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 Steven Nguyen
      21.31  Mathbox for Igor Ieskov
      21.32  Mathbox for OpenAI
      21.33  Mathbox for Stefan O'Rear
      21.34  Mathbox for Noam Pasman
      21.35  Mathbox for Jon Pennant
      21.36  Mathbox for Richard Penner
      21.37  Mathbox for Stanislas Polu
      21.38  Mathbox for Rohan Ridenour
      21.39  Mathbox for Steve Rodriguez
      21.40  Mathbox for Andrew Salmon
      21.41  Mathbox for Alan Sare
      21.42  Mathbox for Eric Schmidt
      21.43  Mathbox for Glauco Siliprandi
      21.44  Mathbox for Saveliy Skresanov
      21.45  Mathbox for Ender Ting
      21.46  Mathbox for Jarvin Udandy
      21.47  Mathbox for Adhemar
      21.48  Mathbox for Alexander van der Vekens
      21.49  Mathbox for Zhi Wang
      21.50  Mathbox for Emmett Weisz
      21.51  Mathbox for David A. Wheeler
      21.52  Mathbox for Kunhao Zheng

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

​​*PART 1  CLASSICAL FIRST-ORDER LOGIC WITH EQUALITY
      ​*1.1  Pre-logic
            *1.1.1  Inferences for assisting proof development   idi 1
      ​​*1.2  Propositional calculus
            1.2.1  Recursively define primitive wffs for propositional calculus   wn 3
            ​*1.2.2  The axioms of propositional calculus   ax-mp 5
            ​*1.2.3  Logical implication   mp2 9
            ​*1.2.4  Logical negation   con4 113
            ​*1.2.5  Logical equivalence   wb 207
            ​*1.2.6  Logical conjunction   wa 396
            ​*1.2.7  Logical disjunction   wo 853
            ​*1.2.8  Mixed connectives   jaao 962
            ​*1.2.9  The conditional operator for propositions   wif 1068
            ​*1.2.10  The weak deduction theorem for propositional calculus   elimh 1088
            1.2.11  Abbreviated conjunction and disjunction of three wff's   w3o 1091
            1.2.12  Logical "nand" (Sheffer stroke)   wnan 1498
            1.2.13  Logical "xor"   wxo 1518
            1.2.14  Logical "nor"   wnor 1535
            1.2.15  True and false constants   wal 1545
                  ​*1.2.15.1  Universal quantifier for use by df-tru   wal 1545
                  ​*1.2.15.2  Equality predicate for use by df-tru   cv 1546
                  1.2.15.3  The true constant   wtru 1548
                  1.2.15.4  The false constant   wfal 1559
            ​*1.2.16  Truth tables   truimtru 1570
                  1.2.16.1  Implication   truimtru 1570
                  1.2.16.2  Negation   nottru 1574
                  1.2.16.3  Equivalence   trubitru 1576
                  1.2.16.4  Conjunction   truantru 1580
                  1.2.16.5  Disjunction   truortru 1584
                  1.2.16.6  Alternative denial   trunantru 1588
                  1.2.16.7  Exclusive disjunction   truxortru 1592
                  1.2.16.8  Joint denial   trunortru 1596
            ​*1.2.17  Half adder and full adder in propositional calculus   whad 1600
                  1.2.17.1  Full adder: sum   whad 1600
                  1.2.17.2  Full adder: carry   wcad 1613
      ​1.3  Other axiomatizations related to classical propositional calculus
            ​*1.3.1  Minimal implicational calculus   minimp 1628
            ​*1.3.2  Implicational Calculus   impsingle 1634
            1.3.3  Derive the Lukasiewicz axioms from Meredith's sole axiom   meredith 1648
            1.3.4  Derive the standard axioms from the Lukasiewicz axioms   luklem1 1665
            ​*1.3.5  Derive Nicod's axiom from the standard axioms   nic-dfim 1676
            1.3.6  Derive the Lukasiewicz axioms from Nicod's axiom   nic-imp 1682
            1.3.7  Derive Nicod's Axiom from Lukasiewicz's First Sheffer Stroke Axiom   lukshef-ax1 1701
            1.3.8  Derive the Lukasiewicz Axioms from the Tarski-Bernays-Wajsberg Axioms   tbw-bijust 1705
            1.3.9  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's First CO Axiom   merco1 1720
            1.3.10  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's Second CO Axiom   merco2 1743
            1.3.11  Derive the Lukasiewicz axioms from the Russell-Bernays Axioms   rb-bijust 1756
            ​*1.3.12  Stoic logic non-modal portion (Chrysippus of Soli)   mptnan 1775
      ​​*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 1786
                  1.4.1.1  Existential quantifier   wex 1786
                  1.4.1.2  Nonfreeness predicate   wnf 1790
            1.4.2  Rule scheme ax-gen (Generalization)   ax-gen 1802
            1.4.3  Axiom scheme ax-4 (Quantified Implication)   ax-4 1816
                  ​*1.4.3.1  The empty domain of discourse   empty 1913
            1.4.4  Axiom scheme ax-5 (Distinctness) - first use of $d   ax-5 1917
            ​*1.4.5  Equality predicate (continued)   weq 1969
            1.4.6  Axiom scheme ax-6 (Existence)   ax-6 1974
            1.4.7  Axiom scheme ax-7 (Equality)   ax-7 2015
            1.4.8  Define proper substitution   sbjust 2072
            1.4.9  Membership predicate   wcel 2119
            1.4.10  Axiom scheme ax-8 (Left Equality for Binary Predicate)   ax-8 2121
            1.4.11  Axiom scheme ax-9 (Right Equality for Binary Predicate)   ax-9 2129
            ​*1.4.12  Logical redundancy of ax-10 , ax-11 , ax-12 , ax-13   ax6dgen 2139
      ​​*1.5  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes)
            1.5.1  Axiom scheme ax-10 (Quantified Negation)   ax-10 2152
            1.5.2  Axiom scheme ax-11 (Quantifier Commutation)   ax-11 2168
            1.5.3  Axiom scheme ax-12 (Substitution)   ax-12 2189
            1.5.4  Axiom scheme ax-13 (Quantified Equality)   ax-13 2380
      ​1.6  Uniqueness and unique existence
            1.6.1  Uniqueness: the at-most-one quantifier   wmo 2541
            1.6.2  Unique existence: the unique existential quantifier   weu 2572
      ​1.7  Other axiomatizations related to classical predicate calculus
            ​*1.7.1  Aristotelian logic: Assertic syllogisms   barbara 2667
            ​*1.7.2  Intuitionistic logic   axia1 2697
​​*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 2712
            2.1.2  Classes   cab 2718
                  2.1.2.1  Class abstractions   cab 2718
                  ​*2.1.2.2  Class equality   df-cleq 2732
                  2.1.2.3  Class membership   df-clel 2815
                  2.1.2.4  Elementary properties of class abstractions   eqabdv 2873
            2.1.3  Class form not-free predicate   wnfc 2887
            2.1.4  Negated equality and membership   wne 2935
                  2.1.4.1  Negated equality   wne 2935
                  2.1.4.2  Negated membership   wnel 3039
            2.1.5  Restricted quantification   wral 3054
                  2.1.5.1  Restricted universal and existential quantification   wral 3054
                  2.1.5.2  Restricted existential uniqueness and at-most-one quantifier   wreu 3343
                  2.1.5.3  Restricted class abstraction   crab 3392
            2.1.6  The universal class   cvv 3432
            ​*2.1.7  Conditional equality (experimental)   wcdeq 3711
            2.1.8  Russell's Paradox   rru 3727
            2.1.9  Proper substitution of classes for sets   wsbc 3730
            2.1.10  Proper substitution of classes for sets into classes   csb 3838
            2.1.11  Define basic set operations and relations   cdif 3887
            2.1.12  Subclasses and subsets   df-ss 3907
            2.1.13  The difference, union, and intersection of two classes   dfdif3 4055
                  2.1.13.1  The difference of two classes   dfdif3 4055
                  2.1.13.2  The union of two classes   elun 4090
                  2.1.13.3  The intersection of two classes   elini 4135
                  2.1.13.4  The symmetric difference of two classes   csymdif 4187
                  2.1.13.5  Combinations of difference, union, and intersection of two classes   unabs 4200
                  2.1.13.6  Class abstractions with difference, union, and intersection of two classes   unabw 4242
                  2.1.13.7  Restricted uniqueness with difference, union, and intersection   reuun2 4260
            2.1.14  The empty set   c0 4268
            ​*2.1.15  The conditional operator for classes   cif 4461
            ​*2.1.16  The weak deduction theorem for set theory   dedth 4520
            2.1.17  Power classes   cpw 4536
            2.1.18  Unordered and ordered pairs   snjust 4561
            2.1.19  The union of a class   cuni 4845
            2.1.20  The intersection of a class   cint 4884
            2.1.21  Indexed union and intersection   ciun 4928
            2.1.22  Disjointness   wdisj 5046
            2.1.23  Binary relations   wbr 5079
            2.1.24  Ordered-pair class abstractions (class builders)   copab 5141
            2.1.25  Functions in maps-to notation   cmpt 5160
            2.1.26  Transitive classes   wtr 5186
      ​2.2  ZF Set Theory - add the Axiom of Replacement
            2.2.1  Introduce the Axiom of Replacement   ax-rep 5206
            2.2.2  Derive the Axiom of Separation   axsepgfromrep 5223
            2.2.3  Derive the Null Set Axiom   axnulALT 5233
            2.2.4  Theorems requiring subset and intersection existence   exnelv 5242
            2.2.5  Theorems requiring empty set existence   class2set 5290
      ​2.3  ZF Set Theory - add the Axiom of Power Sets
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 5301
            2.3.2  Derive the Axiom of Pairing   axprlem1 5359
            2.3.3  Ordered pair theorem   opnz 5420
            2.3.4  Ordered-pair class abstractions (cont.)   opabidw 5473
            2.3.5  Power class of union and intersection   pwin 5516
            2.3.6  The identity relation   cid 5519
            2.3.7  The membership relation (or epsilon relation)   cep 5524
            ​*2.3.8  Partial and total orderings   wpo 5531
            2.3.9  Founded and well-ordering relations   wfr 5575
            2.3.10  Relations   cxp 5623
            2.3.11  The Predecessor Class   cpred 6258
            2.3.12  Well-founded induction (variant)   frpomin 6298
            2.3.13  Well-ordered induction   tz6.26 6305
            2.3.14  Ordinals   word 6316
            2.3.15  Definite description binder (inverted iota)   cio 6446
            2.3.16  Functions   wfun 6486
            2.3.17  Cantor's Theorem   canth 7317
            2.3.18  Restricted iota (description binder)   crio 7319
            2.3.19  Operations   co 7363
                  2.3.19.1  Variable-to-class conversion for operations   caovclg 7555
            2.3.20  Maps-to notation   mpondm0 7603
            2.3.21  Function operation   cof 7625
            2.3.22  Proper subset relation   crpss 7672
      ​2.4  ZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 7685
            2.4.2  Ordinals (continued)   epweon 7725
            2.4.3  Transfinite induction   tfi 7800
            2.4.4  The natural numbers (i.e., finite ordinals)   com 7813
            2.4.5  Peano's postulates   peano1 7836
            2.4.6  Finite induction (for finite ordinals)   find 7842
            2.4.7  Relations and functions (cont.)   dmexg 7848
            2.4.8  First and second members of an ordered pair   c1st 7936
            2.4.9  Induction on Cartesian products   frpoins3xpg 8087
            2.4.10  Ordering on Cartesian products   xpord2lem 8089
            2.4.11  Ordering Ordinal Sequences   orderseqlem 8104
            ​*2.4.12  The support of functions   csupp 8107
            ​*2.4.13  Special maps-to operations   opeliunxp2f 8157
            2.4.14  Function transposition   ctpos 8172
            2.4.15  Curry and uncurry   ccur 8212
            2.4.16  Undefined values   cund 8219
            2.4.17  Well-founded recursion   cfrecs 8227
            2.4.18  Well-ordered recursion   cwrecs 8258
            2.4.19  Functions on ordinals; strictly monotone ordinal functions   iunon 8276
            2.4.20  "Strong" transfinite recursion   crecs 8307
            2.4.21  Recursive definition generator   crdg 8345
            2.4.22  Finite recursion   frfnom 8371
            2.4.23  Ordinal arithmetic   c1o 8395
            2.4.24  Natural number arithmetic   nna0 8537
            2.4.25  Natural addition   cnadd 8598
            2.4.26  Equivalence relations and classes   wer 8637
            2.4.27  The mapping operation   cmap 8770
            2.4.28  Infinite Cartesian products   cixp 8842
            2.4.29  Equinumerosity   cen 8887
            2.4.30  Schroeder-Bernstein Theorem   sbthlem1 9022
            2.4.31  Equinumerosity (cont.)   xpf1o 9074
            2.4.32  Finite sets   dif1enlem 9091
            2.4.33  Pigeonhole Principle   phplem1 9135
            2.4.34  Finite sets (cont.)   onomeneq 9145
            2.4.35  Finitely supported functions   cfsupp 9271
            2.4.36  Finite intersections   cfi 9320
            2.4.37  Hall's marriage theorem   marypha1lem 9343
            2.4.38  Supremum and infimum   csup 9350
            2.4.39  Ordinal isomorphism, Hartogs's theorem   coi 9421
            2.4.40  Hartogs function   char 9468
            2.4.41  Weak dominance   cwdom 9476
      ​2.5  ZF Set Theory - add the Axiom of Regularity
            2.5.1  Introduce the Axiom of Regularity   ax-reg 9504
            2.5.2  Axiom of Infinity equivalents   inf0 9540
      ​2.6  ZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-inf 9557
            2.6.2  Existence of omega (the set of natural numbers)   omex 9562
            2.6.3  Cantor normal form   ccnf 9580
            2.6.4  Transitive closure of a relation   cttrcl 9626
            2.6.5  Transitive closure   trcl 9647
            2.6.6  Set induction (or epsilon induction)   setind 9666
            2.6.7  Well-Founded Induction   frmin 9671
            2.6.8  Well-Founded Recursion   frr3g 9678
            2.6.9  Rank   cr1 9684
            2.6.10  Scott's trick; collection principle; Hilbert's epsilon   scottex 9807
            2.6.11  Disjoint union   cdju 9820
            2.6.12  Cardinal numbers   ccrd 9857
            2.6.13  Axiom of Choice equivalents   wac 10035
            ​*2.6.14  Cardinal number arithmetic   undjudom 10088
            2.6.15  The Ackermann bijection   ackbij2lem1 10138
            2.6.16  Cofinality (without Axiom of Choice)   cflem 10165
            2.6.17  Eight inequivalent definitions of finite set   sornom 10197
            2.6.18  Hereditarily size-limited sets without Choice   itunifval 10336
​​*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 10355
            3.1.2  Introduce the Axiom of Dependent Choice   ax-dc 10366
      ​3.2  ZFC Set Theory - add the Axiom of Choice
            3.2.1  Introduce the Axiom of Choice   ax-ac 10379
            3.2.2  AC equivalents: well-ordering, Zorn's lemma   numthcor 10414
            3.2.3  Cardinal number theorems using Axiom of Choice   cardval 10466
            3.2.4  Cardinal number arithmetic using Axiom of Choice   iunctb 10495
            3.2.5  Cofinality using the Axiom of Choice   alephreg 10503
      ​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 10541
            3.4.2  Derivation of the Axiom of Choice   gchaclem 10599
​​*PART 4  TG (TARSKI-GROTHENDIECK) SET THEORY
      ​4.1  Inaccessibles
            4.1.1  Weakly and strongly inaccessible cardinals   cwina 10603
            4.1.2  Weak universes   cwun 10621
            4.1.3  Tarski classes   ctsk 10669
            4.1.4  Grothendieck universes   cgru 10711
      ​4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
            4.2.1  Introduce the Tarski-Grothendieck Axiom   ax-groth 10744
            4.2.2  Derive the Power Set, Infinity and Choice Axioms   grothpw 10747
            4.2.3  Tarski map function   ctskm 10758
​​*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 10765
            5.1.2  Final derivation of real and complex number postulates   axaddf 11066
            5.1.3  Real and complex number postulates restated as axioms   ax-cnex 11092
      ​5.2  Derive the basic properties from the field axioms
            5.2.1  Some deductions from the field axioms for complex numbers   cnex 11117
            5.2.2  Infinity and the extended real number system   cpnf 11174
            5.2.3  Restate the ordering postulates with extended real "less than"   axlttri 11215
            5.2.4  Ordering on reals   lttr 11220
            5.2.5  Initial properties of the complex numbers   mul12 11309
      ​5.3  Real and complex numbers - basic operations
            5.3.1  Addition   add12 11362
            5.3.2  Subtraction   cmin 11375
            5.3.3  Multiplication   kcnktkm1cn 11579
            5.3.4  Ordering on reals (cont.)   gt0ne0 11613
            5.3.5  Reciprocals   ixi 11777
            5.3.6  Division   cdiv 11805
            5.3.7  Ordering on reals (cont.)   elimgt0 11991
            5.3.8  Completeness Axiom and Suprema   fimaxre 12098
            5.3.9  Imaginary and complex number properties   neg1cn 12142
            5.3.10  Function operation analogue theorems   ofsubeq0 12154
            ​*5.3.11  Indicator Functions   cind 12157
      ​5.4  Integer sets
            5.4.1  Positive integers (as a subset of complex numbers)   cn 12172
            5.4.2  Principle of mathematical induction   nnind 12190
            ​*5.4.3  Decimal representation of numbers   c2 12234
            ​*5.4.4  Some properties of specific numbers   1pneg1e0 12293
            5.4.5  Simple number properties   halfcl 12401
            5.4.6  The Archimedean property   nnunb 12431
            5.4.7  Nonnegative integers (as a subset of complex numbers)   cn0 12435
            ​*5.4.8  Extended nonnegative integers   cxnn0 12508
            5.4.9  Integers (as a subset of complex numbers)   cz 12522
            5.4.10  Decimal arithmetic   cdc 12642
            5.4.11  Upper sets of integers   cuz 12786
            5.4.12  Well-ordering principle for bounded-below sets of integers   uzwo3 12891
            5.4.13  Rational numbers (as a subset of complex numbers)   cq 12896
            5.4.14  Existence of the set of complex numbers   rpnnen1lem2 12925
      ​5.5  Order sets
            5.5.1  Positive reals (as a subset of complex numbers)   crp 12940
            5.5.2  Infinity and the extended real number system (cont.)   cxne 13058
            5.5.3  Supremum and infimum on the extended reals   xrsupexmnf 13255
            5.5.4  Real number intervals   cioo 13296
            5.5.5  Finite intervals of integers   cfz 13459
            ​*5.5.6  Finite intervals of nonnegative integers   elfz2nn0 13570
            5.5.7  Half-open integer ranges   cfzo 13606
      ​5.6  Elementary integer functions
            5.6.1  The floor and ceiling functions   cfl 13747
            5.6.2  The modulo (remainder) operation   cmo 13826
            5.6.3  Miscellaneous theorems about integers   om2uz0i 13907
            5.6.4  Strong induction over upper sets of integers   uzsinds 13947
            5.6.5  Finitely supported functions over the nonnegative integers   fsuppmapnn0fiublem 13950
            5.6.6  The infinite sequence builder "seq" - extension   cseq 13961
            5.6.7  Integer powers   cexp 14021
            5.6.8  Ordered pair theorem for nonnegative integers   nn0le2msqi 14227
            5.6.9  Factorial function   cfa 14233
            5.6.10  The binomial coefficient operation   cbc 14262
            5.6.11  The ` # ` (set size) function   chash 14290
                  5.6.11.1  Proper unordered pairs and triples (sets of size 2 and 3)   hashprlei 14428
                  5.6.11.2  Functions with a domain containing at least two different elements   fundmge2nop0 14462
                  5.6.11.3  Finite induction on the size of the first component of a binary relation   hashdifsnp1 14466
      ​​*5.7  Words over a set
            5.7.1  Definitions and basic theorems   cword 14473
            5.7.2  Last symbol of a word   clsw 14522
            5.7.3  Concatenations of words   cconcat 14530
            5.7.4  Singleton words   cs1 14556
            5.7.5  Concatenations with singleton words   ccatws1cl 14577
            5.7.6  Subwords/substrings   csubstr 14601
            5.7.7  Prefixes of a word   cpfx 14631
            5.7.8  Subwords of subwords   swrdswrdlem 14664
            5.7.9  Subwords and concatenations   pfxcctswrd 14670
            5.7.10  Subwords of concatenations   swrdccatfn 14684
            5.7.11  Splicing words (substring replacement)   csplice 14709
            5.7.12  Reversing words   creverse 14718
            5.7.13  Repeated symbol words   creps 14728
            ​*5.7.14  Cyclical shifts of words   ccsh 14748
            5.7.15  Mapping words by a function   wrdco 14791
            5.7.16  Longer string literals   cs2 14801
      ​​*5.8  Reflexive and transitive closures of relations
            5.8.1  The reflexive and transitive properties of relations   coss12d 14932
            5.8.2  Basic properties of closures   cleq1lem 14942
            5.8.3  Definitions and basic properties of transitive closures   ctcl 14945
            5.8.4  Exponentiation of relations   crelexp 14979
            5.8.5  Reflexive-transitive closure as an indexed union   crtrcl 15015
            ​*5.8.6  Principle of transitive induction   relexpindlem 15023
      ​5.9  Elementary real and complex functions
            5.9.1  The "shift" operation   cshi 15026
            5.9.2  Signum (sgn or sign) function   csgn 15046
            5.9.3  Real and imaginary parts; conjugate   ccj 15056
            5.9.4  Square root; absolute value   csqrt 15193
      ​5.10  Elementary limits and convergence
            5.10.1  Superior limit (lim sup)   clsp 15430
            5.10.2  Limits   cli 15444
            5.10.3  Finite and infinite sums   csu 15646
            5.10.4  The binomial theorem   binomlem 15792
            5.10.5  The inclusion/exclusion principle   incexclem 15799
            5.10.6  Infinite sums (cont.)   isumshft 15802
            5.10.7  Miscellaneous converging and diverging sequences   divrcnv 15815
            5.10.8  Arithmetic series   arisum 15823
            5.10.9  Geometric series   expcnv 15827
            5.10.10  Ratio test for infinite series convergence   cvgrat 15846
            5.10.11  Mertens' theorem   mertenslem1 15847
            5.10.12  Finite and infinite products   prodf 15850
                  5.10.12.1  Product sequences   prodf 15850
                  5.10.12.2  Non-trivial convergence   ntrivcvg 15860
                  5.10.12.3  Complex products   cprod 15866
                  5.10.12.4  Finite products   fprod 15904
                  5.10.12.5  Infinite products   iprodclim 15961
            5.10.13  Falling and Rising Factorial   cfallfac 15967
            5.10.14  Bernoulli polynomials and sums of k-th powers   cbp 16009
      ​5.11  Elementary trigonometry
            5.11.1  The exponential, sine, and cosine functions   ce 16024
                  5.11.1.1  The circle constant (tau = 2 pi)   ctau 16167
            5.11.2  _e is irrational   eirrlem 16169
      ​5.12  Cardinality of real and complex number subsets
            5.12.1  Countability of integers and rationals   xpnnen 16176
            5.12.2  The reals are uncountable   rpnnen2lem1 16179
​​*PART 6  ELEMENTARY NUMBER THEORY
      ​6.1  Elementary properties of divisibility
            6.1.1  Irrationality of square root of 2   sqrt2irrlem 16213
            6.1.2  Some Number sets are chains of proper subsets   nthruc 16217
            6.1.3  The divides relation   cdvds 16219
            ​*6.1.4  Even and odd numbers   evenelz 16303
            6.1.5  The division algorithm   divalglem0 16360
            6.1.6  Bit sequences   cbits 16386
            6.1.7  The greatest common divisor operator   cgcd 16461
            6.1.8  Bézout's identity   bezoutlem1 16506
            6.1.9  Algorithms   nn0seqcvgd 16537
            6.1.10  Euclid's Algorithm   eucalgval2 16548
            ​*6.1.11  The least common multiple   clcm 16555
            ​*6.1.12  Coprimality and Euclid's lemma   coprmgcdb 16616
            6.1.13  Cancellability of congruences   congr 16631
      ​6.2  Elementary prime number theory
            ​*6.2.1  Elementary properties   cprime 16638
            ​*6.2.2  Coprimality and Euclid's lemma (cont.)   coprm 16679
            6.2.3  Properties of the canonical representation of a rational   cnumer 16701
            6.2.4  Euler's theorem   codz 16731
            6.2.5  Arithmetic modulo a prime number   modprm1div 16766
            6.2.6  Pythagorean Triples   coprimeprodsq 16777
            6.2.7  The prime count function   cpc 16805
            6.2.8  Pocklington's theorem   prmpwdvds 16873
            6.2.9  Infinite primes theorem   unbenlem 16877
            6.2.10  Sum of prime reciprocals   prmreclem1 16885
            6.2.11  Fundamental theorem of arithmetic   1arithlem1 16892
            6.2.12  Lagrange's four-square theorem   cgz 16898
            6.2.13  Van der Waerden's theorem   cvdwa 16934
            6.2.14  Ramsey's theorem   cram 16968
            ​*6.2.15  Primorial function   cprmo 17000
            ​*6.2.16  Prime gaps   prmgaplem1 17018
            6.2.17  Decimal arithmetic (cont.)   dec2dvds 17032
            6.2.18  Cyclical shifts of words (cont.)   cshwsidrepsw 17062
            6.2.19  Specific prime numbers   prmlem0 17074
            6.2.20  Very large primes   1259lem1 17099
​PART 7  BASIC STRUCTURES
      ​7.1  Extensible structures
            ​*7.1.1  Basic definitions   cstr 17114
                  7.1.1.1  Extensible structures as structures with components   cstr 17114
                  7.1.1.2  Substitution of components   csts 17131
                  7.1.1.3  Slots   cslot 17149
                  ​*7.1.1.4  Structure component indices   cnx 17161
                  7.1.1.5  Base sets   cbs 17177
                  7.1.1.6  Base set restrictions   cress 17198
            7.1.2  Slot definitions   cplusg 17218
            7.1.3  Definition of the structure product   crest 17381
            7.1.4  Definition of the structure quotient   cordt 17461
      ​7.2  Moore spaces
            7.2.1  Moore closures   mrcflem 17570
            7.2.2  Independent sets in a Moore system   mrisval 17594
            7.2.3  Algebraic closure systems   isacs 17615
​PART 8  BASIC CATEGORY THEORY
      ​8.1  Categories
            8.1.1  Categories   ccat 17628
            8.1.2  Opposite category   coppc 17675
            8.1.3  Monomorphisms and epimorphisms   cmon 17693
            8.1.4  Sections, inverses, isomorphisms   csect 17709
            ​*8.1.5  Isomorphic objects   ccic 17760
            8.1.6  Subcategories   cssc 17772
            8.1.7  Functors   cfunc 17819
            8.1.8  Full & faithful functors   cful 17869
            8.1.9  Natural transformations and the functor category   cnat 17909
            8.1.10  Initial, terminal and zero objects of a category   cinito 17946
      ​8.2  Arrows (disjointified hom-sets)
            8.2.1  Identity and composition for arrows   cida 18018
      ​8.3  Examples of categories
            8.3.1  The category of sets   csetc 18040
            8.3.2  The category of categories   ccatc 18063
            ​*8.3.3  The category of extensible structures   fncnvimaeqv 18084
      ​8.4  Categorical constructions
            8.4.1  Product of categories   cxpc 18132
            8.4.2  Functor evaluation   cevlf 18173
            8.4.3  Hom functor   chof 18212
​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 18395
            9.5.2  Complete lattices   ccla 18462
            9.5.3  Distributive lattices   cdlat 18484
            9.5.4  Subset order structures   cipo 18491
      ​9.6  Posets, directed sets, and lattices as relations
            ​*9.6.1  Posets and lattices as relations   cps 18528
            9.6.2  Directed sets, nets   cdir 18558
      ​9.7  Chains
​PART 10  BASIC ALGEBRAIC STRUCTURES
      ​10.1  Monoids
            ​*10.1.1  Magmas   cplusf 18603
            ​*10.1.2  Identity elements   mgmidmo 18626
            ​*10.1.3  Iterated sums in a magma   gsumvalx 18642
            10.1.4  Magma homomorphisms and submagmas   cmgmhm 18656
            ​*10.1.5  Semigroups   csgrp 18684
            ​*10.1.6  Definition and basic properties of monoids   cmnd 18700
            10.1.7  Monoid homomorphisms and submonoids   cmhm 18747
            ​*10.1.8  Iterated sums in a monoid   gsumvallem2 18800
            10.1.9  Free monoids   cfrmd 18813
                  ​*10.1.9.1  Monoid of endofunctions   cefmnd 18834
            10.1.10  Examples and counterexamples for magmas, semigroups and monoids   mgm2nsgrplem1 18887
      ​10.2  Groups
            10.2.1  Definition and basic properties   cgrp 18907
            ​*10.2.2  Group multiple operation   cmg 19041
            10.2.3  Subgroups and Quotient groups   csubg 19094
            ​*10.2.4  Cyclic monoids and groups   cycsubmel 19173
            10.2.5  Elementary theory of group homomorphisms   cghm 19185
            10.2.6  Isomorphisms of groups   cgim 19230
                  10.2.6.1  The first isomorphism theorem of groups   ghmqusnsglem1 19253
            10.2.7  Group actions   cga 19262
            10.2.8  Centralizers and centers   ccntz 19288
            10.2.9  The opposite group   coppg 19318
            10.2.10  Symmetric groups   csymg 19342
                  ​*10.2.10.1  Definition and basic properties   csymg 19342
                  10.2.10.2  Cayley's theorem   cayleylem1 19385
                  10.2.10.3  Permutations fixing one element   symgfix2 19389
                  ​*10.2.10.4  Transpositions in the symmetric group   cpmtr 19414
                  10.2.10.5  The sign of a permutation   cpsgn 19462
            10.2.11  p-Groups and Sylow groups; Sylow's theorems   cod 19497
            10.2.12  Direct products   clsm 19607
                  10.2.12.1  Direct products (extension)   smndlsmidm 19629
            10.2.13  Free groups   cefg 19679
            10.2.14  Abelian groups   ccmn 19753
                  10.2.14.1  Definition and basic properties   ccmn 19753
                  10.2.14.2  Cyclic groups   ccyg 19850
                  10.2.14.3  Group sum operation   gsumval3a 19876
                  10.2.14.4  Group sums over (ranges of) integers   fsfnn0gsumfsffz 19956
                  10.2.14.5  Internal direct products   cdprd 19968
                  10.2.14.6  The Fundamental Theorem of Abelian Groups   ablfacrplem 20040
            10.2.15  Simple groups   csimpg 20065
                  10.2.15.1  Definition and basic properties   csimpg 20065
                  10.2.15.2  Classification of abelian simple groups   ablsimpnosubgd 20079
            10.2.16  Totally ordered monoids and groups   comnd 20092
      ​10.3  Rings
            10.3.1  Multiplicative Group   cmgp 20119
            ​*10.3.2  Non-unital rings ("rngs")   crng 20131
            ​*10.3.3  Ring unity (multiplicative identity)   cur 20160
            10.3.4  Semirings   csrg 20165
                  ​*10.3.4.1  The binomial theorem for semirings   srgbinomlem1 20205
            10.3.5  Unital rings   crg 20212
            10.3.6  Opposite ring   coppr 20314
            10.3.7  Divisibility   cdsr 20332
            10.3.8  Ring primes   crpm 20410
            10.3.9  Homomorphisms of non-unital rings   crnghm 20412
            10.3.10  Ring homomorphisms   crh 20447
            10.3.11  Nonzero rings and zero rings   cnzr 20491
            10.3.12  Local rings   clring 20517
            10.3.13  Subrings   csubrng 20524
                  10.3.13.1  Subrings of non-unital rings   csubrng 20524
                  10.3.13.2  Subrings of unital rings   csubrg 20548
                  10.3.13.3  Subrings generated by a subset   crgspn 20589
            10.3.14  Categories of rings   crngc 20595
                  ​*10.3.14.1  The category of non-unital rings   crngc 20595
                  ​*10.3.14.2  The category of (unital) rings   cringc 20624
                  10.3.14.3  Subcategories of the category of rings   srhmsubclem1 20656
            10.3.15  Left regular elements and domains   crlreg 20670
      ​10.4  Division rings and fields
            10.4.1  Definition and basic properties   cdr 20708
            10.4.2  Sub-division rings   csdrg 20765
            10.4.3  Absolute value (abstract algebra)   cabv 20787
            10.4.4  Star rings   cstf 20816
            10.4.5  Totally ordered rings and fields   corng 20836
      ​10.5  Left modules
            10.5.1  Definition and basic properties   clmod 20857
            10.5.2  Subspaces and spans in a left module   clss 20928
            10.5.3  Homomorphisms and isomorphisms of left modules   clmhm 21016
            10.5.4  Subspace sum; bases for a left module   clbs 21071
      ​10.6  Vector spaces
            10.6.1  Definition and basic properties   clvec 21099
      ​10.7  Subring algebras and ideals
            10.7.1  Subring algebras   csra 21168
            ​*10.7.2  Left ideals and spans   clidl 21206
            10.7.3  Two-sided ideals and quotient rings   c2idl 21249
                  ​*10.7.3.1  Condition for a non-unital ring to be unital   rngqiprng1elbas 21286
            10.7.4  Principal ideal rings. Divisibility in the integers   clpidl 21320
            10.7.5  Principal ideal domains   cpid 21336
      ​10.8  The complex numbers as an algebraic extensible structure
            10.8.1  Definition and basic properties   cpsmet 21338
            ​*10.8.2  Ring of integers   czring 21428
                  ​*10.8.2.1  Example for a condition for a non-unital ring to be unital   pzriprnglem1 21463
            10.8.3  Algebraic constructions based on the complex numbers   czrh 21481
            10.8.4  Signs as subgroup of the complex numbers   cnmsgnsubg 21559
            10.8.5  Embedding of permutation signs into a ring   zrhpsgnmhm 21566
            10.8.6  The ordered field of real numbers   crefld 21586
      ​10.9  Generalized pre-Hilbert and Hilbert spaces
            10.9.1  Definition and basic properties   cphl 21606
            10.9.2  Orthocomplements and closed subspaces   cocv 21642
            10.9.3  Orthogonal projection and orthonormal bases   cpj 21682
​​*PART 11  BASIC LINEAR ALGEBRA
      ​11.1  Vectors and free modules
            *11.1.1  Direct sum of left modules   cdsmm 21713
            ​*11.1.2  Free modules   cfrlm 21728
            ​*11.1.3  Standard basis (unit vectors)   cuvc 21764
            ​*11.1.4  Independent sets and families   clindf 21786
            11.1.5  Characterization of free modules   lmimlbs 21818
      ​11.2  Associative algebras
            11.2.1  Definition and basic properties   casa 21832
      ​11.3  Abstract multivariate polynomials
            11.3.1  Definition and basic properties   cmps 21886
            11.3.2  Polynomial evaluation   ces 22055
            11.3.3  The "variable selection" function   cslv 22099
            11.3.4  Additional definitions for (multivariate) polynomials   cmhp 22128
            ​*11.3.5  Univariate polynomials   cps1 22167
            11.3.6  Univariate polynomial evaluation   ces1 22306
                  11.3.6.1  Specialization of polynomial evaluation as a ring homomorphism   evls1scafv 22359
      ​​*11.4  Matrices
            *11.4.1  The matrix multiplication   cmmul 22380
            ​*11.4.2  Square matrices   cmat 22397
            ​*11.4.3  The matrix algebra   matmulr 22428
            ​*11.4.4  Matrices of dimension 0 and 1   mat0dimbas0 22456
            ​*11.4.5  The subalgebras of diagonal and scalar matrices   cdmat 22478
            ​*11.4.6  Multiplication of a matrix with a "column vector"   cmvmul 22530
            11.4.7  Replacement functions for a square matrix   cmarrep 22546
            11.4.8  Submatrices   csubma 22566
      ​11.5  The determinant
            11.5.1  Definition and basic properties   cmdat 22574
            11.5.2  Determinants of 2 x 2 -matrices   m2detleiblem1 22614
            11.5.3  The matrix adjugate/adjunct   cmadu 22622
            ​*11.5.4  Laplace expansion of determinants (special case)   symgmatr01lem 22643
            11.5.5  Inverse matrix   invrvald 22666
            ​*11.5.6  Cramer's rule   slesolvec 22669
      ​​*11.6  Polynomial matrices
            11.6.1  Basic properties   pmatring 22682
            ​*11.6.2  Constant polynomial matrices   ccpmat 22693
            ​*11.6.3  Collecting coefficients of polynomial matrices   cdecpmat 22752
            ​*11.6.4  Ring isomorphism between polynomial matrices and polynomials over matrices   cpm2mp 22782
      ​​*11.7  The characteristic polynomial
            *11.7.1  Definition and basic properties   cchpmat 22816
            ​*11.7.2  The characteristic factor function G   fvmptnn04if 22839
            ​*11.7.3  The Cayley-Hamilton theorem   cpmadurid 22857
​PART 12  BASIC TOPOLOGY
      ​12.1  Topology
            ​*12.1.1  Topological spaces   ctop 22883
                  12.1.1.1  Topologies   ctop 22883
                  12.1.1.2  Topologies on sets   ctopon 22900
                  12.1.1.3  Topological spaces   ctps 22922
            12.1.2  Topological bases   ctb 22935
            12.1.3  Examples of topologies   distop 22985
            12.1.4  Closure and interior   ccld 23006
            12.1.5  Neighborhoods   cnei 23087
            12.1.6  Limit points and perfect sets   clp 23124
            12.1.7  Subspace topologies   restrcl 23147
            12.1.8  Order topology   ordtbaslem 23178
            12.1.9  Limits and continuity in topological spaces   ccn 23214
            12.1.10  Separated spaces: T0, T1, T2 (Hausdorff) ...   ct0 23296
            12.1.11  Compactness   ccmp 23376
            12.1.12  Bolzano-Weierstrass theorem   bwth 23400
            12.1.13  Connectedness   cconn 23401
            12.1.14  First- and second-countability   c1stc 23427
            12.1.15  Local topological properties   clly 23454
            12.1.16  Refinements   cref 23492
            12.1.17  Compactly generated spaces   ckgen 23523
            12.1.18  Product topologies   ctx 23550
            12.1.19  Continuous function-builders   cnmptid 23651
            12.1.20  Quotient maps and quotient topology   ckq 23683
            12.1.21  Homeomorphisms   chmeo 23743
      ​12.2  Filters and filter bases
            12.2.1  Filter bases   elmptrab 23817
            12.2.2  Filters   cfil 23835
            12.2.3  Ultrafilters   cufil 23889
            12.2.4  Filter limits   cfm 23923
            12.2.5  Extension by continuity   ccnext 24049
            12.2.6  Topological groups   ctmd 24060
            12.2.7  Infinite group sum on topological groups   ctsu 24116
            12.2.8  Topological rings, fields, vector spaces   ctrg 24146
      ​12.3  Uniform Structures and Spaces
            12.3.1  Uniform structures   cust 24190
            12.3.2  The topology induced by an uniform structure   cutop 24220
            12.3.3  Uniform Spaces   cuss 24243
            12.3.4  Uniform continuity   cucn 24264
            12.3.5  Cauchy filters in uniform spaces   ccfilu 24275
            12.3.6  Complete uniform spaces   ccusp 24286
      ​12.4  Metric spaces
            12.4.1  Pseudometric spaces   ispsmet 24294
            12.4.2  Basic metric space properties   cxms 24307
            12.4.3  Metric space balls   blfvalps 24373
            12.4.4  Open sets of a metric space   mopnval 24428
            12.4.5  Continuity in metric spaces   metcnp3 24530
            12.4.6  The uniform structure generated by a metric   metuval 24539
            12.4.7  Examples of metric spaces   dscmet 24562
            ​*12.4.8  Normed algebraic structures   cnm 24566
            12.4.9  Normed space homomorphisms (bounded linear operators)   cnmo 24695
            12.4.10  Topology on the reals   qtopbaslem 24748
            12.4.11  Topological definitions using the reals   cii 24867
            12.4.12  Path homotopy   chtpy 24959
            12.4.13  The fundamental group   cpco 24992
      ​12.5  Metric subcomplex vector spaces
            12.5.1  Subcomplex modules   cclm 25054
            ​*12.5.2  Subcomplex vector spaces   ccvs 25115
            ​*12.5.3  Normed subcomplex vector spaces   isncvsngp 25141
            12.5.4  Subcomplex pre-Hilbert spaces   ccph 25158
            12.5.5  Convergence and completeness   ccfil 25244
            12.5.6  Baire's Category Theorem   bcthlem1 25316
            12.5.7  Banach spaces and subcomplex Hilbert spaces   ccms 25324
                  12.5.7.1  The complete ordered field of the real numbers   retopn 25371
            12.5.8  Euclidean spaces   crrx 25375
            12.5.9  Minimizing Vector Theorem   minveclem1 25416
            12.5.10  Projection Theorem   pjthlem1 25429
​PART 13  BASIC REAL AND COMPLEX ANALYSIS
      ​13.1  Continuity
            13.1.1  Intermediate value theorem   pmltpclem1 25440
      ​13.2  Integrals
            13.2.1  Lebesgue measure   covol 25454
            13.2.2  Lebesgue integration   cmbf 25606
                  13.2.2.1  Lesbesgue integral   cmbf 25606
                  13.2.2.2  Lesbesgue directed integral   cdit 25838
      ​13.3  Derivatives
            13.3.1  Real and complex differentiation   climc 25854
                  13.3.1.1  Derivatives of functions of one complex or real variable   climc 25854
                  13.3.1.2  Results on real differentiation   dvferm1lem 25976
​PART 14  BASIC REAL AND COMPLEX FUNCTIONS
      ​14.1  Polynomials
            14.1.1  Polynomial degrees   cmdg 26043
            14.1.2  The division algorithm for univariate polynomials   cmn1 26116
            14.1.3  Elementary properties of complex polynomials   cply 26174
            14.1.4  The division algorithm for polynomials   cquot 26281
            14.1.5  Algebraic numbers   caa 26305
            14.1.6  Liouville's approximation theorem   aalioulem1 26323
      ​14.2  Sequences and series
            14.2.1  Taylor polynomials and Taylor's theorem   ctayl 26343
            14.2.2  Uniform convergence   culm 26366
            14.2.3  Power series   pserval 26400
      ​14.3  Basic trigonometry
            14.3.1  The exponential, sine, and cosine functions (cont.)   efcn 26433
            14.3.2  Properties of pi = 3.14159...   pilem1 26441
            14.3.3  Mapping of the exponential function   efgh 26530
            14.3.4  The natural logarithm on complex numbers   clog 26543
            ​*14.3.5  Logarithms to an arbitrary base   clogb 26753
            14.3.6  Theorems of Pythagoras, isosceles triangles, and intersecting chords   angval 26790
            14.3.7  Solutions of quadratic, cubic, and quartic equations   quad2 26828
            14.3.8  Inverse trigonometric functions   casin 26851
            14.3.9  The Birthday Problem   log2ublem1 26935
            14.3.10  Areas in R^2   carea 26944
            14.3.11  More miscellaneous converging sequences   rlimcnp 26954
            14.3.12  Inequality of arithmetic and geometric means   cvxcl 26973
            14.3.13  Euler-Mascheroni constant   cem 26980
            14.3.14  Zeta function   czeta 27001
            14.3.15  Gamma function   clgam 27004
      ​14.4  Basic number theory
            14.4.1  Wilson's theorem   wilthlem1 27056
            14.4.2  The Fundamental Theorem of Algebra   ftalem1 27061
            14.4.3  The Basel problem (ζ(2) = π²/6)   basellem1 27069
            14.4.4  Number-theoretical functions   ccht 27079
            14.4.5  Perfect Number Theorem   mersenne 27215
            14.4.6  Characters of Z/nZ   cdchr 27220
            14.4.7  Bertrand's postulate   bcctr 27263
            ​*14.4.8  Quadratic residues and the Legendre symbol   clgs 27282
            ​*14.4.9  Gauss' Lemma   gausslemma2dlem0a 27344
            14.4.10  Quadratic reciprocity   lgseisenlem1 27363
            14.4.11  All primes 4n+1 are the sum of two squares   2sqlem1 27405
            14.4.12  Chebyshev's Weak Prime Number Theorem, Dirichlet's Theorem   chebbnd1lem1 27457
            14.4.13  The Prime Number Theorem   mudivsum 27518
            14.4.14  Ostrowski's theorem   abvcxp 27603
​PART 15  SURREAL NUMBERS
      ​​*15.1  Sign sequence representation and Alling's axioms
            15.1.1  Definitions and initial properties   csur 27628
            15.1.2  Ordering   ltssolem1 27664
            15.1.3  Birthday Function   bdayfo 27666
            15.1.4  Density   fvnobday 27667
            ​*15.1.5  Full-Eta Property   bdayimaon 27682
      ​15.2  Initial consequences of Alling's axioms
            15.2.1  Ordering Theorems   cles 27733
            15.2.2  Birthday Theorems   bdayfun 27765
      ​​*15.3  Conway cut representation
            15.3.1  Conway cuts   cslts 27774
            15.3.2  Zero and One   c0s 27822
            15.3.3  Cuts and Options   cmade 27839
            15.3.4  Cofinality and coinitiality   cofslts 27935
      ​15.4  Induction and recursion
            15.4.1  Induction and recursion on one variable   cnorec 27954
            15.4.2  Induction and recursion on two variables   cnorec2 27965
      ​15.5  Surreal arithmetic
            15.5.1  Addition   cadds 27976
            15.5.2  Negation and Subtraction   cnegs 28036
            15.5.3  Multiplication   cmuls 28123
            15.5.4  Division   cdivs 28204
            15.5.5  Absolute value   cabss 28254
      ​15.6  Subsystems of surreals
            15.6.1  Ordinal numbers   cons 28268
            15.6.2  Surreal recursive sequences   cseqs 28300
            15.6.3  Natural numbers   cn0s 28329
            15.6.4  Integers   czs 28395
            15.6.5  Dyadic fractions   c2s 28427
            15.6.6  Real numbers   creno 28506
​​*PART 16  ELEMENTARY GEOMETRY
      ​16.1  Definition and Tarski's Axioms of Geometry
            16.1.1  Justification for the congruence notation   tgjustf 28566
      ​16.2  Tarskian Geometry
            16.2.1  Congruence   tgcgrcomimp 28570
            16.2.2  Betweenness   tgbtwntriv2 28580
            16.2.3  Dimension   tglowdim1 28593
            16.2.4  Betweenness and Congruence   tgifscgr 28601
            16.2.5  Congruence of a series of points   ccgrg 28603
            16.2.6  Motions   cismt 28625
            16.2.7  Colinearity   tglng 28639
            16.2.8  Connectivity of betweenness   tgbtwnconn1lem1 28665
            16.2.9  Less-than relation in geometric congruences   cleg 28675
            16.2.10  Rays   chlg 28693
            16.2.11  Lines   btwnlng1 28712
            16.2.12  Point inversions   cmir 28745
            16.2.13  Right angles   crag 28786
            16.2.14  Half-planes   islnopp 28832
            16.2.15  Midpoints and Line Mirroring   cmid 28865
            16.2.16  Congruence of angles   ccgra 28900
            16.2.17  Angle Comparisons   cinag 28928
            16.2.18  Congruence Theorems   tgsas1 28947
            16.2.19  Equilateral triangles   ceqlg 28958
      ​16.3  Properties of geometries
            16.3.1  Isomorphisms between geometries   f1otrgds 28962
      ​16.4  Geometry in Hilbert spaces
            16.4.1  Geometry in the complex plane   cchhllem 28980
            16.4.2  Geometry in Euclidean spaces   cee 28981
                  16.4.2.1  Definition of the Euclidean space   cee 28981
                  16.4.2.2  Tarski's axioms for geometry for the Euclidean space   axdimuniq 29007
                  16.4.2.3  EE^n fulfills Tarski's Axioms   ceeng 29071
​​*PART 17  GRAPH THEORY
      ​*17.1  Vertices and edges
            17.1.1  The edge function extractor for extensible structures   cedgf 29082
            ​*17.1.2  Vertices and indexed edges   cvtx 29090
                  17.1.2.1  Definitions and basic properties   cvtx 29090
                  17.1.2.2  The vertices and edges of a graph represented as ordered pair   opvtxval 29097
                  17.1.2.3  The vertices and edges of a graph represented as extensible structure   funvtxdmge2val 29105
                  17.1.2.4  Representations of graphs without edges   snstrvtxval 29131
                  17.1.2.5  Degenerated cases of representations of graphs   vtxval0 29133
            17.1.3  Edges as range of the edge function   cedg 29141
      ​​*17.2  Undirected graphs
            17.2.1  Undirected hypergraphs   cuhgr 29150
            17.2.2  Undirected pseudographs and multigraphs   cupgr 29174
            ​*17.2.3  Loop-free graphs   umgrislfupgrlem 29216
            17.2.4  Edges as subsets of vertices of graphs   uhgredgiedgb 29220
            ​*17.2.5  Undirected simple graphs   cuspgr 29242
            17.2.6  Examples for graphs   usgr0e 29330
            17.2.7  Subgraphs   csubgr 29361
            17.2.8  Finite undirected simple graphs   cfusgr 29410
            17.2.9  Neighbors, complete graphs and universal vertices   cnbgr 29426
                  17.2.9.1  Neighbors   cnbgr 29426
                  17.2.9.2  Universal vertices   cuvtx 29479
                  17.2.9.3  Complete graphs   ccplgr 29503
            17.2.10  Vertex degree   cvtxdg 29559
            ​*17.2.11  Regular graphs   crgr 29649
      ​​*17.3  Walks, paths and cycles
            *17.3.1  Walks   cewlks 29689
            17.3.2  Walks for loop-free graphs   lfgrwlkprop 29779
            17.3.3  Trails   ctrls 29782
            17.3.4  Paths and simple paths   cpths 29803
            17.3.5  Closed walks   cclwlks 29863
            17.3.6  Circuits and cycles   ccrcts 29877
            ​*17.3.7  Walks as words   cwwlks 29918
            17.3.8  Walks/paths of length 2 (as length 3 strings)   2wlkdlem1 30018
            17.3.9  Walks in regular graphs   rusgrnumwwlkl1 30064
            ​*17.3.10  Closed walks as words   cclwwlk 30076
                  17.3.10.1  Closed walks as words   cclwwlk 30076
                  17.3.10.2  Closed walks of a fixed length as words   cclwwlkn 30119
                  17.3.10.3  Closed walks on a vertex of a fixed length as words   cclwwlknon 30182
            17.3.11  Examples for walks, trails and paths   0ewlk 30209
            17.3.12  Connected graphs   cconngr 30281
      ​17.4  Eulerian paths and the Konigsberg Bridge problem
            ​*17.4.1  Eulerian paths   ceupth 30292
            ​*17.4.2  The Königsberg Bridge problem   konigsbergvtx 30341
      ​17.5  The Friendship Theorem
            17.5.1  Friendship graphs - basics   cfrgr 30353
            17.5.2  The friendship theorem for small graphs   frgr1v 30366
            17.5.3  Theorems according to Mertzios and Unger   2pthfrgrrn 30377
            ​*17.5.4  Huneke's Proof of the Friendship Theorem   frgrncvvdeqlem1 30394
​PART 18  GUIDES AND MISCELLANEA
      ​18.1  Guides (conventions, explanations, and examples)
            ​*18.1.1  Conventions   conventions 30495
            18.1.2  Natural deduction   natded 30498
            ​*18.1.3  Natural deduction examples   ex-natded5.2 30499
            18.1.4  Definitional examples   ex-or 30516
            18.1.5  Other examples   aevdemo 30555
      ​18.2  Humor
            18.2.1  April Fool's theorem   avril1 30558
      ​18.3  (Future - to be reviewed and classified)
            18.3.1  Planar incidence geometry   cplig 30570
            ​*18.3.2  Aliases kept to prevent broken links   dummylink 30583
​​*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 30585
            19.1.2  Abelian groups   cablo 30640
      ​19.2  Complex vector spaces
            19.2.1  Definition and basic properties   cvc 30654
            19.2.2  Examples of complex vector spaces   cnaddabloOLD 30677
      ​19.3  Normed complex vector spaces
            19.3.1  Definition and basic properties   cnv 30680
            19.3.2  Examples of normed complex vector spaces   cnnv 30773
            19.3.3  Induced metric of a normed complex vector space   imsval 30781
            19.3.4  Inner product   cdip 30796
            19.3.5  Subspaces   css 30817
      ​19.4  Operators on complex vector spaces
            19.4.1  Definitions and basic properties   clno 30836
      ​19.5  Inner product (pre-Hilbert) spaces
            19.5.1  Definition and basic properties   ccphlo 30908
            19.5.2  Examples of pre-Hilbert spaces   cncph 30915
            19.5.3  Properties of pre-Hilbert spaces   isph 30918
      ​19.6  Complex Banach spaces
            19.6.1  Definition and basic properties   ccbn 30958
            19.6.2  Examples of complex Banach spaces   cnbn 30965
            19.6.3  Uniform Boundedness Theorem   ubthlem1 30966
            19.6.4  Minimizing Vector Theorem   minvecolem1 30970
      ​19.7  Complex Hilbert spaces
            19.7.1  Definition and basic properties   chlo 30981
            19.7.2  Standard axioms for a complex Hilbert space   hlex 30994
            19.7.3  Examples of complex Hilbert spaces   cnchl 31012
            19.7.4  Hellinger-Toeplitz Theorem   htthlem 31013
​​*PART 20  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      ​20.1  Axiomatization of complex pre-Hilbert spaces
            20.1.1  Basic Hilbert space definitions   chba 31015
            20.1.2  Preliminary ZFC lemmas   df-hnorm 31064
            ​*20.1.3  Derive the Hilbert space axioms from ZFC set theory   axhilex-zf 31077
            ​*20.1.4  Introduce the vector space axioms for a Hilbert space   ax-hilex 31095
            20.1.5  Vector operations   hvmulex 31107
            20.1.6  Inner product postulates for a Hilbert space   ax-hfi 31175
      ​20.2  Inner product and norms
            20.2.1  Inner product   his5 31182
            20.2.2  Norms   dfhnorm2 31218
            20.2.3  Relate Hilbert space to normed complex vector spaces   hilablo 31256
            20.2.4  Bunjakovaskij-Cauchy-Schwarz inequality   bcsiALT 31275
      ​20.3  Cauchy sequences and completeness axiom
            20.3.1  Cauchy sequences and limits   hcau 31280
            20.3.2  Derivation of the completeness axiom from ZF set theory   hilmet 31290
            20.3.3  Completeness postulate for a Hilbert space   ax-hcompl 31298
            20.3.4  Relate Hilbert space to ZFC pre-Hilbert and Hilbert spaces   hhcms 31299
      ​20.4  Subspaces and projections
            20.4.1  Subspaces   df-sh 31303
            20.4.2  Closed subspaces   df-ch 31317
            20.4.3  Orthocomplements   df-oc 31348
            20.4.4  Subspace sum, span, lattice join, lattice supremum   df-shs 31404
            20.4.5  Projection theorem   pjhthlem1 31487
            20.4.6  Projectors   df-pjh 31491
      ​20.5  Properties of Hilbert subspaces
            20.5.1  Orthomodular law   omlsilem 31498
            20.5.2  Projectors (cont.)   pjhtheu2 31512
            20.5.3  Hilbert lattice operations   sh0le 31536
            20.5.4  Span (cont.) and one-dimensional subspaces   spansn0 31637
            20.5.5  Commutes relation for Hilbert lattice elements   df-cm 31679
            20.5.6  Foulis-Holland theorem   fh1 31714
            20.5.7  Quantum Logic Explorer axioms   qlax1i 31723
            20.5.8  Orthogonal subspaces   chscllem1 31733
            20.5.9  Orthoarguesian laws 5OA and 3OA   5oalem1 31750
            20.5.10  Projectors (cont.)   pjorthi 31765
            20.5.11  Mayet's equation E_3   mayete3i 31824
      ​20.6  Operators on Hilbert spaces
            ​*20.6.1  Operator sum, difference, and scalar multiplication   df-hosum 31826
            20.6.2  Zero and identity operators   df-h0op 31844
            20.6.3  Operations on Hilbert space operators   hoaddcl 31854
            20.6.4  Linear, continuous, bounded, Hermitian, unitary operators and norms   df-nmop 31935
            20.6.5  Linear and continuous functionals and norms   df-nmfn 31941
            20.6.6  Adjoint   df-adjh 31945
            20.6.7  Dirac bra-ket notation   df-bra 31946
            20.6.8  Positive operators   df-leop 31948
            20.6.9  Eigenvectors, eigenvalues, spectrum   df-eigvec 31949
            20.6.10  Theorems about operators and functionals   nmopval 31952
            20.6.11  Riesz lemma   riesz3i 32158
            20.6.12  Adjoints (cont.)   cnlnadjlem1 32163
            20.6.13  Quantum computation error bound theorem   unierri 32200
            20.6.14  Dirac bra-ket notation (cont.)   branmfn 32201
            20.6.15  Positive operators (cont.)   leopg 32218
            20.6.16  Projectors as operators   pjhmopi 32242
      ​20.7  States on a Hilbert lattice and Godowski's equation
            20.7.1  States on a Hilbert lattice   df-st 32307
            20.7.2  Godowski's equation   golem1 32367
      ​20.8  Cover relation, atoms, exchange axiom, and modular symmetry
            20.8.1  Covers relation; modular pairs   df-cv 32375
            20.8.2  Atoms   df-at 32434
            20.8.3  Superposition principle   superpos 32450
            20.8.4  Atoms, exchange and covering properties, atomicity   chcv1 32451
            20.8.5  Irreducibility   chirredlem1 32486
            20.8.6  Atoms (cont.)   atcvat3i 32492
            20.8.7  Modular symmetry   mdsymlem1 32499
​PART 21  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      ​21.1  Mathboxes for user contributions
            21.1.1  Mathbox guidelines   mathbox 32538
      ​21.2  Mathbox for Stefan Allan
      ​21.3  Mathbox for Thierry Arnoux
            21.3.1  Propositional Calculus - misc additions   ad11antr 32543
            21.3.2  Predicate Calculus   sbc2iedf 32559
                  21.3.2.1  Predicate Calculus - misc additions   sbc2iedf 32559
                  21.3.2.2  Restricted quantification - misc additions   ralcom4f 32561
                  21.3.2.3  Equality   eqtrb 32568
                  21.3.2.4  Double restricted existential uniqueness quantification   opsbc2ie 32570
                  21.3.2.5  Double restricted existential uniqueness quantification syntax   w2reu 32572
                  21.3.2.6  Substitution (without distinct variables) - misc additions   sbceqbidf 32581
                  21.3.2.7  Existential "at most one" - misc additions   mo5f 32583
                  21.3.2.8  Existential uniqueness - misc additions   reuxfrdf 32585
                  21.3.2.9  Restricted "at most one" - misc additions   rmoxfrd 32587
                  21.3.2.10  Restricted iota (description binder)   riotaeqbidva 32590
            21.3.3  General Set Theory   dmrab 32591
                  21.3.3.1  Class abstractions (a.k.a. class builders)   dmrab 32591
                  21.3.3.2  Image Sets   abrexdomjm 32602
                  21.3.3.3  Set relations and operations - misc additions   nelun 32608
                  21.3.3.4  Unordered pairs   elpreq 32623
                  21.3.3.5  Unordered triples   tpssg 32632
                  21.3.3.6  Conditional operator - misc additions   ifeqeqx 32637
                  21.3.3.7  Set union   uniinn0 32646
                  21.3.3.8  Indexed union - misc additions   cbviunf 32651
                  21.3.3.9  Indexed intersection - misc additions   iinabrex 32665
                  21.3.3.10  Disjointness - misc additions   disjnf 32666
            21.3.4  Relations and Functions   xpdisjres 32694
                  21.3.4.1  Relations - misc additions   xpdisjres 32694
                  21.3.4.2  Functions - misc additions   fconst7v 32719
                  21.3.4.3  Operations - misc additions   mpomptxf 32777
                  21.3.4.4  The mapping operation   elmaprd 32779
                  21.3.4.5  Support of a function   suppovss 32780
                  21.3.4.6  Explicit Functions with one or two points as a domain   cosnopne 32793
                  21.3.4.7  Isomorphisms - misc. additions   gtiso 32800
                  21.3.4.8  Disjointness (additional proof requiring functions)   disjdsct 32802
                  21.3.4.9  First and second members of an ordered pair - misc additions   df1stres 32803
                  21.3.4.10  Countable Sets   snct 32811
            21.3.5  Real and Complex Numbers   sgnval2 32834
                  21.3.5.1  Complex operations - misc. additions   creq0 32835
                  21.3.5.2  Ordering on reals - misc additions   lt2addrd 32849
                  21.3.5.3  Extended reals - misc additions   nn0mnfxrd 32850
                  21.3.5.4  Extended nonnegative integers - misc additions   xnn0gt0 32868
                  21.3.5.5  Real number intervals - misc additions   joiniooico 32873
                  21.3.5.6  Finite intervals of integers - misc additions   uzssico 32883
                  21.3.5.7  Half-open integer ranges - misc additions   iundisjfi 32895
                  21.3.5.8  The ` # ` (set size) function - misc additions   hashunif 32905
                  21.3.5.9  The greatest common divisor operator - misc. additions   elq2 32911
                  21.3.5.10  Integers   nn0split01 32917
                  21.3.5.11  Decimal numbers   dfdec100 32929
            21.3.6  Real and complex functions   sgncl 32930
                  21.3.6.1  Signum (sgn or sign) function - misc. additions   sgncl 32930
                  21.3.6.2  Integer powers - misc. additions   nexple 32943
                  21.3.6.3  Indicator Functions (continued)   indsumin 32947
            ​*21.3.7  Decimal expansion   cdp2 32956
                  ​*21.3.7.1  Decimal point   cdp 32973
                  21.3.7.2  Division in the extended real number system   cxdiv 33002
            21.3.8  Words over a set - misc additions   wrdres 33021
                  21.3.8.1  Splicing words (substring replacement)   splfv3 33044
                  21.3.8.2  Cyclic shift of words   1cshid 33045
            21.3.9  Extensible Structures   ressplusf 33049
                  21.3.9.1  Structure restriction operator   ressplusf 33049
                  21.3.9.2  Posets   ressprs 33052
                  21.3.9.3  Complete lattices   clatp0cl 33062
                  21.3.9.4  Order Theory   cmnt 33064
                  21.3.9.5  Extended reals Structure - misc additions   ax-xrssca 33090
                  21.3.9.6  The extended nonnegative real numbers commutative monoid   xrge00 33100
            21.3.10  Algebra   mndcld 33108
                  21.3.10.1  Monoids   mndcld 33108
                  21.3.10.2  Monoids Homomorphisms   abliso 33122
                  21.3.10.3  Groups - misc additions   grpidcld 33126
                  21.3.10.4  Abelian Groups - misc additions   ablcomd 33133
                  21.3.10.5  Finitely supported group sums - misc additions   gsumsubg 33134
                  21.3.10.6  Group or monoid sums over words   gsumwun 33164
                  21.3.10.7  Centralizers and centers - misc additions   cntzun 33167
                  21.3.10.8  The symmetric group   symgfcoeu 33170
                  21.3.10.9  Transpositions   pmtridf1o 33182
                  21.3.10.10  Permutation Signs   psgnid 33185
                  21.3.10.11  Permutation cycles   ctocyc 33194
                  21.3.10.12  The Alternating Group   evpmval 33233
                  21.3.10.13  Signum in an ordered monoid   csgns 33246
                  21.3.10.14  Fixed points   cfxp 33251
                  21.3.10.15  The Archimedean property for generic ordered algebraic structures   cinftm 33264
                  21.3.10.16  Semiring left modules   cslmd 33288
                  21.3.10.17  Simple groups   prmsimpcyc 33316
                  21.3.10.18  Rings - misc additions   ringrngd 33317
                  21.3.10.19  Subrings generated by a set   elrgspnlem1 33330
                  21.3.10.20  The zero ring   irrednzr 33338
                  21.3.10.21  Localization of rings   cerl 33341
                  21.3.10.22  Integral Domains   domnmuln0rd 33362
                  21.3.10.23  Euclidean Domains   ceuf 33379
                  21.3.10.24  Division Rings   ringinveu 33385
                  21.3.10.25  The field of rational numbers   qfld 33388
                  21.3.10.26  Subfields   subsdrg 33389
                  21.3.10.27  Field of fractions   cfrac 33393
                  21.3.10.28  Field extensions generated by a set   cfldgen 33401
                  21.3.10.29  Ring homomorphisms - misc additions   rhmdvd 33414
                  21.3.10.30  Scalar restriction operation   cresv 33416
                  21.3.10.31  The commutative ring of gaussian integers   gzcrng 33431
                  21.3.10.32  The archimedean ordered field of real numbers   cnfldfld 33432
                  21.3.10.33  The quotient map and quotient modules   qusker 33439
                  21.3.10.34  The ring of integers modulo ` N `   znfermltl 33456
                  21.3.10.35  Independent sets and families   islinds5 33457
                  21.3.10.36  Ring associates, ring units   dvdsruassoi 33474
                  ​*21.3.10.37  Subgroup sum / Sumset / Minkowski sum   elgrplsmsn 33480
                  21.3.10.38  The quotient map   quslsm 33495
                  21.3.10.39  Ideals   lidlmcld 33509
                  21.3.10.40  Prime Ideals   cprmidl 33525
                  21.3.10.41  Maximal Ideals   cmxidl 33549
                  21.3.10.42  The semiring of ideals of a ring   cidlsrg 33590
                  21.3.10.43  Prime Elements   rprmval 33606
                  21.3.10.44  Unique factorization domains   cufd 33628
                  21.3.10.45  The ring of integers   zringidom 33641
                  21.3.10.46  Associative Algebra   assaassd 33645
                  21.3.10.47  Univariate Polynomials   0ringmon1p 33647
                  21.3.10.48  Polynomial quotient and polynomial remainder   q1pdir 33693
                  21.3.10.49  Multivariate Polynomials   psrbasfsupp 33702
                  21.3.10.50  The ring of symmetric polynomials   csply 33746
                  21.3.10.51  The subring algebra   sra1r 33772
                  21.3.10.52  Division Ring Extensions   drgext0g 33781
                  21.3.10.53  Vector Spaces   lvecdimfi 33787
                  21.3.10.54  Vector Space Dimension   cldim 33790
            21.3.11  Field Extensions   cfldext 33829
                  21.3.11.1  Algebraic numbers   cirng 33874
                  21.3.11.2  Algebraic extensions   calgext 33886
                  21.3.11.3  Minimal polynomials   cminply 33890
                  21.3.11.4  Quadratic Field Extensions   rtelextdg2lem 33917
                  21.3.11.5  Towers of quadratic extentions   fldext2chn 33919
            ​*21.3.12  Constructible Numbers   cconstr 33920
                  21.3.12.1  Impossible constructions   2sqr3minply 33971
            21.3.13  Matrices   csmat 33984
                  21.3.13.1  Submatrices   csmat 33984
                  21.3.13.2  Matrix literals   clmat 34002
                  21.3.13.3  Laplace expansion of determinants   mdetpmtr1 34014
            21.3.14  Topology   ist0cld 34024
                  21.3.14.1  Open maps   txomap 34025
                  21.3.14.2  Topology of the unit circle   qtopt1 34026
                  21.3.14.3  Refinements   reff 34030
                  21.3.14.4  Open cover refinement property   ccref 34033
                  21.3.14.5  Lindelöf spaces   cldlf 34043
                  21.3.14.6  Paracompact spaces   cpcmp 34046
                  ​*21.3.14.7  Spectrum of a ring   crspec 34053
                  21.3.14.8  Pseudometrics   cmetid 34077
                  21.3.14.9  Continuity - misc additions   hauseqcn 34089
                  21.3.14.10  Topology of the closed unit interval   elunitge0 34090
                  21.3.14.11  Topology of ` ( RR X. RR ) `   unicls 34094
                  21.3.14.12  Order topology - misc. additions   cnvordtrestixx 34104
                  21.3.14.13  Continuity in topological spaces - misc. additions   mndpluscn 34117
                  21.3.14.14  Topology of the extended nonnegative real numbers ordered monoid   xrge0hmph 34123
                  21.3.14.15  Limits - misc additions   lmlim 34138
                  21.3.14.16  Univariate polynomials   pl1cn 34146
            21.3.15  Uniform Stuctures and Spaces   chcmp 34147
                  21.3.15.1  Hausdorff uniform completion   chcmp 34147
            21.3.16  Topology and algebraic structures   zringnm 34149
                  21.3.16.1  The norm on the ring of the integer numbers   zringnm 34149
                  21.3.16.2  Topological ` ZZ ` -modules   zlm0 34151
                  21.3.16.3  Canonical embedding of the field of the rational numbers into a division ring   cqqh 34161
                  21.3.16.4  Canonical embedding of the real numbers into a complete ordered field   crrh 34184
                  21.3.16.5  Embedding from the extended real numbers into a complete lattice   cxrh 34207
                  21.3.16.6  Canonical embeddings into the ordered field of the real numbers   zrhre 34210
                  ​*21.3.16.7  Topological Manifolds   cmntop 34213
                  21.3.16.8  Extended sum   cesum 34218
            21.3.17  Mixed Function/Constant operation   cofc 34286
            21.3.18  Abstract measure   csiga 34299
                  21.3.18.1  Sigma-Algebra   csiga 34299
                  21.3.18.2  Generated sigma-Algebra   csigagen 34329
                  ​*21.3.18.3  lambda and pi-Systems, Rings of Sets   ispisys 34343
                  21.3.18.4  The Borel algebra on the real numbers   cbrsiga 34372
                  21.3.18.5  Product Sigma-Algebra   csx 34379
                  21.3.18.6  Measures   cmeas 34386
                  21.3.18.7  The counting measure   cntmeas 34417
                  21.3.18.8  The Lebesgue measure - misc additions   voliune 34420
                  21.3.18.9  The Dirac delta measure   cdde 34423
                  21.3.18.10  The 'almost everywhere' relation   cae 34428
                  21.3.18.11  Measurable functions   cmbfm 34440
                  21.3.18.12  Borel Algebra on ` ( RR X. RR ) `   br2base 34460
                  ​*21.3.18.13  Caratheodory's extension theorem   coms 34482
            21.3.19  Integration   itgeq12dv 34517
                  21.3.19.1  Lebesgue integral - misc additions   itgeq12dv 34517
                  21.3.19.2  Bochner integral   citgm 34518
            21.3.20  Euler's partition theorem   oddpwdc 34545
            21.3.21  Sequences defined by strong recursion   csseq 34574
            21.3.22  Fibonacci Numbers   cfib 34587
            21.3.23  Probability   cprb 34598
                  21.3.23.1  Probability Theory   cprb 34598
                  21.3.23.2  Conditional Probabilities   ccprob 34622
                  21.3.23.3  Real-valued Random Variables   crrv 34631
                  21.3.23.4  Preimage set mapping operator   corvc 34647
                  21.3.23.5  Distribution Functions   orvcelval 34660
                  21.3.23.6  Cumulative Distribution Functions   orvclteel 34664
                  21.3.23.7  Probabilities - example   coinfliplem 34670
                  21.3.23.8  Bertrand's Ballot Problem   ballotlemoex 34677
            21.3.24  Signum (sgn or sign) function - misc. additions   fzssfzo 34730
                  21.3.24.1  Operations on words   ccatmulgnn0dir 34733
            21.3.25  Polynomials with real coefficients - misc additions   plymul02 34737
            21.3.26  Descartes's rule of signs   signspval 34743
                  21.3.26.1  Sign changes in a word over real numbers   signspval 34743
                  21.3.26.2  Counting sign changes in a word over real numbers   signslema 34753
            21.3.27  Number Theory   iblidicc 34783
                  21.3.27.1  Representations of a number as sums of integers   crepr 34799
                  21.3.27.2  Vinogradov Trigonometric Sums and the Circle Method   cvts 34826
                  21.3.27.3  The Ternary Goldbach Conjecture: Final Statement   ax-hgt749 34835
            21.3.28  Elementary Geometry   cstrkg2d 34855
                  ​*21.3.28.1  Two-dimensional geometry   cstrkg2d 34855
                  21.3.28.2  Outer Five Segment (not used, no need to move to main)   cafs 34860
            ​*21.3.29  LeftPad Project   clpad 34865
      ​​*21.4  Mathbox for Jonathan Ben-Naim
            21.4.1  First-order logic and set theory   bnj170 34888
            21.4.2  Well founded induction and recursion   bnj110 35047
            21.4.3  The existence of a minimal element in certain classes   bnj69 35199
            21.4.4  Well-founded induction   bnj1204 35201
            21.4.5  Well-founded recursion, part 1 of 3   bnj60 35251
            21.4.6  Well-founded recursion, part 2 of 3   bnj1500 35257
            21.4.7  Well-founded recursion, part 3 of 3   bnj1522 35261
      ​21.5  Mathbox for BTernaryTau
            21.5.1  First-order logic   nfan1c 35262
                  21.5.1.1  Auxiliary axiom schemes   nfan1c 35262
            21.5.2  ZF set theory   exdifsn 35268
                  21.5.2.1  Finitism   prcinf 35304
                  21.5.2.2  Introduce ax-regs   ax-regs 35317
                  21.5.2.3  Derive ax-regs   axregs 35330
                  21.5.2.4  Global choice   gblacfnacd 35331
            21.5.3  Real and complex numbers   zltp1ne 35339
            21.5.4  Graph theory   lfuhgr 35347
                  21.5.4.1  Acyclic graphs   cacycgr 35371
      ​21.6  Mathbox for Mario Carneiro
            21.6.1  Predicate calculus with all distinct variables   ax-7d 35388
            21.6.2  Miscellaneous stuff   quartfull 35394
            21.6.3  Derangements and the Subfactorial   deranglem 35395
            21.6.4  The Erdős-Szekeres theorem   erdszelem1 35420
            21.6.5  The Kuratowski closure-complement theorem   kur14lem1 35435
            21.6.6  Retracts and sections   cretr 35446
            21.6.7  Path-connected and simply connected spaces   cpconn 35448
            21.6.8  Covering maps   ccvm 35484
            21.6.9  Normal numbers   snmlff 35558
            21.6.10  Godel-sets of formulas - part 1   cgoe 35562
            21.6.11  Godel-sets of formulas - part 2   cgon 35661
            21.6.12  Models of ZF   cgze 35675
            ​*21.6.13  Metamath formal systems   cmcn 35689
            21.6.14  Grammatical formal systems   cm0s 35814
            21.6.15  Models of formal systems   cmuv 35834
            21.6.16  Splitting fields   ccpms 35856
            21.6.17  p-adic number fields   czr 35876
      ​​*21.7  Mathbox for Filip Cernatescu
      ​21.8  Mathbox for Paul Chapman
            21.8.1  Real and complex numbers (cont.)   climuzcnv 35900
            21.8.2  Miscellaneous theorems   elfzm12 35904
      ​21.9  Mathbox for Hongxiu Chen
      ​21.10  Mathbox for Adrian Ducourtial
            21.10.1  Propositional calculus   currybi 35917
            21.10.2  Clone theory   ccloneop 35924
      ​21.11  Mathbox for Scott Fenton
            21.11.1  ZFC Axioms in primitive form   axextprim 35930
            21.11.2  Untangled classes   untelirr 35937
            21.11.3  Extra propositional calculus theorems   3jaodd 35944
            21.11.4  Misc. Useful Theorems   nepss 35947
            21.11.5  Properties of real and complex numbers   sqdivzi 35957
            21.11.6  Infinite products   iprodefisumlem 35969
            21.11.7  Factorial limits   faclimlem1 35972
            21.11.8  Greatest common divisor and divisibility   gcd32 35978
            21.11.9  Properties of relationships   dftr6 35980
            21.11.10  Properties of functions and mappings   funpsstri 35995
            21.11.11  Ordinal numbers   elpotr 36008
            21.11.12  Defined equality axioms   axextdfeq 36024
            21.11.13  Hypothesis builders   hbntg 36032
            21.11.14  Well-founded zero, successor, and limits   cwsuc 36037
            21.11.15  Quantifier-free definitions   ctxp 36057
            21.11.16  Alternate ordered pairs   caltop 36185
            21.11.17  Geometry in the Euclidean space   cofs 36211
                  21.11.17.1  Congruence properties   cofs 36211
                  21.11.17.2  Betweenness properties   btwntriv2 36241
                  21.11.17.3  Segment Transportation   ctransport 36258
                  21.11.17.4  Properties relating betweenness and congruence   cifs 36264
                  21.11.17.5  Connectivity of betweenness   btwnconn1lem1 36316
                  21.11.17.6  Segment less than or equal to   csegle 36335
                  21.11.17.7  Outside-of relationship   coutsideof 36348
                  21.11.17.8  Lines and Rays   cline2 36363
            21.11.18  Forward difference   cfwddif 36387
            21.11.19  Rank theorems   rankung 36395
            21.11.20  Hereditarily Finite Sets   chf 36401
      ​21.12  Mathbox for Gino Giotto
            21.12.1  Equality theorems   rmoeqi 36416
                  21.12.1.1  Inference versions   rmoeqi 36416
                  21.12.1.2  Deduction versions   rmoeqdv 36441
            21.12.2  Change bound variables   in-ax8 36453
                  21.12.2.1  Change bound variables and domains   cbvralvw2 36455
                  21.12.2.2  Change bound variables, deduction versions   cbvmodavw 36479
                  21.12.2.3  Change bound variables and domains, deduction versions   cbvrmodavw2 36512
            21.12.3  Study of ax-mulf usage   mpomulnzcnf 36528
      ​21.13  Mathbox for Jeff Hankins
            21.13.1  Miscellany   a1i14 36529
            21.13.2  Basic topological facts   topbnd 36553
            21.13.3  Topology of the real numbers   ivthALT 36564
            21.13.4  Refinements   cfne 36565
            21.13.5  Neighborhood bases determine topologies   neibastop1 36588
            21.13.6  Lattice structure of topologies   topmtcl 36592
            21.13.7  Filter bases   fgmin 36599
            21.13.8  Directed sets, nets   tailfval 36601
      ​21.14  Mathbox for Anthony Hart
            21.14.1  Propositional Calculus   tb-ax1 36612
            21.14.2  Predicate Calculus   nalfal 36632
            21.14.3  Miscellaneous single axioms   meran1 36640
            21.14.4  Connective Symmetry   negsym1 36646
      ​21.15  Mathbox for Chen-Pang He
            21.15.1  Ordinal topology   ontopbas 36657
      ​21.16  Mathbox for Jeff Hoffman
            21.16.1  Inferences for finite induction on generic function values   fveleq 36680
            21.16.2  gdc.mm   nnssi2 36684
      ​21.17  Mathbox for Matthew House
            21.17.1  Relations on well-ordered indexed unions   weiunval 36691
            21.17.2  Axiom of Transitive Containment   axtco 36700
            21.17.3  Transitive closure of a class   tr0elw 36713
            ​*21.17.4  Stronger axioms of regularity   mh-setind 36765
            21.17.5  Short axioms written in primitive symbols   mh-inf3f1 36770
      ​21.18  Mathbox for Asger C. Ipsen
            21.18.1  Continuous nowhere differentiable functions   dnival 36778
      ​​*21.19  Mathbox for BJ
            *21.19.1  Propositional calculus   bj-mp2c 36847
                  *21.19.1.1  Derived rules of inference   bj-mp2c 36847
                  ​*21.19.1.2  A syntactic theorem   bj-0 36849
                  ​*21.19.1.3  Minimal implicational calculus   bj-a1k 36851
                  ​*21.19.1.4  Positive calculus   bj-bisimpl 36864
                  ​*21.19.1.5  Implication and negation   bj-con2com 36872
                  ​*21.19.1.6  Disjunction   bj-jaoi1 36883
                  ​*21.19.1.7  Logical equivalence   bj-dfbi4 36885
                  21.19.1.8  The conditional operator for propositions   bj-consensus 36890
                  ​*21.19.1.9  Propositional calculus: miscellaneous   bj-imbi12 36895
            ​*21.19.2  Modal logic   bj-axdd2 36904
            ​*21.19.3  Provability logic   cprvb 36909
            ​*21.19.4  First-order logic   bj-exexalal 36918
                  21.19.4.1  Universal and existential quantifiers, nonfreeness predicate   bj-exexalal 36918
                  21.19.4.2  Adding ax-gen   bj-genr 36919
                  21.19.4.3  Adding ax-4   bj-almp 36923
                  21.19.4.4  Adding ax-5   bj-spvw 36976
                  21.19.4.5  Equality and substitution   bj-df-sb 36991
                  21.19.4.6  Adding ax-6   bj-spim0 37010
                  21.19.4.7  Adding ax-7   bj-cbvexw 37018
                  21.19.4.8  Membership predicate, ax-8 and ax-9   bj-ax89 37020
                  21.19.4.9  Adding ax-11   bj-alcomexcom 37022
                  21.19.4.10  Adding ax-12   axc11n11 37026
                  ​*21.19.4.11  Really adding ax-12   bj-substax12 37068
                  21.19.4.12  Nonfreeness   wnnf 37070
                  21.19.4.13  Adding ax-13   bj-axc10 37137
                  ​*21.19.4.14  Removing dependencies on ax-13 (and ax-11)   bj-axc10v 37147
                  ​*21.19.4.15  Distinct var metavariables   bj-hbaeb2 37172
                  ​*21.19.4.16  Around ~ equsal   bj-equsal1t 37176
                  ​*21.19.4.17  Some Principia Mathematica proofs   stdpc5t 37181
                  21.19.4.18  Alternate definition of substitution   bj-sbsb 37191
                  21.19.4.19  Lemmas for substitution   bj-sbf3 37193
                  21.19.4.20  Existential uniqueness   bj-eu3f 37195
                  ​*21.19.4.21  First-order logic: miscellaneous   bj-sblem1 37196
            21.19.5  Set theory   eliminable1 37213
                  ​*21.19.5.1  Eliminability of class terms   eliminable1 37213
                  ​*21.19.5.2  Classes without the axiom of extensionality   bj-denoteslem 37225
                  21.19.5.3  Characterization among sets versus among classes   elelb 37251
                  ​*21.19.5.4  The nonfreeness quantifier for classes   bj-nfcsym 37253
                  ​*21.19.5.5  Lemmas for class substitution   bj-sbeqALT 37254
                  21.19.5.6  Removing some axiom requirements and disjoint variable conditions   bj-exlimvmpi 37265
                  ​*21.19.5.7  Class abstractions   bj-elabd2ALT 37279
                  21.19.5.8  Generalized class abstractions   bj-cgab 37287
                  ​*21.19.5.9  Restricted nonfreeness   wrnf 37295
                  ​*21.19.5.10  Russell's paradox   bj-ru1 37297
                  21.19.5.11  Curry's paradox in set theory   currysetlem 37299
                  ​*21.19.5.12  Some disjointness results   bj-n0i 37305
                  ​*21.19.5.13  Complements on direct products   bj-xpimasn 37309
                  ​*21.19.5.14  "Singletonization" and tagging   bj-snsetex 37317
                  ​*21.19.5.15  Tuples of classes   bj-cproj 37344
                  ​*21.19.5.16  Set theory: elementary operations relative to a universe   bj-rcleqf 37379
                  ​*21.19.5.17  Axioms for finite unions   bj-abex 37384
                  ​*21.19.5.18  Set theory: miscellaneous   eleq2w2ALT 37401
                  ​*21.19.5.19  Axioms of separation and replacement   bj-axnul 37426
                  ​*21.19.5.20  Evaluation at a class   bj-evaleq 37430
                  21.19.5.21  Elementwise operations   celwise 37438
                  ​*21.19.5.22  Elementwise intersection (families of sets induced on a subset)   bj-rest00 37440
                  21.19.5.23  Moore collections (complements)   bj-raldifsn 37459
                  21.19.5.24  Maps-to notation for functions with three arguments   bj-0nelmpt 37475
                  ​*21.19.5.25  Currying   csethom 37481
                  ​*21.19.5.26  Setting components of extensible structures   cstrset 37493
            ​*21.19.6  Extended real and complex numbers, real and complex projective lines   bj-nfald 37496
                  21.19.6.1  Complements on class abstractions of ordered pairs and binary relations   bj-nfald 37496
                  ​*21.19.6.2  Identity relation (complements)   bj-opabssvv 37511
                  ​*21.19.6.3  Functionalized identity (diagonal in a Cartesian square)   cdiag2 37533
                  ​*21.19.6.4  Direct image and inverse image   cimdir 37539
                  ​*21.19.6.5  Extended numbers and projective lines as sets   cfractemp 37557
                  ​*21.19.6.6  Addition and opposite   caddcc 37598
                  ​*21.19.6.7  Order relation on the extended reals   cltxr 37602
                  ​*21.19.6.8  Argument, multiplication and inverse   carg 37604
                  21.19.6.9  The canonical bijection from the finite ordinals   ciomnn 37610
                  21.19.6.10  Divisibility   cnnbar 37621
            ​*21.19.7  Monoids   bj-smgrpssmgm 37629
                  ​*21.19.7.1  Finite sums in monoids   cfinsum 37644
            ​*21.19.8  Affine, Euclidean, and Cartesian geometry   bj-fvimacnv0 37647
                  *21.19.8.1  Real vector spaces   bj-fvimacnv0 37647
                  ​*21.19.8.2  Complex numbers (supplements)   bj-subcom 37669
                  ​*21.19.8.3  Barycentric coordinates   bj-bary1lem 37671
            21.19.9  Monoid of endomorphisms   cend 37674
      ​21.20  Mathbox for Jim Kingdon
            21.20.1  Circle constant   taupilem3 37680
            21.20.2  Number theory   dfgcd3 37685
            21.20.3  Real numbers   irrdifflemf 37686
      ​21.21  Mathbox for ML
            21.21.1  Miscellaneous   csbrecsg 37691
            21.21.2  Cartesian exponentiation   cfinxp 37746
            21.21.3  Topology   iunctb2 37766
                  ​*21.21.3.1  Pi-base theorems   pibp16 37776
      ​21.22  Mathbox for Wolf Lammen
            21.22.1  1. Bootstrapping   wl-section-boot 37785
            21.22.2  Implication chains   wl-section-impchain 37809
            21.22.3  Theorems around the conditional operator   wl-ifp-ncond1 37827
            21.22.4  Alternative development of hadd, cadd   wl-df-3xor 37831
            21.22.5  An alternative axiom ~ ax-13   ax-wl-13v 37856
            21.22.6  Bootstrapping set theory with classes   wl-cleq-0 37858
            21.22.7  Other stuff   wl-mps 37879
      ​21.23  Mathbox for Brendan Leahy
      ​21.24  Mathbox for Jeff Madsen
            21.24.1  Logic and set theory   unirep 38082
            21.24.2  Real and complex numbers; integers   filbcmb 38108
            21.24.3  Sequences and sums   sdclem2 38110
            21.24.4  Topology   subspopn 38120
            21.24.5  Metric spaces   metf1o 38123
            21.24.6  Continuous maps and homeomorphisms   constcncf 38130
            21.24.7  Boundedness   ctotbnd 38134
            21.24.8  Isometries   cismty 38166
            21.24.9  Heine-Borel Theorem   heibor1lem 38177
            21.24.10  Banach Fixed Point Theorem   bfplem1 38190
            21.24.11  Euclidean space   crrn 38193
            21.24.12  Intervals (continued)   ismrer1 38206
            21.24.13  Operation properties   cass 38210
            21.24.14  Groups and related structures   cmagm 38216
            21.24.15  Group homomorphism and isomorphism   cghomOLD 38251
            21.24.16  Rings   crngo 38262
            21.24.17  Division Rings   cdrng 38316
            21.24.18  Ring homomorphisms   crngohom 38328
            21.24.19  Commutative rings   ccm2 38357
            21.24.20  Ideals   cidl 38375
            21.24.21  Prime rings and integral domains   cprrng 38414
            21.24.22  Ideal generators   cigen 38427
      ​21.25  Mathbox for Giovanni Mascellani
            ​*21.25.1  Tools for automatic proof building   efald2 38446
            ​*21.25.2  Tseitin axioms   fald 38497
            ​*21.25.3  Equality deductions   iuneq2f 38524
            ​*21.25.4  Miscellanea   orcomdd 38535
      ​21.26  Mathbox for Peter Mazsa
            21.26.1  Notations   cxrn 38542
            21.26.2  Preparatory theorems   el2v1 38597
            21.26.3  Range Cartesian product   df-xrn 38748
            21.26.4  Relations   df-rels 38808
            21.26.5  Quotient map (coset map)   df-qmap 38814
            21.26.6  Lifts, shifts, successor, and predecessor   df-adjliftmap 38823
            21.26.7  Cosets by ` R `   df-coss 38869
            21.26.8  Subset relations   df-ssr 38946
            21.26.9  Reflexivity   df-refs 38958
            21.26.10  Converse reflexivity   df-cnvrefs 38973
            21.26.11  Symmetry   df-syms 38990
            21.26.12  Reflexivity and symmetry   symrefref2 39015
            21.26.13  Transitivity   df-trs 39024
            21.26.14  Equivalence relations   df-eqvrels 39036
            21.26.15  Redundancy   df-redunds 39075
            21.26.16  Domain quotients   df-dmqss 39090
            21.26.17  Equivalence relations on domain quotients   df-ers 39116
            21.26.18  Functions   df-funss 39133
            21.26.19  Disjoints vs. converse functions   df-disjss 39156
            21.26.20  Antisymmetry   df-antisymrel 39231
            21.26.21  Partitions: disjoints on domain quotients   df-parts 39236
            21.26.22  Partition-Equivalence Theorems   disjim 39252
            21.26.23  Type-safe Partition-Equivalence: PetParts, PetErs, Pet2Parts, Pet2Ers   df-petparts 39336
      ​21.27  Mathbox for Rodolfo Medina
            21.27.1  Partitions   prtlem60 39346
      ​​*21.28  Mathbox for Norm Megill
            *21.28.1  Obsolete schemes ax-c4,c5,c7,c10,c11,c11n,c15,c9,c14,c16   ax-c5 39376
            ​*21.28.2  Rederive new axioms ax-4, ax-10, ax-6, ax-12, ax-13 from old   axc5 39386
            ​*21.28.3  Legacy theorems using obsolete axioms   ax5ALT 39400
            21.28.4  Experiments with weak deduction theorem   elimhyps 39454
            21.28.5  Miscellanea   cnaddcom 39465
            21.28.6  Atoms, hyperplanes, and covering in a left vector space (or module)   clsa 39467
            21.28.7  Functionals and kernels of a left vector space (or module)   clfn 39550
            21.28.8  Opposite rings and dual vector spaces   cld 39616
            21.28.9  Ortholattices and orthomodular lattices   cops 39665
            21.28.10  Atomic lattices with covering property   ccvr 39755
            21.28.11  Hilbert lattices   chlt 39843
            21.28.12  Projective geometries based on Hilbert lattices   clln 39984
            21.28.13  Construction of a vector space from a Hilbert lattice   cdlema1N 40284
            21.28.14  Construction of involution and inner product from a Hilbert lattice   clpoN 41973
      ​21.29  Mathbox for metakunt
            21.29.1  Commutative Semiring   ccsrg 42455
            21.29.2  General helpful statements   rhmzrhval 42458
            21.29.3  Some gcd and lcm results   12gcd5e1 42489
            21.29.4  Least common multiple inequality theorem   3factsumint1 42507
            21.29.5  Logarithm inequalities   3exp7 42539
            21.29.6  Miscellaneous results for AKS formalisation   intlewftc 42547
            21.29.7  Sticks and stones   sticksstones1 42632
            21.29.8  Continuation AKS   aks6d1c6lem1 42656
      ​21.30  Mathbox for Steven Nguyen
            21.30.1  Utility theorems   jarrii 42691
            ​*21.30.2  Arithmetic theorems   c0exALT 42737
            21.30.3  Exponents and divisibility   oexpreposd 42800
            21.30.4  Trigonometry and Calculus   tanhalfpim 42827
            ​*21.30.5  Independence of ax-mulcom   cresub 42843
            21.30.6  Structures   sn-base0 42986
            ​*21.30.7  Projective spaces   cprjsp 43052
            21.30.8  Basic reductions for Fermat's Last Theorem   dffltz 43085
            ​*21.30.9  Exemplar theorems   iddii 43115
                  ​*21.30.9.1  Standard replacements of ax-10 , ax-11 , ax-12   nfa1w 43126
      ​21.31  Mathbox for Igor Ieskov
      ​21.32  Mathbox for OpenAI
      ​21.33  Mathbox for Stefan O'Rear
            21.33.1  Additional elementary logic and set theory   moxfr 43142
            21.33.2  Additional theory of functions   imaiinfv 43143
            21.33.3  Additional topology   elrfi 43144
            21.33.4  Characterization of closure operators. Kuratowski closure axioms   ismrcd1 43148
            21.33.5  Algebraic closure systems   cnacs 43152
            21.33.6  Miscellanea 1. Map utilities   constmap 43163
            21.33.7  Miscellanea for polynomials   mptfcl 43170
            21.33.8  Multivariate polynomials over the integers   cmzpcl 43171
            21.33.9  Miscellanea for Diophantine sets 1   coeq0i 43203
            21.33.10  Diophantine sets 1: definitions   cdioph 43205
            21.33.11  Diophantine sets 2 miscellanea   ellz1 43217
            21.33.12  Diophantine sets 2: union and intersection. Monotone Boolean algebra   diophin 43222
            21.33.13  Diophantine sets 3: construction   diophrex 43225
            21.33.14  Diophantine sets 4 miscellanea   2sbcrex 43234
            21.33.15  Diophantine sets 4: Quantification   rexrabdioph 43240
            21.33.16  Diophantine sets 5: Arithmetic sets   rabdiophlem1 43247
            21.33.17  Diophantine sets 6: reusability. renumbering of variables   eldioph4b 43257
            21.33.18  Pigeonhole Principle and cardinality helpers   fphpd 43262
            21.33.19  A non-closed set of reals is infinite   rencldnfilem 43266
            21.33.20  Lagrange's rational approximation theorem   irrapxlem1 43268
            21.33.21  Pell equations 1: A nontrivial solution always exists   pellexlem1 43275
            21.33.22  Pell equations 2: Algebraic number theory of the solution set   csquarenn 43282
            21.33.23  Pell equations 3: characterizing fundamental solution   infmrgelbi 43324
            ​*21.33.24  Logarithm laws generalized to an arbitrary base   reglogcl 43336
            21.33.25  Pell equations 4: the positive solution group is infinite cyclic   pellfund14 43344
            21.33.26  X and Y sequences 1: Definition and recurrence laws   crmx 43346
            21.33.27  Ordering and induction lemmas for the integers   monotuz 43387
            21.33.28  X and Y sequences 2: Order properties   rmxypos 43393
            21.33.29  Congruential equations   congtr 43411
            21.33.30  Alternating congruential equations   acongid 43421
            21.33.31  Additional theorems on integer divisibility   coprmdvdsb 43431
            21.33.32  X and Y sequences 3: Divisibility properties   jm2.18 43434
            21.33.33  X and Y sequences 4: Diophantine representability of Y   jm2.27a 43451
            21.33.34  X and Y sequences 5: Diophantine representability of X, ^, _C   rmxdiophlem 43461
            21.33.35  Uncategorized stuff not associated with a major project   setindtr 43470
            21.33.36  More equivalents of the Axiom of Choice   axac10 43479
            21.33.37  Finitely generated left modules   clfig 43513
            21.33.38  Noetherian left modules I   clnm 43521
            21.33.39  Addenda for structure powers   pwssplit4 43535
            21.33.40  Every set admits a group structure iff choice   unxpwdom3 43541
            21.33.41  Noetherian rings and left modules II   clnr 43555
            21.33.42  Hilbert's Basis Theorem   cldgis 43567
            21.33.43  Additional material on polynomials [DEPRECATED]   cmnc 43577
            21.33.44  Degree and minimal polynomial of algebraic numbers   cdgraa 43586
            21.33.45  Algebraic integers I   citgo 43603
            21.33.46  Endomorphism algebra   cmend 43617
            21.33.47  Cyclic groups and order   idomodle 43637
            21.33.48  Cyclotomic polynomials   ccytp 43643
            21.33.49  Miscellaneous topology   fgraphopab 43649
      ​21.34  Mathbox for Noam Pasman
      ​21.35  Mathbox for Jon Pennant
      ​21.36  Mathbox for Richard Penner
            21.36.1  Set Theory and Ordinal Numbers   uniel 43663
            21.36.2  Natural addition of Cantor normal forms   oawordex2 43772
            21.36.3  Surreal Contributions   abeqabi 43853
            21.36.4  Short Studies   nlimsuc 43886
                  21.36.4.1  Additional work on conditional logical operator   ifpan123g 43904
                  21.36.4.2  Sophisms   rp-fakeimass 43957
                  ​*21.36.4.3  Finite Sets   rp-isfinite5 43962
                  21.36.4.4  General Observations   intabssd 43964
                  21.36.4.5  Infinite Sets   pwelg 44005
                  ​*21.36.4.6  Finite intersection property   fipjust 44010
                  21.36.4.7  RP ADDTO: Subclasses and subsets   rababg 44019
                  21.36.4.8  RP ADDTO: The intersection of a class   elinintab 44020
                  21.36.4.9  RP ADDTO: Theorems requiring subset and intersection existence   elinintrab 44022
                  21.36.4.10  RP ADDTO: Relations   xpinintabd 44025
                  ​*21.36.4.11  RP ADDTO: Functions   elmapintab 44041
                  ​*21.36.4.12  RP ADDTO: Finite induction (for finite ordinals)   cnvcnvintabd 44045
                  21.36.4.13  RP ADDTO: First and second members of an ordered pair   elcnvlem 44046
                  21.36.4.14  RP ADDTO: The reflexive and transitive properties of relations   undmrnresiss 44049
                  21.36.4.15  RP ADDTO: Basic properties of closures   cleq2lem 44053
                  21.36.4.16  RP REPLACE: Definitions and basic properties of transitive closures   trcleq2lemRP 44075
                  ​*21.36.4.17  Additions for square root; absolute value   sqrtcvallem1 44076
            21.36.5  Additional statements on relations and subclasses   al3im 44092
                  21.36.5.1  Transitive relations (not to be confused with transitive classes)   trrelind 44110
                  21.36.5.2  Reflexive closures   crcl 44117
                  ​*21.36.5.3  Finite relationship composition   relexp2 44122
                  21.36.5.4  Transitive closure of a relation   dftrcl3 44165
                  ​*21.36.5.5  Adapted from Frege   frege77d 44191
            ​*21.36.6  Propositions from _Begriffsschrift_   dfxor4 44211
                  *21.36.6.1  _Begriffsschrift_ Chapter I   dfxor4 44211
                  ​*21.36.6.2  _Begriffsschrift_ Notation hints   whe 44217
                  21.36.6.3  _Begriffsschrift_ Chapter II Implication   ax-frege1 44235
                  21.36.6.4  _Begriffsschrift_ Chapter II Implication and Negation   axfrege28 44274
                  ​*21.36.6.5  _Begriffsschrift_ Chapter II with logical equivalence   axfrege52a 44301
                  21.36.6.6  _Begriffsschrift_ Chapter II with equivalence of sets   axfrege52c 44332
                  ​*21.36.6.7  _Begriffsschrift_ Chapter II with equivalence of classes   frege53c 44359
                  ​*21.36.6.8  _Begriffsschrift_ Chapter III Properties hereditary in a sequence   dffrege69 44377
                  ​*21.36.6.9  _Begriffsschrift_ Chapter III Following in a sequence   dffrege76 44384
                  ​*21.36.6.10  _Begriffsschrift_ Chapter III Member of sequence   dffrege99 44407
                  ​*21.36.6.11  _Begriffsschrift_ Chapter III Single-valued procedures   dffrege115 44423
            ​*21.36.7  Exploring Topology via Seifert and Threlfall   enrelmap 44442
                  *21.36.7.1  Equinumerosity of sets of relations and maps   enrelmap 44442
                  ​*21.36.7.2  Generic Pseudoclosure Spaces, Pseudointerior Spaces, and Pseudoneighborhoods   or3or 44468
                  ​*21.36.7.3  Generic Neighborhood Spaces   gneispa 44575
            ​*21.36.8  Exploring Higher Homotopy via Kerodon   k0004lem1 44592
                  *21.36.8.1  Simplicial Sets   k0004lem1 44592
      ​21.37  Mathbox for Stanislas Polu
            21.37.1  IMO Problems   wwlemuld 44601
                  21.37.1.1  IMO 1972 B2   wwlemuld 44601
            ​*21.37.2  INT Inequalities Proof Generator   int-addcomd 44618
            ​*21.37.3  N-Digit Addition Proof Generator   unitadd 44640
            21.37.4  AM-GM (for k = 2,3,4)   gsumws3 44641
      ​21.38  Mathbox for Rohan Ridenour
            21.38.1  Misc   spALT 44646
            21.38.2  Monoid rings   cmnring 44656
            21.38.3  Shorter primitive equivalent of ax-groth   gru0eld 44674
                  21.38.3.1  Grothendieck universes are closed under collection   gru0eld 44674
                  21.38.3.2  Minimal universes   ismnu 44706
                  21.38.3.3  Primitive equivalent of ax-groth   expandan 44733
      ​21.39  Mathbox for Steve Rodriguez
            21.39.1  Miscellanea   nanorxor 44750
            21.39.2  Ratio test for infinite series convergence and divergence   dvgrat 44757
            21.39.3  Multiples   reldvds 44760
            21.39.4  Function operations   caofcan 44768
            21.39.5  Calculus   lhe4.4ex1a 44774
            21.39.6  The generalized binomial coefficient operation   cbcc 44781
            21.39.7  Binomial series   uzmptshftfval 44791
      ​21.40  Mathbox for Andrew Salmon
            21.40.1  Principia Mathematica * 10   pm10.12 44803
            21.40.2  Principia Mathematica * 11   2alanimi 44817
            21.40.3  Predicate Calculus   sbeqal1 44843
            21.40.4  Principia Mathematica * 13 and * 14   pm13.13a 44852
            21.40.5  Set Theory   elnev 44882
            21.40.6  Arithmetic   addcomgi 44900
            21.40.7  Geometry   cplusr 44901
      ​​*21.41  Mathbox for Alan Sare
            21.41.1  Auxiliary theorems for the Virtual Deduction tool   idiALT 44923
            21.41.2  Supplementary unification deductions   bi1imp 44927
            21.41.3  Conventional Metamath proofs, some derived from VD proofs   iidn3 44946
            21.41.4  What is Virtual Deduction?   wvd1 45014
            21.41.5  Virtual Deduction Theorems   df-vd1 45015
            21.41.6  Theorems proved using Virtual Deduction   trsspwALT 45262
            21.41.7  Theorems proved using Virtual Deduction with mmj2 assistance   simplbi2VD 45290
            21.41.8  Virtual Deduction transcriptions of textbook proofs   sb5ALTVD 45357
            21.41.9  Theorems proved using conjunction-form Virtual Deduction   elpwgdedVD 45361
            21.41.10  Theorems with a VD proof in conventional notation derived from a VD proof   suctrALT3 45368
            ​*21.41.11  Theorems with a proof in conventional notation derived from a VD proof   notnotrALT2 45371
      ​21.42  Mathbox for Eric Schmidt
            21.42.1  Miscellany   rspesbcd 45382
            21.42.2  Study of dfbi1ALT   dfbi1ALTa 45384
            21.42.3  Relation-preserving functions   wrelp 45387
            21.42.4  Orbits   orbitex 45400
            21.42.5  Well-founded sets   trwf 45404
            21.42.6  Absoluteness in transitive models   ralabso 45413
            21.42.7  Lemmas for showing axioms hold in models   traxext 45422
            21.42.8  The class of well-founded sets is a model for ZFC   wfaxext 45438
            21.42.9  Permutation models   brpermmodel 45448
      ​21.43  Mathbox for Glauco Siliprandi
            21.43.1  Miscellanea   evth2f 45464
            21.43.2  Functions   fnresdmss 45616
            21.43.3  Ordering on real numbers - Real and complex numbers basic operations   sub2times 45722
            21.43.4  Real intervals   gtnelioc 45937
            21.43.5  Finite sums   fsummulc1f 46017
            21.43.6  Finite multiplication of numbers and finite multiplication of functions   fmul01 46026
            21.43.7  Limits   clim1fr1 46047
                  21.43.7.1  Inferior limit (lim inf)   clsi 46195
                  ​*21.43.7.2  Limits for sequences of extended real numbers   clsxlim 46262
            21.43.8  Trigonometry   coseq0 46308
            21.43.9  Continuous Functions   mulcncff 46314
            21.43.10  Derivatives   dvsinexp 46355
            21.43.11  Integrals   itgsin0pilem1 46394
            21.43.12  Stone Weierstrass theorem - real version   stoweidlem1 46445
            21.43.13  Wallis' product for π   wallispilem1 46509
            21.43.14  Stirling's approximation formula for ` n ` factorial   stirlinglem1 46518
            21.43.15  Dirichlet kernel   dirkerval 46535
            21.43.16  Fourier Series   fourierdlem1 46552
            21.43.17  e is transcendental   elaa2lem 46677
            21.43.18  n-dimensional Euclidean space   rrxtopn 46728
            21.43.19  Basic measure theory   csalg 46752
                  ​*21.43.19.1  σ-Algebras   csalg 46752
                  21.43.19.2  Sum of nonnegative extended reals   csumge0 46806
                  ​*21.43.19.3  Measures   cmea 46893
                  ​*21.43.19.4  Outer measures and Caratheodory's construction   come 46933
                  ​*21.43.19.5  Lebesgue measure on n-dimensional Real numbers   covoln 46980
                  ​*21.43.19.6  Measurable functions   csmblfn 47139
      ​21.44  Mathbox for Saveliy Skresanov
            21.44.1  Ceva's theorem   sigarval 47294
            21.44.2  Simple groups   simpcntrab 47314
      ​21.45  Mathbox for Ender Ting
            21.45.1  Interesting facts   et-ltneverrefl 47315
            21.45.2  Increasing sequences and subsequences   ormklocald 47320
            21.45.3  Scratchpad for number theory   evenwodadd 47333
            21.45.4  Scratchpad for math on real numbers   squeezedltsq 47334
      ​21.46  Mathbox for Jarvin Udandy
      ​21.47  Mathbox for Adhemar
            ​*21.47.1  Minimal implicational calculus   adh-minim 47465
      ​21.48  Mathbox for Alexander van der Vekens
            21.48.1  General auxiliary theorems (1)   n0nsn2el 47489
                  21.48.1.1  Unordered and ordered pairs - extension for singletons   n0nsn2el 47489
                  21.48.1.2  Unordered and ordered pairs - extension for unordered pairs   elprneb 47493
                  21.48.1.3  Unordered and ordered pairs - extension for ordered pairs   oppr 47494
                  21.48.1.4  Relations - extension   eubrv 47499
                  21.48.1.5  Definite description binder (inverted iota) - extension   iota0def 47502
                  21.48.1.6  Functions - extension   fveqvfvv 47504
            21.48.2  Alternative for Russell's definition of a description binder   caiota 47547
            21.48.3  Double restricted existential uniqueness   r19.32 47562
                  21.48.3.1  Restricted quantification (extension)   r19.32 47562
                  21.48.3.2  Restricted uniqueness and "at most one" quantification   reuf1odnf 47571
                  21.48.3.3  Analogs to Existential uniqueness (double quantification)   2reu3 47574
                  21.48.3.4  Additional theorems for double restricted existential uniqueness   2reu8i 47577
            ​*21.48.4  Alternative definitions of function and operation values   wdfat 47580
                  21.48.4.1  Restricted quantification (extension)   ralbinrald 47586
                  21.48.4.2  The universal class (extension)   nvelim 47587
                  21.48.4.3  Introduce the Axiom of Power Sets (extension)   alneu 47588
                  21.48.4.4  Predicate "defined at"   dfateq12d 47590
                  21.48.4.5  Alternative definition of the value of a function   dfafv2 47596
                  21.48.4.6  Alternative definition of the value of an operation   aoveq123d 47642
            ​*21.48.5  Alternative definitions of function values (2)   cafv2 47672
            21.48.6  General auxiliary theorems (2)   an4com24 47732
                  21.48.6.1  Logical conjunction - extension   an4com24 47732
                  21.48.6.2  Abbreviated conjunction and disjunction of three wff's - extension   3an4ancom24 47733
                  21.48.6.3  Negated membership (alternative)   cnelbr 47735
                  21.48.6.4  The empty set - extension   ralralimp 47742
                  21.48.6.5  Indexed union and intersection - extension   otiunsndisjX 47743
                  21.48.6.6  Functions - extension   fvifeq 47744
                  21.48.6.7  Maps-to notation - extension   fvmptrab 47756
                  21.48.6.8  Subtraction - extension   cnambpcma 47758
                  21.48.6.9  Ordering on reals (cont.) - extension   leaddsuble 47761
                  21.48.6.10  Imaginary and complex number properties - extension   readdcnnred 47767
                  21.48.6.11  Nonnegative integers (as a subset of complex numbers) - extension   nn0resubcl 47772
                  21.48.6.12  Integers (as a subset of complex numbers) - extension   zgeltp1eq 47773
                  21.48.6.13  Decimal arithmetic - extension   1t10e1p1e11 47774
                  21.48.6.14  Upper sets of integers - extension   eluzge0nn0 47776
                  21.48.6.15  Infinity and the extended real number system (cont.) - extension   nltle2tri 47777
                  21.48.6.16  Finite intervals of integers - extension   ssfz12 47778
                  21.48.6.17  Half-open integer ranges - extension   fzopred 47787
                  21.48.6.18  The floor and ceiling functions - extension   2ltceilhalf 47796
                  21.48.6.19  The modulo (remainder) operation - extension   fldivmod 47808
                  21.48.6.20  The infinite sequence builder "seq"   smonoord 47841
                  21.48.6.21  Integer powers - extension   2timesltsq 47842
                  21.48.6.22  Finite and infinite sums - extension   fsummsndifre 47844
                  21.48.6.23  The divides relation - extension   nndivides2 47848
                  21.48.6.24  Extensible structures - extension   setsidel 47852
            ​*21.48.7  Preimages of function values   preimafvsnel 47855
            ​*21.48.8  Partitions of real intervals   ciccp 47889
            21.48.9  Shifting functions with an integer range domain   fargshiftfv 47915
            21.48.10  Words over a set (extension)   lswn0 47920
                  21.48.10.1  Last symbol of a word - extension   lswn0 47920
            21.48.11  Unordered pairs   wich 47921
                  21.48.11.1  Interchangeable setvar variables   wich 47921
                  21.48.11.2  Set of unordered pairs   sprid 47950
                  ​*21.48.11.3  Proper (unordered) pairs   prpair 47977
                  21.48.11.4  Set of proper unordered pairs   cprpr 47988
            21.48.12  Number theory (extension)   nprmmul1 48003
                  21.48.12.1  Properties of non-prime numbers   nprmmul1 48003
                  ​*21.48.12.2  Fermat numbers   cfmtno 48006
                  ​*21.48.12.3  Mersenne primes   m2prm 48070
                  21.48.12.4  Proth's theorem   modexp2m1d 48091
                  21.48.12.5  The prime-counting function according to Ján Mináč   nprmdvdsfacm1lem1 48099
                  21.48.12.6  Solutions of quadratic equations   quad1 48112
            ​*21.48.13  Even and odd numbers   ceven 48116
                  21.48.13.1  Definitions and basic properties   ceven 48116
                  21.48.13.2  Alternate definitions using the "divides" relation   dfeven2 48141
                  21.48.13.3  Alternate definitions using the "modulo" operation   dfeven3 48150
                  21.48.13.4  Alternate definitions using the "gcd" operation   iseven5 48156
                  21.48.13.5  Theorems of part 5 revised   zneoALTV 48161
                  21.48.13.6  Theorems of part 6 revised   odd2np1ALTV 48166
                  21.48.13.7  Theorems of AV's mathbox revised   0evenALTV 48180
                  21.48.13.8  Additional theorems   epoo 48195
                  21.48.13.9  Perfect Number Theorem (revised)   perfectALTVlem1 48213
            21.48.14  Number theory (extension 2)   cfppr 48216
                  ​*21.48.14.1  Fermat pseudoprimes   cfppr 48216
                  ​*21.48.14.2  Goldbach's conjectures   cgbe 48237
            21.48.15  Graph theory (extension)   cclnbgr 48310
                  21.48.15.1  Closed neighborhood of a vertex   cclnbgr 48310
                  ​*21.48.15.2  Semiclosed and semiopen neighborhoods (experimental)   dfsclnbgr2 48338
                  21.48.15.3  Induced subgraphs   cisubgr 48352
                  ​*21.48.15.4  Isomorphisms of graphs   cgrisom 48366
                  ​*21.48.15.5  Triangles in graphs   cgrtri 48429
                  ​*21.48.15.6  Star graphs   cstgr 48443
                  ​*21.48.15.7  Local isomorphisms of graphs   cgrlim 48468
                  ​*21.48.15.8  Generalized Petersen graphs   cgpg 48532
                  21.48.15.9  Loop-free graphs - extension   1hegrlfgr 48624
                  21.48.15.10  Walks - extension   cupwlks 48625
                  21.48.15.11  Edges of graphs expressed as sets of unordered pairs   upgredgssspr 48635
            21.48.16  Monoids (extension)   ovn0dmfun 48648
                  21.48.16.1  Auxiliary theorems   ovn0dmfun 48648
                  21.48.16.2  Magmas, Semigroups and Monoids (extension)   plusfreseq 48656
                  21.48.16.3  Examples and counterexamples for magmas, semigroups and monoids (extension)   opmpoismgm 48659
                  21.48.16.4  Group sum operation (extension 1)   gsumsplit2f 48672
            ​*21.48.17  Magmas and internal binary operations (alternate approach)   ccllaw 48675
                  *21.48.17.1  Laws for internal binary operations   ccllaw 48675
                  ​*21.48.17.2  Internal binary operations   cintop 48688
                  21.48.17.3  Alternative definitions for magmas and semigroups   cmgm2 48707
            21.48.18  Rings (extension)   lmod0rng 48721
                  21.48.18.1  Nonzero rings (extension)   lmod0rng 48721
                  21.48.18.2  Ideals as non-unital rings   lidldomn1 48723
                  21.48.18.3  The non-unital ring of even integers   0even 48729
                  21.48.18.4  A constructed not unital ring   cznrnglem 48751
                  ​*21.48.18.5  The category of non-unital rings (alternate definition)   crngcALTV 48755
                  ​*21.48.18.6  The category of (unital) rings (alternate definition)   cringcALTV 48779
            21.48.19  Basic algebraic structures (extension)   eliunxp2 48826
                  21.48.19.1  Auxiliary theorems   eliunxp2 48826
                  21.48.19.2  The binomial coefficient operation (extension)   bcpascm1 48843
                  21.48.19.3  The ` ZZ `-module ` ZZ X. ZZ `   zlmodzxzlmod 48846
                  21.48.19.4  Group sum operation (extension 2)   mgpsumunsn 48853
                  21.48.19.5  Symmetric groups (extension)   exple2lt6 48856
                  21.48.19.6  Divisibility (extension)   invginvrid 48859
                  21.48.19.7  The support of functions (extension)   rmsupp0 48860
                  21.48.19.8  Finitely supported functions (extension)   rmsuppfi 48864
                  21.48.19.9  Left modules (extension)   lmodvsmdi 48871
                  21.48.19.10  Associative algebras (extension)   assaascl0 48873
                  21.48.19.11  Univariate polynomials (extension)   ply1vr1smo 48875
                  21.48.19.12  Univariate polynomials (examples)   linply1 48885
            21.48.20  Linear algebra (extension)   cdmatalt 48888
                  ​*21.48.20.1  The subalgebras of diagonal and scalar matrices (extension)   cdmatalt 48888
                  ​*21.48.20.2  Linear combinations   clinc 48896
                  ​*21.48.20.3  Linear independence   clininds 48932
                  21.48.20.4  Simple left modules and the ` ZZ `-module   lmod1lem1 48979
                  21.48.20.5  Differences between (left) modules and (left) vector spaces   lvecpsslmod 48999
            21.48.21  Complexity theory   suppdm 49002
                  21.48.21.1  Auxiliary theorems   suppdm 49002
                  21.48.21.2  Even and odd integers   nn0onn0ex 49015
                  21.48.21.3  The natural logarithm on complex numbers (extension)   logcxp0 49027
                  21.48.21.4  Division of functions   cfdiv 49029
                  21.48.21.5  Upper bounds   cbigo 49039
                  21.48.21.6  Logarithm to an arbitrary base (extension)   rege1logbrege0 49050
                  ​*21.48.21.7  The binary logarithm   fldivexpfllog2 49057
                  21.48.21.8  Binary length   cblen 49061
                  ​*21.48.21.9  Digits   cdig 49087
                  21.48.21.10  Nonnegative integer as sum of its shifted digits   dignn0flhalflem1 49107
                  21.48.21.11  Algorithms for the multiplication of nonnegative integers   nn0mulfsum 49116
                  ​*21.48.21.12  N-ary functions   cnaryf 49118
                  ​*21.48.21.13  The Ackermann function   citco 49149
            21.48.22  Elementary geometry (extension)   fv1prop 49191
                  21.48.22.1  Auxiliary theorems   fv1prop 49191
                  21.48.22.2  Real euclidean space of dimension 2   rrx2pxel 49203
                  21.48.22.3  Spheres and lines in real Euclidean spaces   cline 49219
      ​21.49  Mathbox for Zhi Wang
            21.49.1  Propositional calculus   pm4.71da 49281
            21.49.2  Predicate calculus with equality   dtrucor3 49290
                  21.49.2.1  Axiom scheme ax-5 (Distinctness)   dtrucor3 49290
            21.49.3  ZF Set Theory - start with the Axiom of Extensionality   ralbidb 49291
                  21.49.3.1  Restricted quantification   ralbidb 49291
                  21.49.3.2  The universal class   reuxfr1dd 49298
                  21.49.3.3  The empty set   ssdisjd 49299
                  21.49.3.4  Unordered and ordered pairs   vsn 49303
                  21.49.3.5  The union of a class   unilbss 49309
                  21.49.3.6  Indexed union and intersection   iuneq0 49310
            21.49.4  ZF Set Theory - add the Axiom of Replacement   inpw 49316
                  21.49.4.1  Theorems requiring subset and intersection existence   inpw 49316
            21.49.5  ZF Set Theory - add the Axiom of Power Sets   opth1neg 49317
                  21.49.5.1  Ordered pair theorem   opth1neg 49317
                  21.49.5.2  Ordered-pair class abstractions (cont.)   brab2dd 49319
                  21.49.5.3  Relations   iinxp 49322
                  21.49.5.4  Functions   mof0 49329
                  21.49.5.5  Operations   ovsng 49349
            21.49.6  ZF Set Theory - add the Axiom of Union   fonex 49358
                  21.49.6.1  Relations and functions (cont.)   fonex 49358
                  21.49.6.2  First and second members of an ordered pair   eloprab1st2nd 49359
                  21.49.6.3  Operations in maps-to notation (continued)   fmpodg 49360
                  21.49.6.4  Function transposition   resinsnlem 49362
                  21.49.6.5  Infinite Cartesian products   ixpv 49381
                  21.49.6.6  Equinumerosity   fvconst0ci 49382
            21.49.7  Order sets   iccin 49387
                  21.49.7.1  Real number intervals   iccin 49387
            21.49.8  Extensible structures   slotresfo 49390
                  21.49.8.1  Basic definitions   slotresfo 49390
            21.49.9  Moore spaces   mreuniss 49391
            ​*21.49.10  Topology   clduni 49392
                  21.49.10.1  Closure and interior   clduni 49392
                  21.49.10.2  Neighborhoods   neircl 49396
                  21.49.10.3  Subspace topologies   restcls2lem 49404
                  21.49.10.4  Limits and continuity in topological spaces   cnneiima 49408
                  21.49.10.5  Topological definitions using the reals   iooii 49409
                  21.49.10.6  Separated sets   sepnsepolem1 49413
                  21.49.10.7  Separated spaces: T0, T1, T2 (Hausdorff) ...   isnrm4 49422
            21.49.11  Preordered sets and directed sets using extensible structures   isprsd 49446
            21.49.12  Posets and lattices using extensible structures   lubeldm2 49447
                  21.49.12.1  Posets   lubeldm2 49447
                  21.49.12.2  Lattices   toslat 49473
                  21.49.12.3  Subset order structures   intubeu 49475
            21.49.13  Rings   elmgpcntrd 49496
                  21.49.13.1  Multiplicative Group   elmgpcntrd 49496
            21.49.14  Associative algebras   asclelbasALT 49497
                  21.49.14.1  Definition and basic properties   asclelbasALT 49497
            21.49.15  Categories   homf0 49500
                  21.49.15.1  Categories   homf0 49500
                  21.49.15.2  Opposite category   oppccatb 49507
                  21.49.15.3  Monomorphisms and epimorphisms   idmon 49511
                  21.49.15.4  Sections, inverses, isomorphisms   sectrcl 49513
                  21.49.15.5  Isomorphic objects   cicfn 49533
                  21.49.15.6  Subcategories   dmdm 49544
                  21.49.15.7  Functors   reldmfunc 49566
                  21.49.15.8  Opposite functors   coppf 49613
                  21.49.15.9  Full & faithful functors   imasubc 49642
                  21.49.15.10  Universal property   upciclem1 49657
                  21.49.15.11  Natural transformations and the functor category   isnatd 49714
                  21.49.15.12  Initial, terminal and zero objects of a category   initoo2 49723
                  21.49.15.13  Product of categories   reldmxpc 49737
                  21.49.15.14  Swap functors   cswapf 49750
                  21.49.15.15  Functor evaluation   oppc1stflem 49778
                  21.49.15.16  Transposed curry functors   cofuswapfcl 49784
                  21.49.15.17  Constant functors   diag1 49795
                  21.49.15.18  Functor composition bifunctors   fucofulem1 49801
                  21.49.15.19  Post-composition functors   postcofval 49855
                  21.49.15.20  Pre-composition functors   precofvallem 49857
            21.49.16  Examples of categories   catcrcl 49886
                  21.49.16.1  The category of categories   catcrcl 49886
                  21.49.16.2  Thin categories   cthinc 49908
                  21.49.16.3  Terminal categories   ctermc 49963
                  21.49.16.4  Preordered sets as thin categories   cprstc 50040
                  21.49.16.5  Monoids as categories   cmndtc 50068
                  21.49.16.6  Categories with at most one object and at most two morphisms   2arwcatlem1 50086
            21.49.17  Kan extensions and related concepts   clan 50096
                  21.49.17.1  Kan extensions   clan 50096
                  21.49.17.2  Limits and colimits   clmd 50134
      ​21.50  Mathbox for Emmett Weisz
            ​*21.50.1  Miscellaneous Theorems   nfintd 50164
            21.50.2  Set Recursion   csetrecs 50174
                  ​*21.50.2.1  Basic Properties of Set Recursion   csetrecs 50174
                  21.50.2.2  Examples and properties of set recursion   elsetrecslem 50190
            ​*21.50.3  Construction of Games and Surreal Numbers   cpg 50200
      ​​*21.51  Mathbox for David A. Wheeler
            21.51.1  Natural deduction   sbidd 50209
            ​*21.51.2  Greater than, greater than or equal to   cge-real 50211
            ​*21.51.3  Hyperbolic trigonometric functions   csinh 50221
            ​*21.51.4  Reciprocal trigonometric functions (sec, csc, cot)   csec 50232
            ​*21.51.5  Identities for "if"   ifnmfalse 50254
            ​*21.51.6  Logarithms generalized to arbitrary base using ` logb `   logb2aval 50255
            ​*21.51.7  Logarithm laws generalized to an arbitrary base - log_   clog- 50256
            ​*21.51.8  Formally define notions such as reflexivity   wreflexive 50258
            ​*21.51.9  Algebra helpers   mvlraddi 50262
            ​*21.51.10  Algebra helper examples   i2linesi 50269
            ​*21.51.11  Formal methods "surprises"   alimp-surprise 50271
            ​*21.51.12  Allsome quantifier   walsi 50277
            ​*21.51.13  Miscellaneous   5m4e1 50288
            21.51.14  Theorems about algebraic numbers   aacllem 50292
      ​21.52  Mathbox for Kunhao Zheng
            21.52.1  Weighted AM-GM inequality   amgmwlem 50293