TOC of Theorem List - Metamath Proof Explorer

Table of Contents Summary

PART 1  CLASSICAL FIRST-ORDER LOGIC WITH EQUALITY
      1.1  Pre-logic
      1.2  Propositional calculus
      1.3  Other axiomatizations related to classical propositional calculus
      1.4  Predicate calculus with equality: Tarski's system S2 (1 rule, 6 schemes)
      1.5  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes)
      1.6  Uniqueness and unique existence
      1.7  Other axiomatizations related to classical predicate calculus
PART 2  ZF (ZERMELO-FRAENKEL) SET THEORY
      2.1  ZF Set Theory - start with the Axiom of Extensionality
      2.2  ZF Set Theory - add the Axiom of Replacement
      2.3  ZF Set Theory - add the Axiom of Power Sets
      2.4  ZF Set Theory - add the Axiom of Union
      2.5  ZF Set Theory - add the Axiom of Regularity
      2.6  ZF Set Theory - add the Axiom of Infinity
PART 3  ZFC (ZERMELO-FRAENKEL WITH CHOICE) SET THEORY
      3.1  ZFC Set Theory - add Countable Choice and Dependent Choice
      3.2  ZFC Set Theory - add the Axiom of Choice
      3.3  ZFC Axioms with no distinct variable requirements
      3.4  The Generalized Continuum Hypothesis
PART 4  TG (TARSKI-GROTHENDIECK) SET THEORY
      4.1  Inaccessibles
      4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
PART 5  REAL AND COMPLEX NUMBERS
      5.1  Construction and axiomatization of real and complex numbers
      5.2  Derive the basic properties from the field axioms
      5.3  Real and complex numbers - basic operations
      5.4  Integer sets
      5.5  Order sets
      5.6  Elementary integer functions
      5.7  Words over a set
      5.8  Reflexive and transitive closures of relations
      5.9  Elementary real and complex functions
      5.10  Elementary limits and convergence
      5.11  Elementary trigonometry
      5.12  Cardinality of real and complex number subsets
PART 6  ELEMENTARY NUMBER THEORY
      6.1  Elementary properties of divisibility
      6.2  Elementary prime number theory
PART 7  BASIC STRUCTURES
      7.1  Extensible structures
      7.2  Moore spaces
PART 8  BASIC CATEGORY THEORY
      8.1  Categories
      8.2  Arrows (disjointified hom-sets)
      8.3  Examples of categories
      8.4  Categorical constructions
PART 9  BASIC ORDER THEORY
      9.1  Dual of an order structure
      9.2  Preordered sets and directed sets
      9.3  Partially ordered sets (posets)
      9.4  Totally ordered sets (tosets)
      9.5  Lattices
      9.6  Posets, directed sets, and lattices as relations
PART 10  BASIC ALGEBRAIC STRUCTURES
      10.1  Monoids
      10.2  Groups
      10.3  Rings
      10.4  Division rings and fields
      10.5  Left modules
      10.6  Vector spaces
      10.7  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
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 Scott Fenton
      21.10  Mathbox for Jeff Hankins
      21.11  Mathbox for Anthony Hart
      21.12  Mathbox for Chen-Pang He
      21.13  Mathbox for Jeff Hoffman
      21.14  Mathbox for Asger C. Ipsen
      21.15  Mathbox for BJ
      21.16  Mathbox for Jim Kingdon
      21.17  Mathbox for ML
      21.18  Mathbox for Wolf Lammen
      21.19  Mathbox for Brendan Leahy
      21.20  Mathbox for Jeff Madsen
      21.21  Mathbox for Giovanni Mascellani
      21.22  Mathbox for Peter Mazsa
      21.23  Mathbox for Rodolfo Medina
      21.24  Mathbox for Norm Megill
      21.25  Mathbox for metakunt
      21.26  Mathbox for Steven Nguyen
      21.27  Mathbox for Igor Ieskov
      21.28  Mathbox for OpenAI
      21.29  Mathbox for Stefan O'Rear
      21.30  Mathbox for Noam Pasman
      21.31  Mathbox for Jon Pennant
      21.32  Mathbox for Richard Penner
      21.33  Mathbox for Stanislas Polu
      21.34  Mathbox for Rohan Ridenour
      21.35  Mathbox for Steve Rodriguez
      21.36  Mathbox for Andrew Salmon
      21.37  Mathbox for Alan Sare
      21.38  Mathbox for Glauco Siliprandi
      21.39  Mathbox for Saveliy Skresanov
      21.40  Mathbox for Ender Ting
      21.41  Mathbox for Jarvin Udandy
      21.42  Mathbox for Adhemar
      21.43  Mathbox for Alexander van der Vekens
      21.44  Mathbox for Zhi Wang
      21.45  Mathbox for Emmett Weisz
      21.46  Mathbox for David A. Wheeler
      21.47  Mathbox for Kunhao Zheng

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

​​*PART 1  CLASSICAL FIRST-ORDER LOGIC WITH EQUALITY
      ​*1.1  Pre-logic
            *1.1.1  Inferences for assisting proof development   idi 1
      ​​*1.2  Propositional calculus
            1.2.1  Recursively define primitive wffs for propositional calculus   wn 3
            ​*1.2.2  The axioms of propositional calculus   ax-mp 5
            ​*1.2.3  Logical implication   mp2 9
            ​*1.2.4  Logical negation   con4 113
            ​*1.2.5  Logical equivalence   wb 205
            ​*1.2.6  Logical conjunction   wa 396
            ​*1.2.7  Logical disjunction   wo 845
            ​*1.2.8  Mixed connectives   jaao 953
            ​*1.2.9  The conditional operator for propositions   wif 1061
            ​*1.2.10  The weak deduction theorem for propositional calculus   elimh 1083
            1.2.11  Abbreviated conjunction and disjunction of three wff's   w3o 1086
            1.2.12  Logical "nand" (Sheffer stroke)   wnan 1489
            1.2.13  Logical "xor"   wxo 1509
            1.2.14  Logical "nor"   wnor 1528
            1.2.15  True and false constants   wal 1539
                  ​*1.2.15.1  Universal quantifier for use by df-tru   wal 1539
                  ​*1.2.15.2  Equality predicate for use by df-tru   cv 1540
                  1.2.15.3  The true constant   wtru 1542
                  1.2.15.4  The false constant   wfal 1553
            ​*1.2.16  Truth tables   truimtru 1564
                  1.2.16.1  Implication   truimtru 1564
                  1.2.16.2  Negation   nottru 1568
                  1.2.16.3  Equivalence   trubitru 1570
                  1.2.16.4  Conjunction   truantru 1574
                  1.2.16.5  Disjunction   truortru 1578
                  1.2.16.6  Alternative denial   trunantru 1582
                  1.2.16.7  Exclusive disjunction   truxortru 1586
                  1.2.16.8  Joint denial   trunortru 1590
            ​*1.2.17  Half adder and full adder in propositional calculus   whad 1594
                  1.2.17.1  Full adder: sum   whad 1594
                  1.2.17.2  Full adder: carry   wcad 1607
      ​1.3  Other axiomatizations related to classical propositional calculus
            ​*1.3.1  Minimal implicational calculus   minimp 1623
            ​*1.3.2  Implicational Calculus   impsingle 1629
            1.3.3  Derive the Lukasiewicz axioms from Meredith's sole axiom   meredith 1643
            1.3.4  Derive the standard axioms from the Lukasiewicz axioms   luklem1 1660
            ​*1.3.5  Derive Nicod's axiom from the standard axioms   nic-dfim 1671
            1.3.6  Derive the Lukasiewicz axioms from Nicod's axiom   nic-imp 1677
            1.3.7  Derive Nicod's Axiom from Lukasiewicz's First Sheffer Stroke Axiom   lukshef-ax1 1696
            1.3.8  Derive the Lukasiewicz Axioms from the Tarski-Bernays-Wajsberg Axioms   tbw-bijust 1700
            1.3.9  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's First CO Axiom   merco1 1715
            1.3.10  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's Second CO Axiom   merco2 1738
            1.3.11  Derive the Lukasiewicz axioms from the Russell-Bernays Axioms   rb-bijust 1751
            ​*1.3.12  Stoic logic non-modal portion (Chrysippus of Soli)   mptnan 1770
      ​​*1.4  Predicate calculus with equality: Tarski's system S2 (1 rule, 6 schemes)
            *1.4.1  Universal quantifier (continued); define "exists" and "not free"   wex 1781
                  1.4.1.1  Existential quantifier   wex 1781
                  1.4.1.2  Nonfreeness predicate   wnf 1785
            1.4.2  Rule scheme ax-gen (Generalization)   ax-gen 1797
            1.4.3  Axiom scheme ax-4 (Quantified Implication)   ax-4 1811
                  ​*1.4.3.1  The empty domain of discourse   empty 1909
            1.4.4  Axiom scheme ax-5 (Distinctness) - first use of $d   ax-5 1913
            ​*1.4.5  Equality predicate (continued)   weq 1966
            1.4.6  Axiom scheme ax-6 (Existence)   ax-6 1971
            1.4.7  Axiom scheme ax-7 (Equality)   ax-7 2011
            1.4.8  Define proper substitution   sbjust 2066
            1.4.9  Membership predicate   wcel 2106
            1.4.10  Axiom scheme ax-8 (Left Equality for Binary Predicate)   ax-8 2108
            1.4.11  Axiom scheme ax-9 (Right Equality for Binary Predicate)   ax-9 2116
            ​*1.4.12  Logical redundancy of ax-10 , ax-11 , ax-12 , ax-13   ax6dgen 2124
      ​​*1.5  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes)
            1.5.1  Axiom scheme ax-10 (Quantified Negation)   ax-10 2137
            1.5.2  Axiom scheme ax-11 (Quantifier Commutation)   ax-11 2154
            1.5.3  Axiom scheme ax-12 (Substitution)   ax-12 2171
            1.5.4  Axiom scheme ax-13 (Quantified Equality)   ax-13 2370
      ​1.6  Uniqueness and unique existence
            1.6.1  Uniqueness: the at-most-one quantifier   wmo 2531
            1.6.2  Unique existence: the unique existential quantifier   weu 2561
      ​1.7  Other axiomatizations related to classical predicate calculus
            ​*1.7.1  Aristotelian logic: Assertic syllogisms   barbara 2657
            ​*1.7.2  Intuitionistic logic   axia1 2687
​​*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 2702
            2.1.2  Classes   cab 2708
                  2.1.2.1  Class abstractions   cab 2708
                  ​*2.1.2.2  Class equality   df-cleq 2723
                  2.1.2.3  Class membership   df-clel 2809
                  2.1.2.4  Elementary properties of class abstractions   eqabdv 2866
            2.1.3  Class form not-free predicate   wnfc 2882
            2.1.4  Negated equality and membership   wne 2939
                  2.1.4.1  Negated equality   wne 2939
                  2.1.4.2  Negated membership   wnel 3045
            2.1.5  Restricted quantification   wral 3060
                  2.1.5.1  Restricted universal and existential quantification   wral 3060
                  2.1.5.2  Restricted existential uniqueness and at-most-one quantifier   wreu 3349
                  2.1.5.3  Restricted class abstraction   crab 3405
            2.1.6  The universal class   cvv 3446
            ​*2.1.7  Conditional equality (experimental)   wcdeq 3724
            2.1.8  Russell's Paradox   rru 3740
            2.1.9  Proper substitution of classes for sets   wsbc 3742
            2.1.10  Proper substitution of classes for sets into classes   csb 3858
            2.1.11  Define basic set operations and relations   cdif 3910
            2.1.12  Subclasses and subsets   df-ss 3930
            2.1.13  The difference, union, and intersection of two classes   dfdif3 4079
                  2.1.13.1  The difference of two classes   dfdif3 4079
                  2.1.13.2  The union of two classes   elun 4113
                  2.1.13.3  The intersection of two classes   elini 4158
                  2.1.13.4  The symmetric difference of two classes   csymdif 4206
                  2.1.13.5  Combinations of difference, union, and intersection of two classes   unabs 4219
                  2.1.13.6  Class abstractions with difference, union, and intersection of two classes   unabw 4262
                  2.1.13.7  Restricted uniqueness with difference, union, and intersection   reuun2 4279
            2.1.14  The empty set   c0 4287
            ​*2.1.15  The conditional operator for classes   cif 4491
            ​*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 4590
            2.1.19  The union of a class   cuni 4870
            2.1.20  The intersection of a class   cint 4912
            2.1.21  Indexed union and intersection   ciun 4959
            2.1.22  Disjointness   wdisj 5075
            2.1.23  Binary relations   wbr 5110
            2.1.24  Ordered-pair class abstractions (class builders)   copab 5172
            2.1.25  Functions in maps-to notation   cmpt 5193
            2.1.26  Transitive classes   wtr 5227
      ​2.2  ZF Set Theory - add the Axiom of Replacement
            2.2.1  Introduce the Axiom of Replacement   ax-rep 5247
            2.2.2  Derive the Axiom of Separation   axsepgfromrep 5259
            2.2.3  Derive the Null Set Axiom   axnulALT 5266
            2.2.4  Theorems requiring subset and intersection existence   nalset 5275
            2.2.5  Theorems requiring empty set existence   class2set 5315
      ​2.3  ZF Set Theory - add the Axiom of Power Sets
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 5325
            2.3.2  Derive the Axiom of Pairing   axprlem1 5383
            2.3.3  Ordered pair theorem   opnz 5435
            2.3.4  Ordered-pair class abstractions (cont.)   opabidw 5486
            2.3.5  Power class of union and intersection   pwin 5532
            2.3.6  The identity relation   cid 5535
            2.3.7  The membership relation (or epsilon relation)   cep 5541
            ​*2.3.8  Partial and total orderings   wpo 5548
            2.3.9  Founded and well-ordering relations   wfr 5590
            2.3.10  Relations   cxp 5636
            2.3.11  The Predecessor Class   cpred 6257
            2.3.12  Well-founded induction (variant)   frpomin 6299
            2.3.13  Well-ordered induction   tz6.26 6306
            2.3.14  Ordinals   word 6321
            2.3.15  Definite description binder (inverted iota)   cio 6451
            2.3.16  Functions   wfun 6495
            2.3.17  Cantor's Theorem   canth 7315
            2.3.18  Restricted iota (description binder)   crio 7317
            2.3.19  Operations   co 7362
                  2.3.19.1  Variable-to-class conversion for operations   caovclg 7551
            2.3.20  Maps-to notation   mpondm0 7599
            2.3.21  Function operation   cof 7620
            2.3.22  Proper subset relation   crpss 7664
      ​2.4  ZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 7677
            2.4.2  Ordinals (continued)   epweon 7714
            2.4.3  Transfinite induction   tfi 7794
            2.4.4  The natural numbers (i.e., finite ordinals)   com 7807
            2.4.5  Peano's postulates   peano1 7830
            2.4.6  Finite induction (for finite ordinals)   find 7838
            2.4.7  Relations and functions (cont.)   dmexg 7845
            2.4.8  First and second members of an ordered pair   c1st 7924
            2.4.9  Induction on Cartesian products   frpoins3xpg 8077
            2.4.10  Ordering on Cartesian products   xpord2lem 8079
            2.4.11  Ordering Ordinal Sequences   orderseqlem 8094
            ​*2.4.12  The support of functions   csupp 8097
            ​*2.4.13  Special maps-to operations   opeliunxp2f 8146
            2.4.14  Function transposition   ctpos 8161
            2.4.15  Curry and uncurry   ccur 8201
            2.4.16  Undefined values   cund 8208
            2.4.17  Well-founded recursion   cfrecs 8216
            2.4.18  Well-ordered recursion   cwrecs 8247
            2.4.19  Functions on ordinals; strictly monotone ordinal functions   iunon 8290
            2.4.20  "Strong" transfinite recursion   crecs 8321
            2.4.21  Recursive definition generator   crdg 8360
            2.4.22  Finite recursion   frfnom 8386
            2.4.23  Ordinal arithmetic   c1o 8410
            2.4.24  Natural number arithmetic   nna0 8556
            2.4.25  Natural addition   cnadd 8616
            2.4.26  Equivalence relations and classes   wer 8652
            2.4.27  The mapping operation   cmap 8772
            2.4.28  Infinite Cartesian products   cixp 8842
            2.4.29  Equinumerosity   cen 8887
            2.4.30  Schroeder-Bernstein Theorem   sbthlem1 9034
            2.4.31  Equinumerosity (cont.)   xpf1o 9090
            2.4.32  Finite sets   dif1enlem 9107
            2.4.33  Pigeonhole Principle   phplem1 9158
            2.4.34  Finite sets (cont.)   onomeneq 9179
            2.4.35  Finitely supported functions   cfsupp 9312
            2.4.36  Finite intersections   cfi 9355
            2.4.37  Hall's marriage theorem   marypha1lem 9378
            2.4.38  Supremum and infimum   csup 9385
            2.4.39  Ordinal isomorphism, Hartogs's theorem   coi 9454
            2.4.40  Hartogs function   char 9501
            2.4.41  Weak dominance   cwdom 9509
      ​2.5  ZF Set Theory - add the Axiom of Regularity
            2.5.1  Introduce the Axiom of Regularity   ax-reg 9537
            2.5.2  Axiom of Infinity equivalents   inf0 9566
      ​2.6  ZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-inf 9583
            2.6.2  Existence of omega (the set of natural numbers)   omex 9588
            2.6.3  Cantor normal form   ccnf 9606
            2.6.4  Transitive closure of a relation   cttrcl 9652
            2.6.5  Transitive closure   trcl 9673
            2.6.6  Well-Founded Induction   frmin 9694
            2.6.7  Well-Founded Recursion   frr3g 9701
            2.6.8  Rank   cr1 9707
            2.6.9  Scott's trick; collection principle; Hilbert's epsilon   scottex 9830
            2.6.10  Disjoint union   cdju 9843
            2.6.11  Cardinal numbers   ccrd 9880
            2.6.12  Axiom of Choice equivalents   wac 10060
            ​*2.6.13  Cardinal number arithmetic   undjudom 10112
            2.6.14  The Ackermann bijection   ackbij2lem1 10164
            2.6.15  Cofinality (without Axiom of Choice)   cflem 10191
            2.6.16  Eight inequivalent definitions of finite set   sornom 10222
            2.6.17  Hereditarily size-limited sets without Choice   itunifval 10361
​​*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 10380
            3.1.2  Introduce the Axiom of Dependent Choice   ax-dc 10391
      ​3.2  ZFC Set Theory - add the Axiom of Choice
            3.2.1  Introduce the Axiom of Choice   ax-ac 10404
            3.2.2  AC equivalents: well-ordering, Zorn's lemma   numthcor 10439
            3.2.3  Cardinal number theorems using Axiom of Choice   cardval 10491
            3.2.4  Cardinal number arithmetic using Axiom of Choice   iunctb 10519
            3.2.5  Cofinality using the Axiom of Choice   alephreg 10527
      ​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 10565
            3.4.2  Derivation of the Axiom of Choice   gchaclem 10623
​​*PART 4  TG (TARSKI-GROTHENDIECK) SET THEORY
      ​4.1  Inaccessibles
            4.1.1  Weakly and strongly inaccessible cardinals   cwina 10627
            4.1.2  Weak universes   cwun 10645
            4.1.3  Tarski classes   ctsk 10693
            4.1.4  Grothendieck universes   cgru 10735
      ​4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
            4.2.1  Introduce the Tarski-Grothendieck Axiom   ax-groth 10768
            4.2.2  Derive the Power Set, Infinity and Choice Axioms   grothpw 10771
            4.2.3  Tarski map function   ctskm 10782
​​*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 10789
            5.1.2  Final derivation of real and complex number postulates   axaddf 11090
            5.1.3  Real and complex number postulates restated as axioms   ax-cnex 11116
      ​5.2  Derive the basic properties from the field axioms
            5.2.1  Some deductions from the field axioms for complex numbers   cnex 11141
            5.2.2  Infinity and the extended real number system   cpnf 11195
            5.2.3  Restate the ordering postulates with extended real "less than"   axlttri 11235
            5.2.4  Ordering on reals   lttr 11240
            5.2.5  Initial properties of the complex numbers   mul12 11329
      ​5.3  Real and complex numbers - basic operations
            5.3.1  Addition   add12 11381
            5.3.2  Subtraction   cmin 11394
            5.3.3  Multiplication   kcnktkm1cn 11595
            5.3.4  Ordering on reals (cont.)   gt0ne0 11629
            5.3.5  Reciprocals   ixi 11793
            5.3.6  Division   cdiv 11821
            5.3.7  Ordering on reals (cont.)   elimgt0 12002
            5.3.8  Completeness Axiom and Suprema   fimaxre 12108
            5.3.9  Imaginary and complex number properties   inelr 12152
            5.3.10  Function operation analogue theorems   ofsubeq0 12159
      ​5.4  Integer sets
            5.4.1  Positive integers (as a subset of complex numbers)   cn 12162
            5.4.2  Principle of mathematical induction   nnind 12180
            ​*5.4.3  Decimal representation of numbers   c2 12217
            ​*5.4.4  Some properties of specific numbers   neg1cn 12276
            5.4.5  Simple number properties   halfcl 12387
            5.4.6  The Archimedean property   nnunb 12418
            5.4.7  Nonnegative integers (as a subset of complex numbers)   cn0 12422
            ​*5.4.8  Extended nonnegative integers   cxnn0 12494
            5.4.9  Integers (as a subset of complex numbers)   cz 12508
            5.4.10  Decimal arithmetic   cdc 12627
            5.4.11  Upper sets of integers   cuz 12772
            5.4.12  Well-ordering principle for bounded-below sets of integers   uzwo3 12877
            5.4.13  Rational numbers (as a subset of complex numbers)   cq 12882
            5.4.14  Existence of the set of complex numbers   rpnnen1lem2 12911
      ​5.5  Order sets
            5.5.1  Positive reals (as a subset of complex numbers)   crp 12924
            5.5.2  Infinity and the extended real number system (cont.)   cxne 13039
            5.5.3  Supremum and infimum on the extended reals   xrsupexmnf 13234
            5.5.4  Real number intervals   cioo 13274
            5.5.5  Finite intervals of integers   cfz 13434
            ​*5.5.6  Finite intervals of nonnegative integers   elfz2nn0 13542
            5.5.7  Half-open integer ranges   cfzo 13577
      ​5.6  Elementary integer functions
            5.6.1  The floor and ceiling functions   cfl 13705
            5.6.2  The modulo (remainder) operation   cmo 13784
            5.6.3  Miscellaneous theorems about integers   om2uz0i 13862
            5.6.4  Strong induction over upper sets of integers   uzsinds 13902
            5.6.5  Finitely supported functions over the nonnegative integers   fsuppmapnn0fiublem 13905
            5.6.6  The infinite sequence builder "seq" - extension   cseq 13916
            5.6.7  Integer powers   cexp 13977
            5.6.8  Ordered pair theorem for nonnegative integers   nn0le2msqi 14177
            5.6.9  Factorial function   cfa 14183
            5.6.10  The binomial coefficient operation   cbc 14212
            5.6.11  The ` # ` (set size) function   chash 14240
                  5.6.11.1  Proper unordered pairs and triples (sets of size 2 and 3)   hashprlei 14379
                  5.6.11.2  Functions with a domain containing at least two different elements   fundmge2nop0 14403
                  5.6.11.3  Finite induction on the size of the first component of a binary relation   hashdifsnp1 14407
      ​​*5.7  Words over a set
            5.7.1  Definitions and basic theorems   cword 14414
            5.7.2  Last symbol of a word   clsw 14462
            5.7.3  Concatenations of words   cconcat 14470
            5.7.4  Singleton words   cs1 14495
            5.7.5  Concatenations with singleton words   ccatws1cl 14516
            5.7.6  Subwords/substrings   csubstr 14540
            5.7.7  Prefixes of a word   cpfx 14570
            5.7.8  Subwords of subwords   swrdswrdlem 14604
            5.7.9  Subwords and concatenations   pfxcctswrd 14610
            5.7.10  Subwords of concatenations   swrdccatfn 14624
            5.7.11  Splicing words (substring replacement)   csplice 14649
            5.7.12  Reversing words   creverse 14658
            5.7.13  Repeated symbol words   creps 14668
            ​*5.7.14  Cyclical shifts of words   ccsh 14688
            5.7.15  Mapping words by a function   wrdco 14732
            5.7.16  Longer string literals   cs2 14742
      ​​*5.8  Reflexive and transitive closures of relations
            5.8.1  The reflexive and transitive properties of relations   coss12d 14869
            5.8.2  Basic properties of closures   cleq1lem 14879
            5.8.3  Definitions and basic properties of transitive closures   ctcl 14882
            5.8.4  Exponentiation of relations   crelexp 14916
            5.8.5  Reflexive-transitive closure as an indexed union   crtrcl 14952
            ​*5.8.6  Principle of transitive induction.   relexpindlem 14960
      ​5.9  Elementary real and complex functions
            5.9.1  The "shift" operation   cshi 14963
            5.9.2  Signum (sgn or sign) function   csgn 14983
            5.9.3  Real and imaginary parts; conjugate   ccj 14993
            5.9.4  Square root; absolute value   csqrt 15130
      ​5.10  Elementary limits and convergence
            5.10.1  Superior limit (lim sup)   clsp 15364
            5.10.2  Limits   cli 15378
            5.10.3  Finite and infinite sums   csu 15582
            5.10.4  The binomial theorem   binomlem 15725
            5.10.5  The inclusion/exclusion principle   incexclem 15732
            5.10.6  Infinite sums (cont.)   isumshft 15735
            5.10.7  Miscellaneous converging and diverging sequences   divrcnv 15748
            5.10.8  Arithmetic series   arisum 15756
            5.10.9  Geometric series   expcnv 15760
            5.10.10  Ratio test for infinite series convergence   cvgrat 15779
            5.10.11  Mertens' theorem   mertenslem1 15780
            5.10.12  Finite and infinite products   prodf 15783
                  5.10.12.1  Product sequences   prodf 15783
                  5.10.12.2  Non-trivial convergence   ntrivcvg 15793
                  5.10.12.3  Complex products   cprod 15799
                  5.10.12.4  Finite products   fprod 15835
                  5.10.12.5  Infinite products   iprodclim 15892
            5.10.13  Falling and Rising Factorial   cfallfac 15898
            5.10.14  Bernoulli polynomials and sums of k-th powers   cbp 15940
      ​5.11  Elementary trigonometry
            5.11.1  The exponential, sine, and cosine functions   ce 15955
                  5.11.1.1  The circle constant (tau = 2 pi)   ctau 16095
            5.11.2  _e is irrational   eirrlem 16097
      ​5.12  Cardinality of real and complex number subsets
            5.12.1  Countability of integers and rationals   xpnnen 16104
            5.12.2  The reals are uncountable   rpnnen2lem1 16107
​​*PART 6  ELEMENTARY NUMBER THEORY
      ​6.1  Elementary properties of divisibility
            6.1.1  Irrationality of square root of 2   sqrt2irrlem 16141
            6.1.2  Some Number sets are chains of proper subsets   nthruc 16145
            6.1.3  The divides relation   cdvds 16147
            ​*6.1.4  Even and odd numbers   evenelz 16229
            6.1.5  The division algorithm   divalglem0 16286
            6.1.6  Bit sequences   cbits 16310
            6.1.7  The greatest common divisor operator   cgcd 16385
            6.1.8  Bézout's identity   bezoutlem1 16431
            6.1.9  Algorithms   nn0seqcvgd 16457
            6.1.10  Euclid's Algorithm   eucalgval2 16468
            ​*6.1.11  The least common multiple   clcm 16475
            ​*6.1.12  Coprimality and Euclid's lemma   coprmgcdb 16536
            6.1.13  Cancellability of congruences   congr 16551
      ​6.2  Elementary prime number theory
            ​*6.2.1  Elementary properties   cprime 16558
            ​*6.2.2  Coprimality and Euclid's lemma (cont.)   coprm 16598
            6.2.3  Properties of the canonical representation of a rational   cnumer 16619
            6.2.4  Euler's theorem   codz 16646
            6.2.5  Arithmetic modulo a prime number   modprm1div 16680
            6.2.6  Pythagorean Triples   coprimeprodsq 16691
            6.2.7  The prime count function   cpc 16719
            6.2.8  Pocklington's theorem   prmpwdvds 16787
            6.2.9  Infinite primes theorem   unbenlem 16791
            6.2.10  Sum of prime reciprocals   prmreclem1 16799
            6.2.11  Fundamental theorem of arithmetic   1arithlem1 16806
            6.2.12  Lagrange's four-square theorem   cgz 16812
            6.2.13  Van der Waerden's theorem   cvdwa 16848
            6.2.14  Ramsey's theorem   cram 16882
            ​*6.2.15  Primorial function   cprmo 16914
            ​*6.2.16  Prime gaps   prmgaplem1 16932
            6.2.17  Decimal arithmetic (cont.)   dec2dvds 16946
            6.2.18  Cyclical shifts of words (cont.)   cshwsidrepsw 16977
            6.2.19  Specific prime numbers   prmlem0 16989
            6.2.20  Very large primes   1259lem1 17014
​PART 7  BASIC STRUCTURES
      ​7.1  Extensible structures
            ​*7.1.1  Basic definitions   cstr 17029
                  7.1.1.1  Extensible structures as structures with components   cstr 17029
                  7.1.1.2  Substitution of components   csts 17046
                  7.1.1.3  Slots   cslot 17064
                  ​*7.1.1.4  Structure component indices   cnx 17076
                  7.1.1.5  Base sets   cbs 17094
                  7.1.1.6  Base set restrictions   cress 17123
            7.1.2  Slot definitions   cplusg 17147
            7.1.3  Definition of the structure product   crest 17316
            7.1.4  Definition of the structure quotient   cordt 17395
      ​7.2  Moore spaces
            7.2.1  Moore closures   mrcflem 17500
            7.2.2  Independent sets in a Moore system   mrisval 17524
            7.2.3  Algebraic closure systems   isacs 17545
​PART 8  BASIC CATEGORY THEORY
      ​8.1  Categories
            8.1.1  Categories   ccat 17558
            8.1.2  Opposite category   coppc 17605
            8.1.3  Monomorphisms and epimorphisms   cmon 17625
            8.1.4  Sections, inverses, isomorphisms   csect 17641
            ​*8.1.5  Isomorphic objects   ccic 17692
            8.1.6  Subcategories   cssc 17704
            8.1.7  Functors   cfunc 17754
            8.1.8  Full & faithful functors   cful 17803
            8.1.9  Natural transformations and the functor category   cnat 17842
            8.1.10  Initial, terminal and zero objects of a category   cinito 17881
      ​8.2  Arrows (disjointified hom-sets)
            8.2.1  Identity and composition for arrows   cida 17953
      ​8.3  Examples of categories
            8.3.1  The category of sets   csetc 17975
            8.3.2  The category of categories   ccatc 17998
            ​*8.3.3  The category of extensible structures   fncnvimaeqv 18021
      ​8.4  Categorical constructions
            8.4.1  Product of categories   cxpc 18070
            8.4.2  Functor evaluation   cevlf 18112
            8.4.3  Hom functor   chof 18151
​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 18334
            9.5.2  Complete lattices   ccla 18401
            9.5.3  Distributive lattices   cdlat 18423
            9.5.4  Subset order structures   cipo 18430
      ​9.6  Posets, directed sets, and lattices as relations
            ​*9.6.1  Posets and lattices as relations   cps 18467
            9.6.2  Directed sets, nets   cdir 18497
​PART 10  BASIC ALGEBRAIC STRUCTURES
      ​10.1  Monoids
            ​*10.1.1  Magmas   cplusf 18508
            ​*10.1.2  Identity elements   mgmidmo 18529
            ​*10.1.3  Iterated sums in a magma   gsumvalx 18545
            ​*10.1.4  Semigroups   csgrp 18559
            ​*10.1.5  Definition and basic properties of monoids   cmnd 18570
            10.1.6  Monoid homomorphisms and submonoids   cmhm 18613
            ​*10.1.7  Iterated sums in a monoid   gsumvallem2 18658
            10.1.8  Free monoids   cfrmd 18671
                  ​*10.1.8.1  Monoid of endofunctions   cefmnd 18692
            10.1.9  Examples and counterexamples for magmas, semigroups and monoids   mgm2nsgrplem1 18742
      ​10.2  Groups
            10.2.1  Definition and basic properties   cgrp 18762
            ​*10.2.2  Group multiple operation   cmg 18886
            10.2.3  Subgroups and Quotient groups   csubg 18936
            ​*10.2.4  Cyclic monoids and groups   cycsubmel 19007
            10.2.5  Elementary theory of group homomorphisms   cghm 19019
            10.2.6  Isomorphisms of groups   cgim 19061
            10.2.7  Group actions   cga 19083
            10.2.8  Centralizers and centers   ccntz 19109
            10.2.9  The opposite group   coppg 19137
            10.2.10  Symmetric groups   csymg 19162
                  ​*10.2.10.1  Definition and basic properties   csymg 19162
                  10.2.10.2  Cayley's theorem   cayleylem1 19208
                  10.2.10.3  Permutations fixing one element   symgfix2 19212
                  ​*10.2.10.4  Transpositions in the symmetric group   cpmtr 19237
                  10.2.10.5  The sign of a permutation   cpsgn 19285
            10.2.11  p-Groups and Sylow groups; Sylow's theorems   cod 19320
            10.2.12  Direct products   clsm 19430
                  10.2.12.1  Direct products (extension)   smndlsmidm 19452
            10.2.13  Free groups   cefg 19502
            10.2.14  Abelian groups   ccmn 19576
                  10.2.14.1  Definition and basic properties   ccmn 19576
                  10.2.14.2  Cyclic groups   ccyg 19668
                  10.2.14.3  Group sum operation   gsumval3a 19694
                  10.2.14.4  Group sums over (ranges of) integers   fsfnn0gsumfsffz 19774
                  10.2.14.5  Internal direct products   cdprd 19786
                  10.2.14.6  The Fundamental Theorem of Abelian Groups   ablfacrplem 19858
            10.2.15  Simple groups   csimpg 19883
                  10.2.15.1  Definition and basic properties   csimpg 19883
                  10.2.15.2  Classification of abelian simple groups   ablsimpnosubgd 19897
      ​10.3  Rings
            10.3.1  Multiplicative Group   cmgp 19910
            ​*10.3.2  Ring unity (multiplicative identity)   cur 19927
            10.3.3  Semirings   csrg 19931
                  ​*10.3.3.1  The binomial theorem for semirings   srgbinomlem1 19971
            10.3.4  Definition and basic properties of unital rings   crg 19978
            10.3.5  Opposite ring   coppr 20062
            10.3.6  Divisibility   cdsr 20081
            10.3.7  Ring primes   crpm 20157
            10.3.8  Ring homomorphisms   crh 20159
            10.3.9  Nonzero rings and zero rings   cnzr 20201
            10.3.10  Local rings   clring 20218
      ​10.4  Division rings and fields
            10.4.1  Definition and basic properties   cdr 20225
            10.4.2  Subrings of a ring   csubrg 20266
                  10.4.2.1  Sub-division rings   csdrg 20309
            10.4.3  Absolute value (abstract algebra)   cabv 20331
            10.4.4  Star rings   cstf 20358
      ​10.5  Left modules
            10.5.1  Definition and basic properties   clmod 20378
            10.5.2  Subspaces and spans in a left module   clss 20449
            10.5.3  Homomorphisms and isomorphisms of left modules   clmhm 20537
            10.5.4  Subspace sum; bases for a left module   clbs 20592
      ​10.6  Vector spaces
            10.6.1  Definition and basic properties   clvec 20620
      ​10.7  Ideals
            10.7.1  The subring algebra; ideals   csra 20688
            10.7.2  Two-sided ideals and quotient rings   c2idl 20760
            10.7.3  Principal ideal rings. Divisibility in the integers   clpidl 20770
            10.7.4  Left regular elements. More kinds of rings   crlreg 20786
      ​10.8  The complex numbers as an algebraic extensible structure
            10.8.1  Definition and basic properties   cpsmet 20817
            ​*10.8.2  Ring of integers   czring 20906
            10.8.3  Algebraic constructions based on the complex numbers   czrh 20937
            10.8.4  Signs as subgroup of the complex numbers   cnmsgnsubg 21018
            10.8.5  Embedding of permutation signs into a ring   zrhpsgnmhm 21025
            10.8.6  The ordered field of real numbers   crefld 21045
      ​10.9  Generalized pre-Hilbert and Hilbert spaces
            10.9.1  Definition and basic properties   cphl 21065
            10.9.2  Orthocomplements and closed subspaces   cocv 21101
            10.9.3  Orthogonal projection and orthonormal bases   cpj 21143
​​*PART 11  BASIC LINEAR ALGEBRA
      ​11.1  Vectors and free modules
            *11.1.1  Direct sum of left modules   cdsmm 21174
            ​*11.1.2  Free modules   cfrlm 21189
            ​*11.1.3  Standard basis (unit vectors)   cuvc 21225
            ​*11.1.4  Independent sets and families   clindf 21247
            11.1.5  Characterization of free modules   lmimlbs 21279
      ​11.2  Associative algebras
            11.2.1  Definition and basic properties   casa 21293
      ​11.3  Abstract multivariate polynomials
            11.3.1  Definition and basic properties   cmps 21343
            11.3.2  Polynomial evaluation   ces 21517
            11.3.3  Additional definitions for (multivariate) polynomials   cslv 21555
            ​*11.3.4  Univariate polynomials   cps1 21583
            11.3.5  Univariate polynomial evaluation   ces1 21716
      ​​*11.4  Matrices
            *11.4.1  The matrix multiplication   cmmul 21769
            ​*11.4.2  Square matrices   cmat 21791
            ​*11.4.3  The matrix algebra   matmulr 21824
            ​*11.4.4  Matrices of dimension 0 and 1   mat0dimbas0 21852
            ​*11.4.5  The subalgebras of diagonal and scalar matrices   cdmat 21874
            ​*11.4.6  Multiplication of a matrix with a "column vector"   cmvmul 21926
            11.4.7  Replacement functions for a square matrix   cmarrep 21942
            11.4.8  Submatrices   csubma 21962
      ​11.5  The determinant
            11.5.1  Definition and basic properties   cmdat 21970
            11.5.2  Determinants of 2 x 2 -matrices   m2detleiblem1 22010
            11.5.3  The matrix adjugate/adjunct   cmadu 22018
            ​*11.5.4  Laplace expansion of determinants (special case)   symgmatr01lem 22039
            11.5.5  Inverse matrix   invrvald 22062
            ​*11.5.6  Cramer's rule   slesolvec 22065
      ​​*11.6  Polynomial matrices
            11.6.1  Basic properties   pmatring 22078
            ​*11.6.2  Constant polynomial matrices   ccpmat 22089
            ​*11.6.3  Collecting coefficients of polynomial matrices   cdecpmat 22148
            ​*11.6.4  Ring isomorphism between polynomial matrices and polynomials over matrices   cpm2mp 22178
      ​​*11.7  The characteristic polynomial
            *11.7.1  Definition and basic properties   cchpmat 22212
            ​*11.7.2  The characteristic factor function G   fvmptnn04if 22235
            ​*11.7.3  The Cayley-Hamilton theorem   cpmadurid 22253
​PART 12  BASIC TOPOLOGY
      ​12.1  Topology
            ​*12.1.1  Topological spaces   ctop 22279
                  12.1.1.1  Topologies   ctop 22279
                  12.1.1.2  Topologies on sets   ctopon 22296
                  12.1.1.3  Topological spaces   ctps 22318
            12.1.2  Topological bases   ctb 22332
            12.1.3  Examples of topologies   distop 22382
            12.1.4  Closure and interior   ccld 22404
            12.1.5  Neighborhoods   cnei 22485
            12.1.6  Limit points and perfect sets   clp 22522
            12.1.7  Subspace topologies   restrcl 22545
            12.1.8  Order topology   ordtbaslem 22576
            12.1.9  Limits and continuity in topological spaces   ccn 22612
            12.1.10  Separated spaces: T0, T1, T2 (Hausdorff) ...   ct0 22694
            12.1.11  Compactness   ccmp 22774
            12.1.12  Bolzano-Weierstrass theorem   bwth 22798
            12.1.13  Connectedness   cconn 22799
            12.1.14  First- and second-countability   c1stc 22825
            12.1.15  Local topological properties   clly 22852
            12.1.16  Refinements   cref 22890
            12.1.17  Compactly generated spaces   ckgen 22921
            12.1.18  Product topologies   ctx 22948
            12.1.19  Continuous function-builders   cnmptid 23049
            12.1.20  Quotient maps and quotient topology   ckq 23081
            12.1.21  Homeomorphisms   chmeo 23141
      ​12.2  Filters and filter bases
            12.2.1  Filter bases   elmptrab 23215
            12.2.2  Filters   cfil 23233
            12.2.3  Ultrafilters   cufil 23287
            12.2.4  Filter limits   cfm 23321
            12.2.5  Extension by continuity   ccnext 23447
            12.2.6  Topological groups   ctmd 23458
            12.2.7  Infinite group sum on topological groups   ctsu 23514
            12.2.8  Topological rings, fields, vector spaces   ctrg 23544
      ​12.3  Uniform Structures and Spaces
            12.3.1  Uniform structures   cust 23588
            12.3.2  The topology induced by an uniform structure   cutop 23619
            12.3.3  Uniform Spaces   cuss 23642
            12.3.4  Uniform continuity   cucn 23664
            12.3.5  Cauchy filters in uniform spaces   ccfilu 23675
            12.3.6  Complete uniform spaces   ccusp 23686
      ​12.4  Metric spaces
            12.4.1  Pseudometric spaces   ispsmet 23694
            12.4.2  Basic metric space properties   cxms 23707
            12.4.3  Metric space balls   blfvalps 23773
            12.4.4  Open sets of a metric space   mopnval 23828
            12.4.5  Continuity in metric spaces   metcnp3 23933
            12.4.6  The uniform structure generated by a metric   metuval 23942
            12.4.7  Examples of metric spaces   dscmet 23965
            ​*12.4.8  Normed algebraic structures   cnm 23969
            12.4.9  Normed space homomorphisms (bounded linear operators)   cnmo 24106
            12.4.10  Topology on the reals   qtopbaslem 24159
            12.4.11  Topological definitions using the reals   cii 24275
            12.4.12  Path homotopy   chtpy 24367
            12.4.13  The fundamental group   cpco 24400
      ​12.5  Metric subcomplex vector spaces
            12.5.1  Subcomplex modules   cclm 24462
            ​*12.5.2  Subcomplex vector spaces   ccvs 24523
            ​*12.5.3  Normed subcomplex vector spaces   isncvsngp 24550
            12.5.4  Subcomplex pre-Hilbert spaces   ccph 24567
            12.5.5  Convergence and completeness   ccfil 24653
            12.5.6  Baire's Category Theorem   bcthlem1 24725
            12.5.7  Banach spaces and subcomplex Hilbert spaces   ccms 24733
                  12.5.7.1  The complete ordered field of the real numbers   retopn 24780
            12.5.8  Euclidean spaces   crrx 24784
            12.5.9  Minimizing Vector Theorem   minveclem1 24825
            12.5.10  Projection Theorem   pjthlem1 24838
​PART 13  BASIC REAL AND COMPLEX ANALYSIS
      ​13.1  Continuity
            13.1.1  Intermediate value theorem   pmltpclem1 24849
      ​13.2  Integrals
            13.2.1  Lebesgue measure   covol 24863
            13.2.2  Lebesgue integration   cmbf 25015
                  13.2.2.1  Lesbesgue integral   cmbf 25015
                  13.2.2.2  Lesbesgue directed integral   cdit 25247
      ​13.3  Derivatives
            13.3.1  Real and complex differentiation   climc 25263
                  13.3.1.1  Derivatives of functions of one complex or real variable   climc 25263
                  13.3.1.2  Results on real differentiation   dvferm1lem 25385
​PART 14  BASIC REAL AND COMPLEX FUNCTIONS
      ​14.1  Polynomials
            14.1.1  Polynomial degrees   cmdg 25452
            14.1.2  The division algorithm for univariate polynomials   cmn1 25527
            14.1.3  Elementary properties of complex polynomials   cply 25582
            14.1.4  The division algorithm for polynomials   cquot 25687
            14.1.5  Algebraic numbers   caa 25711
            14.1.6  Liouville's approximation theorem   aalioulem1 25729
      ​14.2  Sequences and series
            14.2.1  Taylor polynomials and Taylor's theorem   ctayl 25749
            14.2.2  Uniform convergence   culm 25772
            14.2.3  Power series   pserval 25806
      ​14.3  Basic trigonometry
            14.3.1  The exponential, sine, and cosine functions (cont.)   efcn 25839
            14.3.2  Properties of pi = 3.14159...   pilem1 25847
            14.3.3  Mapping of the exponential function   efgh 25934
            14.3.4  The natural logarithm on complex numbers   clog 25947
            ​*14.3.5  Logarithms to an arbitrary base   clogb 26151
            14.3.6  Theorems of Pythagoras, isosceles triangles, and intersecting chords   angval 26188
            14.3.7  Solutions of quadratic, cubic, and quartic equations   quad2 26226
            14.3.8  Inverse trigonometric functions   casin 26249
            14.3.9  The Birthday Problem   log2ublem1 26333
            14.3.10  Areas in R^2   carea 26342
            14.3.11  More miscellaneous converging sequences   rlimcnp 26352
            14.3.12  Inequality of arithmetic and geometric means   cvxcl 26371
            14.3.13  Euler-Mascheroni constant   cem 26378
            14.3.14  Zeta function   czeta 26399
            14.3.15  Gamma function   clgam 26402
      ​14.4  Basic number theory
            14.4.1  Wilson's theorem   wilthlem1 26454
            14.4.2  The Fundamental Theorem of Algebra   ftalem1 26459
            14.4.3  The Basel problem (ζ(2) = π²/6)   basellem1 26467
            14.4.4  Number-theoretical functions   ccht 26477
            14.4.5  Perfect Number Theorem   mersenne 26612
            14.4.6  Characters of Z/nZ   cdchr 26617
            14.4.7  Bertrand's postulate   bcctr 26660
            ​*14.4.8  Quadratic residues and the Legendre symbol   clgs 26679
            ​*14.4.9  Gauss' Lemma   gausslemma2dlem0a 26741
            14.4.10  Quadratic reciprocity   lgseisenlem1 26760
            14.4.11  All primes 4n+1 are the sum of two squares   2sqlem1 26802
            14.4.12  Chebyshev's Weak Prime Number Theorem, Dirichlet's Theorem   chebbnd1lem1 26854
            14.4.13  The Prime Number Theorem   mudivsum 26915
            14.4.14  Ostrowski's theorem   abvcxp 27000
​PART 15  SURREAL NUMBERS
      ​​*15.1  Sign sequence representation and Alling's axioms
            15.1.1  Definitions and initial properties   csur 27025
            15.1.2  Ordering   sltsolem1 27060
            15.1.3  Birthday Function   bdayfo 27062
            15.1.4  Density   fvnobday 27063
            ​*15.1.5  Full-Eta Property   bdayimaon 27078
      ​15.2  Initial consequences of Alling's axioms
            15.2.1  Ordering Theorems   csle 27129
            15.2.2  Birthday Theorems   bdayfun 27155
      ​​*15.3  Conway cut representation
            15.3.1  Conway cuts   csslt 27163
            15.3.2  Zero and One   c0s 27204
            15.3.3  Cuts and Options   cmade 27215
            15.3.4  Cofinality and coinitiality   cofsslt 27280
      ​15.4  Induction and recursion
            15.4.1  Induction and recursion on one variable   cnorec 27292
            15.4.2  Induction and recursion on two variables   cnorec2 27303
      ​15.5  Surreal arithmetic
            15.5.1  Addition   cadds 27314
            15.5.2  Negation and Subtraction   cnegs 27361
            15.5.3  Multiplication   cmuls 27414
​​*PART 16  ELEMENTARY GEOMETRY
      ​16.1  Definition and Tarski's Axioms of Geometry
            16.1.1  Justification for the congruence notation   tgjustf 27478
      ​16.2  Tarskian Geometry
            16.2.1  Congruence   tgcgrcomimp 27482
            16.2.2  Betweenness   tgbtwntriv2 27492
            16.2.3  Dimension   tglowdim1 27505
            16.2.4  Betweenness and Congruence   tgifscgr 27513
            16.2.5  Congruence of a series of points   ccgrg 27515
            16.2.6  Motions   cismt 27537
            16.2.7  Colinearity   tglng 27551
            16.2.8  Connectivity of betweenness   tgbtwnconn1lem1 27577
            16.2.9  Less-than relation in geometric congruences   cleg 27587
            16.2.10  Rays   chlg 27605
            16.2.11  Lines   btwnlng1 27624
            16.2.12  Point inversions   cmir 27657
            16.2.13  Right angles   crag 27698
            16.2.14  Half-planes   islnopp 27744
            16.2.15  Midpoints and Line Mirroring   cmid 27777
            16.2.16  Congruence of angles   ccgra 27812
            16.2.17  Angle Comparisons   cinag 27840
            16.2.18  Congruence Theorems   tgsas1 27859
            16.2.19  Equilateral triangles   ceqlg 27870
      ​16.3  Properties of geometries
            16.3.1  Isomorphisms between geometries   f1otrgds 27874
      ​16.4  Geometry in Hilbert spaces
            16.4.1  Geometry in the complex plane   cchhllem 27898
            16.4.2  Geometry in Euclidean spaces   cee 27900
                  16.4.2.1  Definition of the Euclidean space   cee 27900
                  16.4.2.2  Tarski's axioms for geometry for the Euclidean space   axdimuniq 27925
                  16.4.2.3  EE^n fulfills Tarski's Axioms   ceeng 27989
​​*PART 17  GRAPH THEORY
      ​*17.1  Vertices and edges
            17.1.1  The edge function extractor for extensible structures   cedgf 28000
            ​*17.1.2  Vertices and indexed edges   cvtx 28010
                  17.1.2.1  Definitions and basic properties   cvtx 28010
                  17.1.2.2  The vertices and edges of a graph represented as ordered pair   opvtxval 28017
                  17.1.2.3  The vertices and edges of a graph represented as extensible structure   funvtxdmge2val 28025
                  17.1.2.4  Representations of graphs without edges   snstrvtxval 28051
                  17.1.2.5  Degenerated cases of representations of graphs   vtxval0 28053
            17.1.3  Edges as range of the edge function   cedg 28061
      ​​*17.2  Undirected graphs
            17.2.1  Undirected hypergraphs   cuhgr 28070
            17.2.2  Undirected pseudographs and multigraphs   cupgr 28094
            ​*17.2.3  Loop-free graphs   umgrislfupgrlem 28136
            17.2.4  Edges as subsets of vertices of graphs   uhgredgiedgb 28140
            ​*17.2.5  Undirected simple graphs   cuspgr 28162
            17.2.6  Examples for graphs   usgr0e 28247
            17.2.7  Subgraphs   csubgr 28278
            17.2.8  Finite undirected simple graphs   cfusgr 28327
            17.2.9  Neighbors, complete graphs and universal vertices   cnbgr 28343
                  17.2.9.1  Neighbors   cnbgr 28343
                  17.2.9.2  Universal vertices   cuvtx 28396
                  17.2.9.3  Complete graphs   ccplgr 28420
            17.2.10  Vertex degree   cvtxdg 28476
            ​*17.2.11  Regular graphs   crgr 28566
      ​​*17.3  Walks, paths and cycles
            *17.3.1  Walks   cewlks 28606
            17.3.2  Walks for loop-free graphs   lfgrwlkprop 28698
            17.3.3  Trails   ctrls 28701
            17.3.4  Paths and simple paths   cpths 28723
            17.3.5  Closed walks   cclwlks 28781
            17.3.6  Circuits and cycles   ccrcts 28795
            ​*17.3.7  Walks as words   cwwlks 28833
            17.3.8  Walks/paths of length 2 (as length 3 strings)   2wlkdlem1 28933
            17.3.9  Walks in regular graphs   rusgrnumwwlkl1 28976
            ​*17.3.10  Closed walks as words   cclwwlk 28988
                  17.3.10.1  Closed walks as words   cclwwlk 28988
                  17.3.10.2  Closed walks of a fixed length as words   cclwwlkn 29031
                  17.3.10.3  Closed walks on a vertex of a fixed length as words   cclwwlknon 29094
            17.3.11  Examples for walks, trails and paths   0ewlk 29121
            17.3.12  Connected graphs   cconngr 29193
      ​17.4  Eulerian paths and the Konigsberg Bridge problem
            ​*17.4.1  Eulerian paths   ceupth 29204
            ​*17.4.2  The Königsberg Bridge problem   konigsbergvtx 29253
      ​17.5  The Friendship Theorem
            17.5.1  Friendship graphs - basics   cfrgr 29265
            17.5.2  The friendship theorem for small graphs   frgr1v 29278
            17.5.3  Theorems according to Mertzios and Unger   2pthfrgrrn 29289
            ​*17.5.4  Huneke's Proof of the Friendship Theorem   frgrncvvdeqlem1 29306
​PART 18  GUIDES AND MISCELLANEA
      ​18.1  Guides (conventions, explanations, and examples)
            ​*18.1.1  Conventions   conventions 29407
            18.1.2  Natural deduction   natded 29410
            ​*18.1.3  Natural deduction examples   ex-natded5.2 29411
            18.1.4  Definitional examples   ex-or 29428
            18.1.5  Other examples   aevdemo 29467
      ​18.2  Humor
            18.2.1  April Fool's theorem   avril1 29470
      ​18.3  (Future - to be reviewed and classified)
            18.3.1  Planar incidence geometry   cplig 29479
            ​*18.3.2  Aliases kept to prevent broken links   dummylink 29492
​​*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 29494
            19.1.2  Abelian groups   cablo 29549
      ​19.2  Complex vector spaces
            19.2.1  Definition and basic properties   cvc 29563
            19.2.2  Examples of complex vector spaces   cnaddabloOLD 29586
      ​19.3  Normed complex vector spaces
            19.3.1  Definition and basic properties   cnv 29589
            19.3.2  Examples of normed complex vector spaces   cnnv 29682
            19.3.3  Induced metric of a normed complex vector space   imsval 29690
            19.3.4  Inner product   cdip 29705
            19.3.5  Subspaces   css 29726
      ​19.4  Operators on complex vector spaces
            19.4.1  Definitions and basic properties   clno 29745
      ​19.5  Inner product (pre-Hilbert) spaces
            19.5.1  Definition and basic properties   ccphlo 29817
            19.5.2  Examples of pre-Hilbert spaces   cncph 29824
            19.5.3  Properties of pre-Hilbert spaces   isph 29827
      ​19.6  Complex Banach spaces
            19.6.1  Definition and basic properties   ccbn 29867
            19.6.2  Examples of complex Banach spaces   cnbn 29874
            19.6.3  Uniform Boundedness Theorem   ubthlem1 29875
            19.6.4  Minimizing Vector Theorem   minvecolem1 29879
      ​19.7  Complex Hilbert spaces
            19.7.1  Definition and basic properties   chlo 29890
            19.7.2  Standard axioms for a complex Hilbert space   hlex 29903
            19.7.3  Examples of complex Hilbert spaces   cnchl 29921
            19.7.4  Hellinger-Toeplitz Theorem   htthlem 29922
​​*PART 20  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      ​20.1  Axiomatization of complex pre-Hilbert spaces
            20.1.1  Basic Hilbert space definitions   chba 29924
            20.1.2  Preliminary ZFC lemmas   df-hnorm 29973
            ​*20.1.3  Derive the Hilbert space axioms from ZFC set theory   axhilex-zf 29986
            ​*20.1.4  Introduce the vector space axioms for a Hilbert space   ax-hilex 30004
            20.1.5  Vector operations   hvmulex 30016
            20.1.6  Inner product postulates for a Hilbert space   ax-hfi 30084
      ​20.2  Inner product and norms
            20.2.1  Inner product   his5 30091
            20.2.2  Norms   dfhnorm2 30127
            20.2.3  Relate Hilbert space to normed complex vector spaces   hilablo 30165
            20.2.4  Bunjakovaskij-Cauchy-Schwarz inequality   bcsiALT 30184
      ​20.3  Cauchy sequences and completeness axiom
            20.3.1  Cauchy sequences and limits   hcau 30189
            20.3.2  Derivation of the completeness axiom from ZF set theory   hilmet 30199
            20.3.3  Completeness postulate for a Hilbert space   ax-hcompl 30207
            20.3.4  Relate Hilbert space to ZFC pre-Hilbert and Hilbert spaces   hhcms 30208
      ​20.4  Subspaces and projections
            20.4.1  Subspaces   df-sh 30212
            20.4.2  Closed subspaces   df-ch 30226
            20.4.3  Orthocomplements   df-oc 30257
            20.4.4  Subspace sum, span, lattice join, lattice supremum   df-shs 30313
            20.4.5  Projection theorem   pjhthlem1 30396
            20.4.6  Projectors   df-pjh 30400
      ​20.5  Properties of Hilbert subspaces
            20.5.1  Orthomodular law   omlsilem 30407
            20.5.2  Projectors (cont.)   pjhtheu2 30421
            20.5.3  Hilbert lattice operations   sh0le 30445
            20.5.4  Span (cont.) and one-dimensional subspaces   spansn0 30546
            20.5.5  Commutes relation for Hilbert lattice elements   df-cm 30588
            20.5.6  Foulis-Holland theorem   fh1 30623
            20.5.7  Quantum Logic Explorer axioms   qlax1i 30632
            20.5.8  Orthogonal subspaces   chscllem1 30642
            20.5.9  Orthoarguesian laws 5OA and 3OA   5oalem1 30659
            20.5.10  Projectors (cont.)   pjorthi 30674
            20.5.11  Mayet's equation E_3   mayete3i 30733
      ​20.6  Operators on Hilbert spaces
            ​*20.6.1  Operator sum, difference, and scalar multiplication   df-hosum 30735
            20.6.2  Zero and identity operators   df-h0op 30753
            20.6.3  Operations on Hilbert space operators   hoaddcl 30763
            20.6.4  Linear, continuous, bounded, Hermitian, unitary operators and norms   df-nmop 30844
            20.6.5  Linear and continuous functionals and norms   df-nmfn 30850
            20.6.6  Adjoint   df-adjh 30854
            20.6.7  Dirac bra-ket notation   df-bra 30855
            20.6.8  Positive operators   df-leop 30857
            20.6.9  Eigenvectors, eigenvalues, spectrum   df-eigvec 30858
            20.6.10  Theorems about operators and functionals   nmopval 30861
            20.6.11  Riesz lemma   riesz3i 31067
            20.6.12  Adjoints (cont.)   cnlnadjlem1 31072
            20.6.13  Quantum computation error bound theorem   unierri 31109
            20.6.14  Dirac bra-ket notation (cont.)   branmfn 31110
            20.6.15  Positive operators (cont.)   leopg 31127
            20.6.16  Projectors as operators   pjhmopi 31151
      ​20.7  States on a Hilbert lattice and Godowski's equation
            20.7.1  States on a Hilbert lattice   df-st 31216
            20.7.2  Godowski's equation   golem1 31276
      ​20.8  Cover relation, atoms, exchange axiom, and modular symmetry
            20.8.1  Covers relation; modular pairs   df-cv 31284
            20.8.2  Atoms   df-at 31343
            20.8.3  Superposition principle   superpos 31359
            20.8.4  Atoms, exchange and covering properties, atomicity   chcv1 31360
            20.8.5  Irreducibility   chirredlem1 31395
            20.8.6  Atoms (cont.)   atcvat3i 31401
            20.8.7  Modular symmetry   mdsymlem1 31408
​PART 21  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      ​21.1  Mathboxes for user contributions
            21.1.1  Mathbox guidelines   mathbox 31447
      ​21.2  Mathbox for Stefan Allan
      ​21.3  Mathbox for Thierry Arnoux
            21.3.1  Propositional Calculus - misc additions   bian1d 31452
            21.3.2  Predicate Calculus   sbc2iedf 31459
                  21.3.2.1  Predicate Calculus - misc additions   sbc2iedf 31459
                  21.3.2.2  Restricted quantification - misc additions   ralcom4f 31461
                  21.3.2.3  Equality   eqtrb 31466
                  21.3.2.4  Double restricted existential uniqueness quantification   opsbc2ie 31468
                  21.3.2.5  Double restricted existential uniqueness quantification syntax   w2reu 31470
                  21.3.2.6  Substitution (without distinct variables) - misc additions   sbceqbidf 31479
                  21.3.2.7  Existential "at most one" - misc additions   mo5f 31481
                  21.3.2.8  Existential uniqueness - misc additions   reuxfrdf 31483
                  21.3.2.9  Restricted "at most one" - misc additions   rmoxfrd 31485
                  21.3.2.10  Restricted iota (description binder)   riotaeqbidva 31488
            21.3.3  General Set Theory   dmrab 31489
                  21.3.3.1  Class abstractions (a.k.a. class builders)   dmrab 31489
                  21.3.3.2  Image Sets   abrexdomjm 31497
                  21.3.3.3  Set relations and operations - misc additions   elunsn 31503
                  21.3.3.4  Unordered pairs   eqsnd 31520
                  21.3.3.5  Conditional operator - misc additions   ifeqeqx 31528
                  21.3.3.6  Set union   uniinn0 31536
                  21.3.3.7  Indexed union - misc additions   cbviunf 31541
                  21.3.3.8  Indexed intersection - misc additions   iinabrex 31554
                  21.3.3.9  Disjointness - misc additions   disjnf 31555
            21.3.4  Relations and Functions   xpdisjres 31583
                  21.3.4.1  Relations - misc additions   xpdisjres 31583
                  21.3.4.2  Functions - misc additions   ac6sf2 31606
                  21.3.4.3  Operations - misc additions   mpomptxf 31664
                  21.3.4.4  Support of a function   suppovss 31665
                  21.3.4.5  Explicit Functions with one or two points as a domain   cosnopne 31676
                  21.3.4.6  Isomorphisms - misc. additions   gtiso 31682
                  21.3.4.7  Disjointness (additional proof requiring functions)   disjdsct 31684
                  21.3.4.8  First and second members of an ordered pair - misc additions   df1stres 31685
                  21.3.4.9  Equivalence relations and classes   ecref 31693
                  21.3.4.10  Supremum - misc additions   supssd 31694
                  21.3.4.11  Finite Sets   imafi2 31696
                  21.3.4.12  Countable Sets   snct 31698
            21.3.5  Real and Complex Numbers   creq0 31720
                  21.3.5.1  Complex operations - misc. additions   creq0 31720
                  21.3.5.2  Ordering on reals - misc additions   lt2addrd 31724
                  21.3.5.3  Extended reals - misc additions   xrlelttric 31725
                  21.3.5.4  Extended nonnegative integers - misc additions   xnn0gt0 31742
                  21.3.5.5  Real number intervals - misc additions   joiniooico 31745
                  21.3.5.6  Finite intervals of integers - misc additions   uzssico 31755
                  21.3.5.7  Half-open integer ranges - misc additions   iundisjfi 31767
                  21.3.5.8  The ` # ` (set size) function - misc additions   hashunif 31778
                  21.3.5.9  The greatest common divisor operator - misc. additions   dvdszzq 31781
                  21.3.5.10  Integers   nnindf 31785
                  21.3.5.11  Decimal numbers   dfdec100 31796
            ​*21.3.6  Decimal expansion   cdp2 31797
                  ​*21.3.6.1  Decimal point   cdp 31814
                  21.3.6.2  Division in the extended real number system   cxdiv 31843
            21.3.7  Words over a set - misc additions   wrdfd 31862
                  21.3.7.1  Splicing words (substring replacement)   splfv3 31882
                  21.3.7.2  Cyclic shift of words   1cshid 31883
            21.3.8  Extensible Structures   ressplusf 31887
                  21.3.8.1  Structure restriction operator   ressplusf 31887
                  21.3.8.2  The opposite group   oppgle 31890
                  21.3.8.3  Posets   ressprs 31893
                  21.3.8.4  Complete lattices   clatp0cl 31906
                  21.3.8.5  Order Theory   cmnt 31908
                  21.3.8.6  Extended reals Structure - misc additions   ax-xrssca 31934
                  21.3.8.7  The extended nonnegative real numbers commutative monoid   xrge0base 31946
            21.3.9  Algebra   abliso 31957
                  21.3.9.1  Monoids Homomorphisms   abliso 31957
                  21.3.9.2  Finitely supported group sums - misc additions   gsumsubg 31958
                  21.3.9.3  Centralizers and centers - misc additions   cntzun 31972
                  21.3.9.4  Totally ordered monoids and groups   comnd 31975
                  21.3.9.5  The symmetric group   symgfcoeu 32003
                  21.3.9.6  Transpositions   pmtridf1o 32013
                  21.3.9.7  Permutation Signs   psgnid 32016
                  21.3.9.8  Permutation cycles   ctocyc 32025
                  21.3.9.9  The Alternating Group   evpmval 32064
                  21.3.9.10  Signum in an ordered monoid   csgns 32077
                  21.3.9.11  The Archimedean property for generic ordered algebraic structures   cinftm 32082
                  21.3.9.12  Semiring left modules   cslmd 32105
                  21.3.9.13  Simple groups   prmsimpcyc 32133
                  21.3.9.14  Rings - misc additions   0ringsubrg 32134
                  21.3.9.15  Division Rings   rndrhmcl 32144
                  21.3.9.16  Subfields   sdrgdvcl 32145
                  21.3.9.17  Field extensions generated by a set   cfldgen 32148
                  21.3.9.18  Totally ordered rings and fields   corng 32161
                  21.3.9.19  Ring homomorphisms - misc additions   rhmdvd 32184
                  21.3.9.20  Scalar restriction operation   cresv 32186
                  21.3.9.21  The commutative ring of gaussian integers   gzcrng 32206
                  21.3.9.22  The archimedean ordered field of real numbers   reofld 32207
                  21.3.9.23  The quotient map and quotient modules   qusker 32212
                  21.3.9.24  The ring of integers modulo ` N `   fermltlchr 32226
                  21.3.9.25  Independent sets and families   islinds5 32228
                  ​*21.3.9.26  Subgroup sum / Sumset / Minkowski sum   elgrplsmsn 32244
                  21.3.9.27  The quotient map   qusmul 32259
                  21.3.9.28  Ideals   intlidl 32274
                  21.3.9.29  Prime Ideals   cprmidl 32283
                  21.3.9.30  Maximal Ideals   cmxidl 32304
                  21.3.9.31  The semiring of ideals of a ring   cidlsrg 32318
                  21.3.9.32  Unique factorization domains   cufd 32334
                  21.3.9.33  Associative algebras   asclmulg 32339
                  21.3.9.34  Univariate Polynomials   0ringmon1p 32340
                  21.3.9.35  The subring algebra   sra1r 32369
                  21.3.9.36  Division Ring Extensions   drgext0g 32375
                  21.3.9.37  Vector Spaces   lvecdimfi 32381
                  21.3.9.38  Vector Space Dimension   cldim 32382
            21.3.10  Field Extensions   cfldext 32414
                  21.3.10.1  Algebraic numbers   cirng 32444
                  21.3.10.2  Minimal polynomials   cminply 32453
            21.3.11  Matrices   csmat 32463
                  21.3.11.1  Submatrices   csmat 32463
                  21.3.11.2  Matrix literals   clmat 32481
                  21.3.11.3  Laplace expansion of determinants   mdetpmtr1 32493
            21.3.12  Topology   ist0cld 32503
                  21.3.12.1  Open maps   txomap 32504
                  21.3.12.2  Topology of the unit circle   qtopt1 32505
                  21.3.12.3  Refinements   reff 32509
                  21.3.12.4  Open cover refinement property   ccref 32512
                  21.3.12.5  Lindelöf spaces   cldlf 32522
                  21.3.12.6  Paracompact spaces   cpcmp 32525
                  ​*21.3.12.7  Spectrum of a ring   crspec 32532
                  21.3.12.8  Pseudometrics   cmetid 32556
                  21.3.12.9  Continuity - misc additions   hauseqcn 32568
                  21.3.12.10  Topology of the closed unit interval   elunitge0 32569
                  21.3.12.11  Topology of ` ( RR X. RR ) `   unicls 32573
                  21.3.12.12  Order topology - misc. additions   cnvordtrestixx 32583
                  21.3.12.13  Continuity in topological spaces - misc. additions   mndpluscn 32596
                  21.3.12.14  Topology of the extended nonnegative real numbers ordered monoid   xrge0hmph 32602
                  21.3.12.15  Limits - misc additions   lmlim 32617
                  21.3.12.16  Univariate polynomials   pl1cn 32625
            21.3.13  Uniform Stuctures and Spaces   chcmp 32626
                  21.3.13.1  Hausdorff uniform completion   chcmp 32626
            21.3.14  Topology and algebraic structures   zringnm 32628
                  21.3.14.1  The norm on the ring of the integer numbers   zringnm 32628
                  21.3.14.2  Topological ` ZZ ` -modules   zlm0 32630
                  21.3.14.3  Canonical embedding of the field of the rational numbers into a division ring   cqqh 32642
                  21.3.14.4  Canonical embedding of the real numbers into a complete ordered field   crrh 32663
                  21.3.14.5  Embedding from the extended real numbers into a complete lattice   cxrh 32686
                  21.3.14.6  Canonical embeddings into the ordered field of the real numbers   zrhre 32689
                  ​*21.3.14.7  Topological Manifolds   cmntop 32692
            21.3.15  Real and complex functions   nexple 32697
                  21.3.15.1  Integer powers - misc. additions   nexple 32697
                  21.3.15.2  Indicator Functions   cind 32698
                  21.3.15.3  Extended sum   cesum 32715
            21.3.16  Mixed Function/Constant operation   cofc 32783
            21.3.17  Abstract measure   csiga 32796
                  21.3.17.1  Sigma-Algebra   csiga 32796
                  21.3.17.2  Generated sigma-Algebra   csigagen 32826
                  ​*21.3.17.3  lambda and pi-Systems, Rings of Sets   ispisys 32840
                  21.3.17.4  The Borel algebra on the real numbers   cbrsiga 32869
                  21.3.17.5  Product Sigma-Algebra   csx 32876
                  21.3.17.6  Measures   cmeas 32883
                  21.3.17.7  The counting measure   cntmeas 32914
                  21.3.17.8  The Lebesgue measure - misc additions   voliune 32917
                  21.3.17.9  The Dirac delta measure   cdde 32920
                  21.3.17.10  The 'almost everywhere' relation   cae 32925
                  21.3.17.11  Measurable functions   cmbfm 32937
                  21.3.17.12  Borel Algebra on ` ( RR X. RR ) `   br2base 32958
                  ​*21.3.17.13  Caratheodory's extension theorem   coms 32980
            21.3.18  Integration   itgeq12dv 33015
                  21.3.18.1  Lebesgue integral - misc additions   itgeq12dv 33015
                  21.3.18.2  Bochner integral   citgm 33016
            21.3.19  Euler's partition theorem   oddpwdc 33043
            21.3.20  Sequences defined by strong recursion   csseq 33072
            21.3.21  Fibonacci Numbers   cfib 33085
            21.3.22  Probability   cprb 33096
                  21.3.22.1  Probability Theory   cprb 33096
                  21.3.22.2  Conditional Probabilities   ccprob 33120
                  21.3.22.3  Real-valued Random Variables   crrv 33129
                  21.3.22.4  Preimage set mapping operator   corvc 33144
                  21.3.22.5  Distribution Functions   orvcelval 33157
                  21.3.22.6  Cumulative Distribution Functions   orvclteel 33161
                  21.3.22.7  Probabilities - example   coinfliplem 33167
                  21.3.22.8  Bertrand's Ballot Problem   ballotlemoex 33174
            21.3.23  Signum (sgn or sign) function - misc. additions   sgncl 33227
                  21.3.23.1  Operations on words   ccatmulgnn0dir 33243
            21.3.24  Polynomials with real coefficients - misc additions   plymul02 33247
            21.3.25  Descartes's rule of signs   signspval 33253
                  21.3.25.1  Sign changes in a word over real numbers   signspval 33253
                  21.3.25.2  Counting sign changes in a word over real numbers   signslema 33263
            21.3.26  Number Theory   efcld 33293
                  21.3.26.1  Representations of a number as sums of integers   crepr 33310
                  21.3.26.2  Vinogradov Trigonometric Sums and the Circle Method   cvts 33337
                  21.3.26.3  The Ternary Goldbach Conjecture: Final Statement   ax-hgt749 33346
            21.3.27  Elementary Geometry   cstrkg2d 33366
                  ​*21.3.27.1  Two-dimensional geometry   cstrkg2d 33366
                  21.3.27.2  Outer Five Segment (not used, no need to move to main)   cafs 33371
            ​*21.3.28  LeftPad Project   clpad 33376
      ​​*21.4  Mathbox for Jonathan Ben-Naim
            21.4.1  First-order logic and set theory   bnj170 33399
            21.4.2  Well founded induction and recursion   bnj110 33559
            21.4.3  The existence of a minimal element in certain classes   bnj69 33711
            21.4.4  Well-founded induction   bnj1204 33713
            21.4.5  Well-founded recursion, part 1 of 3   bnj60 33763
            21.4.6  Well-founded recursion, part 2 of 3   bnj1500 33769
            21.4.7  Well-founded recursion, part 3 of 3   bnj1522 33773
      ​21.5  Mathbox for BTernaryTau
            21.5.1  ZF set theory   exdifsn 33774
                  21.5.1.1  Finitism   fineqvrep 33785
            21.5.2  Real and complex numbers   zltp1ne 33789
            21.5.3  Graph theory   lfuhgr 33798
                  21.5.3.1  Acyclic graphs   cacycgr 33823
      ​21.6  Mathbox for Mario Carneiro
            21.6.1  Predicate calculus with all distinct variables   ax-7d 33840
            21.6.2  Miscellaneous stuff   quartfull 33846
            21.6.3  Derangements and the Subfactorial   deranglem 33847
            21.6.4  The Erdős-Szekeres theorem   erdszelem1 33872
            21.6.5  The Kuratowski closure-complement theorem   kur14lem1 33887
            21.6.6  Retracts and sections   cretr 33898
            21.6.7  Path-connected and simply connected spaces   cpconn 33900
            21.6.8  Covering maps   ccvm 33936
            21.6.9  Normal numbers   snmlff 34010
            21.6.10  Godel-sets of formulas - part 1   cgoe 34014
            21.6.11  Godel-sets of formulas - part 2   cgon 34113
            21.6.12  Models of ZF   cgze 34127
            ​*21.6.13  Metamath formal systems   cmcn 34141
            21.6.14  Grammatical formal systems   cm0s 34266
            21.6.15  Models of formal systems   cmuv 34286
            21.6.16  Splitting fields   ccpms 34308
            21.6.17  p-adic number fields   czr 34322
      ​​*21.7  Mathbox for Filip Cernatescu
      ​21.8  Mathbox for Paul Chapman
            21.8.1  Real and complex numbers (cont.)   climuzcnv 34346
            21.8.2  Miscellaneous theorems   elfzm12 34350
      ​21.9  Mathbox for Scott Fenton
            21.9.1  ZFC Axioms in primitive form   axextprim 34359
            21.9.2  Untangled classes   untelirr 34366
            21.9.3  Extra propositional calculus theorems   3jaodd 34373
            21.9.4  Misc. Useful Theorems   nepss 34376
            21.9.5  Properties of real and complex numbers   sqdivzi 34386
            21.9.6  Infinite products   iprodefisumlem 34399
            21.9.7  Factorial limits   faclimlem1 34402
            21.9.8  Greatest common divisor and divisibility   gcd32 34408
            21.9.9  Properties of relationships   dftr6 34410
            21.9.10  Properties of functions and mappings   funpsstri 34426
            21.9.11  Set induction (or epsilon induction)   setinds 34439
            21.9.12  Ordinal numbers   elpotr 34442
            21.9.13  Defined equality axioms   axextdfeq 34458
            21.9.14  Hypothesis builders   hbntg 34466
            21.9.15  Well-founded zero, successor, and limits   cwsuc 34471
            21.9.16  Quantifier-free definitions   ctxp 34491
            21.9.17  Alternate ordered pairs   caltop 34617
            21.9.18  Geometry in the Euclidean space   cofs 34643
                  21.9.18.1  Congruence properties   cofs 34643
                  21.9.18.2  Betweenness properties   btwntriv2 34673
                  21.9.18.3  Segment Transportation   ctransport 34690
                  21.9.18.4  Properties relating betweenness and congruence   cifs 34696
                  21.9.18.5  Connectivity of betweenness   btwnconn1lem1 34748
                  21.9.18.6  Segment less than or equal to   csegle 34767
                  21.9.18.7  Outside-of relationship   coutsideof 34780
                  21.9.18.8  Lines and Rays   cline2 34795
            21.9.19  Forward difference   cfwddif 34819
            21.9.20  Rank theorems   rankung 34827
            21.9.21  Hereditarily Finite Sets   chf 34833
      ​21.10  Mathbox for Jeff Hankins
            21.10.1  Miscellany   a1i14 34848
            21.10.2  Basic topological facts   topbnd 34872
            21.10.3  Topology of the real numbers   ivthALT 34883
            21.10.4  Refinements   cfne 34884
            21.10.5  Neighborhood bases determine topologies   neibastop1 34907
            21.10.6  Lattice structure of topologies   topmtcl 34911
            21.10.7  Filter bases   fgmin 34918
            21.10.8  Directed sets, nets   tailfval 34920
      ​21.11  Mathbox for Anthony Hart
            21.11.1  Propositional Calculus   tb-ax1 34931
            21.11.2  Predicate Calculus   nalfal 34951
            21.11.3  Miscellaneous single axioms   meran1 34959
            21.11.4  Connective Symmetry   negsym1 34965
      ​21.12  Mathbox for Chen-Pang He
            21.12.1  Ordinal topology   ontopbas 34976
      ​21.13  Mathbox for Jeff Hoffman
            21.13.1  Inferences for finite induction on generic function values   fveleq 34999
            21.13.2  gdc.mm   nnssi2 35003
      ​21.14  Mathbox for Asger C. Ipsen
            21.14.1  Continuous nowhere differentiable functions   dnival 35010
      ​​*21.15  Mathbox for BJ
            *21.15.1  Propositional calculus   bj-mp2c 35079
                  *21.15.1.1  Derived rules of inference   bj-mp2c 35079
                  ​*21.15.1.2  A syntactic theorem   bj-0 35081
                  21.15.1.3  Minimal implicational calculus   bj-a1k 35083
                  ​*21.15.1.4  Positive calculus   bj-syl66ib 35094
                  21.15.1.5  Implication and negation   bj-con2com 35100
                  ​*21.15.1.6  Disjunction   bj-jaoi1 35111
                  ​*21.15.1.7  Logical equivalence   bj-dfbi4 35113
                  21.15.1.8  The conditional operator for propositions   bj-consensus 35118
                  ​*21.15.1.9  Propositional calculus: miscellaneous   bj-imbi12 35123
            ​*21.15.2  Modal logic   bj-axdd2 35133
            ​*21.15.3  Provability logic   cprvb 35138
            ​*21.15.4  First-order logic   bj-genr 35147
                  21.15.4.1  Adding ax-gen   bj-genr 35147
                  21.15.4.2  Adding ax-4   bj-2alim 35151
                  21.15.4.3  Adding ax-5   bj-ax12wlem 35184
                  21.15.4.4  Equality and substitution   bj-ssbeq 35193
                  21.15.4.5  Adding ax-6   bj-spimvwt 35209
                  21.15.4.6  Adding ax-7   bj-cbvexw 35216
                  21.15.4.7  Membership predicate, ax-8 and ax-9   bj-ax89 35218
                  21.15.4.8  Adding ax-11   bj-alcomexcom 35221
                  21.15.4.9  Adding ax-12   axc11n11 35223
                  21.15.4.10  Nonfreeness   wnnf 35264
                  21.15.4.11  Adding ax-13   bj-axc10 35324
                  ​*21.15.4.12  Removing dependencies on ax-13 (and ax-11)   bj-axc10v 35334
                  ​*21.15.4.13  Distinct var metavariables   bj-hbaeb2 35359
                  ​*21.15.4.14  Around ~ equsal   bj-equsal1t 35363
                  ​*21.15.4.15  Some Principia Mathematica proofs   stdpc5t 35368
                  21.15.4.16  Alternate definition of substitution   bj-sbsb 35378
                  21.15.4.17  Lemmas for substitution   bj-sbf3 35380
                  21.15.4.18  Existential uniqueness   bj-eu3f 35383
                  ​*21.15.4.19  First-order logic: miscellaneous   bj-sblem1 35384
            21.15.5  Set theory   eliminable1 35401
                  ​*21.15.5.1  Eliminability of class terms   eliminable1 35401
                  ​*21.15.5.2  Classes without the axiom of extensionality   bj-denoteslem 35413
                  21.15.5.3  Characterization among sets versus among classes   elelb 35440
                  ​*21.15.5.4  The nonfreeness quantifier for classes   bj-nfcsym 35442
                  ​*21.15.5.5  Lemmas for class substitution   bj-sbeqALT 35443
                  21.15.5.6  Removing some axiom requirements and disjoint variable conditions   bj-exlimvmpi 35454
                  ​*21.15.5.7  Class abstractions   bj-elabd2ALT 35468
                  21.15.5.8  Generalized class abstractions   bj-cgab 35476
                  ​*21.15.5.9  Restricted nonfreeness   wrnf 35484
                  ​*21.15.5.10  Russell's paradox   bj-ru0 35486
                  21.15.5.11  Curry's paradox in set theory   currysetlem 35489
                  ​*21.15.5.12  Some disjointness results   bj-n0i 35495
                  ​*21.15.5.13  Complements on direct products   bj-xpimasn 35499
                  ​*21.15.5.14  "Singletonization" and tagging   bj-snsetex 35507
                  ​*21.15.5.15  Tuples of classes   bj-cproj 35534
                  ​*21.15.5.16  Set theory: elementary operations relative to a universe   bj-rcleqf 35569
                  ​*21.15.5.17  Axioms for finite unions   bj-abex 35574
                  ​*21.15.5.18  Set theory: miscellaneous   eleq2w2ALT 35591
                  ​*21.15.5.19  Evaluation at a class   bj-evaleq 35616
                  21.15.5.20  Elementwise operations   celwise 35623
                  ​*21.15.5.21  Elementwise intersection (families of sets induced on a subset)   bj-rest00 35625
                  21.15.5.22  Moore collections (complements)   bj-raldifsn 35644
                  21.15.5.23  Maps-to notation for functions with three arguments   bj-0nelmpt 35660
                  ​*21.15.5.24  Currying   csethom 35666
                  ​*21.15.5.25  Setting components of extensible structures   cstrset 35678
            ​*21.15.6  Extended real and complex numbers, real and complex projective lines   bj-nfald 35681
                  21.15.6.1  Complements on class abstractions of ordered pairs and binary relations   bj-nfald 35681
                  ​*21.15.6.2  Identity relation (complements)   bj-opabssvv 35694
                  ​*21.15.6.3  Functionalized identity (diagonal in a Cartesian square)   cdiag2 35716
                  ​*21.15.6.4  Direct image and inverse image   cimdir 35722
                  ​*21.15.6.5  Extended numbers and projective lines as sets   cfractemp 35740
                  ​*21.15.6.6  Addition and opposite   caddcc 35781
                  ​*21.15.6.7  Order relation on the extended reals   cltxr 35785
                  ​*21.15.6.8  Argument, multiplication and inverse   carg 35787
                  21.15.6.9  The canonical bijection from the finite ordinals   ciomnn 35793
                  21.15.6.10  Divisibility   cnnbar 35804
            ​*21.15.7  Monoids   bj-smgrpssmgm 35812
                  ​*21.15.7.1  Finite sums in monoids   cfinsum 35827
            ​*21.15.8  Affine, Euclidean, and Cartesian geometry   bj-fvimacnv0 35830
                  *21.15.8.1  Real vector spaces   bj-fvimacnv0 35830
                  ​*21.15.8.2  Complex numbers (supplements)   bj-subcom 35852
                  ​*21.15.8.3  Barycentric coordinates   bj-bary1lem 35854
            21.15.9  Monoid of endomorphisms   cend 35857
      ​21.16  Mathbox for Jim Kingdon
                  21.16.0.1  Circle constant   taupilem3 35863
                  21.16.0.2  Number theory   dfgcd3 35868
                  21.16.0.3  Real numbers   irrdifflemf 35869
      ​21.17  Mathbox for ML
            21.17.1  Miscellaneous   csbrecsg 35872
            21.17.2  Cartesian exponentiation   cfinxp 35927
            21.17.3  Topology   iunctb2 35947
                  ​*21.17.3.1  Pi-base theorems   pibp16 35957
      ​21.18  Mathbox for Wolf Lammen
            21.18.1  1. Bootstrapping   wl-section-boot 35966
            21.18.2  Implication chains   wl-section-impchain 35990
            21.18.3  Theorems around the conditional operator   wl-ifp-ncond1 36008
            21.18.4  Alternative development of hadd, cadd   wl-df-3xor 36012
            21.18.5  An alternative axiom ~ ax-13   ax-wl-13v 36037
            21.18.6  Other stuff   wl-mps 36039
      ​21.19  Mathbox for Brendan Leahy
      ​21.20  Mathbox for Jeff Madsen
            21.20.1  Logic and set theory   unirep 36245
            21.20.2  Real and complex numbers; integers   filbcmb 36272
            21.20.3  Sequences and sums   sdclem2 36274
            21.20.4  Topology   subspopn 36284
            21.20.5  Metric spaces   metf1o 36287
            21.20.6  Continuous maps and homeomorphisms   constcncf 36294
            21.20.7  Boundedness   ctotbnd 36298
            21.20.8  Isometries   cismty 36330
            21.20.9  Heine-Borel Theorem   heibor1lem 36341
            21.20.10  Banach Fixed Point Theorem   bfplem1 36354
            21.20.11  Euclidean space   crrn 36357
            21.20.12  Intervals (continued)   ismrer1 36370
            21.20.13  Operation properties   cass 36374
            21.20.14  Groups and related structures   cmagm 36380
            21.20.15  Group homomorphism and isomorphism   cghomOLD 36415
            21.20.16  Rings   crngo 36426
            21.20.17  Division Rings   cdrng 36480
            21.20.18  Ring homomorphisms   crnghom 36492
            21.20.19  Commutative rings   ccm2 36521
            21.20.20  Ideals   cidl 36539
            21.20.21  Prime rings and integral domains   cprrng 36578
            21.20.22  Ideal generators   cigen 36591
      ​21.21  Mathbox for Giovanni Mascellani
            ​*21.21.1  Tools for automatic proof building   efald2 36610
            ​*21.21.2  Tseitin axioms   fald 36661
            ​*21.21.3  Equality deductions   iuneq2f 36688
            ​*21.21.4  Miscellanea   orcomdd 36699
      ​21.22  Mathbox for Peter Mazsa
            21.22.1  Notations   cxrn 36706
            21.22.2  Preparatory theorems   el2v1 36749
            21.22.3  Range Cartesian product   df-xrn 36906
            21.22.4  Cosets by ` R `   df-coss 36946
            21.22.5  Relations   df-rels 37020
            21.22.6  Subset relations   df-ssr 37033
            21.22.7  Reflexivity   df-refs 37045
            21.22.8  Converse reflexivity   df-cnvrefs 37060
            21.22.9  Symmetry   df-syms 37077
            21.22.10  Reflexivity and symmetry   symrefref2 37098
            21.22.11  Transitivity   df-trs 37107
            21.22.12  Equivalence relations   df-eqvrels 37119
            21.22.13  Redundancy   df-redunds 37158
            21.22.14  Domain quotients   df-dmqss 37173
            21.22.15  Equivalence relations on domain quotients   df-ers 37198
            21.22.16  Functions   df-funss 37215
            21.22.17  Disjoints vs. converse functions   df-disjss 37238
            21.22.18  Antisymmetry   df-antisymrel 37295
            21.22.19  Partitions: disjoints on domain quotients   df-parts 37300
            21.22.20  Partition-Equivalence Theorems   disjim 37316
      ​21.23  Mathbox for Rodolfo Medina
            21.23.1  Partitions   prtlem60 37388
      ​​*21.24  Mathbox for Norm Megill
            *21.24.1  Obsolete schemes ax-c4,c5,c7,c10,c11,c11n,c15,c9,c14,c16   ax-c5 37418
            ​*21.24.2  Rederive new axioms ax-4, ax-10, ax-6, ax-12, ax-13 from old   axc5 37428
            ​*21.24.3  Legacy theorems using obsolete axioms   ax5ALT 37442
            21.24.4  Experiments with weak deduction theorem   elimhyps 37496
            21.24.5  Miscellanea   cnaddcom 37507
            21.24.6  Atoms, hyperplanes, and covering in a left vector space (or module)   clsa 37509
            21.24.7  Functionals and kernels of a left vector space (or module)   clfn 37592
            21.24.8  Opposite rings and dual vector spaces   cld 37658
            21.24.9  Ortholattices and orthomodular lattices   cops 37707
            21.24.10  Atomic lattices with covering property   ccvr 37797
            21.24.11  Hilbert lattices   chlt 37885
            21.24.12  Projective geometries based on Hilbert lattices   clln 38027
            21.24.13  Construction of a vector space from a Hilbert lattice   cdlema1N 38327
            21.24.14  Construction of involution and inner product from a Hilbert lattice   clpoN 40016
      ​21.25  Mathbox for metakunt
            21.25.1  General helpful statements   leexp1ad 40502
            21.25.2  Some gcd and lcm results   12gcd5e1 40533
            21.25.3  Least common multiple inequality theorem   3factsumint1 40551
            21.25.4  Logarithm inequalities   3exp7 40583
            21.25.5  Miscellaneous results for AKS formalisation   intlewftc 40591
            21.25.6  Sticks and stones   sticksstones1 40627
            21.25.7  Permutation results   metakunt1 40650
            21.25.8  Unused lemmas scheduled for deletion   andiff 40684
      ​21.26  Mathbox for Steven Nguyen
            ​*21.26.1  Exemplar theorems   bicomdALT 40689
            21.26.2  Utility theorems   ioin9i8 40698
            21.26.3  Structures   nelsubginvcld 40739
            ​*21.26.4  Arithmetic theorems   c0exALT 40833
            21.26.5  Exponents and divisibility   oexpreposd 40865
            21.26.6  Real subtraction   cresub 40892
            ​*21.26.7  Projective spaces   cprjsp 40997
            21.26.8  Basic reductions for Fermat's Last Theorem   dffltz 41030
      ​21.27  Mathbox for Igor Ieskov
      ​21.28  Mathbox for OpenAI
      ​21.29  Mathbox for Stefan O'Rear
            21.29.1  Additional elementary logic and set theory   moxfr 41073
            21.29.2  Additional theory of functions   imaiinfv 41074
            21.29.3  Additional topology   elrfi 41075
            21.29.4  Characterization of closure operators. Kuratowski closure axioms   ismrcd1 41079
            21.29.5  Algebraic closure systems   cnacs 41083
            21.29.6  Miscellanea 1. Map utilities   constmap 41094
            21.29.7  Miscellanea for polynomials   mptfcl 41101
            21.29.8  Multivariate polynomials over the integers   cmzpcl 41102
            21.29.9  Miscellanea for Diophantine sets 1   coeq0i 41134
            21.29.10  Diophantine sets 1: definitions   cdioph 41136
            21.29.11  Diophantine sets 2 miscellanea   ellz1 41148
            21.29.12  Diophantine sets 2: union and intersection. Monotone Boolean algebra   diophin 41153
            21.29.13  Diophantine sets 3: construction   diophrex 41156
            21.29.14  Diophantine sets 4 miscellanea   2sbcrex 41165
            21.29.15  Diophantine sets 4: Quantification   rexrabdioph 41175
            21.29.16  Diophantine sets 5: Arithmetic sets   rabdiophlem1 41182
            21.29.17  Diophantine sets 6: reusability. renumbering of variables   eldioph4b 41192
            21.29.18  Pigeonhole Principle and cardinality helpers   fphpd 41197
            21.29.19  A non-closed set of reals is infinite   rencldnfilem 41201
            21.29.20  Lagrange's rational approximation theorem   irrapxlem1 41203
            21.29.21  Pell equations 1: A nontrivial solution always exists   pellexlem1 41210
            21.29.22  Pell equations 2: Algebraic number theory of the solution set   csquarenn 41217
            21.29.23  Pell equations 3: characterizing fundamental solution   infmrgelbi 41259
            ​*21.29.24  Logarithm laws generalized to an arbitrary base   reglogcl 41271
            21.29.25  Pell equations 4: the positive solution group is infinite cyclic   pellfund14 41279
            21.29.26  X and Y sequences 1: Definition and recurrence laws   crmx 41281
            21.29.27  Ordering and induction lemmas for the integers   monotuz 41323
            21.29.28  X and Y sequences 2: Order properties   rmxypos 41329
            21.29.29  Congruential equations   congtr 41347
            21.29.30  Alternating congruential equations   acongid 41357
            21.29.31  Additional theorems on integer divisibility   coprmdvdsb 41367
            21.29.32  X and Y sequences 3: Divisibility properties   jm2.18 41370
            21.29.33  X and Y sequences 4: Diophantine representability of Y   jm2.27a 41387
            21.29.34  X and Y sequences 5: Diophantine representability of X, ^, _C   rmxdiophlem 41397
            21.29.35  Uncategorized stuff not associated with a major project   setindtr 41406
            21.29.36  More equivalents of the Axiom of Choice   axac10 41415
            21.29.37  Finitely generated left modules   clfig 41452
            21.29.38  Noetherian left modules I   clnm 41460
            21.29.39  Addenda for structure powers   pwssplit4 41474
            21.29.40  Every set admits a group structure iff choice   unxpwdom3 41480
            21.29.41  Noetherian rings and left modules II   clnr 41494
            21.29.42  Hilbert's Basis Theorem   cldgis 41506
            21.29.43  Additional material on polynomials [DEPRECATED]   cmnc 41516
            21.29.44  Degree and minimal polynomial of algebraic numbers   cdgraa 41525
            21.29.45  Algebraic integers I   citgo 41542
            21.29.46  Endomorphism algebra   cmend 41560
            21.29.47  Cyclic groups and order   idomrootle 41580
            21.29.48  Cyclotomic polynomials   ccytp 41587
            21.29.49  Miscellaneous topology   fgraphopab 41595
      ​21.30  Mathbox for Noam Pasman
      ​21.31  Mathbox for Jon Pennant
      ​21.32  Mathbox for Richard Penner
            21.32.1  Set Theory and Ordinal Numbers   uniel 41609
            21.32.2  Natural addition of Cantor normal forms   oawordex2 41719
            21.32.3  Surreal Contributions   abeqabi 41802
            21.32.4  Short Studies   nlimsuc 41835
                  21.32.4.1  Additional work on conditional logical operator   ifpan123g 41853
                  21.32.4.2  Sophisms   rp-fakeimass 41906
                  ​*21.32.4.3  Finite Sets   rp-isfinite5 41911
                  21.32.4.4  General Observations   intabssd 41913
                  21.32.4.5  Infinite Sets   pwelg 41954
                  ​*21.32.4.6  Finite intersection property   fipjust 41959
                  21.32.4.7  RP ADDTO: Subclasses and subsets   rababg 41968
                  21.32.4.8  RP ADDTO: The intersection of a class   elinintab 41969
                  21.32.4.9  RP ADDTO: Theorems requiring subset and intersection existence   elinintrab 41971
                  21.32.4.10  RP ADDTO: Relations   xpinintabd 41974
                  ​*21.32.4.11  RP ADDTO: Functions   elmapintab 41990
                  ​*21.32.4.12  RP ADDTO: Finite induction (for finite ordinals)   cnvcnvintabd 41994
                  21.32.4.13  RP ADDTO: First and second members of an ordered pair   elcnvlem 41995
                  21.32.4.14  RP ADDTO: The reflexive and transitive properties of relations   undmrnresiss 41998
                  21.32.4.15  RP ADDTO: Basic properties of closures   cleq2lem 42002
                  21.32.4.16  RP REPLACE: Definitions and basic properties of transitive closures   trcleq2lemRP 42024
                  ​*21.32.4.17  Additions for square root; absolute value   sqrtcvallem1 42025
            21.32.5  Additional statements on relations and subclasses   al3im 42041
                  21.32.5.1  Transitive relations (not to be confused with transitive classes).   trrelind 42059
                  21.32.5.2  Reflexive closures   crcl 42066
                  ​*21.32.5.3  Finite relationship composition.   relexp2 42071
                  21.32.5.4  Transitive closure of a relation   dftrcl3 42114
                  ​*21.32.5.5  Adapted from Frege   frege77d 42140
            ​*21.32.6  Propositions from _Begriffsschrift_   dfxor4 42160
                  *21.32.6.1  _Begriffsschrift_ Chapter I   dfxor4 42160
                  ​*21.32.6.2  _Begriffsschrift_ Notation hints   whe 42166
                  21.32.6.3  _Begriffsschrift_ Chapter II Implication   ax-frege1 42184
                  21.32.6.4  _Begriffsschrift_ Chapter II Implication and Negation   axfrege28 42223
                  ​*21.32.6.5  _Begriffsschrift_ Chapter II with logical equivalence   axfrege52a 42250
                  21.32.6.6  _Begriffsschrift_ Chapter II with equivalence of sets   axfrege52c 42281
                  ​*21.32.6.7  _Begriffsschrift_ Chapter II with equivalence of classes   frege53c 42308
                  ​*21.32.6.8  _Begriffsschrift_ Chapter III Properties hereditary in a sequence   dffrege69 42326
                  ​*21.32.6.9  _Begriffsschrift_ Chapter III Following in a sequence   dffrege76 42333
                  ​*21.32.6.10  _Begriffsschrift_ Chapter III Member of sequence   dffrege99 42356
                  ​*21.32.6.11  _Begriffsschrift_ Chapter III Single-valued procedures   dffrege115 42372
            ​*21.32.7  Exploring Topology via Seifert and Threlfall   enrelmap 42391
                  *21.32.7.1  Equinumerosity of sets of relations and maps   enrelmap 42391
                  ​*21.32.7.2  Generic Pseudoclosure Spaces, Pseudointerior Spaces, and Pseudoneighborhoods   or3or 42417
                  ​*21.32.7.3  Generic Neighborhood Spaces   gneispa 42524
            ​*21.32.8  Exploring Higher Homotopy via Kerodon   k0004lem1 42541
                  *21.32.8.1  Simplicial Sets   k0004lem1 42541
      ​21.33  Mathbox for Stanislas Polu
            21.33.1  IMO Problems   wwlemuld 42550
                  21.33.1.1  IMO 1972 B2   wwlemuld 42550
            ​*21.33.2  INT Inequalities Proof Generator   int-addcomd 42568
            ​*21.33.3  N-Digit Addition Proof Generator   unitadd 42590
            21.33.4  AM-GM (for k = 2,3,4)   gsumws3 42591
      ​21.34  Mathbox for Rohan Ridenour
            21.34.1  Misc   spALT 42596
            21.34.2  Monoid rings   cmnring 42608
            21.34.3  Shorter primitive equivalent of ax-groth   gru0eld 42631
                  21.34.3.1  Grothendieck universes are closed under collection   gru0eld 42631
                  21.34.3.2  Minimal universes   ismnu 42663
                  21.34.3.3  Primitive equivalent of ax-groth   expandan 42690
      ​21.35  Mathbox for Steve Rodriguez
            21.35.1  Miscellanea   nanorxor 42707
            21.35.2  Ratio test for infinite series convergence and divergence   dvgrat 42714
            21.35.3  Multiples   reldvds 42717
            21.35.4  Function operations   caofcan 42725
            21.35.5  Calculus   lhe4.4ex1a 42731
            21.35.6  The generalized binomial coefficient operation   cbcc 42738
            21.35.7  Binomial series   uzmptshftfval 42748
      ​21.36  Mathbox for Andrew Salmon
            21.36.1  Principia Mathematica * 10   pm10.12 42760
            21.36.2  Principia Mathematica * 11   2alanimi 42774
            21.36.3  Predicate Calculus   sbeqal1 42800
            21.36.4  Principia Mathematica * 13 and * 14   pm13.13a 42809
            21.36.5  Set Theory   elnev 42840
            21.36.6  Arithmetic   addcomgi 42858
            21.36.7  Geometry   cplusr 42859
      ​​*21.37  Mathbox for Alan Sare
            21.37.1  Auxiliary theorems for the Virtual Deduction tool   idiALT 42881
            21.37.2  Supplementary unification deductions   bi1imp 42885
            21.37.3  Conventional Metamath proofs, some derived from VD proofs   iidn3 42905
            21.37.4  What is Virtual Deduction?   wvd1 42973
            21.37.5  Virtual Deduction Theorems   df-vd1 42974
            21.37.6  Theorems proved using Virtual Deduction   trsspwALT 43222
            21.37.7  Theorems proved using Virtual Deduction with mmj2 assistance   simplbi2VD 43250
            21.37.8  Virtual Deduction transcriptions of textbook proofs   sb5ALTVD 43317
            21.37.9  Theorems proved using conjunction-form Virtual Deduction   elpwgdedVD 43321
            21.37.10  Theorems with a VD proof in conventional notation derived from a VD proof   suctrALT3 43328
            ​*21.37.11  Theorems with a proof in conventional notation derived from a VD proof   notnotrALT2 43331
      ​21.38  Mathbox for Glauco Siliprandi
            21.38.1  Miscellanea   evth2f 43342
            21.38.2  Functions   feq1dd 43506
            21.38.3  Ordering on real numbers - Real and complex numbers basic operations   sub2times 43627
            21.38.4  Real intervals   gtnelioc 43849
            21.38.5  Finite sums   fsummulc1f 43932
            21.38.6  Finite multiplication of numbers and finite multiplication of functions   fmul01 43941
            21.38.7  Limits   clim1fr1 43962
                  21.38.7.1  Inferior limit (lim inf)   clsi 44112
                  ​*21.38.7.2  Limits for sequences of extended real numbers   clsxlim 44179
            21.38.8  Trigonometry   coseq0 44225
            21.38.9  Continuous Functions   mulcncff 44231
            21.38.10  Derivatives   dvsinexp 44272
            21.38.11  Integrals   itgsin0pilem1 44311
            21.38.12  Stone Weierstrass theorem - real version   stoweidlem1 44362
            21.38.13  Wallis' product for π   wallispilem1 44426
            21.38.14  Stirling's approximation formula for ` n ` factorial   stirlinglem1 44435
            21.38.15  Dirichlet kernel   dirkerval 44452
            21.38.16  Fourier Series   fourierdlem1 44469
            21.38.17  e is transcendental   elaa2lem 44594
            21.38.18  n-dimensional Euclidean space   rrxtopn 44645
            21.38.19  Basic measure theory   csalg 44669
                  ​*21.38.19.1  σ-Algebras   csalg 44669
                  21.38.19.2  Sum of nonnegative extended reals   csumge0 44723
                  ​*21.38.19.3  Measures   cmea 44810
                  ​*21.38.19.4  Outer measures and Caratheodory's construction   come 44850
                  ​*21.38.19.5  Lebesgue measure on n-dimensional Real numbers   covoln 44897
                  ​*21.38.19.6  Measurable functions   csmblfn 45056
      ​21.39  Mathbox for Saveliy Skresanov
            21.39.1  Ceva's theorem   sigarval 45211
            21.39.2  Simple groups   simpcntrab 45231
      ​21.40  Mathbox for Ender Ting
            21.40.1  Increasing sequences and subsequences   et-ltneverrefl 45232
      ​21.41  Mathbox for Jarvin Udandy
      ​21.42  Mathbox for Adhemar
            ​*21.42.1  Minimal implicational calculus   adh-minim 45356
      ​21.43  Mathbox for Alexander van der Vekens
            21.43.1  General auxiliary theorems (1)   eusnsn 45380
                  21.43.1.1  Unordered and ordered pairs - extension for singletons   eusnsn 45380
                  21.43.1.2  Unordered and ordered pairs - extension for unordered pairs   elprneb 45383
                  21.43.1.3  Unordered and ordered pairs - extension for ordered pairs   oppr 45384
                  21.43.1.4  Relations - extension   eubrv 45389
                  21.43.1.5  Definite description binder (inverted iota) - extension   iota0def 45392
                  21.43.1.6  Functions - extension   fveqvfvv 45394
            21.43.2  Alternative for Russell's definition of a description binder   caiota 45435
            21.43.3  Double restricted existential uniqueness   r19.32 45450
                  21.43.3.1  Restricted quantification (extension)   r19.32 45450
                  21.43.3.2  Restricted uniqueness and "at most one" quantification   reuf1odnf 45459
                  21.43.3.3  Analogs to Existential uniqueness (double quantification)   2reu3 45462
                  21.43.3.4  Additional theorems for double restricted existential uniqueness   2reu8i 45465
            ​*21.43.4  Alternative definitions of function and operation values   wdfat 45468
                  21.43.4.1  Restricted quantification (extension)   ralbinrald 45474
                  21.43.4.2  The universal class (extension)   nvelim 45475
                  21.43.4.3  Introduce the Axiom of Power Sets (extension)   alneu 45476
                  21.43.4.4  Predicate "defined at"   dfateq12d 45478
                  21.43.4.5  Alternative definition of the value of a function   dfafv2 45484
                  21.43.4.6  Alternative definition of the value of an operation   aoveq123d 45530
            ​*21.43.5  Alternative definitions of function values (2)   cafv2 45560
            21.43.6  General auxiliary theorems (2)   an4com24 45620
                  21.43.6.1  Logical conjunction - extension   an4com24 45620
                  21.43.6.2  Abbreviated conjunction and disjunction of three wff's - extension   3an4ancom24 45621
                  21.43.6.3  Negated membership (alternative)   cnelbr 45623
                  21.43.6.4  The empty set - extension   ralralimp 45630
                  21.43.6.5  Indexed union and intersection - extension   otiunsndisjX 45631
                  21.43.6.6  Functions - extension   fvifeq 45632
                  21.43.6.7  Maps-to notation - extension   fvmptrab 45644
                  21.43.6.8  Subtraction - extension   cnambpcma 45646
                  21.43.6.9  Ordering on reals (cont.) - extension   leaddsuble 45649
                  21.43.6.10  Imaginary and complex number properties - extension   readdcnnred 45655
                  21.43.6.11  Nonnegative integers (as a subset of complex numbers) - extension   nn0resubcl 45660
                  21.43.6.12  Integers (as a subset of complex numbers) - extension   zgeltp1eq 45661
                  21.43.6.13  Decimal arithmetic - extension   1t10e1p1e11 45662
                  21.43.6.14  Upper sets of integers - extension   eluzge0nn0 45664
                  21.43.6.15  Infinity and the extended real number system (cont.) - extension   nltle2tri 45665
                  21.43.6.16  Finite intervals of integers - extension   ssfz12 45666
                  21.43.6.17  Half-open integer ranges - extension   fzopred 45674
                  21.43.6.18  The modulo (remainder) operation - extension   m1mod0mod1 45681
                  21.43.6.19  The infinite sequence builder "seq"   smonoord 45683
                  21.43.6.20  Finite and infinite sums - extension   fsummsndifre 45684
                  21.43.6.21  Extensible structures - extension   setsidel 45688
            ​*21.43.7  Preimages of function values   preimafvsnel 45691
            ​*21.43.8  Partitions of real intervals   ciccp 45725
            21.43.9  Shifting functions with an integer range domain   fargshiftfv 45751
            21.43.10  Words over a set (extension)   lswn0 45756
                  21.43.10.1  Last symbol of a word - extension   lswn0 45756
            21.43.11  Unordered pairs   wich 45757
                  21.43.11.1  Interchangeable setvar variables   wich 45757
                  21.43.11.2  Set of unordered pairs   sprid 45786
                  ​*21.43.11.3  Proper (unordered) pairs   prpair 45813
                  21.43.11.4  Set of proper unordered pairs   cprpr 45824
            21.43.12  Number theory (extension)   cfmtno 45839
                  ​*21.43.12.1  Fermat numbers   cfmtno 45839
                  ​*21.43.12.2  Mersenne primes   m2prm 45903
                  21.43.12.3  Proth's theorem   modexp2m1d 45924
                  21.43.12.4  Solutions of quadratic equations   quad1 45932
            ​*21.43.13  Even and odd numbers   ceven 45936
                  21.43.13.1  Definitions and basic properties   ceven 45936
                  21.43.13.2  Alternate definitions using the "divides" relation   dfeven2 45961
                  21.43.13.3  Alternate definitions using the "modulo" operation   dfeven3 45970
                  21.43.13.4  Alternate definitions using the "gcd" operation   iseven5 45976
                  21.43.13.5  Theorems of part 5 revised   zneoALTV 45981
                  21.43.13.6  Theorems of part 6 revised   odd2np1ALTV 45986
                  21.43.13.7  Theorems of AV's mathbox revised   0evenALTV 46000
                  21.43.13.8  Additional theorems   epoo 46015
                  21.43.13.9  Perfect Number Theorem (revised)   perfectALTVlem1 46033
            21.43.14  Number theory (extension 2)   cfppr 46036
                  ​*21.43.14.1  Fermat pseudoprimes   cfppr 46036
                  ​*21.43.14.2  Goldbach's conjectures   cgbe 46057
            21.43.15  Graph theory (extension)   cgrisom 46130
                  ​*21.43.15.1  Isomorphic graphs   cgrisom 46130
                  21.43.15.2  Loop-free graphs - extension   1hegrlfgr 46154
                  21.43.15.3  Walks - extension   cupwlks 46155
                  21.43.15.4  Edges of graphs expressed as sets of unordered pairs   upgredgssspr 46165
            21.43.16  Monoids (extension)   ovn0dmfun 46178
                  21.43.16.1  Auxiliary theorems   ovn0dmfun 46178
                  21.43.16.2  Magmas, Semigroups and Monoids (extension)   plusfreseq 46186
                  21.43.16.3  Magma homomorphisms and submagmas   cmgmhm 46191
                  21.43.16.4  Examples and counterexamples for magmas, semigroups and monoids (extension)   opmpoismgm 46221
                  21.43.16.5  Group sum operation (extension 1)   gsumsplit2f 46234
            ​*21.43.17  Magmas and internal binary operations (alternate approach)   ccllaw 46237
                  *21.43.17.1  Laws for internal binary operations   ccllaw 46237
                  ​*21.43.17.2  Internal binary operations   cintop 46250
                  21.43.17.3  Alternative definitions for magmas and semigroups   cmgm2 46269
            21.43.18  Categories (extension)   idfusubc0 46283
                  21.43.18.1  Subcategories (extension)   idfusubc0 46283
            21.43.19  Rings (extension)   lmod0rng 46286
                  21.43.19.1  Nonzero rings (extension)   lmod0rng 46286
                  ​*21.43.19.2  Non-unital rings ("rngs")   crng 46292
                  21.43.19.3  Rng homomorphisms   crngh 46303
                  21.43.19.4  Ring homomorphisms (extension)   rhmfn 46336
                  21.43.19.5  Ideals as non-unital rings   lidldomn1 46339
                  21.43.19.6  The non-unital ring of even integers   0even 46349
                  21.43.19.7  A constructed not unital ring   cznrnglem 46371
                  ​*21.43.19.8  The category of non-unital rings   crngc 46375
                  ​*21.43.19.9  The category of (unital) rings   cringc 46421
                  21.43.19.10  Subcategories of the category of rings   srhmsubclem1 46491
            21.43.20  Basic algebraic structures (extension)   opeliun2xp 46528
                  21.43.20.1  Auxiliary theorems   opeliun2xp 46528
                  21.43.20.2  The binomial coefficient operation (extension)   bcpascm1 46547
                  21.43.20.3  The ` ZZ `-module ` ZZ X. ZZ `   zlmodzxzlmod 46550
                  21.43.20.4  Group sum operation (extension 2)   mgpsumunsn 46557
                  21.43.20.5  Symmetric groups (extension)   exple2lt6 46560
                  21.43.20.6  Divisibility (extension)   invginvrid 46563
                  21.43.20.7  The support of functions (extension)   rmsupp0 46564
                  21.43.20.8  Finitely supported functions (extension)   rmsuppfi 46569
                  21.43.20.9  Left modules (extension)   lmodvsmdi 46578
                  21.43.20.10  Associative algebras (extension)   assaascl0 46580
                  21.43.20.11  Univariate polynomials (extension)   ply1vr1smo 46582
                  21.43.20.12  Univariate polynomials (examples)   linply1 46594
            21.43.21  Linear algebra (extension)   cdmatalt 46597
                  ​*21.43.21.1  The subalgebras of diagonal and scalar matrices (extension)   cdmatalt 46597
                  ​*21.43.21.2  Linear combinations   clinc 46605
                  ​*21.43.21.3  Linear independence   clininds 46641
                  21.43.21.4  Simple left modules and the ` ZZ `-module   lmod1lem1 46688
                  21.43.21.5  Differences between (left) modules and (left) vector spaces   lvecpsslmod 46708
            21.43.22  Complexity theory   suppdm 46711
                  21.43.22.1  Auxiliary theorems   suppdm 46711
                  21.43.22.2  The modulo (remainder) operation (extension)   fldivmod 46724
                  21.43.22.3  Even and odd integers   nn0onn0ex 46729
                  21.43.22.4  The natural logarithm on complex numbers (extension)   logcxp0 46741
                  21.43.22.5  Division of functions   cfdiv 46743
                  21.43.22.6  Upper bounds   cbigo 46753
                  21.43.22.7  Logarithm to an arbitrary base (extension)   rege1logbrege0 46764
                  ​*21.43.22.8  The binary logarithm   fldivexpfllog2 46771
                  21.43.22.9  Binary length   cblen 46775
                  ​*21.43.22.10  Digits   cdig 46801
                  21.43.22.11  Nonnegative integer as sum of its shifted digits   dignn0flhalflem1 46821
                  21.43.22.12  Algorithms for the multiplication of nonnegative integers   nn0mulfsum 46830
                  ​*21.43.22.13  N-ary functions   cnaryf 46832
                  ​*21.43.22.14  The Ackermann function   citco 46863
            21.43.23  Elementary geometry (extension)   fv1prop 46905
                  21.43.23.1  Auxiliary theorems   fv1prop 46905
                  21.43.23.2  Real euclidean space of dimension 2   rrx2pxel 46917
                  21.43.23.3  Spheres and lines in real Euclidean spaces   cline 46933
      ​21.44  Mathbox for Zhi Wang
            21.44.1  Propositional calculus   pm4.71da 46995
            21.44.2  Predicate calculus with equality   dtrucor3 47004
                  21.44.2.1  Axiom scheme ax-5 (Distinctness)   dtrucor3 47004
            21.44.3  ZF Set Theory - start with the Axiom of Extensionality   ralbidb 47005
                  21.44.3.1  Restricted quantification   ralbidb 47005
                  21.44.3.2  The empty set   ssdisjd 47012
                  21.44.3.3  Unordered and ordered pairs   vsn 47016
                  21.44.3.4  The union of a class   unilbss 47022
            21.44.4  ZF Set Theory - add the Axiom of Replacement   inpw 47023
                  21.44.4.1  Theorems requiring subset and intersection existence   inpw 47023
            21.44.5  ZF Set Theory - add the Axiom of Power Sets   mof0 47024
                  21.44.5.1  Functions   mof0 47024
                  21.44.5.2  Operations   fvconstr 47042
            21.44.6  ZF Set Theory - add the Axiom of Union   fvconst0ci 47045
                  21.44.6.1  Equinumerosity   fvconst0ci 47045
            21.44.7  Order sets   iccin 47049
                  21.44.7.1  Real number intervals   iccin 47049
            21.44.8  Moore spaces   mreuniss 47052
            ​*21.44.9  Topology   clduni 47053
                  21.44.9.1  Closure and interior   clduni 47053
                  21.44.9.2  Neighborhoods   neircl 47057
                  21.44.9.3  Subspace topologies   restcls2lem 47065
                  21.44.9.4  Limits and continuity in topological spaces   cnneiima 47069
                  21.44.9.5  Topological definitions using the reals   iooii 47070
                  21.44.9.6  Separated sets   sepnsepolem1 47074
                  21.44.9.7  Separated spaces: T0, T1, T2 (Hausdorff) ...   isnrm4 47083
            21.44.10  Preordered sets and directed sets using extensible structures   isprsd 47108
            21.44.11  Posets and lattices using extensible structures   lubeldm2 47109
                  21.44.11.1  Posets   lubeldm2 47109
                  21.44.11.2  Lattices   toslat 47127
                  21.44.11.3  Subset order structures   intubeu 47129
            21.44.12  Categories   catprslem 47150
                  21.44.12.1  Categories   catprslem 47150
                  21.44.12.2  Monomorphisms and epimorphisms   idmon 47156
                  21.44.12.3  Functors   funcf2lem 47158
            21.44.13  Examples of categories   cthinc 47159
                  21.44.13.1  Thin categories   cthinc 47159
                  21.44.13.2  Preordered sets as thin categories   cprstc 47202
                  21.44.13.3  Monoids as categories   cmndtc 47223
      ​21.45  Mathbox for Emmett Weisz
            ​*21.45.1  Miscellaneous Theorems   nfintd 47238
            21.45.2  Set Recursion   csetrecs 47248
                  ​*21.45.2.1  Basic Properties of Set Recursion   csetrecs 47248
                  21.45.2.2  Examples and properties of set recursion   elsetrecslem 47264
            ​*21.45.3  Construction of Games and Surreal Numbers   cpg 47274
      ​​*21.46  Mathbox for David A. Wheeler
            21.46.1  Natural deduction   sbidd 47283
            ​*21.46.2  Greater than, greater than or equal to.   cge-real 47285
            ​*21.46.3  Hyperbolic trigonometric functions   csinh 47295
            ​*21.46.4  Reciprocal trigonometric functions (sec, csc, cot)   csec 47306
            ​*21.46.5  Identities for "if"   ifnmfalse 47328
            ​*21.46.6  Logarithms generalized to arbitrary base using ` logb `   logb2aval 47329
            ​*21.46.7  Logarithm laws generalized to an arbitrary base - log_   clog- 47330
            ​*21.46.8  Formally define notions such as reflexivity   wreflexive 47332
            ​*21.46.9  Algebra helpers   comraddi 47336
            ​*21.46.10  Algebra helper examples   i2linesi 47345
            ​*21.46.11  Formal methods "surprises"   alimp-surprise 47347
            ​*21.46.12  Allsome quantifier   walsi 47353
            ​*21.46.13  Miscellaneous   5m4e1 47364
            21.46.14  Theorems about algebraic numbers   aacllem 47368
      ​21.47  Mathbox for Kunhao Zheng
            21.47.1  Weighted AM-GM inequality   amgmwlem 47369