TOC of Theorem List - Metamath Proof Explorer

Table of Contents Summary

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

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

​​*PART 1  CLASSICAL FIRST-ORDER LOGIC WITH EQUALITY
      ​*1.1  Pre-logic
            *1.1.1  Inferences for assisting proof development   idi 1
      ​​*1.2  Propositional calculus
            1.2.1  Recursively define primitive wffs for propositional calculus   wn 3
            ​*1.2.2  The axioms of propositional calculus   ax-mp 5
            ​*1.2.3  Logical implication   mp2 9
            ​*1.2.4  Logical negation   con4 113
            ​*1.2.5  Logical equivalence   wb 206
            ​*1.2.6  Logical conjunction   wa 395
            ​*1.2.7  Logical disjunction   wo 848
            ​*1.2.8  Mixed connectives   jaao 957
            ​*1.2.9  The conditional operator for propositions   wif 1063
            ​*1.2.10  The weak deduction theorem for propositional calculus   elimh 1083
            1.2.11  Abbreviated conjunction and disjunction of three wff's   w3o 1086
            1.2.12  Logical "nand" (Sheffer stroke)   wnan 1493
            1.2.13  Logical "xor"   wxo 1513
            1.2.14  Logical "nor"   wnor 1530
            1.2.15  True and false constants   wal 1540
                  ​*1.2.15.1  Universal quantifier for use by df-tru   wal 1540
                  ​*1.2.15.2  Equality predicate for use by df-tru   cv 1541
                  1.2.15.3  The true constant   wtru 1543
                  1.2.15.4  The false constant   wfal 1554
            ​*1.2.16  Truth tables   truimtru 1565
                  1.2.16.1  Implication   truimtru 1565
                  1.2.16.2  Negation   nottru 1569
                  1.2.16.3  Equivalence   trubitru 1571
                  1.2.16.4  Conjunction   truantru 1575
                  1.2.16.5  Disjunction   truortru 1579
                  1.2.16.6  Alternative denial   trunantru 1583
                  1.2.16.7  Exclusive disjunction   truxortru 1587
                  1.2.16.8  Joint denial   trunortru 1591
            ​*1.2.17  Half adder and full adder in propositional calculus   whad 1595
                  1.2.17.1  Full adder: sum   whad 1595
                  1.2.17.2  Full adder: carry   wcad 1608
      ​1.3  Other axiomatizations related to classical propositional calculus
            ​*1.3.1  Minimal implicational calculus   minimp 1623
            ​*1.3.2  Implicational Calculus   impsingle 1629
            1.3.3  Derive the Lukasiewicz axioms from Meredith's sole axiom   meredith 1643
            1.3.4  Derive the standard axioms from the Lukasiewicz axioms   luklem1 1660
            ​*1.3.5  Derive Nicod's axiom from the standard axioms   nic-dfim 1671
            1.3.6  Derive the Lukasiewicz axioms from Nicod's axiom   nic-imp 1677
            1.3.7  Derive Nicod's Axiom from Lukasiewicz's First Sheffer Stroke Axiom   lukshef-ax1 1696
            1.3.8  Derive the Lukasiewicz Axioms from the Tarski-Bernays-Wajsberg Axioms   tbw-bijust 1700
            1.3.9  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's First CO Axiom   merco1 1715
            1.3.10  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's Second CO Axiom   merco2 1738
            1.3.11  Derive the Lukasiewicz axioms from the Russell-Bernays Axioms   rb-bijust 1751
            ​*1.3.12  Stoic logic non-modal portion (Chrysippus of Soli)   mptnan 1770
      ​​*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 1781
                  1.4.1.1  Existential quantifier   wex 1781
                  1.4.1.2  Nonfreeness predicate   wnf 1785
            1.4.2  Rule scheme ax-gen (Generalization)   ax-gen 1797
            1.4.3  Axiom scheme ax-4 (Quantified Implication)   ax-4 1811
                  ​*1.4.3.1  The empty domain of discourse   empty 1908
            1.4.4  Axiom scheme ax-5 (Distinctness) - first use of $d   ax-5 1912
            ​*1.4.5  Equality predicate (continued)   weq 1964
            1.4.6  Axiom scheme ax-6 (Existence)   ax-6 1969
            1.4.7  Axiom scheme ax-7 (Equality)   ax-7 2010
            1.4.8  Define proper substitution   sbjust 2067
            1.4.9  Membership predicate   wcel 2114
            1.4.10  Axiom scheme ax-8 (Left Equality for Binary Predicate)   ax-8 2116
            1.4.11  Axiom scheme ax-9 (Right Equality for Binary Predicate)   ax-9 2124
            ​*1.4.12  Logical redundancy of ax-10 , ax-11 , ax-12 , ax-13   ax6dgen 2134
      ​​*1.5  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes)
            1.5.1  Axiom scheme ax-10 (Quantified Negation)   ax-10 2147
            1.5.2  Axiom scheme ax-11 (Quantifier Commutation)   ax-11 2163
            1.5.3  Axiom scheme ax-12 (Substitution)   ax-12 2185
            1.5.4  Axiom scheme ax-13 (Quantified Equality)   ax-13 2377
      ​1.6  Uniqueness and unique existence
            1.6.1  Uniqueness: the at-most-one quantifier   wmo 2538
            1.6.2  Unique existence: the unique existential quantifier   weu 2569
      ​1.7  Other axiomatizations related to classical predicate calculus
            ​*1.7.1  Aristotelian logic: Assertic syllogisms   barbara 2664
            ​*1.7.2  Intuitionistic logic   axia1 2694
​​*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 2709
            2.1.2  Classes   cab 2715
                  2.1.2.1  Class abstractions   cab 2715
                  ​*2.1.2.2  Class equality   df-cleq 2729
                  2.1.2.3  Class membership   df-clel 2812
                  2.1.2.4  Elementary properties of class abstractions   eqabdv 2870
            2.1.3  Class form not-free predicate   wnfc 2884
            2.1.4  Negated equality and membership   wne 2933
                  2.1.4.1  Negated equality   wne 2933
                  2.1.4.2  Negated membership   wnel 3037
            2.1.5  Restricted quantification   wral 3052
                  2.1.5.1  Restricted universal and existential quantification   wral 3052
                  2.1.5.2  Restricted existential uniqueness and at-most-one quantifier   wreu 3350
                  2.1.5.3  Restricted class abstraction   crab 3401
            2.1.6  The universal class   cvv 3442
            ​*2.1.7  Conditional equality (experimental)   wcdeq 3723
            2.1.8  Russell's Paradox   rru 3739
            2.1.9  Proper substitution of classes for sets   wsbc 3742
            2.1.10  Proper substitution of classes for sets into classes   csb 3851
            2.1.11  Define basic set operations and relations   cdif 3900
            2.1.12  Subclasses and subsets   df-ss 3920
            2.1.13  The difference, union, and intersection of two classes   dfdif3 4071
                  2.1.13.1  The difference of two classes   dfdif3 4071
                  2.1.13.2  The union of two classes   elun 4107
                  2.1.13.3  The intersection of two classes   elini 4153
                  2.1.13.4  The symmetric difference of two classes   csymdif 4206
                  2.1.13.5  Combinations of difference, union, and intersection of two classes   unabs 4219
                  2.1.13.6  Class abstractions with difference, union, and intersection of two classes   unabw 4261
                  2.1.13.7  Restricted uniqueness with difference, union, and intersection   reuun2 4279
            2.1.14  The empty set   c0 4287
            ​*2.1.15  The conditional operator for classes   cif 4481
            ​*2.1.16  The weak deduction theorem for set theory   dedth 4540
            2.1.17  Power classes   cpw 4556
            2.1.18  Unordered and ordered pairs   snjust 4581
            2.1.19  The union of a class   cuni 4865
            2.1.20  The intersection of a class   cint 4904
            2.1.21  Indexed union and intersection   ciun 4948
            2.1.22  Disjointness   wdisj 5067
            2.1.23  Binary relations   wbr 5100
            2.1.24  Ordered-pair class abstractions (class builders)   copab 5162
            2.1.25  Functions in maps-to notation   cmpt 5181
            2.1.26  Transitive classes   wtr 5207
      ​2.2  ZF Set Theory - add the Axiom of Replacement
            2.2.1  Introduce the Axiom of Replacement   ax-rep 5226
            2.2.2  Derive the Axiom of Separation   axsepgfromrep 5241
            2.2.3  Derive the Null Set Axiom   axnulALT 5251
            2.2.4  Theorems requiring subset and intersection existence   nalset 5260
            2.2.5  Theorems requiring empty set existence   class2set 5302
      ​2.3  ZF Set Theory - add the Axiom of Power Sets
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 5312
            2.3.2  Derive the Axiom of Pairing   axprlem1 5370
            2.3.3  Ordered pair theorem   opnz 5429
            2.3.4  Ordered-pair class abstractions (cont.)   opabidw 5480
            2.3.5  Power class of union and intersection   pwin 5523
            2.3.6  The identity relation   cid 5526
            2.3.7  The membership relation (or epsilon relation)   cep 5531
            ​*2.3.8  Partial and total orderings   wpo 5538
            2.3.9  Founded and well-ordering relations   wfr 5582
            2.3.10  Relations   cxp 5630
            2.3.11  The Predecessor Class   cpred 6266
            2.3.12  Well-founded induction (variant)   frpomin 6306
            2.3.13  Well-ordered induction   tz6.26 6313
            2.3.14  Ordinals   word 6324
            2.3.15  Definite description binder (inverted iota)   cio 6454
            2.3.16  Functions   wfun 6494
            2.3.17  Cantor's Theorem   canth 7322
            2.3.18  Restricted iota (description binder)   crio 7324
            2.3.19  Operations   co 7368
                  2.3.19.1  Variable-to-class conversion for operations   caovclg 7560
            2.3.20  Maps-to notation   mpondm0 7608
            2.3.21  Function operation   cof 7630
            2.3.22  Proper subset relation   crpss 7677
      ​2.4  ZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 7690
            2.4.2  Ordinals (continued)   epweon 7730
            2.4.3  Transfinite induction   tfi 7805
            2.4.4  The natural numbers (i.e., finite ordinals)   com 7818
            2.4.5  Peano's postulates   peano1 7841
            2.4.6  Finite induction (for finite ordinals)   find 7847
            2.4.7  Relations and functions (cont.)   dmexg 7853
            2.4.8  First and second members of an ordered pair   c1st 7941
            2.4.9  Induction on Cartesian products   frpoins3xpg 8092
            2.4.10  Ordering on Cartesian products   xpord2lem 8094
            2.4.11  Ordering Ordinal Sequences   orderseqlem 8109
            ​*2.4.12  The support of functions   csupp 8112
            ​*2.4.13  Special maps-to operations   opeliunxp2f 8162
            2.4.14  Function transposition   ctpos 8177
            2.4.15  Curry and uncurry   ccur 8217
            2.4.16  Undefined values   cund 8224
            2.4.17  Well-founded recursion   cfrecs 8232
            2.4.18  Well-ordered recursion   cwrecs 8263
            2.4.19  Functions on ordinals; strictly monotone ordinal functions   iunon 8281
            2.4.20  "Strong" transfinite recursion   crecs 8312
            2.4.21  Recursive definition generator   crdg 8350
            2.4.22  Finite recursion   frfnom 8376
            2.4.23  Ordinal arithmetic   c1o 8400
            2.4.24  Natural number arithmetic   nna0 8542
            2.4.25  Natural addition   cnadd 8603
            2.4.26  Equivalence relations and classes   wer 8642
            2.4.27  The mapping operation   cmap 8775
            2.4.28  Infinite Cartesian products   cixp 8847
            2.4.29  Equinumerosity   cen 8892
            2.4.30  Schroeder-Bernstein Theorem   sbthlem1 9027
            2.4.31  Equinumerosity (cont.)   xpf1o 9079
            2.4.32  Finite sets   dif1enlem 9096
            2.4.33  Pigeonhole Principle   phplem1 9140
            2.4.34  Finite sets (cont.)   onomeneq 9150
            2.4.35  Finitely supported functions   cfsupp 9276
            2.4.36  Finite intersections   cfi 9325
            2.4.37  Hall's marriage theorem   marypha1lem 9348
            2.4.38  Supremum and infimum   csup 9355
            2.4.39  Ordinal isomorphism, Hartogs's theorem   coi 9426
            2.4.40  Hartogs function   char 9473
            2.4.41  Weak dominance   cwdom 9481
      ​2.5  ZF Set Theory - add the Axiom of Regularity
            2.5.1  Introduce the Axiom of Regularity   ax-reg 9509
            2.5.2  Axiom of Infinity equivalents   inf0 9542
      ​2.6  ZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-inf 9559
            2.6.2  Existence of omega (the set of natural numbers)   omex 9564
            2.6.3  Cantor normal form   ccnf 9582
            2.6.4  Transitive closure of a relation   cttrcl 9628
            2.6.5  Transitive closure   trcl 9649
            2.6.6  Set induction (or epsilon induction)   setind 9668
            2.6.7  Well-Founded Induction   frmin 9673
            2.6.8  Well-Founded Recursion   frr3g 9680
            2.6.9  Rank   cr1 9686
            2.6.10  Scott's trick; collection principle; Hilbert's epsilon   scottex 9809
            2.6.11  Disjoint union   cdju 9822
            2.6.12  Cardinal numbers   ccrd 9859
            2.6.13  Axiom of Choice equivalents   wac 10037
            ​*2.6.14  Cardinal number arithmetic   undjudom 10090
            2.6.15  The Ackermann bijection   ackbij2lem1 10140
            2.6.16  Cofinality (without Axiom of Choice)   cflem 10167
            2.6.17  Eight inequivalent definitions of finite set   sornom 10199
            2.6.18  Hereditarily size-limited sets without Choice   itunifval 10338
​​*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 10357
            3.1.2  Introduce the Axiom of Dependent Choice   ax-dc 10368
      ​3.2  ZFC Set Theory - add the Axiom of Choice
            3.2.1  Introduce the Axiom of Choice   ax-ac 10381
            3.2.2  AC equivalents: well-ordering, Zorn's lemma   numthcor 10416
            3.2.3  Cardinal number theorems using Axiom of Choice   cardval 10468
            3.2.4  Cardinal number arithmetic using Axiom of Choice   iunctb 10497
            3.2.5  Cofinality using the Axiom of Choice   alephreg 10505
      ​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 10543
            3.4.2  Derivation of the Axiom of Choice   gchaclem 10601
​​*PART 4  TG (TARSKI-GROTHENDIECK) SET THEORY
      ​4.1  Inaccessibles
            4.1.1  Weakly and strongly inaccessible cardinals   cwina 10605
            4.1.2  Weak universes   cwun 10623
            4.1.3  Tarski classes   ctsk 10671
            4.1.4  Grothendieck universes   cgru 10713
      ​4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
            4.2.1  Introduce the Tarski-Grothendieck Axiom   ax-groth 10746
            4.2.2  Derive the Power Set, Infinity and Choice Axioms   grothpw 10749
            4.2.3  Tarski map function   ctskm 10760
​​*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 10767
            5.1.2  Final derivation of real and complex number postulates   axaddf 11068
            5.1.3  Real and complex number postulates restated as axioms   ax-cnex 11094
      ​5.2  Derive the basic properties from the field axioms
            5.2.1  Some deductions from the field axioms for complex numbers   cnex 11119
            5.2.2  Infinity and the extended real number system   cpnf 11175
            5.2.3  Restate the ordering postulates with extended real "less than"   axlttri 11216
            5.2.4  Ordering on reals   lttr 11221
            5.2.5  Initial properties of the complex numbers   mul12 11310
      ​5.3  Real and complex numbers - basic operations
            5.3.1  Addition   add12 11363
            5.3.2  Subtraction   cmin 11376
            5.3.3  Multiplication   kcnktkm1cn 11580
            5.3.4  Ordering on reals (cont.)   gt0ne0 11614
            5.3.5  Reciprocals   ixi 11778
            5.3.6  Division   cdiv 11806
            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
      ​5.4  Integer sets
            5.4.1  Positive integers (as a subset of complex numbers)   cn 12157
            5.4.2  Principle of mathematical induction   nnind 12175
            ​*5.4.3  Decimal representation of numbers   c2 12212
            ​*5.4.4  Some properties of specific numbers   1pneg1e0 12271
            5.4.5  Simple number properties   halfcl 12379
            5.4.6  The Archimedean property   nnunb 12409
            5.4.7  Nonnegative integers (as a subset of complex numbers)   cn0 12413
            ​*5.4.8  Extended nonnegative integers   cxnn0 12486
            5.4.9  Integers (as a subset of complex numbers)   cz 12500
            5.4.10  Decimal arithmetic   cdc 12619
            5.4.11  Upper sets of integers   cuz 12763
            5.4.12  Well-ordering principle for bounded-below sets of integers   uzwo3 12868
            5.4.13  Rational numbers (as a subset of complex numbers)   cq 12873
            5.4.14  Existence of the set of complex numbers   rpnnen1lem2 12902
      ​5.5  Order sets
            5.5.1  Positive reals (as a subset of complex numbers)   crp 12917
            5.5.2  Infinity and the extended real number system (cont.)   cxne 13035
            5.5.3  Supremum and infimum on the extended reals   xrsupexmnf 13232
            5.5.4  Real number intervals   cioo 13273
            5.5.5  Finite intervals of integers   cfz 13435
            ​*5.5.6  Finite intervals of nonnegative integers   elfz2nn0 13546
            5.5.7  Half-open integer ranges   cfzo 13582
      ​5.6  Elementary integer functions
            5.6.1  The floor and ceiling functions   cfl 13722
            5.6.2  The modulo (remainder) operation   cmo 13801
            5.6.3  Miscellaneous theorems about integers   om2uz0i 13882
            5.6.4  Strong induction over upper sets of integers   uzsinds 13922
            5.6.5  Finitely supported functions over the nonnegative integers   fsuppmapnn0fiublem 13925
            5.6.6  The infinite sequence builder "seq" - extension   cseq 13936
            5.6.7  Integer powers   cexp 13996
            5.6.8  Ordered pair theorem for nonnegative integers   nn0le2msqi 14202
            5.6.9  Factorial function   cfa 14208
            5.6.10  The binomial coefficient operation   cbc 14237
            5.6.11  The ` # ` (set size) function   chash 14265
                  5.6.11.1  Proper unordered pairs and triples (sets of size 2 and 3)   hashprlei 14403
                  5.6.11.2  Functions with a domain containing at least two different elements   fundmge2nop0 14437
                  5.6.11.3  Finite induction on the size of the first component of a binary relation   hashdifsnp1 14441
      ​​*5.7  Words over a set
            5.7.1  Definitions and basic theorems   cword 14448
            5.7.2  Last symbol of a word   clsw 14497
            5.7.3  Concatenations of words   cconcat 14505
            5.7.4  Singleton words   cs1 14531
            5.7.5  Concatenations with singleton words   ccatws1cl 14552
            5.7.6  Subwords/substrings   csubstr 14576
            5.7.7  Prefixes of a word   cpfx 14606
            5.7.8  Subwords of subwords   swrdswrdlem 14639
            5.7.9  Subwords and concatenations   pfxcctswrd 14645
            5.7.10  Subwords of concatenations   swrdccatfn 14659
            5.7.11  Splicing words (substring replacement)   csplice 14684
            5.7.12  Reversing words   creverse 14693
            5.7.13  Repeated symbol words   creps 14703
            ​*5.7.14  Cyclical shifts of words   ccsh 14723
            5.7.15  Mapping words by a function   wrdco 14766
            5.7.16  Longer string literals   cs2 14776
      ​​*5.8  Reflexive and transitive closures of relations
            5.8.1  The reflexive and transitive properties of relations   coss12d 14907
            5.8.2  Basic properties of closures   cleq1lem 14917
            5.8.3  Definitions and basic properties of transitive closures   ctcl 14920
            5.8.4  Exponentiation of relations   crelexp 14954
            5.8.5  Reflexive-transitive closure as an indexed union   crtrcl 14990
            ​*5.8.6  Principle of transitive induction.   relexpindlem 14998
      ​5.9  Elementary real and complex functions
            5.9.1  The "shift" operation   cshi 15001
            5.9.2  Signum (sgn or sign) function   csgn 15021
            5.9.3  Real and imaginary parts; conjugate   ccj 15031
            5.9.4  Square root; absolute value   csqrt 15168
      ​5.10  Elementary limits and convergence
            5.10.1  Superior limit (lim sup)   clsp 15405
            5.10.2  Limits   cli 15419
            5.10.3  Finite and infinite sums   csu 15621
            5.10.4  The binomial theorem   binomlem 15764
            5.10.5  The inclusion/exclusion principle   incexclem 15771
            5.10.6  Infinite sums (cont.)   isumshft 15774
            5.10.7  Miscellaneous converging and diverging sequences   divrcnv 15787
            5.10.8  Arithmetic series   arisum 15795
            5.10.9  Geometric series   expcnv 15799
            5.10.10  Ratio test for infinite series convergence   cvgrat 15818
            5.10.11  Mertens' theorem   mertenslem1 15819
            5.10.12  Finite and infinite products   prodf 15822
                  5.10.12.1  Product sequences   prodf 15822
                  5.10.12.2  Non-trivial convergence   ntrivcvg 15832
                  5.10.12.3  Complex products   cprod 15838
                  5.10.12.4  Finite products   fprod 15876
                  5.10.12.5  Infinite products   iprodclim 15933
            5.10.13  Falling and Rising Factorial   cfallfac 15939
            5.10.14  Bernoulli polynomials and sums of k-th powers   cbp 15981
      ​5.11  Elementary trigonometry
            5.11.1  The exponential, sine, and cosine functions   ce 15996
                  5.11.1.1  The circle constant (tau = 2 pi)   ctau 16139
            5.11.2  _e is irrational   eirrlem 16141
      ​5.12  Cardinality of real and complex number subsets
            5.12.1  Countability of integers and rationals   xpnnen 16148
            5.12.2  The reals are uncountable   rpnnen2lem1 16151
​​*PART 6  ELEMENTARY NUMBER THEORY
      ​6.1  Elementary properties of divisibility
            6.1.1  Irrationality of square root of 2   sqrt2irrlem 16185
            6.1.2  Some Number sets are chains of proper subsets   nthruc 16189
            6.1.3  The divides relation   cdvds 16191
            ​*6.1.4  Even and odd numbers   evenelz 16275
            6.1.5  The division algorithm   divalglem0 16332
            6.1.6  Bit sequences   cbits 16358
            6.1.7  The greatest common divisor operator   cgcd 16433
            6.1.8  Bézout's identity   bezoutlem1 16478
            6.1.9  Algorithms   nn0seqcvgd 16509
            6.1.10  Euclid's Algorithm   eucalgval2 16520
            ​*6.1.11  The least common multiple   clcm 16527
            ​*6.1.12  Coprimality and Euclid's lemma   coprmgcdb 16588
            6.1.13  Cancellability of congruences   congr 16603
      ​6.2  Elementary prime number theory
            ​*6.2.1  Elementary properties   cprime 16610
            ​*6.2.2  Coprimality and Euclid's lemma (cont.)   coprm 16650
            6.2.3  Properties of the canonical representation of a rational   cnumer 16672
            6.2.4  Euler's theorem   codz 16702
            6.2.5  Arithmetic modulo a prime number   modprm1div 16737
            6.2.6  Pythagorean Triples   coprimeprodsq 16748
            6.2.7  The prime count function   cpc 16776
            6.2.8  Pocklington's theorem   prmpwdvds 16844
            6.2.9  Infinite primes theorem   unbenlem 16848
            6.2.10  Sum of prime reciprocals   prmreclem1 16856
            6.2.11  Fundamental theorem of arithmetic   1arithlem1 16863
            6.2.12  Lagrange's four-square theorem   cgz 16869
            6.2.13  Van der Waerden's theorem   cvdwa 16905
            6.2.14  Ramsey's theorem   cram 16939
            ​*6.2.15  Primorial function   cprmo 16971
            ​*6.2.16  Prime gaps   prmgaplem1 16989
            6.2.17  Decimal arithmetic (cont.)   dec2dvds 17003
            6.2.18  Cyclical shifts of words (cont.)   cshwsidrepsw 17033
            6.2.19  Specific prime numbers   prmlem0 17045
            6.2.20  Very large primes   1259lem1 17070
​PART 7  BASIC STRUCTURES
      ​7.1  Extensible structures
            ​*7.1.1  Basic definitions   cstr 17085
                  7.1.1.1  Extensible structures as structures with components   cstr 17085
                  7.1.1.2  Substitution of components   csts 17102
                  7.1.1.3  Slots   cslot 17120
                  ​*7.1.1.4  Structure component indices   cnx 17132
                  7.1.1.5  Base sets   cbs 17148
                  7.1.1.6  Base set restrictions   cress 17169
            7.1.2  Slot definitions   cplusg 17189
            7.1.3  Definition of the structure product   crest 17352
            7.1.4  Definition of the structure quotient   cordt 17432
      ​7.2  Moore spaces
            7.2.1  Moore closures   mrcflem 17541
            7.2.2  Independent sets in a Moore system   mrisval 17565
            7.2.3  Algebraic closure systems   isacs 17586
​PART 8  BASIC CATEGORY THEORY
      ​8.1  Categories
            8.1.1  Categories   ccat 17599
            8.1.2  Opposite category   coppc 17646
            8.1.3  Monomorphisms and epimorphisms   cmon 17664
            8.1.4  Sections, inverses, isomorphisms   csect 17680
            ​*8.1.5  Isomorphic objects   ccic 17731
            8.1.6  Subcategories   cssc 17743
            8.1.7  Functors   cfunc 17790
            8.1.8  Full & faithful functors   cful 17840
            8.1.9  Natural transformations and the functor category   cnat 17880
            8.1.10  Initial, terminal and zero objects of a category   cinito 17917
      ​8.2  Arrows (disjointified hom-sets)
            8.2.1  Identity and composition for arrows   cida 17989
      ​8.3  Examples of categories
            8.3.1  The category of sets   csetc 18011
            8.3.2  The category of categories   ccatc 18034
            ​*8.3.3  The category of extensible structures   fncnvimaeqv 18055
      ​8.4  Categorical constructions
            8.4.1  Product of categories   cxpc 18103
            8.4.2  Functor evaluation   cevlf 18144
            8.4.3  Hom functor   chof 18183
​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 18366
            9.5.2  Complete lattices   ccla 18433
            9.5.3  Distributive lattices   cdlat 18455
            9.5.4  Subset order structures   cipo 18462
      ​9.6  Posets, directed sets, and lattices as relations
            ​*9.6.1  Posets and lattices as relations   cps 18499
            9.6.2  Directed sets, nets   cdir 18529
            9.6.3  Chains   cchn 18540
​PART 10  BASIC ALGEBRAIC STRUCTURES
      ​10.1  Monoids
            ​*10.1.1  Magmas   cplusf 18574
            ​*10.1.2  Identity elements   mgmidmo 18597
            ​*10.1.3  Iterated sums in a magma   gsumvalx 18613
            10.1.4  Magma homomorphisms and submagmas   cmgmhm 18627
            ​*10.1.5  Semigroups   csgrp 18655
            ​*10.1.6  Definition and basic properties of monoids   cmnd 18671
            10.1.7  Monoid homomorphisms and submonoids   cmhm 18718
            ​*10.1.8  Iterated sums in a monoid   gsumvallem2 18771
            10.1.9  Free monoids   cfrmd 18784
                  ​*10.1.9.1  Monoid of endofunctions   cefmnd 18805
            10.1.10  Examples and counterexamples for magmas, semigroups and monoids   mgm2nsgrplem1 18855
      ​10.2  Groups
            10.2.1  Definition and basic properties   cgrp 18875
            ​*10.2.2  Group multiple operation   cmg 19009
            10.2.3  Subgroups and Quotient groups   csubg 19062
            ​*10.2.4  Cyclic monoids and groups   cycsubmel 19141
            10.2.5  Elementary theory of group homomorphisms   cghm 19153
            10.2.6  Isomorphisms of groups   cgim 19198
                  10.2.6.1  The first isomorphism theorem of groups   ghmqusnsglem1 19221
            10.2.7  Group actions   cga 19230
            10.2.8  Centralizers and centers   ccntz 19256
            10.2.9  The opposite group   coppg 19286
            10.2.10  Symmetric groups   csymg 19310
                  ​*10.2.10.1  Definition and basic properties   csymg 19310
                  10.2.10.2  Cayley's theorem   cayleylem1 19353
                  10.2.10.3  Permutations fixing one element   symgfix2 19357
                  ​*10.2.10.4  Transpositions in the symmetric group   cpmtr 19382
                  10.2.10.5  The sign of a permutation   cpsgn 19430
            10.2.11  p-Groups and Sylow groups; Sylow's theorems   cod 19465
            10.2.12  Direct products   clsm 19575
                  10.2.12.1  Direct products (extension)   smndlsmidm 19597
            10.2.13  Free groups   cefg 19647
            10.2.14  Abelian groups   ccmn 19721
                  10.2.14.1  Definition and basic properties   ccmn 19721
                  10.2.14.2  Cyclic groups   ccyg 19818
                  10.2.14.3  Group sum operation   gsumval3a 19844
                  10.2.14.4  Group sums over (ranges of) integers   fsfnn0gsumfsffz 19924
                  10.2.14.5  Internal direct products   cdprd 19936
                  10.2.14.6  The Fundamental Theorem of Abelian Groups   ablfacrplem 20008
            10.2.15  Simple groups   csimpg 20033
                  10.2.15.1  Definition and basic properties   csimpg 20033
                  10.2.15.2  Classification of abelian simple groups   ablsimpnosubgd 20047
            10.2.16  Totally ordered monoids and groups   comnd 20060
      ​10.3  Rings
            10.3.1  Multiplicative Group   cmgp 20087
            ​*10.3.2  Non-unital rings ("rngs")   crng 20099
            ​*10.3.3  Ring unity (multiplicative identity)   cur 20128
            10.3.4  Semirings   csrg 20133
                  ​*10.3.4.1  The binomial theorem for semirings   srgbinomlem1 20173
            10.3.5  Unital rings   crg 20180
            10.3.6  Opposite ring   coppr 20284
            10.3.7  Divisibility   cdsr 20302
            10.3.8  Ring primes   crpm 20380
            10.3.9  Homomorphisms of non-unital rings   crnghm 20382
            10.3.10  Ring homomorphisms   crh 20417
            10.3.11  Nonzero rings and zero rings   cnzr 20457
            10.3.12  Local rings   clring 20483
            10.3.13  Subrings   csubrng 20490
                  10.3.13.1  Subrings of non-unital rings   csubrng 20490
                  10.3.13.2  Subrings of unital rings   csubrg 20514
                  10.3.13.3  Subrings generated by a subset   crgspn 20555
            10.3.14  Categories of rings   crngc 20561
                  ​*10.3.14.1  The category of non-unital rings   crngc 20561
                  ​*10.3.14.2  The category of (unital) rings   cringc 20590
                  10.3.14.3  Subcategories of the category of rings   srhmsubclem1 20622
            10.3.15  Left regular elements and domains   crlreg 20636
      ​10.4  Division rings and fields
            10.4.1  Definition and basic properties   cdr 20674
            10.4.2  Sub-division rings   csdrg 20731
            10.4.3  Absolute value (abstract algebra)   cabv 20753
            10.4.4  Star rings   cstf 20782
            10.4.5  Totally ordered rings and fields   corng 20802
      ​10.5  Left modules
            10.5.1  Definition and basic properties   clmod 20823
            10.5.2  Subspaces and spans in a left module   clss 20894
            10.5.3  Homomorphisms and isomorphisms of left modules   clmhm 20983
            10.5.4  Subspace sum; bases for a left module   clbs 21038
      ​10.6  Vector spaces
            10.6.1  Definition and basic properties   clvec 21066
      ​10.7  Subring algebras and ideals
            10.7.1  Subring algebras   csra 21135
            ​*10.7.2  Left ideals and spans   clidl 21173
            10.7.3  Two-sided ideals and quotient rings   c2idl 21216
                  ​*10.7.3.1  Condition for a non-unital ring to be unital   rngqiprng1elbas 21253
            10.7.4  Principal ideal rings. Divisibility in the integers   clpidl 21287
            10.7.5  Principal ideal domains   cpid 21303
      ​10.8  The complex numbers as an algebraic extensible structure
            10.8.1  Definition and basic properties   cpsmet 21305
            ​*10.8.2  Ring of integers   czring 21413
                  ​*10.8.2.1  Example for a condition for a non-unital ring to be unital   pzriprnglem1 21448
            10.8.3  Algebraic constructions based on the complex numbers   czrh 21466
            10.8.4  Signs as subgroup of the complex numbers   cnmsgnsubg 21544
            10.8.5  Embedding of permutation signs into a ring   zrhpsgnmhm 21551
            10.8.6  The ordered field of real numbers   crefld 21571
      ​10.9  Generalized pre-Hilbert and Hilbert spaces
            10.9.1  Definition and basic properties   cphl 21591
            10.9.2  Orthocomplements and closed subspaces   cocv 21627
            10.9.3  Orthogonal projection and orthonormal bases   cpj 21667
​​*PART 11  BASIC LINEAR ALGEBRA
      ​11.1  Vectors and free modules
            *11.1.1  Direct sum of left modules   cdsmm 21698
            ​*11.1.2  Free modules   cfrlm 21713
            ​*11.1.3  Standard basis (unit vectors)   cuvc 21749
            ​*11.1.4  Independent sets and families   clindf 21771
            11.1.5  Characterization of free modules   lmimlbs 21803
      ​11.2  Associative algebras
            11.2.1  Definition and basic properties   casa 21817
      ​11.3  Abstract multivariate polynomials
            11.3.1  Definition and basic properties   cmps 21872
            11.3.2  Polynomial evaluation   ces 22039
            11.3.3  Additional definitions for (multivariate) polynomials   cslv 22083
            ​*11.3.4  Univariate polynomials   cps1 22127
            11.3.5  Univariate polynomial evaluation   ces1 22269
                  11.3.5.1  Specialization of polynomial evaluation as a ring homomorphism   evls1scafv 22322
      ​​*11.4  Matrices
            *11.4.1  The matrix multiplication   cmmul 22346
            ​*11.4.2  Square matrices   cmat 22363
            ​*11.4.3  The matrix algebra   matmulr 22394
            ​*11.4.4  Matrices of dimension 0 and 1   mat0dimbas0 22422
            ​*11.4.5  The subalgebras of diagonal and scalar matrices   cdmat 22444
            ​*11.4.6  Multiplication of a matrix with a "column vector"   cmvmul 22496
            11.4.7  Replacement functions for a square matrix   cmarrep 22512
            11.4.8  Submatrices   csubma 22532
      ​11.5  The determinant
            11.5.1  Definition and basic properties   cmdat 22540
            11.5.2  Determinants of 2 x 2 -matrices   m2detleiblem1 22580
            11.5.3  The matrix adjugate/adjunct   cmadu 22588
            ​*11.5.4  Laplace expansion of determinants (special case)   symgmatr01lem 22609
            11.5.5  Inverse matrix   invrvald 22632
            ​*11.5.6  Cramer's rule   slesolvec 22635
      ​​*11.6  Polynomial matrices
            11.6.1  Basic properties   pmatring 22648
            ​*11.6.2  Constant polynomial matrices   ccpmat 22659
            ​*11.6.3  Collecting coefficients of polynomial matrices   cdecpmat 22718
            ​*11.6.4  Ring isomorphism between polynomial matrices and polynomials over matrices   cpm2mp 22748
      ​​*11.7  The characteristic polynomial
            *11.7.1  Definition and basic properties   cchpmat 22782
            ​*11.7.2  The characteristic factor function G   fvmptnn04if 22805
            ​*11.7.3  The Cayley-Hamilton theorem   cpmadurid 22823
​PART 12  BASIC TOPOLOGY
      ​12.1  Topology
            ​*12.1.1  Topological spaces   ctop 22849
                  12.1.1.1  Topologies   ctop 22849
                  12.1.1.2  Topologies on sets   ctopon 22866
                  12.1.1.3  Topological spaces   ctps 22888
            12.1.2  Topological bases   ctb 22901
            12.1.3  Examples of topologies   distop 22951
            12.1.4  Closure and interior   ccld 22972
            12.1.5  Neighborhoods   cnei 23053
            12.1.6  Limit points and perfect sets   clp 23090
            12.1.7  Subspace topologies   restrcl 23113
            12.1.8  Order topology   ordtbaslem 23144
            12.1.9  Limits and continuity in topological spaces   ccn 23180
            12.1.10  Separated spaces: T0, T1, T2 (Hausdorff) ...   ct0 23262
            12.1.11  Compactness   ccmp 23342
            12.1.12  Bolzano-Weierstrass theorem   bwth 23366
            12.1.13  Connectedness   cconn 23367
            12.1.14  First- and second-countability   c1stc 23393
            12.1.15  Local topological properties   clly 23420
            12.1.16  Refinements   cref 23458
            12.1.17  Compactly generated spaces   ckgen 23489
            12.1.18  Product topologies   ctx 23516
            12.1.19  Continuous function-builders   cnmptid 23617
            12.1.20  Quotient maps and quotient topology   ckq 23649
            12.1.21  Homeomorphisms   chmeo 23709
      ​12.2  Filters and filter bases
            12.2.1  Filter bases   elmptrab 23783
            12.2.2  Filters   cfil 23801
            12.2.3  Ultrafilters   cufil 23855
            12.2.4  Filter limits   cfm 23889
            12.2.5  Extension by continuity   ccnext 24015
            12.2.6  Topological groups   ctmd 24026
            12.2.7  Infinite group sum on topological groups   ctsu 24082
            12.2.8  Topological rings, fields, vector spaces   ctrg 24112
      ​12.3  Uniform Structures and Spaces
            12.3.1  Uniform structures   cust 24156
            12.3.2  The topology induced by an uniform structure   cutop 24186
            12.3.3  Uniform Spaces   cuss 24209
            12.3.4  Uniform continuity   cucn 24230
            12.3.5  Cauchy filters in uniform spaces   ccfilu 24241
            12.3.6  Complete uniform spaces   ccusp 24252
      ​12.4  Metric spaces
            12.4.1  Pseudometric spaces   ispsmet 24260
            12.4.2  Basic metric space properties   cxms 24273
            12.4.3  Metric space balls   blfvalps 24339
            12.4.4  Open sets of a metric space   mopnval 24394
            12.4.5  Continuity in metric spaces   metcnp3 24496
            12.4.6  The uniform structure generated by a metric   metuval 24505
            12.4.7  Examples of metric spaces   dscmet 24528
            ​*12.4.8  Normed algebraic structures   cnm 24532
            12.4.9  Normed space homomorphisms (bounded linear operators)   cnmo 24661
            12.4.10  Topology on the reals   qtopbaslem 24714
            12.4.11  Topological definitions using the reals   cii 24836
            12.4.12  Path homotopy   chtpy 24934
            12.4.13  The fundamental group   cpco 24968
      ​12.5  Metric subcomplex vector spaces
            12.5.1  Subcomplex modules   cclm 25030
            ​*12.5.2  Subcomplex vector spaces   ccvs 25091
            ​*12.5.3  Normed subcomplex vector spaces   isncvsngp 25117
            12.5.4  Subcomplex pre-Hilbert spaces   ccph 25134
            12.5.5  Convergence and completeness   ccfil 25220
            12.5.6  Baire's Category Theorem   bcthlem1 25292
            12.5.7  Banach spaces and subcomplex Hilbert spaces   ccms 25300
                  12.5.7.1  The complete ordered field of the real numbers   retopn 25347
            12.5.8  Euclidean spaces   crrx 25351
            12.5.9  Minimizing Vector Theorem   minveclem1 25392
            12.5.10  Projection Theorem   pjthlem1 25405
​PART 13  BASIC REAL AND COMPLEX ANALYSIS
      ​13.1  Continuity
            13.1.1  Intermediate value theorem   pmltpclem1 25417
      ​13.2  Integrals
            13.2.1  Lebesgue measure   covol 25431
            13.2.2  Lebesgue integration   cmbf 25583
                  13.2.2.1  Lesbesgue integral   cmbf 25583
                  13.2.2.2  Lesbesgue directed integral   cdit 25815
      ​13.3  Derivatives
            13.3.1  Real and complex differentiation   climc 25831
                  13.3.1.1  Derivatives of functions of one complex or real variable   climc 25831
                  13.3.1.2  Results on real differentiation   dvferm1lem 25956
​PART 14  BASIC REAL AND COMPLEX FUNCTIONS
      ​14.1  Polynomials
            14.1.1  Polynomial degrees   cmdg 26026
            14.1.2  The division algorithm for univariate polynomials   cmn1 26099
            14.1.3  Elementary properties of complex polynomials   cply 26157
            14.1.4  The division algorithm for polynomials   cquot 26266
            14.1.5  Algebraic numbers   caa 26290
            14.1.6  Liouville's approximation theorem   aalioulem1 26308
      ​14.2  Sequences and series
            14.2.1  Taylor polynomials and Taylor's theorem   ctayl 26328
            14.2.2  Uniform convergence   culm 26353
            14.2.3  Power series   pserval 26387
      ​14.3  Basic trigonometry
            14.3.1  The exponential, sine, and cosine functions (cont.)   efcn 26421
            14.3.2  Properties of pi = 3.14159...   pilem1 26429
            14.3.3  Mapping of the exponential function   efgh 26518
            14.3.4  The natural logarithm on complex numbers   clog 26531
            ​*14.3.5  Logarithms to an arbitrary base   clogb 26742
            14.3.6  Theorems of Pythagoras, isosceles triangles, and intersecting chords   angval 26779
            14.3.7  Solutions of quadratic, cubic, and quartic equations   quad2 26817
            14.3.8  Inverse trigonometric functions   casin 26840
            14.3.9  The Birthday Problem   log2ublem1 26924
            14.3.10  Areas in R^2   carea 26933
            14.3.11  More miscellaneous converging sequences   rlimcnp 26943
            14.3.12  Inequality of arithmetic and geometric means   cvxcl 26963
            14.3.13  Euler-Mascheroni constant   cem 26970
            14.3.14  Zeta function   czeta 26991
            14.3.15  Gamma function   clgam 26994
      ​14.4  Basic number theory
            14.4.1  Wilson's theorem   wilthlem1 27046
            14.4.2  The Fundamental Theorem of Algebra   ftalem1 27051
            14.4.3  The Basel problem (ζ(2) = π²/6)   basellem1 27059
            14.4.4  Number-theoretical functions   ccht 27069
            14.4.5  Perfect Number Theorem   mersenne 27206
            14.4.6  Characters of Z/nZ   cdchr 27211
            14.4.7  Bertrand's postulate   bcctr 27254
            ​*14.4.8  Quadratic residues and the Legendre symbol   clgs 27273
            ​*14.4.9  Gauss' Lemma   gausslemma2dlem0a 27335
            14.4.10  Quadratic reciprocity   lgseisenlem1 27354
            14.4.11  All primes 4n+1 are the sum of two squares   2sqlem1 27396
            14.4.12  Chebyshev's Weak Prime Number Theorem, Dirichlet's Theorem   chebbnd1lem1 27448
            14.4.13  The Prime Number Theorem   mudivsum 27509
            14.4.14  Ostrowski's theorem   abvcxp 27594
​PART 15  SURREAL NUMBERS
      ​​*15.1  Sign sequence representation and Alling's axioms
            15.1.1  Definitions and initial properties   csur 27619
            15.1.2  Ordering   ltssolem1 27655
            15.1.3  Birthday Function   bdayfo 27657
            15.1.4  Density   fvnobday 27658
            ​*15.1.5  Full-Eta Property   bdayimaon 27673
      ​15.2  Initial consequences of Alling's axioms
            15.2.1  Ordering Theorems   cles 27724
            15.2.2  Birthday Theorems   bdayfun 27756
      ​​*15.3  Conway cut representation
            15.3.1  Conway cuts   cslts 27765
            15.3.2  Zero and One   c0s 27813
            15.3.3  Cuts and Options   cmade 27830
            15.3.4  Cofinality and coinitiality   cofslts 27926
      ​15.4  Induction and recursion
            15.4.1  Induction and recursion on one variable   cnorec 27945
            15.4.2  Induction and recursion on two variables   cnorec2 27956
      ​15.5  Surreal arithmetic
            15.5.1  Addition   cadds 27967
            15.5.2  Negation and Subtraction   cnegs 28027
            15.5.3  Multiplication   cmuls 28114
            15.5.4  Division   cdivs 28195
            15.5.5  Absolute value   cabss 28245
      ​15.6  Subsystems of surreals
            15.6.1  Ordinal numbers   cons 28259
            15.6.2  Surreal recursive sequences   cseqs 28291
            15.6.3  Natural numbers   cn0s 28320
            15.6.4  Integers   czs 28386
            15.6.5  Dyadic fractions   c2s 28418
            15.6.6  Real numbers   creno 28497
​​*PART 16  ELEMENTARY GEOMETRY
      ​16.1  Definition and Tarski's Axioms of Geometry
            16.1.1  Justification for the congruence notation   tgjustf 28557
      ​16.2  Tarskian Geometry
            16.2.1  Congruence   tgcgrcomimp 28561
            16.2.2  Betweenness   tgbtwntriv2 28571
            16.2.3  Dimension   tglowdim1 28584
            16.2.4  Betweenness and Congruence   tgifscgr 28592
            16.2.5  Congruence of a series of points   ccgrg 28594
            16.2.6  Motions   cismt 28616
            16.2.7  Colinearity   tglng 28630
            16.2.8  Connectivity of betweenness   tgbtwnconn1lem1 28656
            16.2.9  Less-than relation in geometric congruences   cleg 28666
            16.2.10  Rays   chlg 28684
            16.2.11  Lines   btwnlng1 28703
            16.2.12  Point inversions   cmir 28736
            16.2.13  Right angles   crag 28777
            16.2.14  Half-planes   islnopp 28823
            16.2.15  Midpoints and Line Mirroring   cmid 28856
            16.2.16  Congruence of angles   ccgra 28891
            16.2.17  Angle Comparisons   cinag 28919
            16.2.18  Congruence Theorems   tgsas1 28938
            16.2.19  Equilateral triangles   ceqlg 28949
      ​16.3  Properties of geometries
            16.3.1  Isomorphisms between geometries   f1otrgds 28953
      ​16.4  Geometry in Hilbert spaces
            16.4.1  Geometry in the complex plane   cchhllem 28971
            16.4.2  Geometry in Euclidean spaces   cee 28972
                  16.4.2.1  Definition of the Euclidean space   cee 28972
                  16.4.2.2  Tarski's axioms for geometry for the Euclidean space   axdimuniq 28998
                  16.4.2.3  EE^n fulfills Tarski's Axioms   ceeng 29062
​​*PART 17  GRAPH THEORY
      ​*17.1  Vertices and edges
            17.1.1  The edge function extractor for extensible structures   cedgf 29073
            ​*17.1.2  Vertices and indexed edges   cvtx 29081
                  17.1.2.1  Definitions and basic properties   cvtx 29081
                  17.1.2.2  The vertices and edges of a graph represented as ordered pair   opvtxval 29088
                  17.1.2.3  The vertices and edges of a graph represented as extensible structure   funvtxdmge2val 29096
                  17.1.2.4  Representations of graphs without edges   snstrvtxval 29122
                  17.1.2.5  Degenerated cases of representations of graphs   vtxval0 29124
            17.1.3  Edges as range of the edge function   cedg 29132
      ​​*17.2  Undirected graphs
            17.2.1  Undirected hypergraphs   cuhgr 29141
            17.2.2  Undirected pseudographs and multigraphs   cupgr 29165
            ​*17.2.3  Loop-free graphs   umgrislfupgrlem 29207
            17.2.4  Edges as subsets of vertices of graphs   uhgredgiedgb 29211
            ​*17.2.5  Undirected simple graphs   cuspgr 29233
            17.2.6  Examples for graphs   usgr0e 29321
            17.2.7  Subgraphs   csubgr 29352
            17.2.8  Finite undirected simple graphs   cfusgr 29401
            17.2.9  Neighbors, complete graphs and universal vertices   cnbgr 29417
                  17.2.9.1  Neighbors   cnbgr 29417
                  17.2.9.2  Universal vertices   cuvtx 29470
                  17.2.9.3  Complete graphs   ccplgr 29494
            17.2.10  Vertex degree   cvtxdg 29551
            ​*17.2.11  Regular graphs   crgr 29641
      ​​*17.3  Walks, paths and cycles
            *17.3.1  Walks   cewlks 29681
            17.3.2  Walks for loop-free graphs   lfgrwlkprop 29771
            17.3.3  Trails   ctrls 29774
            17.3.4  Paths and simple paths   cpths 29795
            17.3.5  Closed walks   cclwlks 29855
            17.3.6  Circuits and cycles   ccrcts 29869
            ​*17.3.7  Walks as words   cwwlks 29910
            17.3.8  Walks/paths of length 2 (as length 3 strings)   2wlkdlem1 30010
            17.3.9  Walks in regular graphs   rusgrnumwwlkl1 30056
            ​*17.3.10  Closed walks as words   cclwwlk 30068
                  17.3.10.1  Closed walks as words   cclwwlk 30068
                  17.3.10.2  Closed walks of a fixed length as words   cclwwlkn 30111
                  17.3.10.3  Closed walks on a vertex of a fixed length as words   cclwwlknon 30174
            17.3.11  Examples for walks, trails and paths   0ewlk 30201
            17.3.12  Connected graphs   cconngr 30273
      ​17.4  Eulerian paths and the Konigsberg Bridge problem
            ​*17.4.1  Eulerian paths   ceupth 30284
            ​*17.4.2  The Königsberg Bridge problem   konigsbergvtx 30333
      ​17.5  The Friendship Theorem
            17.5.1  Friendship graphs - basics   cfrgr 30345
            17.5.2  The friendship theorem for small graphs   frgr1v 30358
            17.5.3  Theorems according to Mertzios and Unger   2pthfrgrrn 30369
            ​*17.5.4  Huneke's Proof of the Friendship Theorem   frgrncvvdeqlem1 30386
​PART 18  GUIDES AND MISCELLANEA
      ​18.1  Guides (conventions, explanations, and examples)
            ​*18.1.1  Conventions   conventions 30487
            18.1.2  Natural deduction   natded 30490
            ​*18.1.3  Natural deduction examples   ex-natded5.2 30491
            18.1.4  Definitional examples   ex-or 30508
            18.1.5  Other examples   aevdemo 30547
      ​18.2  Humor
            18.2.1  April Fool's theorem   avril1 30550
      ​18.3  (Future - to be reviewed and classified)
            18.3.1  Planar incidence geometry   cplig 30561
            ​*18.3.2  Aliases kept to prevent broken links   dummylink 30574
​​*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 30576
            19.1.2  Abelian groups   cablo 30631
      ​19.2  Complex vector spaces
            19.2.1  Definition and basic properties   cvc 30645
            19.2.2  Examples of complex vector spaces   cnaddabloOLD 30668
      ​19.3  Normed complex vector spaces
            19.3.1  Definition and basic properties   cnv 30671
            19.3.2  Examples of normed complex vector spaces   cnnv 30764
            19.3.3  Induced metric of a normed complex vector space   imsval 30772
            19.3.4  Inner product   cdip 30787
            19.3.5  Subspaces   css 30808
      ​19.4  Operators on complex vector spaces
            19.4.1  Definitions and basic properties   clno 30827
      ​19.5  Inner product (pre-Hilbert) spaces
            19.5.1  Definition and basic properties   ccphlo 30899
            19.5.2  Examples of pre-Hilbert spaces   cncph 30906
            19.5.3  Properties of pre-Hilbert spaces   isph 30909
      ​19.6  Complex Banach spaces
            19.6.1  Definition and basic properties   ccbn 30949
            19.6.2  Examples of complex Banach spaces   cnbn 30956
            19.6.3  Uniform Boundedness Theorem   ubthlem1 30957
            19.6.4  Minimizing Vector Theorem   minvecolem1 30961
      ​19.7  Complex Hilbert spaces
            19.7.1  Definition and basic properties   chlo 30972
            19.7.2  Standard axioms for a complex Hilbert space   hlex 30985
            19.7.3  Examples of complex Hilbert spaces   cnchl 31003
            19.7.4  Hellinger-Toeplitz Theorem   htthlem 31004
​​*PART 20  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      ​20.1  Axiomatization of complex pre-Hilbert spaces
            20.1.1  Basic Hilbert space definitions   chba 31006
            20.1.2  Preliminary ZFC lemmas   df-hnorm 31055
            ​*20.1.3  Derive the Hilbert space axioms from ZFC set theory   axhilex-zf 31068
            ​*20.1.4  Introduce the vector space axioms for a Hilbert space   ax-hilex 31086
            20.1.5  Vector operations   hvmulex 31098
            20.1.6  Inner product postulates for a Hilbert space   ax-hfi 31166
      ​20.2  Inner product and norms
            20.2.1  Inner product   his5 31173
            20.2.2  Norms   dfhnorm2 31209
            20.2.3  Relate Hilbert space to normed complex vector spaces   hilablo 31247
            20.2.4  Bunjakovaskij-Cauchy-Schwarz inequality   bcsiALT 31266
      ​20.3  Cauchy sequences and completeness axiom
            20.3.1  Cauchy sequences and limits   hcau 31271
            20.3.2  Derivation of the completeness axiom from ZF set theory   hilmet 31281
            20.3.3  Completeness postulate for a Hilbert space   ax-hcompl 31289
            20.3.4  Relate Hilbert space to ZFC pre-Hilbert and Hilbert spaces   hhcms 31290
      ​20.4  Subspaces and projections
            20.4.1  Subspaces   df-sh 31294
            20.4.2  Closed subspaces   df-ch 31308
            20.4.3  Orthocomplements   df-oc 31339
            20.4.4  Subspace sum, span, lattice join, lattice supremum   df-shs 31395
            20.4.5  Projection theorem   pjhthlem1 31478
            20.4.6  Projectors   df-pjh 31482
      ​20.5  Properties of Hilbert subspaces
            20.5.1  Orthomodular law   omlsilem 31489
            20.5.2  Projectors (cont.)   pjhtheu2 31503
            20.5.3  Hilbert lattice operations   sh0le 31527
            20.5.4  Span (cont.) and one-dimensional subspaces   spansn0 31628
            20.5.5  Commutes relation for Hilbert lattice elements   df-cm 31670
            20.5.6  Foulis-Holland theorem   fh1 31705
            20.5.7  Quantum Logic Explorer axioms   qlax1i 31714
            20.5.8  Orthogonal subspaces   chscllem1 31724
            20.5.9  Orthoarguesian laws 5OA and 3OA   5oalem1 31741
            20.5.10  Projectors (cont.)   pjorthi 31756
            20.5.11  Mayet's equation E_3   mayete3i 31815
      ​20.6  Operators on Hilbert spaces
            ​*20.6.1  Operator sum, difference, and scalar multiplication   df-hosum 31817
            20.6.2  Zero and identity operators   df-h0op 31835
            20.6.3  Operations on Hilbert space operators   hoaddcl 31845
            20.6.4  Linear, continuous, bounded, Hermitian, unitary operators and norms   df-nmop 31926
            20.6.5  Linear and continuous functionals and norms   df-nmfn 31932
            20.6.6  Adjoint   df-adjh 31936
            20.6.7  Dirac bra-ket notation   df-bra 31937
            20.6.8  Positive operators   df-leop 31939
            20.6.9  Eigenvectors, eigenvalues, spectrum   df-eigvec 31940
            20.6.10  Theorems about operators and functionals   nmopval 31943
            20.6.11  Riesz lemma   riesz3i 32149
            20.6.12  Adjoints (cont.)   cnlnadjlem1 32154
            20.6.13  Quantum computation error bound theorem   unierri 32191
            20.6.14  Dirac bra-ket notation (cont.)   branmfn 32192
            20.6.15  Positive operators (cont.)   leopg 32209
            20.6.16  Projectors as operators   pjhmopi 32233
      ​20.7  States on a Hilbert lattice and Godowski's equation
            20.7.1  States on a Hilbert lattice   df-st 32298
            20.7.2  Godowski's equation   golem1 32358
      ​20.8  Cover relation, atoms, exchange axiom, and modular symmetry
            20.8.1  Covers relation; modular pairs   df-cv 32366
            20.8.2  Atoms   df-at 32425
            20.8.3  Superposition principle   superpos 32441
            20.8.4  Atoms, exchange and covering properties, atomicity   chcv1 32442
            20.8.5  Irreducibility   chirredlem1 32477
            20.8.6  Atoms (cont.)   atcvat3i 32483
            20.8.7  Modular symmetry   mdsymlem1 32490
​PART 21  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      ​21.1  Mathboxes for user contributions
            21.1.1  Mathbox guidelines   mathbox 32529
      ​21.2  Mathbox for Stefan Allan
      ​21.3  Mathbox for Thierry Arnoux
            21.3.1  Propositional Calculus - misc additions   ad11antr 32534
            21.3.2  Predicate Calculus   sbc2iedf 32550
                  21.3.2.1  Predicate Calculus - misc additions   sbc2iedf 32550
                  21.3.2.2  Restricted quantification - misc additions   ralcom4f 32552
                  21.3.2.3  Equality   eqtrb 32559
                  21.3.2.4  Double restricted existential uniqueness quantification   opsbc2ie 32561
                  21.3.2.5  Double restricted existential uniqueness quantification syntax   w2reu 32563
                  21.3.2.6  Substitution (without distinct variables) - misc additions   sbceqbidf 32572
                  21.3.2.7  Existential "at most one" - misc additions   mo5f 32574
                  21.3.2.8  Existential uniqueness - misc additions   reuxfrdf 32576
                  21.3.2.9  Restricted "at most one" - misc additions   rmoxfrd 32578
                  21.3.2.10  Restricted iota (description binder)   riotaeqbidva 32581
            21.3.3  General Set Theory   dmrab 32582
                  21.3.3.1  Class abstractions (a.k.a. class builders)   dmrab 32582
                  21.3.3.2  Image Sets   abrexdomjm 32593
                  21.3.3.3  Set relations and operations - misc additions   nelun 32599
                  21.3.3.4  Unordered pairs   elpreq 32614
                  21.3.3.5  Unordered triples   tpssg 32623
                  21.3.3.6  Conditional operator - misc additions   ifeqeqx 32628
                  21.3.3.7  Set union   uniinn0 32636
                  21.3.3.8  Indexed union - misc additions   cbviunf 32641
                  21.3.3.9  Indexed intersection - misc additions   iinabrex 32655
                  21.3.3.10  Disjointness - misc additions   disjnf 32656
            21.3.4  Relations and Functions   xpdisjres 32684
                  21.3.4.1  Relations - misc additions   xpdisjres 32684
                  21.3.4.2  Functions - misc additions   fconst7v 32709
                  21.3.4.3  Operations - misc additions   mpomptxf 32767
                  21.3.4.4  The mapping operation   elmaprd 32769
                  21.3.4.5  Support of a function   suppovss 32770
                  21.3.4.6  Explicit Functions with one or two points as a domain   cosnopne 32783
                  21.3.4.7  Isomorphisms - misc. additions   gtiso 32790
                  21.3.4.8  Disjointness (additional proof requiring functions)   disjdsct 32792
                  21.3.4.9  First and second members of an ordered pair - misc additions   df1stres 32793
                  21.3.4.10  Countable Sets   snct 32801
            21.3.5  Real and Complex Numbers   sgnval2 32824
                  21.3.5.1  Complex operations - misc. additions   creq0 32825
                  21.3.5.2  Ordering on reals - misc additions   lt2addrd 32840
                  21.3.5.3  Extended reals - misc additions   nn0mnfxrd 32841
                  21.3.5.4  Extended nonnegative integers - misc additions   xnn0gt0 32859
                  21.3.5.5  Real number intervals - misc additions   joiniooico 32864
                  21.3.5.6  Finite intervals of integers - misc additions   uzssico 32874
                  21.3.5.7  Half-open integer ranges - misc additions   iundisjfi 32886
                  21.3.5.8  The ` # ` (set size) function - misc additions   hashunif 32896
                  21.3.5.9  The greatest common divisor operator - misc. additions   elq2 32902
                  21.3.5.10  Integers   nn0split01 32908
                  21.3.5.11  Decimal numbers   dfdec100 32921
            21.3.6  Real and complex functions   sgncl 32922
                  21.3.6.1  Signum (sgn or sign) function - misc. additions   sgncl 32922
                  21.3.6.2  Integer powers - misc. additions   nexple 32935
                  21.3.6.3  Indicator Functions   cind 32939
            ​*21.3.7  Decimal expansion   cdp2 32962
                  ​*21.3.7.1  Decimal point   cdp 32979
                  21.3.7.2  Division in the extended real number system   cxdiv 33008
            21.3.8  Words over a set - misc additions   wrdres 33027
                  21.3.8.1  Splicing words (substring replacement)   splfv3 33050
                  21.3.8.2  Cyclic shift of words   1cshid 33051
            21.3.9  Extensible Structures   ressplusf 33055
                  21.3.9.1  Structure restriction operator   ressplusf 33055
                  21.3.9.2  Posets   ressprs 33058
                  21.3.9.3  Complete lattices   clatp0cl 33068
                  21.3.9.4  Order Theory   cmnt 33070
                  21.3.9.5  Extended reals Structure - misc additions   ax-xrssca 33096
                  21.3.9.6  The extended nonnegative real numbers commutative monoid   xrge00 33106
            21.3.10  Algebra   mndcld 33114
                  21.3.10.1  Monoids   mndcld 33114
                  21.3.10.2  Monoids Homomorphisms   abliso 33128
                  21.3.10.3  Groups - misc additions   grpinvinvd 33132
                  21.3.10.4  Abelian Groups - misc additions   ablcomd 33138
                  21.3.10.5  Finitely supported group sums - misc additions   gsumsubg 33139
                  21.3.10.6  Group or monoid sums over words   gsumwun 33169
                  21.3.10.7  Centralizers and centers - misc additions   cntzun 33172
                  21.3.10.8  The symmetric group   symgfcoeu 33175
                  21.3.10.9  Transpositions   pmtridf1o 33187
                  21.3.10.10  Permutation Signs   psgnid 33190
                  21.3.10.11  Permutation cycles   ctocyc 33199
                  21.3.10.12  The Alternating Group   evpmval 33238
                  21.3.10.13  Signum in an ordered monoid   csgns 33251
                  21.3.10.14  Fixed points   cfxp 33256
                  21.3.10.15  The Archimedean property for generic ordered algebraic structures   cinftm 33269
                  21.3.10.16  Semiring left modules   cslmd 33293
                  21.3.10.17  Simple groups   prmsimpcyc 33321
                  21.3.10.18  Rings - misc additions   ringrngd 33322
                  21.3.10.19  Subrings generated by a set   elrgspnlem1 33335
                  21.3.10.20  The zero ring   irrednzr 33343
                  21.3.10.21  Localization of rings   cerl 33346
                  21.3.10.22  Integral Domains   domnmuln0rd 33367
                  21.3.10.23  Euclidean Domains   ceuf 33381
                  21.3.10.24  Division Rings   ringinveu 33387
                  21.3.10.25  The field of rational numbers   qfld 33390
                  21.3.10.26  Subfields   subsdrg 33391
                  21.3.10.27  Field of fractions   cfrac 33395
                  21.3.10.28  Field extensions generated by a set   cfldgen 33403
                  21.3.10.29  Ring homomorphisms - misc additions   rhmdvd 33416
                  21.3.10.30  Scalar restriction operation   cresv 33418
                  21.3.10.31  The commutative ring of gaussian integers   gzcrng 33433
                  21.3.10.32  The archimedean ordered field of real numbers   cnfldfld 33434
                  21.3.10.33  The quotient map and quotient modules   qusker 33441
                  21.3.10.34  The ring of integers modulo ` N `   znfermltl 33458
                  21.3.10.35  Independent sets and families   islinds5 33459
                  21.3.10.36  Ring associates, ring units   dvdsruassoi 33476
                  ​*21.3.10.37  Subgroup sum / Sumset / Minkowski sum   elgrplsmsn 33482
                  21.3.10.38  The quotient map   quslsm 33497
                  21.3.10.39  Ideals   lidlmcld 33511
                  21.3.10.40  Prime Ideals   cprmidl 33527
                  21.3.10.41  Maximal Ideals   cmxidl 33551
                  21.3.10.42  The semiring of ideals of a ring   cidlsrg 33592
                  21.3.10.43  Prime Elements   rprmval 33608
                  21.3.10.44  Unique factorization domains   cufd 33630
                  21.3.10.45  The ring of integers   zringidom 33643
                  21.3.10.46  Associative Algebra   assaassd 33647
                  21.3.10.47  Univariate Polynomials   0ringmon1p 33649
                  21.3.10.48  Polynomial quotient and polynomial remainder   q1pdir 33695
                  21.3.10.49  Multivariate Polynomials   psrbasfsupp 33704
                  21.3.10.50  The ring of symmetric polynomials   csply 33731
                  21.3.10.51  The subring algebra   sra1r 33757
                  21.3.10.52  Division Ring Extensions   drgext0g 33766
                  21.3.10.53  Vector Spaces   lvecdimfi 33772
                  21.3.10.54  Vector Space Dimension   cldim 33775
            21.3.11  Field Extensions   cfldext 33815
                  21.3.11.1  Algebraic numbers   cirng 33860
                  21.3.11.2  Algebraic extensions   calgext 33872
                  21.3.11.3  Minimal polynomials   cminply 33876
                  21.3.11.4  Quadratic Field Extensions   rtelextdg2lem 33903
                  21.3.11.5  Towers of quadratic extentions   fldext2chn 33905
            ​*21.3.12  Constructible Numbers   cconstr 33906
                  21.3.12.1  Impossible constructions   2sqr3minply 33957
            21.3.13  Matrices   csmat 33970
                  21.3.13.1  Submatrices   csmat 33970
                  21.3.13.2  Matrix literals   clmat 33988
                  21.3.13.3  Laplace expansion of determinants   mdetpmtr1 34000
            21.3.14  Topology   ist0cld 34010
                  21.3.14.1  Open maps   txomap 34011
                  21.3.14.2  Topology of the unit circle   qtopt1 34012
                  21.3.14.3  Refinements   reff 34016
                  21.3.14.4  Open cover refinement property   ccref 34019
                  21.3.14.5  Lindelöf spaces   cldlf 34029
                  21.3.14.6  Paracompact spaces   cpcmp 34032
                  ​*21.3.14.7  Spectrum of a ring   crspec 34039
                  21.3.14.8  Pseudometrics   cmetid 34063
                  21.3.14.9  Continuity - misc additions   hauseqcn 34075
                  21.3.14.10  Topology of the closed unit interval   elunitge0 34076
                  21.3.14.11  Topology of ` ( RR X. RR ) `   unicls 34080
                  21.3.14.12  Order topology - misc. additions   cnvordtrestixx 34090
                  21.3.14.13  Continuity in topological spaces - misc. additions   mndpluscn 34103
                  21.3.14.14  Topology of the extended nonnegative real numbers ordered monoid   xrge0hmph 34109
                  21.3.14.15  Limits - misc additions   lmlim 34124
                  21.3.14.16  Univariate polynomials   pl1cn 34132
            21.3.15  Uniform Stuctures and Spaces   chcmp 34133
                  21.3.15.1  Hausdorff uniform completion   chcmp 34133
            21.3.16  Topology and algebraic structures   zringnm 34135
                  21.3.16.1  The norm on the ring of the integer numbers   zringnm 34135
                  21.3.16.2  Topological ` ZZ ` -modules   zlm0 34137
                  21.3.16.3  Canonical embedding of the field of the rational numbers into a division ring   cqqh 34147
                  21.3.16.4  Canonical embedding of the real numbers into a complete ordered field   crrh 34170
                  21.3.16.5  Embedding from the extended real numbers into a complete lattice   cxrh 34193
                  21.3.16.6  Canonical embeddings into the ordered field of the real numbers   zrhre 34196
                  ​*21.3.16.7  Topological Manifolds   cmntop 34199
                  21.3.16.8  Extended sum   cesum 34204
            21.3.17  Mixed Function/Constant operation   cofc 34272
            21.3.18  Abstract measure   csiga 34285
                  21.3.18.1  Sigma-Algebra   csiga 34285
                  21.3.18.2  Generated sigma-Algebra   csigagen 34315
                  ​*21.3.18.3  lambda and pi-Systems, Rings of Sets   ispisys 34329
                  21.3.18.4  The Borel algebra on the real numbers   cbrsiga 34358
                  21.3.18.5  Product Sigma-Algebra   csx 34365
                  21.3.18.6  Measures   cmeas 34372
                  21.3.18.7  The counting measure   cntmeas 34403
                  21.3.18.8  The Lebesgue measure - misc additions   voliune 34406
                  21.3.18.9  The Dirac delta measure   cdde 34409
                  21.3.18.10  The 'almost everywhere' relation   cae 34414
                  21.3.18.11  Measurable functions   cmbfm 34426
                  21.3.18.12  Borel Algebra on ` ( RR X. RR ) `   br2base 34446
                  ​*21.3.18.13  Caratheodory's extension theorem   coms 34468
            21.3.19  Integration   itgeq12dv 34503
                  21.3.19.1  Lebesgue integral - misc additions   itgeq12dv 34503
                  21.3.19.2  Bochner integral   citgm 34504
            21.3.20  Euler's partition theorem   oddpwdc 34531
            21.3.21  Sequences defined by strong recursion   csseq 34560
            21.3.22  Fibonacci Numbers   cfib 34573
            21.3.23  Probability   cprb 34584
                  21.3.23.1  Probability Theory   cprb 34584
                  21.3.23.2  Conditional Probabilities   ccprob 34608
                  21.3.23.3  Real-valued Random Variables   crrv 34617
                  21.3.23.4  Preimage set mapping operator   corvc 34633
                  21.3.23.5  Distribution Functions   orvcelval 34646
                  21.3.23.6  Cumulative Distribution Functions   orvclteel 34650
                  21.3.23.7  Probabilities - example   coinfliplem 34656
                  21.3.23.8  Bertrand's Ballot Problem   ballotlemoex 34663
            21.3.24  Signum (sgn or sign) function - misc. additions   fzssfzo 34716
                  21.3.24.1  Operations on words   ccatmulgnn0dir 34719
            21.3.25  Polynomials with real coefficients - misc additions   plymul02 34723
            21.3.26  Descartes's rule of signs   signspval 34729
                  21.3.26.1  Sign changes in a word over real numbers   signspval 34729
                  21.3.26.2  Counting sign changes in a word over real numbers   signslema 34739
            21.3.27  Number Theory   iblidicc 34769
                  21.3.27.1  Representations of a number as sums of integers   crepr 34785
                  21.3.27.2  Vinogradov Trigonometric Sums and the Circle Method   cvts 34812
                  21.3.27.3  The Ternary Goldbach Conjecture: Final Statement   ax-hgt749 34821
            21.3.28  Elementary Geometry   cstrkg2d 34841
                  ​*21.3.28.1  Two-dimensional geometry   cstrkg2d 34841
                  21.3.28.2  Outer Five Segment (not used, no need to move to main)   cafs 34846
            ​*21.3.29  LeftPad Project   clpad 34851
      ​​*21.4  Mathbox for Jonathan Ben-Naim
            21.4.1  First-order logic and set theory   bnj170 34874
            21.4.2  Well founded induction and recursion   bnj110 35033
            21.4.3  The existence of a minimal element in certain classes   bnj69 35185
            21.4.4  Well-founded induction   bnj1204 35187
            21.4.5  Well-founded recursion, part 1 of 3   bnj60 35237
            21.4.6  Well-founded recursion, part 2 of 3   bnj1500 35243
            21.4.7  Well-founded recursion, part 3 of 3   bnj1522 35247
      ​21.5  Mathbox for BTernaryTau
            21.5.1  First-order logic   nfan1c 35248
                  21.5.1.1  Auxiliary axiom schemes   nfan1c 35248
            21.5.2  ZF set theory   exdifsn 35254
                  21.5.2.1  Finitism   prcinf 35288
                  21.5.2.2  Introduce ax-regs   ax-regs 35301
                  21.5.2.3  Derive ax-regs   axregs 35314
                  21.5.2.4  Global choice   gblacfnacd 35315
            21.5.3  Real and complex numbers   zltp1ne 35323
            21.5.4  Graph theory   lfuhgr 35331
                  21.5.4.1  Acyclic graphs   cacycgr 35355
      ​21.6  Mathbox for Mario Carneiro
            21.6.1  Predicate calculus with all distinct variables   ax-7d 35372
            21.6.2  Miscellaneous stuff   quartfull 35378
            21.6.3  Derangements and the Subfactorial   deranglem 35379
            21.6.4  The Erdős-Szekeres theorem   erdszelem1 35404
            21.6.5  The Kuratowski closure-complement theorem   kur14lem1 35419
            21.6.6  Retracts and sections   cretr 35430
            21.6.7  Path-connected and simply connected spaces   cpconn 35432
            21.6.8  Covering maps   ccvm 35468
            21.6.9  Normal numbers   snmlff 35542
            21.6.10  Godel-sets of formulas - part 1   cgoe 35546
            21.6.11  Godel-sets of formulas - part 2   cgon 35645
            21.6.12  Models of ZF   cgze 35659
            ​*21.6.13  Metamath formal systems   cmcn 35673
            21.6.14  Grammatical formal systems   cm0s 35798
            21.6.15  Models of formal systems   cmuv 35818
            21.6.16  Splitting fields   ccpms 35840
            21.6.17  p-adic number fields   czr 35860
      ​​*21.7  Mathbox for Filip Cernatescu
      ​21.8  Mathbox for Paul Chapman
            21.8.1  Real and complex numbers (cont.)   climuzcnv 35884
            21.8.2  Miscellaneous theorems   elfzm12 35888
      ​21.9  Mathbox for Hongxiu Chen
      ​21.10  Mathbox for Adrian Ducourtial
            21.10.1  Propositional calculus   currybi 35901
            21.10.2  Clone theory   ccloneop 35908
      ​21.11  Mathbox for Scott Fenton
            21.11.1  ZFC Axioms in primitive form   axextprim 35914
            21.11.2  Untangled classes   untelirr 35921
            21.11.3  Extra propositional calculus theorems   3jaodd 35928
            21.11.4  Misc. Useful Theorems   nepss 35931
            21.11.5  Properties of real and complex numbers   sqdivzi 35941
            21.11.6  Infinite products   iprodefisumlem 35953
            21.11.7  Factorial limits   faclimlem1 35956
            21.11.8  Greatest common divisor and divisibility   gcd32 35962
            21.11.9  Properties of relationships   dftr6 35964
            21.11.10  Properties of functions and mappings   funpsstri 35979
            21.11.11  Ordinal numbers   elpotr 35992
            21.11.12  Defined equality axioms   axextdfeq 36008
            21.11.13  Hypothesis builders   hbntg 36016
            21.11.14  Well-founded zero, successor, and limits   cwsuc 36021
            21.11.15  Quantifier-free definitions   ctxp 36041
            21.11.16  Alternate ordered pairs   caltop 36169
            21.11.17  Geometry in the Euclidean space   cofs 36195
                  21.11.17.1  Congruence properties   cofs 36195
                  21.11.17.2  Betweenness properties   btwntriv2 36225
                  21.11.17.3  Segment Transportation   ctransport 36242
                  21.11.17.4  Properties relating betweenness and congruence   cifs 36248
                  21.11.17.5  Connectivity of betweenness   btwnconn1lem1 36300
                  21.11.17.6  Segment less than or equal to   csegle 36319
                  21.11.17.7  Outside-of relationship   coutsideof 36332
                  21.11.17.8  Lines and Rays   cline2 36347
            21.11.18  Forward difference   cfwddif 36371
            21.11.19  Rank theorems   rankung 36379
            21.11.20  Hereditarily Finite Sets   chf 36385
      ​21.12  Mathbox for Gino Giotto
            21.12.1  Equality theorems.   rmoeqi 36400
                  21.12.1.1  Inference versions.   rmoeqi 36400
                  21.12.1.2  Deduction versions.   rmoeqdv 36425
            21.12.2  Change bound variables.   in-ax8 36437
                  21.12.2.1  Change bound variables and domains.   cbvralvw2 36439
                  21.12.2.2  Change bound variables, deduction versions.   cbvmodavw 36463
                  21.12.2.3  Change bound variables and domains, deduction versions.   cbvrmodavw2 36496
            21.12.3  Study of ax-mulf usage.   mpomulnzcnf 36512
      ​21.13  Mathbox for Jeff Hankins
            21.13.1  Miscellany   a1i14 36513
            21.13.2  Basic topological facts   topbnd 36537
            21.13.3  Topology of the real numbers   ivthALT 36548
            21.13.4  Refinements   cfne 36549
            21.13.5  Neighborhood bases determine topologies   neibastop1 36572
            21.13.6  Lattice structure of topologies   topmtcl 36576
            21.13.7  Filter bases   fgmin 36583
            21.13.8  Directed sets, nets   tailfval 36585
      ​21.14  Mathbox for Anthony Hart
            21.14.1  Propositional Calculus   tb-ax1 36596
            21.14.2  Predicate Calculus   nalfal 36616
            21.14.3  Miscellaneous single axioms   meran1 36624
            21.14.4  Connective Symmetry   negsym1 36630
      ​21.15  Mathbox for Chen-Pang He
            21.15.1  Ordinal topology   ontopbas 36641
      ​21.16  Mathbox for Jeff Hoffman
            21.16.1  Inferences for finite induction on generic function values   fveleq 36664
            21.16.2  gdc.mm   nnssi2 36668
      ​21.17  Mathbox for Matthew House
            21.17.1  Relations on well-ordered indexed unions   weiunval 36675
            ​*21.17.2  Stronger axioms of regularity   exeltr 36684
      ​21.18  Mathbox for Asger C. Ipsen
            21.18.1  Continuous nowhere differentiable functions   dnival 36690
      ​​*21.19  Mathbox for BJ
            *21.19.1  Propositional calculus   bj-mp2c 36759
                  *21.19.1.1  Derived rules of inference   bj-mp2c 36759
                  ​*21.19.1.2  A syntactic theorem   bj-0 36761
                  21.19.1.3  Minimal implicational calculus   bj-a1k 36763
                  ​*21.19.1.4  Positive calculus   bj-bisimpl 36774
                  21.19.1.5  Implication and negation   bj-con2com 36782
                  ​*21.19.1.6  Disjunction   bj-jaoi1 36792
                  ​*21.19.1.7  Logical equivalence   bj-dfbi4 36794
                  21.19.1.8  The conditional operator for propositions   bj-consensus 36799
                  ​*21.19.1.9  Propositional calculus: miscellaneous   bj-imbi12 36804
            ​*21.19.2  Modal logic   bj-axdd2 36813
            ​*21.19.3  Provability logic   cprvb 36818
            ​*21.19.4  First-order logic   bj-genr 36827
                  21.19.4.1  Adding ax-gen   bj-genr 36827
                  21.19.4.2  Adding ax-4   bj-almp 36832
                  21.19.4.3  Adding ax-5   bj-spvw 36871
                  21.19.4.4  Equality and substitution   bj-df-sb 36890
                  21.19.4.5  Adding ax-6   bj-spimvwt 36907
                  21.19.4.6  Adding ax-7   bj-cbvexw 36914
                  21.19.4.7  Membership predicate, ax-8 and ax-9   bj-ax89 36916
                  21.19.4.8  Adding ax-11   bj-alcomexcom 36918
                  21.19.4.9  Adding ax-12   axc11n11 36920
                  ​*21.19.4.10  Really adding ax-12   bj-substax12 36960
                  21.19.4.11  Nonfreeness   wnnf 36962
                  21.19.4.12  Adding ax-13   bj-axc10 37022
                  ​*21.19.4.13  Removing dependencies on ax-13 (and ax-11)   bj-axc10v 37032
                  ​*21.19.4.14  Distinct var metavariables   bj-hbaeb2 37057
                  ​*21.19.4.15  Around ~ equsal   bj-equsal1t 37061
                  ​*21.19.4.16  Some Principia Mathematica proofs   stdpc5t 37066
                  21.19.4.17  Alternate definition of substitution   bj-sbsb 37076
                  21.19.4.18  Lemmas for substitution   bj-sbf3 37078
                  21.19.4.19  Existential uniqueness   bj-eu3f 37080
                  ​*21.19.4.20  First-order logic: miscellaneous   bj-sblem1 37081
            21.19.5  Set theory   eliminable1 37098
                  ​*21.19.5.1  Eliminability of class terms   eliminable1 37098
                  ​*21.19.5.2  Classes without the axiom of extensionality   bj-denoteslem 37110
                  21.19.5.3  Characterization among sets versus among classes   elelb 37136
                  ​*21.19.5.4  The nonfreeness quantifier for classes   bj-nfcsym 37138
                  ​*21.19.5.5  Lemmas for class substitution   bj-sbeqALT 37139
                  21.19.5.6  Removing some axiom requirements and disjoint variable conditions   bj-exlimvmpi 37150
                  ​*21.19.5.7  Class abstractions   bj-elabd2ALT 37164
                  21.19.5.8  Generalized class abstractions   bj-cgab 37172
                  ​*21.19.5.9  Restricted nonfreeness   wrnf 37180
                  ​*21.19.5.10  Russell's paradox   bj-ru1 37182
                  21.19.5.11  Curry's paradox in set theory   currysetlem 37184
                  ​*21.19.5.12  Some disjointness results   bj-n0i 37190
                  ​*21.19.5.13  Complements on direct products   bj-xpimasn 37194
                  ​*21.19.5.14  "Singletonization" and tagging   bj-snsetex 37202
                  ​*21.19.5.15  Tuples of classes   bj-cproj 37229
                  ​*21.19.5.16  Set theory: elementary operations relative to a universe   bj-rcleqf 37264
                  ​*21.19.5.17  Axioms for finite unions   bj-abex 37269
                  ​*21.19.5.18  Set theory: miscellaneous   eleq2w2ALT 37286
                  ​*21.19.5.19  Axioms of separation and replacement   bj-axnul 37311
                  ​*21.19.5.20  Evaluation at a class   bj-evaleq 37315
                  21.19.5.21  Elementwise operations   celwise 37323
                  ​*21.19.5.22  Elementwise intersection (families of sets induced on a subset)   bj-rest00 37325
                  21.19.5.23  Moore collections (complements)   bj-raldifsn 37344
                  21.19.5.24  Maps-to notation for functions with three arguments   bj-0nelmpt 37360
                  ​*21.19.5.25  Currying   csethom 37366
                  ​*21.19.5.26  Setting components of extensible structures   cstrset 37378
            ​*21.19.6  Extended real and complex numbers, real and complex projective lines   bj-nfald 37381
                  21.19.6.1  Complements on class abstractions of ordered pairs and binary relations   bj-nfald 37381
                  ​*21.19.6.2  Identity relation (complements)   bj-opabssvv 37394
                  ​*21.19.6.3  Functionalized identity (diagonal in a Cartesian square)   cdiag2 37416
                  ​*21.19.6.4  Direct image and inverse image   cimdir 37422
                  ​*21.19.6.5  Extended numbers and projective lines as sets   cfractemp 37440
                  ​*21.19.6.6  Addition and opposite   caddcc 37481
                  ​*21.19.6.7  Order relation on the extended reals   cltxr 37485
                  ​*21.19.6.8  Argument, multiplication and inverse   carg 37487
                  21.19.6.9  The canonical bijection from the finite ordinals   ciomnn 37493
                  21.19.6.10  Divisibility   cnnbar 37504
            ​*21.19.7  Monoids   bj-smgrpssmgm 37512
                  ​*21.19.7.1  Finite sums in monoids   cfinsum 37527
            ​*21.19.8  Affine, Euclidean, and Cartesian geometry   bj-fvimacnv0 37530
                  *21.19.8.1  Real vector spaces   bj-fvimacnv0 37530
                  ​*21.19.8.2  Complex numbers (supplements)   bj-subcom 37552
                  ​*21.19.8.3  Barycentric coordinates   bj-bary1lem 37554
            21.19.9  Monoid of endomorphisms   cend 37557
      ​21.20  Mathbox for Jim Kingdon
            21.20.1  Circle constant   taupilem3 37563
            21.20.2  Number theory   dfgcd3 37568
            21.20.3  Real numbers   irrdifflemf 37569
      ​21.21  Mathbox for ML
            21.21.1  Miscellaneous   csbrecsg 37572
            21.21.2  Cartesian exponentiation   cfinxp 37627
            21.21.3  Topology   iunctb2 37647
                  ​*21.21.3.1  Pi-base theorems   pibp16 37657
      ​21.22  Mathbox for Wolf Lammen
            21.22.1  1. Bootstrapping   wl-section-boot 37666
            21.22.2  Implication chains   wl-section-impchain 37690
            21.22.3  Theorems around the conditional operator   wl-ifp-ncond1 37708
            21.22.4  Alternative development of hadd, cadd   wl-df-3xor 37712
            21.22.5  An alternative axiom ~ ax-13   ax-wl-13v 37737
            21.22.6  Bootstrapping set theory with classes   wl-cleq-0 37739
            21.22.7  Other stuff   wl-mps 37751
      ​21.23  Mathbox for Brendan Leahy
      ​21.24  Mathbox for Jeff Madsen
            21.24.1  Logic and set theory   unirep 37954
            21.24.2  Real and complex numbers; integers   filbcmb 37980
            21.24.3  Sequences and sums   sdclem2 37982
            21.24.4  Topology   subspopn 37992
            21.24.5  Metric spaces   metf1o 37995
            21.24.6  Continuous maps and homeomorphisms   constcncf 38002
            21.24.7  Boundedness   ctotbnd 38006
            21.24.8  Isometries   cismty 38038
            21.24.9  Heine-Borel Theorem   heibor1lem 38049
            21.24.10  Banach Fixed Point Theorem   bfplem1 38062
            21.24.11  Euclidean space   crrn 38065
            21.24.12  Intervals (continued)   ismrer1 38078
            21.24.13  Operation properties   cass 38082
            21.24.14  Groups and related structures   cmagm 38088
            21.24.15  Group homomorphism and isomorphism   cghomOLD 38123
            21.24.16  Rings   crngo 38134
            21.24.17  Division Rings   cdrng 38188
            21.24.18  Ring homomorphisms   crngohom 38200
            21.24.19  Commutative rings   ccm2 38229
            21.24.20  Ideals   cidl 38247
            21.24.21  Prime rings and integral domains   cprrng 38286
            21.24.22  Ideal generators   cigen 38299
      ​21.25  Mathbox for Giovanni Mascellani
            ​*21.25.1  Tools for automatic proof building   efald2 38318
            ​*21.25.2  Tseitin axioms   fald 38369
            ​*21.25.3  Equality deductions   iuneq2f 38396
            ​*21.25.4  Miscellanea   orcomdd 38407
      ​21.26  Mathbox for Peter Mazsa
            21.26.1  Notations   cxrn 38414
            21.26.2  Preparatory theorems   el2v1 38469
            21.26.3  Range Cartesian product   df-xrn 38620
            21.26.4  Relations   df-rels 38680
            21.26.5  Quotient map (coset map)   df-qmap 38686
            21.26.6  Lifts, shifts, successor, and predecessor   df-adjliftmap 38695
            21.26.7  Cosets by ` R `   df-coss 38741
            21.26.8  Subset relations   df-ssr 38818
            21.26.9  Reflexivity   df-refs 38830
            21.26.10  Converse reflexivity   df-cnvrefs 38845
            21.26.11  Symmetry   df-syms 38862
            21.26.12  Reflexivity and symmetry   symrefref2 38887
            21.26.13  Transitivity   df-trs 38896
            21.26.14  Equivalence relations   df-eqvrels 38908
            21.26.15  Redundancy   df-redunds 38947
            21.26.16  Domain quotients   df-dmqss 38962
            21.26.17  Equivalence relations on domain quotients   df-ers 38988
            21.26.18  Functions   df-funss 39005
            21.26.19  Disjoints vs. converse functions   df-disjss 39028
            21.26.20  Antisymmetry   df-antisymrel 39103
            21.26.21  Partitions: disjoints on domain quotients   df-parts 39108
            21.26.22  Partition-Equivalence Theorems   disjim 39124
            21.26.23  Type-safe Partition-Equivalence: PetParts, PetErs, Pet2Parts, Pet2Ers   df-petparts 39208
      ​21.27  Mathbox for Rodolfo Medina
            21.27.1  Partitions   prtlem60 39218
      ​​*21.28  Mathbox for Norm Megill
            *21.28.1  Obsolete schemes ax-c4,c5,c7,c10,c11,c11n,c15,c9,c14,c16   ax-c5 39248
            ​*21.28.2  Rederive new axioms ax-4, ax-10, ax-6, ax-12, ax-13 from old   axc5 39258
            ​*21.28.3  Legacy theorems using obsolete axioms   ax5ALT 39272
            21.28.4  Experiments with weak deduction theorem   elimhyps 39326
            21.28.5  Miscellanea   cnaddcom 39337
            21.28.6  Atoms, hyperplanes, and covering in a left vector space (or module)   clsa 39339
            21.28.7  Functionals and kernels of a left vector space (or module)   clfn 39422
            21.28.8  Opposite rings and dual vector spaces   cld 39488
            21.28.9  Ortholattices and orthomodular lattices   cops 39537
            21.28.10  Atomic lattices with covering property   ccvr 39627
            21.28.11  Hilbert lattices   chlt 39715
            21.28.12  Projective geometries based on Hilbert lattices   clln 39856
            21.28.13  Construction of a vector space from a Hilbert lattice   cdlema1N 40156
            21.28.14  Construction of involution and inner product from a Hilbert lattice   clpoN 41845
      ​21.29  Mathbox for metakunt
            21.29.1  Commutative Semiring   ccsrg 42327
            21.29.2  General helpful statements   rhmzrhval 42330
            21.29.3  Some gcd and lcm results   12gcd5e1 42362
            21.29.4  Least common multiple inequality theorem   3factsumint1 42380
            21.29.5  Logarithm inequalities   3exp7 42412
            21.29.6  Miscellaneous results for AKS formalisation   intlewftc 42420
            21.29.7  Sticks and stones   sticksstones1 42505
            21.29.8  Continuation AKS   aks6d1c6lem1 42529
      ​21.30  Mathbox for Steven Nguyen
            21.30.1  Utility theorems   jarrii 42564
            ​*21.30.2  Arithmetic theorems   c0exALT 42611
            21.30.3  Exponents and divisibility   oexpreposd 42681
            21.30.4  Trigonometry and Calculus   tanhalfpim 42708
            ​*21.30.5  Independence of ax-mulcom   cresub 42724
            21.30.6  Structures   sn-base0 42854
            ​*21.30.7  Projective spaces   cprjsp 42948
            21.30.8  Basic reductions for Fermat's Last Theorem   dffltz 42981
            ​*21.30.9  Exemplar theorems   iddii 43011
                  ​*21.30.9.1  Standard replacements of ax-10 , ax-11 , ax-12   nfa1w 43022
      ​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 43038
            21.33.2  Additional theory of functions   imaiinfv 43039
            21.33.3  Additional topology   elrfi 43040
            21.33.4  Characterization of closure operators. Kuratowski closure axioms   ismrcd1 43044
            21.33.5  Algebraic closure systems   cnacs 43048
            21.33.6  Miscellanea 1. Map utilities   constmap 43059
            21.33.7  Miscellanea for polynomials   mptfcl 43066
            21.33.8  Multivariate polynomials over the integers   cmzpcl 43067
            21.33.9  Miscellanea for Diophantine sets 1   coeq0i 43099
            21.33.10  Diophantine sets 1: definitions   cdioph 43101
            21.33.11  Diophantine sets 2 miscellanea   ellz1 43113
            21.33.12  Diophantine sets 2: union and intersection. Monotone Boolean algebra   diophin 43118
            21.33.13  Diophantine sets 3: construction   diophrex 43121
            21.33.14  Diophantine sets 4 miscellanea   2sbcrex 43130
            21.33.15  Diophantine sets 4: Quantification   rexrabdioph 43140
            21.33.16  Diophantine sets 5: Arithmetic sets   rabdiophlem1 43147
            21.33.17  Diophantine sets 6: reusability. renumbering of variables   eldioph4b 43157
            21.33.18  Pigeonhole Principle and cardinality helpers   fphpd 43162
            21.33.19  A non-closed set of reals is infinite   rencldnfilem 43166
            21.33.20  Lagrange's rational approximation theorem   irrapxlem1 43168
            21.33.21  Pell equations 1: A nontrivial solution always exists   pellexlem1 43175
            21.33.22  Pell equations 2: Algebraic number theory of the solution set   csquarenn 43182
            21.33.23  Pell equations 3: characterizing fundamental solution   infmrgelbi 43224
            ​*21.33.24  Logarithm laws generalized to an arbitrary base   reglogcl 43236
            21.33.25  Pell equations 4: the positive solution group is infinite cyclic   pellfund14 43244
            21.33.26  X and Y sequences 1: Definition and recurrence laws   crmx 43246
            21.33.27  Ordering and induction lemmas for the integers   monotuz 43287
            21.33.28  X and Y sequences 2: Order properties   rmxypos 43293
            21.33.29  Congruential equations   congtr 43311
            21.33.30  Alternating congruential equations   acongid 43321
            21.33.31  Additional theorems on integer divisibility   coprmdvdsb 43331
            21.33.32  X and Y sequences 3: Divisibility properties   jm2.18 43334
            21.33.33  X and Y sequences 4: Diophantine representability of Y   jm2.27a 43351
            21.33.34  X and Y sequences 5: Diophantine representability of X, ^, _C   rmxdiophlem 43361
            21.33.35  Uncategorized stuff not associated with a major project   setindtr 43370
            21.33.36  More equivalents of the Axiom of Choice   axac10 43379
            21.33.37  Finitely generated left modules   clfig 43413
            21.33.38  Noetherian left modules I   clnm 43421
            21.33.39  Addenda for structure powers   pwssplit4 43435
            21.33.40  Every set admits a group structure iff choice   unxpwdom3 43441
            21.33.41  Noetherian rings and left modules II   clnr 43455
            21.33.42  Hilbert's Basis Theorem   cldgis 43467
            21.33.43  Additional material on polynomials [DEPRECATED]   cmnc 43477
            21.33.44  Degree and minimal polynomial of algebraic numbers   cdgraa 43486
            21.33.45  Algebraic integers I   citgo 43503
            21.33.46  Endomorphism algebra   cmend 43517
            21.33.47  Cyclic groups and order   idomodle 43537
            21.33.48  Cyclotomic polynomials   ccytp 43543
            21.33.49  Miscellaneous topology   fgraphopab 43549
      ​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 43563
            21.36.2  Natural addition of Cantor normal forms   oawordex2 43672
            21.36.3  Surreal Contributions   abeqabi 43753
            21.36.4  Short Studies   nlimsuc 43786
                  21.36.4.1  Additional work on conditional logical operator   ifpan123g 43804
                  21.36.4.2  Sophisms   rp-fakeimass 43857
                  ​*21.36.4.3  Finite Sets   rp-isfinite5 43862
                  21.36.4.4  General Observations   intabssd 43864
                  21.36.4.5  Infinite Sets   pwelg 43905
                  ​*21.36.4.6  Finite intersection property   fipjust 43910
                  21.36.4.7  RP ADDTO: Subclasses and subsets   rababg 43919
                  21.36.4.8  RP ADDTO: The intersection of a class   elinintab 43920
                  21.36.4.9  RP ADDTO: Theorems requiring subset and intersection existence   elinintrab 43922
                  21.36.4.10  RP ADDTO: Relations   xpinintabd 43925
                  ​*21.36.4.11  RP ADDTO: Functions   elmapintab 43941
                  ​*21.36.4.12  RP ADDTO: Finite induction (for finite ordinals)   cnvcnvintabd 43945
                  21.36.4.13  RP ADDTO: First and second members of an ordered pair   elcnvlem 43946
                  21.36.4.14  RP ADDTO: The reflexive and transitive properties of relations   undmrnresiss 43949
                  21.36.4.15  RP ADDTO: Basic properties of closures   cleq2lem 43953
                  21.36.4.16  RP REPLACE: Definitions and basic properties of transitive closures   trcleq2lemRP 43975
                  ​*21.36.4.17  Additions for square root; absolute value   sqrtcvallem1 43976
            21.36.5  Additional statements on relations and subclasses   al3im 43992
                  21.36.5.1  Transitive relations (not to be confused with transitive classes).   trrelind 44010
                  21.36.5.2  Reflexive closures   crcl 44017
                  ​*21.36.5.3  Finite relationship composition.   relexp2 44022
                  21.36.5.4  Transitive closure of a relation   dftrcl3 44065
                  ​*21.36.5.5  Adapted from Frege   frege77d 44091
            ​*21.36.6  Propositions from _Begriffsschrift_   dfxor4 44111
                  *21.36.6.1  _Begriffsschrift_ Chapter I   dfxor4 44111
                  ​*21.36.6.2  _Begriffsschrift_ Notation hints   whe 44117
                  21.36.6.3  _Begriffsschrift_ Chapter II Implication   ax-frege1 44135
                  21.36.6.4  _Begriffsschrift_ Chapter II Implication and Negation   axfrege28 44174
                  ​*21.36.6.5  _Begriffsschrift_ Chapter II with logical equivalence   axfrege52a 44201
                  21.36.6.6  _Begriffsschrift_ Chapter II with equivalence of sets   axfrege52c 44232
                  ​*21.36.6.7  _Begriffsschrift_ Chapter II with equivalence of classes   frege53c 44259
                  ​*21.36.6.8  _Begriffsschrift_ Chapter III Properties hereditary in a sequence   dffrege69 44277
                  ​*21.36.6.9  _Begriffsschrift_ Chapter III Following in a sequence   dffrege76 44284
                  ​*21.36.6.10  _Begriffsschrift_ Chapter III Member of sequence   dffrege99 44307
                  ​*21.36.6.11  _Begriffsschrift_ Chapter III Single-valued procedures   dffrege115 44323
            ​*21.36.7  Exploring Topology via Seifert and Threlfall   enrelmap 44342
                  *21.36.7.1  Equinumerosity of sets of relations and maps   enrelmap 44342
                  ​*21.36.7.2  Generic Pseudoclosure Spaces, Pseudointerior Spaces, and Pseudoneighborhoods   or3or 44368
                  ​*21.36.7.3  Generic Neighborhood Spaces   gneispa 44475
            ​*21.36.8  Exploring Higher Homotopy via Kerodon   k0004lem1 44492
                  *21.36.8.1  Simplicial Sets   k0004lem1 44492
      ​21.37  Mathbox for Stanislas Polu
            21.37.1  IMO Problems   wwlemuld 44501
                  21.37.1.1  IMO 1972 B2   wwlemuld 44501
            ​*21.37.2  INT Inequalities Proof Generator   int-addcomd 44518
            ​*21.37.3  N-Digit Addition Proof Generator   unitadd 44540
            21.37.4  AM-GM (for k = 2,3,4)   gsumws3 44541
      ​21.38  Mathbox for Rohan Ridenour
            21.38.1  Misc   spALT 44546
            21.38.2  Monoid rings   cmnring 44556
            21.38.3  Shorter primitive equivalent of ax-groth   gru0eld 44574
                  21.38.3.1  Grothendieck universes are closed under collection   gru0eld 44574
                  21.38.3.2  Minimal universes   ismnu 44606
                  21.38.3.3  Primitive equivalent of ax-groth   expandan 44633
      ​21.39  Mathbox for Steve Rodriguez
            21.39.1  Miscellanea   nanorxor 44650
            21.39.2  Ratio test for infinite series convergence and divergence   dvgrat 44657
            21.39.3  Multiples   reldvds 44660
            21.39.4  Function operations   caofcan 44668
            21.39.5  Calculus   lhe4.4ex1a 44674
            21.39.6  The generalized binomial coefficient operation   cbcc 44681
            21.39.7  Binomial series   uzmptshftfval 44691
      ​21.40  Mathbox for Andrew Salmon
            21.40.1  Principia Mathematica * 10   pm10.12 44703
            21.40.2  Principia Mathematica * 11   2alanimi 44717
            21.40.3  Predicate Calculus   sbeqal1 44743
            21.40.4  Principia Mathematica * 13 and * 14   pm13.13a 44752
            21.40.5  Set Theory   elnev 44782
            21.40.6  Arithmetic   addcomgi 44800
            21.40.7  Geometry   cplusr 44801
      ​​*21.41  Mathbox for Alan Sare
            21.41.1  Auxiliary theorems for the Virtual Deduction tool   idiALT 44823
            21.41.2  Supplementary unification deductions   bi1imp 44827
            21.41.3  Conventional Metamath proofs, some derived from VD proofs   iidn3 44846
            21.41.4  What is Virtual Deduction?   wvd1 44914
            21.41.5  Virtual Deduction Theorems   df-vd1 44915
            21.41.6  Theorems proved using Virtual Deduction   trsspwALT 45162
            21.41.7  Theorems proved using Virtual Deduction with mmj2 assistance   simplbi2VD 45190
            21.41.8  Virtual Deduction transcriptions of textbook proofs   sb5ALTVD 45257
            21.41.9  Theorems proved using conjunction-form Virtual Deduction   elpwgdedVD 45261
            21.41.10  Theorems with a VD proof in conventional notation derived from a VD proof   suctrALT3 45268
            ​*21.41.11  Theorems with a proof in conventional notation derived from a VD proof   notnotrALT2 45271
      ​21.42  Mathbox for Eric Schmidt
            21.42.1  Miscellany   rspesbcd 45282
            21.42.2  Study of dfbi1ALT   dfbi1ALTa 45284
            21.42.3  Relation-preserving functions   wrelp 45287
            21.42.4  Orbits   orbitex 45300
            21.42.5  Well-founded sets   trwf 45304
            21.42.6  Absoluteness in transitive models   ralabso 45313
            21.42.7  Lemmas for showing axioms hold in models   traxext 45322
            21.42.8  The class of well-founded sets is a model for ZFC   wfaxext 45338
            21.42.9  Permutation models   brpermmodel 45348
      ​21.43  Mathbox for Glauco Siliprandi
            21.43.1  Miscellanea   evth2f 45364
            21.43.2  Functions   fnresdmss 45516
            21.43.3  Ordering on real numbers - Real and complex numbers basic operations   sub2times 45624
            21.43.4  Real intervals   gtnelioc 45840
            21.43.5  Finite sums   fsummulc1f 45920
            21.43.6  Finite multiplication of numbers and finite multiplication of functions   fmul01 45929
            21.43.7  Limits   clim1fr1 45950
                  21.43.7.1  Inferior limit (lim inf)   clsi 46098
                  ​*21.43.7.2  Limits for sequences of extended real numbers   clsxlim 46165
            21.43.8  Trigonometry   coseq0 46211
            21.43.9  Continuous Functions   mulcncff 46217
            21.43.10  Derivatives   dvsinexp 46258
            21.43.11  Integrals   itgsin0pilem1 46297
            21.43.12  Stone Weierstrass theorem - real version   stoweidlem1 46348
            21.43.13  Wallis' product for π   wallispilem1 46412
            21.43.14  Stirling's approximation formula for ` n ` factorial   stirlinglem1 46421
            21.43.15  Dirichlet kernel   dirkerval 46438
            21.43.16  Fourier Series   fourierdlem1 46455
            21.43.17  e is transcendental   elaa2lem 46580
            21.43.18  n-dimensional Euclidean space   rrxtopn 46631
            21.43.19  Basic measure theory   csalg 46655
                  ​*21.43.19.1  σ-Algebras   csalg 46655
                  21.43.19.2  Sum of nonnegative extended reals   csumge0 46709
                  ​*21.43.19.3  Measures   cmea 46796
                  ​*21.43.19.4  Outer measures and Caratheodory's construction   come 46836
                  ​*21.43.19.5  Lebesgue measure on n-dimensional Real numbers   covoln 46883
                  ​*21.43.19.6  Measurable functions   csmblfn 47042
      ​21.44  Mathbox for Saveliy Skresanov
            21.44.1  Ceva's theorem   sigarval 47197
            21.44.2  Simple groups   simpcntrab 47217
      ​21.45  Mathbox for Ender Ting
            21.45.1  Increasing sequences and subsequences   et-ltneverrefl 47218
            21.45.2  Scratchpad for number theory   evenwodadd 47234
            21.45.3  Scratchpad for math on real numbers   squeezedltsq 47235
      ​21.46  Mathbox for Jarvin Udandy
      ​21.47  Mathbox for Adhemar
            ​*21.47.1  Minimal implicational calculus   adh-minim 47350
      ​21.48  Mathbox for Alexander van der Vekens
            21.48.1  General auxiliary theorems (1)   n0nsn2el 47374
                  21.48.1.1  Unordered and ordered pairs - extension for singletons   n0nsn2el 47374
                  21.48.1.2  Unordered and ordered pairs - extension for unordered pairs   elprneb 47378
                  21.48.1.3  Unordered and ordered pairs - extension for ordered pairs   oppr 47379
                  21.48.1.4  Relations - extension   eubrv 47384
                  21.48.1.5  Definite description binder (inverted iota) - extension   iota0def 47387
                  21.48.1.6  Functions - extension   fveqvfvv 47389
            21.48.2  Alternative for Russell's definition of a description binder   caiota 47432
            21.48.3  Double restricted existential uniqueness   r19.32 47447
                  21.48.3.1  Restricted quantification (extension)   r19.32 47447
                  21.48.3.2  Restricted uniqueness and "at most one" quantification   reuf1odnf 47456
                  21.48.3.3  Analogs to Existential uniqueness (double quantification)   2reu3 47459
                  21.48.3.4  Additional theorems for double restricted existential uniqueness   2reu8i 47462
            ​*21.48.4  Alternative definitions of function and operation values   wdfat 47465
                  21.48.4.1  Restricted quantification (extension)   ralbinrald 47471
                  21.48.4.2  The universal class (extension)   nvelim 47472
                  21.48.4.3  Introduce the Axiom of Power Sets (extension)   alneu 47473
                  21.48.4.4  Predicate "defined at"   dfateq12d 47475
                  21.48.4.5  Alternative definition of the value of a function   dfafv2 47481
                  21.48.4.6  Alternative definition of the value of an operation   aoveq123d 47527
            ​*21.48.5  Alternative definitions of function values (2)   cafv2 47557
            21.48.6  General auxiliary theorems (2)   an4com24 47617
                  21.48.6.1  Logical conjunction - extension   an4com24 47617
                  21.48.6.2  Abbreviated conjunction and disjunction of three wff's - extension   3an4ancom24 47618
                  21.48.6.3  Negated membership (alternative)   cnelbr 47620
                  21.48.6.4  The empty set - extension   ralralimp 47627
                  21.48.6.5  Indexed union and intersection - extension   otiunsndisjX 47628
                  21.48.6.6  Functions - extension   fvifeq 47629
                  21.48.6.7  Maps-to notation - extension   fvmptrab 47641
                  21.48.6.8  Subtraction - extension   cnambpcma 47643
                  21.48.6.9  Ordering on reals (cont.) - extension   leaddsuble 47646
                  21.48.6.10  Imaginary and complex number properties - extension   readdcnnred 47652
                  21.48.6.11  Nonnegative integers (as a subset of complex numbers) - extension   nn0resubcl 47657
                  21.48.6.12  Integers (as a subset of complex numbers) - extension   zgeltp1eq 47658
                  21.48.6.13  Decimal arithmetic - extension   1t10e1p1e11 47659
                  21.48.6.14  Upper sets of integers - extension   eluzge0nn0 47661
                  21.48.6.15  Infinity and the extended real number system (cont.) - extension   nltle2tri 47662
                  21.48.6.16  Finite intervals of integers - extension   ssfz12 47663
                  21.48.6.17  Half-open integer ranges - extension   fzopred 47671
                  21.48.6.18  The floor and ceiling functions - extension   2ltceilhalf 47677
                  21.48.6.19  The modulo (remainder) operation - extension   fldivmod 47687
                  21.48.6.20  The infinite sequence builder "seq"   smonoord 47720
                  21.48.6.21  Finite and infinite sums - extension   fsummsndifre 47721
                  21.48.6.22  Extensible structures - extension   setsidel 47725
            ​*21.48.7  Preimages of function values   preimafvsnel 47728
            ​*21.48.8  Partitions of real intervals   ciccp 47762
            21.48.9  Shifting functions with an integer range domain   fargshiftfv 47788
            21.48.10  Words over a set (extension)   lswn0 47793
                  21.48.10.1  Last symbol of a word - extension   lswn0 47793
            21.48.11  Unordered pairs   wich 47794
                  21.48.11.1  Interchangeable setvar variables   wich 47794
                  21.48.11.2  Set of unordered pairs   sprid 47823
                  ​*21.48.11.3  Proper (unordered) pairs   prpair 47850
                  21.48.11.4  Set of proper unordered pairs   cprpr 47861
            21.48.12  Number theory (extension)   cfmtno 47876
                  ​*21.48.12.1  Fermat numbers   cfmtno 47876
                  ​*21.48.12.2  Mersenne primes   m2prm 47940
                  21.48.12.3  Proth's theorem   modexp2m1d 47961
                  21.48.12.4  Solutions of quadratic equations   quad1 47969
            ​*21.48.13  Even and odd numbers   ceven 47973
                  21.48.13.1  Definitions and basic properties   ceven 47973
                  21.48.13.2  Alternate definitions using the "divides" relation   dfeven2 47998
                  21.48.13.3  Alternate definitions using the "modulo" operation   dfeven3 48007
                  21.48.13.4  Alternate definitions using the "gcd" operation   iseven5 48013
                  21.48.13.5  Theorems of part 5 revised   zneoALTV 48018
                  21.48.13.6  Theorems of part 6 revised   odd2np1ALTV 48023
                  21.48.13.7  Theorems of AV's mathbox revised   0evenALTV 48037
                  21.48.13.8  Additional theorems   epoo 48052
                  21.48.13.9  Perfect Number Theorem (revised)   perfectALTVlem1 48070
            21.48.14  Number theory (extension 2)   cfppr 48073
                  ​*21.48.14.1  Fermat pseudoprimes   cfppr 48073
                  ​*21.48.14.2  Goldbach's conjectures   cgbe 48094
            21.48.15  Graph theory (extension)   cclnbgr 48167
                  21.48.15.1  Closed neighborhood of a vertex   cclnbgr 48167
                  ​*21.48.15.2  Semiclosed and semiopen neighborhoods (experimental)   dfsclnbgr2 48195
                  21.48.15.3  Induced subgraphs   cisubgr 48209
                  ​*21.48.15.4  Isomorphisms of graphs   cgrisom 48223
                  ​*21.48.15.5  Triangles in graphs   cgrtri 48286
                  ​*21.48.15.6  Star graphs   cstgr 48300
                  ​*21.48.15.7  Local isomorphisms of graphs   cgrlim 48325
                  ​*21.48.15.8  Generalized Petersen graphs   cgpg 48389
                  21.48.15.9  Loop-free graphs - extension   1hegrlfgr 48481
                  21.48.15.10  Walks - extension   cupwlks 48482
                  21.48.15.11  Edges of graphs expressed as sets of unordered pairs   upgredgssspr 48492
            21.48.16  Monoids (extension)   ovn0dmfun 48505
                  21.48.16.1  Auxiliary theorems   ovn0dmfun 48505
                  21.48.16.2  Magmas, Semigroups and Monoids (extension)   plusfreseq 48513
                  21.48.16.3  Examples and counterexamples for magmas, semigroups and monoids (extension)   opmpoismgm 48516
                  21.48.16.4  Group sum operation (extension 1)   gsumsplit2f 48529
            ​*21.48.17  Magmas and internal binary operations (alternate approach)   ccllaw 48532
                  *21.48.17.1  Laws for internal binary operations   ccllaw 48532
                  ​*21.48.17.2  Internal binary operations   cintop 48545
                  21.48.17.3  Alternative definitions for magmas and semigroups   cmgm2 48564
            21.48.18  Rings (extension)   lmod0rng 48578
                  21.48.18.1  Nonzero rings (extension)   lmod0rng 48578
                  21.48.18.2  Ideals as non-unital rings   lidldomn1 48580
                  21.48.18.3  The non-unital ring of even integers   0even 48586
                  21.48.18.4  A constructed not unital ring   cznrnglem 48608
                  ​*21.48.18.5  The category of non-unital rings (alternate definition)   crngcALTV 48612
                  ​*21.48.18.6  The category of (unital) rings (alternate definition)   cringcALTV 48636
            21.48.19  Basic algebraic structures (extension)   eliunxp2 48683
                  21.48.19.1  Auxiliary theorems   eliunxp2 48683
                  21.48.19.2  The binomial coefficient operation (extension)   bcpascm1 48700
                  21.48.19.3  The ` ZZ `-module ` ZZ X. ZZ `   zlmodzxzlmod 48703
                  21.48.19.4  Group sum operation (extension 2)   mgpsumunsn 48710
                  21.48.19.5  Symmetric groups (extension)   exple2lt6 48713
                  21.48.19.6  Divisibility (extension)   invginvrid 48716
                  21.48.19.7  The support of functions (extension)   rmsupp0 48717
                  21.48.19.8  Finitely supported functions (extension)   rmsuppfi 48721
                  21.48.19.9  Left modules (extension)   lmodvsmdi 48728
                  21.48.19.10  Associative algebras (extension)   assaascl0 48730
                  21.48.19.11  Univariate polynomials (extension)   ply1vr1smo 48732
                  21.48.19.12  Univariate polynomials (examples)   linply1 48742
            21.48.20  Linear algebra (extension)   cdmatalt 48745
                  ​*21.48.20.1  The subalgebras of diagonal and scalar matrices (extension)   cdmatalt 48745
                  ​*21.48.20.2  Linear combinations   clinc 48753
                  ​*21.48.20.3  Linear independence   clininds 48789
                  21.48.20.4  Simple left modules and the ` ZZ `-module   lmod1lem1 48836
                  21.48.20.5  Differences between (left) modules and (left) vector spaces   lvecpsslmod 48856
            21.48.21  Complexity theory   suppdm 48859
                  21.48.21.1  Auxiliary theorems   suppdm 48859
                  21.48.21.2  Even and odd integers   nn0onn0ex 48872
                  21.48.21.3  The natural logarithm on complex numbers (extension)   logcxp0 48884
                  21.48.21.4  Division of functions   cfdiv 48886
                  21.48.21.5  Upper bounds   cbigo 48896
                  21.48.21.6  Logarithm to an arbitrary base (extension)   rege1logbrege0 48907
                  ​*21.48.21.7  The binary logarithm   fldivexpfllog2 48914
                  21.48.21.8  Binary length   cblen 48918
                  ​*21.48.21.9  Digits   cdig 48944
                  21.48.21.10  Nonnegative integer as sum of its shifted digits   dignn0flhalflem1 48964
                  21.48.21.11  Algorithms for the multiplication of nonnegative integers   nn0mulfsum 48973
                  ​*21.48.21.12  N-ary functions   cnaryf 48975
                  ​*21.48.21.13  The Ackermann function   citco 49006
            21.48.22  Elementary geometry (extension)   fv1prop 49048
                  21.48.22.1  Auxiliary theorems   fv1prop 49048
                  21.48.22.2  Real euclidean space of dimension 2   rrx2pxel 49060
                  21.48.22.3  Spheres and lines in real Euclidean spaces   cline 49076
      ​21.49  Mathbox for Zhi Wang
            21.49.1  Propositional calculus   pm4.71da 49138
            21.49.2  Predicate calculus with equality   dtrucor3 49147
                  21.49.2.1  Axiom scheme ax-5 (Distinctness)   dtrucor3 49147
            21.49.3  ZF Set Theory - start with the Axiom of Extensionality   ralbidb 49148
                  21.49.3.1  Restricted quantification   ralbidb 49148
                  21.49.3.2  The universal class   reuxfr1dd 49155
                  21.49.3.3  The empty set   ssdisjd 49156
                  21.49.3.4  Unordered and ordered pairs   vsn 49160
                  21.49.3.5  The union of a class   unilbss 49166
                  21.49.3.6  Indexed union and intersection   iuneq0 49167
            21.49.4  ZF Set Theory - add the Axiom of Replacement   inpw 49173
                  21.49.4.1  Theorems requiring subset and intersection existence   inpw 49173
            21.49.5  ZF Set Theory - add the Axiom of Power Sets   opth1neg 49174
                  21.49.5.1  Ordered pair theorem   opth1neg 49174
                  21.49.5.2  Ordered-pair class abstractions (cont.)   brab2dd 49176
                  21.49.5.3  Relations   iinxp 49179
                  21.49.5.4  Functions   mof0 49186
                  21.49.5.5  Operations   ovsng 49206
            21.49.6  ZF Set Theory - add the Axiom of Union   fonex 49215
                  21.49.6.1  Relations and functions (cont.)   fonex 49215
                  21.49.6.2  First and second members of an ordered pair   eloprab1st2nd 49216
                  21.49.6.3  Operations in maps-to notation (continued)   fmpodg 49217
                  21.49.6.4  Function transposition   resinsnlem 49219
                  21.49.6.5  Infinite Cartesian products   ixpv 49238
                  21.49.6.6  Equinumerosity   fvconst0ci 49239
            21.49.7  Order sets   iccin 49244
                  21.49.7.1  Real number intervals   iccin 49244
            21.49.8  Extensible structures   slotresfo 49247
                  21.49.8.1  Basic definitions   slotresfo 49247
            21.49.9  Moore spaces   mreuniss 49248
            ​*21.49.10  Topology   clduni 49249
                  21.49.10.1  Closure and interior   clduni 49249
                  21.49.10.2  Neighborhoods   neircl 49253
                  21.49.10.3  Subspace topologies   restcls2lem 49261
                  21.49.10.4  Limits and continuity in topological spaces   cnneiima 49265
                  21.49.10.5  Topological definitions using the reals   iooii 49266
                  21.49.10.6  Separated sets   sepnsepolem1 49270
                  21.49.10.7  Separated spaces: T0, T1, T2 (Hausdorff) ...   isnrm4 49279
            21.49.11  Preordered sets and directed sets using extensible structures   isprsd 49303
            21.49.12  Posets and lattices using extensible structures   lubeldm2 49304
                  21.49.12.1  Posets   lubeldm2 49304
                  21.49.12.2  Lattices   toslat 49330
                  21.49.12.3  Subset order structures   intubeu 49332
            21.49.13  Rings   elmgpcntrd 49353
                  21.49.13.1  Multiplicative Group   elmgpcntrd 49353
            21.49.14  Associative algebras   asclelbasALT 49354
                  21.49.14.1  Definition and basic properties   asclelbasALT 49354
            21.49.15  Categories   homf0 49357
                  21.49.15.1  Categories   homf0 49357
                  21.49.15.2  Opposite category   oppccatb 49364
                  21.49.15.3  Monomorphisms and epimorphisms   idmon 49368
                  21.49.15.4  Sections, inverses, isomorphisms   sectrcl 49370
                  21.49.15.5  Isomorphic objects   cicfn 49390
                  21.49.15.6  Subcategories   dmdm 49401
                  21.49.15.7  Functors   reldmfunc 49423
                  21.49.15.8  Opposite functors   coppf 49470
                  21.49.15.9  Full & faithful functors   imasubc 49499
                  21.49.15.10  Universal property   upciclem1 49514
                  21.49.15.11  Natural transformations and the functor category   isnatd 49571
                  21.49.15.12  Initial, terminal and zero objects of a category   initoo2 49580
                  21.49.15.13  Product of categories   reldmxpc 49594
                  21.49.15.14  Swap functors   cswapf 49607
                  21.49.15.15  Functor evaluation   oppc1stflem 49635
                  21.49.15.16  Transposed curry functors   cofuswapfcl 49641
                  21.49.15.17  Constant functors   diag1 49652
                  21.49.15.18  Functor composition bifunctors   fucofulem1 49658
                  21.49.15.19  Post-composition functors   postcofval 49712
                  21.49.15.20  Pre-composition functors   precofvallem 49714
            21.49.16  Examples of categories   catcrcl 49743
                  21.49.16.1  The category of categories   catcrcl 49743
                  21.49.16.2  Thin categories   cthinc 49765
                  21.49.16.3  Terminal categories   ctermc 49820
                  21.49.16.4  Preordered sets as thin categories   cprstc 49897
                  21.49.16.5  Monoids as categories   cmndtc 49925
                  21.49.16.6  Categories with at most one object and at most two morphisms   2arwcatlem1 49943
            21.49.17  Kan extensions and related concepts   clan 49953
                  21.49.17.1  Kan extensions   clan 49953
                  21.49.17.2  Limits and colimits   clmd 49991
      ​21.50  Mathbox for Emmett Weisz
            ​*21.50.1  Miscellaneous Theorems   nfintd 50021
            21.50.2  Set Recursion   csetrecs 50031
                  ​*21.50.2.1  Basic Properties of Set Recursion   csetrecs 50031
                  21.50.2.2  Examples and properties of set recursion   elsetrecslem 50047
            ​*21.50.3  Construction of Games and Surreal Numbers   cpg 50057
      ​​*21.51  Mathbox for David A. Wheeler
            21.51.1  Natural deduction   sbidd 50066
            ​*21.51.2  Greater than, greater than or equal to.   cge-real 50068
            ​*21.51.3  Hyperbolic trigonometric functions   csinh 50078
            ​*21.51.4  Reciprocal trigonometric functions (sec, csc, cot)   csec 50089
            ​*21.51.5  Identities for "if"   ifnmfalse 50111
            ​*21.51.6  Logarithms generalized to arbitrary base using ` logb `   logb2aval 50112
            ​*21.51.7  Logarithm laws generalized to an arbitrary base - log_   clog- 50113
            ​*21.51.8  Formally define notions such as reflexivity   wreflexive 50115
            ​*21.51.9  Algebra helpers   mvlraddi 50119
            ​*21.51.10  Algebra helper examples   i2linesi 50126
            ​*21.51.11  Formal methods "surprises"   alimp-surprise 50128
            ​*21.51.12  Allsome quantifier   walsi 50134
            ​*21.51.13  Miscellaneous   5m4e1 50145
            21.51.14  Theorems about algebraic numbers   aacllem 50149
      ​21.52  Mathbox for Kunhao Zheng
            21.52.1  Weighted AM-GM inequality   amgmwlem 50150