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 3349
                  2.1.5.3  Restricted class abstraction   crab 3400
            2.1.6  The universal class   cvv 3441
            ​*2.1.7  Conditional equality (experimental)   wcdeq 3722
            2.1.8  Russell's Paradox   rru 3738
            2.1.9  Proper substitution of classes for sets   wsbc 3741
            2.1.10  Proper substitution of classes for sets into classes   csb 3850
            2.1.11  Define basic set operations and relations   cdif 3899
            2.1.12  Subclasses and subsets   df-ss 3919
            2.1.13  The difference, union, and intersection of two classes   dfdif3 4070
                  2.1.13.1  The difference of two classes   dfdif3 4070
                  2.1.13.2  The union of two classes   elun 4106
                  2.1.13.3  The intersection of two classes   elini 4152
                  2.1.13.4  The symmetric difference of two classes   csymdif 4205
                  2.1.13.5  Combinations of difference, union, and intersection of two classes   unabs 4218
                  2.1.13.6  Class abstractions with difference, union, and intersection of two classes   unabw 4260
                  2.1.13.7  Restricted uniqueness with difference, union, and intersection   reuun2 4278
            2.1.14  The empty set   c0 4286
            ​*2.1.15  The conditional operator for classes   cif 4480
            ​*2.1.16  The weak deduction theorem for set theory   dedth 4539
            2.1.17  Power classes   cpw 4555
            2.1.18  Unordered and ordered pairs   snjust 4580
            2.1.19  The union of a class   cuni 4864
            2.1.20  The intersection of a class   cint 4903
            2.1.21  Indexed union and intersection   ciun 4947
            2.1.22  Disjointness   wdisj 5066
            2.1.23  Binary relations   wbr 5099
            2.1.24  Ordered-pair class abstractions (class builders)   copab 5161
            2.1.25  Functions in maps-to notation   cmpt 5180
            2.1.26  Transitive classes   wtr 5206
      ​2.2  ZF Set Theory - add the Axiom of Replacement
            2.2.1  Introduce the Axiom of Replacement   ax-rep 5225
            2.2.2  Derive the Axiom of Separation   axsepgfromrep 5240
            2.2.3  Derive the Null Set Axiom   axnulALT 5250
            2.2.4  Theorems requiring subset and intersection existence   nalset 5259
            2.2.5  Theorems requiring empty set existence   class2set 5301
      ​2.3  ZF Set Theory - add the Axiom of Power Sets
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 5311
            2.3.2  Derive the Axiom of Pairing   axprlem1 5369
            2.3.3  Ordered pair theorem   opnz 5422
            2.3.4  Ordered-pair class abstractions (cont.)   opabidw 5473
            2.3.5  Power class of union and intersection   pwin 5516
            2.3.6  The identity relation   cid 5519
            2.3.7  The membership relation (or epsilon relation)   cep 5524
            ​*2.3.8  Partial and total orderings   wpo 5531
            2.3.9  Founded and well-ordering relations   wfr 5575
            2.3.10  Relations   cxp 5623
            2.3.11  The Predecessor Class   cpred 6259
            2.3.12  Well-founded induction (variant)   frpomin 6299
            2.3.13  Well-ordered induction   tz6.26 6306
            2.3.14  Ordinals   word 6317
            2.3.15  Definite description binder (inverted iota)   cio 6447
            2.3.16  Functions   wfun 6487
            2.3.17  Cantor's Theorem   canth 7314
            2.3.18  Restricted iota (description binder)   crio 7316
            2.3.19  Operations   co 7360
                  2.3.19.1  Variable-to-class conversion for operations   caovclg 7552
            2.3.20  Maps-to notation   mpondm0 7600
            2.3.21  Function operation   cof 7622
            2.3.22  Proper subset relation   crpss 7669
      ​2.4  ZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 7682
            2.4.2  Ordinals (continued)   epweon 7722
            2.4.3  Transfinite induction   tfi 7797
            2.4.4  The natural numbers (i.e., finite ordinals)   com 7810
            2.4.5  Peano's postulates   peano1 7833
            2.4.6  Finite induction (for finite ordinals)   find 7839
            2.4.7  Relations and functions (cont.)   dmexg 7845
            2.4.8  First and second members of an ordered pair   c1st 7933
            2.4.9  Induction on Cartesian products   frpoins3xpg 8084
            2.4.10  Ordering on Cartesian products   xpord2lem 8086
            2.4.11  Ordering Ordinal Sequences   orderseqlem 8101
            ​*2.4.12  The support of functions   csupp 8104
            ​*2.4.13  Special maps-to operations   opeliunxp2f 8154
            2.4.14  Function transposition   ctpos 8169
            2.4.15  Curry and uncurry   ccur 8209
            2.4.16  Undefined values   cund 8216
            2.4.17  Well-founded recursion   cfrecs 8224
            2.4.18  Well-ordered recursion   cwrecs 8255
            2.4.19  Functions on ordinals; strictly monotone ordinal functions   iunon 8273
            2.4.20  "Strong" transfinite recursion   crecs 8304
            2.4.21  Recursive definition generator   crdg 8342
            2.4.22  Finite recursion   frfnom 8368
            2.4.23  Ordinal arithmetic   c1o 8392
            2.4.24  Natural number arithmetic   nna0 8534
            2.4.25  Natural addition   cnadd 8595
            2.4.26  Equivalence relations and classes   wer 8634
            2.4.27  The mapping operation   cmap 8767
            2.4.28  Infinite Cartesian products   cixp 8839
            2.4.29  Equinumerosity   cen 8884
            2.4.30  Schroeder-Bernstein Theorem   sbthlem1 9019
            2.4.31  Equinumerosity (cont.)   xpf1o 9071
            2.4.32  Finite sets   dif1enlem 9088
            2.4.33  Pigeonhole Principle   phplem1 9132
            2.4.34  Finite sets (cont.)   onomeneq 9142
            2.4.35  Finitely supported functions   cfsupp 9268
            2.4.36  Finite intersections   cfi 9317
            2.4.37  Hall's marriage theorem   marypha1lem 9340
            2.4.38  Supremum and infimum   csup 9347
            2.4.39  Ordinal isomorphism, Hartogs's theorem   coi 9418
            2.4.40  Hartogs function   char 9465
            2.4.41  Weak dominance   cwdom 9473
      ​2.5  ZF Set Theory - add the Axiom of Regularity
            2.5.1  Introduce the Axiom of Regularity   ax-reg 9501
            2.5.2  Axiom of Infinity equivalents   inf0 9534
      ​2.6  ZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-inf 9551
            2.6.2  Existence of omega (the set of natural numbers)   omex 9556
            2.6.3  Cantor normal form   ccnf 9574
            2.6.4  Transitive closure of a relation   cttrcl 9620
            2.6.5  Transitive closure   trcl 9641
            2.6.6  Set induction (or epsilon induction)   setind 9660
            2.6.7  Well-Founded Induction   frmin 9665
            2.6.8  Well-Founded Recursion   frr3g 9672
            2.6.9  Rank   cr1 9678
            2.6.10  Scott's trick; collection principle; Hilbert's epsilon   scottex 9801
            2.6.11  Disjoint union   cdju 9814
            2.6.12  Cardinal numbers   ccrd 9851
            2.6.13  Axiom of Choice equivalents   wac 10029
            ​*2.6.14  Cardinal number arithmetic   undjudom 10082
            2.6.15  The Ackermann bijection   ackbij2lem1 10132
            2.6.16  Cofinality (without Axiom of Choice)   cflem 10159
            2.6.17  Eight inequivalent definitions of finite set   sornom 10191
            2.6.18  Hereditarily size-limited sets without Choice   itunifval 10330
​​*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 10349
            3.1.2  Introduce the Axiom of Dependent Choice   ax-dc 10360
      ​3.2  ZFC Set Theory - add the Axiom of Choice
            3.2.1  Introduce the Axiom of Choice   ax-ac 10373
            3.2.2  AC equivalents: well-ordering, Zorn's lemma   numthcor 10408
            3.2.3  Cardinal number theorems using Axiom of Choice   cardval 10460
            3.2.4  Cardinal number arithmetic using Axiom of Choice   iunctb 10489
            3.2.5  Cofinality using the Axiom of Choice   alephreg 10497
      ​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 10535
            3.4.2  Derivation of the Axiom of Choice   gchaclem 10593
​​*PART 4  TG (TARSKI-GROTHENDIECK) SET THEORY
      ​4.1  Inaccessibles
            4.1.1  Weakly and strongly inaccessible cardinals   cwina 10597
            4.1.2  Weak universes   cwun 10615
            4.1.3  Tarski classes   ctsk 10663
            4.1.4  Grothendieck universes   cgru 10705
      ​4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
            4.2.1  Introduce the Tarski-Grothendieck Axiom   ax-groth 10738
            4.2.2  Derive the Power Set, Infinity and Choice Axioms   grothpw 10741
            4.2.3  Tarski map function   ctskm 10752
​​*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 10759
            5.1.2  Final derivation of real and complex number postulates   axaddf 11060
            5.1.3  Real and complex number postulates restated as axioms   ax-cnex 11086
      ​5.2  Derive the basic properties from the field axioms
            5.2.1  Some deductions from the field axioms for complex numbers   cnex 11111
            5.2.2  Infinity and the extended real number system   cpnf 11167
            5.2.3  Restate the ordering postulates with extended real "less than"   axlttri 11208
            5.2.4  Ordering on reals   lttr 11213
            5.2.5  Initial properties of the complex numbers   mul12 11302
      ​5.3  Real and complex numbers - basic operations
            5.3.1  Addition   add12 11355
            5.3.2  Subtraction   cmin 11368
            5.3.3  Multiplication   kcnktkm1cn 11572
            5.3.4  Ordering on reals (cont.)   gt0ne0 11606
            5.3.5  Reciprocals   ixi 11770
            5.3.6  Division   cdiv 11798
            5.3.7  Ordering on reals (cont.)   elimgt0 11983
            5.3.8  Completeness Axiom and Suprema   fimaxre 12090
            5.3.9  Imaginary and complex number properties   neg1cn 12134
            5.3.10  Function operation analogue theorems   ofsubeq0 12146
      ​5.4  Integer sets
            5.4.1  Positive integers (as a subset of complex numbers)   cn 12149
            5.4.2  Principle of mathematical induction   nnind 12167
            ​*5.4.3  Decimal representation of numbers   c2 12204
            ​*5.4.4  Some properties of specific numbers   1pneg1e0 12263
            5.4.5  Simple number properties   halfcl 12371
            5.4.6  The Archimedean property   nnunb 12401
            5.4.7  Nonnegative integers (as a subset of complex numbers)   cn0 12405
            ​*5.4.8  Extended nonnegative integers   cxnn0 12478
            5.4.9  Integers (as a subset of complex numbers)   cz 12492
            5.4.10  Decimal arithmetic   cdc 12611
            5.4.11  Upper sets of integers   cuz 12755
            5.4.12  Well-ordering principle for bounded-below sets of integers   uzwo3 12860
            5.4.13  Rational numbers (as a subset of complex numbers)   cq 12865
            5.4.14  Existence of the set of complex numbers   rpnnen1lem2 12894
      ​5.5  Order sets
            5.5.1  Positive reals (as a subset of complex numbers)   crp 12909
            5.5.2  Infinity and the extended real number system (cont.)   cxne 13027
            5.5.3  Supremum and infimum on the extended reals   xrsupexmnf 13224
            5.5.4  Real number intervals   cioo 13265
            5.5.5  Finite intervals of integers   cfz 13427
            ​*5.5.6  Finite intervals of nonnegative integers   elfz2nn0 13538
            5.5.7  Half-open integer ranges   cfzo 13574
      ​5.6  Elementary integer functions
            5.6.1  The floor and ceiling functions   cfl 13714
            5.6.2  The modulo (remainder) operation   cmo 13793
            5.6.3  Miscellaneous theorems about integers   om2uz0i 13874
            5.6.4  Strong induction over upper sets of integers   uzsinds 13914
            5.6.5  Finitely supported functions over the nonnegative integers   fsuppmapnn0fiublem 13917
            5.6.6  The infinite sequence builder "seq" - extension   cseq 13928
            5.6.7  Integer powers   cexp 13988
            5.6.8  Ordered pair theorem for nonnegative integers   nn0le2msqi 14194
            5.6.9  Factorial function   cfa 14200
            5.6.10  The binomial coefficient operation   cbc 14229
            5.6.11  The ` # ` (set size) function   chash 14257
                  5.6.11.1  Proper unordered pairs and triples (sets of size 2 and 3)   hashprlei 14395
                  5.6.11.2  Functions with a domain containing at least two different elements   fundmge2nop0 14429
                  5.6.11.3  Finite induction on the size of the first component of a binary relation   hashdifsnp1 14433
      ​​*5.7  Words over a set
            5.7.1  Definitions and basic theorems   cword 14440
            5.7.2  Last symbol of a word   clsw 14489
            5.7.3  Concatenations of words   cconcat 14497
            5.7.4  Singleton words   cs1 14523
            5.7.5  Concatenations with singleton words   ccatws1cl 14544
            5.7.6  Subwords/substrings   csubstr 14568
            5.7.7  Prefixes of a word   cpfx 14598
            5.7.8  Subwords of subwords   swrdswrdlem 14631
            5.7.9  Subwords and concatenations   pfxcctswrd 14637
            5.7.10  Subwords of concatenations   swrdccatfn 14651
            5.7.11  Splicing words (substring replacement)   csplice 14676
            5.7.12  Reversing words   creverse 14685
            5.7.13  Repeated symbol words   creps 14695
            ​*5.7.14  Cyclical shifts of words   ccsh 14715
            5.7.15  Mapping words by a function   wrdco 14758
            5.7.16  Longer string literals   cs2 14768
      ​​*5.8  Reflexive and transitive closures of relations
            5.8.1  The reflexive and transitive properties of relations   coss12d 14899
            5.8.2  Basic properties of closures   cleq1lem 14909
            5.8.3  Definitions and basic properties of transitive closures   ctcl 14912
            5.8.4  Exponentiation of relations   crelexp 14946
            5.8.5  Reflexive-transitive closure as an indexed union   crtrcl 14982
            ​*5.8.6  Principle of transitive induction.   relexpindlem 14990
      ​5.9  Elementary real and complex functions
            5.9.1  The "shift" operation   cshi 14993
            5.9.2  Signum (sgn or sign) function   csgn 15013
            5.9.3  Real and imaginary parts; conjugate   ccj 15023
            5.9.4  Square root; absolute value   csqrt 15160
      ​5.10  Elementary limits and convergence
            5.10.1  Superior limit (lim sup)   clsp 15397
            5.10.2  Limits   cli 15411
            5.10.3  Finite and infinite sums   csu 15613
            5.10.4  The binomial theorem   binomlem 15756
            5.10.5  The inclusion/exclusion principle   incexclem 15763
            5.10.6  Infinite sums (cont.)   isumshft 15766
            5.10.7  Miscellaneous converging and diverging sequences   divrcnv 15779
            5.10.8  Arithmetic series   arisum 15787
            5.10.9  Geometric series   expcnv 15791
            5.10.10  Ratio test for infinite series convergence   cvgrat 15810
            5.10.11  Mertens' theorem   mertenslem1 15811
            5.10.12  Finite and infinite products   prodf 15814
                  5.10.12.1  Product sequences   prodf 15814
                  5.10.12.2  Non-trivial convergence   ntrivcvg 15824
                  5.10.12.3  Complex products   cprod 15830
                  5.10.12.4  Finite products   fprod 15868
                  5.10.12.5  Infinite products   iprodclim 15925
            5.10.13  Falling and Rising Factorial   cfallfac 15931
            5.10.14  Bernoulli polynomials and sums of k-th powers   cbp 15973
      ​5.11  Elementary trigonometry
            5.11.1  The exponential, sine, and cosine functions   ce 15988
                  5.11.1.1  The circle constant (tau = 2 pi)   ctau 16131
            5.11.2  _e is irrational   eirrlem 16133
      ​5.12  Cardinality of real and complex number subsets
            5.12.1  Countability of integers and rationals   xpnnen 16140
            5.12.2  The reals are uncountable   rpnnen2lem1 16143
​​*PART 6  ELEMENTARY NUMBER THEORY
      ​6.1  Elementary properties of divisibility
            6.1.1  Irrationality of square root of 2   sqrt2irrlem 16177
            6.1.2  Some Number sets are chains of proper subsets   nthruc 16181
            6.1.3  The divides relation   cdvds 16183
            ​*6.1.4  Even and odd numbers   evenelz 16267
            6.1.5  The division algorithm   divalglem0 16324
            6.1.6  Bit sequences   cbits 16350
            6.1.7  The greatest common divisor operator   cgcd 16425
            6.1.8  Bézout's identity   bezoutlem1 16470
            6.1.9  Algorithms   nn0seqcvgd 16501
            6.1.10  Euclid's Algorithm   eucalgval2 16512
            ​*6.1.11  The least common multiple   clcm 16519
            ​*6.1.12  Coprimality and Euclid's lemma   coprmgcdb 16580
            6.1.13  Cancellability of congruences   congr 16595
      ​6.2  Elementary prime number theory
            ​*6.2.1  Elementary properties   cprime 16602
            ​*6.2.2  Coprimality and Euclid's lemma (cont.)   coprm 16642
            6.2.3  Properties of the canonical representation of a rational   cnumer 16664
            6.2.4  Euler's theorem   codz 16694
            6.2.5  Arithmetic modulo a prime number   modprm1div 16729
            6.2.6  Pythagorean Triples   coprimeprodsq 16740
            6.2.7  The prime count function   cpc 16768
            6.2.8  Pocklington's theorem   prmpwdvds 16836
            6.2.9  Infinite primes theorem   unbenlem 16840
            6.2.10  Sum of prime reciprocals   prmreclem1 16848
            6.2.11  Fundamental theorem of arithmetic   1arithlem1 16855
            6.2.12  Lagrange's four-square theorem   cgz 16861
            6.2.13  Van der Waerden's theorem   cvdwa 16897
            6.2.14  Ramsey's theorem   cram 16931
            ​*6.2.15  Primorial function   cprmo 16963
            ​*6.2.16  Prime gaps   prmgaplem1 16981
            6.2.17  Decimal arithmetic (cont.)   dec2dvds 16995
            6.2.18  Cyclical shifts of words (cont.)   cshwsidrepsw 17025
            6.2.19  Specific prime numbers   prmlem0 17037
            6.2.20  Very large primes   1259lem1 17062
​PART 7  BASIC STRUCTURES
      ​7.1  Extensible structures
            ​*7.1.1  Basic definitions   cstr 17077
                  7.1.1.1  Extensible structures as structures with components   cstr 17077
                  7.1.1.2  Substitution of components   csts 17094
                  7.1.1.3  Slots   cslot 17112
                  ​*7.1.1.4  Structure component indices   cnx 17124
                  7.1.1.5  Base sets   cbs 17140
                  7.1.1.6  Base set restrictions   cress 17161
            7.1.2  Slot definitions   cplusg 17181
            7.1.3  Definition of the structure product   crest 17344
            7.1.4  Definition of the structure quotient   cordt 17424
      ​7.2  Moore spaces
            7.2.1  Moore closures   mrcflem 17533
            7.2.2  Independent sets in a Moore system   mrisval 17557
            7.2.3  Algebraic closure systems   isacs 17578
​PART 8  BASIC CATEGORY THEORY
      ​8.1  Categories
            8.1.1  Categories   ccat 17591
            8.1.2  Opposite category   coppc 17638
            8.1.3  Monomorphisms and epimorphisms   cmon 17656
            8.1.4  Sections, inverses, isomorphisms   csect 17672
            ​*8.1.5  Isomorphic objects   ccic 17723
            8.1.6  Subcategories   cssc 17735
            8.1.7  Functors   cfunc 17782
            8.1.8  Full & faithful functors   cful 17832
            8.1.9  Natural transformations and the functor category   cnat 17872
            8.1.10  Initial, terminal and zero objects of a category   cinito 17909
      ​8.2  Arrows (disjointified hom-sets)
            8.2.1  Identity and composition for arrows   cida 17981
      ​8.3  Examples of categories
            8.3.1  The category of sets   csetc 18003
            8.3.2  The category of categories   ccatc 18026
            ​*8.3.3  The category of extensible structures   fncnvimaeqv 18047
      ​8.4  Categorical constructions
            8.4.1  Product of categories   cxpc 18095
            8.4.2  Functor evaluation   cevlf 18136
            8.4.3  Hom functor   chof 18175
​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 18358
            9.5.2  Complete lattices   ccla 18425
            9.5.3  Distributive lattices   cdlat 18447
            9.5.4  Subset order structures   cipo 18454
      ​9.6  Posets, directed sets, and lattices as relations
            ​*9.6.1  Posets and lattices as relations   cps 18491
            9.6.2  Directed sets, nets   cdir 18521
            9.6.3  Chains   cchn 18532
​PART 10  BASIC ALGEBRAIC STRUCTURES
      ​10.1  Monoids
            ​*10.1.1  Magmas   cplusf 18566
            ​*10.1.2  Identity elements   mgmidmo 18589
            ​*10.1.3  Iterated sums in a magma   gsumvalx 18605
            10.1.4  Magma homomorphisms and submagmas   cmgmhm 18619
            ​*10.1.5  Semigroups   csgrp 18647
            ​*10.1.6  Definition and basic properties of monoids   cmnd 18663
            10.1.7  Monoid homomorphisms and submonoids   cmhm 18710
            ​*10.1.8  Iterated sums in a monoid   gsumvallem2 18763
            10.1.9  Free monoids   cfrmd 18776
                  ​*10.1.9.1  Monoid of endofunctions   cefmnd 18797
            10.1.10  Examples and counterexamples for magmas, semigroups and monoids   mgm2nsgrplem1 18847
      ​10.2  Groups
            10.2.1  Definition and basic properties   cgrp 18867
            ​*10.2.2  Group multiple operation   cmg 19001
            10.2.3  Subgroups and Quotient groups   csubg 19054
            ​*10.2.4  Cyclic monoids and groups   cycsubmel 19133
            10.2.5  Elementary theory of group homomorphisms   cghm 19145
            10.2.6  Isomorphisms of groups   cgim 19190
                  10.2.6.1  The first isomorphism theorem of groups   ghmqusnsglem1 19213
            10.2.7  Group actions   cga 19222
            10.2.8  Centralizers and centers   ccntz 19248
            10.2.9  The opposite group   coppg 19278
            10.2.10  Symmetric groups   csymg 19302
                  ​*10.2.10.1  Definition and basic properties   csymg 19302
                  10.2.10.2  Cayley's theorem   cayleylem1 19345
                  10.2.10.3  Permutations fixing one element   symgfix2 19349
                  ​*10.2.10.4  Transpositions in the symmetric group   cpmtr 19374
                  10.2.10.5  The sign of a permutation   cpsgn 19422
            10.2.11  p-Groups and Sylow groups; Sylow's theorems   cod 19457
            10.2.12  Direct products   clsm 19567
                  10.2.12.1  Direct products (extension)   smndlsmidm 19589
            10.2.13  Free groups   cefg 19639
            10.2.14  Abelian groups   ccmn 19713
                  10.2.14.1  Definition and basic properties   ccmn 19713
                  10.2.14.2  Cyclic groups   ccyg 19810
                  10.2.14.3  Group sum operation   gsumval3a 19836
                  10.2.14.4  Group sums over (ranges of) integers   fsfnn0gsumfsffz 19916
                  10.2.14.5  Internal direct products   cdprd 19928
                  10.2.14.6  The Fundamental Theorem of Abelian Groups   ablfacrplem 20000
            10.2.15  Simple groups   csimpg 20025
                  10.2.15.1  Definition and basic properties   csimpg 20025
                  10.2.15.2  Classification of abelian simple groups   ablsimpnosubgd 20039
            10.2.16  Totally ordered monoids and groups   comnd 20052
      ​10.3  Rings
            10.3.1  Multiplicative Group   cmgp 20079
            ​*10.3.2  Non-unital rings ("rngs")   crng 20091
            ​*10.3.3  Ring unity (multiplicative identity)   cur 20120
            10.3.4  Semirings   csrg 20125
                  ​*10.3.4.1  The binomial theorem for semirings   srgbinomlem1 20165
            10.3.5  Unital rings   crg 20172
            10.3.6  Opposite ring   coppr 20276
            10.3.7  Divisibility   cdsr 20294
            10.3.8  Ring primes   crpm 20372
            10.3.9  Homomorphisms of non-unital rings   crnghm 20374
            10.3.10  Ring homomorphisms   crh 20409
            10.3.11  Nonzero rings and zero rings   cnzr 20449
            10.3.12  Local rings   clring 20475
            10.3.13  Subrings   csubrng 20482
                  10.3.13.1  Subrings of non-unital rings   csubrng 20482
                  10.3.13.2  Subrings of unital rings   csubrg 20506
                  10.3.13.3  Subrings generated by a subset   crgspn 20547
            10.3.14  Categories of rings   crngc 20553
                  ​*10.3.14.1  The category of non-unital rings   crngc 20553
                  ​*10.3.14.2  The category of (unital) rings   cringc 20582
                  10.3.14.3  Subcategories of the category of rings   srhmsubclem1 20614
            10.3.15  Left regular elements and domains   crlreg 20628
      ​10.4  Division rings and fields
            10.4.1  Definition and basic properties   cdr 20666
            10.4.2  Sub-division rings   csdrg 20723
            10.4.3  Absolute value (abstract algebra)   cabv 20745
            10.4.4  Star rings   cstf 20774
            10.4.5  Totally ordered rings and fields   corng 20794
      ​10.5  Left modules
            10.5.1  Definition and basic properties   clmod 20815
            10.5.2  Subspaces and spans in a left module   clss 20886
            10.5.3  Homomorphisms and isomorphisms of left modules   clmhm 20975
            10.5.4  Subspace sum; bases for a left module   clbs 21030
      ​10.6  Vector spaces
            10.6.1  Definition and basic properties   clvec 21058
      ​10.7  Subring algebras and ideals
            10.7.1  Subring algebras   csra 21127
            ​*10.7.2  Left ideals and spans   clidl 21165
            10.7.3  Two-sided ideals and quotient rings   c2idl 21208
                  ​*10.7.3.1  Condition for a non-unital ring to be unital   rngqiprng1elbas 21245
            10.7.4  Principal ideal rings. Divisibility in the integers   clpidl 21279
            10.7.5  Principal ideal domains   cpid 21295
      ​10.8  The complex numbers as an algebraic extensible structure
            10.8.1  Definition and basic properties   cpsmet 21297
            ​*10.8.2  Ring of integers   czring 21405
                  ​*10.8.2.1  Example for a condition for a non-unital ring to be unital   pzriprnglem1 21440
            10.8.3  Algebraic constructions based on the complex numbers   czrh 21458
            10.8.4  Signs as subgroup of the complex numbers   cnmsgnsubg 21536
            10.8.5  Embedding of permutation signs into a ring   zrhpsgnmhm 21543
            10.8.6  The ordered field of real numbers   crefld 21563
      ​10.9  Generalized pre-Hilbert and Hilbert spaces
            10.9.1  Definition and basic properties   cphl 21583
            10.9.2  Orthocomplements and closed subspaces   cocv 21619
            10.9.3  Orthogonal projection and orthonormal bases   cpj 21659
​​*PART 11  BASIC LINEAR ALGEBRA
      ​11.1  Vectors and free modules
            *11.1.1  Direct sum of left modules   cdsmm 21690
            ​*11.1.2  Free modules   cfrlm 21705
            ​*11.1.3  Standard basis (unit vectors)   cuvc 21741
            ​*11.1.4  Independent sets and families   clindf 21763
            11.1.5  Characterization of free modules   lmimlbs 21795
      ​11.2  Associative algebras
            11.2.1  Definition and basic properties   casa 21809
      ​11.3  Abstract multivariate polynomials
            11.3.1  Definition and basic properties   cmps 21864
            11.3.2  Polynomial evaluation   ces 22031
            11.3.3  Additional definitions for (multivariate) polynomials   cslv 22075
            ​*11.3.4  Univariate polynomials   cps1 22119
            11.3.5  Univariate polynomial evaluation   ces1 22261
                  11.3.5.1  Specialization of polynomial evaluation as a ring homomorphism   evls1scafv 22314
      ​​*11.4  Matrices
            *11.4.1  The matrix multiplication   cmmul 22338
            ​*11.4.2  Square matrices   cmat 22355
            ​*11.4.3  The matrix algebra   matmulr 22386
            ​*11.4.4  Matrices of dimension 0 and 1   mat0dimbas0 22414
            ​*11.4.5  The subalgebras of diagonal and scalar matrices   cdmat 22436
            ​*11.4.6  Multiplication of a matrix with a "column vector"   cmvmul 22488
            11.4.7  Replacement functions for a square matrix   cmarrep 22504
            11.4.8  Submatrices   csubma 22524
      ​11.5  The determinant
            11.5.1  Definition and basic properties   cmdat 22532
            11.5.2  Determinants of 2 x 2 -matrices   m2detleiblem1 22572
            11.5.3  The matrix adjugate/adjunct   cmadu 22580
            ​*11.5.4  Laplace expansion of determinants (special case)   symgmatr01lem 22601
            11.5.5  Inverse matrix   invrvald 22624
            ​*11.5.6  Cramer's rule   slesolvec 22627
      ​​*11.6  Polynomial matrices
            11.6.1  Basic properties   pmatring 22640
            ​*11.6.2  Constant polynomial matrices   ccpmat 22651
            ​*11.6.3  Collecting coefficients of polynomial matrices   cdecpmat 22710
            ​*11.6.4  Ring isomorphism between polynomial matrices and polynomials over matrices   cpm2mp 22740
      ​​*11.7  The characteristic polynomial
            *11.7.1  Definition and basic properties   cchpmat 22774
            ​*11.7.2  The characteristic factor function G   fvmptnn04if 22797
            ​*11.7.3  The Cayley-Hamilton theorem   cpmadurid 22815
​PART 12  BASIC TOPOLOGY
      ​12.1  Topology
            ​*12.1.1  Topological spaces   ctop 22841
                  12.1.1.1  Topologies   ctop 22841
                  12.1.1.2  Topologies on sets   ctopon 22858
                  12.1.1.3  Topological spaces   ctps 22880
            12.1.2  Topological bases   ctb 22893
            12.1.3  Examples of topologies   distop 22943
            12.1.4  Closure and interior   ccld 22964
            12.1.5  Neighborhoods   cnei 23045
            12.1.6  Limit points and perfect sets   clp 23082
            12.1.7  Subspace topologies   restrcl 23105
            12.1.8  Order topology   ordtbaslem 23136
            12.1.9  Limits and continuity in topological spaces   ccn 23172
            12.1.10  Separated spaces: T0, T1, T2 (Hausdorff) ...   ct0 23254
            12.1.11  Compactness   ccmp 23334
            12.1.12  Bolzano-Weierstrass theorem   bwth 23358
            12.1.13  Connectedness   cconn 23359
            12.1.14  First- and second-countability   c1stc 23385
            12.1.15  Local topological properties   clly 23412
            12.1.16  Refinements   cref 23450
            12.1.17  Compactly generated spaces   ckgen 23481
            12.1.18  Product topologies   ctx 23508
            12.1.19  Continuous function-builders   cnmptid 23609
            12.1.20  Quotient maps and quotient topology   ckq 23641
            12.1.21  Homeomorphisms   chmeo 23701
      ​12.2  Filters and filter bases
            12.2.1  Filter bases   elmptrab 23775
            12.2.2  Filters   cfil 23793
            12.2.3  Ultrafilters   cufil 23847
            12.2.4  Filter limits   cfm 23881
            12.2.5  Extension by continuity   ccnext 24007
            12.2.6  Topological groups   ctmd 24018
            12.2.7  Infinite group sum on topological groups   ctsu 24074
            12.2.8  Topological rings, fields, vector spaces   ctrg 24104
      ​12.3  Uniform Structures and Spaces
            12.3.1  Uniform structures   cust 24148
            12.3.2  The topology induced by an uniform structure   cutop 24178
            12.3.3  Uniform Spaces   cuss 24201
            12.3.4  Uniform continuity   cucn 24222
            12.3.5  Cauchy filters in uniform spaces   ccfilu 24233
            12.3.6  Complete uniform spaces   ccusp 24244
      ​12.4  Metric spaces
            12.4.1  Pseudometric spaces   ispsmet 24252
            12.4.2  Basic metric space properties   cxms 24265
            12.4.3  Metric space balls   blfvalps 24331
            12.4.4  Open sets of a metric space   mopnval 24386
            12.4.5  Continuity in metric spaces   metcnp3 24488
            12.4.6  The uniform structure generated by a metric   metuval 24497
            12.4.7  Examples of metric spaces   dscmet 24520
            ​*12.4.8  Normed algebraic structures   cnm 24524
            12.4.9  Normed space homomorphisms (bounded linear operators)   cnmo 24653
            12.4.10  Topology on the reals   qtopbaslem 24706
            12.4.11  Topological definitions using the reals   cii 24828
            12.4.12  Path homotopy   chtpy 24926
            12.4.13  The fundamental group   cpco 24960
      ​12.5  Metric subcomplex vector spaces
            12.5.1  Subcomplex modules   cclm 25022
            ​*12.5.2  Subcomplex vector spaces   ccvs 25083
            ​*12.5.3  Normed subcomplex vector spaces   isncvsngp 25109
            12.5.4  Subcomplex pre-Hilbert spaces   ccph 25126
            12.5.5  Convergence and completeness   ccfil 25212
            12.5.6  Baire's Category Theorem   bcthlem1 25284
            12.5.7  Banach spaces and subcomplex Hilbert spaces   ccms 25292
                  12.5.7.1  The complete ordered field of the real numbers   retopn 25339
            12.5.8  Euclidean spaces   crrx 25343
            12.5.9  Minimizing Vector Theorem   minveclem1 25384
            12.5.10  Projection Theorem   pjthlem1 25397
​PART 13  BASIC REAL AND COMPLEX ANALYSIS
      ​13.1  Continuity
            13.1.1  Intermediate value theorem   pmltpclem1 25409
      ​13.2  Integrals
            13.2.1  Lebesgue measure   covol 25423
            13.2.2  Lebesgue integration   cmbf 25575
                  13.2.2.1  Lesbesgue integral   cmbf 25575
                  13.2.2.2  Lesbesgue directed integral   cdit 25807
      ​13.3  Derivatives
            13.3.1  Real and complex differentiation   climc 25823
                  13.3.1.1  Derivatives of functions of one complex or real variable   climc 25823
                  13.3.1.2  Results on real differentiation   dvferm1lem 25948
​PART 14  BASIC REAL AND COMPLEX FUNCTIONS
      ​14.1  Polynomials
            14.1.1  Polynomial degrees   cmdg 26018
            14.1.2  The division algorithm for univariate polynomials   cmn1 26091
            14.1.3  Elementary properties of complex polynomials   cply 26149
            14.1.4  The division algorithm for polynomials   cquot 26258
            14.1.5  Algebraic numbers   caa 26282
            14.1.6  Liouville's approximation theorem   aalioulem1 26300
      ​14.2  Sequences and series
            14.2.1  Taylor polynomials and Taylor's theorem   ctayl 26320
            14.2.2  Uniform convergence   culm 26345
            14.2.3  Power series   pserval 26379
      ​14.3  Basic trigonometry
            14.3.1  The exponential, sine, and cosine functions (cont.)   efcn 26413
            14.3.2  Properties of pi = 3.14159...   pilem1 26421
            14.3.3  Mapping of the exponential function   efgh 26510
            14.3.4  The natural logarithm on complex numbers   clog 26523
            ​*14.3.5  Logarithms to an arbitrary base   clogb 26734
            14.3.6  Theorems of Pythagoras, isosceles triangles, and intersecting chords   angval 26771
            14.3.7  Solutions of quadratic, cubic, and quartic equations   quad2 26809
            14.3.8  Inverse trigonometric functions   casin 26832
            14.3.9  The Birthday Problem   log2ublem1 26916
            14.3.10  Areas in R^2   carea 26925
            14.3.11  More miscellaneous converging sequences   rlimcnp 26935
            14.3.12  Inequality of arithmetic and geometric means   cvxcl 26955
            14.3.13  Euler-Mascheroni constant   cem 26962
            14.3.14  Zeta function   czeta 26983
            14.3.15  Gamma function   clgam 26986
      ​14.4  Basic number theory
            14.4.1  Wilson's theorem   wilthlem1 27038
            14.4.2  The Fundamental Theorem of Algebra   ftalem1 27043
            14.4.3  The Basel problem (ζ(2) = π²/6)   basellem1 27051
            14.4.4  Number-theoretical functions   ccht 27061
            14.4.5  Perfect Number Theorem   mersenne 27198
            14.4.6  Characters of Z/nZ   cdchr 27203
            14.4.7  Bertrand's postulate   bcctr 27246
            ​*14.4.8  Quadratic residues and the Legendre symbol   clgs 27265
            ​*14.4.9  Gauss' Lemma   gausslemma2dlem0a 27327
            14.4.10  Quadratic reciprocity   lgseisenlem1 27346
            14.4.11  All primes 4n+1 are the sum of two squares   2sqlem1 27388
            14.4.12  Chebyshev's Weak Prime Number Theorem, Dirichlet's Theorem   chebbnd1lem1 27440
            14.4.13  The Prime Number Theorem   mudivsum 27501
            14.4.14  Ostrowski's theorem   abvcxp 27586
​PART 15  SURREAL NUMBERS
      ​​*15.1  Sign sequence representation and Alling's axioms
            15.1.1  Definitions and initial properties   csur 27611
            15.1.2  Ordering   sltsolem1 27647
            15.1.3  Birthday Function   bdayfo 27649
            15.1.4  Density   fvnobday 27650
            ​*15.1.5  Full-Eta Property   bdayimaon 27665
      ​15.2  Initial consequences of Alling's axioms
            15.2.1  Ordering Theorems   csle 27716
            15.2.2  Birthday Theorems   bdayfun 27748
      ​​*15.3  Conway cut representation
            15.3.1  Conway cuts   csslt 27757
            15.3.2  Zero and One   c0s 27803
            15.3.3  Cuts and Options   cmade 27820
            15.3.4  Cofinality and coinitiality   cofsslt 27900
      ​15.4  Induction and recursion
            15.4.1  Induction and recursion on one variable   cnorec 27919
            15.4.2  Induction and recursion on two variables   cnorec2 27930
      ​15.5  Surreal arithmetic
            15.5.1  Addition   cadds 27941
            15.5.2  Negation and Subtraction   cnegs 28001
            15.5.3  Multiplication   cmuls 28088
            15.5.4  Division   cdivs 28169
            15.5.5  Absolute value   cabss 28218
      ​15.6  Subsystems of surreals
            15.6.1  Ordinal numbers   cons 28232
            15.6.2  Surreal recursive sequences   cseqs 28264
            15.6.3  Natural numbers   cnn0s 28293
            15.6.4  Integers   czs 28357
            15.6.5  Dyadic fractions   c2s 28389
            15.6.6  Real numbers   creno 28468
​​*PART 16  ELEMENTARY GEOMETRY
      ​16.1  Definition and Tarski's Axioms of Geometry
            16.1.1  Justification for the congruence notation   tgjustf 28528
      ​16.2  Tarskian Geometry
            16.2.1  Congruence   tgcgrcomimp 28532
            16.2.2  Betweenness   tgbtwntriv2 28542
            16.2.3  Dimension   tglowdim1 28555
            16.2.4  Betweenness and Congruence   tgifscgr 28563
            16.2.5  Congruence of a series of points   ccgrg 28565
            16.2.6  Motions   cismt 28587
            16.2.7  Colinearity   tglng 28601
            16.2.8  Connectivity of betweenness   tgbtwnconn1lem1 28627
            16.2.9  Less-than relation in geometric congruences   cleg 28637
            16.2.10  Rays   chlg 28655
            16.2.11  Lines   btwnlng1 28674
            16.2.12  Point inversions   cmir 28707
            16.2.13  Right angles   crag 28748
            16.2.14  Half-planes   islnopp 28794
            16.2.15  Midpoints and Line Mirroring   cmid 28827
            16.2.16  Congruence of angles   ccgra 28862
            16.2.17  Angle Comparisons   cinag 28890
            16.2.18  Congruence Theorems   tgsas1 28909
            16.2.19  Equilateral triangles   ceqlg 28920
      ​16.3  Properties of geometries
            16.3.1  Isomorphisms between geometries   f1otrgds 28924
      ​16.4  Geometry in Hilbert spaces
            16.4.1  Geometry in the complex plane   cchhllem 28942
            16.4.2  Geometry in Euclidean spaces   cee 28943
                  16.4.2.1  Definition of the Euclidean space   cee 28943
                  16.4.2.2  Tarski's axioms for geometry for the Euclidean space   axdimuniq 28969
                  16.4.2.3  EE^n fulfills Tarski's Axioms   ceeng 29033
​​*PART 17  GRAPH THEORY
      ​*17.1  Vertices and edges
            17.1.1  The edge function extractor for extensible structures   cedgf 29044
            ​*17.1.2  Vertices and indexed edges   cvtx 29052
                  17.1.2.1  Definitions and basic properties   cvtx 29052
                  17.1.2.2  The vertices and edges of a graph represented as ordered pair   opvtxval 29059
                  17.1.2.3  The vertices and edges of a graph represented as extensible structure   funvtxdmge2val 29067
                  17.1.2.4  Representations of graphs without edges   snstrvtxval 29093
                  17.1.2.5  Degenerated cases of representations of graphs   vtxval0 29095
            17.1.3  Edges as range of the edge function   cedg 29103
      ​​*17.2  Undirected graphs
            17.2.1  Undirected hypergraphs   cuhgr 29112
            17.2.2  Undirected pseudographs and multigraphs   cupgr 29136
            ​*17.2.3  Loop-free graphs   umgrislfupgrlem 29178
            17.2.4  Edges as subsets of vertices of graphs   uhgredgiedgb 29182
            ​*17.2.5  Undirected simple graphs   cuspgr 29204
            17.2.6  Examples for graphs   usgr0e 29292
            17.2.7  Subgraphs   csubgr 29323
            17.2.8  Finite undirected simple graphs   cfusgr 29372
            17.2.9  Neighbors, complete graphs and universal vertices   cnbgr 29388
                  17.2.9.1  Neighbors   cnbgr 29388
                  17.2.9.2  Universal vertices   cuvtx 29441
                  17.2.9.3  Complete graphs   ccplgr 29465
            17.2.10  Vertex degree   cvtxdg 29522
            ​*17.2.11  Regular graphs   crgr 29612
      ​​*17.3  Walks, paths and cycles
            *17.3.1  Walks   cewlks 29652
            17.3.2  Walks for loop-free graphs   lfgrwlkprop 29742
            17.3.3  Trails   ctrls 29745
            17.3.4  Paths and simple paths   cpths 29766
            17.3.5  Closed walks   cclwlks 29826
            17.3.6  Circuits and cycles   ccrcts 29840
            ​*17.3.7  Walks as words   cwwlks 29881
            17.3.8  Walks/paths of length 2 (as length 3 strings)   2wlkdlem1 29981
            17.3.9  Walks in regular graphs   rusgrnumwwlkl1 30027
            ​*17.3.10  Closed walks as words   cclwwlk 30039
                  17.3.10.1  Closed walks as words   cclwwlk 30039
                  17.3.10.2  Closed walks of a fixed length as words   cclwwlkn 30082
                  17.3.10.3  Closed walks on a vertex of a fixed length as words   cclwwlknon 30145
            17.3.11  Examples for walks, trails and paths   0ewlk 30172
            17.3.12  Connected graphs   cconngr 30244
      ​17.4  Eulerian paths and the Konigsberg Bridge problem
            ​*17.4.1  Eulerian paths   ceupth 30255
            ​*17.4.2  The Königsberg Bridge problem   konigsbergvtx 30304
      ​17.5  The Friendship Theorem
            17.5.1  Friendship graphs - basics   cfrgr 30316
            17.5.2  The friendship theorem for small graphs   frgr1v 30329
            17.5.3  Theorems according to Mertzios and Unger   2pthfrgrrn 30340
            ​*17.5.4  Huneke's Proof of the Friendship Theorem   frgrncvvdeqlem1 30357
​PART 18  GUIDES AND MISCELLANEA
      ​18.1  Guides (conventions, explanations, and examples)
            ​*18.1.1  Conventions   conventions 30458
            18.1.2  Natural deduction   natded 30461
            ​*18.1.3  Natural deduction examples   ex-natded5.2 30462
            18.1.4  Definitional examples   ex-or 30479
            18.1.5  Other examples   aevdemo 30518
      ​18.2  Humor
            18.2.1  April Fool's theorem   avril1 30521
      ​18.3  (Future - to be reviewed and classified)
            18.3.1  Planar incidence geometry   cplig 30532
            ​*18.3.2  Aliases kept to prevent broken links   dummylink 30545
​​*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 30547
            19.1.2  Abelian groups   cablo 30602
      ​19.2  Complex vector spaces
            19.2.1  Definition and basic properties   cvc 30616
            19.2.2  Examples of complex vector spaces   cnaddabloOLD 30639
      ​19.3  Normed complex vector spaces
            19.3.1  Definition and basic properties   cnv 30642
            19.3.2  Examples of normed complex vector spaces   cnnv 30735
            19.3.3  Induced metric of a normed complex vector space   imsval 30743
            19.3.4  Inner product   cdip 30758
            19.3.5  Subspaces   css 30779
      ​19.4  Operators on complex vector spaces
            19.4.1  Definitions and basic properties   clno 30798
      ​19.5  Inner product (pre-Hilbert) spaces
            19.5.1  Definition and basic properties   ccphlo 30870
            19.5.2  Examples of pre-Hilbert spaces   cncph 30877
            19.5.3  Properties of pre-Hilbert spaces   isph 30880
      ​19.6  Complex Banach spaces
            19.6.1  Definition and basic properties   ccbn 30920
            19.6.2  Examples of complex Banach spaces   cnbn 30927
            19.6.3  Uniform Boundedness Theorem   ubthlem1 30928
            19.6.4  Minimizing Vector Theorem   minvecolem1 30932
      ​19.7  Complex Hilbert spaces
            19.7.1  Definition and basic properties   chlo 30943
            19.7.2  Standard axioms for a complex Hilbert space   hlex 30956
            19.7.3  Examples of complex Hilbert spaces   cnchl 30974
            19.7.4  Hellinger-Toeplitz Theorem   htthlem 30975
​​*PART 20  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      ​20.1  Axiomatization of complex pre-Hilbert spaces
            20.1.1  Basic Hilbert space definitions   chba 30977
            20.1.2  Preliminary ZFC lemmas   df-hnorm 31026
            ​*20.1.3  Derive the Hilbert space axioms from ZFC set theory   axhilex-zf 31039
            ​*20.1.4  Introduce the vector space axioms for a Hilbert space   ax-hilex 31057
            20.1.5  Vector operations   hvmulex 31069
            20.1.6  Inner product postulates for a Hilbert space   ax-hfi 31137
      ​20.2  Inner product and norms
            20.2.1  Inner product   his5 31144
            20.2.2  Norms   dfhnorm2 31180
            20.2.3  Relate Hilbert space to normed complex vector spaces   hilablo 31218
            20.2.4  Bunjakovaskij-Cauchy-Schwarz inequality   bcsiALT 31237
      ​20.3  Cauchy sequences and completeness axiom
            20.3.1  Cauchy sequences and limits   hcau 31242
            20.3.2  Derivation of the completeness axiom from ZF set theory   hilmet 31252
            20.3.3  Completeness postulate for a Hilbert space   ax-hcompl 31260
            20.3.4  Relate Hilbert space to ZFC pre-Hilbert and Hilbert spaces   hhcms 31261
      ​20.4  Subspaces and projections
            20.4.1  Subspaces   df-sh 31265
            20.4.2  Closed subspaces   df-ch 31279
            20.4.3  Orthocomplements   df-oc 31310
            20.4.4  Subspace sum, span, lattice join, lattice supremum   df-shs 31366
            20.4.5  Projection theorem   pjhthlem1 31449
            20.4.6  Projectors   df-pjh 31453
      ​20.5  Properties of Hilbert subspaces
            20.5.1  Orthomodular law   omlsilem 31460
            20.5.2  Projectors (cont.)   pjhtheu2 31474
            20.5.3  Hilbert lattice operations   sh0le 31498
            20.5.4  Span (cont.) and one-dimensional subspaces   spansn0 31599
            20.5.5  Commutes relation for Hilbert lattice elements   df-cm 31641
            20.5.6  Foulis-Holland theorem   fh1 31676
            20.5.7  Quantum Logic Explorer axioms   qlax1i 31685
            20.5.8  Orthogonal subspaces   chscllem1 31695
            20.5.9  Orthoarguesian laws 5OA and 3OA   5oalem1 31712
            20.5.10  Projectors (cont.)   pjorthi 31727
            20.5.11  Mayet's equation E_3   mayete3i 31786
      ​20.6  Operators on Hilbert spaces
            ​*20.6.1  Operator sum, difference, and scalar multiplication   df-hosum 31788
            20.6.2  Zero and identity operators   df-h0op 31806
            20.6.3  Operations on Hilbert space operators   hoaddcl 31816
            20.6.4  Linear, continuous, bounded, Hermitian, unitary operators and norms   df-nmop 31897
            20.6.5  Linear and continuous functionals and norms   df-nmfn 31903
            20.6.6  Adjoint   df-adjh 31907
            20.6.7  Dirac bra-ket notation   df-bra 31908
            20.6.8  Positive operators   df-leop 31910
            20.6.9  Eigenvectors, eigenvalues, spectrum   df-eigvec 31911
            20.6.10  Theorems about operators and functionals   nmopval 31914
            20.6.11  Riesz lemma   riesz3i 32120
            20.6.12  Adjoints (cont.)   cnlnadjlem1 32125
            20.6.13  Quantum computation error bound theorem   unierri 32162
            20.6.14  Dirac bra-ket notation (cont.)   branmfn 32163
            20.6.15  Positive operators (cont.)   leopg 32180
            20.6.16  Projectors as operators   pjhmopi 32204
      ​20.7  States on a Hilbert lattice and Godowski's equation
            20.7.1  States on a Hilbert lattice   df-st 32269
            20.7.2  Godowski's equation   golem1 32329
      ​20.8  Cover relation, atoms, exchange axiom, and modular symmetry
            20.8.1  Covers relation; modular pairs   df-cv 32337
            20.8.2  Atoms   df-at 32396
            20.8.3  Superposition principle   superpos 32412
            20.8.4  Atoms, exchange and covering properties, atomicity   chcv1 32413
            20.8.5  Irreducibility   chirredlem1 32448
            20.8.6  Atoms (cont.)   atcvat3i 32454
            20.8.7  Modular symmetry   mdsymlem1 32461
​PART 21  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      ​21.1  Mathboxes for user contributions
            21.1.1  Mathbox guidelines   mathbox 32500
      ​21.2  Mathbox for Stefan Allan
      ​21.3  Mathbox for Thierry Arnoux
            21.3.1  Propositional Calculus - misc additions   ad11antr 32505
            21.3.2  Predicate Calculus   sbc2iedf 32521
                  21.3.2.1  Predicate Calculus - misc additions   sbc2iedf 32521
                  21.3.2.2  Restricted quantification - misc additions   ralcom4f 32523
                  21.3.2.3  Equality   eqtrb 32530
                  21.3.2.4  Double restricted existential uniqueness quantification   opsbc2ie 32532
                  21.3.2.5  Double restricted existential uniqueness quantification syntax   w2reu 32534
                  21.3.2.6  Substitution (without distinct variables) - misc additions   sbceqbidf 32543
                  21.3.2.7  Existential "at most one" - misc additions   mo5f 32545
                  21.3.2.8  Existential uniqueness - misc additions   reuxfrdf 32547
                  21.3.2.9  Restricted "at most one" - misc additions   rmoxfrd 32549
                  21.3.2.10  Restricted iota (description binder)   riotaeqbidva 32552
            21.3.3  General Set Theory   dmrab 32553
                  21.3.3.1  Class abstractions (a.k.a. class builders)   dmrab 32553
                  21.3.3.2  Image Sets   abrexdomjm 32564
                  21.3.3.3  Set relations and operations - misc additions   nelun 32570
                  21.3.3.4  Unordered pairs   elpreq 32585
                  21.3.3.5  Unordered triples   tpssg 32594
                  21.3.3.6  Conditional operator - misc additions   ifeqeqx 32599
                  21.3.3.7  Set union   uniinn0 32607
                  21.3.3.8  Indexed union - misc additions   cbviunf 32612
                  21.3.3.9  Indexed intersection - misc additions   iinabrex 32626
                  21.3.3.10  Disjointness - misc additions   disjnf 32627
            21.3.4  Relations and Functions   xpdisjres 32655
                  21.3.4.1  Relations - misc additions   xpdisjres 32655
                  21.3.4.2  Functions - misc additions   fconst7v 32680
                  21.3.4.3  Operations - misc additions   mpomptxf 32738
                  21.3.4.4  The mapping operation   elmaprd 32740
                  21.3.4.5  Support of a function   suppovss 32741
                  21.3.4.6  Explicit Functions with one or two points as a domain   cosnopne 32754
                  21.3.4.7  Isomorphisms - misc. additions   gtiso 32761
                  21.3.4.8  Disjointness (additional proof requiring functions)   disjdsct 32763
                  21.3.4.9  First and second members of an ordered pair - misc additions   df1stres 32764
                  21.3.4.10  Countable Sets   snct 32772
            21.3.5  Real and Complex Numbers   sgnval2 32795
                  21.3.5.1  Complex operations - misc. additions   creq0 32796
                  21.3.5.2  Ordering on reals - misc additions   lt2addrd 32811
                  21.3.5.3  Extended reals - misc additions   nn0mnfxrd 32812
                  21.3.5.4  Extended nonnegative integers - misc additions   xnn0gt0 32830
                  21.3.5.5  Real number intervals - misc additions   joiniooico 32835
                  21.3.5.6  Finite intervals of integers - misc additions   uzssico 32845
                  21.3.5.7  Half-open integer ranges - misc additions   iundisjfi 32857
                  21.3.5.8  The ` # ` (set size) function - misc additions   hashunif 32867
                  21.3.5.9  The greatest common divisor operator - misc. additions   elq2 32873
                  21.3.5.10  Integers   nn0split01 32879
                  21.3.5.11  Decimal numbers   dfdec100 32892
            21.3.6  Real and complex functions   sgncl 32893
                  21.3.6.1  Signum (sgn or sign) function - misc. additions   sgncl 32893
                  21.3.6.2  Integer powers - misc. additions   nexple 32906
                  21.3.6.3  Indicator Functions   cind 32910
            ​*21.3.7  Decimal expansion   cdp2 32933
                  ​*21.3.7.1  Decimal point   cdp 32950
                  21.3.7.2  Division in the extended real number system   cxdiv 32979
            21.3.8  Words over a set - misc additions   wrdres 32998
                  21.3.8.1  Splicing words (substring replacement)   splfv3 33021
                  21.3.8.2  Cyclic shift of words   1cshid 33022
            21.3.9  Extensible Structures   ressplusf 33026
                  21.3.9.1  Structure restriction operator   ressplusf 33026
                  21.3.9.2  Posets   ressprs 33029
                  21.3.9.3  Complete lattices   clatp0cl 33039
                  21.3.9.4  Order Theory   cmnt 33041
                  21.3.9.5  Extended reals Structure - misc additions   ax-xrssca 33067
                  21.3.9.6  The extended nonnegative real numbers commutative monoid   xrge00 33077
            21.3.10  Algebra   mndcld 33085
                  21.3.10.1  Monoids   mndcld 33085
                  21.3.10.2  Monoids Homomorphisms   abliso 33099
                  21.3.10.3  Groups - misc additions   grpinvinvd 33103
                  21.3.10.4  Abelian Groups - misc additions   ablcomd 33109
                  21.3.10.5  Finitely supported group sums - misc additions   gsumsubg 33110
                  21.3.10.6  Group or monoid sums over words   gsumwun 33139
                  21.3.10.7  Centralizers and centers - misc additions   cntzun 33142
                  21.3.10.8  The symmetric group   symgfcoeu 33145
                  21.3.10.9  Transpositions   pmtridf1o 33157
                  21.3.10.10  Permutation Signs   psgnid 33160
                  21.3.10.11  Permutation cycles   ctocyc 33169
                  21.3.10.12  The Alternating Group   evpmval 33208
                  21.3.10.13  Signum in an ordered monoid   csgns 33221
                  21.3.10.14  Fixed points   cfxp 33226
                  21.3.10.15  The Archimedean property for generic ordered algebraic structures   cinftm 33239
                  21.3.10.16  Semiring left modules   cslmd 33263
                  21.3.10.17  Simple groups   prmsimpcyc 33291
                  21.3.10.18  Rings - misc additions   ringrngd 33292
                  21.3.10.19  Subrings generated by a set   elrgspnlem1 33305
                  21.3.10.20  The zero ring   irrednzr 33313
                  21.3.10.21  Localization of rings   cerl 33316
                  21.3.10.22  Integral Domains   domnmuln0rd 33337
                  21.3.10.23  Euclidean Domains   ceuf 33351
                  21.3.10.24  Division Rings   ringinveu 33357
                  21.3.10.25  The field of rational numbers   qfld 33360
                  21.3.10.26  Subfields   subsdrg 33361
                  21.3.10.27  Field of fractions   cfrac 33365
                  21.3.10.28  Field extensions generated by a set   cfldgen 33373
                  21.3.10.29  Ring homomorphisms - misc additions   rhmdvd 33386
                  21.3.10.30  Scalar restriction operation   cresv 33388
                  21.3.10.31  The commutative ring of gaussian integers   gzcrng 33403
                  21.3.10.32  The archimedean ordered field of real numbers   cnfldfld 33404
                  21.3.10.33  The quotient map and quotient modules   qusker 33411
                  21.3.10.34  The ring of integers modulo ` N `   znfermltl 33428
                  21.3.10.35  Independent sets and families   islinds5 33429
                  21.3.10.36  Ring associates, ring units   dvdsruassoi 33446
                  ​*21.3.10.37  Subgroup sum / Sumset / Minkowski sum   elgrplsmsn 33452
                  21.3.10.38  The quotient map   quslsm 33467
                  21.3.10.39  Ideals   lidlmcld 33481
                  21.3.10.40  Prime Ideals   cprmidl 33497
                  21.3.10.41  Maximal Ideals   cmxidl 33521
                  21.3.10.42  The semiring of ideals of a ring   cidlsrg 33562
                  21.3.10.43  Prime Elements   rprmval 33578
                  21.3.10.44  Unique factorization domains   cufd 33600
                  21.3.10.45  The ring of integers   zringidom 33613
                  21.3.10.46  Associative Algebra   assaassd 33617
                  21.3.10.47  Univariate Polynomials   0ringmon1p 33619
                  21.3.10.48  Polynomial quotient and polynomial remainder   q1pdir 33665
                  21.3.10.49  Multivariate Polynomials   psrbasfsupp 33674
                  21.3.10.50  The ring of symmetric polynomials   csply 33694
                  21.3.10.51  The subring algebra   sra1r 33718
                  21.3.10.52  Division Ring Extensions   drgext0g 33727
                  21.3.10.53  Vector Spaces   lvecdimfi 33733
                  21.3.10.54  Vector Space Dimension   cldim 33736
            21.3.11  Field Extensions   cfldext 33776
                  21.3.11.1  Algebraic numbers   cirng 33821
                  21.3.11.2  Algebraic extensions   calgext 33833
                  21.3.11.3  Minimal polynomials   cminply 33837
                  21.3.11.4  Quadratic Field Extensions   rtelextdg2lem 33864
                  21.3.11.5  Towers of quadratic extentions   fldext2chn 33866
            ​*21.3.12  Constructible Numbers   cconstr 33867
                  21.3.12.1  Impossible constructions   2sqr3minply 33918
            21.3.13  Matrices   csmat 33931
                  21.3.13.1  Submatrices   csmat 33931
                  21.3.13.2  Matrix literals   clmat 33949
                  21.3.13.3  Laplace expansion of determinants   mdetpmtr1 33961
            21.3.14  Topology   ist0cld 33971
                  21.3.14.1  Open maps   txomap 33972
                  21.3.14.2  Topology of the unit circle   qtopt1 33973
                  21.3.14.3  Refinements   reff 33977
                  21.3.14.4  Open cover refinement property   ccref 33980
                  21.3.14.5  Lindelöf spaces   cldlf 33990
                  21.3.14.6  Paracompact spaces   cpcmp 33993
                  ​*21.3.14.7  Spectrum of a ring   crspec 34000
                  21.3.14.8  Pseudometrics   cmetid 34024
                  21.3.14.9  Continuity - misc additions   hauseqcn 34036
                  21.3.14.10  Topology of the closed unit interval   elunitge0 34037
                  21.3.14.11  Topology of ` ( RR X. RR ) `   unicls 34041
                  21.3.14.12  Order topology - misc. additions   cnvordtrestixx 34051
                  21.3.14.13  Continuity in topological spaces - misc. additions   mndpluscn 34064
                  21.3.14.14  Topology of the extended nonnegative real numbers ordered monoid   xrge0hmph 34070
                  21.3.14.15  Limits - misc additions   lmlim 34085
                  21.3.14.16  Univariate polynomials   pl1cn 34093
            21.3.15  Uniform Stuctures and Spaces   chcmp 34094
                  21.3.15.1  Hausdorff uniform completion   chcmp 34094
            21.3.16  Topology and algebraic structures   zringnm 34096
                  21.3.16.1  The norm on the ring of the integer numbers   zringnm 34096
                  21.3.16.2  Topological ` ZZ ` -modules   zlm0 34098
                  21.3.16.3  Canonical embedding of the field of the rational numbers into a division ring   cqqh 34108
                  21.3.16.4  Canonical embedding of the real numbers into a complete ordered field   crrh 34131
                  21.3.16.5  Embedding from the extended real numbers into a complete lattice   cxrh 34154
                  21.3.16.6  Canonical embeddings into the ordered field of the real numbers   zrhre 34157
                  ​*21.3.16.7  Topological Manifolds   cmntop 34160
                  21.3.16.8  Extended sum   cesum 34165
            21.3.17  Mixed Function/Constant operation   cofc 34233
            21.3.18  Abstract measure   csiga 34246
                  21.3.18.1  Sigma-Algebra   csiga 34246
                  21.3.18.2  Generated sigma-Algebra   csigagen 34276
                  ​*21.3.18.3  lambda and pi-Systems, Rings of Sets   ispisys 34290
                  21.3.18.4  The Borel algebra on the real numbers   cbrsiga 34319
                  21.3.18.5  Product Sigma-Algebra   csx 34326
                  21.3.18.6  Measures   cmeas 34333
                  21.3.18.7  The counting measure   cntmeas 34364
                  21.3.18.8  The Lebesgue measure - misc additions   voliune 34367
                  21.3.18.9  The Dirac delta measure   cdde 34370
                  21.3.18.10  The 'almost everywhere' relation   cae 34375
                  21.3.18.11  Measurable functions   cmbfm 34387
                  21.3.18.12  Borel Algebra on ` ( RR X. RR ) `   br2base 34407
                  ​*21.3.18.13  Caratheodory's extension theorem   coms 34429
            21.3.19  Integration   itgeq12dv 34464
                  21.3.19.1  Lebesgue integral - misc additions   itgeq12dv 34464
                  21.3.19.2  Bochner integral   citgm 34465
            21.3.20  Euler's partition theorem   oddpwdc 34492
            21.3.21  Sequences defined by strong recursion   csseq 34521
            21.3.22  Fibonacci Numbers   cfib 34534
            21.3.23  Probability   cprb 34545
                  21.3.23.1  Probability Theory   cprb 34545
                  21.3.23.2  Conditional Probabilities   ccprob 34569
                  21.3.23.3  Real-valued Random Variables   crrv 34578
                  21.3.23.4  Preimage set mapping operator   corvc 34594
                  21.3.23.5  Distribution Functions   orvcelval 34607
                  21.3.23.6  Cumulative Distribution Functions   orvclteel 34611
                  21.3.23.7  Probabilities - example   coinfliplem 34617
                  21.3.23.8  Bertrand's Ballot Problem   ballotlemoex 34624
            21.3.24  Signum (sgn or sign) function - misc. additions   fzssfzo 34677
                  21.3.24.1  Operations on words   ccatmulgnn0dir 34680
            21.3.25  Polynomials with real coefficients - misc additions   plymul02 34684
            21.3.26  Descartes's rule of signs   signspval 34690
                  21.3.26.1  Sign changes in a word over real numbers   signspval 34690
                  21.3.26.2  Counting sign changes in a word over real numbers   signslema 34700
            21.3.27  Number Theory   iblidicc 34730
                  21.3.27.1  Representations of a number as sums of integers   crepr 34746
                  21.3.27.2  Vinogradov Trigonometric Sums and the Circle Method   cvts 34773
                  21.3.27.3  The Ternary Goldbach Conjecture: Final Statement   ax-hgt749 34782
            21.3.28  Elementary Geometry   cstrkg2d 34802
                  ​*21.3.28.1  Two-dimensional geometry   cstrkg2d 34802
                  21.3.28.2  Outer Five Segment (not used, no need to move to main)   cafs 34807
            ​*21.3.29  LeftPad Project   clpad 34812
      ​​*21.4  Mathbox for Jonathan Ben-Naim
            21.4.1  First-order logic and set theory   bnj170 34835
            21.4.2  Well founded induction and recursion   bnj110 34995
            21.4.3  The existence of a minimal element in certain classes   bnj69 35147
            21.4.4  Well-founded induction   bnj1204 35149
            21.4.5  Well-founded recursion, part 1 of 3   bnj60 35199
            21.4.6  Well-founded recursion, part 2 of 3   bnj1500 35205
            21.4.7  Well-founded recursion, part 3 of 3   bnj1522 35209
      ​21.5  Mathbox for BTernaryTau
            21.5.1  First-order logic   nfan1c 35210
                  21.5.1.1  Auxiliary axiom schemes   nfan1c 35210
            21.5.2  ZF set theory   exdifsn 35216
                  21.5.2.1  Finitism   prcinf 35250
                  21.5.2.2  Introduce ax-regs   ax-regs 35263
                  21.5.2.3  Derive ax-regs   axregs 35276
                  21.5.2.4  Global choice   gblacfnacd 35277
            21.5.3  Real and complex numbers   zltp1ne 35285
            21.5.4  Graph theory   lfuhgr 35293
                  21.5.4.1  Acyclic graphs   cacycgr 35317
      ​21.6  Mathbox for Mario Carneiro
            21.6.1  Predicate calculus with all distinct variables   ax-7d 35334
            21.6.2  Miscellaneous stuff   quartfull 35340
            21.6.3  Derangements and the Subfactorial   deranglem 35341
            21.6.4  The Erdős-Szekeres theorem   erdszelem1 35366
            21.6.5  The Kuratowski closure-complement theorem   kur14lem1 35381
            21.6.6  Retracts and sections   cretr 35392
            21.6.7  Path-connected and simply connected spaces   cpconn 35394
            21.6.8  Covering maps   ccvm 35430
            21.6.9  Normal numbers   snmlff 35504
            21.6.10  Godel-sets of formulas - part 1   cgoe 35508
            21.6.11  Godel-sets of formulas - part 2   cgon 35607
            21.6.12  Models of ZF   cgze 35621
            ​*21.6.13  Metamath formal systems   cmcn 35635
            21.6.14  Grammatical formal systems   cm0s 35760
            21.6.15  Models of formal systems   cmuv 35780
            21.6.16  Splitting fields   ccpms 35802
            21.6.17  p-adic number fields   czr 35822
      ​​*21.7  Mathbox for Filip Cernatescu
      ​21.8  Mathbox for Paul Chapman
            21.8.1  Real and complex numbers (cont.)   climuzcnv 35846
            21.8.2  Miscellaneous theorems   elfzm12 35850
      ​21.9  Mathbox for Hongxiu Chen
      ​21.10  Mathbox for Adrian Ducourtial
            21.10.1  Propositional calculus   currybi 35863
            21.10.2  Clone theory   ccloneop 35870
      ​21.11  Mathbox for Scott Fenton
            21.11.1  ZFC Axioms in primitive form   axextprim 35876
            21.11.2  Untangled classes   untelirr 35883
            21.11.3  Extra propositional calculus theorems   3jaodd 35890
            21.11.4  Misc. Useful Theorems   nepss 35893
            21.11.5  Properties of real and complex numbers   sqdivzi 35903
            21.11.6  Infinite products   iprodefisumlem 35915
            21.11.7  Factorial limits   faclimlem1 35918
            21.11.8  Greatest common divisor and divisibility   gcd32 35924
            21.11.9  Properties of relationships   dftr6 35926
            21.11.10  Properties of functions and mappings   funpsstri 35941
            21.11.11  Ordinal numbers   elpotr 35954
            21.11.12  Defined equality axioms   axextdfeq 35970
            21.11.13  Hypothesis builders   hbntg 35978
            21.11.14  Well-founded zero, successor, and limits   cwsuc 35983
            21.11.15  Quantifier-free definitions   ctxp 36003
            21.11.16  Alternate ordered pairs   caltop 36131
            21.11.17  Geometry in the Euclidean space   cofs 36157
                  21.11.17.1  Congruence properties   cofs 36157
                  21.11.17.2  Betweenness properties   btwntriv2 36187
                  21.11.17.3  Segment Transportation   ctransport 36204
                  21.11.17.4  Properties relating betweenness and congruence   cifs 36210
                  21.11.17.5  Connectivity of betweenness   btwnconn1lem1 36262
                  21.11.17.6  Segment less than or equal to   csegle 36281
                  21.11.17.7  Outside-of relationship   coutsideof 36294
                  21.11.17.8  Lines and Rays   cline2 36309
            21.11.18  Forward difference   cfwddif 36333
            21.11.19  Rank theorems   rankung 36341
            21.11.20  Hereditarily Finite Sets   chf 36347
      ​21.12  Mathbox for Gino Giotto
            21.12.1  Equality theorems.   rmoeqi 36362
                  21.12.1.1  Inference versions.   rmoeqi 36362
                  21.12.1.2  Deduction versions.   rmoeqdv 36387
            21.12.2  Change bound variables.   in-ax8 36399
                  21.12.2.1  Change bound variables and domains.   cbvralvw2 36401
                  21.12.2.2  Change bound variables, deduction versions.   cbvmodavw 36425
                  21.12.2.3  Change bound variables and domains, deduction versions.   cbvrmodavw2 36458
            21.12.3  Study of ax-mulf usage.   mpomulnzcnf 36474
      ​21.13  Mathbox for Jeff Hankins
            21.13.1  Miscellany   a1i14 36475
            21.13.2  Basic topological facts   topbnd 36499
            21.13.3  Topology of the real numbers   ivthALT 36510
            21.13.4  Refinements   cfne 36511
            21.13.5  Neighborhood bases determine topologies   neibastop1 36534
            21.13.6  Lattice structure of topologies   topmtcl 36538
            21.13.7  Filter bases   fgmin 36545
            21.13.8  Directed sets, nets   tailfval 36547
      ​21.14  Mathbox for Anthony Hart
            21.14.1  Propositional Calculus   tb-ax1 36558
            21.14.2  Predicate Calculus   nalfal 36578
            21.14.3  Miscellaneous single axioms   meran1 36586
            21.14.4  Connective Symmetry   negsym1 36592
      ​21.15  Mathbox for Chen-Pang He
            21.15.1  Ordinal topology   ontopbas 36603
      ​21.16  Mathbox for Jeff Hoffman
            21.16.1  Inferences for finite induction on generic function values   fveleq 36626
            21.16.2  gdc.mm   nnssi2 36630
      ​21.17  Mathbox for Matthew House
            21.17.1  Relations on well-ordered indexed unions   weiunlem1 36637
      ​21.18  Mathbox for Asger C. Ipsen
            21.18.1  Continuous nowhere differentiable functions   dnival 36646
      ​​*21.19  Mathbox for BJ
            *21.19.1  Propositional calculus   bj-mp2c 36715
                  *21.19.1.1  Derived rules of inference   bj-mp2c 36715
                  ​*21.19.1.2  A syntactic theorem   bj-0 36717
                  21.19.1.3  Minimal implicational calculus   bj-a1k 36719
                  ​*21.19.1.4  Positive calculus   bj-syl66ib 36730
                  21.19.1.5  Implication and negation   bj-con2com 36736
                  ​*21.19.1.6  Disjunction   bj-jaoi1 36746
                  ​*21.19.1.7  Logical equivalence   bj-dfbi4 36748
                  21.19.1.8  The conditional operator for propositions   bj-consensus 36753
                  ​*21.19.1.9  Propositional calculus: miscellaneous   bj-imbi12 36758
            ​*21.19.2  Modal logic   bj-axdd2 36767
            ​*21.19.3  Provability logic   cprvb 36772
            ​*21.19.4  First-order logic   bj-genr 36781
                  21.19.4.1  Adding ax-gen   bj-genr 36781
                  21.19.4.2  Adding ax-4   bj-2alim 36785
                  21.19.4.3  Adding ax-5   bj-ax12wlem 36819
                  21.19.4.4  Equality and substitution   bj-df-sb 36828
                  21.19.4.5  Adding ax-6   bj-spimvwt 36845
                  21.19.4.6  Adding ax-7   bj-cbvexw 36852
                  21.19.4.7  Membership predicate, ax-8 and ax-9   bj-ax89 36854
                  21.19.4.8  Adding ax-11   bj-alcomexcom 36856
                  21.19.4.9  Adding ax-12   axc11n11 36858
                  21.19.4.10  Nonfreeness   wnnf 36899
                  21.19.4.11  Adding ax-13   bj-axc10 36959
                  ​*21.19.4.12  Removing dependencies on ax-13 (and ax-11)   bj-axc10v 36969
                  ​*21.19.4.13  Distinct var metavariables   bj-hbaeb2 36994
                  ​*21.19.4.14  Around ~ equsal   bj-equsal1t 36998
                  ​*21.19.4.15  Some Principia Mathematica proofs   stdpc5t 37003
                  21.19.4.16  Alternate definition of substitution   bj-sbsb 37013
                  21.19.4.17  Lemmas for substitution   bj-sbf3 37015
                  21.19.4.18  Existential uniqueness   bj-eu3f 37017
                  ​*21.19.4.19  First-order logic: miscellaneous   bj-sblem1 37018
            21.19.5  Set theory   eliminable1 37035
                  ​*21.19.5.1  Eliminability of class terms   eliminable1 37035
                  ​*21.19.5.2  Classes without the axiom of extensionality   bj-denoteslem 37047
                  21.19.5.3  Characterization among sets versus among classes   elelb 37073
                  ​*21.19.5.4  The nonfreeness quantifier for classes   bj-nfcsym 37075
                  ​*21.19.5.5  Lemmas for class substitution   bj-sbeqALT 37076
                  21.19.5.6  Removing some axiom requirements and disjoint variable conditions   bj-exlimvmpi 37087
                  ​*21.19.5.7  Class abstractions   bj-elabd2ALT 37101
                  21.19.5.8  Generalized class abstractions   bj-cgab 37109
                  ​*21.19.5.9  Restricted nonfreeness   wrnf 37117
                  ​*21.19.5.10  Russell's paradox   bj-ru1 37119
                  21.19.5.11  Curry's paradox in set theory   currysetlem 37121
                  ​*21.19.5.12  Some disjointness results   bj-n0i 37127
                  ​*21.19.5.13  Complements on direct products   bj-xpimasn 37131
                  ​*21.19.5.14  "Singletonization" and tagging   bj-snsetex 37139
                  ​*21.19.5.15  Tuples of classes   bj-cproj 37166
                  ​*21.19.5.16  Set theory: elementary operations relative to a universe   bj-rcleqf 37201
                  ​*21.19.5.17  Axioms for finite unions   bj-abex 37206
                  ​*21.19.5.18  Set theory: miscellaneous   eleq2w2ALT 37223
                  ​*21.19.5.19  Evaluation at a class   bj-evaleq 37248
                  21.19.5.20  Elementwise operations   celwise 37255
                  ​*21.19.5.21  Elementwise intersection (families of sets induced on a subset)   bj-rest00 37257
                  21.19.5.22  Moore collections (complements)   bj-raldifsn 37276
                  21.19.5.23  Maps-to notation for functions with three arguments   bj-0nelmpt 37292
                  ​*21.19.5.24  Currying   csethom 37298
                  ​*21.19.5.25  Setting components of extensible structures   cstrset 37310
            ​*21.19.6  Extended real and complex numbers, real and complex projective lines   bj-nfald 37313
                  21.19.6.1  Complements on class abstractions of ordered pairs and binary relations   bj-nfald 37313
                  ​*21.19.6.2  Identity relation (complements)   bj-opabssvv 37326
                  ​*21.19.6.3  Functionalized identity (diagonal in a Cartesian square)   cdiag2 37348
                  ​*21.19.6.4  Direct image and inverse image   cimdir 37354
                  ​*21.19.6.5  Extended numbers and projective lines as sets   cfractemp 37372
                  ​*21.19.6.6  Addition and opposite   caddcc 37413
                  ​*21.19.6.7  Order relation on the extended reals   cltxr 37417
                  ​*21.19.6.8  Argument, multiplication and inverse   carg 37419
                  21.19.6.9  The canonical bijection from the finite ordinals   ciomnn 37425
                  21.19.6.10  Divisibility   cnnbar 37436
            ​*21.19.7  Monoids   bj-smgrpssmgm 37444
                  ​*21.19.7.1  Finite sums in monoids   cfinsum 37459
            ​*21.19.8  Affine, Euclidean, and Cartesian geometry   bj-fvimacnv0 37462
                  *21.19.8.1  Real vector spaces   bj-fvimacnv0 37462
                  ​*21.19.8.2  Complex numbers (supplements)   bj-subcom 37484
                  ​*21.19.8.3  Barycentric coordinates   bj-bary1lem 37486
            21.19.9  Monoid of endomorphisms   cend 37489
      ​21.20  Mathbox for Jim Kingdon
            21.20.1  Circle constant   taupilem3 37495
            21.20.2  Number theory   dfgcd3 37500
            21.20.3  Real numbers   irrdifflemf 37501
      ​21.21  Mathbox for ML
            21.21.1  Miscellaneous   csbrecsg 37504
            21.21.2  Cartesian exponentiation   cfinxp 37559
            21.21.3  Topology   iunctb2 37579
                  ​*21.21.3.1  Pi-base theorems   pibp16 37589
      ​21.22  Mathbox for Wolf Lammen
            21.22.1  1. Bootstrapping   wl-section-boot 37598
            21.22.2  Implication chains   wl-section-impchain 37622
            21.22.3  Theorems around the conditional operator   wl-ifp-ncond1 37640
            21.22.4  Alternative development of hadd, cadd   wl-df-3xor 37644
            21.22.5  An alternative axiom ~ ax-13   ax-wl-13v 37669
            21.22.6  Bootstrapping set theory with classes   wl-cleq-0 37671
            21.22.7  Other stuff   wl-mps 37683
      ​21.23  Mathbox for Brendan Leahy
      ​21.24  Mathbox for Jeff Madsen
            21.24.1  Logic and set theory   unirep 37886
            21.24.2  Real and complex numbers; integers   filbcmb 37912
            21.24.3  Sequences and sums   sdclem2 37914
            21.24.4  Topology   subspopn 37924
            21.24.5  Metric spaces   metf1o 37927
            21.24.6  Continuous maps and homeomorphisms   constcncf 37934
            21.24.7  Boundedness   ctotbnd 37938
            21.24.8  Isometries   cismty 37970
            21.24.9  Heine-Borel Theorem   heibor1lem 37981
            21.24.10  Banach Fixed Point Theorem   bfplem1 37994
            21.24.11  Euclidean space   crrn 37997
            21.24.12  Intervals (continued)   ismrer1 38010
            21.24.13  Operation properties   cass 38014
            21.24.14  Groups and related structures   cmagm 38020
            21.24.15  Group homomorphism and isomorphism   cghomOLD 38055
            21.24.16  Rings   crngo 38066
            21.24.17  Division Rings   cdrng 38120
            21.24.18  Ring homomorphisms   crngohom 38132
            21.24.19  Commutative rings   ccm2 38161
            21.24.20  Ideals   cidl 38179
            21.24.21  Prime rings and integral domains   cprrng 38218
            21.24.22  Ideal generators   cigen 38231
      ​21.25  Mathbox for Giovanni Mascellani
            ​*21.25.1  Tools for automatic proof building   efald2 38250
            ​*21.25.2  Tseitin axioms   fald 38301
            ​*21.25.3  Equality deductions   iuneq2f 38328
            ​*21.25.4  Miscellanea   orcomdd 38339
      ​21.26  Mathbox for Peter Mazsa
            21.26.1  Notations   cxrn 38346
            21.26.2  Preparatory theorems   el2v1 38401
            21.26.3  Range Cartesian product   df-xrn 38552
            21.26.4  Relations   df-rels 38612
            21.26.5  Quotient map (coset map)   df-qmap 38618
            21.26.6  Lifts, shifts, successor, and predecessor   df-adjliftmap 38627
            21.26.7  Cosets by ` R `   df-coss 38673
            21.26.8  Subset relations   df-ssr 38750
            21.26.9  Reflexivity   df-refs 38762
            21.26.10  Converse reflexivity   df-cnvrefs 38777
            21.26.11  Symmetry   df-syms 38794
            21.26.12  Reflexivity and symmetry   symrefref2 38819
            21.26.13  Transitivity   df-trs 38828
            21.26.14  Equivalence relations   df-eqvrels 38840
            21.26.15  Redundancy   df-redunds 38879
            21.26.16  Domain quotients   df-dmqss 38894
            21.26.17  Equivalence relations on domain quotients   df-ers 38920
            21.26.18  Functions   df-funss 38937
            21.26.19  Disjoints vs. converse functions   df-disjss 38960
            21.26.20  Antisymmetry   df-antisymrel 39035
            21.26.21  Partitions: disjoints on domain quotients   df-parts 39040
            21.26.22  Pets: Partition-Equivalence Theorems   disjim 39056
            21.26.23  Type-safe Partition-Equivalence: PetParts, PetErs, Pet2Parts, Pet2Ers   df-petparts 39140
      ​21.27  Mathbox for Rodolfo Medina
            21.27.1  Partitions   prtlem60 39150
      ​​*21.28  Mathbox for Norm Megill
            *21.28.1  Obsolete schemes ax-c4,c5,c7,c10,c11,c11n,c15,c9,c14,c16   ax-c5 39180
            ​*21.28.2  Rederive new axioms ax-4, ax-10, ax-6, ax-12, ax-13 from old   axc5 39190
            ​*21.28.3  Legacy theorems using obsolete axioms   ax5ALT 39204
            21.28.4  Experiments with weak deduction theorem   elimhyps 39258
            21.28.5  Miscellanea   cnaddcom 39269
            21.28.6  Atoms, hyperplanes, and covering in a left vector space (or module)   clsa 39271
            21.28.7  Functionals and kernels of a left vector space (or module)   clfn 39354
            21.28.8  Opposite rings and dual vector spaces   cld 39420
            21.28.9  Ortholattices and orthomodular lattices   cops 39469
            21.28.10  Atomic lattices with covering property   ccvr 39559
            21.28.11  Hilbert lattices   chlt 39647
            21.28.12  Projective geometries based on Hilbert lattices   clln 39788
            21.28.13  Construction of a vector space from a Hilbert lattice   cdlema1N 40088
            21.28.14  Construction of involution and inner product from a Hilbert lattice   clpoN 41777
      ​21.29  Mathbox for metakunt
            21.29.1  Commutative Semiring   ccsrg 42259
            21.29.2  General helpful statements   rhmzrhval 42262
            21.29.3  Some gcd and lcm results   12gcd5e1 42294
            21.29.4  Least common multiple inequality theorem   3factsumint1 42312
            21.29.5  Logarithm inequalities   3exp7 42344
            21.29.6  Miscellaneous results for AKS formalisation   intlewftc 42352
            21.29.7  Sticks and stones   sticksstones1 42437
            21.29.8  Continuation AKS   aks6d1c6lem1 42461
      ​21.30  Mathbox for Steven Nguyen
            21.30.1  Utility theorems   jarrii 42496
            ​*21.30.2  Arithmetic theorems   c0exALT 42543
            21.30.3  Exponents and divisibility   oexpreposd 42613
            21.30.4  Trigonometry and Calculus   tanhalfpim 42640
            ​*21.30.5  Independence of ax-mulcom   cresub 42656
            21.30.6  Structures   sn-base0 42786
            ​*21.30.7  Projective spaces   cprjsp 42880
            21.30.8  Basic reductions for Fermat's Last Theorem   dffltz 42913
            ​*21.30.9  Exemplar theorems   iddii 42943
                  ​*21.30.9.1  Standard replacements of ax-10 , ax-11 , ax-12   nfa1w 42954
      ​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 42970
            21.33.2  Additional theory of functions   imaiinfv 42971
            21.33.3  Additional topology   elrfi 42972
            21.33.4  Characterization of closure operators. Kuratowski closure axioms   ismrcd1 42976
            21.33.5  Algebraic closure systems   cnacs 42980
            21.33.6  Miscellanea 1. Map utilities   constmap 42991
            21.33.7  Miscellanea for polynomials   mptfcl 42998
            21.33.8  Multivariate polynomials over the integers   cmzpcl 42999
            21.33.9  Miscellanea for Diophantine sets 1   coeq0i 43031
            21.33.10  Diophantine sets 1: definitions   cdioph 43033
            21.33.11  Diophantine sets 2 miscellanea   ellz1 43045
            21.33.12  Diophantine sets 2: union and intersection. Monotone Boolean algebra   diophin 43050
            21.33.13  Diophantine sets 3: construction   diophrex 43053
            21.33.14  Diophantine sets 4 miscellanea   2sbcrex 43062
            21.33.15  Diophantine sets 4: Quantification   rexrabdioph 43072
            21.33.16  Diophantine sets 5: Arithmetic sets   rabdiophlem1 43079
            21.33.17  Diophantine sets 6: reusability. renumbering of variables   eldioph4b 43089
            21.33.18  Pigeonhole Principle and cardinality helpers   fphpd 43094
            21.33.19  A non-closed set of reals is infinite   rencldnfilem 43098
            21.33.20  Lagrange's rational approximation theorem   irrapxlem1 43100
            21.33.21  Pell equations 1: A nontrivial solution always exists   pellexlem1 43107
            21.33.22  Pell equations 2: Algebraic number theory of the solution set   csquarenn 43114
            21.33.23  Pell equations 3: characterizing fundamental solution   infmrgelbi 43156
            ​*21.33.24  Logarithm laws generalized to an arbitrary base   reglogcl 43168
            21.33.25  Pell equations 4: the positive solution group is infinite cyclic   pellfund14 43176
            21.33.26  X and Y sequences 1: Definition and recurrence laws   crmx 43178
            21.33.27  Ordering and induction lemmas for the integers   monotuz 43219
            21.33.28  X and Y sequences 2: Order properties   rmxypos 43225
            21.33.29  Congruential equations   congtr 43243
            21.33.30  Alternating congruential equations   acongid 43253
            21.33.31  Additional theorems on integer divisibility   coprmdvdsb 43263
            21.33.32  X and Y sequences 3: Divisibility properties   jm2.18 43266
            21.33.33  X and Y sequences 4: Diophantine representability of Y   jm2.27a 43283
            21.33.34  X and Y sequences 5: Diophantine representability of X, ^, _C   rmxdiophlem 43293
            21.33.35  Uncategorized stuff not associated with a major project   setindtr 43302
            21.33.36  More equivalents of the Axiom of Choice   axac10 43311
            21.33.37  Finitely generated left modules   clfig 43345
            21.33.38  Noetherian left modules I   clnm 43353
            21.33.39  Addenda for structure powers   pwssplit4 43367
            21.33.40  Every set admits a group structure iff choice   unxpwdom3 43373
            21.33.41  Noetherian rings and left modules II   clnr 43387
            21.33.42  Hilbert's Basis Theorem   cldgis 43399
            21.33.43  Additional material on polynomials [DEPRECATED]   cmnc 43409
            21.33.44  Degree and minimal polynomial of algebraic numbers   cdgraa 43418
            21.33.45  Algebraic integers I   citgo 43435
            21.33.46  Endomorphism algebra   cmend 43449
            21.33.47  Cyclic groups and order   idomodle 43469
            21.33.48  Cyclotomic polynomials   ccytp 43475
            21.33.49  Miscellaneous topology   fgraphopab 43481
      ​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 43495
            21.36.2  Natural addition of Cantor normal forms   oawordex2 43604
            21.36.3  Surreal Contributions   abeqabi 43685
            21.36.4  Short Studies   nlimsuc 43718
                  21.36.4.1  Additional work on conditional logical operator   ifpan123g 43736
                  21.36.4.2  Sophisms   rp-fakeimass 43789
                  ​*21.36.4.3  Finite Sets   rp-isfinite5 43794
                  21.36.4.4  General Observations   intabssd 43796
                  21.36.4.5  Infinite Sets   pwelg 43837
                  ​*21.36.4.6  Finite intersection property   fipjust 43842
                  21.36.4.7  RP ADDTO: Subclasses and subsets   rababg 43851
                  21.36.4.8  RP ADDTO: The intersection of a class   elinintab 43852
                  21.36.4.9  RP ADDTO: Theorems requiring subset and intersection existence   elinintrab 43854
                  21.36.4.10  RP ADDTO: Relations   xpinintabd 43857
                  ​*21.36.4.11  RP ADDTO: Functions   elmapintab 43873
                  ​*21.36.4.12  RP ADDTO: Finite induction (for finite ordinals)   cnvcnvintabd 43877
                  21.36.4.13  RP ADDTO: First and second members of an ordered pair   elcnvlem 43878
                  21.36.4.14  RP ADDTO: The reflexive and transitive properties of relations   undmrnresiss 43881
                  21.36.4.15  RP ADDTO: Basic properties of closures   cleq2lem 43885
                  21.36.4.16  RP REPLACE: Definitions and basic properties of transitive closures   trcleq2lemRP 43907
                  ​*21.36.4.17  Additions for square root; absolute value   sqrtcvallem1 43908
            21.36.5  Additional statements on relations and subclasses   al3im 43924
                  21.36.5.1  Transitive relations (not to be confused with transitive classes).   trrelind 43942
                  21.36.5.2  Reflexive closures   crcl 43949
                  ​*21.36.5.3  Finite relationship composition.   relexp2 43954
                  21.36.5.4  Transitive closure of a relation   dftrcl3 43997
                  ​*21.36.5.5  Adapted from Frege   frege77d 44023
            ​*21.36.6  Propositions from _Begriffsschrift_   dfxor4 44043
                  *21.36.6.1  _Begriffsschrift_ Chapter I   dfxor4 44043
                  ​*21.36.6.2  _Begriffsschrift_ Notation hints   whe 44049
                  21.36.6.3  _Begriffsschrift_ Chapter II Implication   ax-frege1 44067
                  21.36.6.4  _Begriffsschrift_ Chapter II Implication and Negation   axfrege28 44106
                  ​*21.36.6.5  _Begriffsschrift_ Chapter II with logical equivalence   axfrege52a 44133
                  21.36.6.6  _Begriffsschrift_ Chapter II with equivalence of sets   axfrege52c 44164
                  ​*21.36.6.7  _Begriffsschrift_ Chapter II with equivalence of classes   frege53c 44191
                  ​*21.36.6.8  _Begriffsschrift_ Chapter III Properties hereditary in a sequence   dffrege69 44209
                  ​*21.36.6.9  _Begriffsschrift_ Chapter III Following in a sequence   dffrege76 44216
                  ​*21.36.6.10  _Begriffsschrift_ Chapter III Member of sequence   dffrege99 44239
                  ​*21.36.6.11  _Begriffsschrift_ Chapter III Single-valued procedures   dffrege115 44255
            ​*21.36.7  Exploring Topology via Seifert and Threlfall   enrelmap 44274
                  *21.36.7.1  Equinumerosity of sets of relations and maps   enrelmap 44274
                  ​*21.36.7.2  Generic Pseudoclosure Spaces, Pseudointerior Spaces, and Pseudoneighborhoods   or3or 44300
                  ​*21.36.7.3  Generic Neighborhood Spaces   gneispa 44407
            ​*21.36.8  Exploring Higher Homotopy via Kerodon   k0004lem1 44424
                  *21.36.8.1  Simplicial Sets   k0004lem1 44424
      ​21.37  Mathbox for Stanislas Polu
            21.37.1  IMO Problems   wwlemuld 44433
                  21.37.1.1  IMO 1972 B2   wwlemuld 44433
            ​*21.37.2  INT Inequalities Proof Generator   int-addcomd 44450
            ​*21.37.3  N-Digit Addition Proof Generator   unitadd 44472
            21.37.4  AM-GM (for k = 2,3,4)   gsumws3 44473
      ​21.38  Mathbox for Rohan Ridenour
            21.38.1  Misc   spALT 44478
            21.38.2  Monoid rings   cmnring 44488
            21.38.3  Shorter primitive equivalent of ax-groth   gru0eld 44506
                  21.38.3.1  Grothendieck universes are closed under collection   gru0eld 44506
                  21.38.3.2  Minimal universes   ismnu 44538
                  21.38.3.3  Primitive equivalent of ax-groth   expandan 44565
      ​21.39  Mathbox for Steve Rodriguez
            21.39.1  Miscellanea   nanorxor 44582
            21.39.2  Ratio test for infinite series convergence and divergence   dvgrat 44589
            21.39.3  Multiples   reldvds 44592
            21.39.4  Function operations   caofcan 44600
            21.39.5  Calculus   lhe4.4ex1a 44606
            21.39.6  The generalized binomial coefficient operation   cbcc 44613
            21.39.7  Binomial series   uzmptshftfval 44623
      ​21.40  Mathbox for Andrew Salmon
            21.40.1  Principia Mathematica * 10   pm10.12 44635
            21.40.2  Principia Mathematica * 11   2alanimi 44649
            21.40.3  Predicate Calculus   sbeqal1 44675
            21.40.4  Principia Mathematica * 13 and * 14   pm13.13a 44684
            21.40.5  Set Theory   elnev 44714
            21.40.6  Arithmetic   addcomgi 44732
            21.40.7  Geometry   cplusr 44733
      ​​*21.41  Mathbox for Alan Sare
            21.41.1  Auxiliary theorems for the Virtual Deduction tool   idiALT 44755
            21.41.2  Supplementary unification deductions   bi1imp 44759
            21.41.3  Conventional Metamath proofs, some derived from VD proofs   iidn3 44778
            21.41.4  What is Virtual Deduction?   wvd1 44846
            21.41.5  Virtual Deduction Theorems   df-vd1 44847
            21.41.6  Theorems proved using Virtual Deduction   trsspwALT 45094
            21.41.7  Theorems proved using Virtual Deduction with mmj2 assistance   simplbi2VD 45122
            21.41.8  Virtual Deduction transcriptions of textbook proofs   sb5ALTVD 45189
            21.41.9  Theorems proved using conjunction-form Virtual Deduction   elpwgdedVD 45193
            21.41.10  Theorems with a VD proof in conventional notation derived from a VD proof   suctrALT3 45200
            ​*21.41.11  Theorems with a proof in conventional notation derived from a VD proof   notnotrALT2 45203
      ​21.42  Mathbox for Eric Schmidt
            21.42.1  Miscellany   rspesbcd 45214
            21.42.2  Study of dfbi1ALT   dfbi1ALTa 45216
            21.42.3  Relation-preserving functions   wrelp 45219
            21.42.4  Orbits   orbitex 45232
            21.42.5  Well-founded sets   trwf 45236
            21.42.6  Absoluteness in transitive models   ralabso 45245
            21.42.7  Lemmas for showing axioms hold in models   traxext 45254
            21.42.8  The class of well-founded sets is a model for ZFC   wfaxext 45270
            21.42.9  Permutation models   brpermmodel 45280
      ​21.43  Mathbox for Glauco Siliprandi
            21.43.1  Miscellanea   evth2f 45296
            21.43.2  Functions   fnresdmss 45448
            21.43.3  Ordering on real numbers - Real and complex numbers basic operations   sub2times 45557
            21.43.4  Real intervals   gtnelioc 45773
            21.43.5  Finite sums   fsummulc1f 45853
            21.43.6  Finite multiplication of numbers and finite multiplication of functions   fmul01 45862
            21.43.7  Limits   clim1fr1 45883
                  21.43.7.1  Inferior limit (lim inf)   clsi 46031
                  ​*21.43.7.2  Limits for sequences of extended real numbers   clsxlim 46098
            21.43.8  Trigonometry   coseq0 46144
            21.43.9  Continuous Functions   mulcncff 46150
            21.43.10  Derivatives   dvsinexp 46191
            21.43.11  Integrals   itgsin0pilem1 46230
            21.43.12  Stone Weierstrass theorem - real version   stoweidlem1 46281
            21.43.13  Wallis' product for π   wallispilem1 46345
            21.43.14  Stirling's approximation formula for ` n ` factorial   stirlinglem1 46354
            21.43.15  Dirichlet kernel   dirkerval 46371
            21.43.16  Fourier Series   fourierdlem1 46388
            21.43.17  e is transcendental   elaa2lem 46513
            21.43.18  n-dimensional Euclidean space   rrxtopn 46564
            21.43.19  Basic measure theory   csalg 46588
                  ​*21.43.19.1  σ-Algebras   csalg 46588
                  21.43.19.2  Sum of nonnegative extended reals   csumge0 46642
                  ​*21.43.19.3  Measures   cmea 46729
                  ​*21.43.19.4  Outer measures and Caratheodory's construction   come 46769
                  ​*21.43.19.5  Lebesgue measure on n-dimensional Real numbers   covoln 46816
                  ​*21.43.19.6  Measurable functions   csmblfn 46975
      ​21.44  Mathbox for Saveliy Skresanov
            21.44.1  Ceva's theorem   sigarval 47130
            21.44.2  Simple groups   simpcntrab 47150
      ​21.45  Mathbox for Ender Ting
            21.45.1  Increasing sequences and subsequences   et-ltneverrefl 47151
            21.45.2  Scratchpad for number theory   evenwodadd 47167
            21.45.3  Scratchpad for math on real numbers   squeezedltsq 47168
      ​21.46  Mathbox for Jarvin Udandy
      ​21.47  Mathbox for Adhemar
            ​*21.47.1  Minimal implicational calculus   adh-minim 47283
      ​21.48  Mathbox for Alexander van der Vekens
            21.48.1  General auxiliary theorems (1)   n0nsn2el 47307
                  21.48.1.1  Unordered and ordered pairs - extension for singletons   n0nsn2el 47307
                  21.48.1.2  Unordered and ordered pairs - extension for unordered pairs   elprneb 47311
                  21.48.1.3  Unordered and ordered pairs - extension for ordered pairs   oppr 47312
                  21.48.1.4  Relations - extension   eubrv 47317
                  21.48.1.5  Definite description binder (inverted iota) - extension   iota0def 47320
                  21.48.1.6  Functions - extension   fveqvfvv 47322
            21.48.2  Alternative for Russell's definition of a description binder   caiota 47365
            21.48.3  Double restricted existential uniqueness   r19.32 47380
                  21.48.3.1  Restricted quantification (extension)   r19.32 47380
                  21.48.3.2  Restricted uniqueness and "at most one" quantification   reuf1odnf 47389
                  21.48.3.3  Analogs to Existential uniqueness (double quantification)   2reu3 47392
                  21.48.3.4  Additional theorems for double restricted existential uniqueness   2reu8i 47395
            ​*21.48.4  Alternative definitions of function and operation values   wdfat 47398
                  21.48.4.1  Restricted quantification (extension)   ralbinrald 47404
                  21.48.4.2  The universal class (extension)   nvelim 47405
                  21.48.4.3  Introduce the Axiom of Power Sets (extension)   alneu 47406
                  21.48.4.4  Predicate "defined at"   dfateq12d 47408
                  21.48.4.5  Alternative definition of the value of a function   dfafv2 47414
                  21.48.4.6  Alternative definition of the value of an operation   aoveq123d 47460
            ​*21.48.5  Alternative definitions of function values (2)   cafv2 47490
            21.48.6  General auxiliary theorems (2)   an4com24 47550
                  21.48.6.1  Logical conjunction - extension   an4com24 47550
                  21.48.6.2  Abbreviated conjunction and disjunction of three wff's - extension   3an4ancom24 47551
                  21.48.6.3  Negated membership (alternative)   cnelbr 47553
                  21.48.6.4  The empty set - extension   ralralimp 47560
                  21.48.6.5  Indexed union and intersection - extension   otiunsndisjX 47561
                  21.48.6.6  Functions - extension   fvifeq 47562
                  21.48.6.7  Maps-to notation - extension   fvmptrab 47574
                  21.48.6.8  Subtraction - extension   cnambpcma 47576
                  21.48.6.9  Ordering on reals (cont.) - extension   leaddsuble 47579
                  21.48.6.10  Imaginary and complex number properties - extension   readdcnnred 47585
                  21.48.6.11  Nonnegative integers (as a subset of complex numbers) - extension   nn0resubcl 47590
                  21.48.6.12  Integers (as a subset of complex numbers) - extension   zgeltp1eq 47591
                  21.48.6.13  Decimal arithmetic - extension   1t10e1p1e11 47592
                  21.48.6.14  Upper sets of integers - extension   eluzge0nn0 47594
                  21.48.6.15  Infinity and the extended real number system (cont.) - extension   nltle2tri 47595
                  21.48.6.16  Finite intervals of integers - extension   ssfz12 47596
                  21.48.6.17  Half-open integer ranges - extension   fzopred 47604
                  21.48.6.18  The floor and ceiling functions - extension   2ltceilhalf 47610
                  21.48.6.19  The modulo (remainder) operation - extension   fldivmod 47620
                  21.48.6.20  The infinite sequence builder "seq"   smonoord 47653
                  21.48.6.21  Finite and infinite sums - extension   fsummsndifre 47654
                  21.48.6.22  Extensible structures - extension   setsidel 47658
            ​*21.48.7  Preimages of function values   preimafvsnel 47661
            ​*21.48.8  Partitions of real intervals   ciccp 47695
            21.48.9  Shifting functions with an integer range domain   fargshiftfv 47721
            21.48.10  Words over a set (extension)   lswn0 47726
                  21.48.10.1  Last symbol of a word - extension   lswn0 47726
            21.48.11  Unordered pairs   wich 47727
                  21.48.11.1  Interchangeable setvar variables   wich 47727
                  21.48.11.2  Set of unordered pairs   sprid 47756
                  ​*21.48.11.3  Proper (unordered) pairs   prpair 47783
                  21.48.11.4  Set of proper unordered pairs   cprpr 47794
            21.48.12  Number theory (extension)   cfmtno 47809
                  ​*21.48.12.1  Fermat numbers   cfmtno 47809
                  ​*21.48.12.2  Mersenne primes   m2prm 47873
                  21.48.12.3  Proth's theorem   modexp2m1d 47894
                  21.48.12.4  Solutions of quadratic equations   quad1 47902
            ​*21.48.13  Even and odd numbers   ceven 47906
                  21.48.13.1  Definitions and basic properties   ceven 47906
                  21.48.13.2  Alternate definitions using the "divides" relation   dfeven2 47931
                  21.48.13.3  Alternate definitions using the "modulo" operation   dfeven3 47940
                  21.48.13.4  Alternate definitions using the "gcd" operation   iseven5 47946
                  21.48.13.5  Theorems of part 5 revised   zneoALTV 47951
                  21.48.13.6  Theorems of part 6 revised   odd2np1ALTV 47956
                  21.48.13.7  Theorems of AV's mathbox revised   0evenALTV 47970
                  21.48.13.8  Additional theorems   epoo 47985
                  21.48.13.9  Perfect Number Theorem (revised)   perfectALTVlem1 48003
            21.48.14  Number theory (extension 2)   cfppr 48006
                  ​*21.48.14.1  Fermat pseudoprimes   cfppr 48006
                  ​*21.48.14.2  Goldbach's conjectures   cgbe 48027
            21.48.15  Graph theory (extension)   cclnbgr 48100
                  21.48.15.1  Closed neighborhood of a vertex   cclnbgr 48100
                  ​*21.48.15.2  Semiclosed and semiopen neighborhoods (experimental)   dfsclnbgr2 48128
                  21.48.15.3  Induced subgraphs   cisubgr 48142
                  ​*21.48.15.4  Isomorphisms of graphs   cgrisom 48156
                  ​*21.48.15.5  Triangles in graphs   cgrtri 48219
                  ​*21.48.15.6  Star graphs   cstgr 48233
                  ​*21.48.15.7  Local isomorphisms of graphs   cgrlim 48258
                  ​*21.48.15.8  Generalized Petersen graphs   cgpg 48322
                  21.48.15.9  Loop-free graphs - extension   1hegrlfgr 48414
                  21.48.15.10  Walks - extension   cupwlks 48415
                  21.48.15.11  Edges of graphs expressed as sets of unordered pairs   upgredgssspr 48425
            21.48.16  Monoids (extension)   ovn0dmfun 48438
                  21.48.16.1  Auxiliary theorems   ovn0dmfun 48438
                  21.48.16.2  Magmas, Semigroups and Monoids (extension)   plusfreseq 48446
                  21.48.16.3  Examples and counterexamples for magmas, semigroups and monoids (extension)   opmpoismgm 48449
                  21.48.16.4  Group sum operation (extension 1)   gsumsplit2f 48462
            ​*21.48.17  Magmas and internal binary operations (alternate approach)   ccllaw 48465
                  *21.48.17.1  Laws for internal binary operations   ccllaw 48465
                  ​*21.48.17.2  Internal binary operations   cintop 48478
                  21.48.17.3  Alternative definitions for magmas and semigroups   cmgm2 48497
            21.48.18  Rings (extension)   lmod0rng 48511
                  21.48.18.1  Nonzero rings (extension)   lmod0rng 48511
                  21.48.18.2  Ideals as non-unital rings   lidldomn1 48513
                  21.48.18.3  The non-unital ring of even integers   0even 48519
                  21.48.18.4  A constructed not unital ring   cznrnglem 48541
                  ​*21.48.18.5  The category of non-unital rings (alternate definition)   crngcALTV 48545
                  ​*21.48.18.6  The category of (unital) rings (alternate definition)   cringcALTV 48569
            21.48.19  Basic algebraic structures (extension)   eliunxp2 48616
                  21.48.19.1  Auxiliary theorems   eliunxp2 48616
                  21.48.19.2  The binomial coefficient operation (extension)   bcpascm1 48633
                  21.48.19.3  The ` ZZ `-module ` ZZ X. ZZ `   zlmodzxzlmod 48636
                  21.48.19.4  Group sum operation (extension 2)   mgpsumunsn 48643
                  21.48.19.5  Symmetric groups (extension)   exple2lt6 48646
                  21.48.19.6  Divisibility (extension)   invginvrid 48649
                  21.48.19.7  The support of functions (extension)   rmsupp0 48650
                  21.48.19.8  Finitely supported functions (extension)   rmsuppfi 48654
                  21.48.19.9  Left modules (extension)   lmodvsmdi 48661
                  21.48.19.10  Associative algebras (extension)   assaascl0 48663
                  21.48.19.11  Univariate polynomials (extension)   ply1vr1smo 48665
                  21.48.19.12  Univariate polynomials (examples)   linply1 48675
            21.48.20  Linear algebra (extension)   cdmatalt 48678
                  ​*21.48.20.1  The subalgebras of diagonal and scalar matrices (extension)   cdmatalt 48678
                  ​*21.48.20.2  Linear combinations   clinc 48686
                  ​*21.48.20.3  Linear independence   clininds 48722
                  21.48.20.4  Simple left modules and the ` ZZ `-module   lmod1lem1 48769
                  21.48.20.5  Differences between (left) modules and (left) vector spaces   lvecpsslmod 48789
            21.48.21  Complexity theory   suppdm 48792
                  21.48.21.1  Auxiliary theorems   suppdm 48792
                  21.48.21.2  Even and odd integers   nn0onn0ex 48805
                  21.48.21.3  The natural logarithm on complex numbers (extension)   logcxp0 48817
                  21.48.21.4  Division of functions   cfdiv 48819
                  21.48.21.5  Upper bounds   cbigo 48829
                  21.48.21.6  Logarithm to an arbitrary base (extension)   rege1logbrege0 48840
                  ​*21.48.21.7  The binary logarithm   fldivexpfllog2 48847
                  21.48.21.8  Binary length   cblen 48851
                  ​*21.48.21.9  Digits   cdig 48877
                  21.48.21.10  Nonnegative integer as sum of its shifted digits   dignn0flhalflem1 48897
                  21.48.21.11  Algorithms for the multiplication of nonnegative integers   nn0mulfsum 48906
                  ​*21.48.21.12  N-ary functions   cnaryf 48908
                  ​*21.48.21.13  The Ackermann function   citco 48939
            21.48.22  Elementary geometry (extension)   fv1prop 48981
                  21.48.22.1  Auxiliary theorems   fv1prop 48981
                  21.48.22.2  Real euclidean space of dimension 2   rrx2pxel 48993
                  21.48.22.3  Spheres and lines in real Euclidean spaces   cline 49009
      ​21.49  Mathbox for Zhi Wang
            21.49.1  Propositional calculus   pm4.71da 49071
            21.49.2  Predicate calculus with equality   dtrucor3 49080
                  21.49.2.1  Axiom scheme ax-5 (Distinctness)   dtrucor3 49080
            21.49.3  ZF Set Theory - start with the Axiom of Extensionality   ralbidb 49081
                  21.49.3.1  Restricted quantification   ralbidb 49081
                  21.49.3.2  The universal class   reuxfr1dd 49088
                  21.49.3.3  The empty set   ssdisjd 49089
                  21.49.3.4  Unordered and ordered pairs   vsn 49093
                  21.49.3.5  The union of a class   unilbss 49099
                  21.49.3.6  Indexed union and intersection   iuneq0 49100
            21.49.4  ZF Set Theory - add the Axiom of Replacement   inpw 49106
                  21.49.4.1  Theorems requiring subset and intersection existence   inpw 49106
            21.49.5  ZF Set Theory - add the Axiom of Power Sets   opth1neg 49107
                  21.49.5.1  Ordered pair theorem   opth1neg 49107
                  21.49.5.2  Ordered-pair class abstractions (cont.)   brab2dd 49109
                  21.49.5.3  Relations   iinxp 49112
                  21.49.5.4  Functions   mof0 49119
                  21.49.5.5  Operations   ovsng 49139
            21.49.6  ZF Set Theory - add the Axiom of Union   fonex 49148
                  21.49.6.1  Relations and functions (cont.)   fonex 49148
                  21.49.6.2  First and second members of an ordered pair   eloprab1st2nd 49149
                  21.49.6.3  Operations in maps-to notation (continued)   fmpodg 49150
                  21.49.6.4  Function transposition   resinsnlem 49152
                  21.49.6.5  Infinite Cartesian products   ixpv 49171
                  21.49.6.6  Equinumerosity   fvconst0ci 49172
            21.49.7  Order sets   iccin 49177
                  21.49.7.1  Real number intervals   iccin 49177
            21.49.8  Extensible structures   slotresfo 49180
                  21.49.8.1  Basic definitions   slotresfo 49180
            21.49.9  Moore spaces   mreuniss 49181
            ​*21.49.10  Topology   clduni 49182
                  21.49.10.1  Closure and interior   clduni 49182
                  21.49.10.2  Neighborhoods   neircl 49186
                  21.49.10.3  Subspace topologies   restcls2lem 49194
                  21.49.10.4  Limits and continuity in topological spaces   cnneiima 49198
                  21.49.10.5  Topological definitions using the reals   iooii 49199
                  21.49.10.6  Separated sets   sepnsepolem1 49203
                  21.49.10.7  Separated spaces: T0, T1, T2 (Hausdorff) ...   isnrm4 49212
            21.49.11  Preordered sets and directed sets using extensible structures   isprsd 49236
            21.49.12  Posets and lattices using extensible structures   lubeldm2 49237
                  21.49.12.1  Posets   lubeldm2 49237
                  21.49.12.2  Lattices   toslat 49263
                  21.49.12.3  Subset order structures   intubeu 49265
            21.49.13  Rings   elmgpcntrd 49286
                  21.49.13.1  Multiplicative Group   elmgpcntrd 49286
            21.49.14  Associative algebras   asclelbasALT 49287
                  21.49.14.1  Definition and basic properties   asclelbasALT 49287
            21.49.15  Categories   homf0 49290
                  21.49.15.1  Categories   homf0 49290
                  21.49.15.2  Opposite category   oppccatb 49297
                  21.49.15.3  Monomorphisms and epimorphisms   idmon 49301
                  21.49.15.4  Sections, inverses, isomorphisms   sectrcl 49303
                  21.49.15.5  Isomorphic objects   cicfn 49323
                  21.49.15.6  Subcategories   dmdm 49334
                  21.49.15.7  Functors   reldmfunc 49356
                  21.49.15.8  Opposite functors   coppf 49403
                  21.49.15.9  Full & faithful functors   imasubc 49432
                  21.49.15.10  Universal property   upciclem1 49447
                  21.49.15.11  Natural transformations and the functor category   isnatd 49504
                  21.49.15.12  Initial, terminal and zero objects of a category   initoo2 49513
                  21.49.15.13  Product of categories   reldmxpc 49527
                  21.49.15.14  Swap functors   cswapf 49540
                  21.49.15.15  Functor evaluation   oppc1stflem 49568
                  21.49.15.16  Transposed curry functors   cofuswapfcl 49574
                  21.49.15.17  Constant functors   diag1 49585
                  21.49.15.18  Functor composition bifunctors   fucofulem1 49591
                  21.49.15.19  Post-composition functors   postcofval 49645
                  21.49.15.20  Pre-composition functors   precofvallem 49647
            21.49.16  Examples of categories   catcrcl 49676
                  21.49.16.1  The category of categories   catcrcl 49676
                  21.49.16.2  Thin categories   cthinc 49698
                  21.49.16.3  Terminal categories   ctermc 49753
                  21.49.16.4  Preordered sets as thin categories   cprstc 49830
                  21.49.16.5  Monoids as categories   cmndtc 49858
                  21.49.16.6  Categories with at most one object and at most two morphisms   2arwcatlem1 49876
            21.49.17  Kan extensions and related concepts   clan 49886
                  21.49.17.1  Kan extensions   clan 49886
                  21.49.17.2  Limits and colimits   clmd 49924
      ​21.50  Mathbox for Emmett Weisz
            ​*21.50.1  Miscellaneous Theorems   nfintd 49954
            21.50.2  Set Recursion   csetrecs 49964
                  ​*21.50.2.1  Basic Properties of Set Recursion   csetrecs 49964
                  21.50.2.2  Examples and properties of set recursion   elsetrecslem 49980
            ​*21.50.3  Construction of Games and Surreal Numbers   cpg 49990
      ​​*21.51  Mathbox for David A. Wheeler
            21.51.1  Natural deduction   sbidd 49999
            ​*21.51.2  Greater than, greater than or equal to.   cge-real 50001
            ​*21.51.3  Hyperbolic trigonometric functions   csinh 50011
            ​*21.51.4  Reciprocal trigonometric functions (sec, csc, cot)   csec 50022
            ​*21.51.5  Identities for "if"   ifnmfalse 50044
            ​*21.51.6  Logarithms generalized to arbitrary base using ` logb `   logb2aval 50045
            ​*21.51.7  Logarithm laws generalized to an arbitrary base - log_   clog- 50046
            ​*21.51.8  Formally define notions such as reflexivity   wreflexive 50048
            ​*21.51.9  Algebra helpers   mvlraddi 50052
            ​*21.51.10  Algebra helper examples   i2linesi 50059
            ​*21.51.11  Formal methods "surprises"   alimp-surprise 50061
            ​*21.51.12  Allsome quantifier   walsi 50067
            ​*21.51.13  Miscellaneous   5m4e1 50078
            21.51.14  Theorems about algebraic numbers   aacllem 50082
      ​21.52  Mathbox for Kunhao Zheng
            21.52.1  Weighted AM-GM inequality   amgmwlem 50083