TOC of Theorem List - Metamath Proof Explorer

Table of Contents Summary

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

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

​​*PART 1  CLASSICAL FIRST-ORDER LOGIC WITH EQUALITY
      ​*1.1  Pre-logic
            *1.1.1  Inferences for assisting proof development   idi 1
      ​​*1.2  Propositional calculus
            1.2.1  Recursively define primitive wffs for propositional calculus   wn 3
            ​*1.2.2  The axioms of propositional calculus   ax-mp 5
            ​*1.2.3  Logical implication   mp2 9
            ​*1.2.4  Logical negation   con4 114
            ​*1.2.5  Logical equivalence   wb 209
            ​*1.2.6  Logical conjunction   wa 401
            ​*1.2.7  Logical disjunction   wo 861
            ​*1.2.8  Mixed connectives   jaao 969
            ​*1.2.9  The conditional operator for propositions   wif 1078
            ​*1.2.10  The weak deduction theorem for propositional calculus   elimh 1099
            1.2.11  Abbreviated conjunction and disjunction of three wff's   w3o 1102
            1.2.12  Logical "nand" (Sheffer stroke)   wnan 1521
            1.2.13  Logical "xor"   wxo 1541
            1.2.14  Logical "nor"   wnor 1558
            1.2.15  True and false constants   wal 1568
                  ​*1.2.15.1  Universal quantifier for use by df-tru   wal 1568
                  ​*1.2.15.2  Equality predicate for use by df-tru   cv 1569
                  1.2.15.3  The true constant   wtru 1571
                  1.2.15.4  The false constant   wfal 1582
            ​*1.2.16  Truth tables   truimtru 1593
                  1.2.16.1  Implication   truimtru 1593
                  1.2.16.2  Negation   nottru 1597
                  1.2.16.3  Equivalence   trubitru 1599
                  1.2.16.4  Conjunction   truantru 1603
                  1.2.16.5  Disjunction   truortru 1607
                  1.2.16.6  Alternative denial   trunantru 1611
                  1.2.16.7  Exclusive disjunction   truxortru 1615
                  1.2.16.8  Joint denial   trunortru 1619
            ​*1.2.17  Half adder and full adder in propositional calculus   whad 1623
                  1.2.17.1  Full adder: sum   whad 1623
                  1.2.17.2  Full adder: carry   wcad 1639
      ​1.3  Other axiomatizations related to classical propositional calculus
            ​*1.3.1  Minimal implicational calculus   minimp 1654
            ​*1.3.2  Implicational Calculus   impsingle 1660
            1.3.3  Derive the Lukasiewicz axioms from Meredith's sole axiom   meredith 1674
            1.3.4  Derive the standard axioms from the Lukasiewicz axioms   luklem1 1691
            ​*1.3.5  Derive Nicod's axiom from the standard axioms   nic-dfim 1702
            1.3.6  Derive the Lukasiewicz axioms from Nicod's axiom   nic-imp 1708
            1.3.7  Derive Nicod's Axiom from Lukasiewicz's First Sheffer Stroke Axiom   lukshef-ax1 1727
            1.3.8  Derive the Lukasiewicz Axioms from the Tarski-Bernays-Wajsberg Axioms   tbw-bijust 1731
            1.3.9  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's First CO Axiom   merco1 1746
            1.3.10  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's Second CO Axiom   merco2 1769
            1.3.11  Derive the Lukasiewicz axioms from the Russell-Bernays Axioms   rb-bijust 1782
            ​*1.3.12  Stoic logic non-modal portion (Chrysippus of Soli)   mptnan 1801
      ​​*1.4  Predicate calculus with equality: Tarski's system S2 (1 rule, 6 schemes)
            *1.4.1  Universal quantifier (continued); define "exists" and "not free"   wex 1812
                  1.4.1.1  Existential quantifier   wex 1812
                  1.4.1.2  Nonfreeness predicate   wnf 1816
            1.4.2  Rule scheme ax-gen (Generalization)   ax-gen 1828
            1.4.3  Axiom scheme ax-4 (Quantified Implication)   ax-4 1842
                  ​*1.4.3.1  The empty domain of discourse   empty 1939
            1.4.4  Axiom scheme ax-5 (Distinctness) - first use of $d   ax-5 1943
            ​*1.4.5  Equality predicate (continued)   weq 1995
            1.4.6  Axiom scheme ax-6 (Existence)   ax-6 2000
            1.4.7  Axiom scheme ax-7 (Equality)   ax-7 2041
            1.4.8  Define proper substitution   sbjust 2098
            1.4.9  Membership predicate   wcel 2146
            1.4.10  Axiom scheme ax-8 (Left Equality for Binary Predicate)   ax-8 2148
            1.4.11  Axiom scheme ax-9 (Right Equality for Binary Predicate)   ax-9 2156
            ​*1.4.12  Logical redundancy of ax-10 , ax-11 , ax-12 , ax-13   ax6dgen 2166
      ​​*1.5  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes)
            1.5.1  Axiom scheme ax-10 (Quantified Negation)   ax-10 2179
            1.5.2  Axiom scheme ax-11 (Quantifier Commutation)   ax-11 2195
            1.5.3  Axiom scheme ax-12 (Substitution)   ax-12 2216
            1.5.4  Axiom scheme ax-13 (Quantified Equality)   ax-13 2407
      ​1.6  Uniqueness and unique existence
            1.6.1  Uniqueness: the at-most-one quantifier   wmo 2568
            1.6.2  Unique existence: the unique existential quantifier   weu 2599
      ​1.7  Other axiomatizations related to classical predicate calculus
            ​*1.7.1  Aristotelian logic: Assertic syllogisms   barbara 2693
            ​*1.7.2  Intuitionistic logic   axia1 2723
​​*PART 2  ZF (ZERMELO-FRAENKEL) SET THEORY
      ​2.1  ZF Set Theory - start with the Axiom of Extensionality
            2.1.1  Introduce the Axiom of Extensionality   ax-ext 2738
            2.1.2  Classes   cab 2744
                  2.1.2.1  Class abstractions   cab 2744
                  ​*2.1.2.2  Class equality   df-cleq 2758
                  2.1.2.3  Class membership   df-clel 2841
                  2.1.2.4  Elementary properties of class abstractions   eqabdv 2899
            2.1.3  Class form not-free predicate   wnfc 2913
            2.1.4  Negated equality and membership   wne 2961
                  2.1.4.1  Negated equality   wne 2961
                  2.1.4.2  Negated membership   wnel 3067
            2.1.5  Restricted quantification   wral 3082
                  2.1.5.1  Restricted universal and existential quantification   wral 3082
                  2.1.5.2  Restricted existential uniqueness and at-most-one quantifier   wreu 3370
                  2.1.5.3  Restricted class abstraction   crab 3419
            2.1.6  The universal class   cvv 3458
            ​*2.1.7  Conditional equality (experimental)   wcdeq 3729
            2.1.8  Russell's Paradox   rru 3745
            2.1.9  Proper substitution of classes for sets   wsbc 3747
            2.1.10  Proper substitution of classes for sets into classes   csb 3856
            2.1.11  Define basic set operations and relations   cdif 3905
            2.1.12  Subclasses and subsets   df-ss 3925
            2.1.13  The difference, union, and intersection of two classes   dfdif3 4075
                  2.1.13.1  The difference of two classes   dfdif3 4075
                  2.1.13.2  The union of two classes   elun 4110
                  2.1.13.3  The intersection of two classes   elini 4155
                  2.1.13.4  The symmetric difference of two classes   csymdif 4208
                  2.1.13.5  Combinations of difference, union, and intersection of two classes   unabs 4221
                  2.1.13.6  Class abstractions with difference, union, and intersection of two classes   unabw 4263
                  2.1.13.7  Restricted uniqueness with difference, union, and intersection   reuun2 4281
            2.1.14  The empty set   c0 4289
            ​*2.1.15  The conditional operator for classes   cif 4490
            ​*2.1.16  The weak deduction theorem for set theory   dedth 4549
            2.1.17  Power classes   cpw 4565
            2.1.18  Unordered and ordered pairs   snjust 4591
            2.1.19  The union of a class   cuni 4875
            2.1.20  The intersection of a class   cint 4915
            2.1.21  Indexed union and intersection   ciun 4959
            2.1.22  Disjointness   wdisj 5079
            2.1.23  Binary relations   wbr 5112
            2.1.24  Ordered-pair class abstractions (class builders)   copab 5176
            2.1.25  Functions in maps-to notation   cmpt 5195
            2.1.26  Transitive classes   wtr 5221
      ​2.2  ZF Set Theory - add the Axiom of Replacement
            2.2.1  Introduce the Axiom of Replacement   ax-rep 5241
            2.2.2  Derive the Axiom of Separation   axsepgfromrep 5258
            2.2.3  Derive the Null Set Axiom   axnulALT 5270
            2.2.4  Theorems requiring subset and intersection existence   exnelv 5279
            2.2.5  Theorems requiring empty set existence   class2set 5328
      ​2.3  ZF Set Theory - add the Axiom of Power Sets
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 5339
            2.3.2  Derive the Axiom of Pairing   axprlem1 5397
            2.3.3  Ordered pair theorem   opnz 5458
            2.3.4  Ordered-pair class abstractions (cont.)   opabidw 5511
            2.3.5  Power class of union and intersection   pwin 5555
            2.3.6  The identity relation   cid 5558
            2.3.7  The membership relation (or epsilon relation)   cep 5563
            ​*2.3.8  Partial and total orderings   wpo 5570
            2.3.9  Founded and well-ordering relations   wfr 5614
            2.3.10  Relations   cxp 5662
            2.3.11  The Predecessor Class   cpred 6305
            2.3.12  Well-founded induction (variant)   frpomin 6345
            2.3.13  Well-ordered induction   tz6.26 6352
            2.3.14  Ordinals   word 6363
            2.3.15  Definite description binder (inverted iota)   cio 6494
            2.3.16  Functions   wfun 6534
            2.3.17  Cantor's Theorem   canth 7370
            2.3.18  Restricted iota (description binder)   crio 7372
            2.3.19  Operations   co 7416
                  2.3.19.1  Variable-to-class conversion for operations   caovclg 7608
            2.3.20  Maps-to notation   mpondm0 7656
            2.3.21  Function operation   cof 7678
            2.3.22  Proper subset relation   crpss 7725
      ​2.4  ZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 7738
            2.4.2  Ordinals (continued)   epweon 7776
            2.4.3  Transfinite induction   tfi 7851
            2.4.4  The natural numbers (i.e., finite ordinals)   com 7864
            2.4.5  Peano's postulates   peano1 7887
            2.4.6  Finite induction (for finite ordinals)   find 7894
            2.4.7  Relations and functions (cont.)   dmexg 7900
            2.4.8  First and second members of an ordered pair   c1st 7986
            2.4.9  Induction on Cartesian products   frpoins3xpg 8138
            2.4.10  Ordering on Cartesian products   xpord2lem 8140
            2.4.11  Ordering Ordinal Sequences   orderseqlem 8155
            ​*2.4.12  The support of functions   csupp 8158
            ​*2.4.13  Special maps-to operations   opeliunxp2f 8208
            2.4.14  Function transposition   ctpos 8223
            2.4.15  Curry and uncurry   ccur 8263
            2.4.16  Undefined values   cund 8270
            2.4.17  Well-founded recursion   cfrecs 8279
            2.4.18  Well-ordered recursion   cwrecs 8310
            2.4.19  Functions on ordinals; strictly monotone ordinal functions   iunon 8328
            2.4.20  "Strong" transfinite recursion   crecs 8359
            2.4.21  Recursive definition generator   crdg 8398
            2.4.22  Finite recursion   frfnom 8424
            2.4.23  Ordinal arithmetic   c1o 8448
            2.4.24  Natural number arithmetic   nna0 8592
            2.4.25  Natural addition   cnadd 8653
            2.4.26  Equivalence relations and classes   wer 8693
            2.4.27  The mapping operation   cmap 8826
            2.4.28  Infinite Cartesian products   cixp 8897
            2.4.29  Equinumerosity   cen 8942
            2.4.30  Schroeder-Bernstein Theorem   sbthlem1 9077
            2.4.31  Equinumerosity (cont.)   xpf1o 9129
            2.4.32  Finite sets   dif1enlem 9146
            2.4.33  Pigeonhole Principle   phplem1 9190
            2.4.34  Finite sets (cont.)   onomeneq 9200
            2.4.35  Finitely supported functions   cfsupp 9323
            2.4.36  Finite intersections   cfi 9372
            2.4.37  Hall's marriage theorem   marypha1lem 9395
            2.4.38  Supremum and infimum   csup 9402
            2.4.39  Ordinal isomorphism, Hartogs's theorem   coi 9473
            2.4.40  Hartogs function   char 9520
            2.4.41  Weak dominance   cwdom 9528
      ​2.5  ZF Set Theory - add the Axiom of Regularity
            2.5.1  Introduce the Axiom of Regularity   ax-reg 9556
            2.5.2  Axiom of Infinity equivalents   inf0 9592
      ​2.6  ZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-inf 9609
            2.6.2  Existence of omega (the set of natural numbers)   omex 9614
            2.6.3  Cantor normal form   ccnf 9632
            2.6.4  Transitive closure of a relation   cttrcl 9678
            2.6.5  Transitive closure   trcl 9699
            2.6.6  Set induction (or epsilon induction)   setind 9718
            2.6.7  Well-Founded Induction   frmin 9723
            2.6.8  Well-Founded Recursion   frr3g 9730
            2.6.9  Rank   cr1 9736
            2.6.10  Scott's trick; collection principle; Hilbert's epsilon   cscott 9859
            2.6.11  Disjoint union   cdju 9895
            2.6.12  Cardinal numbers   ccrd 9932
            2.6.13  Axiom of Choice equivalents   wac 10110
            ​*2.6.14  Cardinal number arithmetic   undjudom 10162
            2.6.15  The Ackermann bijection   ackbij2lem1 10212
            2.6.16  Cofinality (without Axiom of Choice)   cflem 10239
            2.6.17  Eight inequivalent definitions of finite set   sornom 10271
            2.6.18  Hereditarily size-limited sets without Choice   itunifval 10410
​​*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 10429
            3.1.2  Introduce the Axiom of Dependent Choice   ax-dc 10440
      ​3.2  ZFC Set Theory - add the Axiom of Choice
            3.2.1  Introduce the Axiom of Choice   ax-ac 10453
            3.2.2  AC equivalents: well-ordering, Zorn's lemma   numthcor 10488
            3.2.3  Cardinal number theorems using Axiom of Choice   cardval 10540
            3.2.4  Cardinal number arithmetic using Axiom of Choice   iunctb 10569
            3.2.5  Cofinality using the Axiom of Choice   alephreg 10577
      ​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 10615
            3.4.2  Derivation of the Axiom of Choice   gchaclem 10673
​​*PART 4  TG (TARSKI-GROTHENDIECK) SET THEORY
      ​4.1  Inaccessibles
            4.1.1  Weakly and strongly inaccessible cardinals   cwina 10677
            4.1.2  Weak universes   cwun 10695
            4.1.3  Tarski classes   ctsk 10743
            4.1.4  Grothendieck universes   cgru 10785
      ​4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
            4.2.1  Introduce the Tarski-Grothendieck Axiom   ax-groth 10818
            4.2.2  Derive the Power Set, Infinity and Choice Axioms   grothpw 10821
            4.2.3  Tarski map function   ctskm 10832
​​*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 10839
            5.1.2  Final derivation of real and complex number postulates   axaddf 11140
            5.1.3  Real and complex number postulates restated as axioms   ax-cnex 11166
      ​5.2  Derive the basic properties from the field axioms
            5.2.1  Some deductions from the field axioms for complex numbers   cnex 11191
            5.2.2  Infinity and the extended real number system   cpnf 11250
            5.2.3  Restate the ordering postulates with extended real "less than"   axlttri 11291
            5.2.4  Ordering on reals   lttr 11296
            5.2.5  Initial properties of the complex numbers   mul12 11385
      ​5.3  Real and complex numbers - basic operations
            5.3.1  Addition   add12 11438
            5.3.2  Subtraction   cmin 11451
            5.3.3  Multiplication   kcnktkm1cn 11655
            5.3.4  Ordering on reals (cont.)   gt0ne0 11689
            5.3.5  Reciprocals   ixi 11853
            5.3.6  Division   cdiv 11881
            5.3.7  Ordering on reals (cont.)   elimgt0 12063
            5.3.8  Completeness Axiom and Suprema   fimaxre 12169
            5.3.9  Imaginary and complex number properties   neg1cn 12213
            5.3.10  Function operation analogue theorems   ofsubeq0 12225
            ​*5.3.11  Indicator Functions   cind 12228
      ​5.4  Integer sets
            5.4.1  Positive integers (as a subset of complex numbers)   cn 12243
            5.4.2  Principle of mathematical induction   nnind 12261
            ​*5.4.3  Decimal representation of numbers   c2 12305
            ​*5.4.4  Some properties of specific numbers   1pneg1e0 12368
            5.4.5  Simple number properties   halfcl 12480
            5.4.6  The Archimedean property   nnunb 12510
            5.4.7  Nonnegative integers (as a subset of complex numbers)   cn0 12514
            ​*5.4.8  Extended nonnegative integers   cxnn0 12587
            5.4.9  Integers (as a subset of complex numbers)   cz 12601
            5.4.10  Decimal arithmetic   cdc 12721
            5.4.11  Upper sets of integers   cuz 12872
            5.4.12  Well-ordering principle for bounded-below sets of integers   uzwo3 12977
            5.4.13  Rational numbers (as a subset of complex numbers)   cq 12982
            5.4.14  Existence of the set of complex numbers   rpnnen1lem2 13011
      ​5.5  Order sets
            5.5.1  Positive reals (as a subset of complex numbers)   crp 13026
            5.5.2  Infinity and the extended real number system (cont.)   cxne 13144
            5.5.3  Supremum and infimum on the extended reals   xrsupexmnf 13341
            5.5.4  Real number intervals   cioo 13382
            5.5.5  Finite intervals of integers   cfz 13545
            ​*5.5.6  Finite intervals of nonnegative integers   elfz2nn0 13657
            5.5.7  Half-open integer ranges   cfzo 13693
      ​5.6  Elementary integer functions
            5.6.1  The floor and ceiling functions   cfl 13834
            5.6.2  The modulo (remainder) operation   cmo 13913
            5.6.3  Miscellaneous theorems about integers   om2uz0i 13994
            5.6.4  Strong induction over upper sets of integers   uzsinds 14034
            5.6.5  Finitely supported functions over the nonnegative integers   fsuppmapnn0fiublem 14037
            5.6.6  The infinite sequence builder "seq" - extension   cseq 14048
            5.6.7  Integer powers   cexp 14108
            5.6.8  Ordered pair theorem for nonnegative integers   nn0le2msqi 14314
            5.6.9  Factorial function   cfa 14320
            5.6.10  The binomial coefficient operation   cbc 14349
            5.6.11  The ` # ` (set size) function   chash 14377
                  5.6.11.1  Proper unordered pairs and triples (sets of size 2 and 3)   hashprlei 14516
                  5.6.11.2  Functions with a domain containing at least two different elements   fundmge2nop0 14550
                  5.6.11.3  Finite induction on the size of the first component of a binary relation   hashdifsnp1 14554
      ​​*5.7  Words over a set
            5.7.1  Definitions and basic theorems   cword 14561
            5.7.2  Last symbol of a word   clsw 14610
            5.7.3  Concatenations of words   cconcat 14618
            5.7.4  Singleton words   cs1 14644
            5.7.5  Concatenations with singleton words   ccatws1cl 14665
            5.7.6  Subwords/substrings   csubstr 14689
            5.7.7  Prefixes of a word   cpfx 14719
            5.7.8  Subwords of subwords   swrdswrdlem 14752
            5.7.9  Subwords and concatenations   pfxcctswrd 14758
            5.7.10  Subwords of concatenations   swrdccatfn 14772
            5.7.11  Splicing words (substring replacement)   csplice 14797
            5.7.12  Reversing words   creverse 14806
            5.7.13  Repeated symbol words   creps 14816
            ​*5.7.14  Cyclical shifts of words   ccsh 14836
            5.7.15  Mapping words by a function   wrdco 14879
            5.7.16  Longer string literals   cs2 14889
      ​​*5.8  Reflexive and transitive closures of relations
            5.8.1  The reflexive and transitive properties of relations   coss12d 15020
            5.8.2  Basic properties of closures   cleq1lem 15030
            5.8.3  Definitions and basic properties of transitive closures   ctcl 15033
            5.8.4  Exponentiation of relations   crelexp 15067
            5.8.5  Reflexive-transitive closure as an indexed union   crtrcl 15103
            ​*5.8.6  Principle of transitive induction   relexpindlem 15111
      ​5.9  Elementary real and complex functions
            5.9.1  The "shift" operation   cshi 15114
            5.9.2  Signum (sgn or sign) function   csgn 15134
            5.9.3  Real and imaginary parts; conjugate   ccj 15158
            5.9.4  Square root; absolute value   csqrt 15295
      ​5.10  Elementary limits and convergence
            5.10.1  Superior limit (lim sup)   clsp 15532
            5.10.2  Limits   cli 15546
            5.10.3  Finite and infinite sums   csu 15748
            5.10.4  The binomial theorem   binomlem 15894
            5.10.5  The inclusion/exclusion principle   incexclem 15901
            5.10.6  Infinite sums (cont.)   isumshft 15904
            5.10.7  Miscellaneous converging and diverging sequences   divrcnv 15917
            5.10.8  Arithmetic series   arisum 15925
            5.10.9  Geometric series   expcnv 15929
            5.10.10  Ratio test for infinite series convergence   cvgrat 15948
            5.10.11  Mertens' theorem   mertenslem1 15949
            5.10.12  Finite and infinite products   prodf 15952
                  5.10.12.1  Product sequences   prodf 15952
                  5.10.12.2  Non-trivial convergence   ntrivcvg 15962
                  5.10.12.3  Complex products   cprod 15968
                  5.10.12.4  Finite products   fprod 16006
                  5.10.12.5  Infinite products   iprodclim 16063
            5.10.13  Falling and Rising Factorial   cfallfac 16069
            5.10.14  Bernoulli polynomials and sums of k-th powers   cbp 16110
      ​5.11  Elementary trigonometry
            5.11.1  The exponential, sine, and cosine functions   ce 16125
                  5.11.1.1  The circle constant (tau = 2 pi)   ctau 16268
            5.11.2  _e is irrational   eirrlem 16270
      ​5.12  Cardinality of real and complex number subsets
            5.12.1  Countability of integers and rationals   xpnnen 16277
            5.12.2  The reals are uncountable   rpnnen2lem1 16280
​​*PART 6  ELEMENTARY NUMBER THEORY
      ​6.1  Elementary properties of divisibility
            6.1.1  Irrationality of square root of 2   sqrt2irrlem 16314
            6.1.2  Some Number sets are chains of proper subsets   nthruc 16318
            6.1.3  The divides relation   cdvds 16320
            ​*6.1.4  Even and odd numbers   evenelz 16404
            6.1.5  The division algorithm   divalglem0 16461
            6.1.6  Bit sequences   cbits 16487
            6.1.7  The greatest common divisor operator   cgcd 16562
            6.1.8  Bézout's identity   bezoutlem1 16607
            6.1.9  Algorithms   nn0seqcvgd 16638
            6.1.10  Euclid's Algorithm   eucalgval2 16649
            ​*6.1.11  The least common multiple   clcm 16656
            ​*6.1.12  Coprimality and Euclid's lemma   coprmgcdb 16717
            6.1.13  Cancellability of congruences   congr 16732
      ​6.2  Elementary prime number theory
            ​*6.2.1  Elementary properties   cprime 16739
            ​*6.2.2  Coprimality and Euclid's lemma (cont.)   coprm 16780
            6.2.3  Properties of the canonical representation of a rational   cnumer 16802
            6.2.4  Euler's theorem   codz 16832
            6.2.5  Arithmetic modulo a prime number   modprm1div 16867
            6.2.6  Pythagorean Triples   coprimeprodsq 16878
            6.2.7  The prime count function   cpc 16906
            6.2.8  Pocklington's theorem   prmpwdvds 16974
            6.2.9  Infinite primes theorem   unbenlem 16978
            6.2.10  Sum of prime reciprocals   prmreclem1 16986
            6.2.11  Fundamental theorem of arithmetic   1arithlem1 16993
            6.2.12  Lagrange's four-square theorem   cgz 16999
            6.2.13  Van der Waerden's theorem   cvdwa 17035
            6.2.14  Ramsey's theorem   cram 17069
            ​*6.2.15  Primorial function   cprmo 17101
            ​*6.2.16  Prime gaps   prmgaplem1 17119
            6.2.17  Decimal arithmetic (cont.)   dec2dvds 17133
            6.2.18  Cyclical shifts of words (cont.)   cshwsidrepsw 17163
            6.2.19  Specific prime numbers   prmlem0 17175
            6.2.20  Very large primes   1259lem1 17201
​PART 7  BASIC STRUCTURES
      ​7.1  Extensible structures
            ​*7.1.1  Basic definitions   cstr 17216
                  7.1.1.1  Extensible structures as structures with components   cstr 17216
                  7.1.1.2  Substitution of components   csts 17233
                  7.1.1.3  Slots   cslot 17251
                  ​*7.1.1.4  Structure component indices   cnx 17263
                  7.1.1.5  Base sets   cbs 17279
                  7.1.1.6  Base set restrictions   cress 17300
            7.1.2  Slot definitions   cplusg 17320
            7.1.3  Definition of the structure product   crest 17483
            7.1.4  Definition of the structure quotient   cordt 17563
      ​7.2  Moore spaces
            7.2.1  Moore closures   mrcflem 17672
            7.2.2  Independent sets in a Moore system   mrisval 17696
            7.2.3  Algebraic closure systems   isacs 17717
​PART 8  BASIC CATEGORY THEORY
      ​8.1  Categories
            8.1.1  Categories   ccat 17730
            8.1.2  Opposite category   coppc 17777
            8.1.3  Monomorphisms and epimorphisms   cmon 17795
            8.1.4  Sections, inverses, isomorphisms   csect 17811
            ​*8.1.5  Isomorphic objects   ccic 17862
            8.1.6  Subcategories   cssc 17874
            8.1.7  Functors   cfunc 17921
            8.1.8  Full & faithful functors   cful 17971
            8.1.9  Natural transformations and the functor category   cnat 18011
            8.1.10  Initial, terminal and zero objects of a category   cinito 18048
      ​8.2  Arrows (disjointified hom-sets)
            8.2.1  Identity and composition for arrows   cida 18120
      ​8.3  Examples of categories
            8.3.1  The category of sets   csetc 18142
            8.3.2  The category of categories   ccatc 18165
            ​*8.3.3  The category of extensible structures   fncnvimaeqv 18186
      ​8.4  Categorical constructions
            8.4.1  Product of categories   cxpc 18234
            8.4.2  Functor evaluation   cevlf 18275
            8.4.3  Hom functor   chof 18314
​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 18497
            9.5.2  Complete lattices   ccla 18564
            9.5.3  Distributive lattices   cdlat 18586
            9.5.4  Subset order structures   cipo 18593
      ​9.6  Posets, directed sets, and lattices as relations
            ​*9.6.1  Posets and lattices as relations   cps 18630
            9.6.2  Directed sets, nets   cdir 18660
      ​9.7  Chains
​PART 10  BASIC ALGEBRAIC STRUCTURES
      ​10.1  Monoids
            ​*10.1.1  Magmas   cplusf 18705
            ​*10.1.2  Identity elements   mgmidmo 18728
            ​*10.1.3  Iterated sums in a magma   gsumvalx 18744
            10.1.4  Magma homomorphisms and submagmas   cmgmhm 18758
            ​*10.1.5  Semigroups   csgrp 18786
            ​*10.1.6  Definition and basic properties of monoids   cmnd 18802
            10.1.7  Monoid homomorphisms and submonoids   cmhm 18849
            ​*10.1.8  Iterated sums in a monoid   gsumvallem2 18903
            10.1.9  Free monoids   cfrmd 18916
                  ​*10.1.9.1  Monoid of endofunctions   cefmnd 18937
            10.1.10  Examples and counterexamples for magmas, semigroups and monoids   mgm2nsgrplem1 18990
      ​10.2  Groups
            10.2.1  Definition and basic properties   cgrp 19010
            ​*10.2.2  Group multiple operation   cmg 19143
            10.2.3  Subgroups and Quotient groups   csubg 19196
            ​*10.2.4  Cyclic monoids and groups   cycsubmel 19281
            10.2.5  Elementary theory of group homomorphisms   cghm 19293
            10.2.6  Isomorphisms of groups   cgim 19337
                  10.2.6.1  The first isomorphism theorem of groups   ghmqusnsglem1 19360
            10.2.7  Group actions   cga 19369
            10.2.8  Centralizers and centers   ccntz 19395
            10.2.9  The opposite group   coppg 19425
            10.2.10  Symmetric groups   csymg 19449
                  ​*10.2.10.1  Definition and basic properties   csymg 19449
                  10.2.10.2  Cayley's theorem   cayleylem1 19492
                  10.2.10.3  Permutations fixing one element   symgfix2 19496
                  ​*10.2.10.4  Transpositions in the symmetric group   cpmtr 19521
                  10.2.10.5  The sign of a permutation   cpsgn 19569
            10.2.11  p-Groups and Sylow groups; Sylow's theorems   cod 19604
            10.2.12  Direct products   clsm 19714
                  10.2.12.1  Direct products (extension)   smndlsmidm 19736
            10.2.13  Free groups   cefg 19786
            10.2.14  Abelian groups   ccmn 19860
                  10.2.14.1  Definition and basic properties   ccmn 19860
                  10.2.14.2  Cyclic groups   ccyg 19957
                  10.2.14.3  Group sum operation   gsumval3a 19983
                  10.2.14.4  Group sums over (ranges of) integers   fsfnn0gsumfsffz 20063
                  10.2.14.5  Internal direct products   cdprd 20075
                  10.2.14.6  The Fundamental Theorem of Abelian Groups   ablfacrplem 20147
            10.2.15  Simple groups   csimpg 20172
                  10.2.15.1  Definition and basic properties   csimpg 20172
                  10.2.15.2  Classification of abelian simple groups   ablsimpnosubgd 20186
            10.2.16  Totally ordered monoids and groups   comnd 20199
      ​10.3  Rings
            10.3.1  Multiplicative Group   cmgp 20226
            ​*10.3.2  Non-unital rings ("rngs")   crng 20240
            ​*10.3.3  Ring unity (multiplicative identity)   cur 20273
            10.3.4  Semirings   csrg 20278
                  ​*10.3.4.1  The binomial theorem for semirings   srgbinomlem1 20318
            10.3.5  Unital rings   crg 20325
            10.3.6  Opposite ring   coppr 20429
            10.3.7  Divisibility   cdsr 20447
            10.3.8  Ring primes   crpm 20525
            10.3.9  Homomorphisms of non-unital rings   crnghm 20527
            10.3.10  Ring homomorphisms   crh 20562
            10.3.11  Nonzero rings and zero rings   cnzr 20624
            10.3.12  Local rings   clring 20652
            10.3.13  Subrings   csubrng 20659
                  10.3.13.1  Subrings of non-unital rings   csubrng 20659
                  10.3.13.2  Subrings of unital rings   csubrg 20683
                  10.3.13.3  Subrings generated by a subset   crgspn 20724
            10.3.14  Categories of rings   crngc 20730
                  ​*10.3.14.1  The category of non-unital rings   crngc 20730
                  ​*10.3.14.2  The category of (unital) rings   cringc 20759
                  10.3.14.3  Subcategories of the category of rings   srhmsubclem1 20791
            10.3.15  Left regular elements and domains   crlreg 20805
      ​10.4  Division rings and fields
            10.4.1  Definition and basic properties   cdr 20842
            10.4.2  Sub-division rings   csdrg 20904
            10.4.3  Absolute value (abstract algebra)   cabv 20926
            10.4.4  Star rings   cstf 20955
            10.4.5  Totally ordered rings and fields   corng 20975
      ​10.5  Left modules
            10.5.1  Definition and basic properties   clmod 20996
            10.5.2  Subspaces and spans in a left module   clss 21067
            10.5.3  Homomorphisms and isomorphisms of left modules   clmhm 21155
            10.5.4  Subspace sum; bases for a left module   clbs 21210
      ​10.6  Vector spaces
            10.6.1  Definition and basic properties   clvec 21238
      ​10.7  Subring algebras and ideals
            10.7.1  Subring algebras   csra 21307
            ​*10.7.2  Left ideals and spans   clidl 21345
            10.7.3  Two-sided ideals and quotient rings   c2idl 21403
                  ​*10.7.3.1  Condition for a non-unital ring to be unital   rngqiprng1elbas 21441
                  10.7.3.2  Prime Ideals   cprmidl 21475
            10.7.4  Principal ideal rings. Divisibility in the integers   clpidl 21503
            10.7.5  Principal ideal domains   cpid 21519
      ​10.8  The complex numbers as an algebraic extensible structure
            10.8.1  Definition and basic properties   cpsmet 21521
            ​*10.8.2  Ring of integers   czring 21611
                  ​*10.8.2.1  Example for a condition for a non-unital ring to be unital   pzriprnglem1 21646
            10.8.3  Algebraic constructions based on the complex numbers   czrh 21664
            10.8.4  Signs as subgroup of the complex numbers   cnmsgnsubg 21742
            10.8.5  Embedding of permutation signs into a ring   zrhpsgnmhm 21749
            10.8.6  The ordered field of real numbers   crefld 21769
      ​10.9  Generalized pre-Hilbert and Hilbert spaces
            10.9.1  Definition and basic properties   cphl 21789
            10.9.2  Orthocomplements and closed subspaces   cocv 21825
            10.9.3  Orthogonal projection and orthonormal bases   cpj 21865
​​*PART 11  BASIC LINEAR ALGEBRA
      ​11.1  Vectors and free modules
            *11.1.1  Direct sum of left modules   cdsmm 21896
            ​*11.1.2  Free modules   cfrlm 21911
            ​*11.1.3  Standard basis (unit vectors)   cuvc 21947
            ​*11.1.4  Independent sets and families   clindf 21969
            11.1.5  Characterization of free modules   lmimlbs 22001
      ​11.2  Associative algebras
            11.2.1  Definition and basic properties   casa 22015
      ​11.3  Abstract multivariate polynomials
            11.3.1  Definition and basic properties   cmps 22069
            11.3.2  Polynomial evaluation   ces 22238
            11.3.3  The "variable selection" function   cslv 22282
            11.3.4  Additional definitions for (multivariate) polynomials   cmhp 22311
            ​*11.3.5  Univariate polynomials   cps1 22350
            11.3.6  Univariate polynomial evaluation   ces1 22488
                  11.3.6.1  Specialization of polynomial evaluation as a ring homomorphism   evls1scafv 22541
      ​​*11.4  Matrices
            *11.4.1  The matrix multiplication   cmmul 22562
            ​*11.4.2  Square matrices   cmat 22579
            ​*11.4.3  The matrix algebra   matmulr 22610
            ​*11.4.4  Matrices of dimension 0 and 1   mat0dimbas0 22638
            ​*11.4.5  The subalgebras of diagonal and scalar matrices   cdmat 22660
            ​*11.4.6  Multiplication of a matrix with a "column vector"   cmvmul 22712
            11.4.7  Replacement functions for a square matrix   cmarrep 22728
            11.4.8  Submatrices   csubma 22748
      ​11.5  The determinant
            11.5.1  Definition and basic properties   cmdat 22756
            11.5.2  Determinants of 2 x 2 -matrices   m2detleiblem1 22796
            11.5.3  The matrix adjugate/adjunct   cmadu 22804
            ​*11.5.4  Laplace expansion of determinants (special case)   symgmatr01lem 22825
            11.5.5  Inverse matrix   invrvald 22848
            ​*11.5.6  Cramer's rule   slesolvec 22851
      ​​*11.6  Polynomial matrices
            11.6.1  Basic properties   pmatring 22864
            ​*11.6.2  Constant polynomial matrices   ccpmat 22875
            ​*11.6.3  Collecting coefficients of polynomial matrices   cdecpmat 22934
            ​*11.6.4  Ring isomorphism between polynomial matrices and polynomials over matrices   cpm2mp 22964
      ​​*11.7  The characteristic polynomial
            *11.7.1  Definition and basic properties   cchpmat 22998
            ​*11.7.2  The characteristic factor function G   fvmptnn04if 23021
            ​*11.7.3  The Cayley-Hamilton theorem   cpmadurid 23039
​PART 12  BASIC TOPOLOGY
      ​12.1  Topology
            ​*12.1.1  Topological spaces   ctop 23065
                  12.1.1.1  Topologies   ctop 23065
                  12.1.1.2  Topologies on sets   ctopon 23082
                  12.1.1.3  Topological spaces   ctps 23104
            12.1.2  Topological bases   ctb 23117
            12.1.3  Examples of topologies   distop 23167
            12.1.4  Closure and interior   ccld 23188
            12.1.5  Neighborhoods   cnei 23269
            12.1.6  Limit points and perfect sets   clp 23306
            12.1.7  Subspace topologies   restrcl 23329
            12.1.8  Order topology   ordtbaslem 23360
            12.1.9  Limits and continuity in topological spaces   ccn 23396
            12.1.10  Separated spaces: T0, T1, T2 (Hausdorff) ...   ct0 23478
            12.1.11  Compactness   ccmp 23558
            12.1.12  Bolzano-Weierstrass theorem   bwth 23582
            12.1.13  Connectedness   cconn 23583
            12.1.14  First- and second-countability   c1stc 23609
            12.1.15  Local topological properties   clly 23636
            12.1.16  Refinements   cref 23674
            12.1.17  Compactly generated spaces   ckgen 23705
            12.1.18  Product topologies   ctx 23732
            12.1.19  Continuous function-builders   cnmptid 23833
            12.1.20  Quotient maps and quotient topology   ckq 23865
            12.1.21  Homeomorphisms   chmeo 23925
      ​12.2  Filters and filter bases
            12.2.1  Filter bases   elmptrab 23999
            12.2.2  Filters   cfil 24017
            12.2.3  Ultrafilters   cufil 24071
            12.2.4  Filter limits   cfm 24105
            12.2.5  Extension by continuity   ccnext 24231
            12.2.6  Topological groups   ctmd 24242
            12.2.7  Infinite group sum on topological groups   ctsu 24298
            12.2.8  Topological rings, fields, vector spaces   ctrg 24328
      ​12.3  Uniform Structures and Spaces
            12.3.1  Uniform structures   cust 24372
            12.3.2  The topology induced by an uniform structure   cutop 24402
            12.3.3  Uniform Spaces   cuss 24425
            12.3.4  Uniform continuity   cucn 24446
            12.3.5  Cauchy filters in uniform spaces   ccfilu 24457
            12.3.6  Complete uniform spaces   ccusp 24468
      ​12.4  Metric spaces
            12.4.1  Pseudometric spaces   ispsmet 24476
            12.4.2  Basic metric space properties   cxms 24489
            12.4.3  Metric space balls   blfvalps 24555
            12.4.4  Open sets of a metric space   mopnval 24610
            12.4.5  Continuity in metric spaces   metcnp3 24712
            12.4.6  The uniform structure generated by a metric   metuval 24721
            12.4.7  Examples of metric spaces   dscmet 24744
            ​*12.4.8  Normed algebraic structures   cnm 24748
            12.4.9  Normed space homomorphisms (bounded linear operators)   cnmo 24877
            12.4.10  Topology on the reals   qtopbaslem 24930
            12.4.11  Topological definitions using the reals   cii 25049
            12.4.12  Path homotopy   chtpy 25141
            12.4.13  The fundamental group   cpco 25174
      ​12.5  Metric subcomplex vector spaces
            12.5.1  Subcomplex modules   cclm 25236
            ​*12.5.2  Subcomplex vector spaces   ccvs 25297
            ​*12.5.3  Normed subcomplex vector spaces   isncvsngp 25323
            12.5.4  Subcomplex pre-Hilbert spaces   ccph 25340
            12.5.5  Convergence and completeness   ccfil 25426
            12.5.6  Baire's Category Theorem   bcthlem1 25498
            12.5.7  Banach spaces and subcomplex Hilbert spaces   ccms 25506
                  12.5.7.1  The complete ordered field of the real numbers   retopn 25553
            12.5.8  Euclidean spaces   crrx 25557
            12.5.9  Minimizing Vector Theorem   minveclem1 25598
            12.5.10  Projection Theorem   pjthlem1 25611
​PART 13  BASIC REAL AND COMPLEX ANALYSIS
      ​13.1  Continuity
            13.1.1  Intermediate value theorem   pmltpclem1 25622
      ​13.2  Integrals
            13.2.1  Lebesgue measure   covol 25636
            13.2.2  Lebesgue integration   cmbf 25788
                  13.2.2.1  Lesbesgue integral   cmbf 25788
                  13.2.2.2  Lesbesgue directed integral   cdit 26020
      ​13.3  Derivatives
            13.3.1  Real and complex differentiation   climc 26036
                  13.3.1.1  Derivatives of functions of one complex or real variable   climc 26036
                  13.3.1.2  Results on real differentiation   dvferm1lem 26158
​PART 14  BASIC REAL AND COMPLEX FUNCTIONS
      ​14.1  Polynomials
            14.1.1  Polynomial degrees   cmdg 26225
            14.1.2  The division algorithm for univariate polynomials   cmn1 26298
            14.1.3  Elementary properties of complex polynomials   cply 26356
            14.1.4  The division algorithm for polynomials   cquot 26466
            14.1.5  Algebraic numbers   caa 26490
            14.1.6  Liouville's approximation theorem   aalioulem1 26510
      ​14.2  Sequences and series
            14.2.1  Taylor polynomials and Taylor's theorem   ctayl 26531
            14.2.2  Uniform convergence   culm 26554
            14.2.3  Power series   pserval 26588
      ​14.3  Basic trigonometry
            14.3.1  The exponential, sine, and cosine functions (cont.)   efcn 26621
            14.3.2  Properties of pi = 3.14159...   pilem1 26629
            14.3.3  Mapping of the exponential function   efgh 26721
            14.3.4  The natural logarithm on complex numbers   clog 26734
            ​*14.3.5  Logarithms to an arbitrary base   clogb 26944
            14.3.6  Theorems of Pythagoras, isosceles triangles, and intersecting chords   angval 26981
            14.3.7  Solutions of quadratic, cubic, and quartic equations   quad2 27019
            14.3.8  Inverse trigonometric functions   casin 27042
            14.3.9  The Birthday Problem   log2ublem1 27126
            14.3.10  Areas in R^2   carea 27135
            14.3.11  More miscellaneous converging sequences   rlimcnp 27145
            14.3.12  Inequality of arithmetic and geometric means   cvxcl 27164
            14.3.13  Euler-Mascheroni constant   cem 27171
            14.3.14  Zeta function   czeta 27192
            14.3.15  Gamma function   clgam 27195
      ​14.4  Basic number theory
            14.4.1  Wilson's theorem   wilthlem1 27247
            14.4.2  The Fundamental Theorem of Algebra   ftalem1 27252
            14.4.3  The Basel problem (ζ(2) = π²/6)   basellem1 27260
            14.4.4  Number-theoretical functions   ccht 27270
            14.4.5  Perfect Number Theorem   mersenne 27406
            14.4.6  Characters of Z/nZ   cdchr 27411
            14.4.7  Bertrand's postulate   bcctr 27454
            ​*14.4.8  Quadratic residues and the Legendre symbol   clgs 27473
            ​*14.4.9  Gauss' Lemma   gausslemma2dlem0a 27535
            14.4.10  Quadratic reciprocity   lgseisenlem1 27554
            14.4.11  All primes 4n+1 are the sum of two squares   2sqlem1 27596
            14.4.12  Chebyshev's Weak Prime Number Theorem, Dirichlet's Theorem   chebbnd1lem1 27648
            14.4.13  The Prime Number Theorem   mudivsum 27709
            14.4.14  Ostrowski's theorem   abvcxp 27794
​PART 15  SURREAL NUMBERS
      ​​*15.1  Sign sequence representation and Alling's axioms
            15.1.1  Definitions and initial properties   csur 27819
            15.1.2  Ordering   ltssolem1 27854
            15.1.3  Birthday Function   bdayfo 27856
            15.1.4  Density   fvnobday 27857
            ​*15.1.5  Full-Eta Property   bdayimaon 27872
      ​15.2  Initial consequences of Alling's axioms
            15.2.1  Ordering Theorems   cles 27923
            15.2.2  Birthday Theorems   bdayfun 27955
      ​​*15.3  Conway cut representation
            15.3.1  Conway cuts   cslts 27965
            15.3.2  Zero and One   c0s 28013
            15.3.3  Cuts and Options   cmade 28030
            15.3.4  Cofinality and coinitiality   cofslts 28126
      ​15.4  Induction and recursion
            15.4.1  Induction and recursion on one variable   cnorec 28145
            15.4.2  Induction and recursion on two variables   cnorec2 28156
      ​15.5  Surreal arithmetic
            15.5.1  Addition   cadds 28167
            15.5.2  Negation and Subtraction   cnegs 28227
            15.5.3  Multiplication   cmuls 28314
            15.5.4  Division   cdivs 28395
            15.5.5  Absolute value   cabss 28445
      ​15.6  Subsystems of surreals
            15.6.1  Ordinal numbers   cons 28459
            15.6.2  Surreal recursive sequences   cseqs 28491
            15.6.3  Natural numbers   cn0s 28520
            15.6.4  Integers   czs 28586
            15.6.5  Dyadic fractions   c2s 28618
            15.6.6  Real numbers   creno 28697
​​*PART 16  ELEMENTARY GEOMETRY
      ​16.1  Definition and Tarski's Axioms of Geometry
            16.1.1  Justification for the congruence notation   tgjustf 28757
      ​16.2  Tarskian Geometry
            16.2.1  Congruence   tgcgrcomimp 28761
            16.2.2  Betweenness   tgbtwntriv2 28771
            16.2.3  Dimension   tglowdim1 28784
            16.2.4  Betweenness and Congruence   tgifscgr 28792
            16.2.5  Congruence of a series of points   ccgrg 28794
            16.2.6  Motions   cismt 28816
            16.2.7  Colinearity   tglng 28830
            16.2.8  Connectivity of betweenness   tgbtwnconn1lem1 28856
            16.2.9  Less-than relation in geometric congruences   cleg 28866
            16.2.10  Rays   chlg 28884
            16.2.11  Lines   btwnlng1 28907
            16.2.12  Point inversions   cmir 28944
            16.2.13  Right angles   crag 28988
            16.2.14  Half-planes   islnopp 29035
            16.2.15  Planes   cplng 29070
            16.2.16  Midpoints and Line Mirroring   cmid 29096
            16.2.17  Congruence of angles   ccgra 29133
            16.2.18  Angle Comparisons   cinag 29167
            16.2.19  Congruence Theorems   tgsas1 29186
            16.2.20  Equilateral triangles   ceqlg 29197
            16.2.21  Parallel lines   cprlng 29201
      ​16.3  Properties of geometries
            16.3.1  Isomorphisms between geometries   f1otrgds 29233
      ​16.4  Geometry in Hilbert spaces
            16.4.1  Geometry in the complex plane   cchhllem 29251
            16.4.2  Geometry in Euclidean spaces   cee 29252
                  16.4.2.1  Definition of the Euclidean space   cee 29252
                  16.4.2.2  Tarski's axioms for geometry for the Euclidean space   axdimuniq 29278
                  16.4.2.3  EE^n fulfills Tarski's Axioms   ceeng 29342
​​*PART 17  GRAPH THEORY
      ​*17.1  Vertices and edges
            17.1.1  The edge function extractor for extensible structures   cedgf 29353
            ​*17.1.2  Vertices and indexed edges   cvtx 29361
                  17.1.2.1  Definitions and basic properties   cvtx 29361
                  17.1.2.2  The vertices and edges of a graph represented as ordered pair   opvtxval 29368
                  17.1.2.3  The vertices and edges of a graph represented as extensible structure   funvtxdmge2val 29376
                  17.1.2.4  Representations of graphs without edges   snstrvtxval 29402
                  17.1.2.5  Degenerated cases of representations of graphs   vtxval0 29404
            17.1.3  Edges as range of the edge function   cedg 29412
      ​​*17.2  Undirected graphs
            17.2.1  Undirected hypergraphs   cuhgr 29421
            17.2.2  Undirected pseudographs and multigraphs   cupgr 29445
            ​*17.2.3  Loop-free graphs   umgrislfupgrlem 29487
            17.2.4  Edges as subsets of vertices of graphs   uhgredgiedgb 29491
            ​*17.2.5  Undirected simple graphs   cuspgr 29513
            17.2.6  Examples for graphs   usgr0e 29601
            17.2.7  Subgraphs   csubgr 29632
            17.2.8  Finite undirected simple graphs   cfusgr 29681
            17.2.9  Neighbors, complete graphs and universal vertices   cnbgr 29697
                  17.2.9.1  Neighbors   cnbgr 29697
                  17.2.9.2  Universal vertices   cuvtx 29750
                  17.2.9.3  Complete graphs   ccplgr 29774
            17.2.10  Vertex degree   cvtxdg 29830
            ​*17.2.11  Regular graphs   crgr 29920
      ​​*17.3  Walks, paths and cycles
            *17.3.1  Walks   cewlks 29960
            17.3.2  Walks for loop-free graphs   lfgrwlkprop 30050
            17.3.3  Trails   ctrls 30053
            17.3.4  Paths and simple paths   cpths 30074
            17.3.5  Closed walks   cclwlks 30134
            17.3.6  Circuits and cycles   ccrcts 30148
            ​*17.3.7  Walks as words   cwwlks 30189
            17.3.8  Walks/paths of length 2 (as length 3 strings)   2wlkdlem1 30289
            17.3.9  Walks in regular graphs   rusgrnumwwlkl1 30335
            ​*17.3.10  Closed walks as words   cclwwlk 30347
                  17.3.10.1  Closed walks as words   cclwwlk 30347
                  17.3.10.2  Closed walks of a fixed length as words   cclwwlkn 30390
                  17.3.10.3  Closed walks on a vertex of a fixed length as words   cclwwlknon 30453
            17.3.11  Examples for walks, trails and paths   0ewlk 30480
            17.3.12  Connected graphs   cconngr 30552
      ​17.4  Eulerian paths and the Konigsberg Bridge problem
            ​*17.4.1  Eulerian paths   ceupth 30563
            ​*17.4.2  The Königsberg Bridge problem   konigsbergvtx 30612
      ​17.5  The Friendship Theorem
            17.5.1  Friendship graphs - basics   cfrgr 30624
            17.5.2  The friendship theorem for small graphs   frgr1v 30637
            17.5.3  Theorems according to Mertzios and Unger   2pthfrgrrn 30648
            ​*17.5.4  Huneke's Proof of the Friendship Theorem   frgrncvvdeqlem1 30665
​PART 18  GUIDES AND MISCELLANEA
      ​18.1  Guides (conventions, explanations, and examples)
            ​*18.1.1  Conventions   conventions 30766
            18.1.2  Natural deduction   natded 30769
            ​*18.1.3  Natural deduction examples   ex-natded5.2 30770
            18.1.4  Definitional examples   ex-or 30787
            18.1.5  Other examples   aevdemo 30826
      ​18.2  Humor
            18.2.1  April Fool's theorem   avril1 30829
      ​18.3  (Future - to be reviewed and classified)
            18.3.1  Planar incidence geometry   cplig 30841
            ​*18.3.2  Aliases kept to prevent broken links   dummylink 30854
​​*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 30856
            19.1.2  Abelian groups   cablo 30911
      ​19.2  Complex vector spaces
            19.2.1  Definition and basic properties   cvc 30925
            19.2.2  Examples of complex vector spaces   cnaddabloOLD 30948
      ​19.3  Normed complex vector spaces
            19.3.1  Definition and basic properties   cnv 30951
            19.3.2  Examples of normed complex vector spaces   cnnv 31044
            19.3.3  Induced metric of a normed complex vector space   imsval 31052
            19.3.4  Inner product   cdip 31067
            19.3.5  Subspaces   css 31088
      ​19.4  Operators on complex vector spaces
            19.4.1  Definitions and basic properties   clno 31107
      ​19.5  Inner product (pre-Hilbert) spaces
            19.5.1  Definition and basic properties   ccphlo 31179
            19.5.2  Examples of pre-Hilbert spaces   cncph 31186
            19.5.3  Properties of pre-Hilbert spaces   isph 31189
      ​19.6  Complex Banach spaces
            19.6.1  Definition and basic properties   ccbn 31229
            19.6.2  Examples of complex Banach spaces   cnbn 31236
            19.6.3  Uniform Boundedness Theorem   ubthlem1 31237
            19.6.4  Minimizing Vector Theorem   minvecolem1 31241
      ​19.7  Complex Hilbert spaces
            19.7.1  Definition and basic properties   chlo 31252
            19.7.2  Standard axioms for a complex Hilbert space   hlex 31265
            19.7.3  Examples of complex Hilbert spaces   cnchl 31283
            19.7.4  Hellinger-Toeplitz Theorem   htthlem 31284
​​*PART 20  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      ​20.1  Axiomatization of complex pre-Hilbert spaces
            20.1.1  Basic Hilbert space definitions   chba 31286
            20.1.2  Preliminary ZFC lemmas   df-hnorm 31335
            ​*20.1.3  Derive the Hilbert space axioms from ZFC set theory   axhilex-zf 31348
            ​*20.1.4  Introduce the vector space axioms for a Hilbert space   ax-hilex 31366
            20.1.5  Vector operations   hvmulex 31378
            20.1.6  Inner product postulates for a Hilbert space   ax-hfi 31446
      ​20.2  Inner product and norms
            20.2.1  Inner product   his5 31453
            20.2.2  Norms   dfhnorm2 31489
            20.2.3  Relate Hilbert space to normed complex vector spaces   hilablo 31527
            20.2.4  Bunjakovaskij-Cauchy-Schwarz inequality   bcsiALT 31546
      ​20.3  Cauchy sequences and completeness axiom
            20.3.1  Cauchy sequences and limits   hcau 31551
            20.3.2  Derivation of the completeness axiom from ZF set theory   hilmet 31561
            20.3.3  Completeness postulate for a Hilbert space   ax-hcompl 31569
            20.3.4  Relate Hilbert space to ZFC pre-Hilbert and Hilbert spaces   hhcms 31570
      ​20.4  Subspaces and projections
            20.4.1  Subspaces   df-sh 31574
            20.4.2  Closed subspaces   df-ch 31588
            20.4.3  Orthocomplements   df-oc 31619
            20.4.4  Subspace sum, span, lattice join, lattice supremum   df-shs 31675
            20.4.5  Projection theorem   pjhthlem1 31758
            20.4.6  Projectors   df-pjh 31762
      ​20.5  Properties of Hilbert subspaces
            20.5.1  Orthomodular law   omlsilem 31769
            20.5.2  Projectors (cont.)   pjhtheu2 31783
            20.5.3  Hilbert lattice operations   sh0le 31807
            20.5.4  Span (cont.) and one-dimensional subspaces   spansn0 31908
            20.5.5  Commutes relation for Hilbert lattice elements   df-cm 31950
            20.5.6  Foulis-Holland theorem   fh1 31985
            20.5.7  Quantum Logic Explorer axioms   qlax1i 31994
            20.5.8  Orthogonal subspaces   chscllem1 32004
            20.5.9  Orthoarguesian laws 5OA and 3OA   5oalem1 32021
            20.5.10  Projectors (cont.)   pjorthi 32036
            20.5.11  Mayet's equation E_3   mayete3i 32095
      ​20.6  Operators on Hilbert spaces
            ​*20.6.1  Operator sum, difference, and scalar multiplication   df-hosum 32097
            20.6.2  Zero and identity operators   df-h0op 32115
            20.6.3  Operations on Hilbert space operators   hoaddcl 32125
            20.6.4  Linear, continuous, bounded, Hermitian, unitary operators and norms   df-nmop 32206
            20.6.5  Linear and continuous functionals and norms   df-nmfn 32212
            20.6.6  Adjoint   df-adjh 32216
            20.6.7  Dirac bra-ket notation   df-bra 32217
            20.6.8  Positive operators   df-leop 32219
            20.6.9  Eigenvectors, eigenvalues, spectrum   df-eigvec 32220
            20.6.10  Theorems about operators and functionals   nmopval 32223
            20.6.11  Riesz lemma   riesz3i 32429
            20.6.12  Adjoints (cont.)   cnlnadjlem1 32434
            20.6.13  Quantum computation error bound theorem   unierri 32471
            20.6.14  Dirac bra-ket notation (cont.)   branmfn 32472
            20.6.15  Positive operators (cont.)   leopg 32489
            20.6.16  Projectors as operators   pjhmopi 32513
      ​20.7  States on a Hilbert lattice and Godowski's equation
            20.7.1  States on a Hilbert lattice   df-st 32578
            20.7.2  Godowski's equation   golem1 32638
      ​20.8  Cover relation, atoms, exchange axiom, and modular symmetry
            20.8.1  Covers relation; modular pairs   df-cv 32646
            20.8.2  Atoms   df-at 32705
            20.8.3  Superposition principle   superpos 32721
            20.8.4  Atoms, exchange and covering properties, atomicity   chcv1 32722
            20.8.5  Irreducibility   chirredlem1 32757
            20.8.6  Atoms (cont.)   atcvat3i 32763
            20.8.7  Modular symmetry   mdsymlem1 32770
​PART 21  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      ​21.1  Mathboxes for user contributions
            21.1.1  Mathbox guidelines   mathbox 32809
      ​21.2  Mathbox for Stefan Allan
      ​21.3  Mathbox for Thierry Arnoux
            21.3.1  Propositional Calculus - misc additions   ad11antr 32814
            21.3.2  Predicate Calculus   sbc2iedf 32827
                  21.3.2.1  Predicate Calculus - misc additions   sbc2iedf 32827
                  21.3.2.2  Restricted quantification - misc additions   ralcom4f 32829
                  21.3.2.3  Equality   eqtrb 32835
                  21.3.2.4  Double restricted existential uniqueness quantification   opsbc2ie 32837
                  21.3.2.5  Double restricted existential uniqueness quantification syntax   w2reu 32839
                  21.3.2.6  Substitution (without distinct variables) - misc additions   sbceqbidf 32848
                  21.3.2.7  Existential "at most one" - misc additions   mo5f 32850
                  21.3.2.8  Existential uniqueness - misc additions   reuxfrdf 32852
                  21.3.2.9  Restricted "at most one" - misc additions   rmoxfrd 32854
                  21.3.2.10  Restricted iota (description binder)   riotaeqbidva 32857
            21.3.3  General Set Theory   dmrab 32858
                  21.3.3.1  Class abstractions (a.k.a. class builders)   dmrab 32858
                  21.3.3.2  Image Sets   abrexdomjm 32868
                  21.3.3.3  Set relations and operations - misc additions   nelun 32874
                  21.3.3.4  Unordered pairs   elpreq 32889
                  21.3.3.5  Unordered triples   tpssg 32898
                  21.3.3.6  Conditional operator - misc additions   ifeqeqx 32903
                  21.3.3.7  Set union   uniinn0 32912
                  21.3.3.8  Indexed union - misc additions   cbviunf 32915
                  21.3.3.9  Indexed intersection - misc additions   iinabrex 32929
                  21.3.3.10  Disjointness - misc additions   disjnf 32930
            21.3.4  Relations and Functions   xpdisjres 32958
                  21.3.4.1  Relations - misc additions   xpdisjres 32958
                  21.3.4.2  Functions - misc additions   fconst7v 32980
                  21.3.4.3  Operations - misc additions   mpomptxf 33038
                  21.3.4.4  The mapping operation   elmaprd 33040
                  21.3.4.5  Support of a function   suppovss 33041
                  21.3.4.6  Explicit Functions with one or two points as a domain   cosnopne 33054
                  21.3.4.7  Isomorphisms - misc. additions   gtiso 33061
                  21.3.4.8  Disjointness (additional proof requiring functions)   disjdsct 33063
                  21.3.4.9  First and second members of an ordered pair - misc additions   df1stres 33064
                  21.3.4.10  Countable Sets   snct 33072
            21.3.5  Real and Complex Numbers   sgnval2 33095
                  21.3.5.1  Complex operations - misc. additions   creq0 33096
                  21.3.5.2  Ordering on reals - misc additions   lt2addrd 33110
                  21.3.5.3  Extended reals - misc additions   nn0mnfxrd 33111
                  21.3.5.4  Extended nonnegative integers - misc additions   xnn0gt0 33129
                  21.3.5.5  Real number intervals - misc additions   joiniooico 33134
                  21.3.5.6  Finite intervals of integers - misc additions   uzssico 33144
                  21.3.5.7  Half-open integer ranges - misc additions   iundisjfi 33156
                  21.3.5.8  The ` # ` (set size) function - misc additions   hashunif 33166
                  21.3.5.9  The greatest common divisor operator - misc. additions   elq2 33171
                  21.3.5.10  Integers   nn0split01 33177
                  21.3.5.11  Decimal numbers   dfdec100 33189
            21.3.6  Real and complex functions   sgnsgn 33190
                  21.3.6.1  Signum (sgn or sign) function - misc. additions   sgnsgn 33190
                  21.3.6.2  Integer powers - misc. additions   nexple 33192
                  21.3.6.3  Indicator Functions (continued)   indsumin 33196
            ​*21.3.7  Decimal expansion   cdp2 33205
                  ​*21.3.7.1  Decimal point   cdp 33222
                  21.3.7.2  Division in the extended real number system   cxdiv 33251
            21.3.8  Words over a set - misc additions   wrdres 33270
                  21.3.8.1  Splicing words (substring replacement)   splfv3 33291
                  21.3.8.2  Cyclic shift of words   1cshid 33292
            21.3.9  Extensible Structures   ressplusf 33296
                  21.3.9.1  Structure restriction operator   ressplusf 33296
                  21.3.9.2  Posets   ressprs 33299
                  21.3.9.3  Complete lattices   clatp0cl 33309
                  21.3.9.4  Order Theory   cmnt 33311
                  21.3.9.5  Extended reals Structure - misc additions   ax-xrssca 33337
                  21.3.9.6  The extended nonnegative real numbers commutative monoid   xrge00 33347
            21.3.10  Algebra   mndcld 33355
                  21.3.10.1  Monoids   mndcld 33355
                  21.3.10.2  Monoids Homomorphisms   abliso 33368
                  21.3.10.3  Groups - misc additions   grpidcld 33372
                  21.3.10.4  Abelian Groups - misc additions   ablcomd 33378
                  21.3.10.5  Finitely supported group sums - misc additions   gsumsubg 33379
                  21.3.10.6  Group or monoid sums over words   gsumwun 33409
                  21.3.10.7  Centralizers and centers - misc additions   cntzun 33412
                  21.3.10.8  The symmetric group   symgfcoeu 33415
                  21.3.10.9  Transpositions   pmtridf1o 33427
                  21.3.10.10  Permutation Signs   psgnid 33430
                  21.3.10.11  Permutation cycles   ctocyc 33439
                  21.3.10.12  The Alternating Group   evpmval 33478
                  21.3.10.13  Signum in an ordered monoid   csgns 33491
                  21.3.10.14  Fixed points   cfxp 33496
                  21.3.10.15  The Archimedean property for generic ordered algebraic structures   cinftm 33509
                  21.3.10.16  Semiring left modules   cslmd 33533
                  21.3.10.17  Simple groups   prmsimpcyc 33561
                  21.3.10.18  Rings - misc additions   ringrngd 33562
                  21.3.10.19  Subrings generated by a set   elrgspnlem1 33575
                  21.3.10.20  The zero ring   irrednzr 33583
                  21.3.10.21  Localization of rings   cerl 33586
                  21.3.10.22  Integral Domains   domnmuln0rd 33610
                  21.3.10.23  Euclidean Domains   ceuf 33624
                  21.3.10.24  Division Rings   rndrhmcl 33630
                  21.3.10.25  The field of rational numbers   qfld 33631
                  21.3.10.26  Subfields   subsdrg 33632
                  21.3.10.27  Field of fractions   cfrac 33636
                  21.3.10.28  Field extensions generated by a set   cfldgen 33644
                  21.3.10.29  Ring homomorphisms - misc additions   rhmdvd 33657
                  21.3.10.30  Scalar restriction operation   cresv 33659
                  21.3.10.31  The commutative ring of gaussian integers   gzcrng 33674
                  21.3.10.32  The archimedean ordered field of real numbers   cnfldfld 33675
                  21.3.10.33  The quotient map and quotient modules   qusker 33682
                  21.3.10.34  The ring of integers modulo ` N `   znfermltl 33694
                  21.3.10.35  Independent sets and families   islinds5 33695
                  21.3.10.36  Ring associates, ring units   dvdsruassoi 33710
                  ​*21.3.10.37  Subgroup sum / Sumset / Minkowski sum   elgrplsmsn 33716
                  21.3.10.38  The quotient map   quslsm 33727
                  21.3.10.39  Ideals   intlidl 33741
                  21.3.10.40  Maximal Ideals   cmxidl 33755
                  21.3.10.41  Local rings   drnglring 33795
                  21.3.10.42  The semiring of ideals of a ring   cidlsrg 33803
                  21.3.10.43  Prime Elements   rprmval 33819
                  21.3.10.44  Unique factorization domains   cufd 33841
                  21.3.10.45  The ring of integers   zringidom 33854
                  21.3.10.46  Associative Algebra   assaassd 33858
                  21.3.10.47  Univariate Polynomials   0ringmon1p 33860
                  21.3.10.48  Polynomial quotient and polynomial remainder   q1pdir 33906
                  21.3.10.49  Multivariate Polynomials   psrbasfsupp 33914
                  21.3.10.50  The ring of symmetric polynomials   csply 33958
                  21.3.10.51  The subring algebra   sra1r 33984
                  21.3.10.52  Division Ring Extensions   drgext0g 33993
                  21.3.10.53  Vector Spaces   lvecdimfi 33999
                  21.3.10.54  Vector Space Dimension   cldim 34002
            21.3.11  Field Extensions   cfldext 34041
                  21.3.11.1  Algebraic numbers   cirng 34086
                  21.3.11.2  Algebraic extensions   calgext 34098
                  21.3.11.3  Minimal polynomials   cminply 34102
                  21.3.11.4  Quadratic Field Extensions   rtelextdg2lem 34129
                  21.3.11.5  Towers of quadratic extentions   fldext2chn 34131
            ​*21.3.12  Constructible Numbers   cconstr 34132
                  21.3.12.1  Impossible constructions   2sqr3minply 34183
            21.3.13  Matrices   csmat 34196
                  21.3.13.1  Submatrices   csmat 34196
                  21.3.13.2  Matrix literals   clmat 34214
                  21.3.13.3  Laplace expansion of determinants   mdetpmtr1 34226
            21.3.14  Topology   ist0cld 34236
                  21.3.14.1  Open maps   txomap 34237
                  21.3.14.2  Topology of the unit circle   qtopt1 34238
                  21.3.14.3  Refinements   reff 34242
                  21.3.14.4  Open cover refinement property   ccref 34245
                  21.3.14.5  Lindelöf spaces   cldlf 34255
                  21.3.14.6  Paracompact spaces   cpcmp 34258
                  ​*21.3.14.7  Spectrum of a ring   crspec 34265
                  21.3.14.8  Pseudometrics   cmetid 34289
                  21.3.14.9  Continuity - misc additions   hauseqcn 34301
                  21.3.14.10  Topology of the closed unit interval   elunitge0 34302
                  21.3.14.11  Topology of ` ( RR X. RR ) `   unicls 34306
                  21.3.14.12  Order topology - misc. additions   cnvordtrestixx 34316
                  21.3.14.13  Continuity in topological spaces - misc. additions   mndpluscn 34329
                  21.3.14.14  Topology of the extended nonnegative real numbers ordered monoid   xrge0hmph 34335
                  21.3.14.15  Limits - misc additions   lmlim 34350
                  21.3.14.16  Univariate polynomials   pl1cn 34358
            21.3.15  Uniform Stuctures and Spaces   chcmp 34359
                  21.3.15.1  Hausdorff uniform completion   chcmp 34359
            21.3.16  Topology and algebraic structures   zringnm 34361
                  21.3.16.1  The norm on the ring of the integer numbers   zringnm 34361
                  21.3.16.2  Topological ` ZZ ` -modules   zlm0 34363
                  21.3.16.3  Canonical embedding of the field of the rational numbers into a division ring   cqqh 34373
                  21.3.16.4  Canonical embedding of the real numbers into a complete ordered field   crrh 34396
                  21.3.16.5  Embedding from the extended real numbers into a complete lattice   cxrh 34419
                  21.3.16.6  Canonical embeddings into the ordered field of the real numbers   zrhre 34422
                  ​*21.3.16.7  Topological Manifolds   cmntop 34425
                  21.3.16.8  Extended sum   cesum 34430
            21.3.17  Mixed Function/Constant operation   cofc 34498
            21.3.18  Abstract measure   csiga 34511
                  21.3.18.1  Sigma-Algebra   csiga 34511
                  21.3.18.2  Generated sigma-Algebra   csigagen 34541
                  ​*21.3.18.3  lambda and pi-Systems, Rings of Sets   ispisys 34555
                  21.3.18.4  The Borel algebra on the real numbers   cbrsiga 34584
                  21.3.18.5  Product Sigma-Algebra   csx 34591
                  21.3.18.6  Measures   cmeas 34598
                  21.3.18.7  The counting measure   cntmeas 34629
                  21.3.18.8  The Lebesgue measure - misc additions   voliune 34632
                  21.3.18.9  The Dirac delta measure   cdde 34635
                  21.3.18.10  The 'almost everywhere' relation   cae 34640
                  21.3.18.11  Measurable functions   cmbfm 34652
                  21.3.18.12  Borel Algebra on ` ( RR X. RR ) `   br2base 34672
                  ​*21.3.18.13  Caratheodory's extension theorem   coms 34694
            21.3.19  Integration   itgeq12dv 34729
                  21.3.19.1  Lebesgue integral - misc additions   itgeq12dv 34729
                  21.3.19.2  Bochner integral   citgm 34730
            21.3.20  Euler's partition theorem   oddpwdc 34757
            21.3.21  Sequences defined by strong recursion   csseq 34786
            21.3.22  Fibonacci Numbers   cfib 34799
            21.3.23  Probability   cprb 34810
                  21.3.23.1  Probability Theory   cprb 34810
                  21.3.23.2  Conditional Probabilities   ccprob 34834
                  21.3.23.3  Real-valued Random Variables   crrv 34843
                  21.3.23.4  Preimage set mapping operator   corvc 34859
                  21.3.23.5  Distribution Functions   orvcelval 34872
                  21.3.23.6  Cumulative Distribution Functions   orvclteel 34876
                  21.3.23.7  Probabilities - example   coinfliplem 34882
                  21.3.23.8  Bertrand's Ballot Problem   ballotlemoex 34889
            21.3.24  Signum (sgn or sign) function - misc. additions   fzssfzo 34942
                  21.3.24.1  Operations on words   ccatmulgnn0dir 34945
            21.3.25  Polynomials with real coefficients - misc additions   plyrecld 34949
            21.3.26  Descartes's rule of signs   signspval 34952
                  21.3.26.1  Sign changes in a word over real numbers   signspval 34952
                  21.3.26.2  Counting sign changes in a word over real numbers   signslema 34962
            21.3.27  Number Theory   iblidicc 34992
                  21.3.27.1  Representations of a number as sums of integers   crepr 35008
                  21.3.27.2  Vinogradov Trigonometric Sums and the Circle Method   cvts 35035
                  21.3.27.3  The Ternary Goldbach Conjecture: Final Statement   ax-hgt749 35044
            21.3.28  Elementary Geometry   cstrkg2d 35064
                  ​*21.3.28.1  Two-dimensional geometry   cstrkg2d 35064
                  21.3.28.2  Morley's Miracle   cgranbtwn 35069
                  21.3.28.3  Outer Five Segment (not used, no need to move to main)   cafs 35072
            ​*21.3.29  LeftPad Project   clpad 35077
      ​​*21.4  Mathbox for Jonathan Ben-Naim
            21.4.1  First-order logic and set theory   bnj170 35100
            21.4.2  Well founded induction and recursion   bnj110 35259
            21.4.3  The existence of a minimal element in certain classes   bnj69 35411
            21.4.4  Well-founded induction   bnj1204 35413
            21.4.5  Well-founded recursion, part 1 of 3   bnj60 35463
            21.4.6  Well-founded recursion, part 2 of 3   bnj1500 35469
            21.4.7  Well-founded recursion, part 3 of 3   bnj1522 35473
      ​21.5  Mathbox for BTernaryTau
            21.5.1  First-order logic   nfan1c 35474
                  21.5.1.1  Auxiliary axiom schemes   nfan1c 35474
            21.5.2  ZF set theory   inv2 35480
                  21.5.2.1  Finitism   prcinf 35538
                  21.5.2.2  Introduce ax-regs   ax-regs 35551
                  21.5.2.3  Derive ax-regs   axregs 35564
                  21.5.2.4  ZFC axioms with reduced distinct variable conditions   axsepg2 35565
                  21.5.2.5  Cardinality without the Axiom of Choice   ckard 35574
                  21.5.2.6  Global choice   gblacfnacd 35598
            21.5.3  Real and complex numbers   zltp1ne 35613
            21.5.4  Graph theory   lfuhgr 35622
                  21.5.4.1  Acyclic graphs   cacycgr 35646
      ​21.6  Mathbox for Mario Carneiro
            21.6.1  Predicate calculus with all distinct variables   ax-7d 35663
            21.6.2  Miscellaneous stuff   quartfull 35669
            21.6.3  Derangements and the Subfactorial   deranglem 35670
            21.6.4  The Erdős-Szekeres theorem   erdszelem1 35695
            21.6.5  The Kuratowski closure-complement theorem   kur14lem1 35710
            21.6.6  Retracts and sections   cretr 35721
            21.6.7  Path-connected and simply connected spaces   cpconn 35723
            21.6.8  Covering maps   ccvm 35759
            21.6.9  Normal numbers   snmlff 35833
            21.6.10  Godel-sets of formulas - part 1   cgoe 35837
            21.6.11  Godel-sets of formulas - part 2   cgon 35936
            21.6.12  Models of ZF   cgze 35950
            ​*21.6.13  Metamath formal systems   cmcn 35964
            21.6.14  Grammatical formal systems   cm0s 36089
            21.6.15  Models of formal systems   cmuv 36109
            21.6.16  Splitting fields   ccpms 36131
            21.6.17  p-adic number fields   czr 36151
      ​​*21.7  Mathbox for Filip Cernatescu
      ​21.8  Mathbox for Paul Chapman
            21.8.1  Real and complex numbers (cont.)   climuzcnv 36175
            21.8.2  Miscellaneous theorems   elfzm12 36179
      ​21.9  Mathbox for Hongxiu Chen
      ​21.10  Mathbox for Adrian Ducourtial
            21.10.1  Propositional calculus   currybi 36192
            21.10.2  Clone theory   ccloneop 36199
      ​21.11  Mathbox for Scott Fenton
            21.11.1  ZFC Axioms in primitive form   axextprim 36205
            21.11.2  Untangled classes   untelirr 36212
            21.11.3  Extra propositional calculus theorems   3jaodd 36219
            21.11.4  Misc. Useful Theorems   nepss 36222
            21.11.5  Properties of real and complex numbers   sqdivzi 36232
            21.11.6  Infinite products   iprodefisumlem 36244
            21.11.7  Factorial limits   faclimlem1 36247
            21.11.8  Greatest common divisor and divisibility   gcd32 36253
            21.11.9  Properties of relationships   dftr6 36255
            21.11.10  Properties of functions and mappings   funpsstri 36270
            21.11.11  Ordinal numbers   elpotr 36283
            21.11.12  Defined equality axioms   axextdfeq 36299
            21.11.13  Hypothesis builders   hbntg 36307
            21.11.14  Well-founded zero, successor, and limits   cwsuc 36312
            21.11.15  Quantifier-free definitions   ctxp 36332
            21.11.16  Alternate ordered pairs   caltop 36460
            21.11.17  Geometry in the Euclidean space   cofs 36486
                  21.11.17.1  Congruence properties   cofs 36486
                  21.11.17.2  Betweenness properties   btwntriv2 36516
                  21.11.17.3  Segment Transportation   ctransport 36533
                  21.11.17.4  Properties relating betweenness and congruence   cifs 36539
                  21.11.17.5  Connectivity of betweenness   btwnconn1lem1 36591
                  21.11.17.6  Segment less than or equal to   csegle 36610
                  21.11.17.7  Outside-of relationship   coutsideof 36623
                  21.11.17.8  Lines and Rays   cline2 36638
            21.11.18  Forward difference   cfwddif 36662
            21.11.19  Rank theorems   rankung 36670
            21.11.20  Hereditarily Finite Sets   chf 36676
            21.11.21  Natural ordinal operations   cnmul 36691
      ​21.12  Mathbox for Gino Giotto
            21.12.1  Equality theorems   rmoeqi 36731
                  21.12.1.1  Inference versions   rmoeqi 36731
                  21.12.1.2  Deduction versions   rmoeqdv 36756
            21.12.2  Change bound variables   in-ax8 36768
                  21.12.2.1  Change bound variables and domains   cbvralvw2 36770
                  21.12.2.2  Change bound variables, deduction versions   cbvmodavw 36794
                  21.12.2.3  Change bound variables and domains, deduction versions   cbvrmodavw2 36827
            21.12.3  Study of ax-mulf usage   mpomulnzcnf 36843
      ​21.13  Mathbox for Jeff Hankins
            21.13.1  Miscellany   a1i14 36844
            21.13.2  Basic topological facts   topbnd 36867
            21.13.3  Topology of the real numbers   ivthALT 36878
            21.13.4  Refinements   cfne 36879
            21.13.5  Neighborhood bases determine topologies   neibastop1 36902
            21.13.6  Lattice structure of topologies   topmtcl 36906
            21.13.7  Filter bases   fgmin 36913
            21.13.8  Directed sets, nets   tailfval 36915
      ​21.14  Mathbox for Anthony Hart
            21.14.1  Propositional Calculus   tb-ax1 36926
            21.14.2  Predicate Calculus   nalfal 36946
            21.14.3  Miscellaneous single axioms   meran1 36954
            21.14.4  Connective Symmetry   negsym1 36960
      ​21.15  Mathbox for Chen-Pang He
            21.15.1  Ordinal topology   ontopbas 36971
      ​21.16  Mathbox for Jeff Hoffman
            21.16.1  Inferences for finite induction on generic function values   fveleq 36994
            21.16.2  gdc.mm   nnssi2 36998
      ​21.17  Mathbox for Matthew House
            21.17.1  Relations on well-ordered indexed unions   weiunval 37005
            21.17.2  Axiom of Transitive Containment   axtco 37014
            21.17.3  Transitive closure of a class   tr0elw 37027
            ​*21.17.4  Stronger axioms of regularity   mh-setind 37079
            21.17.5  Short axioms written in primitive symbols   mh-inf3f1 37084
      ​21.18  Mathbox for Asger C. Ipsen
            21.18.1  Continuous nowhere differentiable functions   dnival 37092
      ​​*21.19  Mathbox for BJ
            *21.19.1  Propositional calculus   bj-mp2c 37161
                  *21.19.1.1  Derived rules of inference   bj-mp2c 37161
                  ​*21.19.1.2  A syntactic theorem   bj-0 37163
                  ​*21.19.1.3  Minimal implicational calculus   bj-poni 37165
                  ​*21.19.1.4  Positive calculus   bj-bisimpl 37177
                  ​*21.19.1.5  Implication and negation   bj-con2com 37185
                  ​*21.19.1.6  Disjunction   bj-jaoi1 37196
                  ​*21.19.1.7  Logical equivalence   bj-dfbi4 37198
                  21.19.1.8  The conditional operator for propositions   bj-consensus 37203
                  ​*21.19.1.9  Propositional calculus: miscellaneous   bj-imbi12 37208
            ​*21.19.2  Modal logic   bj-axdd2 37217
            ​*21.19.3  Provability logic   cprvb 37222
            ​*21.19.4  First-order logic   bj-exexalal 37231
                  21.19.4.1  Universal and existential quantifiers, nonfreeness predicate   bj-exexalal 37231
                  21.19.4.2  Adding ax-gen   bj-genr 37232
                  21.19.4.3  Adding ax-4   bj-almp 37236
                  21.19.4.4  Adding ax-5   bj-spvw 37289
                  21.19.4.5  Equality and substitution   bj-df-sb 37304
                  21.19.4.6  Adding ax-6   bj-spim0 37323
                  21.19.4.7  Adding ax-7   bj-cbvexw 37331
                  21.19.4.8  Membership predicate, ax-8 and ax-9   bj-ax89 37333
                  21.19.4.9  Adding ax-11   bj-alcomexcom 37335
                  21.19.4.10  Adding ax-12   axc11n11 37339
                  ​*21.19.4.11  Really adding ax-12   bj-substax12 37381
                  21.19.4.12  Nonfreeness   wnnf 37383
                  21.19.4.13  Adding ax-13   bj-axc10 37450
                  ​*21.19.4.14  Removing dependencies on ax-13 (and ax-11)   bj-axc10v 37460
                  ​*21.19.4.15  Distinct var metavariables   bj-hbaeb2 37485
                  ​*21.19.4.16  Around ~ equsal   bj-equsal1t 37489
                  ​*21.19.4.17  Some Principia Mathematica proofs   stdpc5t 37494
                  21.19.4.18  Alternate definition of substitution   bj-sbsb 37504
                  21.19.4.19  Lemmas for substitution   bj-sbf3 37506
                  21.19.4.20  Existential uniqueness   bj-eu3f 37508
                  ​*21.19.4.21  First-order logic: miscellaneous   bj-sblem1 37509
            21.19.5  Set theory   eliminable1 37526
                  ​*21.19.5.1  Eliminability of class terms   eliminable1 37526
                  ​*21.19.5.2  Classes without the axiom of extensionality   bj-denoteslem 37538
                  21.19.5.3  Characterization among sets versus among classes   elelb 37564
                  ​*21.19.5.4  The nonfreeness quantifier for classes   bj-nfcsym 37566
                  ​*21.19.5.5  Lemmas for class substitution   bj-sbeqALT 37567
                  21.19.5.6  Removing some axiom requirements and disjoint variable conditions   bj-exlimvmpi 37578
                  ​*21.19.5.7  Class abstractions   bj-elabd2ALT 37593
                  21.19.5.8  Generalized class abstractions   bj-cgab 37601
                  ​*21.19.5.9  Restricted nonfreeness   wrnf 37609
                  ​*21.19.5.10  Russell's paradox   bj-ru1 37611
                  21.19.5.11  Curry's paradox in set theory   currysetlem 37613
                  ​*21.19.5.12  Some disjointness results   bj-n0i 37619
                  ​*21.19.5.13  Complements on direct products   bj-xpimasn 37623
                  ​*21.19.5.14  "Singletonization" and tagging   bj-snsetex 37631
                  ​*21.19.5.15  Tuples of classes   bj-cproj 37658
                  ​*21.19.5.16  Set theory: elementary operations relative to a universe   bj-rcleqf 37693
                  ​*21.19.5.17  Axioms for finite unions   bj-abex 37698
                  ​*21.19.5.18  Set theory: miscellaneous   eleq2w2ALT 37715
                  ​*21.19.5.19  Axioms of separation and replacement   bj-axnul 37741
                  ​*21.19.5.20  Evaluation at a class   bj-evaleq 37745
                  21.19.5.21  Elementwise operations   celwise 37753
                  ​*21.19.5.22  Elementwise intersection (families of sets induced on a subset)   bj-rest00 37755
                  21.19.5.23  Moore collections (complements)   bj-raldifsn 37774
                  21.19.5.24  Maps-to notation for functions with three arguments   bj-0nelmpt 37790
                  ​*21.19.5.25  Currying   csethom 37796
                  ​*21.19.5.26  Setting components of extensible structures   cstrset 37808
            ​*21.19.6  Extended real and complex numbers, real and complex projective lines   bj-nfald 37811
                  21.19.6.1  Complements on class abstractions of ordered pairs and binary relations   bj-nfald 37811
                  ​*21.19.6.2  Identity relation (complements)   bj-opabssvv 37826
                  ​*21.19.6.3  Functionalized identity (diagonal in a Cartesian square)   cdiag2 37848
                  ​*21.19.6.4  Direct image and inverse image   cimdir 37854
                  ​*21.19.6.5  Extended numbers and projective lines as sets   cfractemp 37872
                  ​*21.19.6.6  Addition and opposite   caddcc 37913
                  ​*21.19.6.7  Order relation on the extended reals   cltxr 37917
                  ​*21.19.6.8  Argument, multiplication and inverse   carg 37919
                  21.19.6.9  The canonical bijection from the finite ordinals   ciomnn 37925
                  21.19.6.10  Divisibility   cnnbar 37936
            ​*21.19.7  Monoids   bj-smgrpssmgm 37944
                  ​*21.19.7.1  Finite sums in monoids   cfinsum 37959
            ​*21.19.8  Affine, Euclidean, and Cartesian geometry   bj-fvimacnv0 37962
                  *21.19.8.1  Real vector spaces   bj-fvimacnv0 37962
                  ​*21.19.8.2  Complex numbers (supplements)   bj-subcom 37984
                  ​*21.19.8.3  Barycentric coordinates   bj-bary1lem 37986
            21.19.9  Monoid of endomorphisms   cend 37989
      ​21.20  Mathbox for Jim Kingdon
            21.20.1  Circle constant   taupilem3 37995
            21.20.2  Number theory   dfgcd3 38000
            21.20.3  Real numbers   irrdifflemf 38001
      ​21.21  Mathbox for ML
            21.21.1  Miscellaneous   csbrecsg 38006
            21.21.2  Cartesian exponentiation   cfinxp 38061
            21.21.3  Topology   iunctb2 38081
                  ​*21.21.3.1  Pi-base theorems   pibp16 38091
      ​21.22  Mathbox for Wolf Lammen
            21.22.1  1. Bootstrapping   wl-section-boot 38100
            21.22.2  Implication chains   wl-section-impchain 38124
            21.22.3  Theorems around the conditional operator   wl-ifp-ncond1 38142
            21.22.4  Alternative development of hadd, cadd   wl-df-3xor 38146
            21.22.5  An alternative axiom ~ ax-13   ax-wl-13v 38171
            21.22.6  Bootstrapping set theory with classes   wl-cleq-0 38173
            21.22.7  Other stuff   wl-mps 38194
      ​21.23  Mathbox for Brendan Leahy
      ​21.24  Mathbox for Jeff Madsen
            21.24.1  Logic and set theory   unirep 38397
            21.24.2  Real and complex numbers; integers   filbcmb 38423
            21.24.3  Sequences and sums   sdclem2 38425
            21.24.4  Topology   subspopn 38435
            21.24.5  Metric spaces   metf1o 38438
            21.24.6  Continuous maps and homeomorphisms   constcncf 38445
            21.24.7  Boundedness   ctotbnd 38449
            21.24.8  Isometries   cismty 38481
            21.24.9  Heine-Borel Theorem   heibor1lem 38492
            21.24.10  Banach Fixed Point Theorem   bfplem1 38505
            21.24.11  Euclidean space   crrn 38508
            21.24.12  Intervals (continued)   ismrer1 38521
            21.24.13  Operation properties   cass 38525
            21.24.14  Groups and related structures   cmagm 38531
            21.24.15  Group homomorphism and isomorphism   cghomOLD 38566
            21.24.16  Rings   crngo 38577
            21.24.17  Division Rings   cdrng 38631
            21.24.18  Ring homomorphisms   crngohom 38643
            21.24.19  Commutative rings   ccm2 38672
            21.24.20  Ideals   cidl 38690
            21.24.21  Prime rings and integral domains   cprrng 38729
            21.24.22  Ideal generators   cigen 38742
      ​21.25  Mathbox for Giovanni Mascellani
            ​*21.25.1  Tools for automatic proof building   efald2 38761
            ​*21.25.2  Tseitin axioms   fald 38810
            ​*21.25.3  Equality deductions   iuneq2f 38837
            ​*21.25.4  Miscellanea   orcomdd 38848
      ​21.26  Mathbox for Peter Mazsa
            21.26.1  Notations   cxrn 38855
            21.26.2  Preparatory theorems   el2v1 38910
            21.26.3  Range Cartesian product   df-xrn 39061
            21.26.4  Relations   df-rels 39121
            21.26.5  Quotient map (coset map)   df-qmap 39127
            21.26.6  Lifts, shifts, successor, and predecessor   df-adjliftmap 39136
            21.26.7  Cosets by ` R `   df-coss 39182
            21.26.8  Subset relations   df-ssr 39259
            21.26.9  Reflexivity   df-refs 39271
            21.26.10  Converse reflexivity   df-cnvrefs 39286
            21.26.11  Symmetry   df-syms 39303
            21.26.12  Reflexivity and symmetry   symrefref2 39328
            21.26.13  Transitivity   df-trs 39337
            21.26.14  Equivalence relations   df-eqvrels 39349
            21.26.15  Redundancy   df-redunds 39388
            21.26.16  Domain quotients   df-dmqss 39403
            21.26.17  Equivalence relations on domain quotients   df-ers 39429
            21.26.18  Functions   df-funss 39446
            21.26.19  Disjoints vs. converse functions   df-disjss 39469
            21.26.20  Antisymmetry   df-antisymrel 39544
            21.26.21  Partitions: disjoints on domain quotients   df-parts 39549
            21.26.22  Partition-Equivalence Theorems   disjim 39565
            21.26.23  Type-safe Partition-Equivalence: PetParts, PetErs, Pet2Parts, Pet2Ers   df-petparts 39649
      ​21.27  Mathbox for Rodolfo Medina
            21.27.1  Partitions   prtlem60 39659
      ​​*21.28  Mathbox for Norm Megill
            *21.28.1  Obsolete schemes ax-c4,c5,c7,c10,c11,c11n,c15,c9,c14,c16   ax-c5 39689
            ​*21.28.2  Rederive new axioms ax-4, ax-10, ax-6, ax-12, ax-13 from old   axc5 39699
            ​*21.28.3  Legacy theorems using obsolete axioms   ax5ALT 39713
            21.28.4  Experiments with weak deduction theorem   elimhyps 39767
            21.28.5  Miscellanea   cnaddcom 39778
            21.28.6  Atoms, hyperplanes, and covering in a left vector space (or module)   clsa 39780
            21.28.7  Functionals and kernels of a left vector space (or module)   clfn 39863
            21.28.8  Opposite rings and dual vector spaces   cld 39929
            21.28.9  Ortholattices and orthomodular lattices   cops 39978
            21.28.10  Atomic lattices with covering property   ccvr 40068
            21.28.11  Hilbert lattices   chlt 40156
            21.28.12  Projective geometries based on Hilbert lattices   clln 40297
            21.28.13  Construction of a vector space from a Hilbert lattice   cdlema1N 40597
            21.28.14  Construction of involution and inner product from a Hilbert lattice   clpoN 42286
      ​21.29  Mathbox for metakunt
            21.29.1  Commutative Semiring   ccsrg 42768
            21.29.2  General helpful statements   rhmzrhval 42771
            21.29.3  Some gcd and lcm results   12gcd5e1 42802
            21.29.4  Least common multiple inequality theorem   3factsumint1 42820
            21.29.5  Logarithm inequalities   3exp7 42852
            21.29.6  Miscellaneous results for AKS formalisation   intlewftc 42860
            21.29.7  Sticks and stones   sticksstones1 42945
            21.29.8  Continuation AKS   aks6d1c6lem1 42969
      ​21.30  Mathbox for Luke Murphy
            21.30.1  Solutions of quadratic equations   quadfac 43004
            21.30.2  April Fool's theorem   25or6to4 43005
      ​21.31  Mathbox for Steven Nguyen
            21.31.1  Utility theorems   jarrii 43006
            ​*21.31.2  Arithmetic theorems   c0exALT 43052
            21.31.3  Exponents and divisibility   oexpreposd 43115
            21.31.4  Trigonometry and Calculus   tanhalfpim 43142
            ​*21.31.5  Independence of ax-mulcom   cresub 43158
            21.31.6  Structures   sn-base0 43301
            ​*21.31.7  Projective spaces   cprjsp 43365
            21.31.8  Basic reductions for Fermat's Last Theorem   dffltz 43398
            ​*21.31.9  Exemplar theorems   iddii 43428
                  ​*21.31.9.1  Standard replacements of ax-10 , ax-11 , ax-12   nfa1w 43439
      ​21.32  Mathbox for Igor Ieskov
      ​21.33  Mathbox for OpenAI
      ​21.34  Mathbox for Stefan O'Rear
            21.34.1  Additional elementary logic and set theory   moxfr 43455
            21.34.2  Additional theory of functions   imaiinfv 43456
            21.34.3  Additional topology   elrfi 43457
            21.34.4  Characterization of closure operators. Kuratowski closure axioms   ismrcd1 43461
            21.34.5  Algebraic closure systems   cnacs 43465
            21.34.6  Miscellanea 1. Map utilities   constmap 43476
            21.34.7  Miscellanea for polynomials   mptfcl 43483
            21.34.8  Multivariate polynomials over the integers   cmzpcl 43484
            21.34.9  Miscellanea for Diophantine sets 1   coeq0i 43516
            21.34.10  Diophantine sets 1: definitions   cdioph 43518
            21.34.11  Diophantine sets 2 miscellanea   ellz1 43530
            21.34.12  Diophantine sets 2: union and intersection. Monotone Boolean algebra   diophin 43535
            21.34.13  Diophantine sets 3: construction   diophrex 43538
            21.34.14  Diophantine sets 4 miscellanea   2sbcrex 43547
            21.34.15  Diophantine sets 4: Quantification   rexrabdioph 43553
            21.34.16  Diophantine sets 5: Arithmetic sets   rabdiophlem1 43560
            21.34.17  Diophantine sets 6: reusability. renumbering of variables   eldioph4b 43570
            21.34.18  Pigeonhole Principle and cardinality helpers   fphpd 43575
            21.34.19  A non-closed set of reals is infinite   rencldnfilem 43579
            21.34.20  Lagrange's rational approximation theorem   irrapxlem1 43581
            21.34.21  Pell equations 1: A nontrivial solution always exists   pellexlem1 43588
            21.34.22  Pell equations 2: Algebraic number theory of the solution set   csquarenn 43595
            21.34.23  Pell equations 3: characterizing fundamental solution   infmrgelbi 43637
            ​*21.34.24  Logarithm laws generalized to an arbitrary base   reglogcl 43649
            21.34.25  Pell equations 4: the positive solution group is infinite cyclic   pellfund14 43657
            21.34.26  X and Y sequences 1: Definition and recurrence laws   crmx 43659
            21.34.27  Ordering and induction lemmas for the integers   monotuz 43700
            21.34.28  X and Y sequences 2: Order properties   rmxypos 43706
            21.34.29  Congruential equations   congtr 43724
            21.34.30  Alternating congruential equations   acongid 43734
            21.34.31  Additional theorems on integer divisibility   coprmdvdsb 43744
            21.34.32  X and Y sequences 3: Divisibility properties   jm2.18 43747
            21.34.33  X and Y sequences 4: Diophantine representability of Y   jm2.27a 43764
            21.34.34  X and Y sequences 5: Diophantine representability of X, ^, _C   rmxdiophlem 43774
            21.34.35  Uncategorized stuff not associated with a major project   setindtr 43783
            21.34.36  More equivalents of the Axiom of Choice   axac10 43792
            21.34.37  Finitely generated left modules   clfig 43826
            21.34.38  Noetherian left modules I   clnm 43834
            21.34.39  Addenda for structure powers   pwssplit4 43848
            21.34.40  Every set admits a group structure iff choice   unxpwdom3 43854
            21.34.41  Noetherian rings and left modules II   clnr 43868
            21.34.42  Hilbert's Basis Theorem   cldgis 43880
            21.34.43  Additional material on polynomials [DEPRECATED]   cmnc 43890
            21.34.44  Degree and minimal polynomial of algebraic numbers   cdgraa 43899
            21.34.45  Algebraic integers I   citgo 43916
            21.34.46  Endomorphism algebra   cmend 43930
            21.34.47  Cyclic groups and order   idomodle 43950
            21.34.48  Cyclotomic polynomials   ccytp 43956
            21.34.49  Miscellaneous topology   fgraphopab 43962
      ​21.35  Mathbox for Noam Pasman
      ​21.36  Mathbox for Jon Pennant
      ​21.37  Mathbox for Richard Penner
            21.37.1  Set Theory and Ordinal Numbers   uniel 43976
            21.37.2  Natural addition of Cantor normal forms   oawordex2 44085
            21.37.3  Surreal Contributions   abeqabi 44166
            21.37.4  Short Studies   nlimsuc 44199
                  21.37.4.1  Additional work on conditional logical operator   ifpan123g 44217
                  21.37.4.2  Sophisms   rp-fakeimass 44270
                  ​*21.37.4.3  Finite Sets   rp-isfinite5 44275
                  21.37.4.4  General Observations   intabssd 44277
                  21.37.4.5  Infinite Sets   pwelg 44318
                  ​*21.37.4.6  Finite intersection property   fipjust 44323
                  21.37.4.7  RP ADDTO: Subclasses and subsets   rababg 44332
                  21.37.4.8  RP ADDTO: The intersection of a class   elinintab 44333
                  21.37.4.9  RP ADDTO: Theorems requiring subset and intersection existence   elinintrab 44335
                  21.37.4.10  RP ADDTO: Relations   xpinintabd 44338
                  ​*21.37.4.11  RP ADDTO: Functions   elmapintab 44354
                  ​*21.37.4.12  RP ADDTO: Finite induction (for finite ordinals)   cnvcnvintabd 44358
                  21.37.4.13  RP ADDTO: First and second members of an ordered pair   elcnvlem 44359
                  21.37.4.14  RP ADDTO: The reflexive and transitive properties of relations   undmrnresiss 44362
                  21.37.4.15  RP ADDTO: Basic properties of closures   cleq2lem 44366
                  21.37.4.16  RP REPLACE: Definitions and basic properties of transitive closures   trcleq2lemRP 44388
                  ​*21.37.4.17  Additions for square root; absolute value   sqrtcvallem1 44389
            21.37.5  Additional statements on relations and subclasses   al3im 44405
                  21.37.5.1  Transitive relations (not to be confused with transitive classes)   trrelind 44423
                  21.37.5.2  Reflexive closures   crcl 44430
                  ​*21.37.5.3  Finite relationship composition   relexp2 44435
                  21.37.5.4  Transitive closure of a relation   dftrcl3 44478
                  ​*21.37.5.5  Adapted from Frege   frege77d 44504
            ​*21.37.6  Propositions from _Begriffsschrift_   dfxor4 44524
                  *21.37.6.1  _Begriffsschrift_ Chapter I   dfxor4 44524
                  ​*21.37.6.2  _Begriffsschrift_ Notation hints   whe 44530
                  21.37.6.3  _Begriffsschrift_ Chapter II Implication   ax-frege1 44548
                  21.37.6.4  _Begriffsschrift_ Chapter II Implication and Negation   axfrege28 44587
                  ​*21.37.6.5  _Begriffsschrift_ Chapter II with logical equivalence   axfrege52a 44614
                  21.37.6.6  _Begriffsschrift_ Chapter II with equivalence of sets   axfrege52c 44645
                  ​*21.37.6.7  _Begriffsschrift_ Chapter II with equivalence of classes   frege53c 44672
                  ​*21.37.6.8  _Begriffsschrift_ Chapter III Properties hereditary in a sequence   dffrege69 44690
                  ​*21.37.6.9  _Begriffsschrift_ Chapter III Following in a sequence   dffrege76 44697
                  ​*21.37.6.10  _Begriffsschrift_ Chapter III Member of sequence   dffrege99 44720
                  ​*21.37.6.11  _Begriffsschrift_ Chapter III Single-valued procedures   dffrege115 44736
            ​*21.37.7  Exploring Topology via Seifert and Threlfall   enrelmap 44755
                  *21.37.7.1  Equinumerosity of sets of relations and maps   enrelmap 44755
                  ​*21.37.7.2  Generic Pseudoclosure Spaces, Pseudointerior Spaces, and Pseudoneighborhoods   or3or 44781
                  ​*21.37.7.3  Generic Neighborhood Spaces   gneispa 44888
            ​*21.37.8  Exploring Higher Homotopy via Kerodon   k0004lem1 44905
                  *21.37.8.1  Simplicial Sets   k0004lem1 44905
      ​21.38  Mathbox for Stanislas Polu
            21.38.1  IMO Problems   wwlemuld 44914
                  21.38.1.1  IMO 1972 B2   wwlemuld 44914
            ​*21.38.2  INT Inequalities Proof Generator   int-addcomd 44931
            ​*21.38.3  N-Digit Addition Proof Generator   unitadd 44953
            21.38.4  AM-GM (for k = 2,3,4)   gsumws3 44954
      ​21.39  Mathbox for Rohan Ridenour
            21.39.1  Misc   spALT 44959
            21.39.2  Monoid rings   cmnring 44967
            21.39.3  Shorter primitive equivalent of ax-groth   gru0eld 44985
                  21.39.3.1  Grothendieck universes are closed under collection   gru0eld 44985
                  21.39.3.2  Minimal universes   ismnu 45003
                  21.39.3.3  Primitive equivalent of ax-groth   expandan 45030
      ​21.40  Mathbox for Steve Rodriguez
            21.40.1  Miscellanea   nanorxor 45047
            21.40.2  Ratio test for infinite series convergence and divergence   dvgrat 45054
            21.40.3  Multiples   reldvds 45057
            21.40.4  Function operations   caofcan 45065
            21.40.5  Calculus   lhe4.4ex1a 45071
            21.40.6  The generalized binomial coefficient operation   cbcc 45078
            21.40.7  Binomial series   uzmptshftfval 45088
      ​21.41  Mathbox for Andrew Salmon
            21.41.1  Principia Mathematica * 10   pm10.12 45100
            21.41.2  Principia Mathematica * 11   2alanimi 45114
            21.41.3  Predicate Calculus   sbeqal1 45140
            21.41.4  Principia Mathematica * 13 and * 14   pm13.13a 45149
            21.41.5  Set Theory   elnev 45179
            21.41.6  Arithmetic   addcomgi 45196
            21.41.7  Geometry   cplusr 45197
      ​​*21.42  Mathbox for Alan Sare
            21.42.1  Auxiliary theorems for the Virtual Deduction tool   idiALT 45219
            21.42.2  Supplementary unification deductions   bi1imp 45223
            21.42.3  Conventional Metamath proofs, some derived from VD proofs   iidn3 45242
            21.42.4  What is Virtual Deduction?   wvd1 45310
            21.42.5  Virtual Deduction Theorems   df-vd1 45311
            21.42.6  Theorems proved using Virtual Deduction   trsspwALT 45558
            21.42.7  Theorems proved using Virtual Deduction with mmj2 assistance   simplbi2VD 45586
            21.42.8  Virtual Deduction transcriptions of textbook proofs   sb5ALTVD 45653
            21.42.9  Theorems proved using conjunction-form Virtual Deduction   elpwgdedVD 45657
            21.42.10  Theorems with a VD proof in conventional notation derived from a VD proof   suctrALT3 45664
            ​*21.42.11  Theorems with a proof in conventional notation derived from a VD proof   notnotrALT2 45667
      ​21.43  Mathbox for Eric Schmidt
            21.43.1  Miscellany   rspesbcd 45678
            21.43.2  Study of dfbi1ALT   dfbi1ALTa 45680
            21.43.3  Relation-preserving functions   wrelp 45683
            21.43.4  Orbits   orbitex 45696
            21.43.5  Well-founded sets   trwf 45700
            21.43.6  Absoluteness in transitive models   ralabso 45709
            21.43.7  Lemmas for showing axioms hold in models   traxext 45718
            21.43.8  The class of well-founded sets is a model for ZFC   wfaxext 45734
            21.43.9  Permutation models   brpermmodel 45744
            21.43.10  Isomorphism of finite ordinals and non-negative integers   hashnna 45760
      ​21.44  Mathbox for Glauco Siliprandi
            21.44.1  Miscellanea   evth2f 45767
            21.44.2  Functions   fnresdmss 45918
            21.44.3  Ordering on real numbers - Real and complex numbers basic operations   sub2times 46024
            21.44.4  Real intervals   gtnelioc 46239
            21.44.5  Finite sums   fsummulc1f 46319
            21.44.6  Finite multiplication of numbers and finite multiplication of functions   fmul01 46328
            21.44.7  Limits   clim1fr1 46349
                  21.44.7.1  Inferior limit (lim inf)   clsi 46497
                  ​*21.44.7.2  Limits for sequences of extended real numbers   clsxlim 46564
            21.44.8  Trigonometry   coseq0 46610
            21.44.9  Continuous Functions   mulcncff 46616
            21.44.10  Derivatives   dvsinexp 46657
            21.44.11  Integrals   itgsin0pilem1 46696
            21.44.12  Stone Weierstrass theorem - real version   stoweidlem1 46747
            21.44.13  Wallis' product for π   wallispilem1 46811
            21.44.14  Stirling's approximation formula for ` n ` factorial   stirlinglem1 46820
            21.44.15  Dirichlet kernel   dirkerval 46837
            21.44.16  Fourier Series   fourierdlem1 46854
            21.44.17  e is transcendental   elaa2lem 46979
            21.44.18  n-dimensional Euclidean space   rrxtopn 47030
            21.44.19  Basic measure theory   csalg 47054
                  ​*21.44.19.1  σ-Algebras   csalg 47054
                  21.44.19.2  Sum of nonnegative extended reals   csumge0 47108
                  ​*21.44.19.3  Measures   cmea 47195
                  ​*21.44.19.4  Outer measures and Caratheodory's construction   come 47235
                  ​*21.44.19.5  Lebesgue measure on n-dimensional Real numbers   covoln 47282
                  ​*21.44.19.6  Measurable functions   csmblfn 47441
      ​21.45  Mathbox for Saveliy Skresanov
            21.45.1  Ceva's theorem   sigarval 47596
            21.45.2  Simple groups   simpcntrab 47616
      ​21.46  Mathbox for Ender Ting
            21.46.1  Interesting facts   et-ltneverrefl 47617
            21.46.2  Increasing sequences and subsequences   ormklocald 47622
            21.46.3  Scratchpad for number theory   evenwodadd 47634
            21.46.4  Scratchpad for math on real numbers   squeezedltsq 47635
      ​21.47  Mathbox for Jarvin Udandy
      ​21.48  Mathbox for Adhemar
            ​*21.48.1  Minimal implicational calculus   adh-minim 47770
      ​21.49  Mathbox for Alexander van der Vekens
            21.49.1  General auxiliary theorems (1)   n0nsn2el 47794
                  21.49.1.1  Unordered and ordered pairs - extension for singletons   n0nsn2el 47794
                  21.49.1.2  Unordered and ordered pairs - extension for unordered pairs   elprneb 47798
                  21.49.1.3  Unordered and ordered pairs - extension for ordered pairs   oppr 47799
                  21.49.1.4  Relations - extension   eubrv 47804
                  21.49.1.5  Definite description binder (inverted iota) - extension   iota0def 47807
                  21.49.1.6  Functions - extension   fveqvfvv 47809
            21.49.2  Alternative for Russell's definition of a description binder   caiota 47852
            21.49.3  Double restricted existential uniqueness   r19.32 47867
                  21.49.3.1  Restricted quantification (extension)   r19.32 47867
                  21.49.3.2  Restricted uniqueness and "at most one" quantification   reuf1odnf 47876
                  21.49.3.3  Analogs to Existential uniqueness (double quantification)   2reu3 47879
                  21.49.3.4  Additional theorems for double restricted existential uniqueness   2reu8i 47882
            ​*21.49.4  Alternative definitions of function and operation values   wdfat 47885
                  21.49.4.1  Restricted quantification (extension)   ralbinrald 47891
                  21.49.4.2  The universal class (extension)   nvelim 47892
                  21.49.4.3  Introduce the Axiom of Power Sets (extension)   alneu 47893
                  21.49.4.4  Predicate "defined at"   dfateq12d 47895
                  21.49.4.5  Alternative definition of the value of a function   dfafv2 47901
                  21.49.4.6  Alternative definition of the value of an operation   aoveq123d 47947
            ​*21.49.5  Alternative definitions of function values (2)   cafv2 47977
            21.49.6  General auxiliary theorems (2)   an4com24 48037
                  21.49.6.1  Logical conjunction - extension   an4com24 48037
                  21.49.6.2  Abbreviated conjunction and disjunction of three wff's - extension   3an4ancom24 48038
                  21.49.6.3  Negated membership (alternative)   cnelbr 48040
                  21.49.6.4  The empty set - extension   ralralimp 48047
                  21.49.6.5  Indexed union and intersection - extension   otiunsndisjX 48048
                  21.49.6.6  Functions - extension   fvifeq 48049
                  21.49.6.7  Maps-to notation - extension   fvmptrab 48061
                  21.49.6.8  Subtraction - extension   cnambpcma 48063
                  21.49.6.9  Ordering on reals (cont.) - extension   leaddsuble 48066
                  21.49.6.10  Imaginary and complex number properties - extension   readdcnnred 48072
                  21.49.6.11  Nonnegative integers (as a subset of complex numbers) - extension   nn0resubcl 48077
                  21.49.6.12  Integers (as a subset of complex numbers) - extension   zgeltp1eq 48078
                  21.49.6.13  Decimal arithmetic - extension   1t10e1p1e11 48079
                  21.49.6.14  Upper sets of integers - extension   eluzge0nn0 48081
                  21.49.6.15  Infinity and the extended real number system (cont.) - extension   nltle2tri 48082
                  21.49.6.16  Finite intervals of integers - extension   ssfz12 48083
                  21.49.6.17  Half-open integer ranges - extension   fzopred 48092
                  21.49.6.18  The floor and ceiling functions - extension   2ltceilhalf 48101
                  21.49.6.19  The modulo (remainder) operation - extension   fldivmod 48113
                  21.49.6.20  The infinite sequence builder "seq"   smonoord 48146
                  21.49.6.21  Integer powers - extension   2timesltsq 48147
                  21.49.6.22  Finite and infinite sums - extension   fsummsndifre 48149
                  21.49.6.23  The divides relation - extension   nndivides2 48153
                  21.49.6.24  Extensible structures - extension   setsidel 48157
            ​*21.49.7  Preimages of function values   preimafvsnel 48160
            ​*21.49.8  Partitions of real intervals   ciccp 48194
            21.49.9  Shifting functions with an integer range domain   fargshiftfv 48220
            21.49.10  Words over a set (extension)   lswn0 48225
                  21.49.10.1  Last symbol of a word - extension   lswn0 48225
            21.49.11  Unordered pairs   wich 48226
                  21.49.11.1  Interchangeable setvar variables   wich 48226
                  21.49.11.2  Set of unordered pairs   sprid 48255
                  ​*21.49.11.3  Proper (unordered) pairs   prpair 48282
                  21.49.11.4  Set of proper unordered pairs   cprpr 48293
            21.49.12  Number theory (extension)   nprmmul1 48308
                  21.49.12.1  Properties of non-prime numbers   nprmmul1 48308
                  ​*21.49.12.2  Fermat numbers   cfmtno 48311
                  ​*21.49.12.3  Mersenne primes   m2prm 48375
                  21.49.12.4  Proth's theorem   modexp2m1d 48396
                  21.49.12.5  The prime-counting function according to Ján Mináč   nprmdvdsfacm1lem1 48404
                  21.49.12.6  Solutions of quadratic equations   quad1 48417
            ​*21.49.13  Even and odd numbers   ceven 48421
                  21.49.13.1  Definitions and basic properties   ceven 48421
                  21.49.13.2  Alternate definitions using the "divides" relation   dfeven2 48446
                  21.49.13.3  Alternate definitions using the "modulo" operation   dfeven3 48455
                  21.49.13.4  Alternate definitions using the "gcd" operation   iseven5 48461
                  21.49.13.5  Theorems of part 5 revised   zneoALTV 48466
                  21.49.13.6  Theorems of part 6 revised   odd2np1ALTV 48471
                  21.49.13.7  Theorems of AV's mathbox revised   0evenALTV 48485
                  21.49.13.8  Additional theorems   epoo 48500
                  21.49.13.9  Perfect Number Theorem (revised)   perfectALTVlem1 48518
            21.49.14  Number theory (extension 2)   cfppr 48521
                  ​*21.49.14.1  Fermat pseudoprimes   cfppr 48521
                  ​*21.49.14.2  Goldbach's conjectures   cgbe 48542
            21.49.15  Graph theory (extension)   cclnbgr 48615
                  21.49.15.1  Closed neighborhood of a vertex   cclnbgr 48615
                  ​*21.49.15.2  Semiclosed and semiopen neighborhoods (experimental)   dfsclnbgr2 48643
                  21.49.15.3  Induced subgraphs   cisubgr 48657
                  ​*21.49.15.4  Isomorphisms of graphs   cgrisom 48671
                  ​*21.49.15.5  Triangles in graphs   cgrtri 48734
                  ​*21.49.15.6  Star graphs   cstgr 48748
                  ​*21.49.15.7  Local isomorphisms of graphs   cgrlim 48773
                  ​*21.49.15.8  Generalized Petersen graphs   cgpg 48837
                  21.49.15.9  Loop-free graphs - extension   1hegrlfgr 48929
                  21.49.15.10  Walks - extension   cupwlks 48930
                  21.49.15.11  Edges of graphs expressed as sets of unordered pairs   upgredgssspr 48940
            21.49.16  Monoids (extension)   ovn0dmfun 48953
                  21.49.16.1  Auxiliary theorems   ovn0dmfun 48953
                  21.49.16.2  Magmas, Semigroups and Monoids (extension)   plusfreseq 48961
                  21.49.16.3  Examples and counterexamples for magmas, semigroups and monoids (extension)   opmpoismgm 48964
                  21.49.16.4  Group sum operation (extension 1)   gsumsplit2f 48977
            ​*21.49.17  Magmas and internal binary operations (alternate approach)   ccllaw 48980
                  *21.49.17.1  Laws for internal binary operations   ccllaw 48980
                  ​*21.49.17.2  Internal binary operations   cintop 48993
                  21.49.17.3  Alternative definitions for magmas and semigroups   cmgm2 49012
            21.49.18  Rings (extension)   lmod0rng 49026
                  21.49.18.1  Nonzero rings (extension)   lmod0rng 49026
                  21.49.18.2  Ideals as non-unital rings   lidldomn1 49028
                  21.49.18.3  The non-unital ring of even integers   0even 49034
                  21.49.18.4  A constructed not unital ring   cznrnglem 49056
                  ​*21.49.18.5  The category of non-unital rings (alternate definition)   crngcALTV 49060
                  ​*21.49.18.6  The category of (unital) rings (alternate definition)   cringcALTV 49084
            ​*21.49.19  Prime rings (and integral domains)   cprmrng 49131
            21.49.20  Basic algebraic structures (extension)   eliunxp2 49146
                  21.49.20.1  Auxiliary theorems   eliunxp2 49146
                  21.49.20.2  The binomial coefficient operation (extension)   bcpascm1 49163
                  21.49.20.3  The ` ZZ `-module ` ZZ X. ZZ `   zlmodzxzlmod 49166
                  21.49.20.4  Group sum operation (extension 2)   mgpsumunsn 49173
                  21.49.20.5  Symmetric groups (extension)   exple2lt6 49176
                  21.49.20.6  Divisibility (extension)   invginvrid 49179
                  21.49.20.7  The support of functions (extension)   rmsupp0 49180
                  21.49.20.8  Finitely supported functions (extension)   rmsuppfi 49184
                  21.49.20.9  Left modules (extension)   lmodvsmdi 49191
                  21.49.20.10  Associative algebras (extension)   assaascl0 49193
                  21.49.20.11  Univariate polynomials (extension)   ply1vr1smo 49195
                  21.49.20.12  Univariate polynomials (examples)   linply1 49205
            21.49.21  Linear algebra (extension)   cdmatalt 49208
                  ​*21.49.21.1  The subalgebras of diagonal and scalar matrices (extension)   cdmatalt 49208
                  ​*21.49.21.2  Linear combinations   clinc 49216
                  ​*21.49.21.3  Linear independence   clininds 49252
                  21.49.21.4  Simple left modules and the ` ZZ `-module   lmod1lem1 49299
                  21.49.21.5  Differences between (left) modules and (left) vector spaces   lvecpsslmod 49319
            21.49.22  Complexity theory   suppdm 49322
                  21.49.22.1  Auxiliary theorems   suppdm 49322
                  21.49.22.2  Even and odd integers   nn0onn0ex 49335
                  21.49.22.3  The natural logarithm on complex numbers (extension)   logcxp0 49347
                  21.49.22.4  Division of functions   cfdiv 49349
                  21.49.22.5  Upper bounds   cbigo 49359
                  21.49.22.6  Logarithm to an arbitrary base (extension)   rege1logbrege0 49370
                  ​*21.49.22.7  The binary logarithm   fldivexpfllog2 49377
                  21.49.22.8  Binary length   cblen 49381
                  ​*21.49.22.9  Digits   cdig 49407
                  21.49.22.10  Nonnegative integer as sum of its shifted digits   dignn0flhalflem1 49427
                  21.49.22.11  Algorithms for the multiplication of nonnegative integers   nn0mulfsum 49436
                  ​*21.49.22.12  N-ary functions   cnaryf 49438
                  ​*21.49.22.13  The Ackermann function   citco 49469
            21.49.23  Elementary geometry (extension)   fv1prop 49511
                  21.49.23.1  Auxiliary theorems   fv1prop 49511
                  21.49.23.2  Real euclidean space of dimension 2   rrx2pxel 49523
                  21.49.23.3  Spheres and lines in real Euclidean spaces   cline 49539
      ​21.50  Mathbox for Zhi Wang
            21.50.1  Propositional calculus   logic1 49601
            21.50.2  Predicate calculus with equality   dtrucor3 49609
                  21.50.2.1  Axiom scheme ax-5 (Distinctness)   dtrucor3 49609
            21.50.3  ZF Set Theory - start with the Axiom of Extensionality   ralbidb 49610
                  21.50.3.1  Restricted quantification   ralbidb 49610
                  21.50.3.2  The universal class   reuxfr1dd 49617
                  21.50.3.3  The empty set   ssdisjd 49618
                  21.50.3.4  Unordered and ordered pairs   vsn 49622
                  21.50.3.5  The union of a class   unilbss 49628
                  21.50.3.6  Indexed union and intersection   iuneq0 49629
            21.50.4  ZF Set Theory - add the Axiom of Replacement   inpw 49635
                  21.50.4.1  Theorems requiring subset and intersection existence   inpw 49635
            21.50.5  ZF Set Theory - add the Axiom of Power Sets   opth1neg 49636
                  21.50.5.1  Ordered pair theorem   opth1neg 49636
                  21.50.5.2  Ordered-pair class abstractions (cont.)   brab2dd 49638
                  21.50.5.3  Relations   iinxp 49641
                  21.50.5.4  Functions   mof0 49648
                  21.50.5.5  Operations   ovsng 49668
            21.50.6  ZF Set Theory - add the Axiom of Union   fonex 49677
                  21.50.6.1  Relations and functions (cont.)   fonex 49677
                  21.50.6.2  First and second members of an ordered pair   eloprab1st2nd 49678
                  21.50.6.3  Operations in maps-to notation (continued)   fmpodg 49679
                  21.50.6.4  Function transposition   resinsnlem 49681
                  21.50.6.5  Infinite Cartesian products   ixpv 49700
                  21.50.6.6  Equinumerosity   fvconst0ci 49701
            21.50.7  Order sets   iccin 49706
                  21.50.7.1  Real number intervals   iccin 49706
            21.50.8  Extensible structures   slotresfo 49709
                  21.50.8.1  Basic definitions   slotresfo 49709
            21.50.9  Moore spaces   mreuniss 49710
            ​*21.50.10  Topology   clduni 49711
                  21.50.10.1  Closure and interior   clduni 49711
                  21.50.10.2  Neighborhoods   neircl 49715
                  21.50.10.3  Subspace topologies   restcls2lem 49723
                  21.50.10.4  Limits and continuity in topological spaces   cnneiima 49727
                  21.50.10.5  Topological definitions using the reals   iooii 49728
                  21.50.10.6  Separated sets   sepnsepolem1 49732
                  21.50.10.7  Separated spaces: T0, T1, T2 (Hausdorff) ...   isnrm4 49741
            21.50.11  Preordered sets and directed sets using extensible structures   isprsd 49765
            21.50.12  Posets and lattices using extensible structures   lubeldm2 49766
                  21.50.12.1  Posets   lubeldm2 49766
                  21.50.12.2  Lattices   toslat 49792
                  21.50.12.3  Subset order structures   intubeu 49794
            21.50.13  Rings   elmgpcntrd 49815
                  21.50.13.1  Multiplicative Group   elmgpcntrd 49815
            21.50.14  Associative algebras   asclelbasALT 49816
                  21.50.14.1  Definition and basic properties   asclelbasALT 49816
            21.50.15  Categories   homf0 49819
                  21.50.15.1  Categories   homf0 49819
                  21.50.15.2  Opposite category   oppccatb 49826
                  21.50.15.3  Monomorphisms and epimorphisms   idmon 49830
                  21.50.15.4  Sections, inverses, isomorphisms   sectrcl 49832
                  21.50.15.5  Isomorphic objects   cicfn 49852
                  21.50.15.6  Subcategories   dmdm 49863
                  21.50.15.7  Functors   reldmfunc 49885
                  21.50.15.8  Opposite functors   coppf 49932
                  21.50.15.9  Full & faithful functors   imasubc 49961
                  21.50.15.10  Universal property   upciclem1 49976
                  21.50.15.11  Natural transformations and the functor category   isnatd 50033
                  21.50.15.12  Initial, terminal and zero objects of a category   initoo2 50042
                  21.50.15.13  Product of categories   reldmxpc 50056
                  21.50.15.14  Swap functors   cswapf 50069
                  21.50.15.15  Functor evaluation   oppc1stflem 50097
                  21.50.15.16  Transposed curry functors   cofuswapfcl 50103
                  21.50.15.17  Constant functors   diag1 50114
                  21.50.15.18  Functor composition bifunctors   fucofulem1 50120
                  21.50.15.19  Post-composition functors   postcofval 50174
                  21.50.15.20  Pre-composition functors   precofvallem 50176
            21.50.16  Examples of categories   catcrcl 50205
                  21.50.16.1  The category of categories   catcrcl 50205
                  21.50.16.2  Thin categories   cthinc 50227
                  21.50.16.3  Terminal categories   ctermc 50282
                  21.50.16.4  Preordered sets as thin categories   cprstc 50359
                  21.50.16.5  Monoids as categories   cmndtc 50387
                  21.50.16.6  Categories with at most one object and at most two morphisms   2arwcatlem1 50405
            21.50.17  Kan extensions and related concepts   clan 50415
                  21.50.17.1  Kan extensions   clan 50415
                  21.50.17.2  Limits and colimits   clmd 50453
      ​21.51  Mathbox for Emmett Weisz
            ​*21.51.1  Miscellaneous Theorems   nfintd 50483
            21.51.2  Set Recursion   csetrecs 50493
                  ​*21.51.2.1  Basic Properties of Set Recursion   csetrecs 50493
                  21.51.2.2  Examples and properties of set recursion   elsetrecslem 50509
            ​*21.51.3  Construction of Games and Surreal Numbers   cpg 50519
      ​​*21.52  Mathbox for David A. Wheeler
            21.52.1  Natural deduction   sbidd 50528
            ​*21.52.2  Greater than, greater than or equal to   cge-real 50530
            ​*21.52.3  Hyperbolic trigonometric functions   csinh 50540
            ​*21.52.4  Reciprocal trigonometric functions (sec, csc, cot)   csec 50551
            ​*21.52.5  Identities for "if"   ifnmfalse 50573
            ​*21.52.6  Logarithms generalized to arbitrary base using ` logb `   logb2aval 50574
            ​*21.52.7  Logarithm laws generalized to an arbitrary base - log_   clog- 50575
            ​*21.52.8  Formally define notions such as reflexivity   wreflexive 50577
            ​*21.52.9  Algebra helpers   mvlraddi 50581
            ​*21.52.10  Algebra helper examples   i2linesi 50588
            ​*21.52.11  Formal methods "surprises"   alimp-surprise 50590
            ​*21.52.12  Allsome quantifier   wals 50596
            ​*21.52.13  Allsome one quantifier   walseu 50629
            ​*21.52.14  Miscellaneous   5m4e1 50649
            21.52.15  Theorems about algebraic numbers   aacllem 50653
      ​21.53  Mathbox for Jiamin Zhao
            21.53.1  Cross product and scalar triple product in RR^3   1ne3 50654
      ​21.54  Mathbox for Kunhao Zheng
            21.54.1  Weighted AM-GM inequality   amgmwlem 50681