrices   cdmat 22682
            ​*11.4.6  Multiplication of a matrix with a "column vector"   cmvmul 22734
            11.4.7  Replacement functions for a square matrix   cmarrep 22750
            11.4.8  Submatrices   csubma 22770
      ​11.5  The determinant
            11.5.1  Definition and basic properties   cmdat 22778
            11.5.2  Determinants of 2 x 2 -matrices   m2detleiblem1 22818
            11.5.3  The matrix adjugate/adjunct   cmadu 22826
            ​*11.5.4  Laplace expansion of determinants (special case)   symgmatr01lem 22847
            11.5.5  Inverse matrix   invrvald 22870
            ​*11.5.6  Cramer's rule   slesolvec 22873
      ​​*11.6  Polynomial matrices
            11.6.1  Basic properties   pmatring 22886
            ​*11.6.2  Constant polynomial matrices   ccpmat 22897
            ​*11.6.3  Collecting coefficients of polynomial matrices   cdecpmat 22956
            ​*11.6.4  Ring isomorphism between polynomial matrices and polynomials over matrices   cpm2mp 22986
      ​​*11.7  The characteristic polynomial
            *11.7.1  Definition and basic properties   cchpmat 23020
            ​*11.7.2  The characteristic factor function G   fvmptnn04if 23043
            ​*11.7.3  The Cayley-Hamilton theorem   cpmadurid 23061
​PART 12  BASIC TOPOLOGY
      ​12.1  Topology
            ​*12.1.1  Topological spaces   ctop 23087
                  12.1.1.1  Topologies   ctop 23087
                  12.1.1.2  Topologies on sets   ctopon 23104
                  12.1.1.3  Topological spaces   ctps 23126
            12.1.2  Topological bases   ctb 23139
            12.1.3  Examples of topologies   distop 23189
            12.1.4  Closure and interior   ccld 23210
            12.1.5  Neighborhoods   cnei 23291
            12.1.6  Limit points and perfect sets   clp 23328
            12.1.7  Subspace topologies   restrcl 23351
            12.1.8  Order topology   ordtbaslem 23382
            12.1.9  Limits and continuity in topological spaces   ccn 23418
            12.1.10  Separated spaces: T0, T1, T2 (Hausdorff) ...   ct0 23500
            12.1.11  Compactness   ccmp 23580
            12.1.12  Bolzano-Weierstrass theorem   bwth 23604
            12.1.13  Connectedness   cconn 23605
            12.1.14  First- and second-countability   c1stc 23631
            12.1.15  Local topological properties   clly 23658
            12.1.16  Refinements   cref 23696
            12.1.17  Compactly generated spaces   ckgen 23727
            12.1.18  Product topologies   ctx 23754
            12.1.19  Continuous function-builders   cnmptid 23855
            12.1.20  Quotient maps and quotient topology   ckq 23887
            12.1.21  Homeomorphisms   chmeo 23947
      ​12.2  Filters and filter bases
            12.2.1  Filter bases   elmptrab 24021
            12.2.2  Filters   cfil 24039
            12.2.3  Ultrafilters   cufil 24093
            12.2.4  Filter limits   cfm 24127
            12.2.5  Extension by continuity   ccnext 24253
            12.2.6  Topological groups   ctmd 24264
            12.2.7  Infinite group sum on topological groups   ctsu 24320
            12.2.8  Topological rings, fields, vector spaces   ctrg 24350
      ​12.3  Uniform Structures and Spaces
            12.3.1  Uniform structures   cust 24394
            12.3.2  The topology induced by an uniform structure   cutop 24424
            12.3.3  Uniform Spaces   cuss 24447
            12.3.4  Uniform continuity   cucn 24468
            12.3.5  Cauchy filters in uniform spaces   ccfilu 24479
            12.3.6  Complete uniform spaces   ccusp 24490
      ​12.4  Metric spaces
            12.4.1  Pseudometric spaces   ispsmet 24498
            12.4.2  Basic metric space properties   cxms 24511
            12.4.3  Metric space balls   blfvalps 24577
            12.4.4  Open sets of a metric space   mopnval 24632
            12.4.5  Continuity in metric spaces   metcnp3 24734
            12.4.6  The uniform structure generated by a metric   metuval 24743
            12.4.7  Examples of metric spaces   dscmet 24766
            ​*12.4.8  Normed algebraic structures   cnm 24770
            12.4.9  Normed space homomorphisms (bounded linear operators)   cnmo 24899
            12.4.10  Topology on the reals   qtopbaslem 24952
            12.4.11  Topological definitions using the reals   cii 25071
            12.4.12  Path homotopy   chtpy 25163
            12.4.13  The fundamental group   cpco 25196
      ​12.5  Metric subcomplex vector spaces
            12.5.1  Subcomplex modules   cclm 25258
            ​*12.5.2  Subcomplex vector spaces   ccvs 25319
            ​*12.5.3  Normed subcomplex vector spaces   isncvsngp 25345
            12.5.4  Subcomplex pre-Hilbert spaces   ccph 25362
            12.5.5  Convergence and completeness   ccfil 25448
            12.5.6  Baire's Category Theorem   bcthlem1 25520
            12.5.7  Banach spaces and subcomplex Hilbert spaces   ccms 25528
                  12.5.7.1  The complete ordered field of the real numbers   retopn 25575
            12.5.8  Euclidean spaces   crrx 25579
            12.5.9  Minimizing Vector Theorem   minveclem1 25620
            12.5.10  Projection Theorem   pjthlem1 25633
​PART 13  BASIC REAL AND COMPLEX ANALYSIS
      ​13.1  Continuity
            13.1.1  Intermediate value theorem   pmltpclem1 25644
      ​13.2  Integrals
            13.2.1  Lebesgue measure   covol 25658
            13.2.2  Lebesgue integration   cmbf 25810
                  13.2.2.1  Lesbesgue integral   cmbf 25810
                  13.2.2.2  Lesbesgue directed integral   cdit 26042
      ​13.3  Derivatives
            13.3.1  Real and complex differentiation   climc 26058
                  13.3.1.1  Derivatives of functions of one complex or real variable   climc 26058
                  13.3.1.2  Results on real differentiation   dvferm1lem 26180
​PART 14  BASIC REAL AND COMPLEX FUNCTIONS
      ​14.1  Polynomials
            14.1.1  Polynomial degrees   cmdg 26247
            14.1.2  The division algorithm for univariate polynomials   cmn1 26320
            14.1.3  Elementary properties of complex polynomials   cply 26378
            14.1.4  The division algorithm for polynomials   cquot 26488
            14.1.5  Algebraic numbers   caa 26512
            14.1.6  Liouville's approximation theorem   aalioulem1 26532
      ​14.2  Sequences and series
            14.2.1  Taylor polynomials and Taylor's theorem   ctayl 26553
            14.2.2  Uniform convergence   culm 26576
            14.2.3  Power series   pserval 26610
      ​14.3  Basic trigonometry
            14.3.1  The exponential, sine, and cosine functions (cont.)   efcn 26643
            14.3.2  Properties of pi = 3.14159...   pilem1 26651
            14.3.3  Mapping of the exponential function   efgh 26743
            14.3.4  The natural logarithm on complex numbers   clog 26756
            ​*14.3.5  Logarithms to an arbitrary base   clogb 26966
            14.3.6  Theorems of Pythagoras, isosceles triangles, and intersecting chords   angval 27003
            14.3.7  Solutions of quadratic, cubic, and quartic equations   quad2 27041
            14.3.8  Inverse trigonometric functions   casin 27064
            14.3.9  The Birthday Problem   log2ublem1 27148
            14.3.10  Areas in R^2   carea 27157
            14.3.11  More miscellaneous converging sequences   rlimcnp 27167
            14.3.12  Inequality of arithmetic and geometric means   cvxcl 27186
            14.3.13  Euler-Mascheroni constant   cem 27193
            14.3.14  Zeta function   czeta 27214
            14.3.15  Gamma function   clgam 27217
      ​14.4  Basic number theory
            14.4.1  Wilson's theorem   wilthlem1 27269
            14.4.2  The Fundamental Theorem of Algebra   ftalem1 27274
            14.4.3  The Basel problem (ζ(2) = π²/6)   basellem1 27282
            14.4.4  Number-theoretical functions   ccht 27292
            14.4.5  Perfect Number Theorem   mersenne 27428
            14.4.6  Characters of Z/nZ   cdchr 27433
            14.4.7  Bertrand's postulate   bcctr 27476
            ​*14.4.8  Quadratic residues and the Legendre symbol   clgs 27495
            ​*14.4.9  Gauss' Lemma   gausslemma2dlem0a 27557
            14.4.10  Quadratic reciprocity   lgseisenlem1 27576
            14.4.11  All primes 4n+1 are the sum of two squares   2sqlem1 27618
            14.4.12  Chebyshev's Weak Prime Number Theorem, Dirichlet's Theorem   chebbnd1lem1 27670
            14.4.13  The Prime Number Theorem   mudivsum 27731
            14.4.14  Ostrowski's theorem   abvcxp 27816
​PART 15  SURREAL NUMBERS
      ​​*15.1  Sign sequence representation and Alling's axioms
            15.1.1  Definitions and initial properties   csur 27841
            15.1.2  Ordering   ltssolem1 27876
            15.1.3  Birthday Function   bdayfo 27878
            15.1.4  Density   fvnobday 27879
            ​*15.1.5  Full-Eta Property   bdayimaon 27894
      ​15.2  Initial consequences of Alling's axioms
            15.2.1  Ordering Theorems   cles 27945
            15.2.2  Birthday Theorems   bdayfun 27977
      ​​*15.3  Conway cut representation
            15.3.1  Conway cuts   cslts 27987
            15.3.2  Zero and One   c0s 28035
            15.3.3  Cuts and Options   cmade 28052
            15.3.4  Cofinality and coinitiality   cofslts 28148
      ​15.4  Induction and recursion
            15.4.1  Induction and recursion on one variable   cnorec 28167
            15.4.2  Induction and recursion on two variables   cnorec2 28178
      ​15.5  Surreal arithmetic
            15.5.1  Addition   cadds 28189
            15.5.2  Negation and Subtraction   cnegs 28249
            15.5.3  Multiplication   cmuls 28336
            15.5.4  Division   cdivs 28417
            15.5.5  Absolute value   cabss 28467
      ​15.6  Subsystems of surreals
            15.6.1  Ordinal numbers   cons 28481
            15.6.2  Surreal recursive sequences   cseqs 28513
            15.6.3  Natural numbers   cn0s 28542
            15.6.4  Integers   czs 28608
            15.6.5  Dyadic fractions   c2s 28640
            15.6.6  Real numbers   creno 28719
​​*PART 16  ELEMENTARY GEOMETRY
      ​16.1  Definition and Tarski's Axioms of Geometry
            16.1.1  Justification for the congruence notation   tgjustf 28779
      ​16.2  Tarskian Geometry
            16.2.1  Congruence   tgcgrcomimp 28783
            16.2.2  Betweenness   tgbtwntriv2 28793
            16.2.3  Dimension   tglowdim1 28806
            16.2.4  Betweenness and Congruence   tgifscgr 28814
            16.2.5  Congruence of a series of points   ccgrg 28816
            16.2.6  Motions   cismt 28838
            16.2.7  Colinearity   tglng 28852
            16.2.8  Connectivity of betweenness   tgbtwnconn1lem1 28878
            16.2.9  Less-than relation in geometric congruences   cleg 28888
            16.2.10  Rays   chlg 28906
            16.2.11  Lines   btwnlng1 28929
            16.2.12  Point inversions   cmir 28966
            16.2.13  Right angles   crag 29010
            16.2.14  Half-planes   islnopp 29057
            16.2.15  Planes   cplng 29092
            16.2.16  Midpoints and Line Mirroring   cmid 29118
            16.2.17  Congruence of angles   ccgra 29155
            16.2.18  Angle Comparisons   cinag 29189
            16.2.19  Congruence Theorems   tgsas1 29208
            16.2.20  Equilateral triangles   ceqlg 29219
            16.2.21  Parallel lines   cprlng 29223
      ​16.3  Properties of geometries
            16.3.1  Isomorphisms between geometries   f1otrgds 29255
      ​16.4  Geometry in Hilbert spaces
            16.4.1  Geometry in the complex plane   cchhllem 29273
            16.4.2  Geometry in Euclidean spaces   cee 29274
                  16.4.2.1  Definition of the Euclidean space   cee 29274
                  16.4.2.2  Tarski's axioms for geometry for the Euclidean space   axdimuniq 29300
                  16.4.2.3  EE^n fulfills Tarski's Axioms   ceeng 29364
​​*PART 17  GRAPH THEORY
      ​*17.1  Vertices and edges
            17.1.1  The edge function extractor for extensible structures   cedgf 29375
            ​*17.1.2  Vertices and indexed edges   cvtx 29383
                  17.1.2.1  Definitions and basic properties   cvtx 29383
                  17.1.2.2  The vertices and edges of a graph represented as ordered pair   opvtxval 29390
                  17.1.2.3  The vertices and edges of a graph represented as extensible structure   funvtxdmge2val 29398
                  17.1.2.4  Representations of graphs without edges   snstrvtxval 29424
                  17.1.2.5  Degenerated cases of representations of graphs   vtxval0 29426
            17.1.3  Edges as range of the edge function   cedg 29434
      ​​*17.2  Undirected graphs
            17.2.1  Undirected hypergraphs   cuhgr 29443
            17.2.2  Undirected pseudographs and multigraphs   cupgr 29467
            ​*17.2.3  Loop-free graphs   umgrislfupgrlem 29509
            17.2.4  Edges as subsets of vertices of graphs   uhgredgiedgb 29513
            ​*17.2.5  Undirected simple graphs   cuspgr 29535
            17.2.6  Examples for graphs   usgr0e 29623
            17.2.7  Subgraphs   csubgr 29654
            17.2.8  Finite undirected simple graphs   cfusgr 29703
            17.2.9  Neighbors, complete graphs and universal vertices   cnbgr 29719
                  17.2.9.1  Neighbors   cnbgr 29719
                  17.2.9.2  Universal vertices   cuvtx 29772
                  17.2.9.3  Complete graphs   ccplgr 29796
            17.2.10  Vertex degree   cvtxdg 29852
            ​*17.2.11  Regular graphs   crgr 29942
      ​​*17.3  Walks, paths and cycles
            *17.3.1  Walks   cewlks 29982
            17.3.2  Walks for loop-free graphs   lfgrwlkprop 30072
            17.3.3  Trails   ctrls 30075
            17.3.4  Paths and simple paths   cpths 30096
            17.3.5  Closed walks   cclwlks 30156
            17.3.6  Circuits and cycles   ccrcts 30170
            ​*17.3.7  Walks as words   cwwlks 30211
            17.3.8  Walks/paths of length 2 (as length 3 strings)   2wlkdlem1 30311
            17.3.9  Walks in regular graphs   rusgrnumwwlkl1 30357
            ​*17.3.10  Closed walks as words   cclwwlk 30369
                  17.3.10.1  Closed walks as words   cclwwlk 30369
                  17.3.10.2  Closed walks of a fixed length as words   cclwwlkn 30412
                  17.3.10.3  Closed walks on a vertex of a fixed length as words   cclwwlknon 30475
            17.3.11  Examples for walks, trails and paths   0ewlk 30502
            17.3.12  Connected graphs   cconngr 30574
      ​17.4  Eulerian paths and the Konigsberg Bridge problem
            ​*17.4.1  Eulerian paths   ceupth 30585
            ​*17.4.2  The Königsberg Bridge problem   konigsbergvtx 30634
      ​17.5  The Friendship Theorem
            17.5.1  Friendship graphs - basics   cfrgr 30646
            17.5.2  The friendship theorem for small graphs   frgr1v 30659
            17.5.3  Theorems according to Mertzios and Unger   2pthfrgrrn 30670
            ​*17.5.4  Huneke's Proof of the Friendship Theorem   frgrncvvdeqlem1 30687
​PART 18  GUIDES AND MISCELLANEA
      ​18.1  Guides (conventions, explanations, and examples)
            ​*18.1.1  Conventions   conventions 30788
            18.1.2  Natural deduction   natded 30791
            ​*18.1.3  Natural deduction examples   ex-natded5.2 30792
            18.1.4  Definitional examples   ex-or 30809
            18.1.5  Other examples   aevdemo 30848
      ​18.2  Humor
            18.2.1  April Fool's theorem   avril1 30851
      ​18.3  (Future - to be reviewed and classified)
            18.3.1  Planar incidence geometry   cplig 30863
            ​*18.3.2  Aliases kept to prevent broken links   dummylink 30876
​​*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 30878
            19.1.2  Abelian groups   cablo 30933
      ​19.2  Complex vector spaces
            19.2.1  Definition and basic properties   cvc 30947
            19.2.2  Examples of complex vector spaces   cnaddabloOLD 30970
      ​19.3  Normed complex vector spaces
            19.3.1  Definition and basic properties   cnv 30973
            19.3.2  Examples of normed complex vector spaces   cnnv 31066
            19.3.3  Induced metric of a normed complex vector space   imsval 31074
            19.3.4  Inner product   cdip 31089
            19.3.5  Subspaces   css 31110
      ​19.4  Operators on complex vector spaces
            19.4.1  Definitions and basic properties   clno 31129
      ​19.5  Inner product (pre-Hilbert) spaces
            19.5.1  Definition and basic properties   ccphlo 31201
            19.5.2  Examples of pre-Hilbert spaces   cncph 31208
            19.5.3  Properties of pre-Hilbert spaces   isph 31211
      ​19.6  Complex Banach spaces
            19.6.1  Definition and basic properties   ccbn 31251
            19.6.2  Examples of complex Banach spaces   cnbn 31258
            19.6.3  Uniform Boundedness Theorem   ubthlem1 31259
            19.6.4  Minimizing Vector Theorem   minvecolem1 31263
      ​19.7  Complex Hilbert spaces
            19.7.1  Definition and basic properties   chlo 31274
            19.7.2  Standard axioms for a complex Hilbert space   hlex 31287
            19.7.3  Examples of complex Hilbert spaces   cnchl 31305
            19.7.4  Hellinger-Toeplitz Theorem   htthlem 31306
​​*PART 20  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      ​20.1  Axiomatization of complex pre-Hilbert spaces
            20.1.1  Basic Hilbert space definitions   chba 31308
            20.1.2  Preliminary ZFC lemmas   df-hnorm 31357
            ​*20.1.3  Derive the Hilbert space axioms from ZFC set theory   axhilex-zf 31370
            ​*20.1.4  Introduce the vector space axioms for a Hilbert space   ax-hilex 31388
            20.1.5  Vector operations   hvmulex 31400
            20.1.6  Inner product postulates for a Hilbert space   ax-hfi 31468
      ​20.2  Inner product and norms
            20.2.1  Inner product   his5 31475
            20.2.2  Norms   dfhnorm2 31511
            20.2.3  Relate Hilbert space to normed complex vector spaces   hilablo 31549
            20.2.4  Bunjakovaskij-Cauchy-Schwarz inequality   bcsiALT 31568
      ​20.3  Cauchy sequences and completeness axiom
            20.3.1  Cauchy sequences and limits   hcau 31573
            20.3.2  Derivation of the completeness axiom from ZF set theory   hilmet 31583
            20.3.3  Completeness postulate for a Hilbert space   ax-hcompl 31591
            20.3.4  Relate Hilbert space to ZFC pre-Hilbert and Hilbert spaces   hhcms 31592
      ​20.4  Subspaces and projections
            20.4.1  Subspaces   df-sh 31596
            20.4.2  Closed subspaces   df-ch 31610
            20.4.3  Orthocomplements   df-oc 31641
            20.4.4  Subspace sum, span, lattice join, lattice supremum   df-shs 31697
            20.4.5  Projection theorem   pjhthlem1 31780
            20.4.6  Projectors   df-pjh 31784
      ​20.5  Properties of Hilbert subspaces
            20.5.1  Orthomodular law   omlsilem 31791
            20.5.2  Projectors (cont.)   pjhtheu2 31805
            20.5.3  Hilbert lattice operations   sh0le 31829
            20.5.4  Span (cont.) and one-dimensional subspaces   spansn0 31930
            20.5.5  Commutes relation for Hilbert lattice elements   df-cm 31972
            20.5.6  Foulis-Holland theorem   fh1 32007
            20.5.7  Quantum Logic Explorer axioms   qlax1i 32016
            20.5.8  Orthogonal subspaces   chscllem1 32026
            20.5.9  Orthoarguesian laws 5OA and 3OA   5oalem1 32043
            20.5.10  Projectors (cont.)   pjorthi 32058
            20.5.11  Mayet's equation E_3   mayete3i 32117
      ​20.6  Operators on Hilbert spaces
            ​*20.6.1  Operator sum, difference, and scalar multiplication   df-hosum 32119
            20.6.2  Zero and identity operators   df-h0op 32137
            20.6.3  Operations on Hilbert space operators   hoaddcl 32147
            20.6.4  Linear, continuous, bounded, Hermitian, unitary operators and norms   df-nmop 32228
            20.6.5  Linear and continuous functionals and norms   df-nmfn 32234
            20.6.6  Adjoint   df-adjh 32238
            20.6.7  Dirac bra-ket notation   df-bra 32239
            20.6.8  Positive operators   df-leop 32241
            20.6.9  Eigenvectors, eigenvalues, spectrum   df-eigvec 32242
            20.6.10  Theorems about operators and functionals   nmopval 32245
            20.6.11  Riesz lemma   riesz3i 32451
            20.6.12  Adjoints (cont.)   cnlnadjlem1 32456
            20.6.13  Quantum computation error bound theorem   unierri 32493
            20.6.14  Dirac bra-ket notation (cont.)   branmfn 32494
            20.6.15  Positive operators (cont.)   leopg 32511
            20.6.16  Projectors as operators   pjhmopi 32535
      ​20.7  States on a Hilbert lattice and Godowski's equation
            20.7.1  States on a Hilbert lattice   df-st 32600
            20.7.2  Godowski's equation   golem1 32660
      ​20.8  Cover relation, atoms, exchange axiom, and modular symmetry
            20.8.1  Covers relation; modular pairs   df-cv 32668
            20.8.2  Atoms   df-at 32727
            20.8.3  Superposition principle   superpos 32743
            20.8.4  Atoms, exchange and covering properties, atomicity   chcv1 32744
            20.8.5  Irreducibility   chirredlem1 32779
            20.8.6  Atoms (cont.)   atcvat3i 32785
            20.8.7  Modular symmetry   mdsymlem1 32792
​PART 21  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      ​21.1  Mathboxes for user contributions
            21.1.1  Mathbox guidelines   mathbox 32831
      ​21.2  Mathbox for Stefan Allan
      ​21.3  Mathbox for Thierry Arnoux
            21.3.1  Propositional Calculus - misc additions   ad11antr 32836
            21.3.2  Predicate Calculus   sbc2iedf 32849
                  21.3.2.1  Predicate Calculus - misc additions   sbc2iedf 32849
                  21.3.2.2  Restricted quantification - misc additions   ralcom4f 32851
                  21.3.2.3  Equality   eqtrb 32857
                  21.3.2.4  Double restricted existential uniqueness quantification   opsbc2ie 32859
                  21.3.2.5  Double restricted existential uniqueness quantification syntax   w2reu 32861
                  21.3.2.6  Substitution (without distinct variables) - misc additions   sbceqbidf 32870
                  21.3.2.7  Existential "at most one" - misc additions   mo5f 32872
                  21.3.2.8  Existential uniqueness - misc additions   reuxfrdf 32874
                  21.3.2.9  Restricted "at most one" - misc additions   rmoxfrd 32876
                  21.3.2.10  Restricted iota (description binder)   riotaeqbidva 32879
            21.3.3  General Set Theory   dmrab 32880
                  21.3.3.1  Class abstractions (a.k.a. class builders)   dmrab 32880
                  21.3.3.2  Image Sets   abrexdomjm 32890
                  21.3.3.3  Set relations and operations - misc additions   nelun 32896
                  21.3.3.4  Unordered pairs   elpreq 32911
                  21.3.3.5  Unordered triples   tpssg 32920
                  21.3.3.6  Conditional operator - misc additions   ifeqeqx 32925
                  21.3.3.7  Set union   uniinn0 32934
                  21.3.3.8  Indexed union - misc additions   cbviunf 32937
                  21.3.3.9  Indexed intersection - misc additions   iinabrex 32951
                  21.3.3.10  Disjointness - misc additions   disjnf 32952
            21.3.4  Relations and Functions   xpdisjres 32980
                  21.3.4.1  Relations - misc additions   xpdisjres 32980
                  21.3.4.2  Functions - misc additions   fconst7v 33002
                  21.3.4.3  Operations - misc additions   mpomptxf 33060
                  21.3.4.4  The mapping operation   elmaprd 33062
                  21.3.4.5  Support of a function   suppovss 33063
                  21.3.4.6  Explicit Functions with one or two points as a domain   cosnopne 33076
                  21.3.4.7  Isomorphisms - misc. additions   gtiso 33083
                  21.3.4.8  Disjointness (additional proof requiring functions)   disjdsct 33085
                  21.3.4.9  First and second members of an ordered pair - misc additions   df1stres 33086
                  21.3.4.10  Countable Sets   snct 33094
            21.3.5  Real and Complex Numbers   sgnval2 33117
                  21.3.5.1  Complex operations - misc. additions   creq0 33118
                  21.3.5.2  Ordering on reals - misc additions   lt2addrd 33132
                  21.3.5.3  Extended reals - misc additions   nn0mnfxrd 33133
                  21.3.5.4  Extended nonnegative integers - misc additions   xnn0gt0 33151
                  21.3.5.5  Real number intervals - misc additions   joiniooico 33156
                  21.3.5.6  Finite intervals of integers - misc additions   uzssico 33166
                  21.3.5.7  Half-open integer ranges - misc additions   iundisjfi 33178
                  21.3.5.8  The ` # ` (set size) function - misc additions   hashunif 33188
                  21.3.5.9  The greatest common divisor operator - misc. additions   elq2 33193
                  21.3.5.10  Integers   nn0split01 33199
                  21.3.5.11  Decimal numbers   dfdec100 33211
            21.3.6  Real and complex functions   sgnsgn 33212
                  21.3.6.1  Signum (sgn or sign) function - misc. additions   sgnsgn 33212
                  21.3.6.2  Integer powers - misc. additions   nexple 33214
                  21.3.6.3  Indicator Functions (continued)   indsumin 33218
            ​*21.3.7  Decimal expansion   cdp2 33227
                  ​*21.3.7.1  Decimal point   cdp 33244
                  21.3.7.2  Division in the extended real number system   cxdiv 33273
            21.3.8  Words over a set - misc additions   wrdres 33292
                  21.3.8.1  Splicing words (substring replacement)   splfv3 33309
                  21.3.8.2  Cyclic shift of words   1cshid 33310
            21.3.9  Extensible Structures   ressplusf 33314
                  21.3.9.1  Structure restriction operator   ressplusf 33314
                  21.3.9.2  Posets   ressprs 33317
                  21.3.9.3  Complete lattices   clatp0cl 33327
                  21.3.9.4  Order Theory   cmnt 33329
                  21.3.9.5  Extended reals Structure - misc additions   ax-xrssca 33355
                  21.3.9.6  The extended nonnegative real numbers commutative monoid   xrge00 33365
            21.3.10  Algebra   mndcld 33373
                  21.3.10.1  Monoids   mndcld 33373
                  21.3.10.2  Monoids Homomorphisms   abliso 33386
                  21.3.10.3  Groups - misc additions   grpidcld 33390
                  21.3.10.4  Abelian Groups - misc additions   ablcomd 33396
                  21.3.10.5  Finitely supported group sums - misc additions   gsumsubg 33397
                  21.3.10.6  Group or monoid sums over words   gsumwun 33427
                  21.3.10.7  Centralizers and centers - misc additions   cntzun 33430
                  21.3.10.8  The symmetric group   symgfcoeu 33433
                  21.3.10.9  Transpositions   pmtridf1o 33445
                  21.3.10.10  Permutation Signs   psgnid 33448
                  21.3.10.11  Permutation cycles   ctocyc 33457
                  21.3.10.12  The Alternating Group   evpmval 33496
                  21.3.10.13  Signum in an ordered monoid   csgns 33509
                  21.3.10.14  Fixed points   cfxp 33514
                  21.3.10.15  The Archimedean property for generic ordered algebraic structures   cinftm 33527
                  21.3.10.16  Semiring left modules   cslmd 33551
                  21.3.10.17  Simple groups   prmsimpcyc 33579
                  21.3.10.18  Rings - misc additions   ringrngd 33580
                  21.3.10.19  Subrings generated by a set   elrgspnlem1 33593
                  21.3.10.20  The zero ring   irrednzr 33601
                  21.3.10.21  Localization of rings   cerl 33604
                  21.3.10.22  Integral Domains   domnmuln0rd 33628
                  21.3.10.23  Euclidean Domains   ceuf 33642
                  21.3.10.24  Division Rings   rndrhmcl 33648
                  21.3.10.25  The field of rational numbers   qfld 33649
                  21.3.10.26  Subfields   subsdrg 33650
                  21.3.10.27  Field of fractions   cfrac 33654
                  21.3.10.28  Field extensions generated by a set   cfldgen 33662
                  21.3.10.29  Ring homomorphisms - misc additions   rhmdvd 33675
                  21.3.10.30  Scalar restriction operation   cresv 33677
                  21.3.10.31  The commutative ring of gaussian integers   gzcrng 33692
                  21.3.10.32  The archimedean ordered field of real numbers   cnfldfld 33693
                  21.3.10.33  The quotient map and quotient modules   qusker 33700
                  21.3.10.34  The ring of integers modulo ` N `   znfermltl 33712
                  21.3.10.35  Independent sets and families   islinds5 33713
                  21.3.10.36  Ring associates, ring units   dvdsruassoi 33728
                  ​*21.3.10.37  Subgroup sum / Sumset / Minkowski sum   elgrplsmsn 33734
                  21.3.10.38  The quotient map   quslsm 33745
                  21.3.10.39  Ideals   intlidl 33759
                  21.3.10.40  Maximal Ideals   cmxidl 33773
                  21.3.10.41  Local rings   drnglring 33813
                  21.3.10.42  The semiring of ideals of a ring   cidlsrg 33821
                  21.3.10.43  Prime Elements   rprmval 33837
                  21.3.10.44  Unique factorization domains   cufd 33859
                  21.3.10.45  The ring of integers   zringidom 33872
                  21.3.10.46  Associative Algebra   assaassd 33876
                  21.3.10.47  Univariate Polynomials   0ringmon1p 33878
                  21.3.10.48  Polynomial quotient and polynomial remainder   q1pdir 33924
                  21.3.10.49  Multivariate Polynomials   psrbasfsupp 33932
                  21.3.10.50  The ring of symmetric polynomials   csply 33976
                  21.3.10.51  The subring algebra   sra1r 34002
                  21.3.10.52  Division Ring Extensions   drgext0g 34011
                  21.3.10.53  Vector Spaces   lvecdimfi 34017
                  21.3.10.54  Vector Space Dimension   cldim 34020
            21.3.11  Field Extensions   cfldext 34059
                  21.3.11.1  Algebraic numbers   cirng 34104
                  21.3.11.2  Algebraic extensions   calgext 34116
                  21.3.11.3  Minimal polynomials   cminply 34120
                  21.3.11.4  Quadratic Field Extensions   rtelextdg2lem 34147
                  21.3.11.5  Towers of quadratic extentions   fldext2chn 34149
            ​*21.3.12  Constructible Numbers   cconstr 34150
                  21.3.12.1  Impossible constructions   2sqr3minply 34201
            21.3.13  Matrices   csmat 34214
                  21.3.13.1  Submatrices   csmat 34214
                  21.3.13.2  Matrix literals   clmat 34232
                  21.3.13.3  Laplace expansion of determinants   mdetpmtr1 34244
            21.3.14  Topology   ist0cld 34254
                  21.3.14.1  Open maps   txomap 34255
                  21.3.14.2  Topology of the unit circle   qtopt1 34256
                  21.3.14.3  Refinements   reff 34260
                  21.3.14.4  Open cover refinement property   ccref 34263
                  21.3.14.5  Lindelöf spaces   cldlf 34273
                  21.3.14.6  Paracompact spaces   cpcmp 34276
                  ​*21.3.14.7  Spectrum of a ring   crspec 34283
                  21.3.14.8  Pseudometrics   cmetid 34307
                  21.3.14.9  Continuity - misc additions   hauseqcn 34319
                  21.3.14.10  Topology of the closed unit interval   elunitge0 34320
                  21.3.14.11  Topology of ` ( RR X. RR ) `   unicls 34324
                  21.3.14.12  Order topology - misc. additions   cnvordtrestixx 34334
                  21.3.14.13  Continuity in topological spaces - misc. additions   mndpluscn 34347
                  21.3.14.14  Topology of the extended nonnegative real numbers ordered monoid   xrge0hmph 34353
                  21.3.14.15  Limits - misc additions   lmlim 34368
                  21.3.14.16  Univariate polynomials   pl1cn 34376
            21.3.15  Uniform Stuctures and Spaces   chcmp 34377
                  21.3.15.1  Hausdorff uniform completion   chcmp 34377
            21.3.16  Topology and algebraic structures   zringnm 34379
                  21.3.16.1  The norm on the ring of the integer numbers   zringnm 34379
                  21.3.16.2  Topological ` ZZ ` -modules   zlm0 34381
                  21.3.16.3  Canonical embedding of the field of the rational numbers into a division ring   cqqh 34391
                  21.3.16.4  Canonical embedding of the real numbers into a complete ordered field   crrh 34414
                  21.3.16.5  Embedding from the extended real numbers into a complete lattice   cxrh 34437
                  21.3.16.6  Canonical embeddings into the ordered field of the real numbers   zrhre 34440
                  ​*21.3.16.7  Topological Manifolds   cmntop 34443
                  21.3.16.8  Extended sum   cesum 34448
            21.3.17  Mixed Function/Constant operation   cofc 34516
            21.3.18  Abstract measure   csiga 34529
                  21.3.18.1  Sigma-Algebra   csiga 34529
                  21.3.18.2  Generated sigma-Algebra   csigagen 34559
                  ​*21.3.18.3  lambda and pi-Systems, Rings of Sets   ispisys 34573
                  21.3.18.4  The Borel algebra on the real numbers   cbrsiga 34602
                  21.3.18.5  Product Sigma-Algebra   csx 34609
                  21.3.18.6  Measures   cmeas 34616
                  21.3.18.7  The counting measure   cntmeas 34647
                  21.3.18.8  The Lebesgue measure - misc additions   voliune 34650
                  21.3.18.9  The Dirac delta measure   cdde 34653
                  21.3.18.10  The 'almost everywhere' relation   cae 34658
                  21.3.18.11  Measurable functions   cmbfm 34670
                  21.3.18.12  Borel Algebra on ` ( RR X. RR ) `   br2base 34690
                  ​*21.3.18.13  Caratheodory's extension theorem   coms 34712
            21.3.19  Integration   itgeq12dv 34747
                  21.3.19.1  Lebesgue integral - misc additions   itgeq12dv 34747
                  21.3.19.2  Bochner integral   citgm 34748
            21.3.20  Euler's partition theorem   oddpwdc 34775
            21.3.21  Sequences defined by strong recursion   csseq 34804
            21.3.22  Fibonacci Numbers   cfib 34817
            21.3.23  Probability   cprb 34828
                  21.3.23.1  Probability Theory   cprb 34828
                  21.3.23.2  Conditional Probabilities   ccprob 34852
                  21.3.23.3  Real-valued Random Variables   crrv 34861
                  21.3.23.4  Preimage set mapping operator   corvc 34877
                  21.3.23.5  Distribution Functions   orvcelval 34890
                  21.3.23.6  Cumulative Distribution Functions   orvclteel 34894
                  21.3.23.7  Probabilities - example   coinfliplem 34900
                  21.3.23.8  Bertrand's Ballot Problem   ballotlemoex 34907
            21.3.24  Signum (sgn or sign) function - misc. additions   fzssfzo 34960
                  21.3.24.1  Operations on words   ccatmulgnn0dir 34963
            21.3.25  Polynomials with real coefficients - misc additions   plyrecld 34967
            21.3.26  Descartes's rule of signs   signspval 34970
                  21.3.26.1  Sign changes in a word over real numbers   signspval 34970
                  21.3.26.2  Counting sign changes in a word over real numbers   signslema 34980
            21.3.27  Number Theory   iblidicc 35010
                  21.3.27.1  Representations of a number as sums of integers   crepr 35026
                  21.3.27.2  Vinogradov Trigonometric Sums and the Circle Method   cvts 35053
                  21.3.27.3  The Ternary Goldbach Conjecture: Final Statement   ax-hgt749 35062
            21.3.28  Elementary Geometry   cstrkg2d 35082
                  ​*21.3.28.1  Two-dimensional geometry   cstrkg2d 35082
                  21.3.28.2  Morley's Miracle   cgranbtwn 35087
                  21.3.28.3  Outer Five Segment (not used, no need to move to main)   cafs 35090
            ​*21.3.29  LeftPad Project   clpad 35095
      ​​*21.4  Mathbox for Jonathan Ben-Naim
            21.4.1  First-order logic and set theory   bnj170 35118
            21.4.2  Well founded induction and recursion   bnj110 35277
            21.4.3  The existence of a minimal element in certain classes   bnj69 35429
            21.4.4  Well-founded induction   bnj1204 35431
            21.4.5  Well-founded recursion, part 1 of 3   bnj60 35481
            21.4.6  Well-founded recursion, part 2 of 3   bnj1500 35487
            21.4.7  Well-founded recursion, part 3 of 3   bnj1522 35491
      ​21.5  Mathbox for BTernaryTau
            21.5.1  First-order logic   nfan1c 35492
                  21.5.1.1  Auxiliary axiom schemes   nfan1c 35492
            21.5.2  ZF set theory   inv2 35498
                  21.5.2.1  Finitism   prcinf 35549
                  21.5.2.2  Introduce ax-regs   ax-regs 35562
                  21.5.2.3  Derive ax-regs   axregs 35575
                  21.5.2.4  ZFC axioms with reduced distinct variable conditions   axsepg2 35576
                  21.5.2.5  Cardinality without the Axiom of Choice   ckard 35585
                  21.5.2.6  Global choice   gblacfnacd 35609
            21.5.3  Real and complex numbers   zltp1ne 35624
            21.5.4  Graph theory   lfuhgr 35630
                  21.5.4.1  Acyclic graphs   cacycgr 35654
      ​21.6  Mathbox for Mario Carneiro
            21.6.1  Predicate calculus with all distinct variables   ax-7d 35671
            21.6.2  Miscellaneous stuff   quartfull 35677
            21.6.3  Derangements and the Subfactorial   deranglem 35678
            21.6.4  The Erdős-Szekeres theorem   erdszelem1 35703
            21.6.5  The Kuratowski closure-complement theorem   kur14lem1 35718
            21.6.6  Retracts and sections   cretr 35729
            21.6.7  Path-connected and simply connected spaces   cpconn 35731
            21.6.8  Covering maps   ccvm 35767
            21.6.9  Normal numbers   snmlff 35841
            21.6.10  Godel-sets of formulas - part 1   cgoe 35845
            21.6.11  Godel-sets of formulas - part 2   cgon 35944
            21.6.12  Models of ZF   cgze 35958
            ​*21.6.13  Metamath formal systems   cmcn 35972
            21.6.14  Grammatical formal systems   cm0s 36097
            21.6.15  Models of formal systems   cmuv 36117
            21.6.16  Splitting fields   ccpms 36139
            21.6.17  p-adic number fields   czr 36159
      ​​*21.7  Mathbox for Filip Cernatescu
      ​21.8  Mathbox for Paul Chapman
            21.8.1  Real and complex numbers (cont.)   climuzcnv 36183
            21.8.2  Miscellaneous theorems   elfzm12 36187
      ​21.9  Mathbox for Hongxiu Chen
      ​21.10  Mathbox for Adrian Ducourtial
            21.10.1  Propositional calculus   currybi 36200
            21.10.2  Clone theory   ccloneop 36207
      ​21.11  Mathbox for Scott Fenton
            21.11.1  ZFC Axioms in primitive form   axextprim 36213
            21.11.2  Untangled classes   untelirr 36220
            21.11.3  Extra propositional calculus theorems   3jaodd 36227
            21.11.4  Misc. Useful Theorems   nepss 36230
            21.11.5  Properties of real and complex numbers   sqdivzi 36240
            21.11.6  Infinite products   iprodefisumlem 36252
            21.11.7  Factorial limits   faclimlem1 36255
            21.11.8  Greatest common divisor and divisibility   gcd32 36261
            21.11.9  Properties of relationships   dftr6 36263
            21.11.10  Properties of functions and mappings   funpsstri 36278
            21.11.11  Ordinal numbers   elpotr 36291
            21.11.12  Defined equality axioms   axextdfeq 36307
            21.11.13  Hypothesis builders   hbntg 36315
            21.11.14  Well-founded zero, successor, and limits   cwsuc 36320
            21.11.15  Quantifier-free definitions   ctxp 36340
            21.11.16  Alternate ordered pairs   caltop 36468
            21.11.17  Geometry in the Euclidean space   cofs 36494
                  21.11.17.1  Congruence properties   cofs 36494
                  21.11.17.2  Betweenness properties   btwntriv2 36524
                  21.11.17.3  Segment Transportation   ctransport 36541
                  21.11.17.4  Properties relating betweenness and congruence   cifs 36547
                  21.11.17.5  Connectivity of betweenness   btwnconn1lem1 36599
                  21.11.17.6  Segment less than or equal to   csegle 36618
                  21.11.17.7  Outside-of relationship   coutsideof 36631
                  21.11.17.8  Lines and Rays   cline2 36646
            21.11.18  Forward difference   cfwddif 36670
            21.11.19  Rank theorems   rankung 36678
            21.11.20  Hereditarily Finite Sets   chf 36684
            21.11.21  Natural ordinal operations   cnmul 36699
      ​21.12  Mathbox for Gino Giotto
            21.12.1  Equality theorems   rmoeqi 36739
                  21.12.1.1  Inference versions   rmoeqi 36739
                  21.12.1.2  Deduction versions   rmoeqdv 36764
            21.12.2  Change bound variables   in-ax8 36776
                  21.12.2.1  Change bound variables and domains   cbvralvw2 36778
                  21.12.2.2  Change bound variables, deduction versions   cbvmodavw 36802
                  21.12.2.3  Change bound variables and domains, deduction versions   cbvrmodavw2 36835
            21.12.3  Study of ax-mulf usage   mpomulnzcnf 36851
      ​21.13  Mathbox for Jeff Hankins
            21.13.1  Miscellany   a1i14 36852
            21.13.2  Basic topological facts   topbnd 36875
            21.13.3  Topology of the real numbers   ivthALT 36886
            21.13.4  Refinements   cfne 36887
            21.13.5  Neighborhood bases determine topologies   neibastop1 36910
            21.13.6  Lattice structure of topologies   topmtcl 36914
            21.13.7  Filter bases   fgmin 36921
            21.13.8  Directed sets, nets   tailfval 36923
      ​21.14  Mathbox for Anthony Hart
            21.14.1  Propositional Calculus   tb-ax1 36934
            21.14.2  Predicate Calculus   nalfal 36954
            21.14.3  Miscellaneous single axioms   meran1 36962
            21.14.4  Connective Symmetry   negsym1 36968
      ​21.15  Mathbox for Chen-Pang He
            21.15.1  Ordinal topology   ontopbas 36979
      ​21.16  Mathbox for Jeff Hoffman
            21.16.1  Inferences for finite induction on generic function values   fveleq 37002
            21.16.2  gdc.mm   nnssi2 37006
      ​21.17  Mathbox for Matthew House
            21.17.1  Relations on well-ordered indexed unions   weiunval 37013
            21.17.2  Axiom of Transitive Containment   axtco 37022
            21.17.3  Transitive closure of a class   tr0elw 37035
            ​*21.17.4  Stronger axioms of regularity   mh-setind 37087
            21.17.5  Short axioms written in primitive symbols   mh-inf3f1 37092
      ​21.18  Mathbox for Asger C. Ipsen
            21.18.1  Continuous nowhere differentiable functions   dnival 37100
      ​​*21.19  Mathbox for BJ
            *21.19.1  Propositional calculus   bj-mp2c 37169
                  *21.19.1.1  Derived rules of inference   bj-mp2c 37169
                  ​*21.19.1.2  A syntactic theorem   bj-0 37171
                  ​*21.19.1.3  Minimal implicational calculus   bj-poni 37173
                  ​*21.19.1.4  Positive calculus   bj-bisimpl 37185
                  ​*21.19.1.5  Implication and negation   bj-con2com 37193
                  ​*21.19.1.6  Disjunction   bj-jaoi1 37204
                  ​*21.19.1.7  Logical equivalence   bj-dfbi4 37206
                  21.19.1.8  The conditional operator for propositions   bj-consensus 37211
                  ​*21.19.1.9  Propositional calculus: miscellaneous   bj-imbi12 37216
            ​*21.19.2  Modal logic   bj-axdd2 37225
            ​*21.19.3  Provability logic   cprvb 37230
            ​*21.19.4  First-order logic   bj-exexalal 37239
                  21.19.4.1  Universal and existential quantifiers, nonfreeness predicate   bj-exexalal 37239
                  21.19.4.2  Adding ax-gen   bj-genr 37240
                  21.19.4.3  Adding ax-4   bj-almp 37244
                  21.19.4.4  Adding ax-5   bj-spvw 37297
                  21.19.4.5  Equality and substitution   bj-df-sb 37312
                  21.19.4.6  Adding ax-6   bj-spim0 37331
                  21.19.4.7  Adding ax-7   bj-cbvexw 37339
                  21.19.4.8  Membership predicate, ax-8 and ax-9   bj-ax89 37341
                  21.19.4.9  Adding ax-11   bj-alcomexcom 37343
                  21.19.4.10  Adding ax-12   axc11n11 37347
                  ​*21.19.4.11  Really adding ax-12   bj-substax12 37389
                  21.19.4.12  Nonfreeness   wnnf 37391
                  21.19.4.13  Adding ax-13   bj-axc10 37458
                  ​*21.19.4.14  Removing dependencies on ax-13 (and ax-11)   bj-axc10v 37468
                  ​*21.19.4.15  Distinct var metavariables   bj-hbaeb2 37493
                  ​*21.19.4.16  Around ~ equsal   bj-equsal1t 37497
                  ​*21.19.4.17  Some Principia Mathematica proofs   stdpc5t 37502
                  21.19.4.18  Alternate definition of substitution   bj-sbsb 37512
                  21.19.4.19  Lemmas for substitution   bj-sbf3 37514
                  21.19.4.20  Existential uniqueness   bj-eu3f 37516
                  ​*21.19.4.21  First-order logic: miscellaneous   bj-sblem1 37517
            21.19.5  Set theory   eliminable1 37534
                  ​*21.19.5.1  Eliminability of class terms   eliminable1 37534
                  ​*21.19.5.2  Classes without the axiom of extensionality   bj-denoteslem 37546
                  21.19.5.3  Characterization among sets versus among classes   elelb 37572
                  ​*21.19.5.4  The nonfreeness quantifier for classes   bj-nfcsym 37574
                  ​*21.19.5.5  Lemmas for class substitution   bj-sbeqALT 37575
                  21.19.5.6  Removing some axiom requirements and disjoint variable conditions   bj-exlimvmpi 37586
                  ​*21.19.5.7  Class abstractions   bj-elabd2ALT 37601
                  21.19.5.8  Generalized class abstractions   bj-cgab 37609
                  ​*21.19.5.9  Restricted nonfreeness   wrnf 37617
                  ​*21.19.5.10  Russell's paradox   bj-ru1 37619
                  21.19.5.11  Curry's paradox in set theory   currysetlem 37621
                  ​*21.19.5.12  Some disjointness results   bj-n0i 37627
                  ​*21.19.5.13  Complements on direct products   bj-xpimasn 37631
                  ​*21.19.5.14  "Singletonization" and tagging   bj-snsetex 37639
                  ​*21.19.5.15  Tuples of classes   bj-cproj 37666
                  ​*21.19.5.16  Set theory: elementary operations relative to a universe   bj-rcleqf 37701
                  ​*21.19.5.17  Axioms for finite unions   bj-abex 37706
                  ​*21.19.5.18  Set theory: miscellaneous   eleq2w2ALT 37723
                  ​*21.19.5.19  Axioms of separation and replacement   bj-axnul 37749
                  ​*21.19.5.20  Evaluation at a class   bj-evaleq 37753
                  21.19.5.21  Elementwise operations   celwise 37761
                  ​*21.19.5.22  Elementwise intersection (families of sets induced on a subset)   bj-rest00 37763
                  21.19.5.23  Moore collections (complements)   bj-raldifsn 37782
                  21.19.5.24  Maps-to notation for functions with three arguments   bj-0nelmpt 37798
                  ​*21.19.5.25  Currying   csethom 37804
                  ​*21.19.5.26  Setting components of extensible structures   cstrset 37816
            ​*21.19.6  Extended real and complex numbers, real and complex projective lines   bj-nfald 37819
                  21.19.6.1  Complements on class abstractions of ordered pairs and binary relations   bj-nfald 37819
                  ​*21.19.6.2  Identity relation (complements)   bj-opabssvv 37834
                  ​*21.19.6.3  Functionalized identity (diagonal in a Cartesian square)   cdiag2 37856
                  ​*21.19.6.4  Direct image and inverse image   cimdir 37862
                  ​*21.19.6.5  Extended numbers and projective lines as sets   cfractemp 37880
                  ​*21.19.6.6  Addition and opposite   caddcc 37921
                  ​*21.19.6.7  Order relation on the extended reals   cltxr 37925
                  ​*21.19.6.8  Argument, multiplication and inverse   carg 37927
                  21.19.6.9  The canonical bijection from the finite ordinals   ciomnn 37933
                  21.19.6.10  Divisibility   cnnbar 37944
            ​*21.19.7  Monoids   bj-smgrpssmgm 37952
                  ​*21.19.7.1  Finite sums in monoids   cfinsum 37967
            ​*21.19.8  Affine, Euclidean, and Cartesian geometry   bj-fvimacnv0 37970
                  *21.19.8.1  Real vector spaces   bj-fvimacnv0 37970
                  ​*21.19.8.2  Complex numbers (supplements)   bj-subcom 37992
                  ​*21.19.8.3  Barycentric coordinates   bj-bary1lem 37994
            21.19.9  Monoid of endomorphisms   cend 37997
      ​21.20  Mathbox for Jim Kingdon
            21.20.1  Circle constant   taupilem3 38003
            21.20.2  Number theory   dfgcd3 38008
            21.20.3  Real numbers   irrdifflemf 38009
      ​21.21  Mathbox for ML
            21.21.1  Miscellaneous   csbrecsg 38014
            21.21.2  Cartesian exponentiation   cfinxp 38069
            21.21.3  Topology   iunctb2 38089
                  ​*21.21.3.1  Pi-base theorems   pibp16 38099
      ​21.22  Mathbox for Wolf Lammen
            21.22.1  1. Bootstrapping   wl-section-boot 38108
            21.22.2  Implication chains   wl-section-impchain 38132
            21.22.3  Theorems around the conditional operator   wl-ifp-ncond1 38150
            21.22.4  Alternative development of hadd, cadd   wl-df-3xor 38154
            21.22.5  An alternative axiom ~ ax-13   ax-wl-13v 38179
            21.22.6  Bootstrapping set theory with classes   wl-cleq-0 38181
            21.22.7  Other stuff   wl-mps 38202
      ​21.23  Mathbox for Brendan Leahy
      ​21.24  Mathbox for Jeff Madsen
            21.24.1  Logic and set theory   unirep 38405
            21.24.2  Real and complex numbers; integers   filbcmb 38431
            21.24.3  Sequences and sums   sdclem2 38433
            21.24.4  Topology   subspopn 38443
            21.24.5  Metric spaces   metf1o 38446
            21.24.6  Continuous maps and homeomorphisms   constcncf 38453
            21.24.7  Boundedness   ctotbnd 38457
            21.24.8  Isometries   cismty 38489
            21.24.9  Heine-Borel Theorem   heibor1lem 38500
            21.24.10  Banach Fixed Point Theorem   bfplem1 38513
            21.24.11  Euclidean space   crrn 38516
            21.24.12  Intervals (continued)   ismrer1 38529
            ​*21.24.13  Operation properties   cass 38533
            21.24.14  Groups and related structures   cmagm 38539
            21.24.15  Group homomorphism and isomorphism   cghomOLD 38574
            21.24.16  Rings   crngo 38585
            21.24.17  Division Rings   cdrng 38639
            21.24.18  Ring homomorphisms   crngohom 38651
            21.24.19  Commutative rings   ccm2 38680
            21.24.20  Ideals   cidl 38698
            21.24.21  Prime rings and integral domains   cprrng 38737
            21.24.22  Ideal generators   cigen 38750
      ​21.25  Mathbox for Giovanni Mascellani
            ​*21.25.1  Tools for automatic proof building   efald2 38769
            ​*21.25.2  Tseitin axioms   fald 38818
            ​*21.25.3  Equality deductions   iuneq2f 38845
            ​*21.25.4  Miscellanea   orcomdd 38856
      ​21.26  Mathbox for Peter Mazsa
            21.26.1  Notations   cxrn 38863
            21.26.2  Preparatory theorems   el2v1 38918
            21.26.3  Range Cartesian product   df-xrn 39069
            21.26.4  Relations   df-rels 39129
            21.26.5  Quotient map (coset map)   df-qmap 39135
            21.26.6  Lifts, shifts, successor, and predecessor   df-adjliftmap 39144
            21.26.7  Cosets by ` R `   df-coss 39190
            21.26.8  Subset relations   df-ssr 39267
            21.26.9  Reflexivity   df-refs 39279
            21.26.10  Converse reflexivity   df-cnvrefs 39294
            21.26.11  Symmetry   df-syms 39311
            21.26.12  Reflexivity and symmetry   symrefref2 39336
            21.26.13  Transitivity   df-trs 39345
            21.26.14  Equivalence relations   df-eqvrels 39357
            21.26.15  Redundancy   df-redunds 39396
            21.26.16  Domain quotients   df-dmqss 39411
            21.26.17  Equivalence relations on domain quotients   df-ers 39437
            21.26.18  Functions   df-funss 39454
            21.26.19  Disjoints vs. converse functions   df-disjss 39477
            21.26.20  Antisymmetry   df-antisymrel 39552
            21.26.21  Partitions: disjoints on domain quotients   df-parts 39557
            21.26.22  Partition-Equivalence Theorems   disjim 39573
            21.26.23  Type-safe Partition-Equivalence: PetParts, PetErs, Pet2Parts, Pet2Ers   df-petparts 39657
      ​21.27  Mathbox for Rodolfo Medina
            21.27.1  Partitions   prtlem60 39667
      ​​*21.28  Mathbox for Norm Megill
            *21.28.1  Obsolete schemes ax-c4,c5,c7,c10,c11,c11n,c15,c9,c14,c16   ax-c5 39697
            ​*21.28.2  Rederive new axioms ax-4, ax-10, ax-6, ax-12, ax-13 from old   axc5 39707
            ​*21.28.3  Legacy theorems using obsolete axioms   ax5ALT 39721
            21.28.4  Experiments with weak deduction theorem   elimhyps 39775
            21.28.5  Miscellanea   cnaddcom 39786
            21.28.6  Atoms, hyperplanes, and covering in a left vector space (or module)   clsa 39788
            21.28.7  Functionals and kernels of a left vector space (or module)   clfn 39871
            21.28.8  Opposite rings and dual vector spaces   cld 39937
            21.28.9  Ortholattices and orthomodular lattices   cops 39986
            21.28.10  Atomic lattices with covering property   ccvr 40076
            21.28.11  Hilbert lattices   chlt 40164
            21.28.12  Projective geometries based on Hilbert lattices   clln 40305
            21.28.13  Construction of a vector space from a Hilbert lattice   cdlema1N 40605
            21.28.14  Construction of involution and inner product from a Hilbert lattice   clpoN 42294
      ​21.29  Mathbox for metakunt
            21.29.1  Commutative Semiring   ccsrg 42776
            21.29.2  General helpful statements   rhmzrhval 42779
            21.29.3  Some gcd and lcm results   12gcd5e1 42810
            21.29.4  Least common multiple inequality theorem   3factsumint1 42828
            21.29.5  Logarithm inequalities   3exp7 42860
            21.29.6  Miscellaneous results for AKS formalisation   intlewftc 42868
            21.29.7  Sticks and stones   sticksstones1 42953
            21.29.8  Continuation AKS   aks6d1c6lem1 42977
      ​21.30  Mathbox for Luke Murphy
            21.30.1  Solutions of quadratic equations   quadfac 43012
            21.30.2  April Fool's theorem   25or6to4 43013
      ​21.31  Mathbox for Steven Nguyen
            21.31.1  Utility theorems   jarrii 43014
            ​*21.31.2  Arithmetic theorems   c0exALT 43060
            21.31.3  Exponents and divisibility   oexpreposd 43123
            21.31.4  Trigonometry and Calculus   tanhalfpim 43150
            ​*21.31.5  Independence of ax-mulcom   cresub 43166
            21.31.6  Structures   sn-base0 43309
            ​*21.31.7  Projective spaces   cprjsp 43373
            21.31.8  Basic reductions for Fermat's Last Theorem   dffltz 43406
            ​*21.31.9  Exemplar theorems   iddii 43436
                  ​*21.31.9.1  Standard replacements of ax-10 , ax-11 , ax-12   nfa1w 43447
      ​21.32  Mathbox for Igor Ieskov
      ​21.33  Mathbox for OpenAI
      ​21.34  Mathbox for Stefan O'Rear
            21.34.1  Additional elementary logic and set theory   moxfr 43463
            21.34.2  Additional theory of functions   imaiinfv 43464
            21.34.3  Additional topology   elrfi 43465
            21.34.4  Characterization of closure operators. Kuratowski closure axioms   ismrcd1 43469
            21.34.5  Algebraic closure systems   cnacs 43473
            21.34.6  Miscellanea 1. Map utilities   constmap 43484
            21.34.7  Miscellanea for polynomials   mptfcl 43491
            21.34.8  Multivariate polynomials over the integers   cmzpcl 43492
            21.34.9  Miscellanea for Diophantine sets 1   coeq0i 43524
            21.34.10  Diophantine sets 1: definitions   cdioph 43526
            21.34.11  Diophantine sets 2 miscellanea   ellz1 43538
            21.34.12  Diophantine sets 2: union and intersection. Monotone Boolean algebra   diophin 43543
            21.34.13  Diophantine sets 3: construction   diophrex 43546
            21.34.14  Diophantine sets 4 miscellanea   2sbcrex 43555
            21.34.15  Diophantine sets 4: Quantification   rexrabdioph 43561
            21.34.16  Diophantine sets 5: Arithmetic sets   rabdiophlem1 43568
            21.34.17  Diophantine sets 6: reusability. renumbering of variables   eldioph4b 43578
            21.34.18  Pigeonhole Principle and cardinality helpers   fphpd 43583
            21.34.19  A non-closed set of reals is infinite   rencldnfilem 43587
            21.34.20  Lagrange's rational approximation theorem   irrapxlem1 43589
            21.34.21  Pell equations 1: A nontrivial solution always exists   pellexlem1 43596
            21.34.22  Pell equations 2: Algebraic number theory of the solution set   csquarenn 43603
            21.34.23  Pell equations 3: characterizing fundamental solution   infmrgelbi 43645
            ​*21.34.24  Logarithm laws generalized to an arbitrary base   reglogcl 43657
            21.34.25  Pell equations 4: the positive solution group is infinite cyclic   pellfund14 43665
            21.34.26  X and Y sequences 1: Definition and recurrence laws   crmx 43667
            21.34.27  Ordering and induction lemmas for the integers   monotuz 43708
            21.34.28  X and Y sequences 2: Order properties   rmxypos 43714
            21.34.29  Congruential equations   congtr 43732
            21.34.30  Alternating congruential equations   acongid 43742
            21.34.31  Additional theorems on integer divisibility   coprmdvdsb 43752
            21.34.32  X and Y sequences 3: Divisibility properties   jm2.18 43755
            21.34.33  X and Y sequences 4: Diophantine representability of Y   jm2.27a 43772
            21.34.34  X and Y sequences 5: Diophantine representability of X, ^, _C   rmxdiophlem 43782
            21.34.35  Uncategorized stuff not associated with a major project   setindtr 43791
            21.34.36  More equivalents of the Axiom of Choice   axac10 43800
            21.34.37  Finitely generated left modules   clfig 43834
            21.34.38  Noetherian left modules I   clnm 43842
            21.34.39  Addenda for structure powers   pwssplit4 43856
            21.34.40  Every set admits a group structure iff choice   unxpwdom3 43862
            21.34.41  Noetherian rings and left modules II   clnr 43876
            21.34.42  Hilbert's Basis Theorem   cldgis 43888
            21.34.43  Additional material on polynomials [DEPRECATED]   cmnc 43898
            21.34.44  Degree and minimal polynomial of algebraic numbers   cdgraa 43907
            21.34.45  Algebraic integers I   citgo 43924
            21.34.46  Endomorphism algebra   cmend 43938
            21.34.47  Cyclic groups and order   idomodle 43958
            21.34.48  Cyclotomic polynomials   ccytp 43964
            21.34.49  Miscellaneous topology   fgraphopab 43970
      ​21.35  Mathbox for Noam Pasman
      ​21.36  Mathbox for Jon Pennant
      ​21.37  Mathbox for Richard Penner
            21.37.1  Set Theory and Ordinal Numbers   uniel 43984
            21.37.2  Natural addition of Cantor normal forms   oawordex2 44093
            21.37.3  Surreal Contributions   abeqabi 44174
            21.37.4  Short Studies   nlimsuc 44207
                  21.37.4.1  Additional work on conditional logical operator   ifpan123g 44225
                  21.37.4.2  Sophisms   rp-fakeimass 44278
                  ​*21.37.4.3  Finite Sets   rp-isfinite5 44283
                  21.37.4.4  General Observations   intabssd 44285
                  21.37.4.5  Infinite Sets   pwelg 44326
                  ​*21.37.4.6  Finite intersection property   fipjust 44331
                  21.37.4.7  RP ADDTO: Subclasses and subsets   rababg 44340
                  21.37.4.8  RP ADDTO: The intersection of a class   elinintab 44341
                  21.37.4.9  RP ADDTO: Theorems requiring subset and intersection existence   elinintrab 44343
                  21.37.4.10  RP ADDTO: Relations   xpinintabd 44346
                  ​*21.37.4.11  RP ADDTO: Functions   elmapintab 44362
                  ​*21.37.4.12  RP ADDTO: Finite induction (for finite ordinals)   cnvcnvintabd 44366
                  21.37.4.13  RP ADDTO: First and second members of an ordered pair   elcnvlem 44367
                  21.37.4.14  RP ADDTO: The reflexive and transitive properties of relations   undmrnresiss 44370
                  21.37.4.15  RP ADDTO: Basic properties of closures   cleq2lem 44374
                  21.37.4.16  RP REPLACE: Definitions and basic properties of transitive closures   trcleq2lemRP 44396
                  ​*21.37.4.17  Additions for square root; absolute value   sqrtcvallem1 44397
            21.37.5  Additional statements on relations and subclasses   al3im 44413
                  21.37.5.1  Transitive relations (not to be confused with transitive classes)   trrelind 44431
                  21.37.5.2  Reflexive closures   crcl 44438
                  ​*21.37.5.3  Finite relationship composition   relexp2 44443
                  21.37.5.4  Transitive closure of a relation   dftrcl3 44486
                  ​*21.37.5.5  Adapted from Frege   frege77d 44512
            ​*21.37.6  Propositions from _Begriffsschrift_   dfxor4 44532
                  *21.37.6.1  _Begriffsschrift_ Chapter I   dfxor4 44532
                  ​*21.37.6.2  _Begriffsschrift_ Notation hints   whe 44538
                  21.37.6.3  _Begriffsschrift_ Chapter II Implication   ax-frege1 44556
                  21.37.6.4  _Begriffsschrift_ Chapter II Implication and Negation   axfrege28 44595
                  ​*21.37.6.5  _Begriffsschrift_ Chapter II with logical equivalence   axfrege52a 44622
                  21.37.6.6  _Begriffsschrift_ Chapter II with equivalence of sets   axfrege52c 44653
                  ​*21.37.6.7  _Begriffsschrift_ Chapter II with equivalence of classes   frege53c 44680
                  ​*21.37.6.8  _Begriffsschrift_ Chapter III Properties hereditary in a sequence   dffrege69 44698
                  ​*21.37.6.9  _Begriffsschrift_ Chapter III Following in a sequence   dffrege76 44705
                  ​*21.37.6.10  _Begriffsschrift_ Chapter III Member of sequence   dffrege99 44728
                  ​*21.37.6.11  _Begriffsschrift_ Chapter III Single-valued procedures   dffrege115 44744
            ​*21.37.7  Exploring Topology via Seifert and Threlfall   enrelmap 44763
                  *21.37.7.1  Equinumerosity of sets of relations and maps   enrelmap 44763
                  ​*21.37.7.2  Generic Pseudoclosure Spaces, Pseudointerior Spaces, and Pseudoneighborhoods   or3or 44789
                  ​*21.37.7.3  Generic Neighborhood Spaces   gneispa 44896
            ​*21.37.8  Exploring Higher Homotopy via Kerodon   k0004lem1 44913
                  *21.37.8.1  Simplicial Sets   k0004lem1 44913
      ​21.38  Mathbox for Stanislas Polu
            21.38.1  IMO Problems   wwlemuld 44922
                  21.38.1.1  IMO 1972 B2   wwlemuld 44922
            ​*21.38.2  INT Inequalities Proof Generator   int-addcomd 44939
            ​*21.38.3  N-Digit Addition Proof Generator   unitadd 44961
            21.38.4  AM-GM (for k = 2,3,4)   gsumws3 44962
      ​21.39  Mathbox for Rohan Ridenour
            21.39.1  Misc   spALT 44967
            21.39.2  Monoid rings   cmnring 44975
            21.39.3  Shorter primitive equivalent of ax-groth   gru0eld 44993
                  21.39.3.1  Grothendieck universes are closed under collection   gru0eld 44993
                  21.39.3.2  Minimal universes   ismnu 45011
                  21.39.3.3  Primitive equivalent of ax-groth   expandan 45038
      ​21.40  Mathbox for Steve Rodriguez
            21.40.1  Miscellanea   nanorxor 45055
            21.40.2  Ratio test for infinite series convergence and divergence   dvgrat 45062
            21.40.3  Multiples   reldvds 45065
            21.40.4  Function operations   caofcan 45073
            21.40.5  Calculus   lhe4.4ex1a 45079
            21.40.6  The generalized binomial coefficient operation   cbcc 45086
            21.40.7  Binomial series   uzmptshftfval 45096
      ​21.41  Mathbox for Andrew Salmon
            21.41.1  Principia Mathematica * 10   pm10.12 45108
            21.41.2  Principia Mathematica * 11   2alanimi 45122
            21.41.3  Predicate Calculus   sbeqal1 45148
            21.41.4  Principia Mathematica * 13 and * 14   pm13.13a 45157
            21.41.5  Set Theory   elnev 45187
            21.41.6  Arithmetic   addcomgi 45204
            21.41.7  Geometry   cplusr 45205
      ​​*21.42  Mathbox for Alan Sare
            21.42.1  Auxiliary theorems for the Virtual Deduction tool   idiALT 45227
            21.42.2  Supplementary unification deductions   bi1imp 45231
            21.42.3  Conventional Metamath proofs, some derived from VD proofs   iidn3 45250
            21.42.4  What is Virtual Deduction?   wvd1 45318
            21.42.5  Virtual Deduction Theorems   df-vd1 45319
            21.42.6  Theorems proved using Virtual Deduction   trsspwALT 45566
            21.42.7  Theorems proved using Virtual Deduction with mmj2 assistance   simplbi2VD 45594
            21.42.8  Virtual Deduction transcriptions of textbook proofs   sb5ALTVD 45661
            21.42.9  Theorems proved using conjunction-form Virtual Deduction   elpwgdedVD 45665
            21.42.10  Theorems with a VD proof in conventional notation derived from a VD proof   suctrALT3 45672
            ​*21.42.11  Theorems with a proof in conventional notation derived from a VD proof   notnotrALT2 45675
      ​21.43  Mathbox for Eric Schmidt
            21.43.1  Miscellany   rspesbcd 45686
            21.43.2  Study of dfbi1ALT   dfbi1ALTa 45688
            21.43.3  Relation-preserving functions   wrelp 45691
            21.43.4  Orbits   orbitex 45704
            21.43.5  Well-founded sets   trwf 45708
            21.43.6  Absoluteness in transitive models   ralabso 45717
            21.43.7  Lemmas for showing axioms hold in models   traxext 45726
            21.43.8  The class of well-founded sets is a model for ZFC   wfaxext 45742
            21.43.9  Permutation models   brpermmodel 45752
            21.43.10  Isomorphism of finite ordinals and non-negative integers   hashnna 45768
      ​21.44  Mathbox for Glauco Siliprandi
            21.44.1  Miscellanea   evth2f 45775
            21.44.2  Functions   fnresdmss 45926
            21.44.3  Ordering on real numbers - Real and complex numbers basic operations   sub2times 46032
            21.44.4  Real intervals   gtnelioc 46247
            21.44.5  Finite sums   fsummulc1f 46327
            21.44.6  Finite multiplication of numbers and finite multiplication of functions   fmul01 46336
            21.44.7  Limits   clim1fr1 46357
                  21.44.7.1  Inferior limit (lim inf)   clsi 46505
                  ​*21.44.7.2  Limits for sequences of extended real numbers   clsxlim 46572
            21.44.8  Trigonometry   coseq0 46618
            21.44.9  Continuous Functions   mulcncff 46624
            21.44.10  Derivatives   dvsinexp 46665
            21.44.11  Integrals   itgsin0pilem1 46704
            21.44.12  Stone Weierstrass theorem - real version   stoweidlem1 46755
            21.44.13  Wallis' product for π   wallispilem1 46819
            21.44.14  Stirling's approximation formula for ` n ` factorial   stirlinglem1 46828
            21.44.15  Dirichlet kernel   dirkerval 46845
            21.44.16  Fourier Series   fourierdlem1 46862
            21.44.17  e is transcendental   elaa2lem 46987
            21.44.18  n-dimensional Euclidean space   rrxtopn 47038
            21.44.19  Basic measure theory   csalg 47062
                  ​*21.44.19.1  σ-Algebras   csalg 47062
                  21.44.19.2  Sum of nonnegative extended reals   csumge0 47116
                  ​*21.44.19.3  Measures   cmea 47203
                  ​*21.44.19.4  Outer measures and Caratheodory's construction   come 47243
                  ​*21.44.19.5  Lebesgue measure on n-dimensional Real numbers   covoln 47290
                  ​*21.44.19.6  Measurable functions   csmblfn 47449
      ​21.45  Mathbox for Saveliy Skresanov
            21.45.1  Ceva's theorem   sigarval 47604
            21.45.2  Simple groups   simpcntrab 47624
      ​21.46  Mathbox for Ender Ting
            21.46.1  Interesting facts   et-ltneverrefl 47625
            21.46.2  Increasing sequences and subsequences   ormklocald 47630
            21.46.3  Scratchpad for number theory   evenwodadd 47642
            21.46.4  Scratchpad for math on real numbers   squeezedltsq 47643
      ​21.47  Mathbox for Jarvin Udandy
      ​21.48  Mathbox for Adhemar
            ​*21.48.1  Minimal implicational calculus   adh-minim 47778
      ​21.49  Mathbox for Alexander van der Vekens
            21.49.1  General auxiliary theorems (1)   n0nsn2el 47802
                  21.49.1.1  Unordered and ordered pairs - extension for singletons   n0nsn2el 47802
                  21.49.1.2  Unordered and ordered pairs - extension for unordered pairs   elprneb 47806
                  21.49.1.3  Unordered and ordered pairs - extension for ordered pairs   oppr 47807
                  21.49.1.4  Relations - extension   eubrv 47812
                  21.49.1.5  Definite description binder (inverted iota) - extension   iota0def 47815
                  21.49.1.6  Functions - extension   fveqvfvv 47817
            21.49.2  Alternative for Russell's definition of a description binder   caiota 47860
            21.49.3  Double restricted existential uniqueness   r19.32 47875
                  21.49.3.1  Restricted quantification (extension)   r19.32 47875
                  21.49.3.2  Restricted uniqueness and "at most one" quantification   reuf1odnf 47884
                  21.49.3.3  Analogs to Existential uniqueness (double quantification)   2reu3 47887
                  21.49.3.4  Additional theorems for double restricted existential uniqueness   2reu8i 47890
            ​*21.49.4  Alternative definitions of function and operation values   wdfat 47893
                  21.49.4.1  Restricted quantification (extension)   ralbinrald 47899
                  21.49.4.2  The universal class (extension)   nvelim 47900
                  21.49.4.3  Introduce the Axiom of Power Sets (extension)   alneu 47901
                  21.49.4.4  Predicate "defined at"   dfateq12d 47903
                  21.49.4.5  Alternative definition of the value of a function   dfafv2 47909
                  21.49.4.6  Alternative definition of the value of an operation   aoveq123d 47955
            ​*21.49.5  Alternative definitions of function values (2)   cafv2 47985
            21.49.6  General auxiliary theorems (2)   an4com24 48045
                  21.49.6.1  Logical conjunction - extension   an4com24 48045
                  21.49.6.2  Abbreviated conjunction and disjunction of three wff's - extension   3an4ancom24 48046
                  21.49.6.3  Negated membership (alternative)   cnelbr 48048
                  21.49.6.4  The empty set - extension   ralralimp 48055
                  21.49.6.5  Indexed union and intersection - extension   otiunsndisjX 48056
                  21.49.6.6  Functions - extension   fvifeq 48057
                  21.49.6.7  Maps-to notation - extension   fvmptrab 48069
                  21.49.6.8  Subtraction - extension   cnambpcma 48071
                  21.49.6.9  Ordering on reals (cont.) - extension   leaddsuble 48074
                  21.49.6.10  Imaginary and complex number properties - extension   readdcnnred 48080
                  21.49.6.11  Nonnegative integers (as a subset of complex numbers) - extension   nn0resubcl 48085
                  21.49.6.12  Integers (as a subset of complex numbers) - extension   zgeltp1eq 48086
                  21.49.6.13  Decimal arithmetic - extension   1t10e1p1e11 48087
                  21.49.6.14  Upper sets of integers - extension   eluzge0nn0 48089
                  21.49.6.15  Infinity and the extended real number system (cont.) - extension   nltle2tri 48090
                  21.49.6.16  Finite intervals of integers - extension   ssfz12 48091
                  21.49.6.17  Half-open integer ranges - extension   fzopred 48100
                  21.49.6.18  The floor and ceiling functions - extension   2ltceilhalf 48109
                  21.49.6.19  The modulo (remainder) operation - extension   fldivmod 48121
                  21.49.6.20  The infinite sequence builder "seq"   smonoord 48154
                  21.49.6.21  Integer powers - extension   2timesltsq 48155
                  21.49.6.22  Finite and infinite sums - extension   fsummsndifre 48157
                  21.49.6.23  The divides relation - extension   nndivides2 48161
                  21.49.6.24  Extensible structures - extension   setsidel 48165
            ​*21.49.7  Preimages of function values   preimafvsnel 48168
            ​*21.49.8  Partitions of real intervals   ciccp 48202
            21.49.9  Shifting functions with an integer range domain   fargshiftfv 48228
            21.49.10  Words over a set (extension)   lswn0 48233
                  21.49.10.1  Last symbol of a word - extension   lswn0 48233
            21.49.11  Unordered pairs   wich 48234
                  21.49.11.1  Interchangeable setvar variables   wich 48234
                  21.49.11.2  Set of unordered pairs   sprid 48263
                  ​*21.49.11.3  Proper (unordered) pairs   prpair 48290
                  21.49.11.4  Set of proper unordered pairs   cprpr 48301
            21.49.12  Number theory (extension)   nprmmul1 48316
                  21.49.12.1  Properties of non-prime numbers   nprmmul1 48316
                  ​*21.49.12.2  Fermat numbers   cfmtno 48319
                  ​*21.49.12.3  Mersenne primes   m2prm 48383
                  21.49.12.4  Proth's theorem   modexp2m1d 48404
                  21.49.12.5  The prime-counting function according to Ján Mináč   nprmdvdsfacm1lem1 48412
                  21.49.12.6  Solutions of quadratic equations   quad1 48425
            ​*21.49.13  Even and odd numbers   ceven 48429
                  21.49.13.1  Definitions and basic properties   ceven 48429
                  21.49.13.2  Alternate definitions using the "divides" relation   dfeven2 48454
                  21.49.13.3  Alternate definitions using the "modulo" operation   dfeven3 48463
                  21.49.13.4  Alternate definitions using the "gcd" operation   iseven5 48469
                  21.49.13.5  Theorems of part 5 revised   zneoALTV 48474
                  21.49.13.6  Theorems of part 6 revised   odd2np1ALTV 48479
                  21.49.13.7  Theorems of AV's mathbox revised   0evenALTV 48493
                  21.49.13.8  Additional theorems   epoo 48508
                  21.49.13.9  Perfect Number Theorem (revised)   perfectALTVlem1 48526
            21.49.14  Number theory (extension 2)   cfppr 48529
                  ​*21.49.14.1  Fermat pseudoprimes   cfppr 48529
                  ​*21.49.14.2  Goldbach's conjectures   cgbe 48550
            21.49.15  Graph theory (extension)   cclnbgr 48623
                  21.49.15.1  Closed neighborhood of a vertex   cclnbgr 48623
                  ​*21.49.15.2  Semiclosed and semiopen neighborhoods (experimental)   dfsclnbgr2 48651
                  21.49.15.3  Induced subgraphs   cisubgr 48665
                  ​*21.49.15.4  Isomorphisms of graphs   cgrisom 48679
                  ​*21.49.15.5  Triangles in graphs   cgrtri 48742
                  ​*21.49.15.6  Star graphs   cstgr 48756
                  ​*21.49.15.7  Local isomorphisms of graphs   cgrlim 48781
                  ​*21.49.15.8  Generalized Petersen graphs   cgpg 48845
                  21.49.15.9  Loop-free graphs - extension   1hegrlfgr 48937
                  21.49.15.10  Walks - extension   cupwlks 48938
                  21.49.15.11  Edges of graphs expressed as sets of unordered pairs   upgredgssspr 48948
            21.49.16  Monoids (extension)   ovn0dmfun 48961
                  21.49.16.1  Auxiliary theorems   ovn0dmfun 48961
                  21.49.16.2  Magmas, Semigroups and Monoids (extension)   plusfreseq 48969
                  21.49.16.3  Examples and counterexamples for magmas, semigroups and monoids (extension)   opmpoismgm 48972
                  21.49.16.4  Group sum operation (extension 1)   gsumsplit2f 48985
            ​*21.49.17  Magmas and internal binary operations (alternate approach)   ccllaw 48988
                  *21.49.17.1  Laws for internal binary operations   ccllaw 48988
                  ​*21.49.17.2  Internal binary operations   cintop 49001
                  21.49.17.3  Alternative definitions for magmas and semigroups   cmgm2 49020
            21.49.18  Rings (extension)   lmod0rng 49034
                  21.49.18.1  Nonzero rings (extension)   lmod0rng 49034
                  21.49.18.2  Ideals as non-unital rings   lidldomn1 49036
                  21.49.18.3  The non-unital ring of even integers   0even 49042
                  21.49.18.4  A constructed not unital ring   cznrnglem 49064
                  ​*21.49.18.5  The category of non-unital rings (alternate definition)   crngcALTV 49068
                  ​*21.49.18.6  The category of (unital) rings (alternate definition)   cringcALTV 49092
            ​*21.49.19  Prime rings (and integral domains)   cprmrng 49139
            21.49.20  Basic algebraic structures (extension)   eliunxp2 49154
                  21.49.20.1  Auxiliary theorems   eliunxp2 49154
                  21.49.20.2  The binomial coefficient operation (extension)   bcpascm1 49171
                  21.49.20.3  The ` ZZ `-module ` ZZ X. ZZ `   zlmodzxzlmod 49174
                  21.49.20.4  Group sum operation (extension 2)   mgpsumunsn 49181
                  21.49.20.5  Symmetric groups (extension)   exple2lt6 49184
                  21.49.20.6  Divisibility (extension)   invginvrid 49187
                  21.49.20.7  The support of functions (extension)   rmsupp0 49188
                  21.49.20.8  Finitely supported functions (extension)   rmsuppfi 49192
                  21.49.20.9  Left modules (extension)   lmodvsmdi 49199
                  21.49.20.10  Associative algebras (extension)   assaascl0 49201
                  21.49.20.11  Univariate polynomials (extension)   ply1vr1smo 49203
                  21.49.20.12  Univariate polynomials (examples)   linply1 49213
            21.49.21  Linear algebra (extension)   cdmatalt 49216
                  ​*21.49.21.1  The subalgebras of diagonal and scalar matrices (extension)   cdmatalt 49216
                  ​*21.49.21.2  Linear combinations   clinc 49224
                  ​*21.49.21.3  Linear independence   clininds 49260
                  21.49.21.4  Simple left modules and the ` ZZ `-module   lmod1lem1 49307
                  21.49.21.5  Differences between (left) modules and (left) vector spaces   lvecpsslmod 49327
            21.49.22  Complexity theory   suppdm 49330
                  21.49.22.1  Auxiliary theorems   suppdm 49330
                  21.49.22.2  Even and odd integers   nn0onn0ex 49343
                  21.49.22.3  The natural logarithm on complex numbers (extension)   logcxp0 49355
                  21.49.22.4  Division of functions   cfdiv 49357
                  21.49.22.5  Upper bounds   cbigo 49367
                  21.49.22.6  Logarithm to an arbitrary base (extension)   rege1logbrege0 49378
                  ​*21.49.22.7  The binary logarithm   fldivexpfllog2 49385
                  21.49.22.8  Binary length   cblen 49389
                  ​*21.49.22.9  Digits   cdig 49415
                  21.49.22.10  Nonnegative integer as sum of its shifted digits   dignn0flhalflem1 49435
                  21.49.22.11  Algorithms for the multiplication of nonnegative integers   nn0mulfsum 49444
                  ​*21.49.22.12  N-ary functions   cnaryf 49446
                  ​*21.49.22.13  The Ackermann function   citco 49477
            21.49.23  Elementary geometry (extension)   fv1prop 49519
                  21.49.23.1  Auxiliary theorems   fv1prop 49519
                  21.49.23.2  Real euclidean space of dimension 2   rrx2pxel 49531
                  21.49.23.3  Spheres and lines in real Euclidean spaces   cline 49547
      ​21.50  Mathbox for Zhi Wang
            21.50.1  Propositional calculus   logic1 49609
            21.50.2  Predicate calculus with equality   dtrucor3 49617
                  21.50.2.1  Axiom scheme ax-5 (Distinctness)   dtrucor3 49617
            21.50.3  ZF Set Theory - start with the Axiom of Extensionality   ralbidb 49618
                  21.50.3.1  Restricted quantification   ralbidb 49618
                  21.50.3.2  The universal class   reuxfr1dd 49625
                  21.50.3.3  The empty set   ssdisjd 49626
                  21.50.3.4  Unordered and ordered pairs   vsn 49630
                  21.50.3.5  The union of a class   unilbss 49636
                  21.50.3.6  Indexed union and intersection   iuneq0 49637
            21.50.4  ZF Set Theory - add the Axiom of Replacement   inpw 49643
                  21.50.4.1  Theorems requiring subset and intersection existence   inpw 49643
            21.50.5  ZF Set Theory - add the Axiom of Power Sets   opth1neg 49644
                  21.50.5.1  Ordered pair theorem   opth1neg 49644
                  21.50.5.2  Ordered-pair class abstractions (cont.)   brab2dd 49646
                  21.50.5.3  Relations   iinxp 49649
                  21.50.5.4  Functions   mof0 49656
                  21.50.5.5  Operations   ovsng 49676
            21.50.6  ZF Set Theory - add the Axiom of Union   fonex 49685
                  21.50.6.1  Relations and functions (cont.)   fonex 49685
                  21.50.6.2  First and second members of an ordered pair   eloprab1st2nd 49686
                  21.50.6.3  Operations in maps-to notation (continued)   fmpodg 49687
                  21.50.6.4  Function transposition   resinsnlem 49689
                  21.50.6.5  Infinite Cartesian products   ixpv 49708
                  21.50.6.6  Equinumerosity   fvconst0ci 49709
            21.50.7  Order sets   iccin 49714
                  21.50.7.1  Real number intervals   iccin 49714
            21.50.8  Extensible structures   slotresfo 49717
                  21.50.8.1  Basic definitions   slotresfo 49717
            21.50.9  Moore spaces   mreuniss 49718
            ​*21.50.10  Topology   clduni 49719
                  21.50.10.1  Closure and interior   clduni 49719
                  21.50.10.2  Neighborhoods   neircl 49723
                  21.50.10.3  Subspace topologies   restcls2lem 49731
                  21.50.10.4  Limits and continuity in topological spaces   cnneiima 49735
                  21.50.10.5  Topological definitions using the reals   iooii 49736
                  21.50.10.6  Separated sets   sepnsepolem1 49740
                  21.50.10.7  Separated spaces: T0, T1, T2 (Hausdorff) ...   isnrm4 49749
            21.50.11  Preordered sets and directed sets using extensible structures   isprsd 49773
            21.50.12  Posets and lattices using extensible structures   lubeldm2 49774
                  21.50.12.1  Posets   lubeldm2 49774
                  21.50.12.2  Lattices   toslat 49800
                  21.50.12.3  Subset order structures   intubeu 49802
            21.50.13  Rings   elmgpcntrd 49823
                  21.50.13.1  Multiplicative Group   elmgpcntrd 49823
            21.50.14  Associative algebras   asclelbasALT 49824
                  21.50.14.1  Definition and basic properties   asclelbasALT 49824
            21.50.15  Categories   homf0 49827
                  21.50.15.1  Categories   homf0 49827
                  21.50.15.2  Opposite category   oppccatb 49834
                  21.50.15.3  Monomorphisms and epimorphisms   idmon 49838
                  21.50.15.4  Sections, inverses, isomorphisms   sectrcl 49840
                  21.50.15.5  Isomorphic objects   cicfn 49860
                  21.50.15.6  Subcategories   dmdm 49871
                  21.50.15.7  Functors   reldmfunc 49893
                  21.50.15.8  Opposite functors   coppf 49940
                  21.50.15.9  Full & faithful functors   imasubc 49969
                  21.50.15.10  Universal property   upciclem1 49984
                  21.50.15.11  Natural transformations and the functor category   isnatd 50041
                  21.50.15.12  Initial, terminal and zero objects of a category   initoo2 50050
                  21.50.15.13  Product of categories   reldmxpc 50064
                  21.50.15.14  Swap functors   cswapf 50077
                  21.50.15.15  Functor evaluation   oppc1stflem 50105
                  21.50.15.16  Transposed curry functors   cofuswapfcl 50111
                  21.50.15.17  Constant functors   diag1 50122
                  21.50.15.18  Functor composition bifunctors   fucofulem1 50128
                  21.50.15.19  Post-composition functors   postcofval 50182
                  21.50.15.20  Pre-composition functors   precofvallem 50184
            21.50.16  Examples of categories   catcrcl 50213
                  21.50.16.1  The category of categories   catcrcl 50213
                  21.50.16.2  Thin categories   cthinc 50235
                  21.50.16.3  Terminal categories   ctermc 50290
                  21.50.16.4  Preordered sets as thin categories   cprstc 50367
                  21.50.16.5  Monoids as categories   cmndtc 50395
                  21.50.16.6  Categories with at most one object and at most two morphisms   2arwcatlem1 50413
            21.50.17  Kan extensions and related concepts   clan 50423
                  21.50.17.1  Kan extensions   clan 50423
                  21.50.17.2  Limits and colimits   clmd 50461
      ​21.51  Mathbox for Emmett Weisz
            ​*21.51.1  Miscellaneous Theorems   nfintd 50491
            21.51.2  Set Recursion   csetrecs 50501
                  ​*21.51.2.1  Basic Properties of Set Recursion   csetrecs 50501
                  21.51.2.2  Examples and properties of set recursion   elsetrecslem 50517
            ​*21.51.3  Construction of Games and Surreal Numbers   cpg 50527
      ​​*21.52  Mathbox for David A. Wheeler
            21.52.1  Natural deduction   sbidd 50536
            ​*21.52.2  Greater than, greater than or equal to   cge-real 50538
            ​*21.52.3  Hyperbolic trigonometric functions   csinh 50548
            ​*21.52.4  Reciprocal trigonometric functions (sec, csc, cot)   csec 50559
            ​*21.52.5  Identities for "if"   ifnmfalse 50581
            ​*21.52.6  Logarithms generalized to arbitrary base using ` logb `   logb2aval 50582
            ​*21.52.7  Logarithm laws generalized to an arbitrary base - log_   clog- 50583
            ​*21.52.8  Formally define notions such as reflexivity   wreflexive 50585
            ​*21.52.9  Algebra helpers   mvlraddi 50589
            ​*21.52.10  Algebra helper examples   i2linesi 50596
            ​*21.52.11  Formal methods "surprises"   alimp-surprise 50598
            ​*21.52.12  Allsome quantifier   wals 50604
            ​*21.52.13  Allsome one quantifier   walseu 50637
            ​*21.52.14  Miscellaneous   5m4e1 50657
            21.52.15  Theorems about algebraic numbers   aacllem 50661
      ​21.53  Mathbox for Mingli Yuan
      ​21.54  Mathbox for Jiamin Zhao
            21.54.1  Cross product and scalar triple product in RR^3   1ne3 50663
      ​21.55  Mathbox for Kunhao Zheng
            21.55.1  Weighted AM-GM inequality   amgmwlem 50690