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 3341
                  2.1.5.3  Restricted class abstraction   crab 3390
            2.1.6  The universal class   cvv 3430
            ​*2.1.7  Conditional equality (experimental)   wcdeq 3710
            2.1.8  Russell's Paradox   rru 3726
            2.1.9  Proper substitution of classes for sets   wsbc 3729
            2.1.10  Proper substitution of classes for sets into classes   csb 3838
            2.1.11  Define basic set operations and relations   cdif 3887
            2.1.12  Subclasses and subsets   df-ss 3907
            2.1.13  The difference, union, and intersection of two classes   dfdif3 4058
                  2.1.13.1  The difference of two classes   dfdif3 4058
                  2.1.13.2  The union of two classes   elun 4094
                  2.1.13.3  The intersection of two classes   elini 4140
                  2.1.13.4  The symmetric difference of two classes   csymdif 4193
                  2.1.13.5  Combinations of difference, union, and intersection of two classes   unabs 4206
                  2.1.13.6  Class abstractions with difference, union, and intersection of two classes   unabw 4248
                  2.1.13.7  Restricted uniqueness with difference, union, and intersection   reuun2 4266
            2.1.14  The empty set   c0 4274
            ​*2.1.15  The conditional operator for classes   cif 4467
            ​*2.1.16  The weak deduction theorem for set theory   dedth 4526
            2.1.17  Power classes   cpw 4542
            2.1.18  Unordered and ordered pairs   snjust 4567
            2.1.19  The union of a class   cuni 4851
            2.1.20  The intersection of a class   cint 4890
            2.1.21  Indexed union and intersection   ciun 4934
            2.1.22  Disjointness   wdisj 5053
            2.1.23  Binary relations   wbr 5086
            2.1.24  Ordered-pair class abstractions (class builders)   copab 5148
            2.1.25  Functions in maps-to notation   cmpt 5167
            2.1.26  Transitive classes   wtr 5193
      ​2.2  ZF Set Theory - add the Axiom of Replacement
            2.2.1  Introduce the Axiom of Replacement   ax-rep 5213
            2.2.2  Derive the Axiom of Separation   axsepgfromrep 5230
            2.2.3  Derive the Null Set Axiom   axnulALT 5240
            2.2.4  Theorems requiring subset and intersection existence   exnelv 5249
            2.2.5  Theorems requiring empty set existence   class2set 5297
      ​2.3  ZF Set Theory - add the Axiom of Power Sets
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 5308
            2.3.2  Derive the Axiom of Pairing   axprlem1 5366
            2.3.3  Ordered pair theorem   opnz 5427
            2.3.4  Ordered-pair class abstractions (cont.)   opabidw 5479
            2.3.5  Power class of union and intersection   pwin 5522
            2.3.6  The identity relation   cid 5525
            2.3.7  The membership relation (or epsilon relation)   cep 5530
            ​*2.3.8  Partial and total orderings   wpo 5537
            2.3.9  Founded and well-ordering relations   wfr 5581
            2.3.10  Relations   cxp 5629
            2.3.11  The Predecessor Class   cpred 6265
            2.3.12  Well-founded induction (variant)   frpomin 6305
            2.3.13  Well-ordered induction   tz6.26 6312
            2.3.14  Ordinals   word 6323
            2.3.15  Definite description binder (inverted iota)   cio 6453
            2.3.16  Functions   wfun 6493
            2.3.17  Cantor's Theorem   canth 7321
            2.3.18  Restricted iota (description binder)   crio 7323
            2.3.19  Operations   co 7367
                  2.3.19.1  Variable-to-class conversion for operations   caovclg 7559
            2.3.20  Maps-to notation   mpondm0 7607
            2.3.21  Function operation   cof 7629
            2.3.22  Proper subset relation   crpss 7676
      ​2.4  ZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 7689
            2.4.2  Ordinals (continued)   epweon 7729
            2.4.3  Transfinite induction   tfi 7804
            2.4.4  The natural numbers (i.e., finite ordinals)   com 7817
            2.4.5  Peano's postulates   peano1 7840
            2.4.6  Finite induction (for finite ordinals)   find 7846
            2.4.7  Relations and functions (cont.)   dmexg 7852
            2.4.8  First and second members of an ordered pair   c1st 7940
            2.4.9  Induction on Cartesian products   frpoins3xpg 8090
            2.4.10  Ordering on Cartesian products   xpord2lem 8092
            2.4.11  Ordering Ordinal Sequences   orderseqlem 8107
            ​*2.4.12  The support of functions   csupp 8110
            ​*2.4.13  Special maps-to operations   opeliunxp2f 8160
            2.4.14  Function transposition   ctpos 8175
            2.4.15  Curry and uncurry   ccur 8215
            2.4.16  Undefined values   cund 8222
            2.4.17  Well-founded recursion   cfrecs 8230
            2.4.18  Well-ordered recursion   cwrecs 8261
            2.4.19  Functions on ordinals; strictly monotone ordinal functions   iunon 8279
            2.4.20  "Strong" transfinite recursion   crecs 8310
            2.4.21  Recursive definition generator   crdg 8348
            2.4.22  Finite recursion   frfnom 8374
            2.4.23  Ordinal arithmetic   c1o 8398
            2.4.24  Natural number arithmetic   nna0 8540
            2.4.25  Natural addition   cnadd 8601
            2.4.26  Equivalence relations and classes   wer 8640
            2.4.27  The mapping operation   cmap 8773
            2.4.28  Infinite Cartesian products   cixp 8845
            2.4.29  Equinumerosity   cen 8890
            2.4.30  Schroeder-Bernstein Theorem   sbthlem1 9025
            2.4.31  Equinumerosity (cont.)   xpf1o 9077
            2.4.32  Finite sets   dif1enlem 9094
            2.4.33  Pigeonhole Principle   phplem1 9138
            2.4.34  Finite sets (cont.)   onomeneq 9148
            2.4.35  Finitely supported functions   cfsupp 9274
            2.4.36  Finite intersections   cfi 9323
            2.4.37  Hall's marriage theorem   marypha1lem 9346
            2.4.38  Supremum and infimum   csup 9353
            2.4.39  Ordinal isomorphism, Hartogs's theorem   coi 9424
            2.4.40  Hartogs function   char 9471
            2.4.41  Weak dominance   cwdom 9479
      ​2.5  ZF Set Theory - add the Axiom of Regularity
            2.5.1  Introduce the Axiom of Regularity   ax-reg 9507
            2.5.2  Axiom of Infinity equivalents   inf0 9542
      ​2.6  ZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-inf 9559
            2.6.2  Existence of omega (the set of natural numbers)   omex 9564
            2.6.3  Cantor normal form   ccnf 9582
            2.6.4  Transitive closure of a relation   cttrcl 9628
            2.6.5  Transitive closure   trcl 9649
            2.6.6  Set induction (or epsilon induction)   setind 9668
            2.6.7  Well-Founded Induction   frmin 9673
            2.6.8  Well-Founded Recursion   frr3g 9680
            2.6.9  Rank   cr1 9686
            2.6.10  Scott's trick; collection principle; Hilbert's epsilon   scottex 9809
            2.6.11  Disjoint union   cdju 9822
            2.6.12  Cardinal numbers   ccrd 9859
            2.6.13  Axiom of Choice equivalents   wac 10037
            ​*2.6.14  Cardinal number arithmetic   undjudom 10090
            2.6.15  The Ackermann bijection   ackbij2lem1 10140
            2.6.16  Cofinality (without Axiom of Choice)   cflem 10167
            2.6.17  Eight inequivalent definitions of finite set   sornom 10199
            2.6.18  Hereditarily size-limited sets without Choice   itunifval 10338
​​*PART 3  ZFC (ZERMELO-FRAENKEL WITH CHOICE) SET THEORY
      ​3.1  ZFC Set Theory - add Countable Choice and Dependent Choice
            3.1.1  Introduce the Axiom of Countable Choice   ax-cc 10357
            3.1.2  Introduce the Axiom of Dependent Choice   ax-dc 10368
      ​3.2  ZFC Set Theory - add the Axiom of Choice
            3.2.1  Introduce the Axiom of Choice   ax-ac 10381
            3.2.2  AC equivalents: well-ordering, Zorn's lemma   numthcor 10416
            3.2.3  Cardinal number theorems using Axiom of Choice   cardval 10468
            3.2.4  Cardinal number arithmetic using Axiom of Choice   iunctb 10497
            3.2.5  Cofinality using the Axiom of Choice   alephreg 10505
      ​3.3  ZFC Axioms with no distinct variable requirements
      ​3.4  The Generalized Continuum Hypothesis
            3.4.1  Sets satisfying the Generalized Continuum Hypothesis   cgch 10543
            3.4.2  Derivation of the Axiom of Choice   gchaclem 10601
​​*PART 4  TG (TARSKI-GROTHENDIECK) SET THEORY
      ​4.1  Inaccessibles
            4.1.1  Weakly and strongly inaccessible cardinals   cwina 10605
            4.1.2  Weak universes   cwun 10623
            4.1.3  Tarski classes   ctsk 10671
            4.1.4  Grothendieck universes   cgru 10713
      ​4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
            4.2.1  Introduce the Tarski-Grothendieck Axiom   ax-groth 10746
            4.2.2  Derive the Power Set, Infinity and Choice Axioms   grothpw 10749
            4.2.3  Tarski map function   ctskm 10760
​​*PART 5  REAL AND COMPLEX NUMBERS
      ​5.1  Construction and axiomatization of real and complex numbers
            5.1.1  Dedekind-cut construction of real and complex numbers   cnpi 10767
            5.1.2  Final derivation of real and complex number postulates   axaddf 11068
            5.1.3  Real and complex number postulates restated as axioms   ax-cnex 11094
      ​5.2  Derive the basic properties from the field axioms
            5.2.1  Some deductions from the field axioms for complex numbers   cnex 11119
            5.2.2  Infinity and the extended real number system   cpnf 11176
            5.2.3  Restate the ordering postulates with extended real "less than"   axlttri 11217
            5.2.4  Ordering on reals   lttr 11222
            5.2.5  Initial properties of the complex numbers   mul12 11311
      ​5.3  Real and complex numbers - basic operations
            5.3.1  Addition   add12 11364
            5.3.2  Subtraction   cmin 11377
            5.3.3  Multiplication   kcnktkm1cn 11581
            5.3.4  Ordering on reals (cont.)   gt0ne0 11615
            5.3.5  Reciprocals   ixi 11779
            5.3.6  Division   cdiv 11807
            5.3.7  Ordering on reals (cont.)   elimgt0 11993
            5.3.8  Completeness Axiom and Suprema   fimaxre 12100
            5.3.9  Imaginary and complex number properties   neg1cn 12144
            5.3.10  Function operation analogue theorems   ofsubeq0 12156
            ​*5.3.11  Indicator Functions   cind 12159
      ​5.4  Integer sets
            5.4.1  Positive integers (as a subset of complex numbers)   cn 12174
            5.4.2  Principle of mathematical induction   nnind 12192
            ​*5.4.3  Decimal representation of numbers   c2 12236
            ​*5.4.4  Some properties of specific numbers   1pneg1e0 12295
            5.4.5  Simple number properties   halfcl 12403
            5.4.6  The Archimedean property   nnunb 12433
            5.4.7  Nonnegative integers (as a subset of complex numbers)   cn0 12437
            ​*5.4.8  Extended nonnegative integers   cxnn0 12510
            5.4.9  Integers (as a subset of complex numbers)   cz 12524
            5.4.10  Decimal arithmetic   cdc 12644
            5.4.11  Upper sets of integers   cuz 12788
            5.4.12  Well-ordering principle for bounded-below sets of integers   uzwo3 12893
            5.4.13  Rational numbers (as a subset of complex numbers)   cq 12898
            5.4.14  Existence of the set of complex numbers   rpnnen1lem2 12927
      ​5.5  Order sets
            5.5.1  Positive reals (as a subset of complex numbers)   crp 12942
            5.5.2  Infinity and the extended real number system (cont.)   cxne 13060
            5.5.3  Supremum and infimum on the extended reals   xrsupexmnf 13257
            5.5.4  Real number intervals   cioo 13298
            5.5.5  Finite intervals of integers   cfz 13461
            ​*5.5.6  Finite intervals of nonnegative integers   elfz2nn0 13572
            5.5.7  Half-open integer ranges   cfzo 13608
      ​5.6  Elementary integer functions
            5.6.1  The floor and ceiling functions   cfl 13749
            5.6.2  The modulo (remainder) operation   cmo 13828
            5.6.3  Miscellaneous theorems about integers   om2uz0i 13909
            5.6.4  Strong induction over upper sets of integers   uzsinds 13949
            5.6.5  Finitely supported functions over the nonnegative integers   fsuppmapnn0fiublem 13952
            5.6.6  The infinite sequence builder "seq" - extension   cseq 13963
            5.6.7  Integer powers   cexp 14023
            5.6.8  Ordered pair theorem for nonnegative integers   nn0le2msqi 14229
            5.6.9  Factorial function   cfa 14235
            5.6.10  The binomial coefficient operation   cbc 14264
            5.6.11  The ` # ` (set size) function   chash 14292
                  5.6.11.1  Proper unordered pairs and triples (sets of size 2 and 3)   hashprlei 14430
                  5.6.11.2  Functions with a domain containing at least two different elements   fundmge2nop0 14464
                  5.6.11.3  Finite induction on the size of the first component of a binary relation   hashdifsnp1 14468
      ​​*5.7  Words over a set
            5.7.1  Definitions and basic theorems   cword 14475
            5.7.2  Last symbol of a word   clsw 14524
            5.7.3  Concatenations of words   cconcat 14532
            5.7.4  Singleton words   cs1 14558
            5.7.5  Concatenations with singleton words   ccatws1cl 14579
            5.7.6  Subwords/substrings   csubstr 14603
            5.7.7  Prefixes of a word   cpfx 14633
            5.7.8  Subwords of subwords   swrdswrdlem 14666
            5.7.9  Subwords and concatenations   pfxcctswrd 14672
            5.7.10  Subwords of concatenations   swrdccatfn 14686
            5.7.11  Splicing words (substring replacement)   csplice 14711
            5.7.12  Reversing words   creverse 14720
            5.7.13  Repeated symbol words   creps 14730
            ​*5.7.14  Cyclical shifts of words   ccsh 14750
            5.7.15  Mapping words by a function   wrdco 14793
            5.7.16  Longer string literals   cs2 14803
      ​​*5.8  Reflexive and transitive closures of relations
            5.8.1  The reflexive and transitive properties of relations   coss12d 14934
            5.8.2  Basic properties of closures   cleq1lem 14944
            5.8.3  Definitions and basic properties of transitive closures   ctcl 14947
            5.8.4  Exponentiation of relations   crelexp 14981
            5.8.5  Reflexive-transitive closure as an indexed union   crtrcl 15017
            ​*5.8.6  Principle of transitive induction   relexpindlem 15025
      ​5.9  Elementary real and complex functions
            5.9.1  The "shift" operation   cshi 15028
            5.9.2  Signum (sgn or sign) function   csgn 15048
            5.9.3  Real and imaginary parts; conjugate   ccj 15058
            5.9.4  Square root; absolute value   csqrt 15195
      ​5.10  Elementary limits and convergence
            5.10.1  Superior limit (lim sup)   clsp 15432
            5.10.2  Limits   cli 15446
            5.10.3  Finite and infinite sums   csu 15648
            5.10.4  The binomial theorem   binomlem 15794
            5.10.5  The inclusion/exclusion principle   incexclem 15801
            5.10.6  Infinite sums (cont.)   isumshft 15804
            5.10.7  Miscellaneous converging and diverging sequences   divrcnv 15817
            5.10.8  Arithmetic series   arisum 15825
            5.10.9  Geometric series   expcnv 15829
            5.10.10  Ratio test for infinite series convergence   cvgrat 15848
            5.10.11  Mertens' theorem   mertenslem1 15849
            5.10.12  Finite and infinite products   prodf 15852
                  5.10.12.1  Product sequences   prodf 15852
                  5.10.12.2  Non-trivial convergence   ntrivcvg 15862
                  5.10.12.3  Complex products   cprod 15868
                  5.10.12.4  Finite products   fprod 15906
                  5.10.12.5  Infinite products   iprodclim 15963
            5.10.13  Falling and Rising Factorial   cfallfac 15969
            5.10.14  Bernoulli polynomials and sums of k-th powers   cbp 16011
      ​5.11  Elementary trigonometry
            5.11.1  The exponential, sine, and cosine functions   ce 16026
                  5.11.1.1  The circle constant (tau = 2 pi)   ctau 16169
            5.11.2  _e is irrational   eirrlem 16171
      ​5.12  Cardinality of real and complex number subsets
            5.12.1  Countability of integers and rationals   xpnnen 16178
            5.12.2  The reals are uncountable   rpnnen2lem1 16181
​​*PART 6  ELEMENTARY NUMBER THEORY
      ​6.1  Elementary properties of divisibility
            6.1.1  Irrationality of square root of 2   sqrt2irrlem 16215
            6.1.2  Some Number sets are chains of proper subsets   nthruc 16219
            6.1.3  The divides relation   cdvds 16221
            ​*6.1.4  Even and odd numbers   evenelz 16305
            6.1.5  The division algorithm   divalglem0 16362
            6.1.6  Bit sequences   cbits 16388
            6.1.7  The greatest common divisor operator   cgcd 16463
            6.1.8  Bézout's identity   bezoutlem1 16508
            6.1.9  Algorithms   nn0seqcvgd 16539
            6.1.10  Euclid's Algorithm   eucalgval2 16550
            ​*6.1.11  The least common multiple   clcm 16557
            ​*6.1.12  Coprimality and Euclid's lemma   coprmgcdb 16618
            6.1.13  Cancellability of congruences   congr 16633
      ​6.2  Elementary prime number theory
            ​*6.2.1  Elementary properties   cprime 16640
            ​*6.2.2  Coprimality and Euclid's lemma (cont.)   coprm 16681
            6.2.3  Properties of the canonical representation of a rational   cnumer 16703
            6.2.4  Euler's theorem   codz 16733
            6.2.5  Arithmetic modulo a prime number   modprm1div 16768
            6.2.6  Pythagorean Triples   coprimeprodsq 16779
            6.2.7  The prime count function   cpc 16807
            6.2.8  Pocklington's theorem   prmpwdvds 16875
            6.2.9  Infinite primes theorem   unbenlem 16879
            6.2.10  Sum of prime reciprocals   prmreclem1 16887
            6.2.11  Fundamental theorem of arithmetic   1arithlem1 16894
            6.2.12  Lagrange's four-square theorem   cgz 16900
            6.2.13  Van der Waerden's theorem   cvdwa 16936
            6.2.14  Ramsey's theorem   cram 16970
            ​*6.2.15  Primorial function   cprmo 17002
            ​*6.2.16  Prime gaps   prmgaplem1 17020
            6.2.17  Decimal arithmetic (cont.)   dec2dvds 17034
            6.2.18  Cyclical shifts of words (cont.)   cshwsidrepsw 17064
            6.2.19  Specific prime numbers   prmlem0 17076
            6.2.20  Very large primes   1259lem1 17101
​PART 7  BASIC STRUCTURES
      ​7.1  Extensible structures
            ​*7.1.1  Basic definitions   cstr 17116
                  7.1.1.1  Extensible structures as structures with components   cstr 17116
                  7.1.1.2  Substitution of components   csts 17133
                  7.1.1.3  Slots   cslot 17151
                  ​*7.1.1.4  Structure component indices   cnx 17163
                  7.1.1.5  Base sets   cbs 17179
                  7.1.1.6  Base set restrictions   cress 17200
            7.1.2  Slot definitions   cplusg 17220
            7.1.3  Definition of the structure product   crest 17383
            7.1.4  Definition of the structure quotient   cordt 17463
      ​7.2  Moore spaces
            7.2.1  Moore closures   mrcflem 17572
            7.2.2  Independent sets in a Moore system   mrisval 17596
            7.2.3  Algebraic closure systems   isacs 17617
​PART 8  BASIC CATEGORY THEORY
      ​8.1  Categories
            8.1.1  Categories   ccat 17630
            8.1.2  Opposite category   coppc 17677
            8.1.3  Monomorphisms and epimorphisms   cmon 17695
            8.1.4  Sections, inverses, isomorphisms   csect 17711
            ​*8.1.5  Isomorphic objects   ccic 17762
            8.1.6  Subcategories   cssc 17774
            8.1.7  Functors   cfunc 17821
            8.1.8  Full & faithful functors   cful 17871
            8.1.9  Natural transformations and the functor category   cnat 17911
            8.1.10  Initial, terminal and zero objects of a category   cinito 17948
      ​8.2  Arrows (disjointified hom-sets)
            8.2.1  Identity and composition for arrows   cida 18020
      ​8.3  Examples of categories
            8.3.1  The category of sets   csetc 18042
            8.3.2  The category of categories   ccatc 18065
            ​*8.3.3  The category of extensible structures   fncnvimaeqv 18086
      ​8.4  Categorical constructions
            8.4.1  Product of categories   cxpc 18134
            8.4.2  Functor evaluation   cevlf 18175
            8.4.3  Hom functor   chof 18214
​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 18397
            9.5.2  Complete lattices   ccla 18464
            9.5.3  Distributive lattices   cdlat 18486
            9.5.4  Subset order structures   cipo 18493
      ​9.6  Posets, directed sets, and lattices as relations
            ​*9.6.1  Posets and lattices as relations   cps 18530
            9.6.2  Directed sets, nets   cdir 18560
            9.6.3  Chains   cchn 18571
​PART 10  BASIC ALGEBRAIC STRUCTURES
      ​10.1  Monoids
            ​*10.1.1  Magmas   cplusf 18605
            ​*10.1.2  Identity elements   mgmidmo 18628
            ​*10.1.3  Iterated sums in a magma   gsumvalx 18644
            10.1.4  Magma homomorphisms and submagmas   cmgmhm 18658
            ​*10.1.5  Semigroups   csgrp 18686
            ​*10.1.6  Definition and basic properties of monoids   cmnd 18702
            10.1.7  Monoid homomorphisms and submonoids   cmhm 18749
            ​*10.1.8  Iterated sums in a monoid   gsumvallem2 18802
            10.1.9  Free monoids   cfrmd 18815
                  ​*10.1.9.1  Monoid of endofunctions   cefmnd 18836
            10.1.10  Examples and counterexamples for magmas, semigroups and monoids   mgm2nsgrplem1 18889
      ​10.2  Groups
            10.2.1  Definition and basic properties   cgrp 18909
            ​*10.2.2  Group multiple operation   cmg 19043
            10.2.3  Subgroups and Quotient groups   csubg 19096
            ​*10.2.4  Cyclic monoids and groups   cycsubmel 19175
            10.2.5  Elementary theory of group homomorphisms   cghm 19187
            10.2.6  Isomorphisms of groups   cgim 19232
                  10.2.6.1  The first isomorphism theorem of groups   ghmqusnsglem1 19255
            10.2.7  Group actions   cga 19264
            10.2.8  Centralizers and centers   ccntz 19290
            10.2.9  The opposite group   coppg 19320
            10.2.10  Symmetric groups   csymg 19344
                  ​*10.2.10.1  Definition and basic properties   csymg 19344
                  10.2.10.2  Cayley's theorem   cayleylem1 19387
                  10.2.10.3  Permutations fixing one element   symgfix2 19391
                  ​*10.2.10.4  Transpositions in the symmetric group   cpmtr 19416
                  10.2.10.5  The sign of a permutation   cpsgn 19464
            10.2.11  p-Groups and Sylow groups; Sylow's theorems   cod 19499
            10.2.12  Direct products   clsm 19609
                  10.2.12.1  Direct products (extension)   smndlsmidm 19631
            10.2.13  Free groups   cefg 19681
            10.2.14  Abelian groups   ccmn 19755
                  10.2.14.1  Definition and basic properties   ccmn 19755
                  10.2.14.2  Cyclic groups   ccyg 19852
                  10.2.14.3  Group sum operation   gsumval3a 19878
                  10.2.14.4  Group sums over (ranges of) integers   fsfnn0gsumfsffz 19958
                  10.2.14.5  Internal direct products   cdprd 19970
                  10.2.14.6  The Fundamental Theorem of Abelian Groups   ablfacrplem 20042
            10.2.15  Simple groups   csimpg 20067
                  10.2.15.1  Definition and basic properties   csimpg 20067
                  10.2.15.2  Classification of abelian simple groups   ablsimpnosubgd 20081
            10.2.16  Totally ordered monoids and groups   comnd 20094
      ​10.3  Rings
            10.3.1  Multiplicative Group   cmgp 20121
            ​*10.3.2  Non-unital rings ("rngs")   crng 20133
            ​*10.3.3  Ring unity (multiplicative identity)   cur 20162
            10.3.4  Semirings   csrg 20167
                  ​*10.3.4.1  The binomial theorem for semirings   srgbinomlem1 20207
            10.3.5  Unital rings   crg 20214
            10.3.6  Opposite ring   coppr 20316
            10.3.7  Divisibility   cdsr 20334
            10.3.8  Ring primes   crpm 20412
            10.3.9  Homomorphisms of non-unital rings   crnghm 20414
            10.3.10  Ring homomorphisms   crh 20449
            10.3.11  Nonzero rings and zero rings   cnzr 20489
            10.3.12  Local rings   clring 20515
            10.3.13  Subrings   csubrng 20522
                  10.3.13.1  Subrings of non-unital rings   csubrng 20522
                  10.3.13.2  Subrings of unital rings   csubrg 20546
                  10.3.13.3  Subrings generated by a subset   crgspn 20587
            10.3.14  Categories of rings   crngc 20593
                  ​*10.3.14.1  The category of non-unital rings   crngc 20593
                  ​*10.3.14.2  The category of (unital) rings   cringc 20622
                  10.3.14.3  Subcategories of the category of rings   srhmsubclem1 20654
            10.3.15  Left regular elements and domains   crlreg 20668
      ​10.4  Division rings and fields
            10.4.1  Definition and basic properties   cdr 20706
            10.4.2  Sub-division rings   csdrg 20763
            10.4.3  Absolute value (abstract algebra)   cabv 20785
            10.4.4  Star rings   cstf 20814
            10.4.5  Totally ordered rings and fields   corng 20834
      ​10.5  Left modules
            10.5.1  Definition and basic properties   clmod 20855
            10.5.2  Subspaces and spans in a left module   clss 20926
            10.5.3  Homomorphisms and isomorphisms of left modules   clmhm 21014
            10.5.4  Subspace sum; bases for a left module   clbs 21069
      ​10.6  Vector spaces
            10.6.1  Definition and basic properties   clvec 21097
      ​10.7  Subring algebras and ideals
            10.7.1  Subring algebras   csra 21166
            ​*10.7.2  Left ideals and spans   clidl 21204
            10.7.3  Two-sided ideals and quotient rings   c2idl 21247
                  ​*10.7.3.1  Condition for a non-unital ring to be unital   rngqiprng1elbas 21284
            10.7.4  Principal ideal rings. Divisibility in the integers   clpidl 21318
            10.7.5  Principal ideal domains   cpid 21334
      ​10.8  The complex numbers as an algebraic extensible structure
            10.8.1  Definition and basic properties   cpsmet 21336
            ​*10.8.2  Ring of integers   czring 21426
                  ​*10.8.2.1  Example for a condition for a non-unital ring to be unital   pzriprnglem1 21461
            10.8.3  Algebraic constructions based on the complex numbers   czrh 21479
            10.8.4  Signs as subgroup of the complex numbers   cnmsgnsubg 21557
            10.8.5  Embedding of permutation signs into a ring   zrhpsgnmhm 21564
            10.8.6  The ordered field of real numbers   crefld 21584
      ​10.9  Generalized pre-Hilbert and Hilbert spaces
            10.9.1  Definition and basic properties   cphl 21604
            10.9.2  Orthocomplements and closed subspaces   cocv 21640
            10.9.3  Orthogonal projection and orthonormal bases   cpj 21680
​​*PART 11  BASIC LINEAR ALGEBRA
      ​11.1  Vectors and free modules
            *11.1.1  Direct sum of left modules   cdsmm 21711
            ​*11.1.2  Free modules   cfrlm 21726
            ​*11.1.3  Standard basis (unit vectors)   cuvc 21762
            ​*11.1.4  Independent sets and families   clindf 21784
            11.1.5  Characterization of free modules   lmimlbs 21816
      ​11.2  Associative algebras
            11.2.1  Definition and basic properties   casa 21830
      ​11.3  Abstract multivariate polynomials
            11.3.1  Definition and basic properties   cmps 21884
            11.3.2  Polynomial evaluation   ces 22050
            11.3.3  Additional definitions for (multivariate) polynomials   cslv 22094
            ​*11.3.4  Univariate polynomials   cps1 22138
            11.3.5  Univariate polynomial evaluation   ces1 22278
                  11.3.5.1  Specialization of polynomial evaluation as a ring homomorphism   evls1scafv 22331
      ​​*11.4  Matrices
            *11.4.1  The matrix multiplication   cmmul 22355
            ​*11.4.2  Square matrices   cmat 22372
            ​*11.4.3  The matrix algebra   matmulr 22403
            ​*11.4.4  Matrices of dimension 0 and 1   mat0dimbas0 22431
            ​*11.4.5  The subalgebras of diagonal and scalar matrices   cdmat 22453
            ​*11.4.6  Multiplication of a matrix with a "column vector"   cmvmul 22505
            11.4.7  Replacement functions for a square matrix   cmarrep 22521
            11.4.8  Submatrices   csubma 22541
      ​11.5  The determinant
            11.5.1  Definition and basic properties   cmdat 22549
            11.5.2  Determinants of 2 x 2 -matrices   m2detleiblem1 22589
            11.5.3  The matrix adjugate/adjunct   cmadu 22597
            ​*11.5.4  Laplace expansion of determinants (special case)   symgmatr01lem 22618
            11.5.5  Inverse matrix   invrvald 22641
            ​*11.5.6  Cramer's rule   slesolvec 22644
      ​​*11.6  Polynomial matrices
            11.6.1  Basic properties   pmatring 22657
            ​*11.6.2  Constant polynomial matrices   ccpmat 22668
            ​*11.6.3  Collecting coefficients of polynomial matrices   cdecpmat 22727
            ​*11.6.4  Ring isomorphism between polynomial matrices and polynomials over matrices   cpm2mp 22757
      ​​*11.7  The characteristic polynomial
            *11.7.1  Definition and basic properties   cchpmat 22791
            ​*11.7.2  The characteristic factor function G   fvmptnn04if 22814
            ​*11.7.3  The Cayley-Hamilton theorem   cpmadurid 22832
​PART 12  BASIC TOPOLOGY
      ​12.1  Topology
            ​*12.1.1  Topological spaces   ctop 22858
                  12.1.1.1  Topologies   ctop 22858
                  12.1.1.2  Topologies on sets   ctopon 22875
                  12.1.1.3  Topological spaces   ctps 22897
            12.1.2  Topological bases   ctb 22910
            12.1.3  Examples of topologies   distop 22960
            12.1.4  Closure and interior   ccld 22981
            12.1.5  Neighborhoods   cnei 23062
            12.1.6  Limit points and perfect sets   clp 23099
            12.1.7  Subspace topologies   restrcl 23122
            12.1.8  Order topology   ordtbaslem 23153
            12.1.9  Limits and continuity in topological spaces   ccn 23189
            12.1.10  Separated spaces: T0, T1, T2 (Hausdorff) ...   ct0 23271
            12.1.11  Compactness   ccmp 23351
            12.1.12  Bolzano-Weierstrass theorem   bwth 23375
            12.1.13  Connectedness   cconn 23376
            12.1.14  First- and second-countability   c1stc 23402
            12.1.15  Local topological properties   clly 23429
            12.1.16  Refinements   cref 23467
            12.1.17  Compactly generated spaces   ckgen 23498
            12.1.18  Product topologies   ctx 23525
            12.1.19  Continuous function-builders   cnmptid 23626
            12.1.20  Quotient maps and quotient topology   ckq 23658
            12.1.21  Homeomorphisms   chmeo 23718
      ​12.2  Filters and filter bases
            12.2.1  Filter bases   elmptrab 23792
            12.2.2  Filters   cfil 23810
            12.2.3  Ultrafilters   cufil 23864
            12.2.4  Filter limits   cfm 23898
            12.2.5  Extension by continuity   ccnext 24024
            12.2.6  Topological groups   ctmd 24035
            12.2.7  Infinite group sum on topological groups   ctsu 24091
            12.2.8  Topological rings, fields, vector spaces   ctrg 24121
      ​12.3  Uniform Structures and Spaces
            12.3.1  Uniform structures   cust 24165
            12.3.2  The topology induced by an uniform structure   cutop 24195
            12.3.3  Uniform Spaces   cuss 24218
            12.3.4  Uniform continuity   cucn 24239
            12.3.5  Cauchy filters in uniform spaces   ccfilu 24250
            12.3.6  Complete uniform spaces   ccusp 24261
      ​12.4  Metric spaces
            12.4.1  Pseudometric spaces   ispsmet 24269
            12.4.2  Basic metric space properties   cxms 24282
            12.4.3  Metric space balls   blfvalps 24348
            12.4.4  Open sets of a metric space   mopnval 24403
            12.4.5  Continuity in metric spaces   metcnp3 24505
            12.4.6  The uniform structure generated by a metric   metuval 24514
            12.4.7  Examples of metric spaces   dscmet 24537
            ​*12.4.8  Normed algebraic structures   cnm 24541
            12.4.9  Normed space homomorphisms (bounded linear operators)   cnmo 24670
            12.4.10  Topology on the reals   qtopbaslem 24723
            12.4.11  Topological definitions using the reals   cii 24842
            12.4.12  Path homotopy   chtpy 24934
            12.4.13  The fundamental group   cpco 24967
      ​12.5  Metric subcomplex vector spaces
            12.5.1  Subcomplex modules   cclm 25029
            ​*12.5.2  Subcomplex vector spaces   ccvs 25090
            ​*12.5.3  Normed subcomplex vector spaces   isncvsngp 25116
            12.5.4  Subcomplex pre-Hilbert spaces   ccph 25133
            12.5.5  Convergence and completeness   ccfil 25219
            12.5.6  Baire's Category Theorem   bcthlem1 25291
            12.5.7  Banach spaces and subcomplex Hilbert spaces   ccms 25299
                  12.5.7.1  The complete ordered field of the real numbers   retopn 25346
            12.5.8  Euclidean spaces   crrx 25350
            12.5.9  Minimizing Vector Theorem   minveclem1 25391
            12.5.10  Projection Theorem   pjthlem1 25404
​PART 13  BASIC REAL AND COMPLEX ANALYSIS
      ​13.1  Continuity
            13.1.1  Intermediate value theorem   pmltpclem1 25415
      ​13.2  Integrals
            13.2.1  Lebesgue measure   covol 25429
            13.2.2  Lebesgue integration   cmbf 25581
                  13.2.2.1  Lesbesgue integral   cmbf 25581
                  13.2.2.2  Lesbesgue directed integral   cdit 25813
      ​13.3  Derivatives
            13.3.1  Real and complex differentiation   climc 25829
                  13.3.1.1  Derivatives of functions of one complex or real variable   climc 25829
                  13.3.1.2  Results on real differentiation   dvferm1lem 25951
​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 26256
            14.1.5  Algebraic numbers   caa 26280
            14.1.6  Liouville's approximation theorem   aalioulem1 26298
      ​14.2  Sequences and series
            14.2.1  Taylor polynomials and Taylor's theorem   ctayl 26318
            14.2.2  Uniform convergence   culm 26341
            14.2.3  Power series   pserval 26375
      ​14.3  Basic trigonometry
            14.3.1  The exponential, sine, and cosine functions (cont.)   efcn 26408
            14.3.2  Properties of pi = 3.14159...   pilem1 26416
            14.3.3  Mapping of the exponential function   efgh 26505
            14.3.4  The natural logarithm on complex numbers   clog 26518
            ​*14.3.5  Logarithms to an arbitrary base   clogb 26728
            14.3.6  Theorems of Pythagoras, isosceles triangles, and intersecting chords   angval 26765
            14.3.7  Solutions of quadratic, cubic, and quartic equations   quad2 26803
            14.3.8  Inverse trigonometric functions   casin 26826
            14.3.9  The Birthday Problem   log2ublem1 26910
            14.3.10  Areas in R^2   carea 26919
            14.3.11  More miscellaneous converging sequences   rlimcnp 26929
            14.3.12  Inequality of arithmetic and geometric means   cvxcl 26948
            14.3.13  Euler-Mascheroni constant   cem 26955
            14.3.14  Zeta function   czeta 26976
            14.3.15  Gamma function   clgam 26979
      ​14.4  Basic number theory
            14.4.1  Wilson's theorem   wilthlem1 27031
            14.4.2  The Fundamental Theorem of Algebra   ftalem1 27036
            14.4.3  The Basel problem (ζ(2) = π²/6)   basellem1 27044
            14.4.4  Number-theoretical functions   ccht 27054
            14.4.5  Perfect Number Theorem   mersenne 27190
            14.4.6  Characters of Z/nZ   cdchr 27195
            14.4.7  Bertrand's postulate   bcctr 27238
            ​*14.4.8  Quadratic residues and the Legendre symbol   clgs 27257
            ​*14.4.9  Gauss' Lemma   gausslemma2dlem0a 27319
            14.4.10  Quadratic reciprocity   lgseisenlem1 27338
            14.4.11  All primes 4n+1 are the sum of two squares   2sqlem1 27380
            14.4.12  Chebyshev's Weak Prime Number Theorem, Dirichlet's Theorem   chebbnd1lem1 27432
            14.4.13  The Prime Number Theorem   mudivsum 27493
            14.4.14  Ostrowski's theorem   abvcxp 27578
​PART 15  SURREAL NUMBERS
      ​​*15.1  Sign sequence representation and Alling's axioms
            15.1.1  Definitions and initial properties   csur 27603
            15.1.2  Ordering   ltssolem1 27639
            15.1.3  Birthday Function   bdayfo 27641
            15.1.4  Density   fvnobday 27642
            ​*15.1.5  Full-Eta Property   bdayimaon 27657
      ​15.2  Initial consequences of Alling's axioms
            15.2.1  Ordering Theorems   cles 27708
            15.2.2  Birthday Theorems   bdayfun 27740
      ​​*15.3  Conway cut representation
            15.3.1  Conway cuts   cslts 27749
            15.3.2  Zero and One   c0s 27797
            15.3.3  Cuts and Options   cmade 27814
            15.3.4  Cofinality and coinitiality   cofslts 27910
      ​15.4  Induction and recursion
            15.4.1  Induction and recursion on one variable   cnorec 27929
            15.4.2  Induction and recursion on two variables   cnorec2 27940
      ​15.5  Surreal arithmetic
            15.5.1  Addition   cadds 27951
            15.5.2  Negation and Subtraction   cnegs 28011
            15.5.3  Multiplication   cmuls 28098
            15.5.4  Division   cdivs 28179
            15.5.5  Absolute value   cabss 28229
      ​15.6  Subsystems of surreals
            15.6.1  Ordinal numbers   cons 28243
            15.6.2  Surreal recursive sequences   cseqs 28275
            15.6.3  Natural numbers   cn0s 28304
            15.6.4  Integers   czs 28370
            15.6.5  Dyadic fractions   c2s 28402
            15.6.6  Real numbers   creno 28481
​​*PART 16  ELEMENTARY GEOMETRY
      ​16.1  Definition and Tarski's Axioms of Geometry
            16.1.1  Justification for the congruence notation   tgjustf 28541
      ​16.2  Tarskian Geometry
            16.2.1  Congruence   tgcgrcomimp 28545
            16.2.2  Betweenness   tgbtwntriv2 28555
            16.2.3  Dimension   tglowdim1 28568
            16.2.4  Betweenness and Congruence   tgifscgr 28576
            16.2.5  Congruence of a series of points   ccgrg 28578
            16.2.6  Motions   cismt 28600
            16.2.7  Colinearity   tglng 28614
            16.2.8  Connectivity of betweenness   tgbtwnconn1lem1 28640
            16.2.9  Less-than relation in geometric congruences   cleg 28650
            16.2.10  Rays   chlg 28668
            16.2.11  Lines   btwnlng1 28687
            16.2.12  Point inversions   cmir 28720
            16.2.13  Right angles   crag 28761
            16.2.14  Half-planes   islnopp 28807
            16.2.15  Midpoints and Line Mirroring   cmid 28840
            16.2.16  Congruence of angles   ccgra 28875
            16.2.17  Angle Comparisons   cinag 28903
            16.2.18  Congruence Theorems   tgsas1 28922
            16.2.19  Equilateral triangles   ceqlg 28933
      ​16.3  Properties of geometries
            16.3.1  Isomorphisms between geometries   f1otrgds 28937
      ​16.4  Geometry in Hilbert spaces
            16.4.1  Geometry in the complex plane   cchhllem 28955
            16.4.2  Geometry in Euclidean spaces   cee 28956
                  16.4.2.1  Definition of the Euclidean space   cee 28956
                  16.4.2.2  Tarski's axioms for geometry for the Euclidean space   axdimuniq 28982
                  16.4.2.3  EE^n fulfills Tarski's Axioms   ceeng 29046
​​*PART 17  GRAPH THEORY
      ​*17.1  Vertices and edges
            17.1.1  The edge function extractor for extensible structures   cedgf 29057
            ​*17.1.2  Vertices and indexed edges   cvtx 29065
                  17.1.2.1  Definitions and basic properties   cvtx 29065
                  17.1.2.2  The vertices and edges of a graph represented as ordered pair   opvtxval 29072
                  17.1.2.3  The vertices and edges of a graph represented as extensible structure   funvtxdmge2val 29080
                  17.1.2.4  Representations of graphs without edges   snstrvtxval 29106
                  17.1.2.5  Degenerated cases of representations of graphs   vtxval0 29108
            17.1.3  Edges as range of the edge function   cedg 29116
      ​​*17.2  Undirected graphs
            17.2.1  Undirected hypergraphs   cuhgr 29125
            17.2.2  Undirected pseudographs and multigraphs   cupgr 29149
            ​*17.2.3  Loop-free graphs   umgrislfupgrlem 29191
            17.2.4  Edges as subsets of vertices of graphs   uhgredgiedgb 29195
            ​*17.2.5  Undirected simple graphs   cuspgr 29217
            17.2.6  Examples for graphs   usgr0e 29305
            17.2.7  Subgraphs   csubgr 29336
            17.2.8  Finite undirected simple graphs   cfusgr 29385
            17.2.9  Neighbors, complete graphs and universal vertices   cnbgr 29401
                  17.2.9.1  Neighbors   cnbgr 29401
                  17.2.9.2  Universal vertices   cuvtx 29454
                  17.2.9.3  Complete graphs   ccplgr 29478
            17.2.10  Vertex degree   cvtxdg 29534
            ​*17.2.11  Regular graphs   crgr 29624
      ​​*17.3  Walks, paths and cycles
            *17.3.1  Walks   cewlks 29664
            17.3.2  Walks for loop-free graphs   lfgrwlkprop 29754
            17.3.3  Trails   ctrls 29757
            17.3.4  Paths and simple paths   cpths 29778
            17.3.5  Closed walks   cclwlks 29838
            17.3.6  Circuits and cycles   ccrcts 29852
            ​*17.3.7  Walks as words   cwwlks 29893
            17.3.8  Walks/paths of length 2 (as length 3 strings)   2wlkdlem1 29993
            17.3.9  Walks in regular graphs   rusgrnumwwlkl1 30039
            ​*17.3.10  Closed walks as words   cclwwlk 30051
                  17.3.10.1  Closed walks as words   cclwwlk 30051
                  17.3.10.2  Closed walks of a fixed length as words   cclwwlkn 30094
                  17.3.10.3  Closed walks on a vertex of a fixed length as words   cclwwlknon 30157
            17.3.11  Examples for walks, trails and paths   0ewlk 30184
            17.3.12  Connected graphs   cconngr 30256
      ​17.4  Eulerian paths and the Konigsberg Bridge problem
            ​*17.4.1  Eulerian paths   ceupth 30267
            ​*17.4.2  The Königsberg Bridge problem   konigsbergvtx 30316
      ​17.5  The Friendship Theorem
            17.5.1  Friendship graphs - basics   cfrgr 30328
            17.5.2  The friendship theorem for small graphs   frgr1v 30341
            17.5.3  Theorems according to Mertzios and Unger   2pthfrgrrn 30352
            ​*17.5.4  Huneke's Proof of the Friendship Theorem   frgrncvvdeqlem1 30369
​PART 18  GUIDES AND MISCELLANEA
      ​18.1  Guides (conventions, explanations, and examples)
            ​*18.1.1  Conventions   conventions 30470
            18.1.2  Natural deduction   natded 30473
            ​*18.1.3  Natural deduction examples   ex-natded5.2 30474
            18.1.4  Definitional examples   ex-or 30491
            18.1.5  Other examples   aevdemo 30530
      ​18.2  Humor
            18.2.1  April Fool's theorem   avril1 30533
      ​18.3  (Future - to be reviewed and classified)
            18.3.1  Planar incidence geometry   cplig 30545
            ​*18.3.2  Aliases kept to prevent broken links   dummylink 30558
​​*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 30560
            19.1.2  Abelian groups   cablo 30615
      ​19.2  Complex vector spaces
            19.2.1  Definition and basic properties   cvc 30629
            19.2.2  Examples of complex vector spaces   cnaddabloOLD 30652
      ​19.3  Normed complex vector spaces
            19.3.1  Definition and basic properties   cnv 30655
            19.3.2  Examples of normed complex vector spaces   cnnv 30748
            19.3.3  Induced metric of a normed complex vector space   imsval 30756
            19.3.4  Inner product   cdip 30771
            19.3.5  Subspaces   css 30792
      ​19.4  Operators on complex vector spaces
            19.4.1  Definitions and basic properties   clno 30811
      ​19.5  Inner product (pre-Hilbert) spaces
            19.5.1  Definition and basic properties   ccphlo 30883
            19.5.2  Examples of pre-Hilbert spaces   cncph 30890
            19.5.3  Properties of pre-Hilbert spaces   isph 30893
      ​19.6  Complex Banach spaces
            19.6.1  Definition and basic properties   ccbn 30933
            19.6.2  Examples of complex Banach spaces   cnbn 30940
            19.6.3  Uniform Boundedness Theorem   ubthlem1 30941
            19.6.4  Minimizing Vector Theorem   minvecolem1 30945
      ​19.7  Complex Hilbert spaces
            19.7.1  Definition and basic properties   chlo 30956
            19.7.2  Standard axioms for a complex Hilbert space   hlex 30969
            19.7.3  Examples of complex Hilbert spaces   cnchl 30987
            19.7.4  Hellinger-Toeplitz Theorem   htthlem 30988
​​*PART 20  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      ​20.1  Axiomatization of complex pre-Hilbert spaces
            20.1.1  Basic Hilbert space definitions   chba 30990
            20.1.2  Preliminary ZFC lemmas   df-hnorm 31039
            ​*20.1.3  Derive the Hilbert space axioms from ZFC set theory   axhilex-zf 31052
            ​*20.1.4  Introduce the vector space axioms for a Hilbert space   ax-hilex 31070
            20.1.5  Vector operations   hvmulex 31082
            20.1.6  Inner product postulates for a Hilbert space   ax-hfi 31150
      ​20.2  Inner product and norms
            20.2.1  Inner product   his5 31157
            20.2.2  Norms   dfhnorm2 31193
            20.2.3  Relate Hilbert space to normed complex vector spaces   hilablo 31231
            20.2.4  Bunjakovaskij-Cauchy-Schwarz inequality   bcsiALT 31250
      ​20.3  Cauchy sequences and completeness axiom
            20.3.1  Cauchy sequences and limits   hcau 31255
            20.3.2  Derivation of the completeness axiom from ZF set theory   hilmet 31265
            20.3.3  Completeness postulate for a Hilbert space   ax-hcompl 31273
            20.3.4  Relate Hilbert space to ZFC pre-Hilbert and Hilbert spaces   hhcms 31274
      ​20.4  Subspaces and projections
            20.4.1  Subspaces   df-sh 31278
            20.4.2  Closed subspaces   df-ch 31292
            20.4.3  Orthocomplements   df-oc 31323
            20.4.4  Subspace sum, span, lattice join, lattice supremum   df-shs 31379
            20.4.5  Projection theorem   pjhthlem1 31462
            20.4.6  Projectors   df-pjh 31466
      ​20.5  Properties of Hilbert subspaces
            20.5.1  Orthomodular law   omlsilem 31473
            20.5.2  Projectors (cont.)   pjhtheu2 31487
            20.5.3  Hilbert lattice operations   sh0le 31511
            20.5.4  Span (cont.) and one-dimensional subspaces   spansn0 31612
            20.5.5  Commutes relation for Hilbert lattice elements   df-cm 31654
            20.5.6  Foulis-Holland theorem   fh1 31689
            20.5.7  Quantum Logic Explorer axioms   qlax1i 31698
            20.5.8  Orthogonal subspaces   chscllem1 31708
            20.5.9  Orthoarguesian laws 5OA and 3OA   5oalem1 31725
            20.5.10  Projectors (cont.)   pjorthi 31740
            20.5.11  Mayet's equation E_3   mayete3i 31799
      ​20.6  Operators on Hilbert spaces
            ​*20.6.1  Operator sum, difference, and scalar multiplication   df-hosum 31801
            20.6.2  Zero and identity operators   df-h0op 31819
            20.6.3  Operations on Hilbert space operators   hoaddcl 31829
            20.6.4  Linear, continuous, bounded, Hermitian, unitary operators and norms   df-nmop 31910
            20.6.5  Linear and continuous functionals and norms   df-nmfn 31916
            20.6.6  Adjoint   df-adjh 31920
            20.6.7  Dirac bra-ket notation   df-bra 31921
            20.6.8  Positive operators   df-leop 31923
            20.6.9  Eigenvectors, eigenvalues, spectrum   df-eigvec 31924
            20.6.10  Theorems about operators and functionals   nmopval 31927
            20.6.11  Riesz lemma   riesz3i 32133
            20.6.12  Adjoints (cont.)   cnlnadjlem1 32138
            20.6.13  Quantum computation error bound theorem   unierri 32175
            20.6.14  Dirac bra-ket notation (cont.)   branmfn 32176
            20.6.15  Positive operators (cont.)   leopg 32193
            20.6.16  Projectors as operators   pjhmopi 32217
      ​20.7  States on a Hilbert lattice and Godowski's equation
            20.7.1  States on a Hilbert lattice   df-st 32282
            20.7.2  Godowski's equation   golem1 32342
      ​20.8  Cover relation, atoms, exchange axiom, and modular symmetry
            20.8.1  Covers relation; modular pairs   df-cv 32350
            20.8.2  Atoms   df-at 32409
            20.8.3  Superposition principle   superpos 32425
            20.8.4  Atoms, exchange and covering properties, atomicity   chcv1 32426
            20.8.5  Irreducibility   chirredlem1 32461
            20.8.6  Atoms (cont.)   atcvat3i 32467
            20.8.7  Modular symmetry   mdsymlem1 32474
​PART 21  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      ​21.1  Mathboxes for user contributions
            21.1.1  Mathbox guidelines   mathbox 32513
      ​21.2  Mathbox for Stefan Allan
      ​21.3  Mathbox for Thierry Arnoux
            21.3.1  Propositional Calculus - misc additions   ad11antr 32518
            21.3.2  Predicate Calculus   sbc2iedf 32534
                  21.3.2.1  Predicate Calculus - misc additions   sbc2iedf 32534
                  21.3.2.2  Restricted quantification - misc additions   ralcom4f 32536
                  21.3.2.3  Equality   eqtrb 32543
                  21.3.2.4  Double restricted existential uniqueness quantification   opsbc2ie 32545
                  21.3.2.5  Double restricted existential uniqueness quantification syntax   w2reu 32547
                  21.3.2.6  Substitution (without distinct variables) - misc additions   sbceqbidf 32556
                  21.3.2.7  Existential "at most one" - misc additions   mo5f 32558
                  21.3.2.8  Existential uniqueness - misc additions   reuxfrdf 32560
                  21.3.2.9  Restricted "at most one" - misc additions   rmoxfrd 32562
                  21.3.2.10  Restricted iota (description binder)   riotaeqbidva 32565
            21.3.3  General Set Theory   dmrab 32566
                  21.3.3.1  Class abstractions (a.k.a. class builders)   dmrab 32566
                  21.3.3.2  Image Sets   abrexdomjm 32577
                  21.3.3.3  Set relations and operations - misc additions   nelun 32583
                  21.3.3.4  Unordered pairs   elpreq 32598
                  21.3.3.5  Unordered triples   tpssg 32607
                  21.3.3.6  Conditional operator - misc additions   ifeqeqx 32612
                  21.3.3.7  Set union   uniinn0 32620
                  21.3.3.8  Indexed union - misc additions   cbviunf 32625
                  21.3.3.9  Indexed intersection - misc additions   iinabrex 32639
                  21.3.3.10  Disjointness - misc additions   disjnf 32640
            21.3.4  Relations and Functions   xpdisjres 32668
                  21.3.4.1  Relations - misc additions   xpdisjres 32668
                  21.3.4.2  Functions - misc additions   fconst7v 32693
                  21.3.4.3  Operations - misc additions   mpomptxf 32751
                  21.3.4.4  The mapping operation   elmaprd 32753
                  21.3.4.5  Support of a function   suppovss 32754
                  21.3.4.6  Explicit Functions with one or two points as a domain   cosnopne 32767
                  21.3.4.7  Isomorphisms - misc. additions   gtiso 32774
                  21.3.4.8  Disjointness (additional proof requiring functions)   disjdsct 32776
                  21.3.4.9  First and second members of an ordered pair - misc additions   df1stres 32777
                  21.3.4.10  Countable Sets   snct 32785
            21.3.5  Real and Complex Numbers   sgnval2 32808
                  21.3.5.1  Complex operations - misc. additions   creq0 32809
                  21.3.5.2  Ordering on reals - misc additions   lt2addrd 32823
                  21.3.5.3  Extended reals - misc additions   nn0mnfxrd 32824
                  21.3.5.4  Extended nonnegative integers - misc additions   xnn0gt0 32842
                  21.3.5.5  Real number intervals - misc additions   joiniooico 32847
                  21.3.5.6  Finite intervals of integers - misc additions   uzssico 32857
                  21.3.5.7  Half-open integer ranges - misc additions   iundisjfi 32869
                  21.3.5.8  The ` # ` (set size) function - misc additions   hashunif 32879
                  21.3.5.9  The greatest common divisor operator - misc. additions   elq2 32885
                  21.3.5.10  Integers   nn0split01 32891
                  21.3.5.11  Decimal numbers   dfdec100 32903
            21.3.6  Real and complex functions   sgncl 32904
                  21.3.6.1  Signum (sgn or sign) function - misc. additions   sgncl 32904
                  21.3.6.2  Integer powers - misc. additions   nexple 32917
                  21.3.6.3  Indicator Functions (continued)   indsumin 32921
            ​*21.3.7  Decimal expansion   cdp2 32930
                  ​*21.3.7.1  Decimal point   cdp 32947
                  21.3.7.2  Division in the extended real number system   cxdiv 32976
            21.3.8  Words over a set - misc additions   wrdres 32995
                  21.3.8.1  Splicing words (substring replacement)   splfv3 33018
                  21.3.8.2  Cyclic shift of words   1cshid 33019
            21.3.9  Extensible Structures   ressplusf 33023
                  21.3.9.1  Structure restriction operator   ressplusf 33023
                  21.3.9.2  Posets   ressprs 33026
                  21.3.9.3  Complete lattices   clatp0cl 33036
                  21.3.9.4  Order Theory   cmnt 33038
                  21.3.9.5  Extended reals Structure - misc additions   ax-xrssca 33064
                  21.3.9.6  The extended nonnegative real numbers commutative monoid   xrge00 33074
            21.3.10  Algebra   mndcld 33082
                  21.3.10.1  Monoids   mndcld 33082
                  21.3.10.2  Monoids Homomorphisms   abliso 33096
                  21.3.10.3  Groups - misc additions   grpinvinvd 33100
                  21.3.10.4  Abelian Groups - misc additions   ablcomd 33106
                  21.3.10.5  Finitely supported group sums - misc additions   gsumsubg 33107
                  21.3.10.6  Group or monoid sums over words   gsumwun 33137
                  21.3.10.7  Centralizers and centers - misc additions   cntzun 33140
                  21.3.10.8  The symmetric group   symgfcoeu 33143
                  21.3.10.9  Transpositions   pmtridf1o 33155
                  21.3.10.10  Permutation Signs   psgnid 33158
                  21.3.10.11  Permutation cycles   ctocyc 33167
                  21.3.10.12  The Alternating Group   evpmval 33206
                  21.3.10.13  Signum in an ordered monoid   csgns 33219
                  21.3.10.14  Fixed points   cfxp 33224
                  21.3.10.15  The Archimedean property for generic ordered algebraic structures   cinftm 33237
                  21.3.10.16  Semiring left modules   cslmd 33261
                  21.3.10.17  Simple groups   prmsimpcyc 33289
                  21.3.10.18  Rings - misc additions   ringrngd 33290
                  21.3.10.19  Subrings generated by a set   elrgspnlem1 33303
                  21.3.10.20  The zero ring   irrednzr 33311
                  21.3.10.21  Localization of rings   cerl 33314
                  21.3.10.22  Integral Domains   domnmuln0rd 33335
                  21.3.10.23  Euclidean Domains   ceuf 33349
                  21.3.10.24  Division Rings   ringinveu 33355
                  21.3.10.25  The field of rational numbers   qfld 33358
                  21.3.10.26  Subfields   subsdrg 33359
                  21.3.10.27  Field of fractions   cfrac 33363
                  21.3.10.28  Field extensions generated by a set   cfldgen 33371
                  21.3.10.29  Ring homomorphisms - misc additions   rhmdvd 33384
                  21.3.10.30  Scalar restriction operation   cresv 33386
                  21.3.10.31  The commutative ring of gaussian integers   gzcrng 33401
                  21.3.10.32  The archimedean ordered field of real numbers   cnfldfld 33402
                  21.3.10.33  The quotient map and quotient modules   qusker 33409
                  21.3.10.34  The ring of integers modulo ` N `   znfermltl 33426
                  21.3.10.35  Independent sets and families   islinds5 33427
                  21.3.10.36  Ring associates, ring units   dvdsruassoi 33444
                  ​*21.3.10.37  Subgroup sum / Sumset / Minkowski sum   elgrplsmsn 33450
                  21.3.10.38  The quotient map   quslsm 33465
                  21.3.10.39  Ideals   lidlmcld 33479
                  21.3.10.40  Prime Ideals   cprmidl 33495
                  21.3.10.41  Maximal Ideals   cmxidl 33519
                  21.3.10.42  The semiring of ideals of a ring   cidlsrg 33560
                  21.3.10.43  Prime Elements   rprmval 33576
                  21.3.10.44  Unique factorization domains   cufd 33598
                  21.3.10.45  The ring of integers   zringidom 33611
                  21.3.10.46  Associative Algebra   assaassd 33615
                  21.3.10.47  Univariate Polynomials   0ringmon1p 33617
                  21.3.10.48  Polynomial quotient and polynomial remainder   q1pdir 33663
                  21.3.10.49  Multivariate Polynomials   psrbasfsupp 33672
                  21.3.10.50  The ring of symmetric polynomials   csply 33699
                  21.3.10.51  The subring algebra   sra1r 33725
                  21.3.10.52  Division Ring Extensions   drgext0g 33734
                  21.3.10.53  Vector Spaces   lvecdimfi 33740
                  21.3.10.54  Vector Space Dimension   cldim 33743
            21.3.11  Field Extensions   cfldext 33782
                  21.3.11.1  Algebraic numbers   cirng 33827
                  21.3.11.2  Algebraic extensions   calgext 33839
                  21.3.11.3  Minimal polynomials   cminply 33843
                  21.3.11.4  Quadratic Field Extensions   rtelextdg2lem 33870
                  21.3.11.5  Towers of quadratic extentions   fldext2chn 33872
            ​*21.3.12  Constructible Numbers   cconstr 33873
                  21.3.12.1  Impossible constructions   2sqr3minply 33924
            21.3.13  Matrices   csmat 33937
                  21.3.13.1  Submatrices   csmat 33937
                  21.3.13.2  Matrix literals   clmat 33955
                  21.3.13.3  Laplace expansion of determinants   mdetpmtr1 33967
            21.3.14  Topology   ist0cld 33977
                  21.3.14.1  Open maps   txomap 33978
                  21.3.14.2  Topology of the unit circle   qtopt1 33979
                  21.3.14.3  Refinements   reff 33983
                  21.3.14.4  Open cover refinement property   ccref 33986
                  21.3.14.5  Lindelöf spaces   cldlf 33996
                  21.3.14.6  Paracompact spaces   cpcmp 33999
                  ​*21.3.14.7  Spectrum of a ring   crspec 34006
                  21.3.14.8  Pseudometrics   cmetid 34030
                  21.3.14.9  Continuity - misc additions   hauseqcn 34042
                  21.3.14.10  Topology of the closed unit interval   elunitge0 34043
                  21.3.14.11  Topology of ` ( RR X. RR ) `   unicls 34047
                  21.3.14.12  Order topology - misc. additions   cnvordtrestixx 34057
                  21.3.14.13  Continuity in topological spaces - misc. additions   mndpluscn 34070
                  21.3.14.14  Topology of the extended nonnegative real numbers ordered monoid   xrge0hmph 34076
                  21.3.14.15  Limits - misc additions   lmlim 34091
                  21.3.14.16  Univariate polynomials   pl1cn 34099
            21.3.15  Uniform Stuctures and Spaces   chcmp 34100
                  21.3.15.1  Hausdorff uniform completion   chcmp 34100
            21.3.16  Topology and algebraic structures   zringnm 34102
                  21.3.16.1  The norm on the ring of the integer numbers   zringnm 34102
                  21.3.16.2  Topological ` ZZ ` -modules   zlm0 34104
                  21.3.16.3  Canonical embedding of the field of the rational numbers into a division ring   cqqh 34114
                  21.3.16.4  Canonical embedding of the real numbers into a complete ordered field   crrh 34137
                  21.3.16.5  Embedding from the extended real numbers into a complete lattice   cxrh 34160
                  21.3.16.6  Canonical embeddings into the ordered field of the real numbers   zrhre 34163
                  ​*21.3.16.7  Topological Manifolds   cmntop 34166
                  21.3.16.8  Extended sum   cesum 34171
            21.3.17  Mixed Function/Constant operation   cofc 34239
            21.3.18  Abstract measure   csiga 34252
                  21.3.18.1  Sigma-Algebra   csiga 34252
                  21.3.18.2  Generated sigma-Algebra   csigagen 34282
                  ​*21.3.18.3  lambda and pi-Systems, Rings of Sets   ispisys 34296
                  21.3.18.4  The Borel algebra on the real numbers   cbrsiga 34325
                  21.3.18.5  Product Sigma-Algebra   csx 34332
                  21.3.18.6  Measures   cmeas 34339
                  21.3.18.7  The counting measure   cntmeas 34370
                  21.3.18.8  The Lebesgue measure - misc additions   voliune 34373
                  21.3.18.9  The Dirac delta measure   cdde 34376
                  21.3.18.10  The 'almost everywhere' relation   cae 34381
                  21.3.18.11  Measurable functions   cmbfm 34393
                  21.3.18.12  Borel Algebra on ` ( RR X. RR ) `   br2base 34413
                  ​*21.3.18.13  Caratheodory's extension theorem   coms 34435
            21.3.19  Integration   itgeq12dv 34470
                  21.3.19.1  Lebesgue integral - misc additions   itgeq12dv 34470
                  21.3.19.2  Bochner integral   citgm 34471
            21.3.20  Euler's partition theorem   oddpwdc 34498
            21.3.21  Sequences defined by strong recursion   csseq 34527
            21.3.22  Fibonacci Numbers   cfib 34540
            21.3.23  Probability   cprb 34551
                  21.3.23.1  Probability Theory   cprb 34551
                  21.3.23.2  Conditional Probabilities   ccprob 34575
                  21.3.23.3  Real-valued Random Variables   crrv 34584
                  21.3.23.4  Preimage set mapping operator   corvc 34600
                  21.3.23.5  Distribution Functions   orvcelval 34613
                  21.3.23.6  Cumulative Distribution Functions   orvclteel 34617
                  21.3.23.7  Probabilities - example   coinfliplem 34623
                  21.3.23.8  Bertrand's Ballot Problem   ballotlemoex 34630
            21.3.24  Signum (sgn or sign) function - misc. additions   fzssfzo 34683
                  21.3.24.1  Operations on words   ccatmulgnn0dir 34686
            21.3.25  Polynomials with real coefficients - misc additions   plymul02 34690
            21.3.26  Descartes's rule of signs   signspval 34696
                  21.3.26.1  Sign changes in a word over real numbers   signspval 34696
                  21.3.26.2  Counting sign changes in a word over real numbers   signslema 34706
            21.3.27  Number Theory   iblidicc 34736
                  21.3.27.1  Representations of a number as sums of integers   crepr 34752
                  21.3.27.2  Vinogradov Trigonometric Sums and the Circle Method   cvts 34779
                  21.3.27.3  The Ternary Goldbach Conjecture: Final Statement   ax-hgt749 34788
            21.3.28  Elementary Geometry   cstrkg2d 34808
                  ​*21.3.28.1  Two-dimensional geometry   cstrkg2d 34808
                  21.3.28.2  Outer Five Segment (not used, no need to move to main)   cafs 34813
            ​*21.3.29  LeftPad Project   clpad 34818
      ​​*21.4  Mathbox for Jonathan Ben-Naim
            21.4.1  First-order logic and set theory   bnj170 34841
            21.4.2  Well founded induction and recursion   bnj110 35000
            21.4.3  The existence of a minimal element in certain classes   bnj69 35152
            21.4.4  Well-founded induction   bnj1204 35154
            21.4.5  Well-founded recursion, part 1 of 3   bnj60 35204
            21.4.6  Well-founded recursion, part 2 of 3   bnj1500 35210
            21.4.7  Well-founded recursion, part 3 of 3   bnj1522 35214
      ​21.5  Mathbox for BTernaryTau
            21.5.1  First-order logic   nfan1c 35215
                  21.5.1.1  Auxiliary axiom schemes   nfan1c 35215
            21.5.2  ZF set theory   exdifsn 35221
                  21.5.2.1  Finitism   prcinf 35257
                  21.5.2.2  Introduce ax-regs   ax-regs 35270
                  21.5.2.3  Derive ax-regs   axregs 35283
                  21.5.2.4  Global choice   gblacfnacd 35284
            21.5.3  Real and complex numbers   zltp1ne 35292
            21.5.4  Graph theory   lfuhgr 35300
                  21.5.4.1  Acyclic graphs   cacycgr 35324
      ​21.6  Mathbox for Mario Carneiro
            21.6.1  Predicate calculus with all distinct variables   ax-7d 35341
            21.6.2  Miscellaneous stuff   quartfull 35347
            21.6.3  Derangements and the Subfactorial   deranglem 35348
            21.6.4  The Erdős-Szekeres theorem   erdszelem1 35373
            21.6.5  The Kuratowski closure-complement theorem   kur14lem1 35388
            21.6.6  Retracts and sections   cretr 35399
            21.6.7  Path-connected and simply connected spaces   cpconn 35401
            21.6.8  Covering maps   ccvm 35437
            21.6.9  Normal numbers   snmlff 35511
            21.6.10  Godel-sets of formulas - part 1   cgoe 35515
            21.6.11  Godel-sets of formulas - part 2   cgon 35614
            21.6.12  Models of ZF   cgze 35628
            ​*21.6.13  Metamath formal systems   cmcn 35642
            21.6.14  Grammatical formal systems   cm0s 35767
            21.6.15  Models of formal systems   cmuv 35787
            21.6.16  Splitting fields   ccpms 35809
            21.6.17  p-adic number fields   czr 35829
      ​​*21.7  Mathbox for Filip Cernatescu
      ​21.8  Mathbox for Paul Chapman
            21.8.1  Real and complex numbers (cont.)   climuzcnv 35853
            21.8.2  Miscellaneous theorems   elfzm12 35857
      ​21.9  Mathbox for Hongxiu Chen
      ​21.10  Mathbox for Adrian Ducourtial
            21.10.1  Propositional calculus   currybi 35870
            21.10.2  Clone theory   ccloneop 35877
      ​21.11  Mathbox for Scott Fenton
            21.11.1  ZFC Axioms in primitive form   axextprim 35883
            21.11.2  Untangled classes   untelirr 35890
            21.11.3  Extra propositional calculus theorems   3jaodd 35897
            21.11.4  Misc. Useful Theorems   nepss 35900
            21.11.5  Properties of real and complex numbers   sqdivzi 35910
            21.11.6  Infinite products   iprodefisumlem 35922
            21.11.7  Factorial limits   faclimlem1 35925
            21.11.8  Greatest common divisor and divisibility   gcd32 35931
            21.11.9  Properties of relationships   dftr6 35933
            21.11.10  Properties of functions and mappings   funpsstri 35948
            21.11.11  Ordinal numbers   elpotr 35961
            21.11.12  Defined equality axioms   axextdfeq 35977
            21.11.13  Hypothesis builders   hbntg 35985
            21.11.14  Well-founded zero, successor, and limits   cwsuc 35990
            21.11.15  Quantifier-free definitions   ctxp 36010
            21.11.16  Alternate ordered pairs   caltop 36138
            21.11.17  Geometry in the Euclidean space   cofs 36164
                  21.11.17.1  Congruence properties   cofs 36164
                  21.11.17.2  Betweenness properties   btwntriv2 36194
                  21.11.17.3  Segment Transportation   ctransport 36211
                  21.11.17.4  Properties relating betweenness and congruence   cifs 36217
                  21.11.17.5  Connectivity of betweenness   btwnconn1lem1 36269
                  21.11.17.6  Segment less than or equal to   csegle 36288
                  21.11.17.7  Outside-of relationship   coutsideof 36301
                  21.11.17.8  Lines and Rays   cline2 36316
            21.11.18  Forward difference   cfwddif 36340
            21.11.19  Rank theorems   rankung 36348
            21.11.20  Hereditarily Finite Sets   chf 36354
      ​21.12  Mathbox for Gino Giotto
            21.12.1  Equality theorems   rmoeqi 36369
                  21.12.1.1  Inference versions   rmoeqi 36369
                  21.12.1.2  Deduction versions   rmoeqdv 36394
            21.12.2  Change bound variables   in-ax8 36406
                  21.12.2.1  Change bound variables and domains   cbvralvw2 36408
                  21.12.2.2  Change bound variables, deduction versions   cbvmodavw 36432
                  21.12.2.3  Change bound variables and domains, deduction versions   cbvrmodavw2 36465
            21.12.3  Study of ax-mulf usage   mpomulnzcnf 36481
      ​21.13  Mathbox for Jeff Hankins
            21.13.1  Miscellany   a1i14 36482
            21.13.2  Basic topological facts   topbnd 36506
            21.13.3  Topology of the real numbers   ivthALT 36517
            21.13.4  Refinements   cfne 36518
            21.13.5  Neighborhood bases determine topologies   neibastop1 36541
            21.13.6  Lattice structure of topologies   topmtcl 36545
            21.13.7  Filter bases   fgmin 36552
            21.13.8  Directed sets, nets   tailfval 36554
      ​21.14  Mathbox for Anthony Hart
            21.14.1  Propositional Calculus   tb-ax1 36565
            21.14.2  Predicate Calculus   nalfal 36585
            21.14.3  Miscellaneous single axioms   meran1 36593
            21.14.4  Connective Symmetry   negsym1 36599
      ​21.15  Mathbox for Chen-Pang He
            21.15.1  Ordinal topology   ontopbas 36610
      ​21.16  Mathbox for Jeff Hoffman
            21.16.1  Inferences for finite induction on generic function values   fveleq 36633
            21.16.2  gdc.mm   nnssi2 36637
      ​21.17  Mathbox for Matthew House
            21.17.1  Relations on well-ordered indexed unions   weiunval 36644
            21.17.2  Axiom of Transitive Containment   axtco 36653
            21.17.3  Transitive closure of a class   tr0elw 36666
            ​*21.17.4  Stronger axioms of regularity   mh-setind 36718
            21.17.5  Short axioms written in primitive symbols   mh-inf3f1 36723
      ​21.18  Mathbox for Asger C. Ipsen
            21.18.1  Continuous nowhere differentiable functions   dnival 36731
      ​​*21.19  Mathbox for BJ
            *21.19.1  Propositional calculus   bj-mp2c 36800
                  *21.19.1.1  Derived rules of inference   bj-mp2c 36800
                  ​*21.19.1.2  A syntactic theorem   bj-0 36802
                  ​*21.19.1.3  Minimal implicational calculus   bj-a1k 36804
                  ​*21.19.1.4  Positive calculus   bj-bisimpl 36817
                  ​*21.19.1.5  Implication and negation   bj-con2com 36825
                  ​*21.19.1.6  Disjunction   bj-jaoi1 36836
                  ​*21.19.1.7  Logical equivalence   bj-dfbi4 36838
                  21.19.1.8  The conditional operator for propositions   bj-consensus 36843
                  ​*21.19.1.9  Propositional calculus: miscellaneous   bj-imbi12 36848
            ​*21.19.2  Modal logic   bj-axdd2 36857
            ​*21.19.3  Provability logic   cprvb 36862
            ​*21.19.4  First-order logic   bj-exexalal 36871
                  21.19.4.1  Universal and existential quantifiers, nonfreeness predicate   bj-exexalal 36871
                  21.19.4.2  Adding ax-gen   bj-genr 36872
                  21.19.4.3  Adding ax-4   bj-almp 36876
                  21.19.4.4  Adding ax-5   bj-spvw 36929
                  21.19.4.5  Equality and substitution   bj-df-sb 36944
                  21.19.4.6  Adding ax-6   bj-spim0 36963
                  21.19.4.7  Adding ax-7   bj-cbvexw 36971
                  21.19.4.8  Membership predicate, ax-8 and ax-9   bj-ax89 36973
                  21.19.4.9  Adding ax-11   bj-alcomexcom 36975
                  21.19.4.10  Adding ax-12   axc11n11 36979
                  ​*21.19.4.11  Really adding ax-12   bj-substax12 37021
                  21.19.4.12  Nonfreeness   wnnf 37023
                  21.19.4.13  Adding ax-13   bj-axc10 37090
                  ​*21.19.4.14  Removing dependencies on ax-13 (and ax-11)   bj-axc10v 37100
                  ​*21.19.4.15  Distinct var metavariables   bj-hbaeb2 37125
                  ​*21.19.4.16  Around ~ equsal   bj-equsal1t 37129
                  ​*21.19.4.17  Some Principia Mathematica proofs   stdpc5t 37134
                  21.19.4.18  Alternate definition of substitution   bj-sbsb 37144
                  21.19.4.19  Lemmas for substitution   bj-sbf3 37146
                  21.19.4.20  Existential uniqueness   bj-eu3f 37148
                  ​*21.19.4.21  First-order logic: miscellaneous   bj-sblem1 37149
            21.19.5  Set theory   eliminable1 37166
                  ​*21.19.5.1  Eliminability of class terms   eliminable1 37166
                  ​*21.19.5.2  Classes without the axiom of extensionality   bj-denoteslem 37178
                  21.19.5.3  Characterization among sets versus among classes   elelb 37204
                  ​*21.19.5.4  The nonfreeness quantifier for classes   bj-nfcsym 37206
                  ​*21.19.5.5  Lemmas for class substitution   bj-sbeqALT 37207
                  21.19.5.6  Removing some axiom requirements and disjoint variable conditions   bj-exlimvmpi 37218
                  ​*21.19.5.7  Class abstractions   bj-elabd2ALT 37232
                  21.19.5.8  Generalized class abstractions   bj-cgab 37240
                  ​*21.19.5.9  Restricted nonfreeness   wrnf 37248
                  ​*21.19.5.10  Russell's paradox   bj-ru1 37250
                  21.19.5.11  Curry's paradox in set theory   currysetlem 37252
                  ​*21.19.5.12  Some disjointness results   bj-n0i 37258
                  ​*21.19.5.13  Complements on direct products   bj-xpimasn 37262
                  ​*21.19.5.14  "Singletonization" and tagging   bj-snsetex 37270
                  ​*21.19.5.15  Tuples of classes   bj-cproj 37297
                  ​*21.19.5.16  Set theory: elementary operations relative to a universe   bj-rcleqf 37332
                  ​*21.19.5.17  Axioms for finite unions   bj-abex 37337
                  ​*21.19.5.18  Set theory: miscellaneous   eleq2w2ALT 37354
                  ​*21.19.5.19  Axioms of separation and replacement   bj-axnul 37379
                  ​*21.19.5.20  Evaluation at a class   bj-evaleq 37383
                  21.19.5.21  Elementwise operations   celwise 37391
                  ​*21.19.5.22  Elementwise intersection (families of sets induced on a subset)   bj-rest00 37393
                  21.19.5.23  Moore collections (complements)   bj-raldifsn 37412
                  21.19.5.24  Maps-to notation for functions with three arguments   bj-0nelmpt 37428
                  ​*21.19.5.25  Currying   csethom 37434
                  ​*21.19.5.26  Setting components of extensible structures   cstrset 37446
            ​*21.19.6  Extended real and complex numbers, real and complex projective lines   bj-nfald 37449
                  21.19.6.1  Complements on class abstractions of ordered pairs and binary relations   bj-nfald 37449
                  ​*21.19.6.2  Identity relation (complements)   bj-opabssvv 37464
                  ​*21.19.6.3  Functionalized identity (diagonal in a Cartesian square)   cdiag2 37486
                  ​*21.19.6.4  Direct image and inverse image   cimdir 37492
                  ​*21.19.6.5  Extended numbers and projective lines as sets   cfractemp 37510
                  ​*21.19.6.6  Addition and opposite   caddcc 37551
                  ​*21.19.6.7  Order relation on the extended reals   cltxr 37555
                  ​*21.19.6.8  Argument, multiplication and inverse   carg 37557
                  21.19.6.9  The canonical bijection from the finite ordinals   ciomnn 37563
                  21.19.6.10  Divisibility   cnnbar 37574
            ​*21.19.7  Monoids   bj-smgrpssmgm 37582
                  ​*21.19.7.1  Finite sums in monoids   cfinsum 37597
            ​*21.19.8  Affine, Euclidean, and Cartesian geometry   bj-fvimacnv0 37600
                  *21.19.8.1  Real vector spaces   bj-fvimacnv0 37600
                  ​*21.19.8.2  Complex numbers (supplements)   bj-subcom 37622
                  ​*21.19.8.3  Barycentric coordinates   bj-bary1lem 37624
            21.19.9  Monoid of endomorphisms   cend 37627
      ​21.20  Mathbox for Jim Kingdon
            21.20.1  Circle constant   taupilem3 37633
            21.20.2  Number theory   dfgcd3 37638
            21.20.3  Real numbers   irrdifflemf 37639
      ​21.21  Mathbox for ML
            21.21.1  Miscellaneous   csbrecsg 37644
            21.21.2  Cartesian exponentiation   cfinxp 37699
            21.21.3  Topology   iunctb2 37719
                  ​*21.21.3.1  Pi-base theorems   pibp16 37729
      ​21.22  Mathbox for Wolf Lammen
            21.22.1  1. Bootstrapping   wl-section-boot 37738
            21.22.2  Implication chains   wl-section-impchain 37762
            21.22.3  Theorems around the conditional operator   wl-ifp-ncond1 37780
            21.22.4  Alternative development of hadd, cadd   wl-df-3xor 37784
            21.22.5  An alternative axiom ~ ax-13   ax-wl-13v 37809
            21.22.6  Bootstrapping set theory with classes   wl-cleq-0 37811
            21.22.7  Other stuff   wl-mps 37832
      ​21.23  Mathbox for Brendan Leahy
      ​21.24  Mathbox for Jeff Madsen
            21.24.1  Logic and set theory   unirep 38035
            21.24.2  Real and complex numbers; integers   filbcmb 38061
            21.24.3  Sequences and sums   sdclem2 38063
            21.24.4  Topology   subspopn 38073
            21.24.5  Metric spaces   metf1o 38076
            21.24.6  Continuous maps and homeomorphisms   constcncf 38083
            21.24.7  Boundedness   ctotbnd 38087
            21.24.8  Isometries   cismty 38119
            21.24.9  Heine-Borel Theorem   heibor1lem 38130
            21.24.10  Banach Fixed Point Theorem   bfplem1 38143
            21.24.11  Euclidean space   crrn 38146
            21.24.12  Intervals (continued)   ismrer1 38159
            21.24.13  Operation properties   cass 38163
            21.24.14  Groups and related structures   cmagm 38169
            21.24.15  Group homomorphism and isomorphism   cghomOLD 38204
            21.24.16  Rings   crngo 38215
            21.24.17  Division Rings   cdrng 38269
            21.24.18  Ring homomorphisms   crngohom 38281
            21.24.19  Commutative rings   ccm2 38310
            21.24.20  Ideals   cidl 38328
            21.24.21  Prime rings and integral domains   cprrng 38367
            21.24.22  Ideal generators   cigen 38380
      ​21.25  Mathbox for Giovanni Mascellani
            ​*21.25.1  Tools for automatic proof building   efald2 38399
            ​*21.25.2  Tseitin axioms   fald 38450
            ​*21.25.3  Equality deductions   iuneq2f 38477
            ​*21.25.4  Miscellanea   orcomdd 38488
      ​21.26  Mathbox for Peter Mazsa
            21.26.1  Notations   cxrn 38495
            21.26.2  Preparatory theorems   el2v1 38550
            21.26.3  Range Cartesian product   df-xrn 38701
            21.26.4  Relations   df-rels 38761
            21.26.5  Quotient map (coset map)   df-qmap 38767
            21.26.6  Lifts, shifts, successor, and predecessor   df-adjliftmap 38776
            21.26.7  Cosets by ` R `   df-coss 38822
            21.26.8  Subset relations   df-ssr 38899
            21.26.9  Reflexivity   df-refs 38911
            21.26.10  Converse reflexivity   df-cnvrefs 38926
            21.26.11  Symmetry   df-syms 38943
            21.26.12  Reflexivity and symmetry   symrefref2 38968
            21.26.13  Transitivity   df-trs 38977
            21.26.14  Equivalence relations   df-eqvrels 38989
            21.26.15  Redundancy   df-redunds 39028
            21.26.16  Domain quotients   df-dmqss 39043
            21.26.17  Equivalence relations on domain quotients   df-ers 39069
            21.26.18  Functions   df-funss 39086
            21.26.19  Disjoints vs. converse functions   df-disjss 39109
            21.26.20  Antisymmetry   df-antisymrel 39184
            21.26.21  Partitions: disjoints on domain quotients   df-parts 39189
            21.26.22  Partition-Equivalence Theorems   disjim 39205
            21.26.23  Type-safe Partition-Equivalence: PetParts, PetErs, Pet2Parts, Pet2Ers   df-petparts 39289
      ​21.27  Mathbox for Rodolfo Medina
            21.27.1  Partitions   prtlem60 39299
      ​​*21.28  Mathbox for Norm Megill
            *21.28.1  Obsolete schemes ax-c4,c5,c7,c10,c11,c11n,c15,c9,c14,c16   ax-c5 39329
            ​*21.28.2  Rederive new axioms ax-4, ax-10, ax-6, ax-12, ax-13 from old   axc5 39339
            ​*21.28.3  Legacy theorems using obsolete axioms   ax5ALT 39353
            21.28.4  Experiments with weak deduction theorem   elimhyps 39407
            21.28.5  Miscellanea   cnaddcom 39418
            21.28.6  Atoms, hyperplanes, and covering in a left vector space (or module)   clsa 39420
            21.28.7  Functionals and kernels of a left vector space (or module)   clfn 39503
            21.28.8  Opposite rings and dual vector spaces   cld 39569
            21.28.9  Ortholattices and orthomodular lattices   cops 39618
            21.28.10  Atomic lattices with covering property   ccvr 39708
            21.28.11  Hilbert lattices   chlt 39796
            21.28.12  Projective geometries based on Hilbert lattices   clln 39937
            21.28.13  Construction of a vector space from a Hilbert lattice   cdlema1N 40237
            21.28.14  Construction of involution and inner product from a Hilbert lattice   clpoN 41926
      ​21.29  Mathbox for metakunt
            21.29.1  Commutative Semiring   ccsrg 42408
            21.29.2  General helpful statements   rhmzrhval 42411
            21.29.3  Some gcd and lcm results   12gcd5e1 42442
            21.29.4  Least common multiple inequality theorem   3factsumint1 42460
            21.29.5  Logarithm inequalities   3exp7 42492
            21.29.6  Miscellaneous results for AKS formalisation   intlewftc 42500
            21.29.7  Sticks and stones   sticksstones1 42585
            21.29.8  Continuation AKS   aks6d1c6lem1 42609
      ​21.30  Mathbox for Steven Nguyen
            21.30.1  Utility theorems   jarrii 42644
            ​*21.30.2  Arithmetic theorems   c0exALT 42691
            21.30.3  Exponents and divisibility   oexpreposd 42754
            21.30.4  Trigonometry and Calculus   tanhalfpim 42781
            ​*21.30.5  Independence of ax-mulcom   cresub 42797
            21.30.6  Structures   sn-base0 42940
            ​*21.30.7  Projective spaces   cprjsp 43034
            21.30.8  Basic reductions for Fermat's Last Theorem   dffltz 43067
            ​*21.30.9  Exemplar theorems   iddii 43097
                  ​*21.30.9.1  Standard replacements of ax-10 , ax-11 , ax-12   nfa1w 43108
      ​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 43124
            21.33.2  Additional theory of functions   imaiinfv 43125
            21.33.3  Additional topology   elrfi 43126
            21.33.4  Characterization of closure operators. Kuratowski closure axioms   ismrcd1 43130
            21.33.5  Algebraic closure systems   cnacs 43134
            21.33.6  Miscellanea 1. Map utilities   constmap 43145
            21.33.7  Miscellanea for polynomials   mptfcl 43152
            21.33.8  Multivariate polynomials over the integers   cmzpcl 43153
            21.33.9  Miscellanea for Diophantine sets 1   coeq0i 43185
            21.33.10  Diophantine sets 1: definitions   cdioph 43187
            21.33.11  Diophantine sets 2 miscellanea   ellz1 43199
            21.33.12  Diophantine sets 2: union and intersection. Monotone Boolean algebra   diophin 43204
            21.33.13  Diophantine sets 3: construction   diophrex 43207
            21.33.14  Diophantine sets 4 miscellanea   2sbcrex 43216
            21.33.15  Diophantine sets 4: Quantification   rexrabdioph 43222
            21.33.16  Diophantine sets 5: Arithmetic sets   rabdiophlem1 43229
            21.33.17  Diophantine sets 6: reusability. renumbering of variables   eldioph4b 43239
            21.33.18  Pigeonhole Principle and cardinality helpers   fphpd 43244
            21.33.19  A non-closed set of reals is infinite   rencldnfilem 43248
            21.33.20  Lagrange's rational approximation theorem   irrapxlem1 43250
            21.33.21  Pell equations 1: A nontrivial solution always exists   pellexlem1 43257
            21.33.22  Pell equations 2: Algebraic number theory of the solution set   csquarenn 43264
            21.33.23  Pell equations 3: characterizing fundamental solution   infmrgelbi 43306
            ​*21.33.24  Logarithm laws generalized to an arbitrary base   reglogcl 43318
            21.33.25  Pell equations 4: the positive solution group is infinite cyclic   pellfund14 43326
            21.33.26  X and Y sequences 1: Definition and recurrence laws   crmx 43328
            21.33.27  Ordering and induction lemmas for the integers   monotuz 43369
            21.33.28  X and Y sequences 2: Order properties   rmxypos 43375
            21.33.29  Congruential equations   congtr 43393
            21.33.30  Alternating congruential equations   acongid 43403
            21.33.31  Additional theorems on integer divisibility   coprmdvdsb 43413
            21.33.32  X and Y sequences 3: Divisibility properties   jm2.18 43416
            21.33.33  X and Y sequences 4: Diophantine representability of Y   jm2.27a 43433
            21.33.34  X and Y sequences 5: Diophantine representability of X, ^, _C   rmxdiophlem 43443
            21.33.35  Uncategorized stuff not associated with a major project   setindtr 43452
            21.33.36  More equivalents of the Axiom of Choice   axac10 43461
            21.33.37  Finitely generated left modules   clfig 43495
            21.33.38  Noetherian left modules I   clnm 43503
            21.33.39  Addenda for structure powers   pwssplit4 43517
            21.33.40  Every set admits a group structure iff choice   unxpwdom3 43523
            21.33.41  Noetherian rings and left modules II   clnr 43537
            21.33.42  Hilbert's Basis Theorem   cldgis 43549
            21.33.43  Additional material on polynomials [DEPRECATED]   cmnc 43559
            21.33.44  Degree and minimal polynomial of algebraic numbers   cdgraa 43568
            21.33.45  Algebraic integers I   citgo 43585
            21.33.46  Endomorphism algebra   cmend 43599
            21.33.47  Cyclic groups and order   idomodle 43619
            21.33.48  Cyclotomic polynomials   ccytp 43625
            21.33.49  Miscellaneous topology   fgraphopab 43631
      ​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 43645
            21.36.2  Natural addition of Cantor normal forms   oawordex2 43754
            21.36.3  Surreal Contributions   abeqabi 43835
            21.36.4  Short Studies   nlimsuc 43868
                  21.36.4.1  Additional work on conditional logical operator   ifpan123g 43886
                  21.36.4.2  Sophisms   rp-fakeimass 43939
                  ​*21.36.4.3  Finite Sets   rp-isfinite5 43944
                  21.36.4.4  General Observations   intabssd 43946
                  21.36.4.5  Infinite Sets   pwelg 43987
                  ​*21.36.4.6  Finite intersection property   fipjust 43992
                  21.36.4.7  RP ADDTO: Subclasses and subsets   rababg 44001
                  21.36.4.8  RP ADDTO: The intersection of a class   elinintab 44002
                  21.36.4.9  RP ADDTO: Theorems requiring subset and intersection existence   elinintrab 44004
                  21.36.4.10  RP ADDTO: Relations   xpinintabd 44007
                  ​*21.36.4.11  RP ADDTO: Functions   elmapintab 44023
                  ​*21.36.4.12  RP ADDTO: Finite induction (for finite ordinals)   cnvcnvintabd 44027
                  21.36.4.13  RP ADDTO: First and second members of an ordered pair   elcnvlem 44028
                  21.36.4.14  RP ADDTO: The reflexive and transitive properties of relations   undmrnresiss 44031
                  21.36.4.15  RP ADDTO: Basic properties of closures   cleq2lem 44035
                  21.36.4.16  RP REPLACE: Definitions and basic properties of transitive closures   trcleq2lemRP 44057
                  ​*21.36.4.17  Additions for square root; absolute value   sqrtcvallem1 44058
            21.36.5  Additional statements on relations and subclasses   al3im 44074
                  21.36.5.1  Transitive relations (not to be confused with transitive classes)   trrelind 44092
                  21.36.5.2  Reflexive closures   crcl 44099
                  ​*21.36.5.3  Finite relationship composition   relexp2 44104
                  21.36.5.4  Transitive closure of a relation   dftrcl3 44147
                  ​*21.36.5.5  Adapted from Frege   frege77d 44173
            ​*21.36.6  Propositions from _Begriffsschrift_   dfxor4 44193
                  *21.36.6.1  _Begriffsschrift_ Chapter I   dfxor4 44193
                  ​*21.36.6.2  _Begriffsschrift_ Notation hints   whe 44199
                  21.36.6.3  _Begriffsschrift_ Chapter II Implication   ax-frege1 44217
                  21.36.6.4  _Begriffsschrift_ Chapter II Implication and Negation   axfrege28 44256
                  ​*21.36.6.5  _Begriffsschrift_ Chapter II with logical equivalence   axfrege52a 44283
                  21.36.6.6  _Begriffsschrift_ Chapter II with equivalence of sets   axfrege52c 44314
                  ​*21.36.6.7  _Begriffsschrift_ Chapter II with equivalence of classes   frege53c 44341
                  ​*21.36.6.8  _Begriffsschrift_ Chapter III Properties hereditary in a sequence   dffrege69 44359
                  ​*21.36.6.9  _Begriffsschrift_ Chapter III Following in a sequence   dffrege76 44366
                  ​*21.36.6.10  _Begriffsschrift_ Chapter III Member of sequence   dffrege99 44389
                  ​*21.36.6.11  _Begriffsschrift_ Chapter III Single-valued procedures   dffrege115 44405
            ​*21.36.7  Exploring Topology via Seifert and Threlfall   enrelmap 44424
                  *21.36.7.1  Equinumerosity of sets of relations and maps   enrelmap 44424
                  ​*21.36.7.2  Generic Pseudoclosure Spaces, Pseudointerior Spaces, and Pseudoneighborhoods   or3or 44450
                  ​*21.36.7.3  Generic Neighborhood Spaces   gneispa 44557
            ​*21.36.8  Exploring Higher Homotopy via Kerodon   k0004lem1 44574
                  *21.36.8.1  Simplicial Sets   k0004lem1 44574
      ​21.37  Mathbox for Stanislas Polu
            21.37.1  IMO Problems   wwlemuld 44583
                  21.37.1.1  IMO 1972 B2   wwlemuld 44583
            ​*21.37.2  INT Inequalities Proof Generator   int-addcomd 44600
            ​*21.37.3  N-Digit Addition Proof Generator   unitadd 44622
            21.37.4  AM-GM (for k = 2,3,4)   gsumws3 44623
      ​21.38  Mathbox for Rohan Ridenour
            21.38.1  Misc   spALT 44628
            21.38.2  Monoid rings   cmnring 44638
            21.38.3  Shorter primitive equivalent of ax-groth   gru0eld 44656
                  21.38.3.1  Grothendieck universes are closed under collection   gru0eld 44656
                  21.38.3.2  Minimal universes   ismnu 44688
                  21.38.3.3  Primitive equivalent of ax-groth   expandan 44715
      ​21.39  Mathbox for Steve Rodriguez
            21.39.1  Miscellanea   nanorxor 44732
            21.39.2  Ratio test for infinite series convergence and divergence   dvgrat 44739
            21.39.3  Multiples   reldvds 44742
            21.39.4  Function operations   caofcan 44750
            21.39.5  Calculus   lhe4.4ex1a 44756
            21.39.6  The generalized binomial coefficient operation   cbcc 44763
            21.39.7  Binomial series   uzmptshftfval 44773
      ​21.40  Mathbox for Andrew Salmon
            21.40.1  Principia Mathematica * 10   pm10.12 44785
            21.40.2  Principia Mathematica * 11   2alanimi 44799
            21.40.3  Predicate Calculus   sbeqal1 44825
            21.40.4  Principia Mathematica * 13 and * 14   pm13.13a 44834
            21.40.5  Set Theory   elnev 44864
            21.40.6  Arithmetic   addcomgi 44882
            21.40.7  Geometry   cplusr 44883
      ​​*21.41  Mathbox for Alan Sare
            21.41.1  Auxiliary theorems for the Virtual Deduction tool   idiALT 44905
            21.41.2  Supplementary unification deductions   bi1imp 44909
            21.41.3  Conventional Metamath proofs, some derived from VD proofs   iidn3 44928
            21.41.4  What is Virtual Deduction?   wvd1 44996
            21.41.5  Virtual Deduction Theorems   df-vd1 44997
            21.41.6  Theorems proved using Virtual Deduction   trsspwALT 45244
            21.41.7  Theorems proved using Virtual Deduction with mmj2 assistance   simplbi2VD 45272
            21.41.8  Virtual Deduction transcriptions of textbook proofs   sb5ALTVD 45339
            21.41.9  Theorems proved using conjunction-form Virtual Deduction   elpwgdedVD 45343
            21.41.10  Theorems with a VD proof in conventional notation derived from a VD proof   suctrALT3 45350
            ​*21.41.11  Theorems with a proof in conventional notation derived from a VD proof   notnotrALT2 45353
      ​21.42  Mathbox for Eric Schmidt
            21.42.1  Miscellany   rspesbcd 45364
            21.42.2  Study of dfbi1ALT   dfbi1ALTa 45366
            21.42.3  Relation-preserving functions   wrelp 45369
            21.42.4  Orbits   orbitex 45382
            21.42.5  Well-founded sets   trwf 45386
            21.42.6  Absoluteness in transitive models   ralabso 45395
            21.42.7  Lemmas for showing axioms hold in models   traxext 45404
            21.42.8  The class of well-founded sets is a model for ZFC   wfaxext 45420
            21.42.9  Permutation models   brpermmodel 45430
      ​21.43  Mathbox for Glauco Siliprandi
            21.43.1  Miscellanea   evth2f 45446
            21.43.2  Functions   fnresdmss 45598
            21.43.3  Ordering on real numbers - Real and complex numbers basic operations   sub2times 45706
            21.43.4  Real intervals   gtnelioc 45921
            21.43.5  Finite sums   fsummulc1f 46001
            21.43.6  Finite multiplication of numbers and finite multiplication of functions   fmul01 46010
            21.43.7  Limits   clim1fr1 46031
                  21.43.7.1  Inferior limit (lim inf)   clsi 46179
                  ​*21.43.7.2  Limits for sequences of extended real numbers   clsxlim 46246
            21.43.8  Trigonometry   coseq0 46292
            21.43.9  Continuous Functions   mulcncff 46298
            21.43.10  Derivatives   dvsinexp 46339
            21.43.11  Integrals   itgsin0pilem1 46378
            21.43.12  Stone Weierstrass theorem - real version   stoweidlem1 46429
            21.43.13  Wallis' product for π   wallispilem1 46493
            21.43.14  Stirling's approximation formula for ` n ` factorial   stirlinglem1 46502
            21.43.15  Dirichlet kernel   dirkerval 46519
            21.43.16  Fourier Series   fourierdlem1 46536
            21.43.17  e is transcendental   elaa2lem 46661
            21.43.18  n-dimensional Euclidean space   rrxtopn 46712
            21.43.19  Basic measure theory   csalg 46736
                  ​*21.43.19.1  σ-Algebras   csalg 46736
                  21.43.19.2  Sum of nonnegative extended reals   csumge0 46790
                  ​*21.43.19.3  Measures   cmea 46877
                  ​*21.43.19.4  Outer measures and Caratheodory's construction   come 46917
                  ​*21.43.19.5  Lebesgue measure on n-dimensional Real numbers   covoln 46964
                  ​*21.43.19.6  Measurable functions   csmblfn 47123
      ​21.44  Mathbox for Saveliy Skresanov
            21.44.1  Ceva's theorem   sigarval 47278
            21.44.2  Simple groups   simpcntrab 47298
      ​21.45  Mathbox for Ender Ting
            21.45.1  Increasing sequences and subsequences   et-ltneverrefl 47299
            21.45.2  Scratchpad for number theory   evenwodadd 47315
            21.45.3  Scratchpad for math on real numbers   squeezedltsq 47316
      ​21.46  Mathbox for Jarvin Udandy
      ​21.47  Mathbox for Adhemar
            ​*21.47.1  Minimal implicational calculus   adh-minim 47443
      ​21.48  Mathbox for Alexander van der Vekens
            21.48.1  General auxiliary theorems (1)   n0nsn2el 47467
                  21.48.1.1  Unordered and ordered pairs - extension for singletons   n0nsn2el 47467
                  21.48.1.2  Unordered and ordered pairs - extension for unordered pairs   elprneb 47471
                  21.48.1.3  Unordered and ordered pairs - extension for ordered pairs   oppr 47472
                  21.48.1.4  Relations - extension   eubrv 47477
                  21.48.1.5  Definite description binder (inverted iota) - extension   iota0def 47480
                  21.48.1.6  Functions - extension   fveqvfvv 47482
            21.48.2  Alternative for Russell's definition of a description binder   caiota 47525
            21.48.3  Double restricted existential uniqueness   r19.32 47540
                  21.48.3.1  Restricted quantification (extension)   r19.32 47540
                  21.48.3.2  Restricted uniqueness and "at most one" quantification   reuf1odnf 47549
                  21.48.3.3  Analogs to Existential uniqueness (double quantification)   2reu3 47552
                  21.48.3.4  Additional theorems for double restricted existential uniqueness   2reu8i 47555
            ​*21.48.4  Alternative definitions of function and operation values   wdfat 47558
                  21.48.4.1  Restricted quantification (extension)   ralbinrald 47564
                  21.48.4.2  The universal class (extension)   nvelim 47565
                  21.48.4.3  Introduce the Axiom of Power Sets (extension)   alneu 47566
                  21.48.4.4  Predicate "defined at"   dfateq12d 47568
                  21.48.4.5  Alternative definition of the value of a function   dfafv2 47574
                  21.48.4.6  Alternative definition of the value of an operation   aoveq123d 47620
            ​*21.48.5  Alternative definitions of function values (2)   cafv2 47650
            21.48.6  General auxiliary theorems (2)   an4com24 47710
                  21.48.6.1  Logical conjunction - extension   an4com24 47710
                  21.48.6.2  Abbreviated conjunction and disjunction of three wff's - extension   3an4ancom24 47711
                  21.48.6.3  Negated membership (alternative)   cnelbr 47713
                  21.48.6.4  The empty set - extension   ralralimp 47720
                  21.48.6.5  Indexed union and intersection - extension   otiunsndisjX 47721
                  21.48.6.6  Functions - extension   fvifeq 47722
                  21.48.6.7  Maps-to notation - extension   fvmptrab 47734
                  21.48.6.8  Subtraction - extension   cnambpcma 47736
                  21.48.6.9  Ordering on reals (cont.) - extension   leaddsuble 47739
                  21.48.6.10  Imaginary and complex number properties - extension   readdcnnred 47745
                  21.48.6.11  Nonnegative integers (as a subset of complex numbers) - extension   nn0resubcl 47750
                  21.48.6.12  Integers (as a subset of complex numbers) - extension   zgeltp1eq 47751
                  21.48.6.13  Decimal arithmetic - extension   1t10e1p1e11 47752
                  21.48.6.14  Upper sets of integers - extension   eluzge0nn0 47754
                  21.48.6.15  Infinity and the extended real number system (cont.) - extension   nltle2tri 47755
                  21.48.6.16  Finite intervals of integers - extension   ssfz12 47756
                  21.48.6.17  Half-open integer ranges - extension   fzopred 47765
                  21.48.6.18  The floor and ceiling functions - extension   2ltceilhalf 47774
                  21.48.6.19  The modulo (remainder) operation - extension   fldivmod 47786
                  21.48.6.20  The infinite sequence builder "seq"   smonoord 47819
                  21.48.6.21  Integer powers - extension   2timesltsq 47820
                  21.48.6.22  Finite and infinite sums - extension   fsummsndifre 47822
                  21.48.6.23  The divides relation - extension   nndivides2 47826
                  21.48.6.24  Extensible structures - extension   setsidel 47830
            ​*21.48.7  Preimages of function values   preimafvsnel 47833
            ​*21.48.8  Partitions of real intervals   ciccp 47867
            21.48.9  Shifting functions with an integer range domain   fargshiftfv 47893
            21.48.10  Words over a set (extension)   lswn0 47898
                  21.48.10.1  Last symbol of a word - extension   lswn0 47898
            21.48.11  Unordered pairs   wich 47899
                  21.48.11.1  Interchangeable setvar variables   wich 47899
                  21.48.11.2  Set of unordered pairs   sprid 47928
                  ​*21.48.11.3  Proper (unordered) pairs   prpair 47955
                  21.48.11.4  Set of proper unordered pairs   cprpr 47966
            21.48.12  Number theory (extension)   nprmmul1 47981
                  21.48.12.1  Properties of non-prime numbers   nprmmul1 47981
                  ​*21.48.12.2  Fermat numbers   cfmtno 47984
                  ​*21.48.12.3  Mersenne primes   m2prm 48048
                  21.48.12.4  Proth's theorem   modexp2m1d 48069
                  21.48.12.5  The prime-counting function according to Ján Mináč   nprmdvdsfacm1lem1 48077
                  21.48.12.6  Solutions of quadratic equations   quad1 48090
            ​*21.48.13  Even and odd numbers   ceven 48094
                  21.48.13.1  Definitions and basic properties   ceven 48094
                  21.48.13.2  Alternate definitions using the "divides" relation   dfeven2 48119
                  21.48.13.3  Alternate definitions using the "modulo" operation   dfeven3 48128
                  21.48.13.4  Alternate definitions using the "gcd" operation   iseven5 48134
                  21.48.13.5  Theorems of part 5 revised   zneoALTV 48139
                  21.48.13.6  Theorems of part 6 revised   odd2np1ALTV 48144
                  21.48.13.7  Theorems of AV's mathbox revised   0evenALTV 48158
                  21.48.13.8  Additional theorems   epoo 48173
                  21.48.13.9  Perfect Number Theorem (revised)   perfectALTVlem1 48191
            21.48.14  Number theory (extension 2)   cfppr 48194
                  ​*21.48.14.1  Fermat pseudoprimes   cfppr 48194
                  ​*21.48.14.2  Goldbach's conjectures   cgbe 48215
            21.48.15  Graph theory (extension)   cclnbgr 48288
                  21.48.15.1  Closed neighborhood of a vertex   cclnbgr 48288
                  ​*21.48.15.2  Semiclosed and semiopen neighborhoods (experimental)   dfsclnbgr2 48316
                  21.48.15.3  Induced subgraphs   cisubgr 48330
                  ​*21.48.15.4  Isomorphisms of graphs   cgrisom 48344
                  ​*21.48.15.5  Triangles in graphs   cgrtri 48407
                  ​*21.48.15.6  Star graphs   cstgr 48421
                  ​*21.48.15.7  Local isomorphisms of graphs   cgrlim 48446
                  ​*21.48.15.8  Generalized Petersen graphs   cgpg 48510
                  21.48.15.9  Loop-free graphs - extension   1hegrlfgr 48602
                  21.48.15.10  Walks - extension   cupwlks 48603
                  21.48.15.11  Edges of graphs expressed as sets of unordered pairs   upgredgssspr 48613
            21.48.16  Monoids (extension)   ovn0dmfun 48626
                  21.48.16.1  Auxiliary theorems   ovn0dmfun 48626
                  21.48.16.2  Magmas, Semigroups and Monoids (extension)   plusfreseq 48634
                  21.48.16.3  Examples and counterexamples for magmas, semigroups and monoids (extension)   opmpoismgm 48637
                  21.48.16.4  Group sum operation (extension 1)   gsumsplit2f 48650
            ​*21.48.17  Magmas and internal binary operations (alternate approach)   ccllaw 48653
                  *21.48.17.1  Laws for internal binary operations   ccllaw 48653
                  ​*21.48.17.2  Internal binary operations   cintop 48666
                  21.48.17.3  Alternative definitions for magmas and semigroups   cmgm2 48685
            21.48.18  Rings (extension)   lmod0rng 48699
                  21.48.18.1  Nonzero rings (extension)   lmod0rng 48699
                  21.48.18.2  Ideals as non-unital rings   lidldomn1 48701
                  21.48.18.3  The non-unital ring of even integers   0even 48707
                  21.48.18.4  A constructed not unital ring   cznrnglem 48729
                  ​*21.48.18.5  The category of non-unital rings (alternate definition)   crngcALTV 48733
                  ​*21.48.18.6  The category of (unital) rings (alternate definition)   cringcALTV 48757
            21.48.19  Basic algebraic structures (extension)   eliunxp2 48804
                  21.48.19.1  Auxiliary theorems   eliunxp2 48804
                  21.48.19.2  The binomial coefficient operation (extension)   bcpascm1 48821
                  21.48.19.3  The ` ZZ `-module ` ZZ X. ZZ `   zlmodzxzlmod 48824
                  21.48.19.4  Group sum operation (extension 2)   mgpsumunsn 48831
                  21.48.19.5  Symmetric groups (extension)   exple2lt6 48834
                  21.48.19.6  Divisibility (extension)   invginvrid 48837
                  21.48.19.7  The support of functions (extension)   rmsupp0 48838
                  21.48.19.8  Finitely supported functions (extension)   rmsuppfi 48842
                  21.48.19.9  Left modules (extension)   lmodvsmdi 48849
                  21.48.19.10  Associative algebras (extension)   assaascl0 48851
                  21.48.19.11  Univariate polynomials (extension)   ply1vr1smo 48853
                  21.48.19.12  Univariate polynomials (examples)   linply1 48863
            21.48.20  Linear algebra (extension)   cdmatalt 48866
                  ​*21.48.20.1  The subalgebras of diagonal and scalar matrices (extension)   cdmatalt 48866
                  ​*21.48.20.2  Linear combinations   clinc 48874
                  ​*21.48.20.3  Linear independence   clininds 48910
                  21.48.20.4  Simple left modules and the ` ZZ `-module   lmod1lem1 48957
                  21.48.20.5  Differences between (left) modules and (left) vector spaces   lvecpsslmod 48977
            21.48.21  Complexity theory   suppdm 48980
                  21.48.21.1  Auxiliary theorems   suppdm 48980
                  21.48.21.2  Even and odd integers   nn0onn0ex 48993
                  21.48.21.3  The natural logarithm on complex numbers (extension)   logcxp0 49005
                  21.48.21.4  Division of functions   cfdiv 49007
                  21.48.21.5  Upper bounds   cbigo 49017
                  21.48.21.6  Logarithm to an arbitrary base (extension)   rege1logbrege0 49028
                  ​*21.48.21.7  The binary logarithm   fldivexpfllog2 49035
                  21.48.21.8  Binary length   cblen 49039
                  ​*21.48.21.9  Digits   cdig 49065
                  21.48.21.10  Nonnegative integer as sum of its shifted digits   dignn0flhalflem1 49085
                  21.48.21.11  Algorithms for the multiplication of nonnegative integers   nn0mulfsum 49094
                  ​*21.48.21.12  N-ary functions   cnaryf 49096
                  ​*21.48.21.13  The Ackermann function   citco 49127
            21.48.22  Elementary geometry (extension)   fv1prop 49169
                  21.48.22.1  Auxiliary theorems   fv1prop 49169
                  21.48.22.2  Real euclidean space of dimension 2   rrx2pxel 49181
                  21.48.22.3  Spheres and lines in real Euclidean spaces   cline 49197
      ​21.49  Mathbox for Zhi Wang
            21.49.1  Propositional calculus   pm4.71da 49259
            21.49.2  Predicate calculus with equality   dtrucor3 49268
                  21.49.2.1  Axiom scheme ax-5 (Distinctness)   dtrucor3 49268
            21.49.3  ZF Set Theory - start with the Axiom of Extensionality   ralbidb 49269
                  21.49.3.1  Restricted quantification   ralbidb 49269
                  21.49.3.2  The universal class   reuxfr1dd 49276
                  21.49.3.3  The empty set   ssdisjd 49277
                  21.49.3.4  Unordered and ordered pairs   vsn 49281
                  21.49.3.5  The union of a class   unilbss 49287
                  21.49.3.6  Indexed union and intersection   iuneq0 49288
            21.49.4  ZF Set Theory - add the Axiom of Replacement   inpw 49294
                  21.49.4.1  Theorems requiring subset and intersection existence   inpw 49294
            21.49.5  ZF Set Theory - add the Axiom of Power Sets   opth1neg 49295
                  21.49.5.1  Ordered pair theorem   opth1neg 49295
                  21.49.5.2  Ordered-pair class abstractions (cont.)   brab2dd 49297
                  21.49.5.3  Relations   iinxp 49300
                  21.49.5.4  Functions   mof0 49307
                  21.49.5.5  Operations   ovsng 49327
            21.49.6  ZF Set Theory - add the Axiom of Union   fonex 49336
                  21.49.6.1  Relations and functions (cont.)   fonex 49336
                  21.49.6.2  First and second members of an ordered pair   eloprab1st2nd 49337
                  21.49.6.3  Operations in maps-to notation (continued)   fmpodg 49338
                  21.49.6.4  Function transposition   resinsnlem 49340
                  21.49.6.5  Infinite Cartesian products   ixpv 49359
                  21.49.6.6  Equinumerosity   fvconst0ci 49360
            21.49.7  Order sets   iccin 49365
                  21.49.7.1  Real number intervals   iccin 49365
            21.49.8  Extensible structures   slotresfo 49368
                  21.49.8.1  Basic definitions   slotresfo 49368
            21.49.9  Moore spaces   mreuniss 49369
            ​*21.49.10  Topology   clduni 49370
                  21.49.10.1  Closure and interior   clduni 49370
                  21.49.10.2  Neighborhoods   neircl 49374
                  21.49.10.3  Subspace topologies   restcls2lem 49382
                  21.49.10.4  Limits and continuity in topological spaces   cnneiima 49386
                  21.49.10.5  Topological definitions using the reals   iooii 49387
                  21.49.10.6  Separated sets   sepnsepolem1 49391
                  21.49.10.7  Separated spaces: T0, T1, T2 (Hausdorff) ...   isnrm4 49400
            21.49.11  Preordered sets and directed sets using extensible structures   isprsd 49424
            21.49.12  Posets and lattices using extensible structures   lubeldm2 49425
                  21.49.12.1  Posets   lubeldm2 49425
                  21.49.12.2  Lattices   toslat 49451
                  21.49.12.3  Subset order structures   intubeu 49453
            21.49.13  Rings   elmgpcntrd 49474
                  21.49.13.1  Multiplicative Group   elmgpcntrd 49474
            21.49.14  Associative algebras   asclelbasALT 49475
                  21.49.14.1  Definition and basic properties   asclelbasALT 49475
            21.49.15  Categories   homf0 49478
                  21.49.15.1  Categories   homf0 49478
                  21.49.15.2  Opposite category   oppccatb 49485
                  21.49.15.3  Monomorphisms and epimorphisms   idmon 49489
                  21.49.15.4  Sections, inverses, isomorphisms   sectrcl 49491
                  21.49.15.5  Isomorphic objects   cicfn 49511
                  21.49.15.6  Subcategories   dmdm 49522
                  21.49.15.7  Functors   reldmfunc 49544
                  21.49.15.8  Opposite functors   coppf 49591
                  21.49.15.9  Full & faithful functors   imasubc 49620
                  21.49.15.10  Universal property   upciclem1 49635
                  21.49.15.11  Natural transformations and the functor category   isnatd 49692
                  21.49.15.12  Initial, terminal and zero objects of a category   initoo2 49701
                  21.49.15.13  Product of categories   reldmxpc 49715
                  21.49.15.14  Swap functors   cswapf 49728
                  21.49.15.15  Functor evaluation   oppc1stflem 49756
                  21.49.15.16  Transposed curry functors   cofuswapfcl 49762
                  21.49.15.17  Constant functors   diag1 49773
                  21.49.15.18  Functor composition bifunctors   fucofulem1 49779
                  21.49.15.19  Post-composition functors   postcofval 49833
                  21.49.15.20  Pre-composition functors   precofvallem 49835
            21.49.16  Examples of categories   catcrcl 49864
                  21.49.16.1  The category of categories   catcrcl 49864
                  21.49.16.2  Thin categories   cthinc 49886
                  21.49.16.3  Terminal categories   ctermc 49941
                  21.49.16.4  Preordered sets as thin categories   cprstc 50018
                  21.49.16.5  Monoids as categories   cmndtc 50046
                  21.49.16.6  Categories with at most one object and at most two morphisms   2arwcatlem1 50064
            21.49.17  Kan extensions and related concepts   clan 50074
                  21.49.17.1  Kan extensions   clan 50074
                  21.49.17.2  Limits and colimits   clmd 50112
      ​21.50  Mathbox for Emmett Weisz
            ​*21.50.1  Miscellaneous Theorems   nfintd 50142
            21.50.2  Set Recursion   csetrecs 50152
                  ​*21.50.2.1  Basic Properties of Set Recursion   csetrecs 50152
                  21.50.2.2  Examples and properties of set recursion   elsetrecslem 50168
            ​*21.50.3  Construction of Games and Surreal Numbers   cpg 50178
      ​​*21.51  Mathbox for David A. Wheeler
            21.51.1  Natural deduction   sbidd 50187
            ​*21.51.2  Greater than, greater than or equal to   cge-real 50189
            ​*21.51.3  Hyperbolic trigonometric functions   csinh 50199
            ​*21.51.4  Reciprocal trigonometric functions (sec, csc, cot)   csec 50210
            ​*21.51.5  Identities for "if"   ifnmfalse 50232
            ​*21.51.6  Logarithms generalized to arbitrary base using ` logb `   logb2aval 50233
            ​*21.51.7  Logarithm laws generalized to an arbitrary base - log_   clog- 50234
            ​*21.51.8  Formally define notions such as reflexivity   wreflexive 50236
            ​*21.51.9  Algebra helpers   mvlraddi 50240
            ​*21.51.10  Algebra helper examples   i2linesi 50247
            ​*21.51.11  Formal methods "surprises"   alimp-surprise 50249
            ​*21.51.12  Allsome quantifier   walsi 50255
            ​*21.51.13  Miscellaneous   5m4e1 50266
            21.51.14  Theorems about algebraic numbers   aacllem 50270
      ​21.52  Mathbox for Kunhao Zheng
            21.52.1  Weighted AM-GM inequality   amgmwlem 50271