TOC of Theorem List - Metamath Proof Explorer

Table of Contents Summary

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

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

​​*PART 1  CLASSICAL FIRST-ORDER LOGIC WITH EQUALITY
      ​*1.1  Pre-logic
            *1.1.1  Inferences for assisting proof development   idi 1
      ​​*1.2  Propositional calculus
            1.2.1  Recursively define primitive wffs for propositional calculus   wn 3
            ​*1.2.2  The axioms of propositional calculus   ax-mp 5
            ​*1.2.3  Logical implication   mp2 9
            ​*1.2.4  Logical negation   con4 113
            ​*1.2.5  Logical equivalence   wb 206
            ​*1.2.6  Logical conjunction   wa 395
            ​*1.2.7  Logical disjunction   wo 846
            ​*1.2.8  Mixed connectives   jaao 955
            ​*1.2.9  The conditional operator for propositions   wif 1063
            ​*1.2.10  The weak deduction theorem for propositional calculus   elimh 1083
            1.2.11  Abbreviated conjunction and disjunction of three wff's   w3o 1086
            1.2.12  Logical "nand" (Sheffer stroke)   wnan 1488
            1.2.13  Logical "xor"   wxo 1508
            1.2.14  Logical "nor"   wnor 1525
            1.2.15  True and false constants   wal 1535
                  ​*1.2.15.1  Universal quantifier for use by df-tru   wal 1535
                  ​*1.2.15.2  Equality predicate for use by df-tru   cv 1536
                  1.2.15.3  The true constant   wtru 1538
                  1.2.15.4  The false constant   wfal 1549
            ​*1.2.16  Truth tables   truimtru 1560
                  1.2.16.1  Implication   truimtru 1560
                  1.2.16.2  Negation   nottru 1564
                  1.2.16.3  Equivalence   trubitru 1566
                  1.2.16.4  Conjunction   truantru 1570
                  1.2.16.5  Disjunction   truortru 1574
                  1.2.16.6  Alternative denial   trunantru 1578
                  1.2.16.7  Exclusive disjunction   truxortru 1582
                  1.2.16.8  Joint denial   trunortru 1586
            ​*1.2.17  Half adder and full adder in propositional calculus   whad 1590
                  1.2.17.1  Full adder: sum   whad 1590
                  1.2.17.2  Full adder: carry   wcad 1603
      ​1.3  Other axiomatizations related to classical propositional calculus
            ​*1.3.1  Minimal implicational calculus   minimp 1619
            ​*1.3.2  Implicational Calculus   impsingle 1625
            1.3.3  Derive the Lukasiewicz axioms from Meredith's sole axiom   meredith 1639
            1.3.4  Derive the standard axioms from the Lukasiewicz axioms   luklem1 1656
            ​*1.3.5  Derive Nicod's axiom from the standard axioms   nic-dfim 1667
            1.3.6  Derive the Lukasiewicz axioms from Nicod's axiom   nic-imp 1673
            1.3.7  Derive Nicod's Axiom from Lukasiewicz's First Sheffer Stroke Axiom   lukshef-ax1 1692
            1.3.8  Derive the Lukasiewicz Axioms from the Tarski-Bernays-Wajsberg Axioms   tbw-bijust 1696
            1.3.9  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's First CO Axiom   merco1 1711
            1.3.10  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's Second CO Axiom   merco2 1734
            1.3.11  Derive the Lukasiewicz axioms from the Russell-Bernays Axioms   rb-bijust 1747
            ​*1.3.12  Stoic logic non-modal portion (Chrysippus of Soli)   mptnan 1766
      ​​*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 1777
                  1.4.1.1  Existential quantifier   wex 1777
                  1.4.1.2  Nonfreeness predicate   wnf 1781
            1.4.2  Rule scheme ax-gen (Generalization)   ax-gen 1793
            1.4.3  Axiom scheme ax-4 (Quantified Implication)   ax-4 1807
                  ​*1.4.3.1  The empty domain of discourse   empty 1905
            1.4.4  Axiom scheme ax-5 (Distinctness) - first use of $d   ax-5 1909
            ​*1.4.5  Equality predicate (continued)   weq 1962
            1.4.6  Axiom scheme ax-6 (Existence)   ax-6 1967
            1.4.7  Axiom scheme ax-7 (Equality)   ax-7 2007
            1.4.8  Define proper substitution   sbjust 2063
            1.4.9  Membership predicate   wcel 2108
            1.4.10  Axiom scheme ax-8 (Left Equality for Binary Predicate)   ax-8 2110
            1.4.11  Axiom scheme ax-9 (Right Equality for Binary Predicate)   ax-9 2118
            ​*1.4.12  Logical redundancy of ax-10 , ax-11 , ax-12 , ax-13   ax6dgen 2128
      ​​*1.5  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes)
            1.5.1  Axiom scheme ax-10 (Quantified Negation)   ax-10 2141
            1.5.2  Axiom scheme ax-11 (Quantifier Commutation)   ax-11 2158
            1.5.3  Axiom scheme ax-12 (Substitution)   ax-12 2178
            1.5.4  Axiom scheme ax-13 (Quantified Equality)   ax-13 2380
      ​1.6  Uniqueness and unique existence
            1.6.1  Uniqueness: the at-most-one quantifier   wmo 2541
            1.6.2  Unique existence: the unique existential quantifier   weu 2571
      ​1.7  Other axiomatizations related to classical predicate calculus
            ​*1.7.1  Aristotelian logic: Assertic syllogisms   barbara 2666
            ​*1.7.2  Intuitionistic logic   axia1 2696
​​*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 2711
            2.1.2  Classes   cab 2717
                  2.1.2.1  Class abstractions   cab 2717
                  ​*2.1.2.2  Class equality   df-cleq 2732
                  2.1.2.3  Class membership   df-clel 2819
                  2.1.2.4  Elementary properties of class abstractions   eqabdv 2878
            2.1.3  Class form not-free predicate   wnfc 2893
            2.1.4  Negated equality and membership   wne 2946
                  2.1.4.1  Negated equality   wne 2946
                  2.1.4.2  Negated membership   wnel 3052
            2.1.5  Restricted quantification   wral 3067
                  2.1.5.1  Restricted universal and existential quantification   wral 3067
                  2.1.5.2  Restricted existential uniqueness and at-most-one quantifier   wreu 3386
                  2.1.5.3  Restricted class abstraction   crab 3443
            2.1.6  The universal class   cvv 3488
            ​*2.1.7  Conditional equality (experimental)   wcdeq 3785
            2.1.8  Russell's Paradox   rru 3801
            2.1.9  Proper substitution of classes for sets   wsbc 3804
            2.1.10  Proper substitution of classes for sets into classes   csb 3921
            2.1.11  Define basic set operations and relations   cdif 3973
            2.1.12  Subclasses and subsets   df-ss 3993
            2.1.13  The difference, union, and intersection of two classes   dfdif3 4140
                  2.1.13.1  The difference of two classes   dfdif3 4140
                  2.1.13.2  The union of two classes   elun 4176
                  2.1.13.3  The intersection of two classes   elini 4222
                  2.1.13.4  The symmetric difference of two classes   csymdif 4271
                  2.1.13.5  Combinations of difference, union, and intersection of two classes   unabs 4284
                  2.1.13.6  Class abstractions with difference, union, and intersection of two classes   unabw 4326
                  2.1.13.7  Restricted uniqueness with difference, union, and intersection   reuun2 4344
            2.1.14  The empty set   c0 4352
            ​*2.1.15  The conditional operator for classes   cif 4548
            ​*2.1.16  The weak deduction theorem for set theory   dedth 4606
            2.1.17  Power classes   cpw 4622
            2.1.18  Unordered and ordered pairs   snjust 4647
            2.1.19  The union of a class   cuni 4931
            2.1.20  The intersection of a class   cint 4970
            2.1.21  Indexed union and intersection   ciun 5015
            2.1.22  Disjointness   wdisj 5133
            2.1.23  Binary relations   wbr 5166
            2.1.24  Ordered-pair class abstractions (class builders)   copab 5228
            2.1.25  Functions in maps-to notation   cmpt 5249
            2.1.26  Transitive classes   wtr 5283
      ​2.2  ZF Set Theory - add the Axiom of Replacement
            2.2.1  Introduce the Axiom of Replacement   ax-rep 5303
            2.2.2  Derive the Axiom of Separation   axsepgfromrep 5315
            2.2.3  Derive the Null Set Axiom   axnulALT 5322
            2.2.4  Theorems requiring subset and intersection existence   nalset 5331
            2.2.5  Theorems requiring empty set existence   class2set 5373
      ​2.3  ZF Set Theory - add the Axiom of Power Sets
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 5383
            2.3.2  Derive the Axiom of Pairing   axprlem1 5441
            2.3.3  Ordered pair theorem   opnz 5493
            2.3.4  Ordered-pair class abstractions (cont.)   opabidw 5543
            2.3.5  Power class of union and intersection   pwin 5589
            2.3.6  The identity relation   cid 5592
            2.3.7  The membership relation (or epsilon relation)   cep 5598
            ​*2.3.8  Partial and total orderings   wpo 5605
            2.3.9  Founded and well-ordering relations   wfr 5647
            2.3.10  Relations   cxp 5693
            2.3.11  The Predecessor Class   cpred 6326
            2.3.12  Well-founded induction (variant)   frpomin 6367
            2.3.13  Well-ordered induction   tz6.26 6374
            2.3.14  Ordinals   word 6389
            2.3.15  Definite description binder (inverted iota)   cio 6518
            2.3.16  Functions   wfun 6562
            2.3.17  Cantor's Theorem   canth 7396
            2.3.18  Restricted iota (description binder)   crio 7398
            2.3.19  Operations   co 7443
                  2.3.19.1  Variable-to-class conversion for operations   caovclg 7636
            2.3.20  Maps-to notation   mpondm0 7684
            2.3.21  Function operation   cof 7706
            2.3.22  Proper subset relation   crpss 7751
      ​2.4  ZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 7764
            2.4.2  Ordinals (continued)   epweon 7804
            2.4.3  Transfinite induction   tfi 7884
            2.4.4  The natural numbers (i.e., finite ordinals)   com 7897
            2.4.5  Peano's postulates   peano1 7921
            2.4.6  Finite induction (for finite ordinals)   find 7929
            2.4.7  Relations and functions (cont.)   dmexg 7935
            2.4.8  First and second members of an ordered pair   c1st 8022
            2.4.9  Induction on Cartesian products   frpoins3xpg 8175
            2.4.10  Ordering on Cartesian products   xpord2lem 8177
            2.4.11  Ordering Ordinal Sequences   orderseqlem 8192
            ​*2.4.12  The support of functions   csupp 8195
            ​*2.4.13  Special maps-to operations   opeliunxp2f 8245
            2.4.14  Function transposition   ctpos 8260
            2.4.15  Curry and uncurry   ccur 8300
            2.4.16  Undefined values   cund 8307
            2.4.17  Well-founded recursion   cfrecs 8315
            2.4.18  Well-ordered recursion   cwrecs 8346
            2.4.19  Functions on ordinals; strictly monotone ordinal functions   iunon 8389
            2.4.20  "Strong" transfinite recursion   crecs 8420
            2.4.21  Recursive definition generator   crdg 8459
            2.4.22  Finite recursion   frfnom 8485
            2.4.23  Ordinal arithmetic   c1o 8509
            2.4.24  Natural number arithmetic   nna0 8654
            2.4.25  Natural addition   cnadd 8715
            2.4.26  Equivalence relations and classes   wer 8754
            2.4.27  The mapping operation   cmap 8878
            2.4.28  Infinite Cartesian products   cixp 8949
            2.4.29  Equinumerosity   cen 8994
            2.4.30  Schroeder-Bernstein Theorem   sbthlem1 9143
            2.4.31  Equinumerosity (cont.)   xpf1o 9199
            2.4.32  Finite sets   dif1enlem 9216
            2.4.33  Pigeonhole Principle   phplem1 9264
            2.4.34  Finite sets (cont.)   onomeneq 9285
            2.4.35  Finitely supported functions   cfsupp 9425
            2.4.36  Finite intersections   cfi 9473
            2.4.37  Hall's marriage theorem   marypha1lem 9496
            2.4.38  Supremum and infimum   csup 9503
            2.4.39  Ordinal isomorphism, Hartogs's theorem   coi 9572
            2.4.40  Hartogs function   char 9619
            2.4.41  Weak dominance   cwdom 9627
      ​2.5  ZF Set Theory - add the Axiom of Regularity
            2.5.1  Introduce the Axiom of Regularity   ax-reg 9655
            2.5.2  Axiom of Infinity equivalents   inf0 9684
      ​2.6  ZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-inf 9701
            2.6.2  Existence of omega (the set of natural numbers)   omex 9706
            2.6.3  Cantor normal form   ccnf 9724
            2.6.4  Transitive closure of a relation   cttrcl 9770
            2.6.5  Transitive closure   trcl 9791
            2.6.6  Well-Founded Induction   frmin 9812
            2.6.7  Well-Founded Recursion   frr3g 9819
            2.6.8  Rank   cr1 9825
            2.6.9  Scott's trick; collection principle; Hilbert's epsilon   scottex 9948
            2.6.10  Disjoint union   cdju 9961
            2.6.11  Cardinal numbers   ccrd 9998
            2.6.12  Axiom of Choice equivalents   wac 10178
            ​*2.6.13  Cardinal number arithmetic   undjudom 10231
            2.6.14  The Ackermann bijection   ackbij2lem1 10281
            2.6.15  Cofinality (without Axiom of Choice)   cflem 10308
            2.6.16  Eight inequivalent definitions of finite set   sornom 10340
            2.6.17  Hereditarily size-limited sets without Choice   itunifval 10479
​​*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 10498
            3.1.2  Introduce the Axiom of Dependent Choice   ax-dc 10509
      ​3.2  ZFC Set Theory - add the Axiom of Choice
            3.2.1  Introduce the Axiom of Choice   ax-ac 10522
            3.2.2  AC equivalents: well-ordering, Zorn's lemma   numthcor 10557
            3.2.3  Cardinal number theorems using Axiom of Choice   cardval 10609
            3.2.4  Cardinal number arithmetic using Axiom of Choice   iunctb 10637
            3.2.5  Cofinality using the Axiom of Choice   alephreg 10645
      ​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 10683
            3.4.2  Derivation of the Axiom of Choice   gchaclem 10741
​​*PART 4  TG (TARSKI-GROTHENDIECK) SET THEORY
      ​4.1  Inaccessibles
            4.1.1  Weakly and strongly inaccessible cardinals   cwina 10745
            4.1.2  Weak universes   cwun 10763
            4.1.3  Tarski classes   ctsk 10811
            4.1.4  Grothendieck universes   cgru 10853
      ​4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
            4.2.1  Introduce the Tarski-Grothendieck Axiom   ax-groth 10886
            4.2.2  Derive the Power Set, Infinity and Choice Axioms   grothpw 10889
            4.2.3  Tarski map function   ctskm 10900
​​*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 10907
            5.1.2  Final derivation of real and complex number postulates   axaddf 11208
            5.1.3  Real and complex number postulates restated as axioms   ax-cnex 11234
      ​5.2  Derive the basic properties from the field axioms
            5.2.1  Some deductions from the field axioms for complex numbers   cnex 11259
            5.2.2  Infinity and the extended real number system   cpnf 11315
            5.2.3  Restate the ordering postulates with extended real "less than"   axlttri 11355
            5.2.4  Ordering on reals   lttr 11360
            5.2.5  Initial properties of the complex numbers   mul12 11449
      ​5.3  Real and complex numbers - basic operations
            5.3.1  Addition   add12 11501
            5.3.2  Subtraction   cmin 11514
            5.3.3  Multiplication   kcnktkm1cn 11715
            5.3.4  Ordering on reals (cont.)   gt0ne0 11749
            5.3.5  Reciprocals   ixi 11913
            5.3.6  Division   cdiv 11941
            5.3.7  Ordering on reals (cont.)   elimgt0 12126
            5.3.8  Completeness Axiom and Suprema   fimaxre 12233
            5.3.9  Imaginary and complex number properties   inelr 12277
            5.3.10  Function operation analogue theorems   ofsubeq0 12284
      ​5.4  Integer sets
            5.4.1  Positive integers (as a subset of complex numbers)   cn 12287
            5.4.2  Principle of mathematical induction   nnind 12305
            ​*5.4.3  Decimal representation of numbers   c2 12342
            ​*5.4.4  Some properties of specific numbers   neg1cn 12401
            5.4.5  Simple number properties   halfcl 12512
            5.4.6  The Archimedean property   nnunb 12543
            5.4.7  Nonnegative integers (as a subset of complex numbers)   cn0 12547
            ​*5.4.8  Extended nonnegative integers   cxnn0 12619
            5.4.9  Integers (as a subset of complex numbers)   cz 12633
            5.4.10  Decimal arithmetic   cdc 12752
            5.4.11  Upper sets of integers   cuz 12897
            5.4.12  Well-ordering principle for bounded-below sets of integers   uzwo3 13002
            5.4.13  Rational numbers (as a subset of complex numbers)   cq 13007
            5.4.14  Existence of the set of complex numbers   rpnnen1lem2 13036
      ​5.5  Order sets
            5.5.1  Positive reals (as a subset of complex numbers)   crp 13051
            5.5.2  Infinity and the extended real number system (cont.)   cxne 13166
            5.5.3  Supremum and infimum on the extended reals   xrsupexmnf 13361
            5.5.4  Real number intervals   cioo 13401
            5.5.5  Finite intervals of integers   cfz 13561
            ​*5.5.6  Finite intervals of nonnegative integers   elfz2nn0 13669
            5.5.7  Half-open integer ranges   cfzo 13705
      ​5.6  Elementary integer functions
            5.6.1  The floor and ceiling functions   cfl 13835
            5.6.2  The modulo (remainder) operation   cmo 13914
            5.6.3  Miscellaneous theorems about integers   om2uz0i 13992
            5.6.4  Strong induction over upper sets of integers   uzsinds 14032
            5.6.5  Finitely supported functions over the nonnegative integers   fsuppmapnn0fiublem 14035
            5.6.6  The infinite sequence builder "seq" - extension   cseq 14046
            5.6.7  Integer powers   cexp 14106
            5.6.8  Ordered pair theorem for nonnegative integers   nn0le2msqi 14310
            5.6.9  Factorial function   cfa 14316
            5.6.10  The binomial coefficient operation   cbc 14345
            5.6.11  The ` # ` (set size) function   chash 14373
                  5.6.11.1  Proper unordered pairs and triples (sets of size 2 and 3)   hashprlei 14511
                  5.6.11.2  Functions with a domain containing at least two different elements   fundmge2nop0 14545
                  5.6.11.3  Finite induction on the size of the first component of a binary relation   hashdifsnp1 14549
      ​​*5.7  Words over a set
            5.7.1  Definitions and basic theorems   cword 14556
            5.7.2  Last symbol of a word   clsw 14604
            5.7.3  Concatenations of words   cconcat 14612
            5.7.4  Singleton words   cs1 14637
            5.7.5  Concatenations with singleton words   ccatws1cl 14658
            5.7.6  Subwords/substrings   csubstr 14682
            5.7.7  Prefixes of a word   cpfx 14712
            5.7.8  Subwords of subwords   swrdswrdlem 14746
            5.7.9  Subwords and concatenations   pfxcctswrd 14752
            5.7.10  Subwords of concatenations   swrdccatfn 14766
            5.7.11  Splicing words (substring replacement)   csplice 14791
            5.7.12  Reversing words   creverse 14800
            5.7.13  Repeated symbol words   creps 14810
            ​*5.7.14  Cyclical shifts of words   ccsh 14830
            5.7.15  Mapping words by a function   wrdco 14874
            5.7.16  Longer string literals   cs2 14884
      ​​*5.8  Reflexive and transitive closures of relations
            5.8.1  The reflexive and transitive properties of relations   coss12d 15015
            5.8.2  Basic properties of closures   cleq1lem 15025
            5.8.3  Definitions and basic properties of transitive closures   ctcl 15028
            5.8.4  Exponentiation of relations   crelexp 15062
            5.8.5  Reflexive-transitive closure as an indexed union   crtrcl 15098
            ​*5.8.6  Principle of transitive induction.   relexpindlem 15106
      ​5.9  Elementary real and complex functions
            5.9.1  The "shift" operation   cshi 15109
            5.9.2  Signum (sgn or sign) function   csgn 15129
            5.9.3  Real and imaginary parts; conjugate   ccj 15139
            5.9.4  Square root; absolute value   csqrt 15276
      ​5.10  Elementary limits and convergence
            5.10.1  Superior limit (lim sup)   clsp 15510
            5.10.2  Limits   cli 15524
            5.10.3  Finite and infinite sums   csu 15728
            5.10.4  The binomial theorem   binomlem 15871
            5.10.5  The inclusion/exclusion principle   incexclem 15878
            5.10.6  Infinite sums (cont.)   isumshft 15881
            5.10.7  Miscellaneous converging and diverging sequences   divrcnv 15894
            5.10.8  Arithmetic series   arisum 15902
            5.10.9  Geometric series   expcnv 15906
            5.10.10  Ratio test for infinite series convergence   cvgrat 15925
            5.10.11  Mertens' theorem   mertenslem1 15926
            5.10.12  Finite and infinite products   prodf 15929
                  5.10.12.1  Product sequences   prodf 15929
                  5.10.12.2  Non-trivial convergence   ntrivcvg 15939
                  5.10.12.3  Complex products   cprod 15945
                  5.10.12.4  Finite products   fprod 15983
                  5.10.12.5  Infinite products   iprodclim 16040
            5.10.13  Falling and Rising Factorial   cfallfac 16046
            5.10.14  Bernoulli polynomials and sums of k-th powers   cbp 16088
      ​5.11  Elementary trigonometry
            5.11.1  The exponential, sine, and cosine functions   ce 16103
                  5.11.1.1  The circle constant (tau = 2 pi)   ctau 16244
            5.11.2  _e is irrational   eirrlem 16246
      ​5.12  Cardinality of real and complex number subsets
            5.12.1  Countability of integers and rationals   xpnnen 16253
            5.12.2  The reals are uncountable   rpnnen2lem1 16256
​​*PART 6  ELEMENTARY NUMBER THEORY
      ​6.1  Elementary properties of divisibility
            6.1.1  Irrationality of square root of 2   sqrt2irrlem 16290
            6.1.2  Some Number sets are chains of proper subsets   nthruc 16294
            6.1.3  The divides relation   cdvds 16296
            ​*6.1.4  Even and odd numbers   evenelz 16378
            6.1.5  The division algorithm   divalglem0 16435
            6.1.6  Bit sequences   cbits 16459
            6.1.7  The greatest common divisor operator   cgcd 16534
            6.1.8  Bézout's identity   bezoutlem1 16580
            6.1.9  Algorithms   nn0seqcvgd 16611
            6.1.10  Euclid's Algorithm   eucalgval2 16622
            ​*6.1.11  The least common multiple   clcm 16629
            ​*6.1.12  Coprimality and Euclid's lemma   coprmgcdb 16690
            6.1.13  Cancellability of congruences   congr 16705
      ​6.2  Elementary prime number theory
            ​*6.2.1  Elementary properties   cprime 16712
            ​*6.2.2  Coprimality and Euclid's lemma (cont.)   coprm 16752
            6.2.3  Properties of the canonical representation of a rational   cnumer 16774
            6.2.4  Euler's theorem   codz 16804
            6.2.5  Arithmetic modulo a prime number   modprm1div 16838
            6.2.6  Pythagorean Triples   coprimeprodsq 16849
            6.2.7  The prime count function   cpc 16877
            6.2.8  Pocklington's theorem   prmpwdvds 16945
            6.2.9  Infinite primes theorem   unbenlem 16949
            6.2.10  Sum of prime reciprocals   prmreclem1 16957
            6.2.11  Fundamental theorem of arithmetic   1arithlem1 16964
            6.2.12  Lagrange's four-square theorem   cgz 16970
            6.2.13  Van der Waerden's theorem   cvdwa 17006
            6.2.14  Ramsey's theorem   cram 17040
            ​*6.2.15  Primorial function   cprmo 17072
            ​*6.2.16  Prime gaps   prmgaplem1 17090
            6.2.17  Decimal arithmetic (cont.)   dec2dvds 17104
            6.2.18  Cyclical shifts of words (cont.)   cshwsidrepsw 17135
            6.2.19  Specific prime numbers   prmlem0 17147
            6.2.20  Very large primes   1259lem1 17172
​PART 7  BASIC STRUCTURES
      ​7.1  Extensible structures
            ​*7.1.1  Basic definitions   cstr 17187
                  7.1.1.1  Extensible structures as structures with components   cstr 17187
                  7.1.1.2  Substitution of components   csts 17204
                  7.1.1.3  Slots   cslot 17222
                  ​*7.1.1.4  Structure component indices   cnx 17234
                  7.1.1.5  Base sets   cbs 17252
                  7.1.1.6  Base set restrictions   cress 17281
            7.1.2  Slot definitions   cplusg 17305
            7.1.3  Definition of the structure product   crest 17474
            7.1.4  Definition of the structure quotient   cordt 17553
      ​7.2  Moore spaces
            7.2.1  Moore closures   mrcflem 17658
            7.2.2  Independent sets in a Moore system   mrisval 17682
            7.2.3  Algebraic closure systems   isacs 17703
​PART 8  BASIC CATEGORY THEORY
      ​8.1  Categories
            8.1.1  Categories   ccat 17716
            8.1.2  Opposite category   coppc 17763
            8.1.3  Monomorphisms and epimorphisms   cmon 17783
            8.1.4  Sections, inverses, isomorphisms   csect 17799
            ​*8.1.5  Isomorphic objects   ccic 17850
            8.1.6  Subcategories   cssc 17862
            8.1.7  Functors   cfunc 17912
            8.1.8  Full & faithful functors   cful 17963
            8.1.9  Natural transformations and the functor category   cnat 18003
            8.1.10  Initial, terminal and zero objects of a category   cinito 18042
      ​8.2  Arrows (disjointified hom-sets)
            8.2.1  Identity and composition for arrows   cida 18114
      ​8.3  Examples of categories
            8.3.1  The category of sets   csetc 18136
            8.3.2  The category of categories   ccatc 18159
            ​*8.3.3  The category of extensible structures   fncnvimaeqv 18182
      ​8.4  Categorical constructions
            8.4.1  Product of categories   cxpc 18231
            8.4.2  Functor evaluation   cevlf 18273
            8.4.3  Hom functor   chof 18312
​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 18495
            9.5.2  Complete lattices   ccla 18562
            9.5.3  Distributive lattices   cdlat 18584
            9.5.4  Subset order structures   cipo 18591
      ​9.6  Posets, directed sets, and lattices as relations
            ​*9.6.1  Posets and lattices as relations   cps 18628
            9.6.2  Directed sets, nets   cdir 18658
​PART 10  BASIC ALGEBRAIC STRUCTURES
      ​10.1  Monoids
            ​*10.1.1  Magmas   cplusf 18669
            ​*10.1.2  Identity elements   mgmidmo 18692
            ​*10.1.3  Iterated sums in a magma   gsumvalx 18708
            10.1.4  Magma homomorphisms and submagmas   cmgmhm 18722
            ​*10.1.5  Semigroups   csgrp 18750
            ​*10.1.6  Definition and basic properties of monoids   cmnd 18766
            10.1.7  Monoid homomorphisms and submonoids   cmhm 18810
            ​*10.1.8  Iterated sums in a monoid   gsumvallem2 18863
            10.1.9  Free monoids   cfrmd 18876
                  ​*10.1.9.1  Monoid of endofunctions   cefmnd 18897
            10.1.10  Examples and counterexamples for magmas, semigroups and monoids   mgm2nsgrplem1 18947
      ​10.2  Groups
            10.2.1  Definition and basic properties   cgrp 18967
            ​*10.2.2  Group multiple operation   cmg 19101
            10.2.3  Subgroups and Quotient groups   csubg 19154
            ​*10.2.4  Cyclic monoids and groups   cycsubmel 19234
            10.2.5  Elementary theory of group homomorphisms   cghm 19246
            10.2.6  Isomorphisms of groups   cgim 19291
                  10.2.6.1  The first isomorphism theorem of groups   ghmqusnsglem1 19314
            10.2.7  Group actions   cga 19323
            10.2.8  Centralizers and centers   ccntz 19349
            10.2.9  The opposite group   coppg 19379
            10.2.10  Symmetric groups   csymg 19404
                  ​*10.2.10.1  Definition and basic properties   csymg 19404
                  10.2.10.2  Cayley's theorem   cayleylem1 19448
                  10.2.10.3  Permutations fixing one element   symgfix2 19452
                  ​*10.2.10.4  Transpositions in the symmetric group   cpmtr 19477
                  10.2.10.5  The sign of a permutation   cpsgn 19525
            10.2.11  p-Groups and Sylow groups; Sylow's theorems   cod 19560
            10.2.12  Direct products   clsm 19670
                  10.2.12.1  Direct products (extension)   smndlsmidm 19692
            10.2.13  Free groups   cefg 19742
            10.2.14  Abelian groups   ccmn 19816
                  10.2.14.1  Definition and basic properties   ccmn 19816
                  10.2.14.2  Cyclic groups   ccyg 19913
                  10.2.14.3  Group sum operation   gsumval3a 19939
                  10.2.14.4  Group sums over (ranges of) integers   fsfnn0gsumfsffz 20019
                  10.2.14.5  Internal direct products   cdprd 20031
                  10.2.14.6  The Fundamental Theorem of Abelian Groups   ablfacrplem 20103
            10.2.15  Simple groups   csimpg 20128
                  10.2.15.1  Definition and basic properties   csimpg 20128
                  10.2.15.2  Classification of abelian simple groups   ablsimpnosubgd 20142
      ​10.3  Rings
            10.3.1  Multiplicative Group   cmgp 20155
            ​*10.3.2  Non-unital rings ("rngs")   crng 20173
            ​*10.3.3  Ring unity (multiplicative identity)   cur 20202
            10.3.4  Semirings   csrg 20207
                  ​*10.3.4.1  The binomial theorem for semirings   srgbinomlem1 20247
            10.3.5  Unital rings   crg 20254
            10.3.6  Opposite ring   coppr 20353
            10.3.7  Divisibility   cdsr 20374
            10.3.8  Ring primes   crpm 20452
            10.3.9  Homomorphisms of non-unital rings   crnghm 20454
            10.3.10  Ring homomorphisms   crh 20489
            10.3.11  Nonzero rings and zero rings   cnzr 20532
            10.3.12  Local rings   clring 20558
            10.3.13  Subrings   csubrng 20565
                  10.3.13.1  Subrings of non-unital rings   csubrng 20565
                  10.3.13.2  Subrings of unital rings   csubrg 20589
            10.3.14  Categories of rings   crngc 20632
                  ​*10.3.14.1  The category of non-unital rings   crngc 20632
                  ​*10.3.14.2  The category of (unital) rings   cringc 20661
                  10.3.14.3  Subcategories of the category of rings   srhmsubclem1 20693
            10.3.15  Left regular elements and domains   crlreg 20707
      ​10.4  Division rings and fields
            10.4.1  Definition and basic properties   cdr 20745
            10.4.2  Sub-division rings   csdrg 20803
            10.4.3  Absolute value (abstract algebra)   cabv 20825
            10.4.4  Star rings   cstf 20854
      ​10.5  Left modules
            10.5.1  Definition and basic properties   clmod 20874
            10.5.2  Subspaces and spans in a left module   clss 20946
            10.5.3  Homomorphisms and isomorphisms of left modules   clmhm 21035
            10.5.4  Subspace sum; bases for a left module   clbs 21090
      ​10.6  Vector spaces
            10.6.1  Definition and basic properties   clvec 21118
      ​10.7  Subring algebras and ideals
            10.7.1  Subring algebras   csra 21187
            ​*10.7.2  Left ideals and spans   clidl 21233
            10.7.3  Two-sided ideals and quotient rings   c2idl 21276
                  ​*10.7.3.1  Condition for a non-unital ring to be unital   rngqiprng1elbas 21313
            10.7.4  Principal ideal rings. Divisibility in the integers   clpidl 21347
            10.7.5  Principal ideal domains   cpid 21363
      ​10.8  The complex numbers as an algebraic extensible structure
            10.8.1  Definition and basic properties   cpsmet 21365
            ​*10.8.2  Ring of integers   czring 21474
                  ​*10.8.2.1  Example for a condition for a non-unital ring to be unital   pzriprnglem1 21509
            10.8.3  Algebraic constructions based on the complex numbers   czrh 21527
            10.8.4  Signs as subgroup of the complex numbers   cnmsgnsubg 21612
            10.8.5  Embedding of permutation signs into a ring   zrhpsgnmhm 21619
            10.8.6  The ordered field of real numbers   crefld 21639
      ​10.9  Generalized pre-Hilbert and Hilbert spaces
            10.9.1  Definition and basic properties   cphl 21659
            10.9.2  Orthocomplements and closed subspaces   cocv 21695
            10.9.3  Orthogonal projection and orthonormal bases   cpj 21737
​​*PART 11  BASIC LINEAR ALGEBRA
      ​11.1  Vectors and free modules
            *11.1.1  Direct sum of left modules   cdsmm 21768
            ​*11.1.2  Free modules   cfrlm 21783
            ​*11.1.3  Standard basis (unit vectors)   cuvc 21819
            ​*11.1.4  Independent sets and families   clindf 21841
            11.1.5  Characterization of free modules   lmimlbs 21873
      ​11.2  Associative algebras
            11.2.1  Definition and basic properties   casa 21887
      ​11.3  Abstract multivariate polynomials
            11.3.1  Definition and basic properties   cmps 21940
            11.3.2  Polynomial evaluation   ces 22112
            11.3.3  Additional definitions for (multivariate) polynomials   cslv 22148
            ​*11.3.4  Univariate polynomials   cps1 22189
            11.3.5  Univariate polynomial evaluation   ces1 22330
                  11.3.5.1  Specialization of polynomial evaluation as a ring homomorphism   evls1scafv 22383
      ​​*11.4  Matrices
            *11.4.1  The matrix multiplication   cmmul 22407
            ​*11.4.2  Square matrices   cmat 22424
            ​*11.4.3  The matrix algebra   matmulr 22457
            ​*11.4.4  Matrices of dimension 0 and 1   mat0dimbas0 22485
            ​*11.4.5  The subalgebras of diagonal and scalar matrices   cdmat 22507
            ​*11.4.6  Multiplication of a matrix with a "column vector"   cmvmul 22559
            11.4.7  Replacement functions for a square matrix   cmarrep 22575
            11.4.8  Submatrices   csubma 22595
      ​11.5  The determinant
            11.5.1  Definition and basic properties   cmdat 22603
            11.5.2  Determinants of 2 x 2 -matrices   m2detleiblem1 22643
            11.5.3  The matrix adjugate/adjunct   cmadu 22651
            ​*11.5.4  Laplace expansion of determinants (special case)   symgmatr01lem 22672
            11.5.5  Inverse matrix   invrvald 22695
            ​*11.5.6  Cramer's rule   slesolvec 22698
      ​​*11.6  Polynomial matrices
            11.6.1  Basic properties   pmatring 22711
            ​*11.6.2  Constant polynomial matrices   ccpmat 22722
            ​*11.6.3  Collecting coefficients of polynomial matrices   cdecpmat 22781
            ​*11.6.4  Ring isomorphism between polynomial matrices and polynomials over matrices   cpm2mp 22811
      ​​*11.7  The characteristic polynomial
            *11.7.1  Definition and basic properties   cchpmat 22845
            ​*11.7.2  The characteristic factor function G   fvmptnn04if 22868
            ​*11.7.3  The Cayley-Hamilton theorem   cpmadurid 22886
​PART 12  BASIC TOPOLOGY
      ​12.1  Topology
            ​*12.1.1  Topological spaces   ctop 22912
                  12.1.1.1  Topologies   ctop 22912
                  12.1.1.2  Topologies on sets   ctopon 22929
                  12.1.1.3  Topological spaces   ctps 22951
            12.1.2  Topological bases   ctb 22965
            12.1.3  Examples of topologies   distop 23015
            12.1.4  Closure and interior   ccld 23037
            12.1.5  Neighborhoods   cnei 23118
            12.1.6  Limit points and perfect sets   clp 23155
            12.1.7  Subspace topologies   restrcl 23178
            12.1.8  Order topology   ordtbaslem 23209
            12.1.9  Limits and continuity in topological spaces   ccn 23245
            12.1.10  Separated spaces: T0, T1, T2 (Hausdorff) ...   ct0 23327
            12.1.11  Compactness   ccmp 23407
            12.1.12  Bolzano-Weierstrass theorem   bwth 23431
            12.1.13  Connectedness   cconn 23432
            12.1.14  First- and second-countability   c1stc 23458
            12.1.15  Local topological properties   clly 23485
            12.1.16  Refinements   cref 23523
            12.1.17  Compactly generated spaces   ckgen 23554
            12.1.18  Product topologies   ctx 23581
            12.1.19  Continuous function-builders   cnmptid 23682
            12.1.20  Quotient maps and quotient topology   ckq 23714
            12.1.21  Homeomorphisms   chmeo 23774
      ​12.2  Filters and filter bases
            12.2.1  Filter bases   elmptrab 23848
            12.2.2  Filters   cfil 23866
            12.2.3  Ultrafilters   cufil 23920
            12.2.4  Filter limits   cfm 23954
            12.2.5  Extension by continuity   ccnext 24080
            12.2.6  Topological groups   ctmd 24091
            12.2.7  Infinite group sum on topological groups   ctsu 24147
            12.2.8  Topological rings, fields, vector spaces   ctrg 24177
      ​12.3  Uniform Structures and Spaces
            12.3.1  Uniform structures   cust 24221
            12.3.2  The topology induced by an uniform structure   cutop 24252
            12.3.3  Uniform Spaces   cuss 24275
            12.3.4  Uniform continuity   cucn 24297
            12.3.5  Cauchy filters in uniform spaces   ccfilu 24308
            12.3.6  Complete uniform spaces   ccusp 24319
      ​12.4  Metric spaces
            12.4.1  Pseudometric spaces   ispsmet 24327
            12.4.2  Basic metric space properties   cxms 24340
            12.4.3  Metric space balls   blfvalps 24406
            12.4.4  Open sets of a metric space   mopnval 24461
            12.4.5  Continuity in metric spaces   metcnp3 24566
            12.4.6  The uniform structure generated by a metric   metuval 24575
            12.4.7  Examples of metric spaces   dscmet 24598
            ​*12.4.8  Normed algebraic structures   cnm 24602
            12.4.9  Normed space homomorphisms (bounded linear operators)   cnmo 24739
            12.4.10  Topology on the reals   qtopbaslem 24792
            12.4.11  Topological definitions using the reals   cii 24912
            12.4.12  Path homotopy   chtpy 25010
            12.4.13  The fundamental group   cpco 25044
      ​12.5  Metric subcomplex vector spaces
            12.5.1  Subcomplex modules   cclm 25106
            ​*12.5.2  Subcomplex vector spaces   ccvs 25167
            ​*12.5.3  Normed subcomplex vector spaces   isncvsngp 25194
            12.5.4  Subcomplex pre-Hilbert spaces   ccph 25211
            12.5.5  Convergence and completeness   ccfil 25297
            12.5.6  Baire's Category Theorem   bcthlem1 25369
            12.5.7  Banach spaces and subcomplex Hilbert spaces   ccms 25377
                  12.5.7.1  The complete ordered field of the real numbers   retopn 25424
            12.5.8  Euclidean spaces   crrx 25428
            12.5.9  Minimizing Vector Theorem   minveclem1 25469
            12.5.10  Projection Theorem   pjthlem1 25482
​PART 13  BASIC REAL AND COMPLEX ANALYSIS
      ​13.1  Continuity
            13.1.1  Intermediate value theorem   pmltpclem1 25494
      ​13.2  Integrals
            13.2.1  Lebesgue measure   covol 25508
            13.2.2  Lebesgue integration   cmbf 25660
                  13.2.2.1  Lesbesgue integral   cmbf 25660
                  13.2.2.2  Lesbesgue directed integral   cdit 25893
      ​13.3  Derivatives
            13.3.1  Real and complex differentiation   climc 25909
                  13.3.1.1  Derivatives of functions of one complex or real variable   climc 25909
                  13.3.1.2  Results on real differentiation   dvferm1lem 26034
​PART 14  BASIC REAL AND COMPLEX FUNCTIONS
      ​14.1  Polynomials
            14.1.1  Polynomial degrees   cmdg 26104
            14.1.2  The division algorithm for univariate polynomials   cmn1 26177
            14.1.3  Elementary properties of complex polynomials   cply 26235
            14.1.4  The division algorithm for polynomials   cquot 26342
            14.1.5  Algebraic numbers   caa 26366
            14.1.6  Liouville's approximation theorem   aalioulem1 26384
      ​14.2  Sequences and series
            14.2.1  Taylor polynomials and Taylor's theorem   ctayl 26404
            14.2.2  Uniform convergence   culm 26429
            14.2.3  Power series   pserval 26463
      ​14.3  Basic trigonometry
            14.3.1  The exponential, sine, and cosine functions (cont.)   efcn 26497
            14.3.2  Properties of pi = 3.14159...   pilem1 26505
            14.3.3  Mapping of the exponential function   efgh 26593
            14.3.4  The natural logarithm on complex numbers   clog 26606
            ​*14.3.5  Logarithms to an arbitrary base   clogb 26817
            14.3.6  Theorems of Pythagoras, isosceles triangles, and intersecting chords   angval 26854
            14.3.7  Solutions of quadratic, cubic, and quartic equations   quad2 26892
            14.3.8  Inverse trigonometric functions   casin 26915
            14.3.9  The Birthday Problem   log2ublem1 26999
            14.3.10  Areas in R^2   carea 27008
            14.3.11  More miscellaneous converging sequences   rlimcnp 27018
            14.3.12  Inequality of arithmetic and geometric means   cvxcl 27038
            14.3.13  Euler-Mascheroni constant   cem 27045
            14.3.14  Zeta function   czeta 27066
            14.3.15  Gamma function   clgam 27069
      ​14.4  Basic number theory
            14.4.1  Wilson's theorem   wilthlem1 27121
            14.4.2  The Fundamental Theorem of Algebra   ftalem1 27126
            14.4.3  The Basel problem (ζ(2) = π²/6)   basellem1 27134
            14.4.4  Number-theoretical functions   ccht 27144
            14.4.5  Perfect Number Theorem   mersenne 27281
            14.4.6  Characters of Z/nZ   cdchr 27286
            14.4.7  Bertrand's postulate   bcctr 27329
            ​*14.4.8  Quadratic residues and the Legendre symbol   clgs 27348
            ​*14.4.9  Gauss' Lemma   gausslemma2dlem0a 27410
            14.4.10  Quadratic reciprocity   lgseisenlem1 27429
            14.4.11  All primes 4n+1 are the sum of two squares   2sqlem1 27471
            14.4.12  Chebyshev's Weak Prime Number Theorem, Dirichlet's Theorem   chebbnd1lem1 27523
            14.4.13  The Prime Number Theorem   mudivsum 27584
            14.4.14  Ostrowski's theorem   abvcxp 27669
​PART 15  SURREAL NUMBERS
      ​​*15.1  Sign sequence representation and Alling's axioms
            15.1.1  Definitions and initial properties   csur 27694
            15.1.2  Ordering   sltsolem1 27730
            15.1.3  Birthday Function   bdayfo 27732
            15.1.4  Density   fvnobday 27733
            ​*15.1.5  Full-Eta Property   bdayimaon 27748
      ​15.2  Initial consequences of Alling's axioms
            15.2.1  Ordering Theorems   csle 27799
            15.2.2  Birthday Theorems   bdayfun 27827
      ​​*15.3  Conway cut representation
            15.3.1  Conway cuts   csslt 27835
            15.3.2  Zero and One   c0s 27877
            15.3.3  Cuts and Options   cmade 27891
            15.3.4  Cofinality and coinitiality   cofsslt 27962
      ​15.4  Induction and recursion
            15.4.1  Induction and recursion on one variable   cnorec 27980
            15.4.2  Induction and recursion on two variables   cnorec2 27991
      ​15.5  Surreal arithmetic
            15.5.1  Addition   cadds 28002
            15.5.2  Negation and Subtraction   cnegs 28061
            15.5.3  Multiplication   cmuls 28142
            15.5.4  Division   cdivs 28223
            15.5.5  Absolute value   cabss 28271
      ​15.6  Subsystems of surreals
            15.6.1  Ordinal numbers   cons 28284
            15.6.2  Surreal recursive sequences   cseqs 28299
            15.6.3  Natural numbers   cnn0s 28328
            15.6.4  Integers   czs 28374
            15.6.5  Dyadic fractions   c2s 28404
            15.6.6  Real numbers   creno 28435
​​*PART 16  ELEMENTARY GEOMETRY
      ​16.1  Definition and Tarski's Axioms of Geometry
            16.1.1  Justification for the congruence notation   tgjustf 28491
      ​16.2  Tarskian Geometry
            16.2.1  Congruence   tgcgrcomimp 28495
            16.2.2  Betweenness   tgbtwntriv2 28505
            16.2.3  Dimension   tglowdim1 28518
            16.2.4  Betweenness and Congruence   tgifscgr 28526
            16.2.5  Congruence of a series of points   ccgrg 28528
            16.2.6  Motions   cismt 28550
            16.2.7  Colinearity   tglng 28564
            16.2.8  Connectivity of betweenness   tgbtwnconn1lem1 28590
            16.2.9  Less-than relation in geometric congruences   cleg 28600
            16.2.10  Rays   chlg 28618
            16.2.11  Lines   btwnlng1 28637
            16.2.12  Point inversions   cmir 28670
            16.2.13  Right angles   crag 28711
            16.2.14  Half-planes   islnopp 28757
            16.2.15  Midpoints and Line Mirroring   cmid 28790
            16.2.16  Congruence of angles   ccgra 28825
            16.2.17  Angle Comparisons   cinag 28853
            16.2.18  Congruence Theorems   tgsas1 28872
            16.2.19  Equilateral triangles   ceqlg 28883
      ​16.3  Properties of geometries
            16.3.1  Isomorphisms between geometries   f1otrgds 28887
      ​16.4  Geometry in Hilbert spaces
            16.4.1  Geometry in the complex plane   cchhllem 28911
            16.4.2  Geometry in Euclidean spaces   cee 28913
                  16.4.2.1  Definition of the Euclidean space   cee 28913
                  16.4.2.2  Tarski's axioms for geometry for the Euclidean space   axdimuniq 28938
                  16.4.2.3  EE^n fulfills Tarski's Axioms   ceeng 29002
​​*PART 17  GRAPH THEORY
      ​*17.1  Vertices and edges
            17.1.1  The edge function extractor for extensible structures   cedgf 29013
            ​*17.1.2  Vertices and indexed edges   cvtx 29023
                  17.1.2.1  Definitions and basic properties   cvtx 29023
                  17.1.2.2  The vertices and edges of a graph represented as ordered pair   opvtxval 29030
                  17.1.2.3  The vertices and edges of a graph represented as extensible structure   funvtxdmge2val 29038
                  17.1.2.4  Representations of graphs without edges   snstrvtxval 29064
                  17.1.2.5  Degenerated cases of representations of graphs   vtxval0 29066
            17.1.3  Edges as range of the edge function   cedg 29074
      ​​*17.2  Undirected graphs
            17.2.1  Undirected hypergraphs   cuhgr 29083
            17.2.2  Undirected pseudographs and multigraphs   cupgr 29107
            ​*17.2.3  Loop-free graphs   umgrislfupgrlem 29149
            17.2.4  Edges as subsets of vertices of graphs   uhgredgiedgb 29153
            ​*17.2.5  Undirected simple graphs   cuspgr 29175
            17.2.6  Examples for graphs   usgr0e 29263
            17.2.7  Subgraphs   csubgr 29294
            17.2.8  Finite undirected simple graphs   cfusgr 29343
            17.2.9  Neighbors, complete graphs and universal vertices   cnbgr 29359
                  17.2.9.1  Neighbors   cnbgr 29359
                  17.2.9.2  Universal vertices   cuvtx 29412
                  17.2.9.3  Complete graphs   ccplgr 29436
            17.2.10  Vertex degree   cvtxdg 29493
            ​*17.2.11  Regular graphs   crgr 29583
      ​​*17.3  Walks, paths and cycles
            *17.3.1  Walks   cewlks 29623
            17.3.2  Walks for loop-free graphs   lfgrwlkprop 29715
            17.3.3  Trails   ctrls 29718
            17.3.4  Paths and simple paths   cpths 29740
            17.3.5  Closed walks   cclwlks 29798
            17.3.6  Circuits and cycles   ccrcts 29812
            ​*17.3.7  Walks as words   cwwlks 29850
            17.3.8  Walks/paths of length 2 (as length 3 strings)   2wlkdlem1 29950
            17.3.9  Walks in regular graphs   rusgrnumwwlkl1 29993
            ​*17.3.10  Closed walks as words   cclwwlk 30005
                  17.3.10.1  Closed walks as words   cclwwlk 30005
                  17.3.10.2  Closed walks of a fixed length as words   cclwwlkn 30048
                  17.3.10.3  Closed walks on a vertex of a fixed length as words   cclwwlknon 30111
            17.3.11  Examples for walks, trails and paths   0ewlk 30138
            17.3.12  Connected graphs   cconngr 30210
      ​17.4  Eulerian paths and the Konigsberg Bridge problem
            ​*17.4.1  Eulerian paths   ceupth 30221
            ​*17.4.2  The Königsberg Bridge problem   konigsbergvtx 30270
      ​17.5  The Friendship Theorem
            17.5.1  Friendship graphs - basics   cfrgr 30282
            17.5.2  The friendship theorem for small graphs   frgr1v 30295
            17.5.3  Theorems according to Mertzios and Unger   2pthfrgrrn 30306
            ​*17.5.4  Huneke's Proof of the Friendship Theorem   frgrncvvdeqlem1 30323
​PART 18  GUIDES AND MISCELLANEA
      ​18.1  Guides (conventions, explanations, and examples)
            ​*18.1.1  Conventions   conventions 30424
            18.1.2  Natural deduction   natded 30427
            ​*18.1.3  Natural deduction examples   ex-natded5.2 30428
            18.1.4  Definitional examples   ex-or 30445
            18.1.5  Other examples   aevdemo 30484
      ​18.2  Humor
            18.2.1  April Fool's theorem   avril1 30487
      ​18.3  (Future - to be reviewed and classified)
            18.3.1  Planar incidence geometry   cplig 30498
            ​*18.3.2  Aliases kept to prevent broken links   dummylink 30511
​​*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 30513
            19.1.2  Abelian groups   cablo 30568
      ​19.2  Complex vector spaces
            19.2.1  Definition and basic properties   cvc 30582
            19.2.2  Examples of complex vector spaces   cnaddabloOLD 30605
      ​19.3  Normed complex vector spaces
            19.3.1  Definition and basic properties   cnv 30608
            19.3.2  Examples of normed complex vector spaces   cnnv 30701
            19.3.3  Induced metric of a normed complex vector space   imsval 30709
            19.3.4  Inner product   cdip 30724
            19.3.5  Subspaces   css 30745
      ​19.4  Operators on complex vector spaces
            19.4.1  Definitions and basic properties   clno 30764
      ​19.5  Inner product (pre-Hilbert) spaces
            19.5.1  Definition and basic properties   ccphlo 30836
            19.5.2  Examples of pre-Hilbert spaces   cncph 30843
            19.5.3  Properties of pre-Hilbert spaces   isph 30846
      ​19.6  Complex Banach spaces
            19.6.1  Definition and basic properties   ccbn 30886
            19.6.2  Examples of complex Banach spaces   cnbn 30893
            19.6.3  Uniform Boundedness Theorem   ubthlem1 30894
            19.6.4  Minimizing Vector Theorem   minvecolem1 30898
      ​19.7  Complex Hilbert spaces
            19.7.1  Definition and basic properties   chlo 30909
            19.7.2  Standard axioms for a complex Hilbert space   hlex 30922
            19.7.3  Examples of complex Hilbert spaces   cnchl 30940
            19.7.4  Hellinger-Toeplitz Theorem   htthlem 30941
​​*PART 20  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      ​20.1  Axiomatization of complex pre-Hilbert spaces
            20.1.1  Basic Hilbert space definitions   chba 30943
            20.1.2  Preliminary ZFC lemmas   df-hnorm 30992
            ​*20.1.3  Derive the Hilbert space axioms from ZFC set theory   axhilex-zf 31005
            ​*20.1.4  Introduce the vector space axioms for a Hilbert space   ax-hilex 31023
            20.1.5  Vector operations   hvmulex 31035
            20.1.6  Inner product postulates for a Hilbert space   ax-hfi 31103
      ​20.2  Inner product and norms
            20.2.1  Inner product   his5 31110
            20.2.2  Norms   dfhnorm2 31146
            20.2.3  Relate Hilbert space to normed complex vector spaces   hilablo 31184
            20.2.4  Bunjakovaskij-Cauchy-Schwarz inequality   bcsiALT 31203
      ​20.3  Cauchy sequences and completeness axiom
            20.3.1  Cauchy sequences and limits   hcau 31208
            20.3.2  Derivation of the completeness axiom from ZF set theory   hilmet 31218
            20.3.3  Completeness postulate for a Hilbert space   ax-hcompl 31226
            20.3.4  Relate Hilbert space to ZFC pre-Hilbert and Hilbert spaces   hhcms 31227
      ​20.4  Subspaces and projections
            20.4.1  Subspaces   df-sh 31231
            20.4.2  Closed subspaces   df-ch 31245
            20.4.3  Orthocomplements   df-oc 31276
            20.4.4  Subspace sum, span, lattice join, lattice supremum   df-shs 31332
            20.4.5  Projection theorem   pjhthlem1 31415
            20.4.6  Projectors   df-pjh 31419
      ​20.5  Properties of Hilbert subspaces
            20.5.1  Orthomodular law   omlsilem 31426
            20.5.2  Projectors (cont.)   pjhtheu2 31440
            20.5.3  Hilbert lattice operations   sh0le 31464
            20.5.4  Span (cont.) and one-dimensional subspaces   spansn0 31565
            20.5.5  Commutes relation for Hilbert lattice elements   df-cm 31607
            20.5.6  Foulis-Holland theorem   fh1 31642
            20.5.7  Quantum Logic Explorer axioms   qlax1i 31651
            20.5.8  Orthogonal subspaces   chscllem1 31661
            20.5.9  Orthoarguesian laws 5OA and 3OA   5oalem1 31678
            20.5.10  Projectors (cont.)   pjorthi 31693
            20.5.11  Mayet's equation E_3   mayete3i 31752
      ​20.6  Operators on Hilbert spaces
            ​*20.6.1  Operator sum, difference, and scalar multiplication   df-hosum 31754
            20.6.2  Zero and identity operators   df-h0op 31772
            20.6.3  Operations on Hilbert space operators   hoaddcl 31782
            20.6.4  Linear, continuous, bounded, Hermitian, unitary operators and norms   df-nmop 31863
            20.6.5  Linear and continuous functionals and norms   df-nmfn 31869
            20.6.6  Adjoint   df-adjh 31873
            20.6.7  Dirac bra-ket notation   df-bra 31874
            20.6.8  Positive operators   df-leop 31876
            20.6.9  Eigenvectors, eigenvalues, spectrum   df-eigvec 31877
            20.6.10  Theorems about operators and functionals   nmopval 31880
            20.6.11  Riesz lemma   riesz3i 32086
            20.6.12  Adjoints (cont.)   cnlnadjlem1 32091
            20.6.13  Quantum computation error bound theorem   unierri 32128
            20.6.14  Dirac bra-ket notation (cont.)   branmfn 32129
            20.6.15  Positive operators (cont.)   leopg 32146
            20.6.16  Projectors as operators   pjhmopi 32170
      ​20.7  States on a Hilbert lattice and Godowski's equation
            20.7.1  States on a Hilbert lattice   df-st 32235
            20.7.2  Godowski's equation   golem1 32295
      ​20.8  Cover relation, atoms, exchange axiom, and modular symmetry
            20.8.1  Covers relation; modular pairs   df-cv 32303
            20.8.2  Atoms   df-at 32362
            20.8.3  Superposition principle   superpos 32378
            20.8.4  Atoms, exchange and covering properties, atomicity   chcv1 32379
            20.8.5  Irreducibility   chirredlem1 32414
            20.8.6  Atoms (cont.)   atcvat3i 32420
            20.8.7  Modular symmetry   mdsymlem1 32427
​PART 21  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      ​21.1  Mathboxes for user contributions
            21.1.1  Mathbox guidelines   mathbox 32466
      ​21.2  Mathbox for Stefan Allan
      ​21.3  Mathbox for Thierry Arnoux
            21.3.1  Propositional Calculus - misc additions   an42ds 32471
            21.3.2  Predicate Calculus   sbc2iedf 32486
                  21.3.2.1  Predicate Calculus - misc additions   sbc2iedf 32486
                  21.3.2.2  Restricted quantification - misc additions   ralcom4f 32488
                  21.3.2.3  Equality   eqtrb 32494
                  21.3.2.4  Double restricted existential uniqueness quantification   opsbc2ie 32496
                  21.3.2.5  Double restricted existential uniqueness quantification syntax   w2reu 32498
                  21.3.2.6  Substitution (without distinct variables) - misc additions   sbceqbidf 32507
                  21.3.2.7  Existential "at most one" - misc additions   mo5f 32509
                  21.3.2.8  Existential uniqueness - misc additions   reuxfrdf 32511
                  21.3.2.9  Restricted "at most one" - misc additions   rmoxfrd 32513
                  21.3.2.10  Restricted iota (description binder)   riotaeqbidva 32516
            21.3.3  General Set Theory   dmrab 32517
                  21.3.3.1  Class abstractions (a.k.a. class builders)   dmrab 32517
                  21.3.3.2  Image Sets   abrexdomjm 32527
                  21.3.3.3  Set relations and operations - misc additions   elunsn 32533
                  21.3.3.4  Unordered pairs   elpreq 32550
                  21.3.3.5  Conditional operator - misc additions   ifeqeqx 32557
                  21.3.3.6  Set union   uniinn0 32565
                  21.3.3.7  Indexed union - misc additions   cbviunf 32570
                  21.3.3.8  Indexed intersection - misc additions   iinabrex 32583
                  21.3.3.9  Disjointness - misc additions   disjnf 32584
            21.3.4  Relations and Functions   xpdisjres 32612
                  21.3.4.1  Relations - misc additions   xpdisjres 32612
                  21.3.4.2  Functions - misc additions   feq2dd 32634
                  21.3.4.3  Operations - misc additions   mpomptxf 32687
                  21.3.4.4  Support of a function   suppovss 32689
                  21.3.4.5  Explicit Functions with one or two points as a domain   cosnopne 32698
                  21.3.4.6  Isomorphisms - misc. additions   gtiso 32704
                  21.3.4.7  Disjointness (additional proof requiring functions)   disjdsct 32706
                  21.3.4.8  First and second members of an ordered pair - misc additions   df1stres 32707
                  21.3.4.9  Supremum - misc additions   supssd 32715
                  21.3.4.10  Finite Sets   imafi2 32717
                  21.3.4.11  Countable Sets   snct 32719
            21.3.5  Real and Complex Numbers   creq0 32741
                  21.3.5.1  Complex operations - misc. additions   creq0 32741
                  21.3.5.2  Ordering on reals - misc additions   lt2addrd 32750
                  21.3.5.3  Extended reals - misc additions   xrlelttric 32751
                  21.3.5.4  Extended nonnegative integers - misc additions   xnn0gt0 32768
                  21.3.5.5  Real number intervals - misc additions   joiniooico 32771
                  21.3.5.6  Finite intervals of integers - misc additions   uzssico 32781
                  21.3.5.7  Half-open integer ranges - misc additions   iundisjfi 32793
                  21.3.5.8  The ` # ` (set size) function - misc additions   hashunif 32805
                  21.3.5.9  The greatest common divisor operator - misc. additions   znumd 32808
                  21.3.5.10  Integers   nn0split01 32813
                  21.3.5.11  Decimal numbers   dfdec100 32826
            ​*21.3.6  Decimal expansion   cdp2 32827
                  ​*21.3.6.1  Decimal point   cdp 32844
                  21.3.6.2  Division in the extended real number system   cxdiv 32873
            21.3.7  Words over a set - misc additions   wrdfd 32892
                  21.3.7.1  Splicing words (substring replacement)   splfv3 32917
                  21.3.7.2  Cyclic shift of words   1cshid 32918
            21.3.8  Extensible Structures   ressplusf 32922
                  21.3.8.1  Structure restriction operator   ressplusf 32922
                  21.3.8.2  The opposite group   oppgle 32925
                  21.3.8.3  Posets   ressprs 32928
                  21.3.8.4  Complete lattices   clatp0cl 32941
                  21.3.8.5  Order Theory   cmnt 32943
                  21.3.8.6  Chains   cchn 32969
                  21.3.8.7  Extended reals Structure - misc additions   ax-xrssca 32979
                  21.3.8.8  The extended nonnegative real numbers commutative monoid   xrge0base 32989
            21.3.9  Algebra   mndcld 33000
                  21.3.9.1  Monoids   mndcld 33000
                  21.3.9.2  Monoids Homomorphisms   abliso 33014
                  21.3.9.3  Groups - misc additions   grpsubcld 33018
                  21.3.9.4  Finitely supported group sums - misc additions   gsumsubg 33021
                  21.3.9.5  Centralizers and centers - misc additions   cntzun 33036
                  21.3.9.6  Totally ordered monoids and groups   comnd 33039
                  21.3.9.7  The symmetric group   symgfcoeu 33067
                  21.3.9.8  Transpositions   pmtridf1o 33079
                  21.3.9.9  Permutation Signs   psgnid 33082
                  21.3.9.10  Permutation cycles   ctocyc 33091
                  21.3.9.11  The Alternating Group   evpmval 33130
                  21.3.9.12  Signum in an ordered monoid   csgns 33143
                  21.3.9.13  The Archimedean property for generic ordered algebraic structures   cinftm 33148
                  21.3.9.14  Semiring left modules   cslmd 33171
                  21.3.9.15  Simple groups   prmsimpcyc 33199
                  21.3.9.16  Rings - misc additions   cringmul32d 33200
                  21.3.9.17  The zero ring   irrednzr 33214
                  21.3.9.18  Localization of rings   cerl 33217
                  21.3.9.19  Integral Domains   domnmuln0rd 33238
                  21.3.9.20  Euclidean Domains   ceuf 33249
                  21.3.9.21  Division Rings   ringinveu 33255
                  21.3.9.22  Subfields   sdrgdvcl 33258
                  21.3.9.23  Field of fractions   cfrac 33261
                  21.3.9.24  Field extensions generated by a set   cfldgen 33269
                  21.3.9.25  Totally ordered rings and fields   corng 33282
                  21.3.9.26  Ring homomorphisms - misc additions   rhmdvd 33305
                  21.3.9.27  Scalar restriction operation   cresv 33307
                  21.3.9.28  The commutative ring of gaussian integers   gzcrng 33327
                  21.3.9.29  The archimedean ordered field of real numbers   cnfldfld 33328
                  21.3.9.30  The quotient map and quotient modules   qusker 33334
                  21.3.9.31  The ring of integers modulo ` N `   znfermltl 33351
                  21.3.9.32  Independent sets and families   islinds5 33352
                  21.3.9.33  Ring associates, ring units   dvdsruassoi 33369
                  ​*21.3.9.34  Subgroup sum / Sumset / Minkowski sum   elgrplsmsn 33375
                  21.3.9.35  The quotient map   quslsm 33390
                  21.3.9.36  Ideals   lidlmcld 33404
                  21.3.9.37  Prime Ideals   cprmidl 33420
                  21.3.9.38  Maximal Ideals   cmxidl 33444
                  21.3.9.39  The semiring of ideals of a ring   cidlsrg 33485
                  21.3.9.40  Prime Elements   rprmval 33501
                  21.3.9.41  Unique factorization domains   cufd 33523
                  21.3.9.42  The ring of integers   zringidom 33536
                  21.3.9.43  Univariate Polynomials   0ringmon1p 33540
                  21.3.9.44  Polynomial quotient and polynomial remainder   q1pdir 33580
                  21.3.9.45  The subring algebra   sra1r 33589
                  21.3.9.46  Division Ring Extensions   drgext0g 33596
                  21.3.9.47  Vector Spaces   lvecdimfi 33602
                  21.3.9.48  Vector Space Dimension   cldim 33603
            21.3.10  Field Extensions   cfldext 33643
                  21.3.10.1  Algebraic numbers   cirng 33675
                  21.3.10.2  Minimal polynomials   cminply 33684
                  21.3.10.3  Quadratic Field Extensions   rtelextdg2lem 33709
                  21.3.10.4  Towers of quadratic extentions   fldext2chn 33711
            ​*21.3.11  Constructible Numbers   cconstr 33712
                  21.3.11.1  Impossible constructions   2sqr3minply 33730
            21.3.12  Matrices   csmat 33731
                  21.3.12.1  Submatrices   csmat 33731
                  21.3.12.2  Matrix literals   clmat 33749
                  21.3.12.3  Laplace expansion of determinants   mdetpmtr1 33761
            21.3.13  Topology   ist0cld 33771
                  21.3.13.1  Open maps   txomap 33772
                  21.3.13.2  Topology of the unit circle   qtopt1 33773
                  21.3.13.3  Refinements   reff 33777
                  21.3.13.4  Open cover refinement property   ccref 33780
                  21.3.13.5  Lindelöf spaces   cldlf 33790
                  21.3.13.6  Paracompact spaces   cpcmp 33793
                  ​*21.3.13.7  Spectrum of a ring   crspec 33800
                  21.3.13.8  Pseudometrics   cmetid 33824
                  21.3.13.9  Continuity - misc additions   hauseqcn 33836
                  21.3.13.10  Topology of the closed unit interval   elunitge0 33837
                  21.3.13.11  Topology of ` ( RR X. RR ) `   unicls 33841
                  21.3.13.12  Order topology - misc. additions   cnvordtrestixx 33851
                  21.3.13.13  Continuity in topological spaces - misc. additions   mndpluscn 33864
                  21.3.13.14  Topology of the extended nonnegative real numbers ordered monoid   xrge0hmph 33870
                  21.3.13.15  Limits - misc additions   lmlim 33885
                  21.3.13.16  Univariate polynomials   pl1cn 33893
            21.3.14  Uniform Stuctures and Spaces   chcmp 33894
                  21.3.14.1  Hausdorff uniform completion   chcmp 33894
            21.3.15  Topology and algebraic structures   zringnm 33896
                  21.3.15.1  The norm on the ring of the integer numbers   zringnm 33896
                  21.3.15.2  Topological ` ZZ ` -modules   zlm0 33898
                  21.3.15.3  Canonical embedding of the field of the rational numbers into a division ring   cqqh 33910
                  21.3.15.4  Canonical embedding of the real numbers into a complete ordered field   crrh 33931
                  21.3.15.5  Embedding from the extended real numbers into a complete lattice   cxrh 33954
                  21.3.15.6  Canonical embeddings into the ordered field of the real numbers   zrhre 33957
                  ​*21.3.15.7  Topological Manifolds   cmntop 33960
            21.3.16  Real and complex functions   nexple 33965
                  21.3.16.1  Integer powers - misc. additions   nexple 33965
                  21.3.16.2  Indicator Functions   cind 33966
                  21.3.16.3  Extended sum   cesum 33983
            21.3.17  Mixed Function/Constant operation   cofc 34051
            21.3.18  Abstract measure   csiga 34064
                  21.3.18.1  Sigma-Algebra   csiga 34064
                  21.3.18.2  Generated sigma-Algebra   csigagen 34094
                  ​*21.3.18.3  lambda and pi-Systems, Rings of Sets   ispisys 34108
                  21.3.18.4  The Borel algebra on the real numbers   cbrsiga 34137
                  21.3.18.5  Product Sigma-Algebra   csx 34144
                  21.3.18.6  Measures   cmeas 34151
                  21.3.18.7  The counting measure   cntmeas 34182
                  21.3.18.8  The Lebesgue measure - misc additions   voliune 34185
                  21.3.18.9  The Dirac delta measure   cdde 34188
                  21.3.18.10  The 'almost everywhere' relation   cae 34193
                  21.3.18.11  Measurable functions   cmbfm 34205
                  21.3.18.12  Borel Algebra on ` ( RR X. RR ) `   br2base 34226
                  ​*21.3.18.13  Caratheodory's extension theorem   coms 34248
            21.3.19  Integration   itgeq12dv 34283
                  21.3.19.1  Lebesgue integral - misc additions   itgeq12dv 34283
                  21.3.19.2  Bochner integral   citgm 34284
            21.3.20  Euler's partition theorem   oddpwdc 34311
            21.3.21  Sequences defined by strong recursion   csseq 34340
            21.3.22  Fibonacci Numbers   cfib 34353
            21.3.23  Probability   cprb 34364
                  21.3.23.1  Probability Theory   cprb 34364
                  21.3.23.2  Conditional Probabilities   ccprob 34388
                  21.3.23.3  Real-valued Random Variables   crrv 34397
                  21.3.23.4  Preimage set mapping operator   corvc 34412
                  21.3.23.5  Distribution Functions   orvcelval 34425
                  21.3.23.6  Cumulative Distribution Functions   orvclteel 34429
                  21.3.23.7  Probabilities - example   coinfliplem 34435
                  21.3.23.8  Bertrand's Ballot Problem   ballotlemoex 34442
            21.3.24  Signum (sgn or sign) function - misc. additions   sgncl 34495
                  21.3.24.1  Operations on words   ccatmulgnn0dir 34511
            21.3.25  Polynomials with real coefficients - misc additions   plymul02 34515
            21.3.26  Descartes's rule of signs   signspval 34521
                  21.3.26.1  Sign changes in a word over real numbers   signspval 34521
                  21.3.26.2  Counting sign changes in a word over real numbers   signslema 34531
            21.3.27  Number Theory   iblidicc 34561
                  21.3.27.1  Representations of a number as sums of integers   crepr 34577
                  21.3.27.2  Vinogradov Trigonometric Sums and the Circle Method   cvts 34604
                  21.3.27.3  The Ternary Goldbach Conjecture: Final Statement   ax-hgt749 34613
            21.3.28  Elementary Geometry   cstrkg2d 34633
                  ​*21.3.28.1  Two-dimensional geometry   cstrkg2d 34633
                  21.3.28.2  Outer Five Segment (not used, no need to move to main)   cafs 34638
            ​*21.3.29  LeftPad Project   clpad 34643
      ​​*21.4  Mathbox for Jonathan Ben-Naim
            21.4.1  First-order logic and set theory   bnj170 34666
            21.4.2  Well founded induction and recursion   bnj110 34826
            21.4.3  The existence of a minimal element in certain classes   bnj69 34978
            21.4.4  Well-founded induction   bnj1204 34980
            21.4.5  Well-founded recursion, part 1 of 3   bnj60 35030
            21.4.6  Well-founded recursion, part 2 of 3   bnj1500 35036
            21.4.7  Well-founded recursion, part 3 of 3   bnj1522 35040
      ​21.5  Mathbox for BTernaryTau
            21.5.1  First-order logic   nfan1c 35041
                  21.5.1.1  Auxiliary axiom schemes   nfan1c 35041
            21.5.2  ZF set theory   exdifsn 35047
                  21.5.2.1  Finitism   prcinf 35062
                  21.5.2.2  Global choice   gblacfnacd 35067
            21.5.3  Real and complex numbers   zltp1ne 35069
            21.5.4  Graph theory   lfuhgr 35077
                  21.5.4.1  Acyclic graphs   cacycgr 35102
      ​21.6  Mathbox for Mario Carneiro
            21.6.1  Predicate calculus with all distinct variables   ax-7d 35119
            21.6.2  Miscellaneous stuff   quartfull 35125
            21.6.3  Derangements and the Subfactorial   deranglem 35126
            21.6.4  The Erdős-Szekeres theorem   erdszelem1 35151
            21.6.5  The Kuratowski closure-complement theorem   kur14lem1 35166
            21.6.6  Retracts and sections   cretr 35177
            21.6.7  Path-connected and simply connected spaces   cpconn 35179
            21.6.8  Covering maps   ccvm 35215
            21.6.9  Normal numbers   snmlff 35289
            21.6.10  Godel-sets of formulas - part 1   cgoe 35293
            21.6.11  Godel-sets of formulas - part 2   cgon 35392
            21.6.12  Models of ZF   cgze 35406
            ​*21.6.13  Metamath formal systems   cmcn 35420
            21.6.14  Grammatical formal systems   cm0s 35545
            21.6.15  Models of formal systems   cmuv 35565
            21.6.16  Splitting fields   ccpms 35587
            21.6.17  p-adic number fields   czr 35607
      ​​*21.7  Mathbox for Filip Cernatescu
      ​21.8  Mathbox for Paul Chapman
            21.8.1  Real and complex numbers (cont.)   climuzcnv 35631
            21.8.2  Miscellaneous theorems   elfzm12 35635
      ​21.9  Mathbox for Hongxiu Chen
      ​21.10  Mathbox for Adrian Ducourtial
            21.10.1  Propositional calculus   currybi 35648
            21.10.2  Clone theory   ccloneop 35649
      ​21.11  Mathbox for Scott Fenton
            21.11.1  ZFC Axioms in primitive form   axextprim 35655
            21.11.2  Untangled classes   untelirr 35662
            21.11.3  Extra propositional calculus theorems   3jaodd 35669
            21.11.4  Misc. Useful Theorems   nepss 35672
            21.11.5  Properties of real and complex numbers   sqdivzi 35682
            21.11.6  Infinite products   iprodefisumlem 35694
            21.11.7  Factorial limits   faclimlem1 35697
            21.11.8  Greatest common divisor and divisibility   gcd32 35703
            21.11.9  Properties of relationships   dftr6 35705
            21.11.10  Properties of functions and mappings   funpsstri 35721
            21.11.11  Set induction (or epsilon induction)   setinds 35734
            21.11.12  Ordinal numbers   elpotr 35737
            21.11.13  Defined equality axioms   axextdfeq 35753
            21.11.14  Hypothesis builders   hbntg 35761
            21.11.15  Well-founded zero, successor, and limits   cwsuc 35766
            21.11.16  Quantifier-free definitions   ctxp 35786
            21.11.17  Alternate ordered pairs   caltop 35912
            21.11.18  Geometry in the Euclidean space   cofs 35938
                  21.11.18.1  Congruence properties   cofs 35938
                  21.11.18.2  Betweenness properties   btwntriv2 35968
                  21.11.18.3  Segment Transportation   ctransport 35985
                  21.11.18.4  Properties relating betweenness and congruence   cifs 35991
                  21.11.18.5  Connectivity of betweenness   btwnconn1lem1 36043
                  21.11.18.6  Segment less than or equal to   csegle 36062
                  21.11.18.7  Outside-of relationship   coutsideof 36075
                  21.11.18.8  Lines and Rays   cline2 36090
            21.11.19  Forward difference   cfwddif 36114
            21.11.20  Rank theorems   rankung 36122
            21.11.21  Hereditarily Finite Sets   chf 36128
      ​21.12  Mathbox for Gino Giotto
            21.12.1  Equality theorems.   rmoeqi 36143
                  21.12.1.1  Inference versions.   rmoeqi 36143
                  21.12.1.2  Deduction versions.   rmoeqdv 36168
            21.12.2  Change bound variables.   in-ax8 36182
                  21.12.2.1  Change bound variables and domains.   cbvralvw2 36184
                  21.12.2.2  Change bound variables, deduction versions.   cbvmodavw 36208
                  21.12.2.3  Change bound variables and domains, deduction versions.   cbvrmodavw2 36241
            21.12.3  Study of ax-mulf usage.   mpomulnzcnf 36257
      ​21.13  Mathbox for Jeff Hankins
            21.13.1  Miscellany   a1i14 36258
            21.13.2  Basic topological facts   topbnd 36282
            21.13.3  Topology of the real numbers   ivthALT 36293
            21.13.4  Refinements   cfne 36294
            21.13.5  Neighborhood bases determine topologies   neibastop1 36317
            21.13.6  Lattice structure of topologies   topmtcl 36321
            21.13.7  Filter bases   fgmin 36328
            21.13.8  Directed sets, nets   tailfval 36330
      ​21.14  Mathbox for Anthony Hart
            21.14.1  Propositional Calculus   tb-ax1 36341
            21.14.2  Predicate Calculus   nalfal 36361
            21.14.3  Miscellaneous single axioms   meran1 36369
            21.14.4  Connective Symmetry   negsym1 36375
      ​21.15  Mathbox for Chen-Pang He
            21.15.1  Ordinal topology   ontopbas 36386
      ​21.16  Mathbox for Jeff Hoffman
            21.16.1  Inferences for finite induction on generic function values   fveleq 36409
            21.16.2  gdc.mm   nnssi2 36413
      ​21.17  Mathbox for Matthew House
            21.17.1  Relations on well-ordered indexed unions   weiunlem1 36420
      ​21.18  Mathbox for Asger C. Ipsen
            21.18.1  Continuous nowhere differentiable functions   dnival 36429
      ​​*21.19  Mathbox for BJ
            *21.19.1  Propositional calculus   bj-mp2c 36498
                  *21.19.1.1  Derived rules of inference   bj-mp2c 36498
                  ​*21.19.1.2  A syntactic theorem   bj-0 36500
                  21.19.1.3  Minimal implicational calculus   bj-a1k 36502
                  ​*21.19.1.4  Positive calculus   bj-syl66ib 36513
                  21.19.1.5  Implication and negation   bj-con2com 36519
                  ​*21.19.1.6  Disjunction   bj-jaoi1 36529
                  ​*21.19.1.7  Logical equivalence   bj-dfbi4 36531
                  21.19.1.8  The conditional operator for propositions   bj-consensus 36536
                  ​*21.19.1.9  Propositional calculus: miscellaneous   bj-imbi12 36541
            ​*21.19.2  Modal logic   bj-axdd2 36550
            ​*21.19.3  Provability logic   cprvb 36555
            ​*21.19.4  First-order logic   bj-genr 36564
                  21.19.4.1  Adding ax-gen   bj-genr 36564
                  21.19.4.2  Adding ax-4   bj-2alim 36568
                  21.19.4.3  Adding ax-5   bj-ax12wlem 36602
                  21.19.4.4  Equality and substitution   bj-ssbeq 36611
                  21.19.4.5  Adding ax-6   bj-spimvwt 36627
                  21.19.4.6  Adding ax-7   bj-cbvexw 36634
                  21.19.4.7  Membership predicate, ax-8 and ax-9   bj-ax89 36636
                  21.19.4.8  Adding ax-11   bj-alcomexcom 36638
                  21.19.4.9  Adding ax-12   axc11n11 36640
                  21.19.4.10  Nonfreeness   wnnf 36681
                  21.19.4.11  Adding ax-13   bj-axc10 36741
                  ​*21.19.4.12  Removing dependencies on ax-13 (and ax-11)   bj-axc10v 36751
                  ​*21.19.4.13  Distinct var metavariables   bj-hbaeb2 36776
                  ​*21.19.4.14  Around ~ equsal   bj-equsal1t 36780
                  ​*21.19.4.15  Some Principia Mathematica proofs   stdpc5t 36785
                  21.19.4.16  Alternate definition of substitution   bj-sbsb 36795
                  21.19.4.17  Lemmas for substitution   bj-sbf3 36797
                  21.19.4.18  Existential uniqueness   bj-eu3f 36799
                  ​*21.19.4.19  First-order logic: miscellaneous   bj-sblem1 36800
            21.19.5  Set theory   eliminable1 36817
                  ​*21.19.5.1  Eliminability of class terms   eliminable1 36817
                  ​*21.19.5.2  Classes without the axiom of extensionality   bj-denoteslem 36829
                  21.19.5.3  Characterization among sets versus among classes   elelb 36855
                  ​*21.19.5.4  The nonfreeness quantifier for classes   bj-nfcsym 36857
                  ​*21.19.5.5  Lemmas for class substitution   bj-sbeqALT 36858
                  21.19.5.6  Removing some axiom requirements and disjoint variable conditions   bj-exlimvmpi 36869
                  ​*21.19.5.7  Class abstractions   bj-elabd2ALT 36883
                  21.19.5.8  Generalized class abstractions   bj-cgab 36891
                  ​*21.19.5.9  Restricted nonfreeness   wrnf 36899
                  ​*21.19.5.10  Russell's paradox   bj-ru1 36901
                  21.19.5.11  Curry's paradox in set theory   currysetlem 36903
                  ​*21.19.5.12  Some disjointness results   bj-n0i 36909
                  ​*21.19.5.13  Complements on direct products   bj-xpimasn 36913
                  ​*21.19.5.14  "Singletonization" and tagging   bj-snsetex 36921
                  ​*21.19.5.15  Tuples of classes   bj-cproj 36948
                  ​*21.19.5.16  Set theory: elementary operations relative to a universe   bj-rcleqf 36983
                  ​*21.19.5.17  Axioms for finite unions   bj-abex 36988
                  ​*21.19.5.18  Set theory: miscellaneous   eleq2w2ALT 37005
                  ​*21.19.5.19  Evaluation at a class   bj-evaleq 37030
                  21.19.5.20  Elementwise operations   celwise 37037
                  ​*21.19.5.21  Elementwise intersection (families of sets induced on a subset)   bj-rest00 37039
                  21.19.5.22  Moore collections (complements)   bj-raldifsn 37058
                  21.19.5.23  Maps-to notation for functions with three arguments   bj-0nelmpt 37074
                  ​*21.19.5.24  Currying   csethom 37080
                  ​*21.19.5.25  Setting components of extensible structures   cstrset 37092
            ​*21.19.6  Extended real and complex numbers, real and complex projective lines   bj-nfald 37095
                  21.19.6.1  Complements on class abstractions of ordered pairs and binary relations   bj-nfald 37095
                  ​*21.19.6.2  Identity relation (complements)   bj-opabssvv 37108
                  ​*21.19.6.3  Functionalized identity (diagonal in a Cartesian square)   cdiag2 37130
                  ​*21.19.6.4  Direct image and inverse image   cimdir 37136
                  ​*21.19.6.5  Extended numbers and projective lines as sets   cfractemp 37154
                  ​*21.19.6.6  Addition and opposite   caddcc 37195
                  ​*21.19.6.7  Order relation on the extended reals   cltxr 37199
                  ​*21.19.6.8  Argument, multiplication and inverse   carg 37201
                  21.19.6.9  The canonical bijection from the finite ordinals   ciomnn 37207
                  21.19.6.10  Divisibility   cnnbar 37218
            ​*21.19.7  Monoids   bj-smgrpssmgm 37226
                  ​*21.19.7.1  Finite sums in monoids   cfinsum 37241
            ​*21.19.8  Affine, Euclidean, and Cartesian geometry   bj-fvimacnv0 37244
                  *21.19.8.1  Real vector spaces   bj-fvimacnv0 37244
                  ​*21.19.8.2  Complex numbers (supplements)   bj-subcom 37266
                  ​*21.19.8.3  Barycentric coordinates   bj-bary1lem 37268
            21.19.9  Monoid of endomorphisms   cend 37271
      ​21.20  Mathbox for Jim Kingdon
                  21.20.0.1  Circle constant   taupilem3 37277
                  21.20.0.2  Number theory   dfgcd3 37282
                  21.20.0.3  Real numbers   irrdifflemf 37283
      ​21.21  Mathbox for ML
            21.21.1  Miscellaneous   csbrecsg 37286
            21.21.2  Cartesian exponentiation   cfinxp 37341
            21.21.3  Topology   iunctb2 37361
                  ​*21.21.3.1  Pi-base theorems   pibp16 37371
      ​21.22  Mathbox for Wolf Lammen
            21.22.1  1. Bootstrapping   wl-section-boot 37380
            21.22.2  Implication chains   wl-section-impchain 37404
            21.22.3  Theorems around the conditional operator   wl-ifp-ncond1 37422
            21.22.4  Alternative development of hadd, cadd   wl-df-3xor 37426
            21.22.5  An alternative axiom ~ ax-13   ax-wl-13v 37451
            21.22.6  Other stuff   wl-mps 37453
      ​21.23  Mathbox for Brendan Leahy
      ​21.24  Mathbox for Jeff Madsen
            21.24.1  Logic and set theory   unirep 37666
            21.24.2  Real and complex numbers; integers   filbcmb 37692
            21.24.3  Sequences and sums   sdclem2 37694
            21.24.4  Topology   subspopn 37704
            21.24.5  Metric spaces   metf1o 37707
            21.24.6  Continuous maps and homeomorphisms   constcncf 37714
            21.24.7  Boundedness   ctotbnd 37718
            21.24.8  Isometries   cismty 37750
            21.24.9  Heine-Borel Theorem   heibor1lem 37761
            21.24.10  Banach Fixed Point Theorem   bfplem1 37774
            21.24.11  Euclidean space   crrn 37777
            21.24.12  Intervals (continued)   ismrer1 37790
            21.24.13  Operation properties   cass 37794
            21.24.14  Groups and related structures   cmagm 37800
            21.24.15  Group homomorphism and isomorphism   cghomOLD 37835
            21.24.16  Rings   crngo 37846
            21.24.17  Division Rings   cdrng 37900
            21.24.18  Ring homomorphisms   crngohom 37912
            21.24.19  Commutative rings   ccm2 37941
            21.24.20  Ideals   cidl 37959
            21.24.21  Prime rings and integral domains   cprrng 37998
            21.24.22  Ideal generators   cigen 38011
      ​21.25  Mathbox for Giovanni Mascellani
            ​*21.25.1  Tools for automatic proof building   efald2 38030
            ​*21.25.2  Tseitin axioms   fald 38081
            ​*21.25.3  Equality deductions   iuneq2f 38108
            ​*21.25.4  Miscellanea   orcomdd 38119
      ​21.26  Mathbox for Peter Mazsa
            21.26.1  Notations   cxrn 38126
            21.26.2  Preparatory theorems   el2v1 38169
            21.26.3  Range Cartesian product   df-xrn 38319
            21.26.4  Cosets by ` R `   df-coss 38359
            21.26.5  Relations   df-rels 38433
            21.26.6  Subset relations   df-ssr 38446
            21.26.7  Reflexivity   df-refs 38458
            21.26.8  Converse reflexivity   df-cnvrefs 38473
            21.26.9  Symmetry   df-syms 38490
            21.26.10  Reflexivity and symmetry   symrefref2 38511
            21.26.11  Transitivity   df-trs 38520
            21.26.12  Equivalence relations   df-eqvrels 38532
            21.26.13  Redundancy   df-redunds 38571
            21.26.14  Domain quotients   df-dmqss 38586
            21.26.15  Equivalence relations on domain quotients   df-ers 38611
            21.26.16  Functions   df-funss 38628
            21.26.17  Disjoints vs. converse functions   df-disjss 38651
            21.26.18  Antisymmetry   df-antisymrel 38708
            21.26.19  Partitions: disjoints on domain quotients   df-parts 38713
            21.26.20  Partition-Equivalence Theorems   disjim 38729
      ​21.27  Mathbox for Rodolfo Medina
            21.27.1  Partitions   prtlem60 38801
      ​​*21.28  Mathbox for Norm Megill
            *21.28.1  Obsolete schemes ax-c4,c5,c7,c10,c11,c11n,c15,c9,c14,c16   ax-c5 38831
            ​*21.28.2  Rederive new axioms ax-4, ax-10, ax-6, ax-12, ax-13 from old   axc5 38841
            ​*21.28.3  Legacy theorems using obsolete axioms   ax5ALT 38855
            21.28.4  Experiments with weak deduction theorem   elimhyps 38909
            21.28.5  Miscellanea   cnaddcom 38920
            21.28.6  Atoms, hyperplanes, and covering in a left vector space (or module)   clsa 38922
            21.28.7  Functionals and kernels of a left vector space (or module)   clfn 39005
            21.28.8  Opposite rings and dual vector spaces   cld 39071
            21.28.9  Ortholattices and orthomodular lattices   cops 39120
            21.28.10  Atomic lattices with covering property   ccvr 39210
            21.28.11  Hilbert lattices   chlt 39298
            21.28.12  Projective geometries based on Hilbert lattices   clln 39440
            21.28.13  Construction of a vector space from a Hilbert lattice   cdlema1N 39740
            21.28.14  Construction of involution and inner product from a Hilbert lattice   clpoN 41429
      ​21.29  Mathbox for metakunt
            21.29.1  Commutative Semiring   ccsrg 41915
            21.29.2  General helpful statements   rhmzrhval 41918
            21.29.3  Some gcd and lcm results   12gcd5e1 41952
            21.29.4  Least common multiple inequality theorem   3factsumint1 41970
            21.29.5  Logarithm inequalities   3exp7 42002
            21.29.6  Miscellaneous results for AKS formalisation   intlewftc 42010
            21.29.7  Sticks and stones   sticksstones1 42095
            21.29.8  Continuation AKS   aks6d1c6lem1 42119
            21.29.9  Permutation results   metakunt1 42154
            21.29.10  Unused lemmas scheduled for deletion   fac2xp3 42188
      ​21.30  Mathbox for Steven Nguyen
            21.30.1  Utility theorems   intnanrt 42192
            ​*21.30.2  Arithmetic theorems   c0exALT 42239
            21.30.3  Exponents and divisibility   oexpreposd 42301
            21.30.4  Trigonometry   tanhalfpim 42329
            21.30.5  Real subtraction   cresub 42333
            21.30.6  Structures   nelsubginvcld 42443
            ​*21.30.7  Projective spaces   cprjsp 42548
            21.30.8  Basic reductions for Fermat's Last Theorem   dffltz 42581
            ​*21.30.9  Exemplar theorems   iddii 42611
                  ​*21.30.9.1  Standard replacements of ax-10 , ax-11 , ax-12   nfa1w 42622
      ​21.31  Mathbox for Igor Ieskov
      ​21.32  Mathbox for OpenAI
      ​21.33  Mathbox for Stefan O'Rear
            21.33.1  Additional elementary logic and set theory   moxfr 42640
            21.33.2  Additional theory of functions   imaiinfv 42641
            21.33.3  Additional topology   elrfi 42642
            21.33.4  Characterization of closure operators. Kuratowski closure axioms   ismrcd1 42646
            21.33.5  Algebraic closure systems   cnacs 42650
            21.33.6  Miscellanea 1. Map utilities   constmap 42661
            21.33.7  Miscellanea for polynomials   mptfcl 42668
            21.33.8  Multivariate polynomials over the integers   cmzpcl 42669
            21.33.9  Miscellanea for Diophantine sets 1   coeq0i 42701
            21.33.10  Diophantine sets 1: definitions   cdioph 42703
            21.33.11  Diophantine sets 2 miscellanea   ellz1 42715
            21.33.12  Diophantine sets 2: union and intersection. Monotone Boolean algebra   diophin 42720
            21.33.13  Diophantine sets 3: construction   diophrex 42723
            21.33.14  Diophantine sets 4 miscellanea   2sbcrex 42732
            21.33.15  Diophantine sets 4: Quantification   rexrabdioph 42742
            21.33.16  Diophantine sets 5: Arithmetic sets   rabdiophlem1 42749
            21.33.17  Diophantine sets 6: reusability. renumbering of variables   eldioph4b 42759
            21.33.18  Pigeonhole Principle and cardinality helpers   fphpd 42764
            21.33.19  A non-closed set of reals is infinite   rencldnfilem 42768
            21.33.20  Lagrange's rational approximation theorem   irrapxlem1 42770
            21.33.21  Pell equations 1: A nontrivial solution always exists   pellexlem1 42777
            21.33.22  Pell equations 2: Algebraic number theory of the solution set   csquarenn 42784
            21.33.23  Pell equations 3: characterizing fundamental solution   infmrgelbi 42826
            ​*21.33.24  Logarithm laws generalized to an arbitrary base   reglogcl 42838
            21.33.25  Pell equations 4: the positive solution group is infinite cyclic   pellfund14 42846
            21.33.26  X and Y sequences 1: Definition and recurrence laws   crmx 42848
            21.33.27  Ordering and induction lemmas for the integers   monotuz 42890
            21.33.28  X and Y sequences 2: Order properties   rmxypos 42896
            21.33.29  Congruential equations   congtr 42914
            21.33.30  Alternating congruential equations   acongid 42924
            21.33.31  Additional theorems on integer divisibility   coprmdvdsb 42934
            21.33.32  X and Y sequences 3: Divisibility properties   jm2.18 42937
            21.33.33  X and Y sequences 4: Diophantine representability of Y   jm2.27a 42954
            21.33.34  X and Y sequences 5: Diophantine representability of X, ^, _C   rmxdiophlem 42964
            21.33.35  Uncategorized stuff not associated with a major project   setindtr 42973
            21.33.36  More equivalents of the Axiom of Choice   axac10 42982
            21.33.37  Finitely generated left modules   clfig 43019
            21.33.38  Noetherian left modules I   clnm 43027
            21.33.39  Addenda for structure powers   pwssplit4 43041
            21.33.40  Every set admits a group structure iff choice   unxpwdom3 43047
            21.33.41  Noetherian rings and left modules II   clnr 43061
            21.33.42  Hilbert's Basis Theorem   cldgis 43073
            21.33.43  Additional material on polynomials [DEPRECATED]   cmnc 43083
            21.33.44  Degree and minimal polynomial of algebraic numbers   cdgraa 43092
            21.33.45  Algebraic integers I   citgo 43109
            21.33.46  Endomorphism algebra   cmend 43127
            21.33.47  Cyclic groups and order   idomodle 43147
            21.33.48  Cyclotomic polynomials   ccytp 43153
            21.33.49  Miscellaneous topology   fgraphopab 43159
      ​21.34  Mathbox for Noam Pasman
      ​21.35  Mathbox for Jon Pennant
      ​21.36  Mathbox for Richard Penner
            21.36.1  Set Theory and Ordinal Numbers   uniel 43173
            21.36.2  Natural addition of Cantor normal forms   oawordex2 43283
            21.36.3  Surreal Contributions   abeqabi 43365
            21.36.4  Short Studies   nlimsuc 43398
                  21.36.4.1  Additional work on conditional logical operator   ifpan123g 43416
                  21.36.4.2  Sophisms   rp-fakeimass 43469
                  ​*21.36.4.3  Finite Sets   rp-isfinite5 43474
                  21.36.4.4  General Observations   intabssd 43476
                  21.36.4.5  Infinite Sets   pwelg 43517
                  ​*21.36.4.6  Finite intersection property   fipjust 43522
                  21.36.4.7  RP ADDTO: Subclasses and subsets   rababg 43531
                  21.36.4.8  RP ADDTO: The intersection of a class   elinintab 43532
                  21.36.4.9  RP ADDTO: Theorems requiring subset and intersection existence   elinintrab 43534
                  21.36.4.10  RP ADDTO: Relations   xpinintabd 43537
                  ​*21.36.4.11  RP ADDTO: Functions   elmapintab 43553
                  ​*21.36.4.12  RP ADDTO: Finite induction (for finite ordinals)   cnvcnvintabd 43557
                  21.36.4.13  RP ADDTO: First and second members of an ordered pair   elcnvlem 43558
                  21.36.4.14  RP ADDTO: The reflexive and transitive properties of relations   undmrnresiss 43561
                  21.36.4.15  RP ADDTO: Basic properties of closures   cleq2lem 43565
                  21.36.4.16  RP REPLACE: Definitions and basic properties of transitive closures   trcleq2lemRP 43587
                  ​*21.36.4.17  Additions for square root; absolute value   sqrtcvallem1 43588
            21.36.5  Additional statements on relations and subclasses   al3im 43604
                  21.36.5.1  Transitive relations (not to be confused with transitive classes).   trrelind 43622
                  21.36.5.2  Reflexive closures   crcl 43629
                  ​*21.36.5.3  Finite relationship composition.   relexp2 43634
                  21.36.5.4  Transitive closure of a relation   dftrcl3 43677
                  ​*21.36.5.5  Adapted from Frege   frege77d 43703
            ​*21.36.6  Propositions from _Begriffsschrift_   dfxor4 43723
                  *21.36.6.1  _Begriffsschrift_ Chapter I   dfxor4 43723
                  ​*21.36.6.2  _Begriffsschrift_ Notation hints   whe 43729
                  21.36.6.3  _Begriffsschrift_ Chapter II Implication   ax-frege1 43747
                  21.36.6.4  _Begriffsschrift_ Chapter II Implication and Negation   axfrege28 43786
                  ​*21.36.6.5  _Begriffsschrift_ Chapter II with logical equivalence   axfrege52a 43813
                  21.36.6.6  _Begriffsschrift_ Chapter II with equivalence of sets   axfrege52c 43844
                  ​*21.36.6.7  _Begriffsschrift_ Chapter II with equivalence of classes   frege53c 43871
                  ​*21.36.6.8  _Begriffsschrift_ Chapter III Properties hereditary in a sequence   dffrege69 43889
                  ​*21.36.6.9  _Begriffsschrift_ Chapter III Following in a sequence   dffrege76 43896
                  ​*21.36.6.10  _Begriffsschrift_ Chapter III Member of sequence   dffrege99 43919
                  ​*21.36.6.11  _Begriffsschrift_ Chapter III Single-valued procedures   dffrege115 43935
            ​*21.36.7  Exploring Topology via Seifert and Threlfall   enrelmap 43954
                  *21.36.7.1  Equinumerosity of sets of relations and maps   enrelmap 43954
                  ​*21.36.7.2  Generic Pseudoclosure Spaces, Pseudointerior Spaces, and Pseudoneighborhoods   or3or 43980
                  ​*21.36.7.3  Generic Neighborhood Spaces   gneispa 44087
            ​*21.36.8  Exploring Higher Homotopy via Kerodon   k0004lem1 44104
                  *21.36.8.1  Simplicial Sets   k0004lem1 44104
      ​21.37  Mathbox for Stanislas Polu
            21.37.1  IMO Problems   wwlemuld 44113
                  21.37.1.1  IMO 1972 B2   wwlemuld 44113
            ​*21.37.2  INT Inequalities Proof Generator   int-addcomd 44130
            ​*21.37.3  N-Digit Addition Proof Generator   unitadd 44152
            21.37.4  AM-GM (for k = 2,3,4)   gsumws3 44153
      ​21.38  Mathbox for Rohan Ridenour
            21.38.1  Misc   spALT 44158
            21.38.2  Monoid rings   cmnring 44170
            21.38.3  Shorter primitive equivalent of ax-groth   gru0eld 44193
                  21.38.3.1  Grothendieck universes are closed under collection   gru0eld 44193
                  21.38.3.2  Minimal universes   ismnu 44225
                  21.38.3.3  Primitive equivalent of ax-groth   expandan 44252
      ​21.39  Mathbox for Steve Rodriguez
            21.39.1  Miscellanea   nanorxor 44269
            21.39.2  Ratio test for infinite series convergence and divergence   dvgrat 44276
            21.39.3  Multiples   reldvds 44279
            21.39.4  Function operations   caofcan 44287
            21.39.5  Calculus   lhe4.4ex1a 44293
            21.39.6  The generalized binomial coefficient operation   cbcc 44300
            21.39.7  Binomial series   uzmptshftfval 44310
      ​21.40  Mathbox for Andrew Salmon
            21.40.1  Principia Mathematica * 10   pm10.12 44322
            21.40.2  Principia Mathematica * 11   2alanimi 44336
            21.40.3  Predicate Calculus   sbeqal1 44362
            21.40.4  Principia Mathematica * 13 and * 14   pm13.13a 44371
            21.40.5  Set Theory   elnev 44402
            21.40.6  Arithmetic   addcomgi 44420
            21.40.7  Geometry   cplusr 44421
      ​​*21.41  Mathbox for Alan Sare
            21.41.1  Auxiliary theorems for the Virtual Deduction tool   idiALT 44443
            21.41.2  Supplementary unification deductions   bi1imp 44447
            21.41.3  Conventional Metamath proofs, some derived from VD proofs   iidn3 44467
            21.41.4  What is Virtual Deduction?   wvd1 44535
            21.41.5  Virtual Deduction Theorems   df-vd1 44536
            21.41.6  Theorems proved using Virtual Deduction   trsspwALT 44784
            21.41.7  Theorems proved using Virtual Deduction with mmj2 assistance   simplbi2VD 44812
            21.41.8  Virtual Deduction transcriptions of textbook proofs   sb5ALTVD 44879
            21.41.9  Theorems proved using conjunction-form Virtual Deduction   elpwgdedVD 44883
            21.41.10  Theorems with a VD proof in conventional notation derived from a VD proof   suctrALT3 44890
            ​*21.41.11  Theorems with a proof in conventional notation derived from a VD proof   notnotrALT2 44893
      ​21.42  Mathbox for Glauco Siliprandi
            21.42.1  Miscellanea   evth2f 44904
            21.42.2  Functions   feq1dd 45063
            21.42.3  Ordering on real numbers - Real and complex numbers basic operations   sub2times 45176
            21.42.4  Real intervals   gtnelioc 45398
            21.42.5  Finite sums   fsummulc1f 45481
            21.42.6  Finite multiplication of numbers and finite multiplication of functions   fmul01 45490
            21.42.7  Limits   clim1fr1 45511
                  21.42.7.1  Inferior limit (lim inf)   clsi 45661
                  ​*21.42.7.2  Limits for sequences of extended real numbers   clsxlim 45728
            21.42.8  Trigonometry   coseq0 45774
            21.42.9  Continuous Functions   mulcncff 45780
            21.42.10  Derivatives   dvsinexp 45821
            21.42.11  Integrals   itgsin0pilem1 45860
            21.42.12  Stone Weierstrass theorem - real version   stoweidlem1 45911
            21.42.13  Wallis' product for π   wallispilem1 45975
            21.42.14  Stirling's approximation formula for ` n ` factorial   stirlinglem1 45984
            21.42.15  Dirichlet kernel   dirkerval 46001
            21.42.16  Fourier Series   fourierdlem1 46018
            21.42.17  e is transcendental   elaa2lem 46143
            21.42.18  n-dimensional Euclidean space   rrxtopn 46194
            21.42.19  Basic measure theory   csalg 46218
                  ​*21.42.19.1  σ-Algebras   csalg 46218
                  21.42.19.2  Sum of nonnegative extended reals   csumge0 46272
                  ​*21.42.19.3  Measures   cmea 46359
                  ​*21.42.19.4  Outer measures and Caratheodory's construction   come 46399
                  ​*21.42.19.5  Lebesgue measure on n-dimensional Real numbers   covoln 46446
                  ​*21.42.19.6  Measurable functions   csmblfn 46605
      ​21.43  Mathbox for Saveliy Skresanov
            21.43.1  Ceva's theorem   sigarval 46760
            21.43.2  Simple groups   simpcntrab 46780
      ​21.44  Mathbox for Ender Ting
            21.44.1  Increasing sequences and subsequences   et-ltneverrefl 46781
      ​21.45  Mathbox for Jarvin Udandy
      ​21.46  Mathbox for Adhemar
            ​*21.46.1  Minimal implicational calculus   adh-minim 46905
      ​21.47  Mathbox for Alexander van der Vekens
            21.47.1  General auxiliary theorems (1)   n0nsn2el 46929
                  21.47.1.1  Unordered and ordered pairs - extension for singletons   n0nsn2el 46929
                  21.47.1.2  Unordered and ordered pairs - extension for unordered pairs   elprneb 46933
                  21.47.1.3  Unordered and ordered pairs - extension for ordered pairs   oppr 46934
                  21.47.1.4  Relations - extension   eubrv 46939
                  21.47.1.5  Definite description binder (inverted iota) - extension   iota0def 46942
                  21.47.1.6  Functions - extension   fveqvfvv 46944
            21.47.2  Alternative for Russell's definition of a description binder   caiota 46987
            21.47.3  Double restricted existential uniqueness   r19.32 47002
                  21.47.3.1  Restricted quantification (extension)   r19.32 47002
                  21.47.3.2  Restricted uniqueness and "at most one" quantification   reuf1odnf 47011
                  21.47.3.3  Analogs to Existential uniqueness (double quantification)   2reu3 47014
                  21.47.3.4  Additional theorems for double restricted existential uniqueness   2reu8i 47017
            ​*21.47.4  Alternative definitions of function and operation values   wdfat 47020
                  21.47.4.1  Restricted quantification (extension)   ralbinrald 47026
                  21.47.4.2  The universal class (extension)   nvelim 47027
                  21.47.4.3  Introduce the Axiom of Power Sets (extension)   alneu 47028
                  21.47.4.4  Predicate "defined at"   dfateq12d 47030
                  21.47.4.5  Alternative definition of the value of a function   dfafv2 47036
                  21.47.4.6  Alternative definition of the value of an operation   aoveq123d 47082
            ​*21.47.5  Alternative definitions of function values (2)   cafv2 47112
            21.47.6  General auxiliary theorems (2)   an4com24 47172
                  21.47.6.1  Logical conjunction - extension   an4com24 47172
                  21.47.6.2  Abbreviated conjunction and disjunction of three wff's - extension   3an4ancom24 47173
                  21.47.6.3  Negated membership (alternative)   cnelbr 47175
                  21.47.6.4  The empty set - extension   ralralimp 47182
                  21.47.6.5  Indexed union and intersection - extension   otiunsndisjX 47183
                  21.47.6.6  Functions - extension   fvifeq 47184
                  21.47.6.7  Maps-to notation - extension   fvmptrab 47196
                  21.47.6.8  Subtraction - extension   cnambpcma 47198
                  21.47.6.9  Ordering on reals (cont.) - extension   leaddsuble 47201
                  21.47.6.10  Imaginary and complex number properties - extension   readdcnnred 47207
                  21.47.6.11  Nonnegative integers (as a subset of complex numbers) - extension   nn0resubcl 47212
                  21.47.6.12  Integers (as a subset of complex numbers) - extension   zgeltp1eq 47213
                  21.47.6.13  Decimal arithmetic - extension   1t10e1p1e11 47214
                  21.47.6.14  Upper sets of integers - extension   eluzge0nn0 47216
                  21.47.6.15  Infinity and the extended real number system (cont.) - extension   nltle2tri 47217
                  21.47.6.16  Finite intervals of integers - extension   ssfz12 47218
                  21.47.6.17  Half-open integer ranges - extension   fzopred 47226
                  21.47.6.18  The modulo (remainder) operation - extension   m1mod0mod1 47232
                  21.47.6.19  The infinite sequence builder "seq"   smonoord 47234
                  21.47.6.20  Finite and infinite sums - extension   fsummsndifre 47235
                  21.47.6.21  Extensible structures - extension   setsidel 47239
            ​*21.47.7  Preimages of function values   preimafvsnel 47242
            ​*21.47.8  Partitions of real intervals   ciccp 47276
            21.47.9  Shifting functions with an integer range domain   fargshiftfv 47302
            21.47.10  Words over a set (extension)   lswn0 47307
                  21.47.10.1  Last symbol of a word - extension   lswn0 47307
            21.47.11  Unordered pairs   wich 47308
                  21.47.11.1  Interchangeable setvar variables   wich 47308
                  21.47.11.2  Set of unordered pairs   sprid 47337
                  ​*21.47.11.3  Proper (unordered) pairs   prpair 47364
                  21.47.11.4  Set of proper unordered pairs   cprpr 47375
            21.47.12  Number theory (extension)   cfmtno 47390
                  ​*21.47.12.1  Fermat numbers   cfmtno 47390
                  ​*21.47.12.2  Mersenne primes   m2prm 47454
                  21.47.12.3  Proth's theorem   modexp2m1d 47475
                  21.47.12.4  Solutions of quadratic equations   quad1 47483
            ​*21.47.13  Even and odd numbers   ceven 47487
                  21.47.13.1  Definitions and basic properties   ceven 47487
                  21.47.13.2  Alternate definitions using the "divides" relation   dfeven2 47512
                  21.47.13.3  Alternate definitions using the "modulo" operation   dfeven3 47521
                  21.47.13.4  Alternate definitions using the "gcd" operation   iseven5 47527
                  21.47.13.5  Theorems of part 5 revised   zneoALTV 47532
                  21.47.13.6  Theorems of part 6 revised   odd2np1ALTV 47537
                  21.47.13.7  Theorems of AV's mathbox revised   0evenALTV 47551
                  21.47.13.8  Additional theorems   epoo 47566
                  21.47.13.9  Perfect Number Theorem (revised)   perfectALTVlem1 47584
            21.47.14  Number theory (extension 2)   cfppr 47587
                  ​*21.47.14.1  Fermat pseudoprimes   cfppr 47587
                  ​*21.47.14.2  Goldbach's conjectures   cgbe 47608
            21.47.15  Graph theory (extension)   cclnbgr 47681
                  21.47.15.1  Closed neighborhood of a vertex   cclnbgr 47681
                  ​*21.47.15.2  Semiclosed and semiopen neighborhoods (experimental)   dfsclnbgr2 47707
                  21.47.15.3  Induced subgraphs   cisubgr 47721
                  ​*21.47.15.4  Isomorphisms of graphs   cgrisom 47733
                  ​*21.47.15.5  Triangles in graphs   cgrtri 47777
                  ​*21.47.15.6  Local isomorphisms of graphs   cgrlim 47789
                  21.47.15.7  Loop-free graphs - extension   1hegrlfgr 47844
                  21.47.15.8  Walks - extension   cupwlks 47845
                  21.47.15.9  Edges of graphs expressed as sets of unordered pairs   upgredgssspr 47855
            21.47.16  Monoids (extension)   ovn0dmfun 47868
                  21.47.16.1  Auxiliary theorems   ovn0dmfun 47868
                  21.47.16.2  Magmas, Semigroups and Monoids (extension)   plusfreseq 47876
                  21.47.16.3  Examples and counterexamples for magmas, semigroups and monoids (extension)   opmpoismgm 47879
                  21.47.16.4  Group sum operation (extension 1)   gsumsplit2f 47892
            ​*21.47.17  Magmas and internal binary operations (alternate approach)   ccllaw 47895
                  *21.47.17.1  Laws for internal binary operations   ccllaw 47895
                  ​*21.47.17.2  Internal binary operations   cintop 47908
                  21.47.17.3  Alternative definitions for magmas and semigroups   cmgm2 47927
            21.47.18  Rings (extension)   lmod0rng 47941
                  21.47.18.1  Nonzero rings (extension)   lmod0rng 47941
                  21.47.18.2  Ideals as non-unital rings   lidldomn1 47943
                  21.47.18.3  The non-unital ring of even integers   0even 47949
                  21.47.18.4  A constructed not unital ring   cznrnglem 47971
                  ​*21.47.18.5  The category of non-unital rings (alternate definition)   crngcALTV 47975
                  ​*21.47.18.6  The category of (unital) rings (alternate definition)   cringcALTV 47999
            21.47.19  Basic algebraic structures (extension)   opeliun2xp 48046
                  21.47.19.1  Auxiliary theorems   opeliun2xp 48046
                  21.47.19.2  The binomial coefficient operation (extension)   bcpascm1 48065
                  21.47.19.3  The ` ZZ `-module ` ZZ X. ZZ `   zlmodzxzlmod 48068
                  21.47.19.4  Group sum operation (extension 2)   mgpsumunsn 48075
                  21.47.19.5  Symmetric groups (extension)   exple2lt6 48078
                  21.47.19.6  Divisibility (extension)   invginvrid 48081
                  21.47.19.7  The support of functions (extension)   rmsupp0 48082
                  21.47.19.8  Finitely supported functions (extension)   rmsuppfi 48087
                  21.47.19.9  Left modules (extension)   lmodvsmdi 48096
                  21.47.19.10  Associative algebras (extension)   assaascl0 48098
                  21.47.19.11  Univariate polynomials (extension)   ply1vr1smo 48100
                  21.47.19.12  Univariate polynomials (examples)   linply1 48111
            21.47.20  Linear algebra (extension)   cdmatalt 48114
                  ​*21.47.20.1  The subalgebras of diagonal and scalar matrices (extension)   cdmatalt 48114
                  ​*21.47.20.2  Linear combinations   clinc 48122
                  ​*21.47.20.3  Linear independence   clininds 48158
                  21.47.20.4  Simple left modules and the ` ZZ `-module   lmod1lem1 48205
                  21.47.20.5  Differences between (left) modules and (left) vector spaces   lvecpsslmod 48225
            21.47.21  Complexity theory   suppdm 48228
                  21.47.21.1  Auxiliary theorems   suppdm 48228
                  21.47.21.2  The modulo (remainder) operation (extension)   fldivmod 48241
                  21.47.21.3  Even and odd integers   nn0onn0ex 48246
                  21.47.21.4  The natural logarithm on complex numbers (extension)   logcxp0 48258
                  21.47.21.5  Division of functions   cfdiv 48260
                  21.47.21.6  Upper bounds   cbigo 48270
                  21.47.21.7  Logarithm to an arbitrary base (extension)   rege1logbrege0 48281
                  ​*21.47.21.8  The binary logarithm   fldivexpfllog2 48288
                  21.47.21.9  Binary length   cblen 48292
                  ​*21.47.21.10  Digits   cdig 48318
                  21.47.21.11  Nonnegative integer as sum of its shifted digits   dignn0flhalflem1 48338
                  21.47.21.12  Algorithms for the multiplication of nonnegative integers   nn0mulfsum 48347
                  ​*21.47.21.13  N-ary functions   cnaryf 48349
                  ​*21.47.21.14  The Ackermann function   citco 48380
            21.47.22  Elementary geometry (extension)   fv1prop 48422
                  21.47.22.1  Auxiliary theorems   fv1prop 48422
                  21.47.22.2  Real euclidean space of dimension 2   rrx2pxel 48434
                  21.47.22.3  Spheres and lines in real Euclidean spaces   cline 48450
      ​21.48  Mathbox for Zhi Wang
            21.48.1  Propositional calculus   pm4.71da 48512
            21.48.2  Predicate calculus with equality   dtrucor3 48521
                  21.48.2.1  Axiom scheme ax-5 (Distinctness)   dtrucor3 48521
            21.48.3  ZF Set Theory - start with the Axiom of Extensionality   ralbidb 48522
                  21.48.3.1  Restricted quantification   ralbidb 48522
                  21.48.3.2  The empty set   ssdisjd 48528
                  21.48.3.3  Unordered and ordered pairs   vsn 48532
                  21.48.3.4  The union of a class   unilbss 48538
            21.48.4  ZF Set Theory - add the Axiom of Replacement   inpw 48539
                  21.48.4.1  Theorems requiring subset and intersection existence   inpw 48539
            21.48.5  ZF Set Theory - add the Axiom of Power Sets   mof0 48540
                  21.48.5.1  Functions   mof0 48540
                  21.48.5.2  Operations   fvconstr 48558
            21.48.6  ZF Set Theory - add the Axiom of Union   fvconst0ci 48561
                  21.48.6.1  Equinumerosity   fvconst0ci 48561
            21.48.7  Order sets   iccin 48565
                  21.48.7.1  Real number intervals   iccin 48565
            21.48.8  Moore spaces   mreuniss 48568
            ​*21.48.9  Topology   clduni 48569
                  21.48.9.1  Closure and interior   clduni 48569
                  21.48.9.2  Neighborhoods   neircl 48573
                  21.48.9.3  Subspace topologies   restcls2lem 48581
                  21.48.9.4  Limits and continuity in topological spaces   cnneiima 48585
                  21.48.9.5  Topological definitions using the reals   iooii 48586
                  21.48.9.6  Separated sets   sepnsepolem1 48590
                  21.48.9.7  Separated spaces: T0, T1, T2 (Hausdorff) ...   isnrm4 48599
            21.48.10  Preordered sets and directed sets using extensible structures   isprsd 48624
            21.48.11  Posets and lattices using extensible structures   lubeldm2 48625
                  21.48.11.1  Posets   lubeldm2 48625
                  21.48.11.2  Lattices   toslat 48643
                  21.48.11.3  Subset order structures   intubeu 48645
            21.48.12  Categories   catprslem 48666
                  21.48.12.1  Categories   catprslem 48666
                  21.48.12.2  Monomorphisms and epimorphisms   idmon 48672
                  21.48.12.3  Functors   funcf2lem 48674
            21.48.13  Examples of categories   cthinc 48675
                  21.48.13.1  Thin categories   cthinc 48675
                  21.48.13.2  Preordered sets as thin categories   cprstc 48718
                  21.48.13.3  Monoids as categories   cmndtc 48739
      ​21.49  Mathbox for Emmett Weisz
            ​*21.49.1  Miscellaneous Theorems   nfintd 48754
            21.49.2  Set Recursion   csetrecs 48764
                  ​*21.49.2.1  Basic Properties of Set Recursion   csetrecs 48764
                  21.49.2.2  Examples and properties of set recursion   elsetrecslem 48780
            ​*21.49.3  Construction of Games and Surreal Numbers   cpg 48790
      ​​*21.50  Mathbox for David A. Wheeler
            21.50.1  Natural deduction   sbidd 48799
            ​*21.50.2  Greater than, greater than or equal to.   cge-real 48801
            ​*21.50.3  Hyperbolic trigonometric functions   csinh 48811
            ​*21.50.4  Reciprocal trigonometric functions (sec, csc, cot)   csec 48822
            ​*21.50.5  Identities for "if"   ifnmfalse 48844
            ​*21.50.6  Logarithms generalized to arbitrary base using ` logb `   logb2aval 48845
            ​*21.50.7  Logarithm laws generalized to an arbitrary base - log_   clog- 48846
            ​*21.50.8  Formally define notions such as reflexivity   wreflexive 48848
            ​*21.50.9  Algebra helpers   comraddi 48852
            ​*21.50.10  Algebra helper examples   i2linesi 48861
            ​*21.50.11  Formal methods "surprises"   alimp-surprise 48863
            ​*21.50.12  Allsome quantifier   walsi 48869
            ​*21.50.13  Miscellaneous   5m4e1 48880
            21.50.14  Theorems about algebraic numbers   aacllem 48884
      ​21.51  Mathbox for Kunhao Zheng
            21.51.1  Weighted AM-GM inequality   amgmwlem 48885