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 2536
            1.6.2  Unique existence: the unique existential quantifier   weu 2566
      ​1.7  Other axiomatizations related to classical predicate calculus
            ​*1.7.1  Aristotelian logic: Assertic syllogisms   barbara 2662
            ​*1.7.2  Intuitionistic logic   axia1 2692
​​*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 2707
            2.1.2  Classes   cab 2713
                  2.1.2.1  Class abstractions   cab 2713
                  ​*2.1.2.2  Class equality   df-cleq 2728
                  2.1.2.3  Class membership   df-clel 2814
                  2.1.2.4  Elementary properties of class abstractions   abbi2dv 2871
            2.1.3  Class form not-free predicate   wnfc 2887
            2.1.4  Negated equality and membership   wne 2943
                  2.1.4.1  Negated equality   wne 2943
                  2.1.4.2  Negated membership   wnel 3049
            2.1.5  Restricted quantification   wral 3064
                  2.1.5.1  Restricted universal and existential quantification   wral 3064
                  2.1.5.2  Restricted existential uniqueness and at-most-one quantifier   wreu 3351
                  2.1.5.3  Restricted class abstraction   crab 3407
            2.1.6  The universal class   cvv 3445
            ​*2.1.7  Conditional equality (experimental)   wcdeq 3721
            2.1.8  Russell's Paradox   rru 3737
            2.1.9  Proper substitution of classes for sets   wsbc 3739
            2.1.10  Proper substitution of classes for sets into classes   csb 3855
            2.1.11  Define basic set operations and relations   cdif 3907
            2.1.12  Subclasses and subsets   df-ss 3927
            2.1.13  The difference, union, and intersection of two classes   dfdif3 4074
                  2.1.13.1  The difference of two classes   dfdif3 4074
                  2.1.13.2  The union of two classes   elun 4108
                  2.1.13.3  The intersection of two classes   elini 4153
                  2.1.13.4  The symmetric difference of two classes   csymdif 4201
                  2.1.13.5  Combinations of difference, union, and intersection of two classes   unabs 4214
                  2.1.13.6  Class abstractions with difference, union, and intersection of two classes   unabw 4257
                  2.1.13.7  Restricted uniqueness with difference, union, and intersection   reuun2 4274
            2.1.14  The empty set   c0 4282
            ​*2.1.15  The conditional operator for classes   cif 4486
            ​*2.1.16  The weak deduction theorem for set theory   dedth 4544
            2.1.17  Power classes   cpw 4560
            2.1.18  Unordered and ordered pairs   snjust 4585
            2.1.19  The union of a class   cuni 4865
            2.1.20  The intersection of a class   cint 4907
            2.1.21  Indexed union and intersection   ciun 4954
            2.1.22  Disjointness   wdisj 5070
            2.1.23  Binary relations   wbr 5105
            2.1.24  Ordered-pair class abstractions (class builders)   copab 5167
            2.1.25  Functions in maps-to notation   cmpt 5188
            2.1.26  Transitive classes   wtr 5222
      ​2.2  ZF Set Theory - add the Axiom of Replacement
            2.2.1  Introduce the Axiom of Replacement   ax-rep 5242
            2.2.2  Derive the Axiom of Separation   axsepgfromrep 5254
            2.2.3  Derive the Null Set Axiom   axnulALT 5261
            2.2.4  Theorems requiring subset and intersection existence   nalset 5270
            2.2.5  Theorems requiring empty set existence   class2set 5310
      ​2.3  ZF Set Theory - add the Axiom of Power Sets
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 5320
            2.3.2  Derive the Axiom of Pairing   axprlem1 5378
            2.3.3  Ordered pair theorem   opnz 5430
            2.3.4  Ordered-pair class abstractions (cont.)   opabidw 5481
            2.3.5  Power class of union and intersection   pwin 5527
            2.3.6  The identity relation   cid 5530
            2.3.7  The membership relation (or epsilon relation)   cep 5536
            ​*2.3.8  Partial and total orderings   wpo 5543
            2.3.9  Founded and well-ordering relations   wfr 5585
            2.3.10  Relations   cxp 5631
            2.3.11  The Predecessor Class   cpred 6252
            2.3.12  Well-founded induction (variant)   frpomin 6294
            2.3.13  Well-ordered induction   tz6.26 6301
            2.3.14  Ordinals   word 6316
            2.3.15  Definite description binder (inverted iota)   cio 6446
            2.3.16  Functions   wfun 6490
            2.3.17  Cantor's Theorem   canth 7310
            2.3.18  Restricted iota (description binder)   crio 7312
            2.3.19  Operations   co 7357
                  2.3.19.1  Variable-to-class conversion for operations   caovclg 7546
            2.3.20  Maps-to notation   mpondm0 7594
            2.3.21  Function operation   cof 7615
            2.3.22  Proper subset relation   crpss 7659
      ​2.4  ZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 7672
            2.4.2  Ordinals (continued)   epweon 7709
            2.4.3  Transfinite induction   tfi 7789
            2.4.4  The natural numbers (i.e., finite ordinals)   com 7802
            2.4.5  Peano's postulates   peano1 7825
            2.4.6  Finite induction (for finite ordinals)   find 7833
            2.4.7  Relations and functions (cont.)   dmexg 7840
            2.4.8  First and second members of an ordered pair   c1st 7919
            2.4.9  Induction on Cartesian products   frpoins3xpg 8072
            2.4.10  Ordering on Cartesian products   xpord2lem 8074
            2.4.11  Ordering Ordinal Sequences   orderseqlem 8089
            ​*2.4.12  The support of functions   csupp 8092
            ​*2.4.13  Special maps-to operations   opeliunxp2f 8141
            2.4.14  Function transposition   ctpos 8156
            2.4.15  Curry and uncurry   ccur 8196
            2.4.16  Undefined values   cund 8203
            2.4.17  Well-founded recursion   cfrecs 8211
            2.4.18  Well-ordered recursion   cwrecs 8242
            2.4.19  Functions on ordinals; strictly monotone ordinal functions   iunon 8285
            2.4.20  "Strong" transfinite recursion   crecs 8316
            2.4.21  Recursive definition generator   crdg 8355
            2.4.22  Finite recursion   frfnom 8381
            2.4.23  Ordinal arithmetic   c1o 8405
            2.4.24  Natural number arithmetic   nna0 8551
            2.4.25  Natural addition   cnadd 8611
            2.4.26  Equivalence relations and classes   wer 8645
            2.4.27  The mapping operation   cmap 8765
            2.4.28  Infinite Cartesian products   cixp 8835
            2.4.29  Equinumerosity   cen 8880
            2.4.30  Schroeder-Bernstein Theorem   sbthlem1 9027
            2.4.31  Equinumerosity (cont.)   xpf1o 9083
            2.4.32  Finite sets   dif1enlem 9100
            2.4.33  Pigeonhole Principle   phplem1 9151
            2.4.34  Finite sets (cont.)   onomeneq 9172
            2.4.35  Finitely supported functions   cfsupp 9305
            2.4.36  Finite intersections   cfi 9346
            2.4.37  Hall's marriage theorem   marypha1lem 9369
            2.4.38  Supremum and infimum   csup 9376
            2.4.39  Ordinal isomorphism, Hartogs's theorem   coi 9445
            2.4.40  Hartogs function   char 9492
            2.4.41  Weak dominance   cwdom 9500
      ​2.5  ZF Set Theory - add the Axiom of Regularity
            2.5.1  Introduce the Axiom of Regularity   ax-reg 9528
            2.5.2  Axiom of Infinity equivalents   inf0 9557
      ​2.6  ZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-inf 9574
            2.6.2  Existence of omega (the set of natural numbers)   omex 9579
            2.6.3  Cantor normal form   ccnf 9597
            2.6.4  Transitive closure of a relation   cttrcl 9643
            2.6.5  Transitive closure   trcl 9664
            2.6.6  Well-Founded Induction   frmin 9685
            2.6.7  Well-Founded Recursion   frr3g 9692
            2.6.8  Rank   cr1 9698
            2.6.9  Scott's trick; collection principle; Hilbert's epsilon   scottex 9821
            2.6.10  Disjoint union   cdju 9834
            2.6.11  Cardinal numbers   ccrd 9871
            2.6.12  Axiom of Choice equivalents   wac 10051
            ​*2.6.13  Cardinal number arithmetic   undjudom 10103
            2.6.14  The Ackermann bijection   ackbij2lem1 10155
            2.6.15  Cofinality (without Axiom of Choice)   cflem 10182
            2.6.16  Eight inequivalent definitions of finite set   sornom 10213
            2.6.17  Hereditarily size-limited sets without Choice   itunifval 10352
​​*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 10371
            3.1.2  Introduce the Axiom of Dependent Choice   ax-dc 10382
      ​3.2  ZFC Set Theory - add the Axiom of Choice
            3.2.1  Introduce the Axiom of Choice   ax-ac 10395
            3.2.2  AC equivalents: well-ordering, Zorn's lemma   numthcor 10430
            3.2.3  Cardinal number theorems using Axiom of Choice   cardval 10482
            3.2.4  Cardinal number arithmetic using Axiom of Choice   iunctb 10510
            3.2.5  Cofinality using the Axiom of Choice   alephreg 10518
      ​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 10556
            3.4.2  Derivation of the Axiom of Choice   gchaclem 10614
​​*PART 4  TG (TARSKI-GROTHENDIECK) SET THEORY
      ​4.1  Inaccessibles
            4.1.1  Weakly and strongly inaccessible cardinals   cwina 10618
            4.1.2  Weak universes   cwun 10636
            4.1.3  Tarski classes   ctsk 10684
            4.1.4  Grothendieck universes   cgru 10726
      ​4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
            4.2.1  Introduce the Tarski-Grothendieck Axiom   ax-groth 10759
            4.2.2  Derive the Power Set, Infinity and Choice Axioms   grothpw 10762
            4.2.3  Tarski map function   ctskm 10773
​​*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 10780
            5.1.2  Final derivation of real and complex number postulates   axaddf 11081
            5.1.3  Real and complex number postulates restated as axioms   ax-cnex 11107
      ​5.2  Derive the basic properties from the field axioms
            5.2.1  Some deductions from the field axioms for complex numbers   cnex 11132
            5.2.2  Infinity and the extended real number system   cpnf 11186
            5.2.3  Restate the ordering postulates with extended real "less than"   axlttri 11226
            5.2.4  Ordering on reals   lttr 11231
            5.2.5  Initial properties of the complex numbers   mul12 11320
      ​5.3  Real and complex numbers - basic operations
            5.3.1  Addition   add12 11372
            5.3.2  Subtraction   cmin 11385
            5.3.3  Multiplication   kcnktkm1cn 11586
            5.3.4  Ordering on reals (cont.)   gt0ne0 11620
            5.3.5  Reciprocals   ixi 11784
            5.3.6  Division   cdiv 11812
            5.3.7  Ordering on reals (cont.)   elimgt0 11993
            5.3.8  Completeness Axiom and Suprema   fimaxre 12099
            5.3.9  Imaginary and complex number properties   inelr 12143
            5.3.10  Function operation analogue theorems   ofsubeq0 12150
      ​5.4  Integer sets
            5.4.1  Positive integers (as a subset of complex numbers)   cn 12153
            5.4.2  Principle of mathematical induction   nnind 12171
            ​*5.4.3  Decimal representation of numbers   c2 12208
            ​*5.4.4  Some properties of specific numbers   neg1cn 12267
            5.4.5  Simple number properties   halfcl 12378
            5.4.6  The Archimedean property   nnunb 12409
            5.4.7  Nonnegative integers (as a subset of complex numbers)   cn0 12413
            ​*5.4.8  Extended nonnegative integers   cxnn0 12485
            5.4.9  Integers (as a subset of complex numbers)   cz 12499
            5.4.10  Decimal arithmetic   cdc 12618
            5.4.11  Upper sets of integers   cuz 12763
            5.4.12  Well-ordering principle for bounded-below sets of integers   uzwo3 12868
            5.4.13  Rational numbers (as a subset of complex numbers)   cq 12873
            5.4.14  Existence of the set of complex numbers   rpnnen1lem2 12902
      ​5.5  Order sets
            5.5.1  Positive reals (as a subset of complex numbers)   crp 12915
            5.5.2  Infinity and the extended real number system (cont.)   cxne 13030
            5.5.3  Supremum and infimum on the extended reals   xrsupexmnf 13224
            5.5.4  Real number intervals   cioo 13264
            5.5.5  Finite intervals of integers   cfz 13424
            ​*5.5.6  Finite intervals of nonnegative integers   elfz2nn0 13532
            5.5.7  Half-open integer ranges   cfzo 13567
      ​5.6  Elementary integer functions
            5.6.1  The floor and ceiling functions   cfl 13695
            5.6.2  The modulo (remainder) operation   cmo 13774
            5.6.3  Miscellaneous theorems about integers   om2uz0i 13852
            5.6.4  Strong induction over upper sets of integers   uzsinds 13892
            5.6.5  Finitely supported functions over the nonnegative integers   fsuppmapnn0fiublem 13895
            5.6.6  The infinite sequence builder "seq" - extension   cseq 13906
            5.6.7  Integer powers   cexp 13967
            5.6.8  Ordered pair theorem for nonnegative integers   nn0le2msqi 14167
            5.6.9  Factorial function   cfa 14173
            5.6.10  The binomial coefficient operation   cbc 14202
            5.6.11  The ` # ` (set size) function   chash 14230
                  5.6.11.1  Proper unordered pairs and triples (sets of size 2 and 3)   hashprlei 14367
                  5.6.11.2  Functions with a domain containing at least two different elements   fundmge2nop0 14391
                  5.6.11.3  Finite induction on the size of the first component of a binary relation   hashdifsnp1 14395
      ​​*5.7  Words over a set
            5.7.1  Definitions and basic theorems   cword 14402
            5.7.2  Last symbol of a word   clsw 14450
            5.7.3  Concatenations of words   cconcat 14458
            5.7.4  Singleton words   cs1 14483
            5.7.5  Concatenations with singleton words   ccatws1cl 14504
            5.7.6  Subwords/substrings   csubstr 14528
            5.7.7  Prefixes of a word   cpfx 14558
            5.7.8  Subwords of subwords   swrdswrdlem 14592
            5.7.9  Subwords and concatenations   pfxcctswrd 14598
            5.7.10  Subwords of concatenations   swrdccatfn 14612
            5.7.11  Splicing words (substring replacement)   csplice 14637
            5.7.12  Reversing words   creverse 14646
            5.7.13  Repeated symbol words   creps 14656
            ​*5.7.14  Cyclical shifts of words   ccsh 14676
            5.7.15  Mapping words by a function   wrdco 14720
            5.7.16  Longer string literals   cs2 14730
      ​​*5.8  Reflexive and transitive closures of relations
            5.8.1  The reflexive and transitive properties of relations   coss12d 14857
            5.8.2  Basic properties of closures   cleq1lem 14867
            5.8.3  Definitions and basic properties of transitive closures   ctcl 14870
            5.8.4  Exponentiation of relations   crelexp 14904
            5.8.5  Reflexive-transitive closure as an indexed union   crtrcl 14940
            ​*5.8.6  Principle of transitive induction.   relexpindlem 14948
      ​5.9  Elementary real and complex functions
            5.9.1  The "shift" operation   cshi 14951
            5.9.2  Signum (sgn or sign) function   csgn 14971
            5.9.3  Real and imaginary parts; conjugate   ccj 14981
            5.9.4  Square root; absolute value   csqrt 15118
      ​5.10  Elementary limits and convergence
            5.10.1  Superior limit (lim sup)   clsp 15352
            5.10.2  Limits   cli 15366
            5.10.3  Finite and infinite sums   csu 15570
            5.10.4  The binomial theorem   binomlem 15714
            5.10.5  The inclusion/exclusion principle   incexclem 15721
            5.10.6  Infinite sums (cont.)   isumshft 15724
            5.10.7  Miscellaneous converging and diverging sequences   divrcnv 15737
            5.10.8  Arithmetic series   arisum 15745
            5.10.9  Geometric series   expcnv 15749
            5.10.10  Ratio test for infinite series convergence   cvgrat 15768
            5.10.11  Mertens' theorem   mertenslem1 15769
            5.10.12  Finite and infinite products   prodf 15772
                  5.10.12.1  Product sequences   prodf 15772
                  5.10.12.2  Non-trivial convergence   ntrivcvg 15782
                  5.10.12.3  Complex products   cprod 15788
                  5.10.12.4  Finite products   fprod 15824
                  5.10.12.5  Infinite products   iprodclim 15881
            5.10.13  Falling and Rising Factorial   cfallfac 15887
            5.10.14  Bernoulli polynomials and sums of k-th powers   cbp 15929
      ​5.11  Elementary trigonometry
            5.11.1  The exponential, sine, and cosine functions   ce 15944
                  5.11.1.1  The circle constant (tau = 2 pi)   ctau 16084
            5.11.2  _e is irrational   eirrlem 16086
      ​5.12  Cardinality of real and complex number subsets
            5.12.1  Countability of integers and rationals   xpnnen 16093
            5.12.2  The reals are uncountable   rpnnen2lem1 16096
​​*PART 6  ELEMENTARY NUMBER THEORY
      ​6.1  Elementary properties of divisibility
            6.1.1  Irrationality of square root of 2   sqrt2irrlem 16130
            6.1.2  Some Number sets are chains of proper subsets   nthruc 16134
            6.1.3  The divides relation   cdvds 16136
            ​*6.1.4  Even and odd numbers   evenelz 16218
            6.1.5  The division algorithm   divalglem0 16275
            6.1.6  Bit sequences   cbits 16299
            6.1.7  The greatest common divisor operator   cgcd 16374
            6.1.8  Bézout's identity   bezoutlem1 16420
            6.1.9  Algorithms   nn0seqcvgd 16446
            6.1.10  Euclid's Algorithm   eucalgval2 16457
            ​*6.1.11  The least common multiple   clcm 16464
            ​*6.1.12  Coprimality and Euclid's lemma   coprmgcdb 16525
            6.1.13  Cancellability of congruences   congr 16540
      ​6.2  Elementary prime number theory
            ​*6.2.1  Elementary properties   cprime 16547
            ​*6.2.2  Coprimality and Euclid's lemma (cont.)   coprm 16587
            6.2.3  Properties of the canonical representation of a rational   cnumer 16608
            6.2.4  Euler's theorem   codz 16635
            6.2.5  Arithmetic modulo a prime number   modprm1div 16669
            6.2.6  Pythagorean Triples   coprimeprodsq 16680
            6.2.7  The prime count function   cpc 16708
            6.2.8  Pocklington's theorem   prmpwdvds 16776
            6.2.9  Infinite primes theorem   unbenlem 16780
            6.2.10  Sum of prime reciprocals   prmreclem1 16788
            6.2.11  Fundamental theorem of arithmetic   1arithlem1 16795
            6.2.12  Lagrange's four-square theorem   cgz 16801
            6.2.13  Van der Waerden's theorem   cvdwa 16837
            6.2.14  Ramsey's theorem   cram 16871
            ​*6.2.15  Primorial function   cprmo 16903
            ​*6.2.16  Prime gaps   prmgaplem1 16921
            6.2.17  Decimal arithmetic (cont.)   dec2dvds 16935
            6.2.18  Cyclical shifts of words (cont.)   cshwsidrepsw 16966
            6.2.19  Specific prime numbers   prmlem0 16978
            6.2.20  Very large primes   1259lem1 17003
​PART 7  BASIC STRUCTURES
      ​7.1  Extensible structures
            ​*7.1.1  Basic definitions   cstr 17018
                  7.1.1.1  Extensible structures as structures with components   cstr 17018
                  7.1.1.2  Substitution of components   csts 17035
                  7.1.1.3  Slots   cslot 17053
                  ​*7.1.1.4  Structure component indices   cnx 17065
                  7.1.1.5  Base sets   cbs 17083
                  7.1.1.6  Base set restrictions   cress 17112
            7.1.2  Slot definitions   cplusg 17133
            7.1.3  Definition of the structure product   crest 17302
            7.1.4  Definition of the structure quotient   cordt 17381
      ​7.2  Moore spaces
            7.2.1  Moore closures   mrcflem 17486
            7.2.2  Independent sets in a Moore system   mrisval 17510
            7.2.3  Algebraic closure systems   isacs 17531
​PART 8  BASIC CATEGORY THEORY
      ​8.1  Categories
            8.1.1  Categories   ccat 17544
            8.1.2  Opposite category   coppc 17591
            8.1.3  Monomorphisms and epimorphisms   cmon 17611
            8.1.4  Sections, inverses, isomorphisms   csect 17627
            ​*8.1.5  Isomorphic objects   ccic 17678
            8.1.6  Subcategories   cssc 17690
            8.1.7  Functors   cfunc 17740
            8.1.8  Full & faithful functors   cful 17789
            8.1.9  Natural transformations and the functor category   cnat 17828
            8.1.10  Initial, terminal and zero objects of a category   cinito 17867
      ​8.2  Arrows (disjointified hom-sets)
            8.2.1  Identity and composition for arrows   cida 17939
      ​8.3  Examples of categories
            8.3.1  The category of sets   csetc 17961
            8.3.2  The category of categories   ccatc 17984
            ​*8.3.3  The category of extensible structures   fncnvimaeqv 18007
      ​8.4  Categorical constructions
            8.4.1  Product of categories   cxpc 18056
            8.4.2  Functor evaluation   cevlf 18098
            8.4.3  Hom functor   chof 18137
​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 18320
            9.5.2  Complete lattices   ccla 18387
            9.5.3  Distributive lattices   cdlat 18409
            9.5.4  Subset order structures   cipo 18416
      ​9.6  Posets, directed sets, and lattices as relations
            ​*9.6.1  Posets and lattices as relations   cps 18453
            9.6.2  Directed sets, nets   cdir 18483
​PART 10  BASIC ALGEBRAIC STRUCTURES
      ​10.1  Monoids
            ​*10.1.1  Magmas   cplusf 18494
            ​*10.1.2  Identity elements   mgmidmo 18515
            ​*10.1.3  Iterated sums in a magma   gsumvalx 18531
            ​*10.1.4  Semigroups   csgrp 18545
            ​*10.1.5  Definition and basic properties of monoids   cmnd 18556
            10.1.6  Monoid homomorphisms and submonoids   cmhm 18599
            ​*10.1.7  Iterated sums in a monoid   gsumvallem2 18644
            10.1.8  Free monoids   cfrmd 18657
                  ​*10.1.8.1  Monoid of endofunctions   cefmnd 18678
            10.1.9  Examples and counterexamples for magmas, semigroups and monoids   mgm2nsgrplem1 18728
      ​10.2  Groups
            10.2.1  Definition and basic properties   cgrp 18748
            ​*10.2.2  Group multiple operation   cmg 18872
            10.2.3  Subgroups and Quotient groups   csubg 18922
            ​*10.2.4  Cyclic monoids and groups   cycsubmel 18993
            10.2.5  Elementary theory of group homomorphisms   cghm 19005
            10.2.6  Isomorphisms of groups   cgim 19047
            10.2.7  Group actions   cga 19069
            10.2.8  Centralizers and centers   ccntz 19095
            10.2.9  The opposite group   coppg 19123
            10.2.10  Symmetric groups   csymg 19148
                  ​*10.2.10.1  Definition and basic properties   csymg 19148
                  10.2.10.2  Cayley's theorem   cayleylem1 19194
                  10.2.10.3  Permutations fixing one element   symgfix2 19198
                  ​*10.2.10.4  Transpositions in the symmetric group   cpmtr 19223
                  10.2.10.5  The sign of a permutation   cpsgn 19271
            10.2.11  p-Groups and Sylow groups; Sylow's theorems   cod 19306
            10.2.12  Direct products   clsm 19416
                  10.2.12.1  Direct products (extension)   smndlsmidm 19438
            10.2.13  Free groups   cefg 19488
            10.2.14  Abelian groups   ccmn 19562
                  10.2.14.1  Definition and basic properties   ccmn 19562
                  10.2.14.2  Cyclic groups   ccyg 19654
                  10.2.14.3  Group sum operation   gsumval3a 19680
                  10.2.14.4  Group sums over (ranges of) integers   fsfnn0gsumfsffz 19760
                  10.2.14.5  Internal direct products   cdprd 19772
                  10.2.14.6  The Fundamental Theorem of Abelian Groups   ablfacrplem 19844
            10.2.15  Simple groups   csimpg 19869
                  10.2.15.1  Definition and basic properties   csimpg 19869
                  10.2.15.2  Classification of abelian simple groups   ablsimpnosubgd 19883
      ​10.3  Rings
            10.3.1  Multiplicative Group   cmgp 19896
            ​*10.3.2  Ring unity (multiplicative identity)   cur 19913
            10.3.3  Semirings   csrg 19917
                  ​*10.3.3.1  The binomial theorem for semirings   srgbinomlem1 19957
            10.3.4  Definition and basic properties of unital rings   crg 19964
            10.3.5  Opposite ring   coppr 20048
            10.3.6  Divisibility   cdsr 20067
            10.3.7  Ring primes   crpm 20141
            10.3.8  Ring homomorphisms   crh 20143
      ​10.4  Division rings and fields
            10.4.1  Definition and basic properties   cdr 20185
            10.4.2  Subrings of a ring   csubrg 20218
                  10.4.2.1  Sub-division rings   csdrg 20259
            10.4.3  Absolute value (abstract algebra)   cabv 20275
            10.4.4  Star rings   cstf 20302
      ​10.5  Left modules
            10.5.1  Definition and basic properties   clmod 20322
            10.5.2  Subspaces and spans in a left module   clss 20392
            10.5.3  Homomorphisms and isomorphisms of left modules   clmhm 20480
            10.5.4  Subspace sum; bases for a left module   clbs 20535
      ​10.6  Vector spaces
            10.6.1  Definition and basic properties   clvec 20563
      ​10.7  Ideals
            10.7.1  The subring algebra; ideals   csra 20629
            10.7.2  Two-sided ideals and quotient rings   c2idl 20701
            10.7.3  Principal ideal rings. Divisibility in the integers   clpidl 20711
            10.7.4  Nonzero rings and zero rings   cnzr 20727
            10.7.5  Left regular elements. More kinds of rings   crlreg 20749
      ​10.8  The complex numbers as an algebraic extensible structure
            10.8.1  Definition and basic properties   cpsmet 20780
            ​*10.8.2  Ring of integers   czring 20869
            10.8.3  Algebraic constructions based on the complex numbers   czrh 20900
            10.8.4  Signs as subgroup of the complex numbers   cnmsgnsubg 20981
            10.8.5  Embedding of permutation signs into a ring   zrhpsgnmhm 20988
            10.8.6  The ordered field of real numbers   crefld 21008
      ​10.9  Generalized pre-Hilbert and Hilbert spaces
            10.9.1  Definition and basic properties   cphl 21028
            10.9.2  Orthocomplements and closed subspaces   cocv 21064
            10.9.3  Orthogonal projection and orthonormal bases   cpj 21106
​​*PART 11  BASIC LINEAR ALGEBRA
      ​11.1  Vectors and free modules
            *11.1.1  Direct sum of left modules   cdsmm 21137
            ​*11.1.2  Free modules   cfrlm 21152
            ​*11.1.3  Standard basis (unit vectors)   cuvc 21188
            ​*11.1.4  Independent sets and families   clindf 21210
            11.1.5  Characterization of free modules   lmimlbs 21242
      ​11.2  Associative algebras
            11.2.1  Definition and basic properties   casa 21256
      ​11.3  Abstract multivariate polynomials
            11.3.1  Definition and basic properties   cmps 21306
            11.3.2  Polynomial evaluation   ces 21480
            11.3.3  Additional definitions for (multivariate) polynomials   cslv 21518
            ​*11.3.4  Univariate polynomials   cps1 21546
            11.3.5  Univariate polynomial evaluation   ces1 21679
      ​​*11.4  Matrices
            *11.4.1  The matrix multiplication   cmmul 21732
            ​*11.4.2  Square matrices   cmat 21754
            ​*11.4.3  The matrix algebra   matmulr 21787
            ​*11.4.4  Matrices of dimension 0 and 1   mat0dimbas0 21815
            ​*11.4.5  The subalgebras of diagonal and scalar matrices   cdmat 21837
            ​*11.4.6  Multiplication of a matrix with a "column vector"   cmvmul 21889
            11.4.7  Replacement functions for a square matrix   cmarrep 21905
            11.4.8  Submatrices   csubma 21925
      ​11.5  The determinant
            11.5.1  Definition and basic properties   cmdat 21933
            11.5.2  Determinants of 2 x 2 -matrices   m2detleiblem1 21973
            11.5.3  The matrix adjugate/adjunct   cmadu 21981
            ​*11.5.4  Laplace expansion of determinants (special case)   symgmatr01lem 22002
            11.5.5  Inverse matrix   invrvald 22025
            ​*11.5.6  Cramer's rule   slesolvec 22028
      ​​*11.6  Polynomial matrices
            11.6.1  Basic properties   pmatring 22041
            ​*11.6.2  Constant polynomial matrices   ccpmat 22052
            ​*11.6.3  Collecting coefficients of polynomial matrices   cdecpmat 22111
            ​*11.6.4  Ring isomorphism between polynomial matrices and polynomials over matrices   cpm2mp 22141
      ​​*11.7  The characteristic polynomial
            *11.7.1  Definition and basic properties   cchpmat 22175
            ​*11.7.2  The characteristic factor function G   fvmptnn04if 22198
            ​*11.7.3  The Cayley-Hamilton theorem   cpmadurid 22216
​PART 12  BASIC TOPOLOGY
      ​12.1  Topology
            ​*12.1.1  Topological spaces   ctop 22242
                  12.1.1.1  Topologies   ctop 22242
                  12.1.1.2  Topologies on sets   ctopon 22259
                  12.1.1.3  Topological spaces   ctps 22281
            12.1.2  Topological bases   ctb 22295
            12.1.3  Examples of topologies   distop 22345
            12.1.4  Closure and interior   ccld 22367
            12.1.5  Neighborhoods   cnei 22448
            12.1.6  Limit points and perfect sets   clp 22485
            12.1.7  Subspace topologies   restrcl 22508
            12.1.8  Order topology   ordtbaslem 22539
            12.1.9  Limits and continuity in topological spaces   ccn 22575
            12.1.10  Separated spaces: T0, T1, T2 (Hausdorff) ...   ct0 22657
            12.1.11  Compactness   ccmp 22737
            12.1.12  Bolzano-Weierstrass theorem   bwth 22761
            12.1.13  Connectedness   cconn 22762
            12.1.14  First- and second-countability   c1stc 22788
            12.1.15  Local topological properties   clly 22815
            12.1.16  Refinements   cref 22853
            12.1.17  Compactly generated spaces   ckgen 22884
            12.1.18  Product topologies   ctx 22911
            12.1.19  Continuous function-builders   cnmptid 23012
            12.1.20  Quotient maps and quotient topology   ckq 23044
            12.1.21  Homeomorphisms   chmeo 23104
      ​12.2  Filters and filter bases
            12.2.1  Filter bases   elmptrab 23178
            12.2.2  Filters   cfil 23196
            12.2.3  Ultrafilters   cufil 23250
            12.2.4  Filter limits   cfm 23284
            12.2.5  Extension by continuity   ccnext 23410
            12.2.6  Topological groups   ctmd 23421
            12.2.7  Infinite group sum on topological groups   ctsu 23477
            12.2.8  Topological rings, fields, vector spaces   ctrg 23507
      ​12.3  Uniform Structures and Spaces
            12.3.1  Uniform structures   cust 23551
            12.3.2  The topology induced by an uniform structure   cutop 23582
            12.3.3  Uniform Spaces   cuss 23605
            12.3.4  Uniform continuity   cucn 23627
            12.3.5  Cauchy filters in uniform spaces   ccfilu 23638
            12.3.6  Complete uniform spaces   ccusp 23649
      ​12.4  Metric spaces
            12.4.1  Pseudometric spaces   ispsmet 23657
            12.4.2  Basic metric space properties   cxms 23670
            12.4.3  Metric space balls   blfvalps 23736
            12.4.4  Open sets of a metric space   mopnval 23791
            12.4.5  Continuity in metric spaces   metcnp3 23896
            12.4.6  The uniform structure generated by a metric   metuval 23905
            12.4.7  Examples of metric spaces   dscmet 23928
            ​*12.4.8  Normed algebraic structures   cnm 23932
            12.4.9  Normed space homomorphisms (bounded linear operators)   cnmo 24069
            12.4.10  Topology on the reals   qtopbaslem 24122
            12.4.11  Topological definitions using the reals   cii 24238
            12.4.12  Path homotopy   chtpy 24330
            12.4.13  The fundamental group   cpco 24363
      ​12.5  Metric subcomplex vector spaces
            12.5.1  Subcomplex modules   cclm 24425
            ​*12.5.2  Subcomplex vector spaces   ccvs 24486
            ​*12.5.3  Normed subcomplex vector spaces   isncvsngp 24513
            12.5.4  Subcomplex pre-Hilbert spaces   ccph 24530
            12.5.5  Convergence and completeness   ccfil 24616
            12.5.6  Baire's Category Theorem   bcthlem1 24688
            12.5.7  Banach spaces and subcomplex Hilbert spaces   ccms 24696
                  12.5.7.1  The complete ordered field of the real numbers   retopn 24743
            12.5.8  Euclidean spaces   crrx 24747
            12.5.9  Minimizing Vector Theorem   minveclem1 24788
            12.5.10  Projection Theorem   pjthlem1 24801
​PART 13  BASIC REAL AND COMPLEX ANALYSIS
      ​13.1  Continuity
            13.1.1  Intermediate value theorem   pmltpclem1 24812
      ​13.2  Integrals
            13.2.1  Lebesgue measure   covol 24826
            13.2.2  Lebesgue integration   cmbf 24978
                  13.2.2.1  Lesbesgue integral   cmbf 24978
                  13.2.2.2  Lesbesgue directed integral   cdit 25210
      ​13.3  Derivatives
            13.3.1  Real and complex differentiation   climc 25226
                  13.3.1.1  Derivatives of functions of one complex or real variable   climc 25226
                  13.3.1.2  Results on real differentiation   dvferm1lem 25348
​PART 14  BASIC REAL AND COMPLEX FUNCTIONS
      ​14.1  Polynomials
            14.1.1  Polynomial degrees   cmdg 25415
            14.1.2  The division algorithm for univariate polynomials   cmn1 25490
            14.1.3  Elementary properties of complex polynomials   cply 25545
            14.1.4  The division algorithm for polynomials   cquot 25650
            14.1.5  Algebraic numbers   caa 25674
            14.1.6  Liouville's approximation theorem   aalioulem1 25692
      ​14.2  Sequences and series
            14.2.1  Taylor polynomials and Taylor's theorem   ctayl 25712
            14.2.2  Uniform convergence   culm 25735
            14.2.3  Power series   pserval 25769
      ​14.3  Basic trigonometry
            14.3.1  The exponential, sine, and cosine functions (cont.)   efcn 25802
            14.3.2  Properties of pi = 3.14159...   pilem1 25810
            14.3.3  Mapping of the exponential function   efgh 25897
            14.3.4  The natural logarithm on complex numbers   clog 25910
            ​*14.3.5  Logarithms to an arbitrary base   clogb 26114
            14.3.6  Theorems of Pythagoras, isosceles triangles, and intersecting chords   angval 26151
            14.3.7  Solutions of quadratic, cubic, and quartic equations   quad2 26189
            14.3.8  Inverse trigonometric functions   casin 26212
            14.3.9  The Birthday Problem   log2ublem1 26296
            14.3.10  Areas in R^2   carea 26305
            14.3.11  More miscellaneous converging sequences   rlimcnp 26315
            14.3.12  Inequality of arithmetic and geometric means   cvxcl 26334
            14.3.13  Euler-Mascheroni constant   cem 26341
            14.3.14  Zeta function   czeta 26362
            14.3.15  Gamma function   clgam 26365
      ​14.4  Basic number theory
            14.4.1  Wilson's theorem   wilthlem1 26417
            14.4.2  The Fundamental Theorem of Algebra   ftalem1 26422
            14.4.3  The Basel problem (ζ(2) = π²/6)   basellem1 26430
            14.4.4  Number-theoretical functions   ccht 26440
            14.4.5  Perfect Number Theorem   mersenne 26575
            14.4.6  Characters of Z/nZ   cdchr 26580
            14.4.7  Bertrand's postulate   bcctr 26623
            ​*14.4.8  Quadratic residues and the Legendre symbol   clgs 26642
            ​*14.4.9  Gauss' Lemma   gausslemma2dlem0a 26704
            14.4.10  Quadratic reciprocity   lgseisenlem1 26723
            14.4.11  All primes 4n+1 are the sum of two squares   2sqlem1 26765
            14.4.12  Chebyshev's Weak Prime Number Theorem, Dirichlet's Theorem   chebbnd1lem1 26817
            14.4.13  The Prime Number Theorem   mudivsum 26878
            14.4.14  Ostrowski's theorem   abvcxp 26963
​PART 15  SURREAL NUMBERS
      ​​*15.1  Sign sequence representation and Alling's axioms
            15.1.1  Definitions and initial properties   csur 26988
            15.1.2  Ordering   sltsolem1 27023
            15.1.3  Birthday Function   bdayfo 27025
            15.1.4  Density   fvnobday 27026
            ​*15.1.5  Full-Eta Property   bdayimaon 27041
      ​15.2  Initial consequences of Alling's axioms
            15.2.1  Ordering Theorems   csle 27092
            15.2.2  Birthday Theorems   bdayfun 27112
      ​​*15.3  Conway cut representation
            15.3.1  Conway cuts   csslt 27120
            15.3.2  Zero and One   c0s 27161
            15.3.3  Cuts and Options   cmade 27172
            15.3.4  Cofinality and coinitiality   cofsslt 27237
      ​15.4  Induction and recursion
            15.4.1  Induction and recursion on one variable   cnorec 27249
            15.4.2  Induction and recursion on two variables   cnorec2 27260
      ​15.5  Surreal arithmetic
            15.5.1  Addition   cadds 27271
            15.5.2  Negation and Subtraction   cnegs 27318
​​*PART 16  ELEMENTARY GEOMETRY
      ​16.1  Definition and Tarski's Axioms of Geometry
            16.1.1  Justification for the congruence notation   tgjustf 27415
      ​16.2  Tarskian Geometry
            16.2.1  Congruence   tgcgrcomimp 27419
            16.2.2  Betweenness   tgbtwntriv2 27429
            16.2.3  Dimension   tglowdim1 27442
            16.2.4  Betweenness and Congruence   tgifscgr 27450
            16.2.5  Congruence of a series of points   ccgrg 27452
            16.2.6  Motions   cismt 27474
            16.2.7  Colinearity   tglng 27488
            16.2.8  Connectivity of betweenness   tgbtwnconn1lem1 27514
            16.2.9  Less-than relation in geometric congruences   cleg 27524
            16.2.10  Rays   chlg 27542
            16.2.11  Lines   btwnlng1 27561
            16.2.12  Point inversions   cmir 27594
            16.2.13  Right angles   crag 27635
            16.2.14  Half-planes   islnopp 27681
            16.2.15  Midpoints and Line Mirroring   cmid 27714
            16.2.16  Congruence of angles   ccgra 27749
            16.2.17  Angle Comparisons   cinag 27777
            16.2.18  Congruence Theorems   tgsas1 27796
            16.2.19  Equilateral triangles   ceqlg 27807
      ​16.3  Properties of geometries
            16.3.1  Isomorphisms between geometries   f1otrgds 27811
      ​16.4  Geometry in Hilbert spaces
            16.4.1  Geometry in the complex plane   cchhllem 27835
            16.4.2  Geometry in Euclidean spaces   cee 27837
                  16.4.2.1  Definition of the Euclidean space   cee 27837
                  16.4.2.2  Tarski's axioms for geometry for the Euclidean space   axdimuniq 27862
                  16.4.2.3  EE^n fulfills Tarski's Axioms   ceeng 27926
​​*PART 17  GRAPH THEORY
      ​*17.1  Vertices and edges
            17.1.1  The edge function extractor for extensible structures   cedgf 27937
            ​*17.1.2  Vertices and indexed edges   cvtx 27947
                  17.1.2.1  Definitions and basic properties   cvtx 27947
                  17.1.2.2  The vertices and edges of a graph represented as ordered pair   opvtxval 27954
                  17.1.2.3  The vertices and edges of a graph represented as extensible structure   funvtxdmge2val 27962
                  17.1.2.4  Representations of graphs without edges   snstrvtxval 27988
                  17.1.2.5  Degenerated cases of representations of graphs   vtxval0 27990
            17.1.3  Edges as range of the edge function   cedg 27998
      ​​*17.2  Undirected graphs
            17.2.1  Undirected hypergraphs   cuhgr 28007
            17.2.2  Undirected pseudographs and multigraphs   cupgr 28031
            ​*17.2.3  Loop-free graphs   umgrislfupgrlem 28073
            17.2.4  Edges as subsets of vertices of graphs   uhgredgiedgb 28077
            ​*17.2.5  Undirected simple graphs   cuspgr 28099
            17.2.6  Examples for graphs   usgr0e 28184
            17.2.7  Subgraphs   csubgr 28215
            17.2.8  Finite undirected simple graphs   cfusgr 28264
            17.2.9  Neighbors, complete graphs and universal vertices   cnbgr 28280
                  17.2.9.1  Neighbors   cnbgr 28280
                  17.2.9.2  Universal vertices   cuvtx 28333
                  17.2.9.3  Complete graphs   ccplgr 28357
            17.2.10  Vertex degree   cvtxdg 28413
            ​*17.2.11  Regular graphs   crgr 28503
      ​​*17.3  Walks, paths and cycles
            *17.3.1  Walks   cewlks 28543
            17.3.2  Walks for loop-free graphs   lfgrwlkprop 28635
            17.3.3  Trails   ctrls 28638
            17.3.4  Paths and simple paths   cpths 28660
            17.3.5  Closed walks   cclwlks 28718
            17.3.6  Circuits and cycles   ccrcts 28732
            ​*17.3.7  Walks as words   cwwlks 28770
            17.3.8  Walks/paths of length 2 (as length 3 strings)   2wlkdlem1 28870
            17.3.9  Walks in regular graphs   rusgrnumwwlkl1 28913
            ​*17.3.10  Closed walks as words   cclwwlk 28925
                  17.3.10.1  Closed walks as words   cclwwlk 28925
                  17.3.10.2  Closed walks of a fixed length as words   cclwwlkn 28968
                  17.3.10.3  Closed walks on a vertex of a fixed length as words   cclwwlknon 29031
            17.3.11  Examples for walks, trails and paths   0ewlk 29058
            17.3.12  Connected graphs   cconngr 29130
      ​17.4  Eulerian paths and the Konigsberg Bridge problem
            ​*17.4.1  Eulerian paths   ceupth 29141
            ​*17.4.2  The Königsberg Bridge problem   konigsbergvtx 29190
      ​17.5  The Friendship Theorem
            17.5.1  Friendship graphs - basics   cfrgr 29202
            17.5.2  The friendship theorem for small graphs   frgr1v 29215
            17.5.3  Theorems according to Mertzios and Unger   2pthfrgrrn 29226
            ​*17.5.4  Huneke's Proof of the Friendship Theorem   frgrncvvdeqlem1 29243
​PART 18  GUIDES AND MISCELLANEA
      ​18.1  Guides (conventions, explanations, and examples)
            ​*18.1.1  Conventions   conventions 29344
            18.1.2  Natural deduction   natded 29347
            ​*18.1.3  Natural deduction examples   ex-natded5.2 29348
            18.1.4  Definitional examples   ex-or 29365
            18.1.5  Other examples   aevdemo 29404
      ​18.2  Humor
            18.2.1  April Fool's theorem   avril1 29407
      ​18.3  (Future - to be reviewed and classified)
            18.3.1  Planar incidence geometry   cplig 29416
            ​*18.3.2  Aliases kept to prevent broken links   dummylink 29429
​​*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 29431
            19.1.2  Abelian groups   cablo 29486
      ​19.2  Complex vector spaces
            19.2.1  Definition and basic properties   cvc 29500
            19.2.2  Examples of complex vector spaces   cnaddabloOLD 29523
      ​19.3  Normed complex vector spaces
            19.3.1  Definition and basic properties   cnv 29526
            19.3.2  Examples of normed complex vector spaces   cnnv 29619
            19.3.3  Induced metric of a normed complex vector space   imsval 29627
            19.3.4  Inner product   cdip 29642
            19.3.5  Subspaces   css 29663
      ​19.4  Operators on complex vector spaces
            19.4.1  Definitions and basic properties   clno 29682
      ​19.5  Inner product (pre-Hilbert) spaces
            19.5.1  Definition and basic properties   ccphlo 29754
            19.5.2  Examples of pre-Hilbert spaces   cncph 29761
            19.5.3  Properties of pre-Hilbert spaces   isph 29764
      ​19.6  Complex Banach spaces
            19.6.1  Definition and basic properties   ccbn 29804
            19.6.2  Examples of complex Banach spaces   cnbn 29811
            19.6.3  Uniform Boundedness Theorem   ubthlem1 29812
            19.6.4  Minimizing Vector Theorem   minvecolem1 29816
      ​19.7  Complex Hilbert spaces
            19.7.1  Definition and basic properties   chlo 29827
            19.7.2  Standard axioms for a complex Hilbert space   hlex 29840
            19.7.3  Examples of complex Hilbert spaces   cnchl 29858
            19.7.4  Hellinger-Toeplitz Theorem   htthlem 29859
​​*PART 20  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      ​20.1  Axiomatization of complex pre-Hilbert spaces
            20.1.1  Basic Hilbert space definitions   chba 29861
            20.1.2  Preliminary ZFC lemmas   df-hnorm 29910
            ​*20.1.3  Derive the Hilbert space axioms from ZFC set theory   axhilex-zf 29923
            ​*20.1.4  Introduce the vector space axioms for a Hilbert space   ax-hilex 29941
            20.1.5  Vector operations   hvmulex 29953
            20.1.6  Inner product postulates for a Hilbert space   ax-hfi 30021
      ​20.2  Inner product and norms
            20.2.1  Inner product   his5 30028
            20.2.2  Norms   dfhnorm2 30064
            20.2.3  Relate Hilbert space to normed complex vector spaces   hilablo 30102
            20.2.4  Bunjakovaskij-Cauchy-Schwarz inequality   bcsiALT 30121
      ​20.3  Cauchy sequences and completeness axiom
            20.3.1  Cauchy sequences and limits   hcau 30126
            20.3.2  Derivation of the completeness axiom from ZF set theory   hilmet 30136
            20.3.3  Completeness postulate for a Hilbert space   ax-hcompl 30144
            20.3.4  Relate Hilbert space to ZFC pre-Hilbert and Hilbert spaces   hhcms 30145
      ​20.4  Subspaces and projections
            20.4.1  Subspaces   df-sh 30149
            20.4.2  Closed subspaces   df-ch 30163
            20.4.3  Orthocomplements   df-oc 30194
            20.4.4  Subspace sum, span, lattice join, lattice supremum   df-shs 30250
            20.4.5  Projection theorem   pjhthlem1 30333
            20.4.6  Projectors   df-pjh 30337
      ​20.5  Properties of Hilbert subspaces
            20.5.1  Orthomodular law   omlsilem 30344
            20.5.2  Projectors (cont.)   pjhtheu2 30358
            20.5.3  Hilbert lattice operations   sh0le 30382
            20.5.4  Span (cont.) and one-dimensional subspaces   spansn0 30483
            20.5.5  Commutes relation for Hilbert lattice elements   df-cm 30525
            20.5.6  Foulis-Holland theorem   fh1 30560
            20.5.7  Quantum Logic Explorer axioms   qlax1i 30569
            20.5.8  Orthogonal subspaces   chscllem1 30579
            20.5.9  Orthoarguesian laws 5OA and 3OA   5oalem1 30596
            20.5.10  Projectors (cont.)   pjorthi 30611
            20.5.11  Mayet's equation E_3   mayete3i 30670
      ​20.6  Operators on Hilbert spaces
            ​*20.6.1  Operator sum, difference, and scalar multiplication   df-hosum 30672
            20.6.2  Zero and identity operators   df-h0op 30690
            20.6.3  Operations on Hilbert space operators   hoaddcl 30700
            20.6.4  Linear, continuous, bounded, Hermitian, unitary operators and norms   df-nmop 30781
            20.6.5  Linear and continuous functionals and norms   df-nmfn 30787
            20.6.6  Adjoint   df-adjh 30791
            20.6.7  Dirac bra-ket notation   df-bra 30792
            20.6.8  Positive operators   df-leop 30794
            20.6.9  Eigenvectors, eigenvalues, spectrum   df-eigvec 30795
            20.6.10  Theorems about operators and functionals   nmopval 30798
            20.6.11  Riesz lemma   riesz3i 31004
            20.6.12  Adjoints (cont.)   cnlnadjlem1 31009
            20.6.13  Quantum computation error bound theorem   unierri 31046
            20.6.14  Dirac bra-ket notation (cont.)   branmfn 31047
            20.6.15  Positive operators (cont.)   leopg 31064
            20.6.16  Projectors as operators   pjhmopi 31088
      ​20.7  States on a Hilbert lattice and Godowski's equation
            20.7.1  States on a Hilbert lattice   df-st 31153
            20.7.2  Godowski's equation   golem1 31213
      ​20.8  Cover relation, atoms, exchange axiom, and modular symmetry
            20.8.1  Covers relation; modular pairs   df-cv 31221
            20.8.2  Atoms   df-at 31280
            20.8.3  Superposition principle   superpos 31296
            20.8.4  Atoms, exchange and covering properties, atomicity   chcv1 31297
            20.8.5  Irreducibility   chirredlem1 31332
            20.8.6  Atoms (cont.)   atcvat3i 31338
            20.8.7  Modular symmetry   mdsymlem1 31345
​PART 21  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      ​21.1  Mathboxes for user contributions
            21.1.1  Mathbox guidelines   mathbox 31384
      ​21.2  Mathbox for Stefan Allan
      ​21.3  Mathbox for Thierry Arnoux
            21.3.1  Propositional Calculus - misc additions   bian1d 31389
            21.3.2  Predicate Calculus   sbc2iedf 31396
                  21.3.2.1  Predicate Calculus - misc additions   sbc2iedf 31396
                  21.3.2.2  Restricted quantification - misc additions   ralcom4f 31398
                  21.3.2.3  Equality   eqtrb 31403
                  21.3.2.4  Double restricted existential uniqueness quantification   opsbc2ie 31404
                  21.3.2.5  Double restricted existential uniqueness quantification syntax   w2reu 31406
                  21.3.2.6  Substitution (without distinct variables) - misc additions   sbceqbidf 31415
                  21.3.2.7  Existential "at most one" - misc additions   mo5f 31417
                  21.3.2.8  Existential uniqueness - misc additions   reuxfrdf 31419
                  21.3.2.9  Restricted "at most one" - misc additions   rmoxfrd 31421
                  21.3.2.10  Restricted iota (description binder)   riotaeqbidva 31424
            21.3.3  General Set Theory   dmrab 31425
                  21.3.3.1  Class abstractions (a.k.a. class builders)   dmrab 31425
                  21.3.3.2  Image Sets   abrexdomjm 31433
                  21.3.3.3  Set relations and operations - misc additions   elunsn 31439
                  21.3.3.4  Unordered pairs   eqsnd 31456
                  21.3.3.5  Conditional operator - misc additions   ifeqeqx 31464
                  21.3.3.6  Set union   uniinn0 31469
                  21.3.3.7  Indexed union - misc additions   cbviunf 31474
                  21.3.3.8  Indexed intersection - misc additions   iinabrex 31487
                  21.3.3.9  Disjointness - misc additions   disjnf 31488
            21.3.4  Relations and Functions   xpdisjres 31516
                  21.3.4.1  Relations - misc additions   xpdisjres 31516
                  21.3.4.2  Functions - misc additions   ac6sf2 31539
                  21.3.4.3  Operations - misc additions   mpomptxf 31597
                  21.3.4.4  Explicit Functions with one or two points as a domain   cosnopne 31608
                  21.3.4.5  Isomorphisms - misc. additions   gtiso 31614
                  21.3.4.6  Disjointness (additional proof requiring functions)   disjdsct 31616
                  21.3.4.7  First and second members of an ordered pair - misc additions   df1stres 31617
                  21.3.4.8  Equivalence relations and classes   ecref 31625
                  21.3.4.9  Supremum - misc additions   supssd 31626
                  21.3.4.10  Finite Sets   imafi2 31628
                  21.3.4.11  Countable Sets   snct 31630
            21.3.5  Real and Complex Numbers   creq0 31652
                  21.3.5.1  Complex operations - misc. additions   creq0 31652
                  21.3.5.2  Ordering on reals - misc additions   lt2addrd 31656
                  21.3.5.3  Extended reals - misc additions   xrlelttric 31657
                  21.3.5.4  Extended nonnegative integers - misc additions   xnn0gt0 31674
                  21.3.5.5  Real number intervals - misc additions   joiniooico 31677
                  21.3.5.6  Finite intervals of integers - misc additions   uzssico 31687
                  21.3.5.7  Half-open integer ranges - misc additions   iundisjfi 31699
                  21.3.5.8  The ` # ` (set size) function - misc additions   hashunif 31708
                  21.3.5.9  The greatest common divisor operator - misc. additions   dvdszzq 31711
                  21.3.5.10  Integers   nnindf 31715
                  21.3.5.11  Decimal numbers   dfdec100 31726
            ​*21.3.6  Decimal expansion   cdp2 31727
                  ​*21.3.6.1  Decimal point   cdp 31744
                  21.3.6.2  Division in the extended real number system   cxdiv 31773
            21.3.7  Words over a set - misc additions   wrdfd 31792
                  21.3.7.1  Splicing words (substring replacement)   splfv3 31812
                  21.3.7.2  Cyclic shift of words   1cshid 31813
            21.3.8  Extensible Structures   ressplusf 31817
                  21.3.8.1  Structure restriction operator   ressplusf 31817
                  21.3.8.2  The opposite group   oppgle 31820
                  21.3.8.3  Posets   ressprs 31823
                  21.3.8.4  Complete lattices   clatp0cl 31836
                  21.3.8.5  Order Theory   cmnt 31838
                  21.3.8.6  Extended reals Structure - misc additions   ax-xrssca 31864
                  21.3.8.7  The extended nonnegative real numbers commutative monoid   xrge0base 31876
            21.3.9  Algebra   abliso 31887
                  21.3.9.1  Monoids Homomorphisms   abliso 31887
                  21.3.9.2  Finitely supported group sums - misc additions   gsumsubg 31888
                  21.3.9.3  Centralizers and centers - misc additions   cntzun 31902
                  21.3.9.4  Totally ordered monoids and groups   comnd 31905
                  21.3.9.5  The symmetric group   symgfcoeu 31933
                  21.3.9.6  Transpositions   pmtridf1o 31943
                  21.3.9.7  Permutation Signs   psgnid 31946
                  21.3.9.8  Permutation cycles   ctocyc 31955
                  21.3.9.9  The Alternating Group   evpmval 31994
                  21.3.9.10  Signum in an ordered monoid   csgns 32007
                  21.3.9.11  The Archimedean property for generic ordered algebraic structures   cinftm 32012
                  21.3.9.12  Semiring left modules   cslmd 32035
                  21.3.9.13  Simple groups   prmsimpcyc 32063
                  21.3.9.14  Rings - misc additions   0ringsubrg 32064
                  21.3.9.15  Subfields   sdrgdvcl 32076
                  21.3.9.16  Field extensions generated by a set   cfldgen 32079
                  21.3.9.17  Totally ordered rings and fields   corng 32090
                  21.3.9.18  Ring homomorphisms - misc additions   rhmdvd 32113
                  21.3.9.19  Scalar restriction operation   cresv 32115
                  21.3.9.20  The commutative ring of gaussian integers   gzcrng 32135
                  21.3.9.21  The archimedean ordered field of real numbers   reofld 32136
                  21.3.9.22  The quotient map and quotient modules   qusker 32141
                  21.3.9.23  The ring of integers modulo ` N `   fermltlchr 32154
                  21.3.9.24  Independent sets and families   islinds5 32156
                  ​*21.3.9.25  Subgroup sum / Sumset / Minkowski sum   elgrplsmsn 32171
                  21.3.9.26  The quotient map   quslsm 32186
                  21.3.9.27  Ideals   intlidl 32199
                  21.3.9.28  Prime Ideals   cprmidl 32207
                  21.3.9.29  Maximal Ideals   cmxidl 32228
                  21.3.9.30  The semiring of ideals of a ring   cidlsrg 32242
                  21.3.9.31  Unique factorization domains   cufd 32258
                  21.3.9.32  Associative algebras   asclmulg 32263
                  21.3.9.33  Univariate Polynomials   0ringmon1p 32264
                  21.3.9.34  The subring algebra   sra1r 32285
                  21.3.9.35  Division Ring Extensions   drgext0g 32291
                  21.3.9.36  Vector Spaces   lvecdimfi 32297
                  21.3.9.37  Vector Space Dimension   cldim 32298
            21.3.10  Field Extensions   cfldext 32327
                  21.3.10.1  Algebraic numbers   cirng 32357
                  21.3.10.2  Minimal polynomials   cminply 32366
            21.3.11  Matrices   csmat 32374
                  21.3.11.1  Submatrices   csmat 32374
                  21.3.11.2  Matrix literals   clmat 32392
                  21.3.11.3  Laplace expansion of determinants   mdetpmtr1 32404
            21.3.12  Topology   ist0cld 32414
                  21.3.12.1  Open maps   txomap 32415
                  21.3.12.2  Topology of the unit circle   qtopt1 32416
                  21.3.12.3  Refinements   reff 32420
                  21.3.12.4  Open cover refinement property   ccref 32423
                  21.3.12.5  Lindelöf spaces   cldlf 32433
                  21.3.12.6  Paracompact spaces   cpcmp 32436
                  ​*21.3.12.7  Spectrum of a ring   crspec 32443
                  21.3.12.8  Pseudometrics   cmetid 32467
                  21.3.12.9  Continuity - misc additions   hauseqcn 32479
                  21.3.12.10  Topology of the closed unit interval   elunitge0 32480
                  21.3.12.11  Topology of ` ( RR X. RR ) `   unicls 32484
                  21.3.12.12  Order topology - misc. additions   cnvordtrestixx 32494
                  21.3.12.13  Continuity in topological spaces - misc. additions   mndpluscn 32507
                  21.3.12.14  Topology of the extended nonnegative real numbers ordered monoid   xrge0hmph 32513
                  21.3.12.15  Limits - misc additions   lmlim 32528
                  21.3.12.16  Univariate polynomials   pl1cn 32536
            21.3.13  Uniform Stuctures and Spaces   chcmp 32537
                  21.3.13.1  Hausdorff uniform completion   chcmp 32537
            21.3.14  Topology and algebraic structures   zringnm 32539
                  21.3.14.1  The norm on the ring of the integer numbers   zringnm 32539
                  21.3.14.2  Topological ` ZZ ` -modules   zlm0 32541
                  21.3.14.3  Canonical embedding of the field of the rational numbers into a division ring   cqqh 32553
                  21.3.14.4  Canonical embedding of the real numbers into a complete ordered field   crrh 32574
                  21.3.14.5  Embedding from the extended real numbers into a complete lattice   cxrh 32597
                  21.3.14.6  Canonical embeddings into the ordered field of the real numbers   zrhre 32600
                  ​*21.3.14.7  Topological Manifolds   cmntop 32603
            21.3.15  Real and complex functions   nexple 32608
                  21.3.15.1  Integer powers - misc. additions   nexple 32608
                  21.3.15.2  Indicator Functions   cind 32609
                  21.3.15.3  Extended sum   cesum 32626
            21.3.16  Mixed Function/Constant operation   cofc 32694
            21.3.17  Abstract measure   csiga 32707
                  21.3.17.1  Sigma-Algebra   csiga 32707
                  21.3.17.2  Generated sigma-Algebra   csigagen 32737
                  ​*21.3.17.3  lambda and pi-Systems, Rings of Sets   ispisys 32751
                  21.3.17.4  The Borel algebra on the real numbers   cbrsiga 32780
                  21.3.17.5  Product Sigma-Algebra   csx 32787
                  21.3.17.6  Measures   cmeas 32794
                  21.3.17.7  The counting measure   cntmeas 32825
                  21.3.17.8  The Lebesgue measure - misc additions   voliune 32828
                  21.3.17.9  The Dirac delta measure   cdde 32831
                  21.3.17.10  The 'almost everywhere' relation   cae 32836
                  21.3.17.11  Measurable functions   cmbfm 32848
                  21.3.17.12  Borel Algebra on ` ( RR X. RR ) `   br2base 32869
                  ​*21.3.17.13  Caratheodory's extension theorem   coms 32891
            21.3.18  Integration   itgeq12dv 32926
                  21.3.18.1  Lebesgue integral - misc additions   itgeq12dv 32926
                  21.3.18.2  Bochner integral   citgm 32927
            21.3.19  Euler's partition theorem   oddpwdc 32954
            21.3.20  Sequences defined by strong recursion   csseq 32983
            21.3.21  Fibonacci Numbers   cfib 32996
            21.3.22  Probability   cprb 33007
                  21.3.22.1  Probability Theory   cprb 33007
                  21.3.22.2  Conditional Probabilities   ccprob 33031
                  21.3.22.3  Real-valued Random Variables   crrv 33040
                  21.3.22.4  Preimage set mapping operator   corvc 33055
                  21.3.22.5  Distribution Functions   orvcelval 33068
                  21.3.22.6  Cumulative Distribution Functions   orvclteel 33072
                  21.3.22.7  Probabilities - example   coinfliplem 33078
                  21.3.22.8  Bertrand's Ballot Problem   ballotlemoex 33085
            21.3.23  Signum (sgn or sign) function - misc. additions   sgncl 33138
                  21.3.23.1  Operations on words   ccatmulgnn0dir 33154
            21.3.24  Polynomials with real coefficients - misc additions   plymul02 33158
            21.3.25  Descartes's rule of signs   signspval 33164
                  21.3.25.1  Sign changes in a word over real numbers   signspval 33164
                  21.3.25.2  Counting sign changes in a word over real numbers   signslema 33174
            21.3.26  Number Theory   efcld 33204
                  21.3.26.1  Representations of a number as sums of integers   crepr 33221
                  21.3.26.2  Vinogradov Trigonometric Sums and the Circle Method   cvts 33248
                  21.3.26.3  The Ternary Goldbach Conjecture: Final Statement   ax-hgt749 33257
            21.3.27  Elementary Geometry   cstrkg2d 33277
                  ​*21.3.27.1  Two-dimensional geometry   cstrkg2d 33277
                  21.3.27.2  Outer Five Segment (not used, no need to move to main)   cafs 33282
            ​*21.3.28  LeftPad Project   clpad 33287
      ​​*21.4  Mathbox for Jonathan Ben-Naim
            21.4.1  First-order logic and set theory   bnj170 33310
            21.4.2  Well founded induction and recursion   bnj110 33470
            21.4.3  The existence of a minimal element in certain classes   bnj69 33622
            21.4.4  Well-founded induction   bnj1204 33624
            21.4.5  Well-founded recursion, part 1 of 3   bnj60 33674
            21.4.6  Well-founded recursion, part 2 of 3   bnj1500 33680
            21.4.7  Well-founded recursion, part 3 of 3   bnj1522 33684
      ​21.5  Mathbox for BTernaryTau
            21.5.1  ZF set theory   exdifsn 33685
                  21.5.1.1  Finitism   fineqvrep 33696
            21.5.2  Real and complex numbers   zltp1ne 33700
            21.5.3  Graph theory   lfuhgr 33711
                  21.5.3.1  Acyclic graphs   cacycgr 33736
      ​21.6  Mathbox for Mario Carneiro
            21.6.1  Predicate calculus with all distinct variables   ax-7d 33753
            21.6.2  Miscellaneous stuff   quartfull 33759
            21.6.3  Derangements and the Subfactorial   deranglem 33760
            21.6.4  The Erdős-Szekeres theorem   erdszelem1 33785
            21.6.5  The Kuratowski closure-complement theorem   kur14lem1 33800
            21.6.6  Retracts and sections   cretr 33811
            21.6.7  Path-connected and simply connected spaces   cpconn 33813
            21.6.8  Covering maps   ccvm 33849
            21.6.9  Normal numbers   snmlff 33923
            21.6.10  Godel-sets of formulas - part 1   cgoe 33927
            21.6.11  Godel-sets of formulas - part 2   cgon 34026
            21.6.12  Models of ZF   cgze 34040
            ​*21.6.13  Metamath formal systems   cmcn 34054
            21.6.14  Grammatical formal systems   cm0s 34179
            21.6.15  Models of formal systems   cmuv 34199
            21.6.16  Splitting fields   ccpms 34221
            21.6.17  p-adic number fields   czr 34235
      ​​*21.7  Mathbox for Filip Cernatescu
      ​21.8  Mathbox for Paul Chapman
            21.8.1  Real and complex numbers (cont.)   climuzcnv 34259
            21.8.2  Miscellaneous theorems   elfzm12 34263
      ​21.9  Mathbox for Scott Fenton
            21.9.1  ZFC Axioms in primitive form   axextprim 34272
            21.9.2  Untangled classes   untelirr 34279
            21.9.3  Extra propositional calculus theorems   3jaodd 34286
            21.9.4  Misc. Useful Theorems   nepss 34289
            21.9.5  Properties of real and complex numbers   sqdivzi 34300
            21.9.6  Infinite products   iprodefisumlem 34313
            21.9.7  Factorial limits   faclimlem1 34316
            21.9.8  Greatest common divisor and divisibility   gcd32 34322
            21.9.9  Properties of relationships   dftr6 34324
            21.9.10  Properties of functions and mappings   funpsstri 34340
            21.9.11  Set induction (or epsilon induction)   setinds 34353
            21.9.12  Ordinal numbers   elpotr 34356
            21.9.13  Defined equality axioms   axextdfeq 34372
            21.9.14  Hypothesis builders   hbntg 34380
            21.9.15  Well-founded zero, successor, and limits   cwsuc 34385
            21.9.16  Surreal numbers - multiplication   cmuls 34405
            21.9.17  Quantifier-free definitions   ctxp 34415
            21.9.18  Alternate ordered pairs   caltop 34541
            21.9.19  Geometry in the Euclidean space   cofs 34567
                  21.9.19.1  Congruence properties   cofs 34567
                  21.9.19.2  Betweenness properties   btwntriv2 34597
                  21.9.19.3  Segment Transportation   ctransport 34614
                  21.9.19.4  Properties relating betweenness and congruence   cifs 34620
                  21.9.19.5  Connectivity of betweenness   btwnconn1lem1 34672
                  21.9.19.6  Segment less than or equal to   csegle 34691
                  21.9.19.7  Outside-of relationship   coutsideof 34704
                  21.9.19.8  Lines and Rays   cline2 34719
            21.9.20  Forward difference   cfwddif 34743
            21.9.21  Rank theorems   rankung 34751
            21.9.22  Hereditarily Finite Sets   chf 34757
      ​21.10  Mathbox for Jeff Hankins
            21.10.1  Miscellany   a1i14 34772
            21.10.2  Basic topological facts   topbnd 34796
            21.10.3  Topology of the real numbers   ivthALT 34807
            21.10.4  Refinements   cfne 34808
            21.10.5  Neighborhood bases determine topologies   neibastop1 34831
            21.10.6  Lattice structure of topologies   topmtcl 34835
            21.10.7  Filter bases   fgmin 34842
            21.10.8  Directed sets, nets   tailfval 34844
      ​21.11  Mathbox for Anthony Hart
            21.11.1  Propositional Calculus   tb-ax1 34855
            21.11.2  Predicate Calculus   nalfal 34875
            21.11.3  Miscellaneous single axioms   meran1 34883
            21.11.4  Connective Symmetry   negsym1 34889
      ​21.12  Mathbox for Chen-Pang He
            21.12.1  Ordinal topology   ontopbas 34900
      ​21.13  Mathbox for Jeff Hoffman
            21.13.1  Inferences for finite induction on generic function values   fveleq 34923
            21.13.2  gdc.mm   nnssi2 34927
      ​21.14  Mathbox for Asger C. Ipsen
            21.14.1  Continuous nowhere differentiable functions   dnival 34934
      ​​*21.15  Mathbox for BJ
            *21.15.1  Propositional calculus   bj-mp2c 35003
                  *21.15.1.1  Derived rules of inference   bj-mp2c 35003
                  ​*21.15.1.2  A syntactic theorem   bj-0 35005
                  21.15.1.3  Minimal implicational calculus   bj-a1k 35007
                  ​*21.15.1.4  Positive calculus   bj-syl66ib 35018
                  21.15.1.5  Implication and negation   bj-con2com 35024
                  ​*21.15.1.6  Disjunction   bj-jaoi1 35035
                  ​*21.15.1.7  Logical equivalence   bj-dfbi4 35037
                  21.15.1.8  The conditional operator for propositions   bj-consensus 35042
                  ​*21.15.1.9  Propositional calculus: miscellaneous   bj-imbi12 35047
            ​*21.15.2  Modal logic   bj-axdd2 35057
            ​*21.15.3  Provability logic   cprvb 35062
            ​*21.15.4  First-order logic   bj-genr 35071
                  21.15.4.1  Adding ax-gen   bj-genr 35071
                  21.15.4.2  Adding ax-4   bj-2alim 35075
                  21.15.4.3  Adding ax-5   bj-ax12wlem 35108
                  21.15.4.4  Equality and substitution   bj-ssbeq 35117
                  21.15.4.5  Adding ax-6   bj-spimvwt 35133
                  21.15.4.6  Adding ax-7   bj-cbvexw 35140
                  21.15.4.7  Membership predicate, ax-8 and ax-9   bj-ax89 35142
                  21.15.4.8  Adding ax-11   bj-alcomexcom 35145
                  21.15.4.9  Adding ax-12   axc11n11 35147
                  21.15.4.10  Nonfreeness   wnnf 35188
                  21.15.4.11  Adding ax-13   bj-axc10 35248
                  ​*21.15.4.12  Removing dependencies on ax-13 (and ax-11)   bj-axc10v 35258
                  ​*21.15.4.13  Distinct var metavariables   bj-hbaeb2 35283
                  ​*21.15.4.14  Around ~ equsal   bj-equsal1t 35287
                  ​*21.15.4.15  Some Principia Mathematica proofs   stdpc5t 35292
                  21.15.4.16  Alternate definition of substitution   bj-sbsb 35302
                  21.15.4.17  Lemmas for substitution   bj-sbf3 35304
                  21.15.4.18  Existential uniqueness   bj-eu3f 35307
                  ​*21.15.4.19  First-order logic: miscellaneous   bj-sblem1 35308
            21.15.5  Set theory   eliminable1 35325
                  ​*21.15.5.1  Eliminability of class terms   eliminable1 35325
                  ​*21.15.5.2  Classes without the axiom of extensionality   bj-denoteslem 35337
                  21.15.5.3  Characterization among sets versus among classes   elelb 35364
                  ​*21.15.5.4  The nonfreeness quantifier for classes   bj-nfcsym 35366
                  ​*21.15.5.5  Lemmas for class substitution   bj-sbeqALT 35367
                  21.15.5.6  Removing some axiom requirements and disjoint variable conditions   bj-exlimvmpi 35378
                  ​*21.15.5.7  Class abstractions   bj-elabd2ALT 35395
                  21.15.5.8  Generalized class abstractions   bj-cgab 35403
                  ​*21.15.5.9  Restricted nonfreeness   wrnf 35411
                  ​*21.15.5.10  Russell's paradox   bj-ru0 35413
                  21.15.5.11  Curry's paradox in set theory   currysetlem 35416
                  ​*21.15.5.12  Some disjointness results   bj-n0i 35422
                  ​*21.15.5.13  Complements on direct products   bj-xpimasn 35426
                  ​*21.15.5.14  "Singletonization" and tagging   bj-snsetex 35434
                  ​*21.15.5.15  Tuples of classes   bj-cproj 35461
                  ​*21.15.5.16  Set theory: elementary operations relative to a universe   bj-rcleqf 35496
                  ​*21.15.5.17  Axioms for finite unions   bj-abex 35501
                  ​*21.15.5.18  Set theory: miscellaneous   eleq2w2ALT 35518
                  ​*21.15.5.19  Evaluation at a class   bj-evaleq 35543
                  21.15.5.20  Elementwise operations   celwise 35550
                  ​*21.15.5.21  Elementwise intersection (families of sets induced on a subset)   bj-rest00 35552
                  21.15.5.22  Moore collections (complements)   bj-raldifsn 35571
                  21.15.5.23  Maps-to notation for functions with three arguments   bj-0nelmpt 35587
                  ​*21.15.5.24  Currying   csethom 35593
                  ​*21.15.5.25  Setting components of extensible structures   cstrset 35605
            ​*21.15.6  Extended real and complex numbers, real and complex projective lines   bj-nfald 35608
                  21.15.6.1  Complements on class abstractions of ordered pairs and binary relations   bj-nfald 35608
                  ​*21.15.6.2  Identity relation (complements)   bj-opabssvv 35621
                  ​*21.15.6.3  Functionalized identity (diagonal in a Cartesian square)   cdiag2 35643
                  ​*21.15.6.4  Direct image and inverse image   cimdir 35649
                  ​*21.15.6.5  Extended numbers and projective lines as sets   cfractemp 35667
                  ​*21.15.6.6  Addition and opposite   caddcc 35708
                  ​*21.15.6.7  Order relation on the extended reals   cltxr 35712
                  ​*21.15.6.8  Argument, multiplication and inverse   carg 35714
                  21.15.6.9  The canonical bijection from the finite ordinals   ciomnn 35720
                  21.15.6.10  Divisibility   cnnbar 35731
            ​*21.15.7  Monoids   bj-smgrpssmgm 35739
                  ​*21.15.7.1  Finite sums in monoids   cfinsum 35754
            ​*21.15.8  Affine, Euclidean, and Cartesian geometry   bj-fvimacnv0 35757
                  *21.15.8.1  Real vector spaces   bj-fvimacnv0 35757
                  ​*21.15.8.2  Complex numbers (supplements)   bj-subcom 35779
                  ​*21.15.8.3  Barycentric coordinates   bj-bary1lem 35781
            21.15.9  Monoid of endomorphisms   cend 35784
      ​21.16  Mathbox for Jim Kingdon
                  21.16.0.1  Circle constant   taupilem3 35790
                  21.16.0.2  Number theory   dfgcd3 35795
                  21.16.0.3  Real numbers   irrdifflemf 35796
      ​21.17  Mathbox for ML
            21.17.1  Miscellaneous   csbrecsg 35799
            21.17.2  Cartesian exponentiation   cfinxp 35854
            21.17.3  Topology   iunctb2 35874
                  ​*21.17.3.1  Pi-base theorems   pibp16 35884
      ​21.18  Mathbox for Wolf Lammen
            21.18.1  1. Bootstrapping   wl-section-boot 35893
            21.18.2  Implication chains   wl-section-impchain 35917
            21.18.3  Theorems around the conditional operator   wl-ifp-ncond1 35935
            21.18.4  Alternative development of hadd, cadd   wl-df-3xor 35939
            21.18.5  An alternative axiom ~ ax-13   ax-wl-13v 35964
            21.18.6  Other stuff   wl-mps 35966
      ​21.19  Mathbox for Brendan Leahy
      ​21.20  Mathbox for Jeff Madsen
            21.20.1  Logic and set theory   unirep 36172
            21.20.2  Real and complex numbers; integers   filbcmb 36199
            21.20.3  Sequences and sums   sdclem2 36201
            21.20.4  Topology   subspopn 36211
            21.20.5  Metric spaces   metf1o 36214
            21.20.6  Continuous maps and homeomorphisms   constcncf 36221
            21.20.7  Boundedness   ctotbnd 36225
            21.20.8  Isometries   cismty 36257
            21.20.9  Heine-Borel Theorem   heibor1lem 36268
            21.20.10  Banach Fixed Point Theorem   bfplem1 36281
            21.20.11  Euclidean space   crrn 36284
            21.20.12  Intervals (continued)   ismrer1 36297
            21.20.13  Operation properties   cass 36301
            21.20.14  Groups and related structures   cmagm 36307
            21.20.15  Group homomorphism and isomorphism   cghomOLD 36342
            21.20.16  Rings   crngo 36353
            21.20.17  Division Rings   cdrng 36407
            21.20.18  Ring homomorphisms   crnghom 36419
            21.20.19  Commutative rings   ccm2 36448
            21.20.20  Ideals   cidl 36466
            21.20.21  Prime rings and integral domains   cprrng 36505
            21.20.22  Ideal generators   cigen 36518
      ​21.21  Mathbox for Giovanni Mascellani
            ​*21.21.1  Tools for automatic proof building   efald2 36537
            ​*21.21.2  Tseitin axioms   fald 36588
            ​*21.21.3  Equality deductions   iuneq2f 36615
            ​*21.21.4  Miscellanea   orcomdd 36626
      ​21.22  Mathbox for Peter Mazsa
            21.22.1  Notations   cxrn 36633
            21.22.2  Preparatory theorems   el2v1 36676
            21.22.3  Range Cartesian product   df-xrn 36833
            21.22.4  Cosets by ` R `   df-coss 36873
            21.22.5  Relations   df-rels 36947
            21.22.6  Subset relations   df-ssr 36960
            21.22.7  Reflexivity   df-refs 36972
            21.22.8  Converse reflexivity   df-cnvrefs 36987
            21.22.9  Symmetry   df-syms 37004
            21.22.10  Reflexivity and symmetry   symrefref2 37025
            21.22.11  Transitivity   df-trs 37034
            21.22.12  Equivalence relations   df-eqvrels 37046
            21.22.13  Redundancy   df-redunds 37085
            21.22.14  Domain quotients   df-dmqss 37100
            21.22.15  Equivalence relations on domain quotients   df-ers 37125
            21.22.16  Functions   df-funss 37142
            21.22.17  Disjoints vs. converse functions   df-disjss 37165
            21.22.18  Antisymmetry   df-antisymrel 37222
            21.22.19  Partitions: disjoints on domain quotients   df-parts 37227
            21.22.20  Partition-Equivalence Theorems   disjim 37243
      ​21.23  Mathbox for Rodolfo Medina
            21.23.1  Partitions   prtlem60 37315
      ​​*21.24  Mathbox for Norm Megill
            *21.24.1  Obsolete schemes ax-c4,c5,c7,c10,c11,c11n,c15,c9,c14,c16   ax-c5 37345
            ​*21.24.2  Rederive new axioms ax-4, ax-10, ax-6, ax-12, ax-13 from old   axc5 37355
            ​*21.24.3  Legacy theorems using obsolete axioms   ax5ALT 37369
            21.24.4  Experiments with weak deduction theorem   elimhyps 37423
            21.24.5  Miscellanea   cnaddcom 37434
            21.24.6  Atoms, hyperplanes, and covering in a left vector space (or module)   clsa 37436
            21.24.7  Functionals and kernels of a left vector space (or module)   clfn 37519
            21.24.8  Opposite rings and dual vector spaces   cld 37585
            21.24.9  Ortholattices and orthomodular lattices   cops 37634
            21.24.10  Atomic lattices with covering property   ccvr 37724
            21.24.11  Hilbert lattices   chlt 37812
            21.24.12  Projective geometries based on Hilbert lattices   clln 37954
            21.24.13  Construction of a vector space from a Hilbert lattice   cdlema1N 38254
            21.24.14  Construction of involution and inner product from a Hilbert lattice   clpoN 39943
      ​21.25  Mathbox for metakunt
            21.25.1  General helpful statements   leexp1ad 40429
            21.25.2  Some gcd and lcm results   12gcd5e1 40460
            21.25.3  Least common multiple inequality theorem   3factsumint1 40478
            21.25.4  Logarithm inequalities   3exp7 40510
            21.25.5  Miscellaneous results for AKS formalisation   intlewftc 40518
            21.25.6  Sticks and stones   sticksstones1 40554
            21.25.7  Permutation results   metakunt1 40577
            21.25.8  Unused lemmas scheduled for deletion   andiff 40611
      ​21.26  Mathbox for Steven Nguyen
            ​*21.26.1  Miscellaneous theorems   bicomdALT 40616
            21.26.2  Utility theorems   ioin9i8 40625
            21.26.3  Structures   ressbasssg 40668
            ​*21.26.4  Arithmetic theorems   c0exALT 40761
            21.26.5  Exponents and divisibility   oexpreposd 40793
            21.26.6  Real subtraction   cresub 40820
            ​*21.26.7  Projective spaces   cprjsp 40925
            21.26.8  Basic reductions for Fermat's Last Theorem   dffltz 40958
      ​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 41001
            21.29.2  Additional theory of functions   imaiinfv 41002
            21.29.3  Additional topology   elrfi 41003
            21.29.4  Characterization of closure operators. Kuratowski closure axioms   ismrcd1 41007
            21.29.5  Algebraic closure systems   cnacs 41011
            21.29.6  Miscellanea 1. Map utilities   constmap 41022
            21.29.7  Miscellanea for polynomials   mptfcl 41029
            21.29.8  Multivariate polynomials over the integers   cmzpcl 41030
            21.29.9  Miscellanea for Diophantine sets 1   coeq0i 41062
            21.29.10  Diophantine sets 1: definitions   cdioph 41064
            21.29.11  Diophantine sets 2 miscellanea   ellz1 41076
            21.29.12  Diophantine sets 2: union and intersection. Monotone Boolean algebra   diophin 41081
            21.29.13  Diophantine sets 3: construction   diophrex 41084
            21.29.14  Diophantine sets 4 miscellanea   2sbcrex 41093
            21.29.15  Diophantine sets 4: Quantification   rexrabdioph 41103
            21.29.16  Diophantine sets 5: Arithmetic sets   rabdiophlem1 41110
            21.29.17  Diophantine sets 6: reusability. renumbering of variables   eldioph4b 41120
            21.29.18  Pigeonhole Principle and cardinality helpers   fphpd 41125
            21.29.19  A non-closed set of reals is infinite   rencldnfilem 41129
            21.29.20  Lagrange's rational approximation theorem   irrapxlem1 41131
            21.29.21  Pell equations 1: A nontrivial solution always exists   pellexlem1 41138
            21.29.22  Pell equations 2: Algebraic number theory of the solution set   csquarenn 41145
            21.29.23  Pell equations 3: characterizing fundamental solution   infmrgelbi 41187
            ​*21.29.24  Logarithm laws generalized to an arbitrary base   reglogcl 41199
            21.29.25  Pell equations 4: the positive solution group is infinite cyclic   pellfund14 41207
            21.29.26  X and Y sequences 1: Definition and recurrence laws   crmx 41209
            21.29.27  Ordering and induction lemmas for the integers   monotuz 41251
            21.29.28  X and Y sequences 2: Order properties   rmxypos 41257
            21.29.29  Congruential equations   congtr 41275
            21.29.30  Alternating congruential equations   acongid 41285
            21.29.31  Additional theorems on integer divisibility   coprmdvdsb 41295
            21.29.32  X and Y sequences 3: Divisibility properties   jm2.18 41298
            21.29.33  X and Y sequences 4: Diophantine representability of Y   jm2.27a 41315
            21.29.34  X and Y sequences 5: Diophantine representability of X, ^, _C   rmxdiophlem 41325
            21.29.35  Uncategorized stuff not associated with a major project   setindtr 41334
            21.29.36  More equivalents of the Axiom of Choice   axac10 41343
            21.29.37  Finitely generated left modules   clfig 41380
            21.29.38  Noetherian left modules I   clnm 41388
            21.29.39  Addenda for structure powers   pwssplit4 41402
            21.29.40  Every set admits a group structure iff choice   unxpwdom3 41408
            21.29.41  Noetherian rings and left modules II   clnr 41422
            21.29.42  Hilbert's Basis Theorem   cldgis 41434
            21.29.43  Additional material on polynomials [DEPRECATED]   cmnc 41444
            21.29.44  Degree and minimal polynomial of algebraic numbers   cdgraa 41453
            21.29.45  Algebraic integers I   citgo 41470
            21.29.46  Endomorphism algebra   cmend 41488
            21.29.47  Cyclic groups and order   idomrootle 41508
            21.29.48  Cyclotomic polynomials   ccytp 41515
            21.29.49  Miscellaneous topology   fgraphopab 41523
      ​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 41537
            21.32.2  Natural addition of Cantor normal forms   oawordex2 41646
            21.32.3  Surreal Contributions   abeqabi 41670
            21.32.4  Short Studies   nlimsuc 41703
                  21.32.4.1  Additional work on conditional logical operator   ifpan123g 41721
                  21.32.4.2  Sophisms   rp-fakeimass 41774
                  ​*21.32.4.3  Finite Sets   rp-isfinite5 41779
                  21.32.4.4  General Observations   intabssd 41781
                  21.32.4.5  Infinite Sets   pwelg 41822
                  ​*21.32.4.6  Finite intersection property   fipjust 41827
                  21.32.4.7  RP ADDTO: Subclasses and subsets   rababg 41836
                  21.32.4.8  RP ADDTO: The intersection of a class   elinintab 41837
                  21.32.4.9  RP ADDTO: Theorems requiring subset and intersection existence   elinintrab 41839
                  21.32.4.10  RP ADDTO: Relations   xpinintabd 41842
                  ​*21.32.4.11  RP ADDTO: Functions   elmapintab 41858
                  ​*21.32.4.12  RP ADDTO: Finite induction (for finite ordinals)   cnvcnvintabd 41862
                  21.32.4.13  RP ADDTO: First and second members of an ordered pair   elcnvlem 41863
                  21.32.4.14  RP ADDTO: The reflexive and transitive properties of relations   undmrnresiss 41866
                  21.32.4.15  RP ADDTO: Basic properties of closures   cleq2lem 41870
                  21.32.4.16  RP REPLACE: Definitions and basic properties of transitive closures   trcleq2lemRP 41892
                  ​*21.32.4.17  Additions for square root; absolute value   sqrtcvallem1 41893
            21.32.5  Additional statements on relations and subclasses   al3im 41909
                  21.32.5.1  Transitive relations (not to be confused with transitive classes).   trrelind 41927
                  21.32.5.2  Reflexive closures   crcl 41934
                  ​*21.32.5.3  Finite relationship composition.   relexp2 41939
                  21.32.5.4  Transitive closure of a relation   dftrcl3 41982
                  ​*21.32.5.5  Adapted from Frege   frege77d 42008
            ​*21.32.6  Propositions from _Begriffsschrift_   dfxor4 42028
                  *21.32.6.1  _Begriffsschrift_ Chapter I   dfxor4 42028
                  ​*21.32.6.2  _Begriffsschrift_ Notation hints   whe 42034
                  21.32.6.3  _Begriffsschrift_ Chapter II Implication   ax-frege1 42052
                  21.32.6.4  _Begriffsschrift_ Chapter II Implication and Negation   axfrege28 42091
                  ​*21.32.6.5  _Begriffsschrift_ Chapter II with logical equivalence   axfrege52a 42118
                  21.32.6.6  _Begriffsschrift_ Chapter II with equivalence of sets   axfrege52c 42149
                  ​*21.32.6.7  _Begriffsschrift_ Chapter II with equivalence of classes   frege53c 42176
                  ​*21.32.6.8  _Begriffsschrift_ Chapter III Properties hereditary in a sequence   dffrege69 42194
                  ​*21.32.6.9  _Begriffsschrift_ Chapter III Following in a sequence   dffrege76 42201
                  ​*21.32.6.10  _Begriffsschrift_ Chapter III Member of sequence   dffrege99 42224
                  ​*21.32.6.11  _Begriffsschrift_ Chapter III Single-valued procedures   dffrege115 42240
            ​*21.32.7  Exploring Topology via Seifert and Threlfall   enrelmap 42259
                  *21.32.7.1  Equinumerosity of sets of relations and maps   enrelmap 42259
                  ​*21.32.7.2  Generic Pseudoclosure Spaces, Pseudointerior Spaces, and Pseudoneighborhoods   or3or 42285
                  ​*21.32.7.3  Generic Neighborhood Spaces   gneispa 42392
            ​*21.32.8  Exploring Higher Homotopy via Kerodon   k0004lem1 42409
                  *21.32.8.1  Simplicial Sets   k0004lem1 42409
      ​21.33  Mathbox for Stanislas Polu
            21.33.1  IMO Problems   wwlemuld 42418
                  21.33.1.1  IMO 1972 B2   wwlemuld 42418
            ​*21.33.2  INT Inequalities Proof Generator   int-addcomd 42436
            ​*21.33.3  N-Digit Addition Proof Generator   unitadd 42458
            21.33.4  AM-GM (for k = 2,3,4)   gsumws3 42459
      ​21.34  Mathbox for Rohan Ridenour
            21.34.1  Misc   spALT 42464
            21.34.2  Monoid rings   cmnring 42476
            21.34.3  Shorter primitive equivalent of ax-groth   gru0eld 42499
                  21.34.3.1  Grothendieck universes are closed under collection   gru0eld 42499
                  21.34.3.2  Minimal universes   ismnu 42531
                  21.34.3.3  Primitive equivalent of ax-groth   expandan 42558
      ​21.35  Mathbox for Steve Rodriguez
            21.35.1  Miscellanea   nanorxor 42575
            21.35.2  Ratio test for infinite series convergence and divergence   dvgrat 42582
            21.35.3  Multiples   reldvds 42585
            21.35.4  Function operations   caofcan 42593
            21.35.5  Calculus   lhe4.4ex1a 42599
            21.35.6  The generalized binomial coefficient operation   cbcc 42606
            21.35.7  Binomial series   uzmptshftfval 42616
      ​21.36  Mathbox for Andrew Salmon
            21.36.1  Principia Mathematica * 10   pm10.12 42628
            21.36.2  Principia Mathematica * 11   2alanimi 42642
            21.36.3  Predicate Calculus   sbeqal1 42668
            21.36.4  Principia Mathematica * 13 and * 14   pm13.13a 42677
            21.36.5  Set Theory   elnev 42708
            21.36.6  Arithmetic   addcomgi 42726
            21.36.7  Geometry   cplusr 42727
      ​​*21.37  Mathbox for Alan Sare
            21.37.1  Auxiliary theorems for the Virtual Deduction tool   idiALT 42749
            21.37.2  Supplementary unification deductions   bi1imp 42753
            21.37.3  Conventional Metamath proofs, some derived from VD proofs   iidn3 42773
            21.37.4  What is Virtual Deduction?   wvd1 42841
            21.37.5  Virtual Deduction Theorems   df-vd1 42842
            21.37.6  Theorems proved using Virtual Deduction   trsspwALT 43090
            21.37.7  Theorems proved using Virtual Deduction with mmj2 assistance   simplbi2VD 43118
            21.37.8  Virtual Deduction transcriptions of textbook proofs   sb5ALTVD 43185
            21.37.9  Theorems proved using conjunction-form Virtual Deduction   elpwgdedVD 43189
            21.37.10  Theorems with a VD proof in conventional notation derived from a VD proof   suctrALT3 43196
            ​*21.37.11  Theorems with a proof in conventional notation derived from a VD proof   notnotrALT2 43199
      ​21.38  Mathbox for Glauco Siliprandi
            21.38.1  Miscellanea   evth2f 43210
            21.38.2  Functions   feq1dd 43374
            21.38.3  Ordering on real numbers - Real and complex numbers basic operations   sub2times 43496
            21.38.4  Real intervals   gtnelioc 43719
            21.38.5  Finite sums   fsummulc1f 43802
            21.38.6  Finite multiplication of numbers and finite multiplication of functions   fmul01 43811
            21.38.7  Limits   clim1fr1 43832
                  21.38.7.1  Inferior limit (lim inf)   clsi 43982
                  ​*21.38.7.2  Limits for sequences of extended real numbers   clsxlim 44049
            21.38.8  Trigonometry   coseq0 44095
            21.38.9  Continuous Functions   mulcncff 44101
            21.38.10  Derivatives   dvsinexp 44142
            21.38.11  Integrals   itgsin0pilem1 44181
            21.38.12  Stone Weierstrass theorem - real version   stoweidlem1 44232
            21.38.13  Wallis' product for π   wallispilem1 44296
            21.38.14  Stirling's approximation formula for ` n ` factorial   stirlinglem1 44305
            21.38.15  Dirichlet kernel   dirkerval 44322
            21.38.16  Fourier Series   fourierdlem1 44339
            21.38.17  e is transcendental   elaa2lem 44464
            21.38.18  n-dimensional Euclidean space   rrxtopn 44515
            21.38.19  Basic measure theory   csalg 44539
                  ​*21.38.19.1  σ-Algebras   csalg 44539
                  21.38.19.2  Sum of nonnegative extended reals   csumge0 44593
                  ​*21.38.19.3  Measures   cmea 44680
                  ​*21.38.19.4  Outer measures and Caratheodory's construction   come 44720
                  ​*21.38.19.5  Lebesgue measure on n-dimensional Real numbers   covoln 44767
                  ​*21.38.19.6  Measurable functions   csmblfn 44926
      ​21.39  Mathbox for Saveliy Skresanov
            21.39.1  Ceva's theorem   sigarval 45081
            21.39.2  Simple groups   simpcntrab 45101
      ​21.40  Mathbox for Ender Ting
            21.40.1  Increasing sequences and subsequences   et-ltneverrefl 45102
      ​21.41  Mathbox for Jarvin Udandy
      ​21.42  Mathbox for Adhemar
            ​*21.42.1  Minimal implicational calculus   adh-minim 45226
      ​21.43  Mathbox for Alexander van der Vekens
            21.43.1  General auxiliary theorems (1)   eusnsn 45250
                  21.43.1.1  Unordered and ordered pairs - extension for singletons   eusnsn 45250
                  21.43.1.2  Unordered and ordered pairs - extension for unordered pairs   elprneb 45253
                  21.43.1.3  Unordered and ordered pairs - extension for ordered pairs   oppr 45254
                  21.43.1.4  Relations - extension   eubrv 45259
                  21.43.1.5  Definite description binder (inverted iota) - extension   iota0def 45262
                  21.43.1.6  Functions - extension   fveqvfvv 45264
            21.43.2  Alternative for Russell's definition of a description binder   caiota 45305
            21.43.3  Double restricted existential uniqueness   r19.32 45320
                  21.43.3.1  Restricted quantification (extension)   r19.32 45320
                  21.43.3.2  Restricted uniqueness and "at most one" quantification   reuf1odnf 45329
                  21.43.3.3  Analogs to Existential uniqueness (double quantification)   2reu3 45332
                  21.43.3.4  Additional theorems for double restricted existential uniqueness   2reu8i 45335
            ​*21.43.4  Alternative definitions of function and operation values   wdfat 45338
                  21.43.4.1  Restricted quantification (extension)   ralbinrald 45344
                  21.43.4.2  The universal class (extension)   nvelim 45345
                  21.43.4.3  Introduce the Axiom of Power Sets (extension)   alneu 45346
                  21.43.4.4  Predicate "defined at"   dfateq12d 45348
                  21.43.4.5  Alternative definition of the value of a function   dfafv2 45354
                  21.43.4.6  Alternative definition of the value of an operation   aoveq123d 45400
            ​*21.43.5  Alternative definitions of function values (2)   cafv2 45430
            21.43.6  General auxiliary theorems (2)   an4com24 45490
                  21.43.6.1  Logical conjunction - extension   an4com24 45490
                  21.43.6.2  Abbreviated conjunction and disjunction of three wff's - extension   3an4ancom24 45491
                  21.43.6.3  Negated membership (alternative)   cnelbr 45493
                  21.43.6.4  The empty set - extension   ralralimp 45500
                  21.43.6.5  Indexed union and intersection - extension   otiunsndisjX 45501
                  21.43.6.6  Functions - extension   fvifeq 45502
                  21.43.6.7  Maps-to notation - extension   fvmptrab 45514
                  21.43.6.8  Subtraction - extension   cnambpcma 45516
                  21.43.6.9  Ordering on reals (cont.) - extension   leaddsuble 45519
                  21.43.6.10  Imaginary and complex number properties - extension   readdcnnred 45525
                  21.43.6.11  Nonnegative integers (as a subset of complex numbers) - extension   nn0resubcl 45530
                  21.43.6.12  Integers (as a subset of complex numbers) - extension   zgeltp1eq 45531
                  21.43.6.13  Decimal arithmetic - extension   1t10e1p1e11 45532
                  21.43.6.14  Upper sets of integers - extension   eluzge0nn0 45534
                  21.43.6.15  Infinity and the extended real number system (cont.) - extension   nltle2tri 45535
                  21.43.6.16  Finite intervals of integers - extension   ssfz12 45536
                  21.43.6.17  Half-open integer ranges - extension   fzopred 45544
                  21.43.6.18  The modulo (remainder) operation - extension   m1mod0mod1 45551
                  21.43.6.19  The infinite sequence builder "seq"   smonoord 45553
                  21.43.6.20  Finite and infinite sums - extension   fsummsndifre 45554
                  21.43.6.21  Extensible structures - extension   setsidel 45558
            ​*21.43.7  Preimages of function values   preimafvsnel 45561
            ​*21.43.8  Partitions of real intervals   ciccp 45595
            21.43.9  Shifting functions with an integer range domain   fargshiftfv 45621
            21.43.10  Words over a set (extension)   lswn0 45626
                  21.43.10.1  Last symbol of a word - extension   lswn0 45626
            21.43.11  Unordered pairs   wich 45627
                  21.43.11.1  Interchangeable setvar variables   wich 45627
                  21.43.11.2  Set of unordered pairs   sprid 45656
                  ​*21.43.11.3  Proper (unordered) pairs   prpair 45683
                  21.43.11.4  Set of proper unordered pairs   cprpr 45694
            21.43.12  Number theory (extension)   cfmtno 45709
                  ​*21.43.12.1  Fermat numbers   cfmtno 45709
                  ​*21.43.12.2  Mersenne primes   m2prm 45773
                  21.43.12.3  Proth's theorem   modexp2m1d 45794
                  21.43.12.4  Solutions of quadratic equations   quad1 45802
            ​*21.43.13  Even and odd numbers   ceven 45806
                  21.43.13.1  Definitions and basic properties   ceven 45806
                  21.43.13.2  Alternate definitions using the "divides" relation   dfeven2 45831
                  21.43.13.3  Alternate definitions using the "modulo" operation   dfeven3 45840
                  21.43.13.4  Alternate definitions using the "gcd" operation   iseven5 45846
                  21.43.13.5  Theorems of part 5 revised   zneoALTV 45851
                  21.43.13.6  Theorems of part 6 revised   odd2np1ALTV 45856
                  21.43.13.7  Theorems of AV's mathbox revised   0evenALTV 45870
                  21.43.13.8  Additional theorems   epoo 45885
                  21.43.13.9  Perfect Number Theorem (revised)   perfectALTVlem1 45903
            21.43.14  Number theory (extension 2)   cfppr 45906
                  ​*21.43.14.1  Fermat pseudoprimes   cfppr 45906
                  ​*21.43.14.2  Goldbach's conjectures   cgbe 45927
            21.43.15  Graph theory (extension)   cgrisom 46000
                  ​*21.43.15.1  Isomorphic graphs   cgrisom 46000
                  21.43.15.2  Loop-free graphs - extension   1hegrlfgr 46024
                  21.43.15.3  Walks - extension   cupwlks 46025
                  21.43.15.4  Edges of graphs expressed as sets of unordered pairs   upgredgssspr 46035
            21.43.16  Monoids (extension)   ovn0dmfun 46048
                  21.43.16.1  Auxiliary theorems   ovn0dmfun 46048
                  21.43.16.2  Magmas, Semigroups and Monoids (extension)   plusfreseq 46056
                  21.43.16.3  Magma homomorphisms and submagmas   cmgmhm 46061
                  21.43.16.4  Examples and counterexamples for magmas, semigroups and monoids (extension)   opmpoismgm 46091
                  21.43.16.5  Group sum operation (extension 1)   gsumsplit2f 46104
            ​*21.43.17  Magmas and internal binary operations (alternate approach)   ccllaw 46107
                  *21.43.17.1  Laws for internal binary operations   ccllaw 46107
                  ​*21.43.17.2  Internal binary operations   cintop 46120
                  21.43.17.3  Alternative definitions for magmas and semigroups   cmgm2 46139
            21.43.18  Categories (extension)   idfusubc0 46153
                  21.43.18.1  Subcategories (extension)   idfusubc0 46153
            21.43.19  Rings (extension)   lmod0rng 46156
                  21.43.19.1  Nonzero rings (extension)   lmod0rng 46156
                  ​*21.43.19.2  Non-unital rings ("rngs")   crng 46162
                  21.43.19.3  Rng homomorphisms   crngh 46173
                  21.43.19.4  Ring homomorphisms (extension)   rhmfn 46206
                  21.43.19.5  Ideals as non-unital rings   lidldomn1 46209
                  21.43.19.6  The non-unital ring of even integers   0even 46219
                  21.43.19.7  A constructed not unital ring   cznrnglem 46241
                  ​*21.43.19.8  The category of non-unital rings   crngc 46245
                  ​*21.43.19.9  The category of (unital) rings   cringc 46291
                  21.43.19.10  Subcategories of the category of rings   srhmsubclem1 46361
            21.43.20  Basic algebraic structures (extension)   opeliun2xp 46398
                  21.43.20.1  Auxiliary theorems   opeliun2xp 46398
                  21.43.20.2  The binomial coefficient operation (extension)   bcpascm1 46417
                  21.43.20.3  The ` ZZ `-module ` ZZ X. ZZ `   zlmodzxzlmod 46420
                  21.43.20.4  Group sum operation (extension 2)   mgpsumunsn 46427
                  21.43.20.5  Symmetric groups (extension)   exple2lt6 46430
                  21.43.20.6  Divisibility (extension)   invginvrid 46433
                  21.43.20.7  The support of functions (extension)   rmsupp0 46434
                  21.43.20.8  Finitely supported functions (extension)   rmsuppfi 46439
                  21.43.20.9  Left modules (extension)   lmodvsmdi 46448
                  21.43.20.10  Associative algebras (extension)   assaascl0 46450
                  21.43.20.11  Univariate polynomials (extension)   ply1vr1smo 46452
                  21.43.20.12  Univariate polynomials (examples)   linply1 46464
            21.43.21  Linear algebra (extension)   cdmatalt 46467
                  ​*21.43.21.1  The subalgebras of diagonal and scalar matrices (extension)   cdmatalt 46467
                  ​*21.43.21.2  Linear combinations   clinc 46475
                  ​*21.43.21.3  Linear independence   clininds 46511
                  21.43.21.4  Simple left modules and the ` ZZ `-module   lmod1lem1 46558
                  21.43.21.5  Differences between (left) modules and (left) vector spaces   lvecpsslmod 46578
            21.43.22  Complexity theory   suppdm 46581
                  21.43.22.1  Auxiliary theorems   suppdm 46581
                  21.43.22.2  The modulo (remainder) operation (extension)   fldivmod 46594
                  21.43.22.3  Even and odd integers   nn0onn0ex 46599
                  21.43.22.4  The natural logarithm on complex numbers (extension)   logcxp0 46611
                  21.43.22.5  Division of functions   cfdiv 46613
                  21.43.22.6  Upper bounds   cbigo 46623
                  21.43.22.7  Logarithm to an arbitrary base (extension)   rege1logbrege0 46634
                  ​*21.43.22.8  The binary logarithm   fldivexpfllog2 46641
                  21.43.22.9  Binary length   cblen 46645
                  ​*21.43.22.10  Digits   cdig 46671
                  21.43.22.11  Nonnegative integer as sum of its shifted digits   dignn0flhalflem1 46691
                  21.43.22.12  Algorithms for the multiplication of nonnegative integers   nn0mulfsum 46700
                  ​*21.43.22.13  N-ary functions   cnaryf 46702
                  ​*21.43.22.14  The Ackermann function   citco 46733
            21.43.23  Elementary geometry (extension)   fv1prop 46775
                  21.43.23.1  Auxiliary theorems   fv1prop 46775
                  21.43.23.2  Real euclidean space of dimension 2   rrx2pxel 46787
                  21.43.23.3  Spheres and lines in real Euclidean spaces   cline 46803
      ​21.44  Mathbox for Zhi Wang
            21.44.1  Propositional calculus   pm4.71da 46865
            21.44.2  Predicate calculus with equality   dtrucor3 46874
                  21.44.2.1  Axiom scheme ax-5 (Distinctness)   dtrucor3 46874
            21.44.3  ZF Set Theory - start with the Axiom of Extensionality   ralbidb 46875
                  21.44.3.1  Restricted quantification   ralbidb 46875
                  21.44.3.2  The empty set   ssdisjd 46882
                  21.44.3.3  Unordered and ordered pairs   vsn 46886
                  21.44.3.4  The union of a class   unilbss 46892
            21.44.4  ZF Set Theory - add the Axiom of Replacement   inpw 46893
                  21.44.4.1  Theorems requiring subset and intersection existence   inpw 46893
            21.44.5  ZF Set Theory - add the Axiom of Power Sets   mof0 46894
                  21.44.5.1  Functions   mof0 46894
                  21.44.5.2  Operations   fvconstr 46912
            21.44.6  ZF Set Theory - add the Axiom of Union   fvconst0ci 46915
                  21.44.6.1  Equinumerosity   fvconst0ci 46915
            21.44.7  Order sets   iccin 46919
                  21.44.7.1  Real number intervals   iccin 46919
            21.44.8  Moore spaces   mreuniss 46922
            ​*21.44.9  Topology   clduni 46923
                  21.44.9.1  Closure and interior   clduni 46923
                  21.44.9.2  Neighborhoods   neircl 46927
                  21.44.9.3  Subspace topologies   restcls2lem 46935
                  21.44.9.4  Limits and continuity in topological spaces   cnneiima 46939
                  21.44.9.5  Topological definitions using the reals   iooii 46940
                  21.44.9.6  Separated sets   sepnsepolem1 46944
                  21.44.9.7  Separated spaces: T0, T1, T2 (Hausdorff) ...   isnrm4 46953
            21.44.10  Preordered sets and directed sets using extensible structures   isprsd 46978
            21.44.11  Posets and lattices using extensible structures   lubeldm2 46979
                  21.44.11.1  Posets   lubeldm2 46979
                  21.44.11.2  Lattices   toslat 46997
                  21.44.11.3  Subset order structures   intubeu 46999
            21.44.12  Categories   catprslem 47020
                  21.44.12.1  Categories   catprslem 47020
                  21.44.12.2  Monomorphisms and epimorphisms   idmon 47026
                  21.44.12.3  Functors   funcf2lem 47028
            21.44.13  Examples of categories   cthinc 47029
                  21.44.13.1  Thin categories   cthinc 47029
                  21.44.13.2  Preordered sets as thin categories   cprstc 47072
                  21.44.13.3  Monoids as categories   cmndtc 47093
      ​21.45  Mathbox for Emmett Weisz
            ​*21.45.1  Miscellaneous Theorems   nfintd 47108
            21.45.2  Set Recursion   csetrecs 47118
                  ​*21.45.2.1  Basic Properties of Set Recursion   csetrecs 47118
                  21.45.2.2  Examples and properties of set recursion   elsetrecslem 47134
            ​*21.45.3  Construction of Games and Surreal Numbers   cpg 47144
      ​​*21.46  Mathbox for David A. Wheeler
            21.46.1  Natural deduction   sbidd 47153
            ​*21.46.2  Greater than, greater than or equal to.   cge-real 47155
            ​*21.46.3  Hyperbolic trigonometric functions   csinh 47165
            ​*21.46.4  Reciprocal trigonometric functions (sec, csc, cot)   csec 47176
            ​*21.46.5  Identities for "if"   ifnmfalse 47198
            ​*21.46.6  Logarithms generalized to arbitrary base using ` logb `   logb2aval 47199
            ​*21.46.7  Logarithm laws generalized to an arbitrary base - log_   clog- 47200
            ​*21.46.8  Formally define notions such as reflexivity   wreflexive 47202
            ​*21.46.9  Algebra helpers   comraddi 47206
            ​*21.46.10  Algebra helper examples   i2linesi 47215
            ​*21.46.11  Formal methods "surprises"   alimp-surprise 47217
            ​*21.46.12  Allsome quantifier   walsi 47223
            ​*21.46.13  Miscellaneous   5m4e1 47234
            21.46.14  Theorems about algebraic numbers   aacllem 47238
      ​21.47  Mathbox for Kunhao Zheng
            21.47.1  Weighted AM-GM inequality   amgmwlem 47239