TOC of Theorem List - Metamath Proof Explorer

Table of Contents Summary

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

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

​​*PART 1  CLASSICAL FIRST-ORDER LOGIC WITH EQUALITY
      ​*1.1  Pre-logic
            *1.1.1  Inferences for assisting proof development   idi 1
      ​​*1.2  Propositional calculus
            1.2.1  Recursively define primitive wffs for propositional calculus   wn 3
            ​*1.2.2  The axioms of propositional calculus   ax-mp 5
            ​*1.2.3  Logical implication   mp2 9
            ​*1.2.4  Logical negation   con4 113
            ​*1.2.5  Logical equivalence   wb 206
            ​*1.2.6  Logical conjunction   wa 395
            ​*1.2.7  Logical disjunction   wo 848
            ​*1.2.8  Mixed connectives   jaao 957
            ​*1.2.9  The conditional operator for propositions   wif 1063
            ​*1.2.10  The weak deduction theorem for propositional calculus   elimh 1083
            1.2.11  Abbreviated conjunction and disjunction of three wff's   w3o 1086
            1.2.12  Logical "nand" (Sheffer stroke)   wnan 1490
            1.2.13  Logical "xor"   wxo 1510
            1.2.14  Logical "nor"   wnor 1527
            1.2.15  True and false constants   wal 1537
                  ​*1.2.15.1  Universal quantifier for use by df-tru   wal 1537
                  ​*1.2.15.2  Equality predicate for use by df-tru   cv 1538
                  1.2.15.3  The true constant   wtru 1540
                  1.2.15.4  The false constant   wfal 1551
            ​*1.2.16  Truth tables   truimtru 1562
                  1.2.16.1  Implication   truimtru 1562
                  1.2.16.2  Negation   nottru 1566
                  1.2.16.3  Equivalence   trubitru 1568
                  1.2.16.4  Conjunction   truantru 1572
                  1.2.16.5  Disjunction   truortru 1576
                  1.2.16.6  Alternative denial   trunantru 1580
                  1.2.16.7  Exclusive disjunction   truxortru 1584
                  1.2.16.8  Joint denial   trunortru 1588
            ​*1.2.17  Half adder and full adder in propositional calculus   whad 1592
                  1.2.17.1  Full adder: sum   whad 1592
                  1.2.17.2  Full adder: carry   wcad 1605
      ​1.3  Other axiomatizations related to classical propositional calculus
            ​*1.3.1  Minimal implicational calculus   minimp 1620
            ​*1.3.2  Implicational Calculus   impsingle 1626
            1.3.3  Derive the Lukasiewicz axioms from Meredith's sole axiom   meredith 1640
            1.3.4  Derive the standard axioms from the Lukasiewicz axioms   luklem1 1657
            ​*1.3.5  Derive Nicod's axiom from the standard axioms   nic-dfim 1668
            1.3.6  Derive the Lukasiewicz axioms from Nicod's axiom   nic-imp 1674
            1.3.7  Derive Nicod's Axiom from Lukasiewicz's First Sheffer Stroke Axiom   lukshef-ax1 1693
            1.3.8  Derive the Lukasiewicz Axioms from the Tarski-Bernays-Wajsberg Axioms   tbw-bijust 1697
            1.3.9  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's First CO Axiom   merco1 1712
            1.3.10  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's Second CO Axiom   merco2 1735
            1.3.11  Derive the Lukasiewicz axioms from the Russell-Bernays Axioms   rb-bijust 1748
            ​*1.3.12  Stoic logic non-modal portion (Chrysippus of Soli)   mptnan 1767
      ​​*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 1778
                  1.4.1.1  Existential quantifier   wex 1778
                  1.4.1.2  Nonfreeness predicate   wnf 1782
            1.4.2  Rule scheme ax-gen (Generalization)   ax-gen 1794
            1.4.3  Axiom scheme ax-4 (Quantified Implication)   ax-4 1808
                  ​*1.4.3.1  The empty domain of discourse   empty 1906
            1.4.4  Axiom scheme ax-5 (Distinctness) - first use of $d   ax-5 1910
            ​*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 2157
            1.5.3  Axiom scheme ax-12 (Substitution)   ax-12 2177
            1.5.4  Axiom scheme ax-13 (Quantified Equality)   ax-13 2377
      ​1.6  Uniqueness and unique existence
            1.6.1  Uniqueness: the at-most-one quantifier   wmo 2538
            1.6.2  Unique existence: the unique existential quantifier   weu 2568
      ​1.7  Other axiomatizations related to classical predicate calculus
            ​*1.7.1  Aristotelian logic: Assertic syllogisms   barbara 2663
            ​*1.7.2  Intuitionistic logic   axia1 2693
​​*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 2708
            2.1.2  Classes   cab 2714
                  2.1.2.1  Class abstractions   cab 2714
                  ​*2.1.2.2  Class equality   df-cleq 2729
                  2.1.2.3  Class membership   df-clel 2816
                  2.1.2.4  Elementary properties of class abstractions   eqabdv 2875
            2.1.3  Class form not-free predicate   wnfc 2890
            2.1.4  Negated equality and membership   wne 2940
                  2.1.4.1  Negated equality   wne 2940
                  2.1.4.2  Negated membership   wnel 3046
            2.1.5  Restricted quantification   wral 3061
                  2.1.5.1  Restricted universal and existential quantification   wral 3061
                  2.1.5.2  Restricted existential uniqueness and at-most-one quantifier   wreu 3378
                  2.1.5.3  Restricted class abstraction   crab 3436
            2.1.6  The universal class   cvv 3481
            ​*2.1.7  Conditional equality (experimental)   wcdeq 3775
            2.1.8  Russell's Paradox   rru 3791
            2.1.9  Proper substitution of classes for sets   wsbc 3794
            2.1.10  Proper substitution of classes for sets into classes   csb 3911
            2.1.11  Define basic set operations and relations   cdif 3963
            2.1.12  Subclasses and subsets   df-ss 3983
            2.1.13  The difference, union, and intersection of two classes   dfdif3 4130
                  2.1.13.1  The difference of two classes   dfdif3 4130
                  2.1.13.2  The union of two classes   elun 4166
                  2.1.13.3  The intersection of two classes   elini 4212
                  2.1.13.4  The symmetric difference of two classes   csymdif 4261
                  2.1.13.5  Combinations of difference, union, and intersection of two classes   unabs 4274
                  2.1.13.6  Class abstractions with difference, union, and intersection of two classes   unabw 4316
                  2.1.13.7  Restricted uniqueness with difference, union, and intersection   reuun2 4334
            2.1.14  The empty set   c0 4342
            ​*2.1.15  The conditional operator for classes   cif 4534
            ​*2.1.16  The weak deduction theorem for set theory   dedth 4592
            2.1.17  Power classes   cpw 4608
            2.1.18  Unordered and ordered pairs   snjust 4633
            2.1.19  The union of a class   cuni 4915
            2.1.20  The intersection of a class   cint 4954
            2.1.21  Indexed union and intersection   ciun 4999
            2.1.22  Disjointness   wdisj 5118
            2.1.23  Binary relations   wbr 5151
            2.1.24  Ordered-pair class abstractions (class builders)   copab 5213
            2.1.25  Functions in maps-to notation   cmpt 5234
            2.1.26  Transitive classes   wtr 5268
      ​2.2  ZF Set Theory - add the Axiom of Replacement
            2.2.1  Introduce the Axiom of Replacement   ax-rep 5288
            2.2.2  Derive the Axiom of Separation   axsepgfromrep 5303
            2.2.3  Derive the Null Set Axiom   axnulALT 5313
            2.2.4  Theorems requiring subset and intersection existence   nalset 5322
            2.2.5  Theorems requiring empty set existence   class2set 5364
      ​2.3  ZF Set Theory - add the Axiom of Power Sets
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 5374
            2.3.2  Derive the Axiom of Pairing   axprlem1 5432
            2.3.3  Ordered pair theorem   opnz 5487
            2.3.4  Ordered-pair class abstractions (cont.)   opabidw 5538
            2.3.5  Power class of union and intersection   pwin 5583
            2.3.6  The identity relation   cid 5586
            2.3.7  The membership relation (or epsilon relation)   cep 5592
            ​*2.3.8  Partial and total orderings   wpo 5599
            2.3.9  Founded and well-ordering relations   wfr 5642
            2.3.10  Relations   cxp 5691
            2.3.11  The Predecessor Class   cpred 6328
            2.3.12  Well-founded induction (variant)   frpomin 6369
            2.3.13  Well-ordered induction   tz6.26 6376
            2.3.14  Ordinals   word 6391
            2.3.15  Definite description binder (inverted iota)   cio 6520
            2.3.16  Functions   wfun 6563
            2.3.17  Cantor's Theorem   canth 7392
            2.3.18  Restricted iota (description binder)   crio 7394
            2.3.19  Operations   co 7438
                  2.3.19.1  Variable-to-class conversion for operations   caovclg 7632
            2.3.20  Maps-to notation   mpondm0 7680
            2.3.21  Function operation   cof 7702
            2.3.22  Proper subset relation   crpss 7748
      ​2.4  ZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 7761
            2.4.2  Ordinals (continued)   epweon 7801
            2.4.3  Transfinite induction   tfi 7881
            2.4.4  The natural numbers (i.e., finite ordinals)   com 7894
            2.4.5  Peano's postulates   peano1 7918
            2.4.6  Finite induction (for finite ordinals)   find 7925
            2.4.7  Relations and functions (cont.)   dmexg 7931
            2.4.8  First and second members of an ordered pair   c1st 8020
            2.4.9  Induction on Cartesian products   frpoins3xpg 8173
            2.4.10  Ordering on Cartesian products   xpord2lem 8175
            2.4.11  Ordering Ordinal Sequences   orderseqlem 8190
            ​*2.4.12  The support of functions   csupp 8193
            ​*2.4.13  Special maps-to operations   opeliunxp2f 8243
            2.4.14  Function transposition   ctpos 8258
            2.4.15  Curry and uncurry   ccur 8298
            2.4.16  Undefined values   cund 8305
            2.4.17  Well-founded recursion   cfrecs 8313
            2.4.18  Well-ordered recursion   cwrecs 8344
            2.4.19  Functions on ordinals; strictly monotone ordinal functions   iunon 8387
            2.4.20  "Strong" transfinite recursion   crecs 8418
            2.4.21  Recursive definition generator   crdg 8457
            2.4.22  Finite recursion   frfnom 8483
            2.4.23  Ordinal arithmetic   c1o 8507
            2.4.24  Natural number arithmetic   nna0 8650
            2.4.25  Natural addition   cnadd 8711
            2.4.26  Equivalence relations and classes   wer 8750
            2.4.27  The mapping operation   cmap 8874
            2.4.28  Infinite Cartesian products   cixp 8945
            2.4.29  Equinumerosity   cen 8990
            2.4.30  Schroeder-Bernstein Theorem   sbthlem1 9131
            2.4.31  Equinumerosity (cont.)   xpf1o 9187
            2.4.32  Finite sets   dif1enlem 9204
            2.4.33  Pigeonhole Principle   phplem1 9251
            2.4.34  Finite sets (cont.)   onomeneq 9272
            2.4.35  Finitely supported functions   cfsupp 9408
            2.4.36  Finite intersections   cfi 9457
            2.4.37  Hall's marriage theorem   marypha1lem 9480
            2.4.38  Supremum and infimum   csup 9487
            2.4.39  Ordinal isomorphism, Hartogs's theorem   coi 9556
            2.4.40  Hartogs function   char 9603
            2.4.41  Weak dominance   cwdom 9611
      ​2.5  ZF Set Theory - add the Axiom of Regularity
            2.5.1  Introduce the Axiom of Regularity   ax-reg 9639
            2.5.2  Axiom of Infinity equivalents   inf0 9668
      ​2.6  ZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-inf 9685
            2.6.2  Existence of omega (the set of natural numbers)   omex 9690
            2.6.3  Cantor normal form   ccnf 9708
            2.6.4  Transitive closure of a relation   cttrcl 9754
            2.6.5  Transitive closure   trcl 9775
            2.6.6  Well-Founded Induction   frmin 9796
            2.6.7  Well-Founded Recursion   frr3g 9803
            2.6.8  Rank   cr1 9809
            2.6.9  Scott's trick; collection principle; Hilbert's epsilon   scottex 9932
            2.6.10  Disjoint union   cdju 9945
            2.6.11  Cardinal numbers   ccrd 9982
            2.6.12  Axiom of Choice equivalents   wac 10162
            ​*2.6.13  Cardinal number arithmetic   undjudom 10215
            2.6.14  The Ackermann bijection   ackbij2lem1 10265
            2.6.15  Cofinality (without Axiom of Choice)   cflem 10292
            2.6.16  Eight inequivalent definitions of finite set   sornom 10324
            2.6.17  Hereditarily size-limited sets without Choice   itunifval 10463
​​*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 10482
            3.1.2  Introduce the Axiom of Dependent Choice   ax-dc 10493
      ​3.2  ZFC Set Theory - add the Axiom of Choice
            3.2.1  Introduce the Axiom of Choice   ax-ac 10506
            3.2.2  AC equivalents: well-ordering, Zorn's lemma   numthcor 10541
            3.2.3  Cardinal number theorems using Axiom of Choice   cardval 10593
            3.2.4  Cardinal number arithmetic using Axiom of Choice   iunctb 10621
            3.2.5  Cofinality using the Axiom of Choice   alephreg 10629
      ​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 10667
            3.4.2  Derivation of the Axiom of Choice   gchaclem 10725
​​*PART 4  TG (TARSKI-GROTHENDIECK) SET THEORY
      ​4.1  Inaccessibles
            4.1.1  Weakly and strongly inaccessible cardinals   cwina 10729
            4.1.2  Weak universes   cwun 10747
            4.1.3  Tarski classes   ctsk 10795
            4.1.4  Grothendieck universes   cgru 10837
      ​4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
            4.2.1  Introduce the Tarski-Grothendieck Axiom   ax-groth 10870
            4.2.2  Derive the Power Set, Infinity and Choice Axioms   grothpw 10873
            4.2.3  Tarski map function   ctskm 10884
​​*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 10891
            5.1.2  Final derivation of real and complex number postulates   axaddf 11192
            5.1.3  Real and complex number postulates restated as axioms   ax-cnex 11218
      ​5.2  Derive the basic properties from the field axioms
            5.2.1  Some deductions from the field axioms for complex numbers   cnex 11243
            5.2.2  Infinity and the extended real number system   cpnf 11299
            5.2.3  Restate the ordering postulates with extended real "less than"   axlttri 11339
            5.2.4  Ordering on reals   lttr 11344
            5.2.5  Initial properties of the complex numbers   mul12 11433
      ​5.3  Real and complex numbers - basic operations
            5.3.1  Addition   add12 11486
            5.3.2  Subtraction   cmin 11499
            5.3.3  Multiplication   kcnktkm1cn 11701
            5.3.4  Ordering on reals (cont.)   gt0ne0 11735
            5.3.5  Reciprocals   ixi 11899
            5.3.6  Division   cdiv 11927
            5.3.7  Ordering on reals (cont.)   elimgt0 12112
            5.3.8  Completeness Axiom and Suprema   fimaxre 12219
            5.3.9  Imaginary and complex number properties   inelr 12263
            5.3.10  Function operation analogue theorems   ofsubeq0 12270
      ​5.4  Integer sets
            5.4.1  Positive integers (as a subset of complex numbers)   cn 12273
            5.4.2  Principle of mathematical induction   nnind 12291
            ​*5.4.3  Decimal representation of numbers   c2 12328
            ​*5.4.4  Some properties of specific numbers   neg1cn 12387
            5.4.5  Simple number properties   halfcl 12498
            5.4.6  The Archimedean property   nnunb 12529
            5.4.7  Nonnegative integers (as a subset of complex numbers)   cn0 12533
            ​*5.4.8  Extended nonnegative integers   cxnn0 12606
            5.4.9  Integers (as a subset of complex numbers)   cz 12620
            5.4.10  Decimal arithmetic   cdc 12740
            5.4.11  Upper sets of integers   cuz 12885
            5.4.12  Well-ordering principle for bounded-below sets of integers   uzwo3 12992
            5.4.13  Rational numbers (as a subset of complex numbers)   cq 12997
            5.4.14  Existence of the set of complex numbers   rpnnen1lem2 13026
      ​5.5  Order sets
            5.5.1  Positive reals (as a subset of complex numbers)   crp 13041
            5.5.2  Infinity and the extended real number system (cont.)   cxne 13158
            5.5.3  Supremum and infimum on the extended reals   xrsupexmnf 13353
            5.5.4  Real number intervals   cioo 13393
            5.5.5  Finite intervals of integers   cfz 13553
            ​*5.5.6  Finite intervals of nonnegative integers   elfz2nn0 13664
            5.5.7  Half-open integer ranges   cfzo 13700
      ​5.6  Elementary integer functions
            5.6.1  The floor and ceiling functions   cfl 13836
            5.6.2  The modulo (remainder) operation   cmo 13915
            5.6.3  Miscellaneous theorems about integers   om2uz0i 13994
            5.6.4  Strong induction over upper sets of integers   uzsinds 14034
            5.6.5  Finitely supported functions over the nonnegative integers   fsuppmapnn0fiublem 14037
            5.6.6  The infinite sequence builder "seq" - extension   cseq 14048
            5.6.7  Integer powers   cexp 14108
            5.6.8  Ordered pair theorem for nonnegative integers   nn0le2msqi 14312
            5.6.9  Factorial function   cfa 14318
            5.6.10  The binomial coefficient operation   cbc 14347
            5.6.11  The ` # ` (set size) function   chash 14375
                  5.6.11.1  Proper unordered pairs and triples (sets of size 2 and 3)   hashprlei 14513
                  5.6.11.2  Functions with a domain containing at least two different elements   fundmge2nop0 14547
                  5.6.11.3  Finite induction on the size of the first component of a binary relation   hashdifsnp1 14551
      ​​*5.7  Words over a set
            5.7.1  Definitions and basic theorems   cword 14558
            5.7.2  Last symbol of a word   clsw 14606
            5.7.3  Concatenations of words   cconcat 14614
            5.7.4  Singleton words   cs1 14639
            5.7.5  Concatenations with singleton words   ccatws1cl 14660
            5.7.6  Subwords/substrings   csubstr 14684
            5.7.7  Prefixes of a word   cpfx 14714
            5.7.8  Subwords of subwords   swrdswrdlem 14748
            5.7.9  Subwords and concatenations   pfxcctswrd 14754
            5.7.10  Subwords of concatenations   swrdccatfn 14768
            5.7.11  Splicing words (substring replacement)   csplice 14793
            5.7.12  Reversing words   creverse 14802
            5.7.13  Repeated symbol words   creps 14812
            ​*5.7.14  Cyclical shifts of words   ccsh 14832
            5.7.15  Mapping words by a function   wrdco 14876
            5.7.16  Longer string literals   cs2 14886
      ​​*5.8  Reflexive and transitive closures of relations
            5.8.1  The reflexive and transitive properties of relations   coss12d 15017
            5.8.2  Basic properties of closures   cleq1lem 15027
            5.8.3  Definitions and basic properties of transitive closures   ctcl 15030
            5.8.4  Exponentiation of relations   crelexp 15064
            5.8.5  Reflexive-transitive closure as an indexed union   crtrcl 15100
            ​*5.8.6  Principle of transitive induction.   relexpindlem 15108
      ​5.9  Elementary real and complex functions
            5.9.1  The "shift" operation   cshi 15111
            5.9.2  Signum (sgn or sign) function   csgn 15131
            5.9.3  Real and imaginary parts; conjugate   ccj 15141
            5.9.4  Square root; absolute value   csqrt 15278
      ​5.10  Elementary limits and convergence
            5.10.1  Superior limit (lim sup)   clsp 15512
            5.10.2  Limits   cli 15526
            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 16379
            6.1.5  The division algorithm   divalglem0 16436
            6.1.6  Bit sequences   cbits 16462
            6.1.7  The greatest common divisor operator   cgcd 16537
            6.1.8  Bézout's identity   bezoutlem1 16582
            6.1.9  Algorithms   nn0seqcvgd 16613
            6.1.10  Euclid's Algorithm   eucalgval2 16624
            ​*6.1.11  The least common multiple   clcm 16631
            ​*6.1.12  Coprimality and Euclid's lemma   coprmgcdb 16692
            6.1.13  Cancellability of congruences   congr 16707
      ​6.2  Elementary prime number theory
            ​*6.2.1  Elementary properties   cprime 16714
            ​*6.2.2  Coprimality and Euclid's lemma (cont.)   coprm 16754
            6.2.3  Properties of the canonical representation of a rational   cnumer 16776
            6.2.4  Euler's theorem   codz 16806
            6.2.5  Arithmetic modulo a prime number   modprm1div 16840
            6.2.6  Pythagorean Triples   coprimeprodsq 16851
            6.2.7  The prime count function   cpc 16879
            6.2.8  Pocklington's theorem   prmpwdvds 16947
            6.2.9  Infinite primes theorem   unbenlem 16951
            6.2.10  Sum of prime reciprocals   prmreclem1 16959
            6.2.11  Fundamental theorem of arithmetic   1arithlem1 16966
            6.2.12  Lagrange's four-square theorem   cgz 16972
            6.2.13  Van der Waerden's theorem   cvdwa 17008
            6.2.14  Ramsey's theorem   cram 17042
            ​*6.2.15  Primorial function   cprmo 17074
            ​*6.2.16  Prime gaps   prmgaplem1 17092
            6.2.17  Decimal arithmetic (cont.)   dec2dvds 17106
            6.2.18  Cyclical shifts of words (cont.)   cshwsidrepsw 17137
            6.2.19  Specific prime numbers   prmlem0 17149
            6.2.20  Very large primes   1259lem1 17174
​PART 7  BASIC STRUCTURES
      ​7.1  Extensible structures
            ​*7.1.1  Basic definitions   cstr 17189
                  7.1.1.1  Extensible structures as structures with components   cstr 17189
                  7.1.1.2  Substitution of components   csts 17206
                  7.1.1.3  Slots   cslot 17224
                  ​*7.1.1.4  Structure component indices   cnx 17236
                  7.1.1.5  Base sets   cbs 17254
                  7.1.1.6  Base set restrictions   cress 17283
            7.1.2  Slot definitions   cplusg 17307
            7.1.3  Definition of the structure product   crest 17476
            7.1.4  Definition of the structure quotient   cordt 17555
      ​7.2  Moore spaces
            7.2.1  Moore closures   mrcflem 17660
            7.2.2  Independent sets in a Moore system   mrisval 17684
            7.2.3  Algebraic closure systems   isacs 17705
​PART 8  BASIC CATEGORY THEORY
      ​8.1  Categories
            8.1.1  Categories   ccat 17718
            8.1.2  Opposite category   coppc 17765
            8.1.3  Monomorphisms and epimorphisms   cmon 17785
            8.1.4  Sections, inverses, isomorphisms   csect 17801
            ​*8.1.5  Isomorphic objects   ccic 17852
            8.1.6  Subcategories   cssc 17864
            8.1.7  Functors   cfunc 17914
            8.1.8  Full & faithful functors   cful 17965
            8.1.9  Natural transformations and the functor category   cnat 18005
            8.1.10  Initial, terminal and zero objects of a category   cinito 18044
      ​8.2  Arrows (disjointified hom-sets)
            8.2.1  Identity and composition for arrows   cida 18116
      ​8.3  Examples of categories
            8.3.1  The category of sets   csetc 18138
            8.3.2  The category of categories   ccatc 18161
            ​*8.3.3  The category of extensible structures   fncnvimaeqv 18184
      ​8.4  Categorical constructions
            8.4.1  Product of categories   cxpc 18233
            8.4.2  Functor evaluation   cevlf 18275
            8.4.3  Hom functor   chof 18314
​PART 9  BASIC ORDER THEORY
      ​9.1  Dual of an order structure
      ​9.2  Preordered sets and directed sets
      ​9.3  Partially ordered sets (posets)
      ​9.4  Totally ordered sets (tosets)
      ​9.5  Lattices
            9.5.1  Lattices   clat 18498
            9.5.2  Complete lattices   ccla 18565
            9.5.3  Distributive lattices   cdlat 18587
            9.5.4  Subset order structures   cipo 18594
      ​9.6  Posets, directed sets, and lattices as relations
            ​*9.6.1  Posets and lattices as relations   cps 18631
            9.6.2  Directed sets, nets   cdir 18661
​PART 10  BASIC ALGEBRAIC STRUCTURES
      ​10.1  Monoids
            ​*10.1.1  Magmas   cplusf 18672
            ​*10.1.2  Identity elements   mgmidmo 18695
            ​*10.1.3  Iterated sums in a magma   gsumvalx 18711
            10.1.4  Magma homomorphisms and submagmas   cmgmhm 18725
            ​*10.1.5  Semigroups   csgrp 18753
            ​*10.1.6  Definition and basic properties of monoids   cmnd 18769
            10.1.7  Monoid homomorphisms and submonoids   cmhm 18816
            ​*10.1.8  Iterated sums in a monoid   gsumvallem2 18869
            10.1.9  Free monoids   cfrmd 18882
                  ​*10.1.9.1  Monoid of endofunctions   cefmnd 18903
            10.1.10  Examples and counterexamples for magmas, semigroups and monoids   mgm2nsgrplem1 18953
      ​10.2  Groups
            10.2.1  Definition and basic properties   cgrp 18973
            ​*10.2.2  Group multiple operation   cmg 19107
            10.2.3  Subgroups and Quotient groups   csubg 19160
            ​*10.2.4  Cyclic monoids and groups   cycsubmel 19240
            10.2.5  Elementary theory of group homomorphisms   cghm 19252
            10.2.6  Isomorphisms of groups   cgim 19297
                  10.2.6.1  The first isomorphism theorem of groups   ghmqusnsglem1 19320
            10.2.7  Group actions   cga 19329
            10.2.8  Centralizers and centers   ccntz 19355
            10.2.9  The opposite group   coppg 19385
            10.2.10  Symmetric groups   csymg 19410
                  ​*10.2.10.1  Definition and basic properties   csymg 19410
                  10.2.10.2  Cayley's theorem   cayleylem1 19454
                  10.2.10.3  Permutations fixing one element   symgfix2 19458
                  ​*10.2.10.4  Transpositions in the symmetric group   cpmtr 19483
                  10.2.10.5  The sign of a permutation   cpsgn 19531
            10.2.11  p-Groups and Sylow groups; Sylow's theorems   cod 19566
            10.2.12  Direct products   clsm 19676
                  10.2.12.1  Direct products (extension)   smndlsmidm 19698
            10.2.13  Free groups   cefg 19748
            10.2.14  Abelian groups   ccmn 19822
                  10.2.14.1  Definition and basic properties   ccmn 19822
                  10.2.14.2  Cyclic groups   ccyg 19919
                  10.2.14.3  Group sum operation   gsumval3a 19945
                  10.2.14.4  Group sums over (ranges of) integers   fsfnn0gsumfsffz 20025
                  10.2.14.5  Internal direct products   cdprd 20037
                  10.2.14.6  The Fundamental Theorem of Abelian Groups   ablfacrplem 20109
            10.2.15  Simple groups   csimpg 20134
                  10.2.15.1  Definition and basic properties   csimpg 20134
                  10.2.15.2  Classification of abelian simple groups   ablsimpnosubgd 20148
      ​10.3  Rings
            10.3.1  Multiplicative Group   cmgp 20161
            ​*10.3.2  Non-unital rings ("rngs")   crng 20179
            ​*10.3.3  Ring unity (multiplicative identity)   cur 20208
            10.3.4  Semirings   csrg 20213
                  ​*10.3.4.1  The binomial theorem for semirings   srgbinomlem1 20253
            10.3.5  Unital rings   crg 20260
            10.3.6  Opposite ring   coppr 20359
            10.3.7  Divisibility   cdsr 20380
            10.3.8  Ring primes   crpm 20458
            10.3.9  Homomorphisms of non-unital rings   crnghm 20460
            10.3.10  Ring homomorphisms   crh 20495
            10.3.11  Nonzero rings and zero rings   cnzr 20538
            10.3.12  Local rings   clring 20564
            10.3.13  Subrings   csubrng 20571
                  10.3.13.1  Subrings of non-unital rings   csubrng 20571
                  10.3.13.2  Subrings of unital rings   csubrg 20595
                  10.3.13.3  Subrings generated by a subset   crgspn 20636
            10.3.14  Categories of rings   crngc 20642
                  ​*10.3.14.1  The category of non-unital rings   crngc 20642
                  ​*10.3.14.2  The category of (unital) rings   cringc 20671
                  10.3.14.3  Subcategories of the category of rings   srhmsubclem1 20703
            10.3.15  Left regular elements and domains   crlreg 20717
      ​10.4  Division rings and fields
            10.4.1  Definition and basic properties   cdr 20755
            10.4.2  Sub-division rings   csdrg 20813
            10.4.3  Absolute value (abstract algebra)   cabv 20835
            10.4.4  Star rings   cstf 20864
      ​10.5  Left modules
            10.5.1  Definition and basic properties   clmod 20884
            10.5.2  Subspaces and spans in a left module   clss 20956
            10.5.3  Homomorphisms and isomorphisms of left modules   clmhm 21045
            10.5.4  Subspace sum; bases for a left module   clbs 21100
      ​10.6  Vector spaces
            10.6.1  Definition and basic properties   clvec 21128
      ​10.7  Subring algebras and ideals
            10.7.1  Subring algebras   csra 21197
            ​*10.7.2  Left ideals and spans   clidl 21243
            10.7.3  Two-sided ideals and quotient rings   c2idl 21286
                  ​*10.7.3.1  Condition for a non-unital ring to be unital   rngqiprng1elbas 21323
            10.7.4  Principal ideal rings. Divisibility in the integers   clpidl 21357
            10.7.5  Principal ideal domains   cpid 21373
      ​10.8  The complex numbers as an algebraic extensible structure
            10.8.1  Definition and basic properties   cpsmet 21375
            ​*10.8.2  Ring of integers   czring 21484
                  ​*10.8.2.1  Example for a condition for a non-unital ring to be unital   pzriprnglem1 21519
            10.8.3  Algebraic constructions based on the complex numbers   czrh 21537
            10.8.4  Signs as subgroup of the complex numbers   cnmsgnsubg 21622
            10.8.5  Embedding of permutation signs into a ring   zrhpsgnmhm 21629
            10.8.6  The ordered field of real numbers   crefld 21649
      ​10.9  Generalized pre-Hilbert and Hilbert spaces
            10.9.1  Definition and basic properties   cphl 21669
            10.9.2  Orthocomplements and closed subspaces   cocv 21705
            10.9.3  Orthogonal projection and orthonormal bases   cpj 21747
​​*PART 11  BASIC LINEAR ALGEBRA
      ​11.1  Vectors and free modules
            *11.1.1  Direct sum of left modules   cdsmm 21778
            ​*11.1.2  Free modules   cfrlm 21793
            ​*11.1.3  Standard basis (unit vectors)   cuvc 21829
            ​*11.1.4  Independent sets and families   clindf 21851
            11.1.5  Characterization of free modules   lmimlbs 21883
      ​11.2  Associative algebras
            11.2.1  Definition and basic properties   casa 21897
      ​11.3  Abstract multivariate polynomials
            11.3.1  Definition and basic properties   cmps 21951
            11.3.2  Polynomial evaluation   ces 22123
            11.3.3  Additional definitions for (multivariate) polynomials   cslv 22159
            ​*11.3.4  Univariate polynomials   cps1 22201
            11.3.5  Univariate polynomial evaluation   ces1 22342
                  11.3.5.1  Specialization of polynomial evaluation as a ring homomorphism   evls1scafv 22395
      ​​*11.4  Matrices
            *11.4.1  The matrix multiplication   cmmul 22419
            ​*11.4.2  Square matrices   cmat 22436
            ​*11.4.3  The matrix algebra   matmulr 22469
            ​*11.4.4  Matrices of dimension 0 and 1   mat0dimbas0 22497
            ​*11.4.5  The subalgebras of diagonal and scalar matrices   cdmat 22519
            ​*11.4.6  Multiplication of a matrix with a "column vector"   cmvmul 22571
            11.4.7  Replacement functions for a square matrix   cmarrep 22587
            11.4.8  Submatrices   csubma 22607
      ​11.5  The determinant
            11.5.1  Definition and basic properties   cmdat 22615
            11.5.2  Determinants of 2 x 2 -matrices   m2detleiblem1 22655
            11.5.3  The matrix adjugate/adjunct   cmadu 22663
            ​*11.5.4  Laplace expansion of determinants (special case)   symgmatr01lem 22684
            11.5.5  Inverse matrix   invrvald 22707
            ​*11.5.6  Cramer's rule   slesolvec 22710
      ​​*11.6  Polynomial matrices
            11.6.1  Basic properties   pmatring 22723
            ​*11.6.2  Constant polynomial matrices   ccpmat 22734
            ​*11.6.3  Collecting coefficients of polynomial matrices   cdecpmat 22793
            ​*11.6.4  Ring isomorphism between polynomial matrices and polynomials over matrices   cpm2mp 22823
      ​​*11.7  The characteristic polynomial
            *11.7.1  Definition and basic properties   cchpmat 22857
            ​*11.7.2  The characteristic factor function G   fvmptnn04if 22880
            ​*11.7.3  The Cayley-Hamilton theorem   cpmadurid 22898
​PART 12  BASIC TOPOLOGY
      ​12.1  Topology
            ​*12.1.1  Topological spaces   ctop 22924
                  12.1.1.1  Topologies   ctop 22924
                  12.1.1.2  Topologies on sets   ctopon 22941
                  12.1.1.3  Topological spaces   ctps 22963
            12.1.2  Topological bases   ctb 22977
            12.1.3  Examples of topologies   distop 23027
            12.1.4  Closure and interior   ccld 23049
            12.1.5  Neighborhoods   cnei 23130
            12.1.6  Limit points and perfect sets   clp 23167
            12.1.7  Subspace topologies   restrcl 23190
            12.1.8  Order topology   ordtbaslem 23221
            12.1.9  Limits and continuity in topological spaces   ccn 23257
            12.1.10  Separated spaces: T0, T1, T2 (Hausdorff) ...   ct0 23339
            12.1.11  Compactness   ccmp 23419
            12.1.12  Bolzano-Weierstrass theorem   bwth 23443
            12.1.13  Connectedness   cconn 23444
            12.1.14  First- and second-countability   c1stc 23470
            12.1.15  Local topological properties   clly 23497
            12.1.16  Refinements   cref 23535
            12.1.17  Compactly generated spaces   ckgen 23566
            12.1.18  Product topologies   ctx 23593
            12.1.19  Continuous function-builders   cnmptid 23694
            12.1.20  Quotient maps and quotient topology   ckq 23726
            12.1.21  Homeomorphisms   chmeo 23786
      ​12.2  Filters and filter bases
            12.2.1  Filter bases   elmptrab 23860
            12.2.2  Filters   cfil 23878
            12.2.3  Ultrafilters   cufil 23932
            12.2.4  Filter limits   cfm 23966
            12.2.5  Extension by continuity   ccnext 24092
            12.2.6  Topological groups   ctmd 24103
            12.2.7  Infinite group sum on topological groups   ctsu 24159
            12.2.8  Topological rings, fields, vector spaces   ctrg 24189
      ​12.3  Uniform Structures and Spaces
            12.3.1  Uniform structures   cust 24233
            12.3.2  The topology induced by an uniform structure   cutop 24264
            12.3.3  Uniform Spaces   cuss 24287
            12.3.4  Uniform continuity   cucn 24309
            12.3.5  Cauchy filters in uniform spaces   ccfilu 24320
            12.3.6  Complete uniform spaces   ccusp 24331
      ​12.4  Metric spaces
            12.4.1  Pseudometric spaces   ispsmet 24339
            12.4.2  Basic metric space properties   cxms 24352
            12.4.3  Metric space balls   blfvalps 24418
            12.4.4  Open sets of a metric space   mopnval 24473
            12.4.5  Continuity in metric spaces   metcnp3 24578
            12.4.6  The uniform structure generated by a metric   metuval 24587
            12.4.7  Examples of metric spaces   dscmet 24610
            ​*12.4.8  Normed algebraic structures   cnm 24614
            12.4.9  Normed space homomorphisms (bounded linear operators)   cnmo 24751
            12.4.10  Topology on the reals   qtopbaslem 24804
            12.4.11  Topological definitions using the reals   cii 24926
            12.4.12  Path homotopy   chtpy 25024
            12.4.13  The fundamental group   cpco 25058
      ​12.5  Metric subcomplex vector spaces
            12.5.1  Subcomplex modules   cclm 25120
            ​*12.5.2  Subcomplex vector spaces   ccvs 25181
            ​*12.5.3  Normed subcomplex vector spaces   isncvsngp 25208
            12.5.4  Subcomplex pre-Hilbert spaces   ccph 25225
            12.5.5  Convergence and completeness   ccfil 25311
            12.5.6  Baire's Category Theorem   bcthlem1 25383
            12.5.7  Banach spaces and subcomplex Hilbert spaces   ccms 25391
                  12.5.7.1  The complete ordered field of the real numbers   retopn 25438
            12.5.8  Euclidean spaces   crrx 25442
            12.5.9  Minimizing Vector Theorem   minveclem1 25483
            12.5.10  Projection Theorem   pjthlem1 25496
​PART 13  BASIC REAL AND COMPLEX ANALYSIS
      ​13.1  Continuity
            13.1.1  Intermediate value theorem   pmltpclem1 25508
      ​13.2  Integrals
            13.2.1  Lebesgue measure   covol 25522
            13.2.2  Lebesgue integration   cmbf 25674
                  13.2.2.1  Lesbesgue integral   cmbf 25674
                  13.2.2.2  Lesbesgue directed integral   cdit 25907
      ​13.3  Derivatives
            13.3.1  Real and complex differentiation   climc 25923
                  13.3.1.1  Derivatives of functions of one complex or real variable   climc 25923
                  13.3.1.2  Results on real differentiation   dvferm1lem 26048
​PART 14  BASIC REAL AND COMPLEX FUNCTIONS
      ​14.1  Polynomials
            14.1.1  Polynomial degrees   cmdg 26118
            14.1.2  The division algorithm for univariate polynomials   cmn1 26191
            14.1.3  Elementary properties of complex polynomials   cply 26249
            14.1.4  The division algorithm for polynomials   cquot 26358
            14.1.5  Algebraic numbers   caa 26382
            14.1.6  Liouville's approximation theorem   aalioulem1 26400
      ​14.2  Sequences and series
            14.2.1  Taylor polynomials and Taylor's theorem   ctayl 26420
            14.2.2  Uniform convergence   culm 26445
            14.2.3  Power series   pserval 26479
      ​14.3  Basic trigonometry
            14.3.1  The exponential, sine, and cosine functions (cont.)   efcn 26513
            14.3.2  Properties of pi = 3.14159...   pilem1 26521
            14.3.3  Mapping of the exponential function   efgh 26609
            14.3.4  The natural logarithm on complex numbers   clog 26622
            ​*14.3.5  Logarithms to an arbitrary base   clogb 26833
            14.3.6  Theorems of Pythagoras, isosceles triangles, and intersecting chords   angval 26870
            14.3.7  Solutions of quadratic, cubic, and quartic equations   quad2 26908
            14.3.8  Inverse trigonometric functions   casin 26931
            14.3.9  The Birthday Problem   log2ublem1 27015
            14.3.10  Areas in R^2   carea 27024
            14.3.11  More miscellaneous converging sequences   rlimcnp 27034
            14.3.12  Inequality of arithmetic and geometric means   cvxcl 27054
            14.3.13  Euler-Mascheroni constant   cem 27061
            14.3.14  Zeta function   czeta 27082
            14.3.15  Gamma function   clgam 27085
      ​14.4  Basic number theory
            14.4.1  Wilson's theorem   wilthlem1 27137
            14.4.2  The Fundamental Theorem of Algebra   ftalem1 27142
            14.4.3  The Basel problem (ζ(2) = π²/6)   basellem1 27150
            14.4.4  Number-theoretical functions   ccht 27160
            14.4.5  Perfect Number Theorem   mersenne 27297
            14.4.6  Characters of Z/nZ   cdchr 27302
            14.4.7  Bertrand's postulate   bcctr 27345
            ​*14.4.8  Quadratic residues and the Legendre symbol   clgs 27364
            ​*14.4.9  Gauss' Lemma   gausslemma2dlem0a 27426
            14.4.10  Quadratic reciprocity   lgseisenlem1 27445
            14.4.11  All primes 4n+1 are the sum of two squares   2sqlem1 27487
            14.4.12  Chebyshev's Weak Prime Number Theorem, Dirichlet's Theorem   chebbnd1lem1 27539
            14.4.13  The Prime Number Theorem   mudivsum 27600
            14.4.14  Ostrowski's theorem   abvcxp 27685
​PART 15  SURREAL NUMBERS
      ​​*15.1  Sign sequence representation and Alling's axioms
            15.1.1  Definitions and initial properties   csur 27710
            15.1.2  Ordering   sltsolem1 27746
            15.1.3  Birthday Function   bdayfo 27748
            15.1.4  Density   fvnobday 27749
            ​*15.1.5  Full-Eta Property   bdayimaon 27764
      ​15.2  Initial consequences of Alling's axioms
            15.2.1  Ordering Theorems   csle 27815
            15.2.2  Birthday Theorems   bdayfun 27843
      ​​*15.3  Conway cut representation
            15.3.1  Conway cuts   csslt 27851
            15.3.2  Zero and One   c0s 27893
            15.3.3  Cuts and Options   cmade 27907
            15.3.4  Cofinality and coinitiality   cofsslt 27978
      ​15.4  Induction and recursion
            15.4.1  Induction and recursion on one variable   cnorec 27996
            15.4.2  Induction and recursion on two variables   cnorec2 28007
      ​15.5  Surreal arithmetic
            15.5.1  Addition   cadds 28018
            15.5.2  Negation and Subtraction   cnegs 28077
            15.5.3  Multiplication   cmuls 28158
            15.5.4  Division   cdivs 28239
            15.5.5  Absolute value   cabss 28287
      ​15.6  Subsystems of surreals
            15.6.1  Ordinal numbers   cons 28300
            15.6.2  Surreal recursive sequences   cseqs 28315
            15.6.3  Natural numbers   cnn0s 28344
            15.6.4  Integers   czs 28390
            15.6.5  Dyadic fractions   c2s 28420
            15.6.6  Real numbers   creno 28451
​​*PART 16  ELEMENTARY GEOMETRY
      ​16.1  Definition and Tarski's Axioms of Geometry
            16.1.1  Justification for the congruence notation   tgjustf 28507
      ​16.2  Tarskian Geometry
            16.2.1  Congruence   tgcgrcomimp 28511
            16.2.2  Betweenness   tgbtwntriv2 28521
            16.2.3  Dimension   tglowdim1 28534
            16.2.4  Betweenness and Congruence   tgifscgr 28542
            16.2.5  Congruence of a series of points   ccgrg 28544
            16.2.6  Motions   cismt 28566
            16.2.7  Colinearity   tglng 28580
            16.2.8  Connectivity of betweenness   tgbtwnconn1lem1 28606
            16.2.9  Less-than relation in geometric congruences   cleg 28616
            16.2.10  Rays   chlg 28634
            16.2.11  Lines   btwnlng1 28653
            16.2.12  Point inversions   cmir 28686
            16.2.13  Right angles   crag 28727
            16.2.14  Half-planes   islnopp 28773
            16.2.15  Midpoints and Line Mirroring   cmid 28806
            16.2.16  Congruence of angles   ccgra 28841
            16.2.17  Angle Comparisons   cinag 28869
            16.2.18  Congruence Theorems   tgsas1 28888
            16.2.19  Equilateral triangles   ceqlg 28899
      ​16.3  Properties of geometries
            16.3.1  Isomorphisms between geometries   f1otrgds 28903
      ​16.4  Geometry in Hilbert spaces
            16.4.1  Geometry in the complex plane   cchhllem 28927
            16.4.2  Geometry in Euclidean spaces   cee 28929
                  16.4.2.1  Definition of the Euclidean space   cee 28929
                  16.4.2.2  Tarski's axioms for geometry for the Euclidean space   axdimuniq 28954
                  16.4.2.3  EE^n fulfills Tarski's Axioms   ceeng 29018
​​*PART 17  GRAPH THEORY
      ​*17.1  Vertices and edges
            17.1.1  The edge function extractor for extensible structures   cedgf 29029
            ​*17.1.2  Vertices and indexed edges   cvtx 29039
                  17.1.2.1  Definitions and basic properties   cvtx 29039
                  17.1.2.2  The vertices and edges of a graph represented as ordered pair   opvtxval 29046
                  17.1.2.3  The vertices and edges of a graph represented as extensible structure   funvtxdmge2val 29054
                  17.1.2.4  Representations of graphs without edges   snstrvtxval 29080
                  17.1.2.5  Degenerated cases of representations of graphs   vtxval0 29082
            17.1.3  Edges as range of the edge function   cedg 29090
      ​​*17.2  Undirected graphs
            17.2.1  Undirected hypergraphs   cuhgr 29099
            17.2.2  Undirected pseudographs and multigraphs   cupgr 29123
            ​*17.2.3  Loop-free graphs   umgrislfupgrlem 29165
            17.2.4  Edges as subsets of vertices of graphs   uhgredgiedgb 29169
            ​*17.2.5  Undirected simple graphs   cuspgr 29191
            17.2.6  Examples for graphs   usgr0e 29279
            17.2.7  Subgraphs   csubgr 29310
            17.2.8  Finite undirected simple graphs   cfusgr 29359
            17.2.9  Neighbors, complete graphs and universal vertices   cnbgr 29375
                  17.2.9.1  Neighbors   cnbgr 29375
                  17.2.9.2  Universal vertices   cuvtx 29428
                  17.2.9.3  Complete graphs   ccplgr 29452
            17.2.10  Vertex degree   cvtxdg 29509
            ​*17.2.11  Regular graphs   crgr 29599
      ​​*17.3  Walks, paths and cycles
            *17.3.1  Walks   cewlks 29639
            17.3.2  Walks for loop-free graphs   lfgrwlkprop 29731
            17.3.3  Trails   ctrls 29734
            17.3.4  Paths and simple paths   cpths 29756
            17.3.5  Closed walks   cclwlks 29816
            17.3.6  Circuits and cycles   ccrcts 29830
            ​*17.3.7  Walks as words   cwwlks 29871
            17.3.8  Walks/paths of length 2 (as length 3 strings)   2wlkdlem1 29971
            17.3.9  Walks in regular graphs   rusgrnumwwlkl1 30014
            ​*17.3.10  Closed walks as words   cclwwlk 30026
                  17.3.10.1  Closed walks as words   cclwwlk 30026
                  17.3.10.2  Closed walks of a fixed length as words   cclwwlkn 30069
                  17.3.10.3  Closed walks on a vertex of a fixed length as words   cclwwlknon 30132
            17.3.11  Examples for walks, trails and paths   0ewlk 30159
            17.3.12  Connected graphs   cconngr 30231
      ​17.4  Eulerian paths and the Konigsberg Bridge problem
            ​*17.4.1  Eulerian paths   ceupth 30242
            ​*17.4.2  The Königsberg Bridge problem   konigsbergvtx 30291
      ​17.5  The Friendship Theorem
            17.5.1  Friendship graphs - basics   cfrgr 30303
            17.5.2  The friendship theorem for small graphs   frgr1v 30316
            17.5.3  Theorems according to Mertzios and Unger   2pthfrgrrn 30327
            ​*17.5.4  Huneke's Proof of the Friendship Theorem   frgrncvvdeqlem1 30344
​PART 18  GUIDES AND MISCELLANEA
      ​18.1  Guides (conventions, explanations, and examples)
            ​*18.1.1  Conventions   conventions 30445
            18.1.2  Natural deduction   natded 30448
            ​*18.1.3  Natural deduction examples   ex-natded5.2 30449
            18.1.4  Definitional examples   ex-or 30466
            18.1.5  Other examples   aevdemo 30505
      ​18.2  Humor
            18.2.1  April Fool's theorem   avril1 30508
      ​18.3  (Future - to be reviewed and classified)
            18.3.1  Planar incidence geometry   cplig 30519
            ​*18.3.2  Aliases kept to prevent broken links   dummylink 30532
​​*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 30534
            19.1.2  Abelian groups   cablo 30589
      ​19.2  Complex vector spaces
            19.2.1  Definition and basic properties   cvc 30603
            19.2.2  Examples of complex vector spaces   cnaddabloOLD 30626
      ​19.3  Normed complex vector spaces
            19.3.1  Definition and basic properties   cnv 30629
            19.3.2  Examples of normed complex vector spaces   cnnv 30722
            19.3.3  Induced metric of a normed complex vector space   imsval 30730
            19.3.4  Inner product   cdip 30745
            19.3.5  Subspaces   css 30766
      ​19.4  Operators on complex vector spaces
            19.4.1  Definitions and basic properties   clno 30785
      ​19.5  Inner product (pre-Hilbert) spaces
            19.5.1  Definition and basic properties   ccphlo 30857
            19.5.2  Examples of pre-Hilbert spaces   cncph 30864
            19.5.3  Properties of pre-Hilbert spaces   isph 30867
      ​19.6  Complex Banach spaces
            19.6.1  Definition and basic properties   ccbn 30907
            19.6.2  Examples of complex Banach spaces   cnbn 30914
            19.6.3  Uniform Boundedness Theorem   ubthlem1 30915
            19.6.4  Minimizing Vector Theorem   minvecolem1 30919
      ​19.7  Complex Hilbert spaces
            19.7.1  Definition and basic properties   chlo 30930
            19.7.2  Standard axioms for a complex Hilbert space   hlex 30943
            19.7.3  Examples of complex Hilbert spaces   cnchl 30961
            19.7.4  Hellinger-Toeplitz Theorem   htthlem 30962
​​*PART 20  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      ​20.1  Axiomatization of complex pre-Hilbert spaces
            20.1.1  Basic Hilbert space definitions   chba 30964
            20.1.2  Preliminary ZFC lemmas   df-hnorm 31013
            ​*20.1.3  Derive the Hilbert space axioms from ZFC set theory   axhilex-zf 31026
            ​*20.1.4  Introduce the vector space axioms for a Hilbert space   ax-hilex 31044
            20.1.5  Vector operations   hvmulex 31056
            20.1.6  Inner product postulates for a Hilbert space   ax-hfi 31124
      ​20.2  Inner product and norms
            20.2.1  Inner product   his5 31131
            20.2.2  Norms   dfhnorm2 31167
            20.2.3  Relate Hilbert space to normed complex vector spaces   hilablo 31205
            20.2.4  Bunjakovaskij-Cauchy-Schwarz inequality   bcsiALT 31224
      ​20.3  Cauchy sequences and completeness axiom
            20.3.1  Cauchy sequences and limits   hcau 31229
            20.3.2  Derivation of the completeness axiom from ZF set theory   hilmet 31239
            20.3.3  Completeness postulate for a Hilbert space   ax-hcompl 31247
            20.3.4  Relate Hilbert space to ZFC pre-Hilbert and Hilbert spaces   hhcms 31248
      ​20.4  Subspaces and projections
            20.4.1  Subspaces   df-sh 31252
            20.4.2  Closed subspaces   df-ch 31266
            20.4.3  Orthocomplements   df-oc 31297
            20.4.4  Subspace sum, span, lattice join, lattice supremum   df-shs 31353
            20.4.5  Projection theorem   pjhthlem1 31436
            20.4.6  Projectors   df-pjh 31440
      ​20.5  Properties of Hilbert subspaces
            20.5.1  Orthomodular law   omlsilem 31447
            20.5.2  Projectors (cont.)   pjhtheu2 31461
            20.5.3  Hilbert lattice operations   sh0le 31485
            20.5.4  Span (cont.) and one-dimensional subspaces   spansn0 31586
            20.5.5  Commutes relation for Hilbert lattice elements   df-cm 31628
            20.5.6  Foulis-Holland theorem   fh1 31663
            20.5.7  Quantum Logic Explorer axioms   qlax1i 31672
            20.5.8  Orthogonal subspaces   chscllem1 31682
            20.5.9  Orthoarguesian laws 5OA and 3OA   5oalem1 31699
            20.5.10  Projectors (cont.)   pjorthi 31714
            20.5.11  Mayet's equation E_3   mayete3i 31773
      ​20.6  Operators on Hilbert spaces
            ​*20.6.1  Operator sum, difference, and scalar multiplication   df-hosum 31775
            20.6.2  Zero and identity operators   df-h0op 31793
            20.6.3  Operations on Hilbert space operators   hoaddcl 31803
            20.6.4  Linear, continuous, bounded, Hermitian, unitary operators and norms   df-nmop 31884
            20.6.5  Linear and continuous functionals and norms   df-nmfn 31890
            20.6.6  Adjoint   df-adjh 31894
            20.6.7  Dirac bra-ket notation   df-bra 31895
            20.6.8  Positive operators   df-leop 31897
            20.6.9  Eigenvectors, eigenvalues, spectrum   df-eigvec 31898
            20.6.10  Theorems about operators and functionals   nmopval 31901
            20.6.11  Riesz lemma   riesz3i 32107
            20.6.12  Adjoints (cont.)   cnlnadjlem1 32112
            20.6.13  Quantum computation error bound theorem   unierri 32149
            20.6.14  Dirac bra-ket notation (cont.)   branmfn 32150
            20.6.15  Positive operators (cont.)   leopg 32167
            20.6.16  Projectors as operators   pjhmopi 32191
      ​20.7  States on a Hilbert lattice and Godowski's equation
            20.7.1  States on a Hilbert lattice   df-st 32256
            20.7.2  Godowski's equation   golem1 32316
      ​20.8  Cover relation, atoms, exchange axiom, and modular symmetry
            20.8.1  Covers relation; modular pairs   df-cv 32324
            20.8.2  Atoms   df-at 32383
            20.8.3  Superposition principle   superpos 32399
            20.8.4  Atoms, exchange and covering properties, atomicity   chcv1 32400
            20.8.5  Irreducibility   chirredlem1 32435
            20.8.6  Atoms (cont.)   atcvat3i 32441
            20.8.7  Modular symmetry   mdsymlem1 32448
​PART 21  SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
      ​21.1  Mathboxes for user contributions
            21.1.1  Mathbox guidelines   mathbox 32487
      ​21.2  Mathbox for Stefan Allan
      ​21.3  Mathbox for Thierry Arnoux
            21.3.1  Propositional Calculus - misc additions   ad11antr 32492
            21.3.2  Predicate Calculus   sbc2iedf 32509
                  21.3.2.1  Predicate Calculus - misc additions   sbc2iedf 32509
                  21.3.2.2  Restricted quantification - misc additions   ralcom4f 32511
                  21.3.2.3  Equality   eqtrb 32517
                  21.3.2.4  Double restricted existential uniqueness quantification   opsbc2ie 32519
                  21.3.2.5  Double restricted existential uniqueness quantification syntax   w2reu 32521
                  21.3.2.6  Substitution (without distinct variables) - misc additions   sbceqbidf 32530
                  21.3.2.7  Existential "at most one" - misc additions   mo5f 32532
                  21.3.2.8  Existential uniqueness - misc additions   reuxfrdf 32534
                  21.3.2.9  Restricted "at most one" - misc additions   rmoxfrd 32536
                  21.3.2.10  Restricted iota (description binder)   riotaeqbidva 32539
            21.3.3  General Set Theory   dmrab 32540
                  21.3.3.1  Class abstractions (a.k.a. class builders)   dmrab 32540
                  21.3.3.2  Image Sets   abrexdomjm 32550
                  21.3.3.3  Set relations and operations - misc additions   nelun 32556
                  21.3.3.4  Unordered pairs   elpreq 32571
                  21.3.3.5  Conditional operator - misc additions   ifeqeqx 32578
                  21.3.3.6  Set union   uniinn0 32586
                  21.3.3.7  Indexed union - misc additions   cbviunf 32591
                  21.3.3.8  Indexed intersection - misc additions   iinabrex 32604
                  21.3.3.9  Disjointness - misc additions   disjnf 32605
            21.3.4  Relations and Functions   xpdisjres 32633
                  21.3.4.1  Relations - misc additions   xpdisjres 32633
                  21.3.4.2  Functions - misc additions   feq2dd 32654
                  21.3.4.3  Operations - misc additions   mpomptxf 32708
                  21.3.4.4  Support of a function   suppovss 32710
                  21.3.4.5  Explicit Functions with one or two points as a domain   cosnopne 32723
                  21.3.4.6  Isomorphisms - misc. additions   gtiso 32730
                  21.3.4.7  Disjointness (additional proof requiring functions)   disjdsct 32732
                  21.3.4.8  First and second members of an ordered pair - misc additions   df1stres 32733
                  21.3.4.9  Supremum - misc additions   supssd 32741
                  21.3.4.10  Finite Sets   imafi2 32743
                  21.3.4.11  Countable Sets   snct 32745
            21.3.5  Real and Complex Numbers   creq0 32767
                  21.3.5.1  Complex operations - misc. additions   creq0 32767
                  21.3.5.2  Ordering on reals - misc additions   lt2addrd 32776
                  21.3.5.3  Extended reals - misc additions   xrlelttric 32777
                  21.3.5.4  Extended nonnegative integers - misc additions   xnn0gt0 32794
                  21.3.5.5  Real number intervals - misc additions   joiniooico 32797
                  21.3.5.6  Finite intervals of integers - misc additions   uzssico 32807
                  21.3.5.7  Half-open integer ranges - misc additions   iundisjfi 32818
                  21.3.5.8  The ` # ` (set size) function - misc additions   hashunif 32830
                  21.3.5.9  The greatest common divisor operator - misc. additions   znumd 32833
                  21.3.5.10  Integers   nn0split01 32838
                  21.3.5.11  Decimal numbers   dfdec100 32851
            21.3.6  Real and complex functions   nexple 32852
                  21.3.6.1  Integer powers - misc. additions   nexple 32852
                  21.3.6.2  Indicator Functions   cind 32853
            ​*21.3.7  Decimal expansion   cdp2 32870
                  ​*21.3.7.1  Decimal point   cdp 32887
                  21.3.7.2  Division in the extended real number system   cxdiv 32916
            21.3.8  Words over a set - misc additions   wrdfd 32935
                  21.3.8.1  Splicing words (substring replacement)   splfv3 32960
                  21.3.8.2  Cyclic shift of words   1cshid 32961
            21.3.9  Extensible Structures   ressplusf 32965
                  21.3.9.1  Structure restriction operator   ressplusf 32965
                  21.3.9.2  The opposite group   oppgle 32968
                  21.3.9.3  Posets   ressprs 32971
                  21.3.9.4  Complete lattices   clatp0cl 32983
                  21.3.9.5  Order Theory   cmnt 32985
                  21.3.9.6  Chains   cchn 33011
                  21.3.9.7  Extended reals Structure - misc additions   ax-xrssca 33021
                  21.3.9.8  The extended nonnegative real numbers commutative monoid   xrge0base 33031
            21.3.10  Algebra   mndcld 33042
                  21.3.10.1  Monoids   mndcld 33042
                  21.3.10.2  Monoids Homomorphisms   abliso 33056
                  21.3.10.3  Groups - misc additions   grpsubcld 33060
                  21.3.10.4  Finitely supported group sums - misc additions   gsumsubg 33064
                  21.3.10.5  Group or monoid sums over words   gsumwun 33083
                  21.3.10.6  Centralizers and centers - misc additions   cntzun 33086
                  21.3.10.7  Totally ordered monoids and groups   comnd 33089
                  21.3.10.8  The symmetric group   symgfcoeu 33117
                  21.3.10.9  Transpositions   pmtridf1o 33129
                  21.3.10.10  Permutation Signs   psgnid 33132
                  21.3.10.11  Permutation cycles   ctocyc 33141
                  21.3.10.12  The Alternating Group   evpmval 33180
                  21.3.10.13  Signum in an ordered monoid   csgns 33193
                  21.3.10.14  The Archimedean property for generic ordered algebraic structures   cinftm 33198
                  21.3.10.15  Semiring left modules   cslmd 33221
                  21.3.10.16  Simple groups   prmsimpcyc 33249
                  21.3.10.17  Rings - misc additions   cringmul32d 33250
                  21.3.10.18  Subrings generated by a set   elrgspnlem1 33264
                  21.3.10.19  The zero ring   irrednzr 33269
                  21.3.10.20  Localization of rings   cerl 33272
                  21.3.10.21  Integral Domains   domnmuln0rd 33293
                  21.3.10.22  Euclidean Domains   ceuf 33304
                  21.3.10.23  Division Rings   ringinveu 33310
                  21.3.10.24  Subfields   sdrgdvcl 33313
                  21.3.10.25  Field of fractions   cfrac 33316
                  21.3.10.26  Field extensions generated by a set   cfldgen 33324
                  21.3.10.27  Totally ordered rings and fields   corng 33337
                  21.3.10.28  Ring homomorphisms - misc additions   rhmdvd 33360
                  21.3.10.29  Scalar restriction operation   cresv 33362
                  21.3.10.30  The commutative ring of gaussian integers   gzcrng 33382
                  21.3.10.31  The archimedean ordered field of real numbers   cnfldfld 33383
                  21.3.10.32  The quotient map and quotient modules   qusker 33389
                  21.3.10.33  The ring of integers modulo ` N `   znfermltl 33406
                  21.3.10.34  Independent sets and families   islinds5 33407
                  21.3.10.35  Ring associates, ring units   dvdsruassoi 33424
                  ​*21.3.10.36  Subgroup sum / Sumset / Minkowski sum   elgrplsmsn 33430
                  21.3.10.37  The quotient map   quslsm 33445
                  21.3.10.38  Ideals   lidlmcld 33459
                  21.3.10.39  Prime Ideals   cprmidl 33475
                  21.3.10.40  Maximal Ideals   cmxidl 33499
                  21.3.10.41  The semiring of ideals of a ring   cidlsrg 33540
                  21.3.10.42  Prime Elements   rprmval 33556
                  21.3.10.43  Unique factorization domains   cufd 33578
                  21.3.10.44  The ring of integers   zringidom 33591
                  21.3.10.45  Univariate Polynomials   0ringmon1p 33595
                  21.3.10.46  Polynomial quotient and polynomial remainder   q1pdir 33635
                  21.3.10.47  The subring algebra   sra1r 33644
                  21.3.10.48  Division Ring Extensions   drgext0g 33651
                  21.3.10.49  Vector Spaces   lvecdimfi 33657
                  21.3.10.50  Vector Space Dimension   cldim 33658
            21.3.11  Field Extensions   cfldext 33698
                  21.3.11.1  Algebraic numbers   cirng 33730
                  21.3.11.2  Minimal polynomials   cminply 33739
                  21.3.11.3  Quadratic Field Extensions   rtelextdg2lem 33764
                  21.3.11.4  Towers of quadratic extentions   fldext2chn 33766
            ​*21.3.12  Constructible Numbers   cconstr 33767
                  21.3.12.1  Impossible constructions   2sqr3minply 33785
            21.3.13  Matrices   csmat 33786
                  21.3.13.1  Submatrices   csmat 33786
                  21.3.13.2  Matrix literals   clmat 33804
                  21.3.13.3  Laplace expansion of determinants   mdetpmtr1 33816
            21.3.14  Topology   ist0cld 33826
                  21.3.14.1  Open maps   txomap 33827
                  21.3.14.2  Topology of the unit circle   qtopt1 33828
                  21.3.14.3  Refinements   reff 33832
                  21.3.14.4  Open cover refinement property   ccref 33835
                  21.3.14.5  Lindelöf spaces   cldlf 33845
                  21.3.14.6  Paracompact spaces   cpcmp 33848
                  ​*21.3.14.7  Spectrum of a ring   crspec 33855
                  21.3.14.8  Pseudometrics   cmetid 33879
                  21.3.14.9  Continuity - misc additions   hauseqcn 33891
                  21.3.14.10  Topology of the closed unit interval   elunitge0 33892
                  21.3.14.11  Topology of ` ( RR X. RR ) `   unicls 33896
                  21.3.14.12  Order topology - misc. additions   cnvordtrestixx 33906
                  21.3.14.13  Continuity in topological spaces - misc. additions   mndpluscn 33919
                  21.3.14.14  Topology of the extended nonnegative real numbers ordered monoid   xrge0hmph 33925
                  21.3.14.15  Limits - misc additions   lmlim 33940
                  21.3.14.16  Univariate polynomials   pl1cn 33948
            21.3.15  Uniform Stuctures and Spaces   chcmp 33949
                  21.3.15.1  Hausdorff uniform completion   chcmp 33949
            21.3.16  Topology and algebraic structures   zringnm 33951
                  21.3.16.1  The norm on the ring of the integer numbers   zringnm 33951
                  21.3.16.2  Topological ` ZZ ` -modules   zlm0 33953
                  21.3.16.3  Canonical embedding of the field of the rational numbers into a division ring   cqqh 33965
                  21.3.16.4  Canonical embedding of the real numbers into a complete ordered field   crrh 33988
                  21.3.16.5  Embedding from the extended real numbers into a complete lattice   cxrh 34011
                  21.3.16.6  Canonical embeddings into the ordered field of the real numbers   zrhre 34014
                  ​*21.3.16.7  Topological Manifolds   cmntop 34017
                  21.3.16.8  Extended sum   cesum 34022
            21.3.17  Mixed Function/Constant operation   cofc 34090
            21.3.18  Abstract measure   csiga 34103
                  21.3.18.1  Sigma-Algebra   csiga 34103
                  21.3.18.2  Generated sigma-Algebra   csigagen 34133
                  ​*21.3.18.3  lambda and pi-Systems, Rings of Sets   ispisys 34147
                  21.3.18.4  The Borel algebra on the real numbers   cbrsiga 34176
                  21.3.18.5  Product Sigma-Algebra   csx 34183
                  21.3.18.6  Measures   cmeas 34190
                  21.3.18.7  The counting measure   cntmeas 34221
                  21.3.18.8  The Lebesgue measure - misc additions   voliune 34224
                  21.3.18.9  The Dirac delta measure   cdde 34227
                  21.3.18.10  The 'almost everywhere' relation   cae 34232
                  21.3.18.11  Measurable functions   cmbfm 34244
                  21.3.18.12  Borel Algebra on ` ( RR X. RR ) `   br2base 34265
                  ​*21.3.18.13  Caratheodory's extension theorem   coms 34287
            21.3.19  Integration   itgeq12dv 34322
                  21.3.19.1  Lebesgue integral - misc additions   itgeq12dv 34322
                  21.3.19.2  Bochner integral   citgm 34323
            21.3.20  Euler's partition theorem   oddpwdc 34350
            21.3.21  Sequences defined by strong recursion   csseq 34379
            21.3.22  Fibonacci Numbers   cfib 34392
            21.3.23  Probability   cprb 34403
                  21.3.23.1  Probability Theory   cprb 34403
                  21.3.23.2  Conditional Probabilities   ccprob 34427
                  21.3.23.3  Real-valued Random Variables   crrv 34436
                  21.3.23.4  Preimage set mapping operator   corvc 34451
                  21.3.23.5  Distribution Functions   orvcelval 34464
                  21.3.23.6  Cumulative Distribution Functions   orvclteel 34468
                  21.3.23.7  Probabilities - example   coinfliplem 34474
                  21.3.23.8  Bertrand's Ballot Problem   ballotlemoex 34481
            21.3.24  Signum (sgn or sign) function - misc. additions   sgncl 34534
                  21.3.24.1  Operations on words   ccatmulgnn0dir 34550
            21.3.25  Polynomials with real coefficients - misc additions   plymul02 34554
            21.3.26  Descartes's rule of signs   signspval 34560
                  21.3.26.1  Sign changes in a word over real numbers   signspval 34560
                  21.3.26.2  Counting sign changes in a word over real numbers   signslema 34570
            21.3.27  Number Theory   iblidicc 34600
                  21.3.27.1  Representations of a number as sums of integers   crepr 34616
                  21.3.27.2  Vinogradov Trigonometric Sums and the Circle Method   cvts 34643
                  21.3.27.3  The Ternary Goldbach Conjecture: Final Statement   ax-hgt749 34652
            21.3.28  Elementary Geometry   cstrkg2d 34672
                  ​*21.3.28.1  Two-dimensional geometry   cstrkg2d 34672
                  21.3.28.2  Outer Five Segment (not used, no need to move to main)   cafs 34677
            ​*21.3.29  LeftPad Project   clpad 34682
      ​​*21.4  Mathbox for Jonathan Ben-Naim
            21.4.1  First-order logic and set theory   bnj170 34705
            21.4.2  Well founded induction and recursion   bnj110 34865
            21.4.3  The existence of a minimal element in certain classes   bnj69 35017
            21.4.4  Well-founded induction   bnj1204 35019
            21.4.5  Well-founded recursion, part 1 of 3   bnj60 35069
            21.4.6  Well-founded recursion, part 2 of 3   bnj1500 35075
            21.4.7  Well-founded recursion, part 3 of 3   bnj1522 35079
      ​21.5  Mathbox for BTernaryTau
            21.5.1  First-order logic   nfan1c 35080
                  21.5.1.1  Auxiliary axiom schemes   nfan1c 35080
            21.5.2  ZF set theory   exdifsn 35086
                  21.5.2.1  Finitism   prcinf 35101
                  21.5.2.2  Global choice   gblacfnacd 35106
            21.5.3  Real and complex numbers   zltp1ne 35108
            21.5.4  Graph theory   lfuhgr 35116
                  21.5.4.1  Acyclic graphs   cacycgr 35140
      ​21.6  Mathbox for Mario Carneiro
            21.6.1  Predicate calculus with all distinct variables   ax-7d 35157
            21.6.2  Miscellaneous stuff   quartfull 35163
            21.6.3  Derangements and the Subfactorial   deranglem 35164
            21.6.4  The Erdős-Szekeres theorem   erdszelem1 35189
            21.6.5  The Kuratowski closure-complement theorem   kur14lem1 35204
            21.6.6  Retracts and sections   cretr 35215
            21.6.7  Path-connected and simply connected spaces   cpconn 35217
            21.6.8  Covering maps   ccvm 35253
            21.6.9  Normal numbers   snmlff 35327
            21.6.10  Godel-sets of formulas - part 1   cgoe 35331
            21.6.11  Godel-sets of formulas - part 2   cgon 35430
            21.6.12  Models of ZF   cgze 35444
            ​*21.6.13  Metamath formal systems   cmcn 35458
            21.6.14  Grammatical formal systems   cm0s 35583
            21.6.15  Models of formal systems   cmuv 35603
            21.6.16  Splitting fields   ccpms 35625
            21.6.17  p-adic number fields   czr 35645
      ​​*21.7  Mathbox for Filip Cernatescu
      ​21.8  Mathbox for Paul Chapman
            21.8.1  Real and complex numbers (cont.)   climuzcnv 35669
            21.8.2  Miscellaneous theorems   elfzm12 35673
      ​21.9  Mathbox for Hongxiu Chen
      ​21.10  Mathbox for Adrian Ducourtial
            21.10.1  Propositional calculus   currybi 35686
            21.10.2  Clone theory   ccloneop 35688
      ​21.11  Mathbox for Scott Fenton
            21.11.1  ZFC Axioms in primitive form   axextprim 35694
            21.11.2  Untangled classes   untelirr 35701
            21.11.3  Extra propositional calculus theorems   3jaodd 35708
            21.11.4  Misc. Useful Theorems   nepss 35711
            21.11.5  Properties of real and complex numbers   sqdivzi 35721
            21.11.6  Infinite products   iprodefisumlem 35733
            21.11.7  Factorial limits   faclimlem1 35736
            21.11.8  Greatest common divisor and divisibility   gcd32 35742
            21.11.9  Properties of relationships   dftr6 35744
            21.11.10  Properties of functions and mappings   funpsstri 35760
            21.11.11  Set induction (or epsilon induction)   setinds 35773
            21.11.12  Ordinal numbers   elpotr 35776
            21.11.13  Defined equality axioms   axextdfeq 35792
            21.11.14  Hypothesis builders   hbntg 35800
            21.11.15  Well-founded zero, successor, and limits   cwsuc 35805
            21.11.16  Quantifier-free definitions   ctxp 35825
            21.11.17  Alternate ordered pairs   caltop 35951
            21.11.18  Geometry in the Euclidean space   cofs 35977
                  21.11.18.1  Congruence properties   cofs 35977
                  21.11.18.2  Betweenness properties   btwntriv2 36007
                  21.11.18.3  Segment Transportation   ctransport 36024
                  21.11.18.4  Properties relating betweenness and congruence   cifs 36030
                  21.11.18.5  Connectivity of betweenness   btwnconn1lem1 36082
                  21.11.18.6  Segment less than or equal to   csegle 36101
                  21.11.18.7  Outside-of relationship   coutsideof 36114
                  21.11.18.8  Lines and Rays   cline2 36129
            21.11.19  Forward difference   cfwddif 36153
            21.11.20  Rank theorems   rankung 36161
            21.11.21  Hereditarily Finite Sets   chf 36167
      ​21.12  Mathbox for Gino Giotto
            21.12.1  Equality theorems.   rmoeqi 36182
                  21.12.1.1  Inference versions.   rmoeqi 36182
                  21.12.1.2  Deduction versions.   rmoeqdv 36207
            21.12.2  Change bound variables.   in-ax8 36219
                  21.12.2.1  Change bound variables and domains.   cbvralvw2 36221
                  21.12.2.2  Change bound variables, deduction versions.   cbvmodavw 36245
                  21.12.2.3  Change bound variables and domains, deduction versions.   cbvrmodavw2 36278
            21.12.3  Study of ax-mulf usage.   mpomulnzcnf 36294
      ​21.13  Mathbox for Jeff Hankins
            21.13.1  Miscellany   a1i14 36295
            21.13.2  Basic topological facts   topbnd 36319
            21.13.3  Topology of the real numbers   ivthALT 36330
            21.13.4  Refinements   cfne 36331
            21.13.5  Neighborhood bases determine topologies   neibastop1 36354
            21.13.6  Lattice structure of topologies   topmtcl 36358
            21.13.7  Filter bases   fgmin 36365
            21.13.8  Directed sets, nets   tailfval 36367
      ​21.14  Mathbox for Anthony Hart
            21.14.1  Propositional Calculus   tb-ax1 36378
            21.14.2  Predicate Calculus   nalfal 36398
            21.14.3  Miscellaneous single axioms   meran1 36406
            21.14.4  Connective Symmetry   negsym1 36412
      ​21.15  Mathbox for Chen-Pang He
            21.15.1  Ordinal topology   ontopbas 36423
      ​21.16  Mathbox for Jeff Hoffman
            21.16.1  Inferences for finite induction on generic function values   fveleq 36446
            21.16.2  gdc.mm   nnssi2 36450
      ​21.17  Mathbox for Matthew House
            21.17.1  Relations on well-ordered indexed unions   weiunlem1 36457
      ​21.18  Mathbox for Asger C. Ipsen
            21.18.1  Continuous nowhere differentiable functions   dnival 36466
      ​​*21.19  Mathbox for BJ
            *21.19.1  Propositional calculus   bj-mp2c 36535
                  *21.19.1.1  Derived rules of inference   bj-mp2c 36535
                  ​*21.19.1.2  A syntactic theorem   bj-0 36537
                  21.19.1.3  Minimal implicational calculus   bj-a1k 36539
                  ​*21.19.1.4  Positive calculus   bj-syl66ib 36550
                  21.19.1.5  Implication and negation   bj-con2com 36556
                  ​*21.19.1.6  Disjunction   bj-jaoi1 36566
                  ​*21.19.1.7  Logical equivalence   bj-dfbi4 36568
                  21.19.1.8  The conditional operator for propositions   bj-consensus 36573
                  ​*21.19.1.9  Propositional calculus: miscellaneous   bj-imbi12 36578
            ​*21.19.2  Modal logic   bj-axdd2 36587
            ​*21.19.3  Provability logic   cprvb 36592
            ​*21.19.4  First-order logic   bj-genr 36601
                  21.19.4.1  Adding ax-gen   bj-genr 36601
                  21.19.4.2  Adding ax-4   bj-2alim 36605
                  21.19.4.3  Adding ax-5   bj-ax12wlem 36639
                  21.19.4.4  Equality and substitution   bj-ssbeq 36648
                  21.19.4.5  Adding ax-6   bj-spimvwt 36664
                  21.19.4.6  Adding ax-7   bj-cbvexw 36671
                  21.19.4.7  Membership predicate, ax-8 and ax-9   bj-ax89 36673
                  21.19.4.8  Adding ax-11   bj-alcomexcom 36675
                  21.19.4.9  Adding ax-12   axc11n11 36677
                  21.19.4.10  Nonfreeness   wnnf 36718
                  21.19.4.11  Adding ax-13   bj-axc10 36778
                  ​*21.19.4.12  Removing dependencies on ax-13 (and ax-11)   bj-axc10v 36788
                  ​*21.19.4.13  Distinct var metavariables   bj-hbaeb2 36813
                  ​*21.19.4.14  Around ~ equsal   bj-equsal1t 36817
                  ​*21.19.4.15  Some Principia Mathematica proofs   stdpc5t 36822
                  21.19.4.16  Alternate definition of substitution   bj-sbsb 36832
                  21.19.4.17  Lemmas for substitution   bj-sbf3 36834
                  21.19.4.18  Existential uniqueness   bj-eu3f 36836
                  ​*21.19.4.19  First-order logic: miscellaneous   bj-sblem1 36837
            21.19.5  Set theory   eliminable1 36854
                  ​*21.19.5.1  Eliminability of class terms   eliminable1 36854
                  ​*21.19.5.2  Classes without the axiom of extensionality   bj-denoteslem 36866
                  21.19.5.3  Characterization among sets versus among classes   elelb 36892
                  ​*21.19.5.4  The nonfreeness quantifier for classes   bj-nfcsym 36894
                  ​*21.19.5.5  Lemmas for class substitution   bj-sbeqALT 36895
                  21.19.5.6  Removing some axiom requirements and disjoint variable conditions   bj-exlimvmpi 36906
                  ​*21.19.5.7  Class abstractions   bj-elabd2ALT 36920
                  21.19.5.8  Generalized class abstractions   bj-cgab 36928
                  ​*21.19.5.9  Restricted nonfreeness   wrnf 36936
                  ​*21.19.5.10  Russell's paradox   bj-ru1 36938
                  21.19.5.11  Curry's paradox in set theory   currysetlem 36940
                  ​*21.19.5.12  Some disjointness results   bj-n0i 36946
                  ​*21.19.5.13  Complements on direct products   bj-xpimasn 36950
                  ​*21.19.5.14  "Singletonization" and tagging   bj-snsetex 36958
                  ​*21.19.5.15  Tuples of classes   bj-cproj 36985
                  ​*21.19.5.16  Set theory: elementary operations relative to a universe   bj-rcleqf 37020
                  ​*21.19.5.17  Axioms for finite unions   bj-abex 37025
                  ​*21.19.5.18  Set theory: miscellaneous   eleq2w2ALT 37042
                  ​*21.19.5.19  Evaluation at a class   bj-evaleq 37067
                  21.19.5.20  Elementwise operations   celwise 37074
                  ​*21.19.5.21  Elementwise intersection (families of sets induced on a subset)   bj-rest00 37076
                  21.19.5.22  Moore collections (complements)   bj-raldifsn 37095
                  21.19.5.23  Maps-to notation for functions with three arguments   bj-0nelmpt 37111
                  ​*21.19.5.24  Currying   csethom 37117
                  ​*21.19.5.25  Setting components of extensible structures   cstrset 37129
            ​*21.19.6  Extended real and complex numbers, real and complex projective lines   bj-nfald 37132
                  21.19.6.1  Complements on class abstractions of ordered pairs and binary relations   bj-nfald 37132
                  ​*21.19.6.2  Identity relation (complements)   bj-opabssvv 37145
                  ​*21.19.6.3  Functionalized identity (diagonal in a Cartesian square)   cdiag2 37167
                  ​*21.19.6.4  Direct image and inverse image   cimdir 37173
                  ​*21.19.6.5  Extended numbers and projective lines as sets   cfractemp 37191
                  ​*21.19.6.6  Addition and opposite   caddcc 37232
                  ​*21.19.6.7  Order relation on the extended reals   cltxr 37236
                  ​*21.19.6.8  Argument, multiplication and inverse   carg 37238
                  21.19.6.9  The canonical bijection from the finite ordinals   ciomnn 37244
                  21.19.6.10  Divisibility   cnnbar 37255
            ​*21.19.7  Monoids   bj-smgrpssmgm 37263
                  ​*21.19.7.1  Finite sums in monoids   cfinsum 37278
            ​*21.19.8  Affine, Euclidean, and Cartesian geometry   bj-fvimacnv0 37281
                  *21.19.8.1  Real vector spaces   bj-fvimacnv0 37281
                  ​*21.19.8.2  Complex numbers (supplements)   bj-subcom 37303
                  ​*21.19.8.3  Barycentric coordinates   bj-bary1lem 37305
            21.19.9  Monoid of endomorphisms   cend 37308
      ​21.20  Mathbox for Jim Kingdon
                  21.20.0.1  Circle constant   taupilem3 37314
                  21.20.0.2  Number theory   dfgcd3 37319
                  21.20.0.3  Real numbers   irrdifflemf 37320
      ​21.21  Mathbox for ML
            21.21.1  Miscellaneous   csbrecsg 37323
            21.21.2  Cartesian exponentiation   cfinxp 37378
            21.21.3  Topology   iunctb2 37398
                  ​*21.21.3.1  Pi-base theorems   pibp16 37408
      ​21.22  Mathbox for Wolf Lammen
            21.22.1  1. Bootstrapping   wl-section-boot 37417
            21.22.2  Implication chains   wl-section-impchain 37441
            21.22.3  Theorems around the conditional operator   wl-ifp-ncond1 37459
            21.22.4  Alternative development of hadd, cadd   wl-df-3xor 37463
            21.22.5  An alternative axiom ~ ax-13   ax-wl-13v 37488
            21.22.6  Bootstrapping set theory with classes   wl-cleq-0 37490
            21.22.7  Other stuff   wl-mps 37502
      ​21.23  Mathbox for Brendan Leahy
      ​21.24  Mathbox for Jeff Madsen
            21.24.1  Logic and set theory   unirep 37715
            21.24.2  Real and complex numbers; integers   filbcmb 37741
            21.24.3  Sequences and sums   sdclem2 37743
            21.24.4  Topology   subspopn 37753
            21.24.5  Metric spaces   metf1o 37756
            21.24.6  Continuous maps and homeomorphisms   constcncf 37763
            21.24.7  Boundedness   ctotbnd 37767
            21.24.8  Isometries   cismty 37799
            21.24.9  Heine-Borel Theorem   heibor1lem 37810
            21.24.10  Banach Fixed Point Theorem   bfplem1 37823
            21.24.11  Euclidean space   crrn 37826
            21.24.12  Intervals (continued)   ismrer1 37839
            21.24.13  Operation properties   cass 37843
            21.24.14  Groups and related structures   cmagm 37849
            21.24.15  Group homomorphism and isomorphism   cghomOLD 37884
            21.24.16  Rings   crngo 37895
            21.24.17  Division Rings   cdrng 37949
            21.24.18  Ring homomorphisms   crngohom 37961
            21.24.19  Commutative rings   ccm2 37990
            21.24.20  Ideals   cidl 38008
            21.24.21  Prime rings and integral domains   cprrng 38047
            21.24.22  Ideal generators   cigen 38060
      ​21.25  Mathbox for Giovanni Mascellani
            ​*21.25.1  Tools for automatic proof building   efald2 38079
            ​*21.25.2  Tseitin axioms   fald 38130
            ​*21.25.3  Equality deductions   iuneq2f 38157
            ​*21.25.4  Miscellanea   orcomdd 38168
      ​21.26  Mathbox for Peter Mazsa
            21.26.1  Notations   cxrn 38175
            21.26.2  Preparatory theorems   el2v1 38218
            21.26.3  Range Cartesian product   df-xrn 38367
            21.26.4  Cosets by ` R `   df-coss 38407
            21.26.5  Relations   df-rels 38481
            21.26.6  Subset relations   df-ssr 38494
            21.26.7  Reflexivity   df-refs 38506
            21.26.8  Converse reflexivity   df-cnvrefs 38521
            21.26.9  Symmetry   df-syms 38538
            21.26.10  Reflexivity and symmetry   symrefref2 38559
            21.26.11  Transitivity   df-trs 38568
            21.26.12  Equivalence relations   df-eqvrels 38580
            21.26.13  Redundancy   df-redunds 38619
            21.26.14  Domain quotients   df-dmqss 38634
            21.26.15  Equivalence relations on domain quotients   df-ers 38659
            21.26.16  Functions   df-funss 38676
            21.26.17  Disjoints vs. converse functions   df-disjss 38699
            21.26.18  Antisymmetry   df-antisymrel 38756
            21.26.19  Partitions: disjoints on domain quotients   df-parts 38761
            21.26.20  Partition-Equivalence Theorems   disjim 38777
      ​21.27  Mathbox for Rodolfo Medina
            21.27.1  Partitions   prtlem60 38849
      ​​*21.28  Mathbox for Norm Megill
            *21.28.1  Obsolete schemes ax-c4,c5,c7,c10,c11,c11n,c15,c9,c14,c16   ax-c5 38879
            ​*21.28.2  Rederive new axioms ax-4, ax-10, ax-6, ax-12, ax-13 from old   axc5 38889
            ​*21.28.3  Legacy theorems using obsolete axioms   ax5ALT 38903
            21.28.4  Experiments with weak deduction theorem   elimhyps 38957
            21.28.5  Miscellanea   cnaddcom 38968
            21.28.6  Atoms, hyperplanes, and covering in a left vector space (or module)   clsa 38970
            21.28.7  Functionals and kernels of a left vector space (or module)   clfn 39053
            21.28.8  Opposite rings and dual vector spaces   cld 39119
            21.28.9  Ortholattices and orthomodular lattices   cops 39168
            21.28.10  Atomic lattices with covering property   ccvr 39258
            21.28.11  Hilbert lattices   chlt 39346
            21.28.12  Projective geometries based on Hilbert lattices   clln 39488
            21.28.13  Construction of a vector space from a Hilbert lattice   cdlema1N 39788
            21.28.14  Construction of involution and inner product from a Hilbert lattice   clpoN 41477
      ​21.29  Mathbox for metakunt
            21.29.1  Commutative Semiring   ccsrg 41963
            21.29.2  General helpful statements   rhmzrhval 41966
            21.29.3  Some gcd and lcm results   12gcd5e1 41999
            21.29.4  Least common multiple inequality theorem   3factsumint1 42017
            21.29.5  Logarithm inequalities   3exp7 42049
            21.29.6  Miscellaneous results for AKS formalisation   intlewftc 42057
            21.29.7  Sticks and stones   sticksstones1 42142
            21.29.8  Continuation AKS   aks6d1c6lem1 42166
            21.29.9  Permutation results   metakunt1 42201
            21.29.10  Unused lemmas scheduled for deletion   fac2xp3 42235
      ​21.30  Mathbox for Steven Nguyen
            21.30.1  Utility theorems   intnanrt 42239
            ​*21.30.2  Arithmetic theorems   c0exALT 42286
            21.30.3  Exponents and divisibility   oexpreposd 42350
            21.30.4  Trigonometry and Calculus   tanhalfpim 42378
            21.30.5  Real subtraction   cresub 42388
            21.30.6  Structures   sn-base0 42498
            ​*21.30.7  Projective spaces   cprjsp 42604
            21.30.8  Basic reductions for Fermat's Last Theorem   dffltz 42637
            ​*21.30.9  Exemplar theorems   iddii 42667
                  ​*21.30.9.1  Standard replacements of ax-10 , ax-11 , ax-12   nfa1w 42678
      ​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 42696
            21.33.2  Additional theory of functions   imaiinfv 42697
            21.33.3  Additional topology   elrfi 42698
            21.33.4  Characterization of closure operators. Kuratowski closure axioms   ismrcd1 42702
            21.33.5  Algebraic closure systems   cnacs 42706
            21.33.6  Miscellanea 1. Map utilities   constmap 42717
            21.33.7  Miscellanea for polynomials   mptfcl 42724
            21.33.8  Multivariate polynomials over the integers   cmzpcl 42725
            21.33.9  Miscellanea for Diophantine sets 1   coeq0i 42757
            21.33.10  Diophantine sets 1: definitions   cdioph 42759
            21.33.11  Diophantine sets 2 miscellanea   ellz1 42771
            21.33.12  Diophantine sets 2: union and intersection. Monotone Boolean algebra   diophin 42776
            21.33.13  Diophantine sets 3: construction   diophrex 42779
            21.33.14  Diophantine sets 4 miscellanea   2sbcrex 42788
            21.33.15  Diophantine sets 4: Quantification   rexrabdioph 42798
            21.33.16  Diophantine sets 5: Arithmetic sets   rabdiophlem1 42805
            21.33.17  Diophantine sets 6: reusability. renumbering of variables   eldioph4b 42815
            21.33.18  Pigeonhole Principle and cardinality helpers   fphpd 42820
            21.33.19  A non-closed set of reals is infinite   rencldnfilem 42824
            21.33.20  Lagrange's rational approximation theorem   irrapxlem1 42826
            21.33.21  Pell equations 1: A nontrivial solution always exists   pellexlem1 42833
            21.33.22  Pell equations 2: Algebraic number theory of the solution set   csquarenn 42840
            21.33.23  Pell equations 3: characterizing fundamental solution   infmrgelbi 42882
            ​*21.33.24  Logarithm laws generalized to an arbitrary base   reglogcl 42894
            21.33.25  Pell equations 4: the positive solution group is infinite cyclic   pellfund14 42902
            21.33.26  X and Y sequences 1: Definition and recurrence laws   crmx 42904
            21.33.27  Ordering and induction lemmas for the integers   monotuz 42946
            21.33.28  X and Y sequences 2: Order properties   rmxypos 42952
            21.33.29  Congruential equations   congtr 42970
            21.33.30  Alternating congruential equations   acongid 42980
            21.33.31  Additional theorems on integer divisibility   coprmdvdsb 42990
            21.33.32  X and Y sequences 3: Divisibility properties   jm2.18 42993
            21.33.33  X and Y sequences 4: Diophantine representability of Y   jm2.27a 43010
            21.33.34  X and Y sequences 5: Diophantine representability of X, ^, _C   rmxdiophlem 43020
            21.33.35  Uncategorized stuff not associated with a major project   setindtr 43029
            21.33.36  More equivalents of the Axiom of Choice   axac10 43038
            21.33.37  Finitely generated left modules   clfig 43072
            21.33.38  Noetherian left modules I   clnm 43080
            21.33.39  Addenda for structure powers   pwssplit4 43094
            21.33.40  Every set admits a group structure iff choice   unxpwdom3 43100
            21.33.41  Noetherian rings and left modules II   clnr 43114
            21.33.42  Hilbert's Basis Theorem   cldgis 43126
            21.33.43  Additional material on polynomials [DEPRECATED]   cmnc 43136
            21.33.44  Degree and minimal polynomial of algebraic numbers   cdgraa 43145
            21.33.45  Algebraic integers I   citgo 43162
            21.33.46  Endomorphism algebra   cmend 43176
            21.33.47  Cyclic groups and order   idomodle 43196
            21.33.48  Cyclotomic polynomials   ccytp 43202
            21.33.49  Miscellaneous topology   fgraphopab 43208
      ​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 43222
            21.36.2  Natural addition of Cantor normal forms   oawordex2 43332
            21.36.3  Surreal Contributions   abeqabi 43414
            21.36.4  Short Studies   nlimsuc 43447
                  21.36.4.1  Additional work on conditional logical operator   ifpan123g 43465
                  21.36.4.2  Sophisms   rp-fakeimass 43518
                  ​*21.36.4.3  Finite Sets   rp-isfinite5 43523
                  21.36.4.4  General Observations   intabssd 43525
                  21.36.4.5  Infinite Sets   pwelg 43566
                  ​*21.36.4.6  Finite intersection property   fipjust 43571
                  21.36.4.7  RP ADDTO: Subclasses and subsets   rababg 43580
                  21.36.4.8  RP ADDTO: The intersection of a class   elinintab 43581
                  21.36.4.9  RP ADDTO: Theorems requiring subset and intersection existence   elinintrab 43583
                  21.36.4.10  RP ADDTO: Relations   xpinintabd 43586
                  ​*21.36.4.11  RP ADDTO: Functions   elmapintab 43602
                  ​*21.36.4.12  RP ADDTO: Finite induction (for finite ordinals)   cnvcnvintabd 43606
                  21.36.4.13  RP ADDTO: First and second members of an ordered pair   elcnvlem 43607
                  21.36.4.14  RP ADDTO: The reflexive and transitive properties of relations   undmrnresiss 43610
                  21.36.4.15  RP ADDTO: Basic properties of closures   cleq2lem 43614
                  21.36.4.16  RP REPLACE: Definitions and basic properties of transitive closures   trcleq2lemRP 43636
                  ​*21.36.4.17  Additions for square root; absolute value   sqrtcvallem1 43637
            21.36.5  Additional statements on relations and subclasses   al3im 43653
                  21.36.5.1  Transitive relations (not to be confused with transitive classes).   trrelind 43671
                  21.36.5.2  Reflexive closures   crcl 43678
                  ​*21.36.5.3  Finite relationship composition.   relexp2 43683
                  21.36.5.4  Transitive closure of a relation   dftrcl3 43726
                  ​*21.36.5.5  Adapted from Frege   frege77d 43752
            ​*21.36.6  Propositions from _Begriffsschrift_   dfxor4 43772
                  *21.36.6.1  _Begriffsschrift_ Chapter I   dfxor4 43772
                  ​*21.36.6.2  _Begriffsschrift_ Notation hints   whe 43778
                  21.36.6.3  _Begriffsschrift_ Chapter II Implication   ax-frege1 43796
                  21.36.6.4  _Begriffsschrift_ Chapter II Implication and Negation   axfrege28 43835
                  ​*21.36.6.5  _Begriffsschrift_ Chapter II with logical equivalence   axfrege52a 43862
                  21.36.6.6  _Begriffsschrift_ Chapter II with equivalence of sets   axfrege52c 43893
                  ​*21.36.6.7  _Begriffsschrift_ Chapter II with equivalence of classes   frege53c 43920
                  ​*21.36.6.8  _Begriffsschrift_ Chapter III Properties hereditary in a sequence   dffrege69 43938
                  ​*21.36.6.9  _Begriffsschrift_ Chapter III Following in a sequence   dffrege76 43945
                  ​*21.36.6.10  _Begriffsschrift_ Chapter III Member of sequence   dffrege99 43968
                  ​*21.36.6.11  _Begriffsschrift_ Chapter III Single-valued procedures   dffrege115 43984
            ​*21.36.7  Exploring Topology via Seifert and Threlfall   enrelmap 44003
                  *21.36.7.1  Equinumerosity of sets of relations and maps   enrelmap 44003
                  ​*21.36.7.2  Generic Pseudoclosure Spaces, Pseudointerior Spaces, and Pseudoneighborhoods   or3or 44029
                  ​*21.36.7.3  Generic Neighborhood Spaces   gneispa 44136
            ​*21.36.8  Exploring Higher Homotopy via Kerodon   k0004lem1 44153
                  *21.36.8.1  Simplicial Sets   k0004lem1 44153
      ​21.37  Mathbox for Stanislas Polu
            21.37.1  IMO Problems   wwlemuld 44162
                  21.37.1.1  IMO 1972 B2   wwlemuld 44162
            ​*21.37.2  INT Inequalities Proof Generator   int-addcomd 44179
            ​*21.37.3  N-Digit Addition Proof Generator   unitadd 44201
            21.37.4  AM-GM (for k = 2,3,4)   gsumws3 44202
      ​21.38  Mathbox for Rohan Ridenour
            21.38.1  Misc   spALT 44207
            21.38.2  Monoid rings   cmnring 44218
            21.38.3  Shorter primitive equivalent of ax-groth   gru0eld 44241
                  21.38.3.1  Grothendieck universes are closed under collection   gru0eld 44241
                  21.38.3.2  Minimal universes   ismnu 44273
                  21.38.3.3  Primitive equivalent of ax-groth   expandan 44300
      ​21.39  Mathbox for Steve Rodriguez
            21.39.1  Miscellanea   nanorxor 44317
            21.39.2  Ratio test for infinite series convergence and divergence   dvgrat 44324
            21.39.3  Multiples   reldvds 44327
            21.39.4  Function operations   caofcan 44335
            21.39.5  Calculus   lhe4.4ex1a 44341
            21.39.6  The generalized binomial coefficient operation   cbcc 44348
            21.39.7  Binomial series   uzmptshftfval 44358
      ​21.40  Mathbox for Andrew Salmon
            21.40.1  Principia Mathematica * 10   pm10.12 44370
            21.40.2  Principia Mathematica * 11   2alanimi 44384
            21.40.3  Predicate Calculus   sbeqal1 44410
            21.40.4  Principia Mathematica * 13 and * 14   pm13.13a 44419
            21.40.5  Set Theory   elnev 44450
            21.40.6  Arithmetic   addcomgi 44468
            21.40.7  Geometry   cplusr 44469
      ​​*21.41  Mathbox for Alan Sare
            21.41.1  Auxiliary theorems for the Virtual Deduction tool   idiALT 44491
            21.41.2  Supplementary unification deductions   bi1imp 44495
            21.41.3  Conventional Metamath proofs, some derived from VD proofs   iidn3 44514
            21.41.4  What is Virtual Deduction?   wvd1 44582
            21.41.5  Virtual Deduction Theorems   df-vd1 44583
            21.41.6  Theorems proved using Virtual Deduction   trsspwALT 44831
            21.41.7  Theorems proved using Virtual Deduction with mmj2 assistance   simplbi2VD 44859
            21.41.8  Virtual Deduction transcriptions of textbook proofs   sb5ALTVD 44926
            21.41.9  Theorems proved using conjunction-form Virtual Deduction   elpwgdedVD 44930
            21.41.10  Theorems with a VD proof in conventional notation derived from a VD proof   suctrALT3 44937
            ​*21.41.11  Theorems with a proof in conventional notation derived from a VD proof   notnotrALT2 44940
      ​21.42  Mathbox for Eric Schmidt
      ​21.43  Mathbox for Glauco Siliprandi
            21.43.1  Miscellanea   evth2f 44983
            21.43.2  Functions   fnresdmss 45140
            21.43.3  Ordering on real numbers - Real and complex numbers basic operations   sub2times 45252
            21.43.4  Real intervals   gtnelioc 45473
            21.43.5  Finite sums   fsummulc1f 45555
            21.43.6  Finite multiplication of numbers and finite multiplication of functions   fmul01 45564
            21.43.7  Limits   clim1fr1 45585
                  21.43.7.1  Inferior limit (lim inf)   clsi 45735
                  ​*21.43.7.2  Limits for sequences of extended real numbers   clsxlim 45802
            21.43.8  Trigonometry   coseq0 45848
            21.43.9  Continuous Functions   mulcncff 45854
            21.43.10  Derivatives   dvsinexp 45895
            21.43.11  Integrals   itgsin0pilem1 45934
            21.43.12  Stone Weierstrass theorem - real version   stoweidlem1 45985
            21.43.13  Wallis' product for π   wallispilem1 46049
            21.43.14  Stirling's approximation formula for ` n ` factorial   stirlinglem1 46058
            21.43.15  Dirichlet kernel   dirkerval 46075
            21.43.16  Fourier Series   fourierdlem1 46092
            21.43.17  e is transcendental   elaa2lem 46217
            21.43.18  n-dimensional Euclidean space   rrxtopn 46268
            21.43.19  Basic measure theory   csalg 46292
                  ​*21.43.19.1  σ-Algebras   csalg 46292
                  21.43.19.2  Sum of nonnegative extended reals   csumge0 46346
                  ​*21.43.19.3  Measures   cmea 46433
                  ​*21.43.19.4  Outer measures and Caratheodory's construction   come 46473
                  ​*21.43.19.5  Lebesgue measure on n-dimensional Real numbers   covoln 46520
                  ​*21.43.19.6  Measurable functions   csmblfn 46679
      ​21.44  Mathbox for Saveliy Skresanov
            21.44.1  Ceva's theorem   sigarval 46834
            21.44.2  Simple groups   simpcntrab 46854
      ​21.45  Mathbox for Ender Ting
            21.45.1  Increasing sequences and subsequences   et-ltneverrefl 46855
      ​21.46  Mathbox for Jarvin Udandy
      ​21.47  Mathbox for Adhemar
            ​*21.47.1  Minimal implicational calculus   adh-minim 46979
      ​21.48  Mathbox for Alexander van der Vekens
            21.48.1  General auxiliary theorems (1)   n0nsn2el 47003
                  21.48.1.1  Unordered and ordered pairs - extension for singletons   n0nsn2el 47003
                  21.48.1.2  Unordered and ordered pairs - extension for unordered pairs   elprneb 47007
                  21.48.1.3  Unordered and ordered pairs - extension for ordered pairs   oppr 47008
                  21.48.1.4  Relations - extension   eubrv 47013
                  21.48.1.5  Definite description binder (inverted iota) - extension   iota0def 47016
                  21.48.1.6  Functions - extension   fveqvfvv 47018
            21.48.2  Alternative for Russell's definition of a description binder   caiota 47061
            21.48.3  Double restricted existential uniqueness   r19.32 47076
                  21.48.3.1  Restricted quantification (extension)   r19.32 47076
                  21.48.3.2  Restricted uniqueness and "at most one" quantification   reuf1odnf 47085
                  21.48.3.3  Analogs to Existential uniqueness (double quantification)   2reu3 47088
                  21.48.3.4  Additional theorems for double restricted existential uniqueness   2reu8i 47091
            ​*21.48.4  Alternative definitions of function and operation values   wdfat 47094
                  21.48.4.1  Restricted quantification (extension)   ralbinrald 47100
                  21.48.4.2  The universal class (extension)   nvelim 47101
                  21.48.4.3  Introduce the Axiom of Power Sets (extension)   alneu 47102
                  21.48.4.4  Predicate "defined at"   dfateq12d 47104
                  21.48.4.5  Alternative definition of the value of a function   dfafv2 47110
                  21.48.4.6  Alternative definition of the value of an operation   aoveq123d 47156
            ​*21.48.5  Alternative definitions of function values (2)   cafv2 47186
            21.48.6  General auxiliary theorems (2)   an4com24 47246
                  21.48.6.1  Logical conjunction - extension   an4com24 47246
                  21.48.6.2  Abbreviated conjunction and disjunction of three wff's - extension   3an4ancom24 47247
                  21.48.6.3  Negated membership (alternative)   cnelbr 47249
                  21.48.6.4  The empty set - extension   ralralimp 47256
                  21.48.6.5  Indexed union and intersection - extension   otiunsndisjX 47257
                  21.48.6.6  Functions - extension   fvifeq 47258
                  21.48.6.7  Maps-to notation - extension   fvmptrab 47270
                  21.48.6.8  Subtraction - extension   cnambpcma 47272
                  21.48.6.9  Ordering on reals (cont.) - extension   leaddsuble 47275
                  21.48.6.10  Imaginary and complex number properties - extension   readdcnnred 47281
                  21.48.6.11  Nonnegative integers (as a subset of complex numbers) - extension   nn0resubcl 47286
                  21.48.6.12  Integers (as a subset of complex numbers) - extension   zgeltp1eq 47287
                  21.48.6.13  Decimal arithmetic - extension   1t10e1p1e11 47288
                  21.48.6.14  Upper sets of integers - extension   eluzge0nn0 47290
                  21.48.6.15  Infinity and the extended real number system (cont.) - extension   nltle2tri 47291
                  21.48.6.16  Finite intervals of integers - extension   ssfz12 47292
                  21.48.6.17  Half-open integer ranges - extension   fzopred 47300
                  21.48.6.18  The modulo (remainder) operation - extension   fldivmod 47306
                  21.48.6.19  The infinite sequence builder "seq"   smonoord 47324
                  21.48.6.20  Finite and infinite sums - extension   fsummsndifre 47325
                  21.48.6.21  Extensible structures - extension   setsidel 47329
            ​*21.48.7  Preimages of function values   preimafvsnel 47332
            ​*21.48.8  Partitions of real intervals   ciccp 47366
            21.48.9  Shifting functions with an integer range domain   fargshiftfv 47392
            21.48.10  Words over a set (extension)   lswn0 47397
                  21.48.10.1  Last symbol of a word - extension   lswn0 47397
            21.48.11  Unordered pairs   wich 47398
                  21.48.11.1  Interchangeable setvar variables   wich 47398
                  21.48.11.2  Set of unordered pairs   sprid 47427
                  ​*21.48.11.3  Proper (unordered) pairs   prpair 47454
                  21.48.11.4  Set of proper unordered pairs   cprpr 47465
            21.48.12  Number theory (extension)   cfmtno 47480
                  ​*21.48.12.1  Fermat numbers   cfmtno 47480
                  ​*21.48.12.2  Mersenne primes   m2prm 47544
                  21.48.12.3  Proth's theorem   modexp2m1d 47565
                  21.48.12.4  Solutions of quadratic equations   quad1 47573
            ​*21.48.13  Even and odd numbers   ceven 47577
                  21.48.13.1  Definitions and basic properties   ceven 47577
                  21.48.13.2  Alternate definitions using the "divides" relation   dfeven2 47602
                  21.48.13.3  Alternate definitions using the "modulo" operation   dfeven3 47611
                  21.48.13.4  Alternate definitions using the "gcd" operation   iseven5 47617
                  21.48.13.5  Theorems of part 5 revised   zneoALTV 47622
                  21.48.13.6  Theorems of part 6 revised   odd2np1ALTV 47627
                  21.48.13.7  Theorems of AV's mathbox revised   0evenALTV 47641
                  21.48.13.8  Additional theorems   epoo 47656
                  21.48.13.9  Perfect Number Theorem (revised)   perfectALTVlem1 47674
            21.48.14  Number theory (extension 2)   cfppr 47677
                  ​*21.48.14.1  Fermat pseudoprimes   cfppr 47677
                  ​*21.48.14.2  Goldbach's conjectures   cgbe 47698
            21.48.15  Graph theory (extension)   cclnbgr 47771
                  21.48.15.1  Closed neighborhood of a vertex   cclnbgr 47771
                  ​*21.48.15.2  Semiclosed and semiopen neighborhoods (experimental)   dfsclnbgr2 47798
                  21.48.15.3  Induced subgraphs   cisubgr 47812
                  ​*21.48.15.4  Isomorphisms of graphs   cgrisom 47826
                  ​*21.48.15.5  Triangles in graphs   cgrtri 47870
                  ​*21.48.15.6  Star graphs   cstgr 47884
                  ​*21.48.15.7  Local isomorphisms of graphs   cgrlim 47909
                  ​*21.48.15.8  Generalized Petersen graphs   cgpg 47965
                  21.48.15.9  Loop-free graphs - extension   1hegrlfgr 48014
                  21.48.15.10  Walks - extension   cupwlks 48015
                  21.48.15.11  Edges of graphs expressed as sets of unordered pairs   upgredgssspr 48025
            21.48.16  Monoids (extension)   ovn0dmfun 48038
                  21.48.16.1  Auxiliary theorems   ovn0dmfun 48038
                  21.48.16.2  Magmas, Semigroups and Monoids (extension)   plusfreseq 48046
                  21.48.16.3  Examples and counterexamples for magmas, semigroups and monoids (extension)   opmpoismgm 48049
                  21.48.16.4  Group sum operation (extension 1)   gsumsplit2f 48062
            ​*21.48.17  Magmas and internal binary operations (alternate approach)   ccllaw 48065
                  *21.48.17.1  Laws for internal binary operations   ccllaw 48065
                  ​*21.48.17.2  Internal binary operations   cintop 48078
                  21.48.17.3  Alternative definitions for magmas and semigroups   cmgm2 48097
            21.48.18  Rings (extension)   lmod0rng 48111
                  21.48.18.1  Nonzero rings (extension)   lmod0rng 48111
                  21.48.18.2  Ideals as non-unital rings   lidldomn1 48113
                  21.48.18.3  The non-unital ring of even integers   0even 48119
                  21.48.18.4  A constructed not unital ring   cznrnglem 48141
                  ​*21.48.18.5  The category of non-unital rings (alternate definition)   crngcALTV 48145
                  ​*21.48.18.6  The category of (unital) rings (alternate definition)   cringcALTV 48169
            21.48.19  Basic algebraic structures (extension)   opeliun2xp 48216
                  21.48.19.1  Auxiliary theorems   opeliun2xp 48216
                  21.48.19.2  The binomial coefficient operation (extension)   bcpascm1 48234
                  21.48.19.3  The ` ZZ `-module ` ZZ X. ZZ `   zlmodzxzlmod 48237
                  21.48.19.4  Group sum operation (extension 2)   mgpsumunsn 48244
                  21.48.19.5  Symmetric groups (extension)   exple2lt6 48247
                  21.48.19.6  Divisibility (extension)   invginvrid 48250
                  21.48.19.7  The support of functions (extension)   rmsupp0 48251
                  21.48.19.8  Finitely supported functions (extension)   rmsuppfi 48255
                  21.48.19.9  Left modules (extension)   lmodvsmdi 48262
                  21.48.19.10  Associative algebras (extension)   assaascl0 48264
                  21.48.19.11  Univariate polynomials (extension)   ply1vr1smo 48266
                  21.48.19.12  Univariate polynomials (examples)   linply1 48277
            21.48.20  Linear algebra (extension)   cdmatalt 48280
                  ​*21.48.20.1  The subalgebras of diagonal and scalar matrices (extension)   cdmatalt 48280
                  ​*21.48.20.2  Linear combinations   clinc 48288
                  ​*21.48.20.3  Linear independence   clininds 48324
                  21.48.20.4  Simple left modules and the ` ZZ `-module   lmod1lem1 48371
                  21.48.20.5  Differences between (left) modules and (left) vector spaces   lvecpsslmod 48391
            21.48.21  Complexity theory   suppdm 48394
                  21.48.21.1  Auxiliary theorems   suppdm 48394
                  21.48.21.2  The modulo (remainder) operation (extension)   mod0mul 48407
                  21.48.21.3  Even and odd integers   nn0onn0ex 48411
                  21.48.21.4  The natural logarithm on complex numbers (extension)   logcxp0 48423
                  21.48.21.5  Division of functions   cfdiv 48425
                  21.48.21.6  Upper bounds   cbigo 48435
                  21.48.21.7  Logarithm to an arbitrary base (extension)   rege1logbrege0 48446
                  ​*21.48.21.8  The binary logarithm   fldivexpfllog2 48453
                  21.48.21.9  Binary length   cblen 48457
                  ​*21.48.21.10  Digits   cdig 48483
                  21.48.21.11  Nonnegative integer as sum of its shifted digits   dignn0flhalflem1 48503
                  21.48.21.12  Algorithms for the multiplication of nonnegative integers   nn0mulfsum 48512
                  ​*21.48.21.13  N-ary functions   cnaryf 48514
                  ​*21.48.21.14  The Ackermann function   citco 48545
            21.48.22  Elementary geometry (extension)   fv1prop 48587
                  21.48.22.1  Auxiliary theorems   fv1prop 48587
                  21.48.22.2  Real euclidean space of dimension 2   rrx2pxel 48599
                  21.48.22.3  Spheres and lines in real Euclidean spaces   cline 48615
      ​21.49  Mathbox for Zhi Wang
            21.49.1  Propositional calculus   pm4.71da 48677
            21.49.2  Predicate calculus with equality   dtrucor3 48686
                  21.49.2.1  Axiom scheme ax-5 (Distinctness)   dtrucor3 48686
            21.49.3  ZF Set Theory - start with the Axiom of Extensionality   ralbidb 48687
                  21.49.3.1  Restricted quantification   ralbidb 48687
                  21.49.3.2  The universal class   reuxfr1dd 48693
                  21.49.3.3  The empty set   ssdisjd 48694
                  21.49.3.4  Unordered and ordered pairs   vsn 48698
                  21.49.3.5  The union of a class   unilbss 48704
            21.49.4  ZF Set Theory - add the Axiom of Replacement   inpw 48705
                  21.49.4.1  Theorems requiring subset and intersection existence   inpw 48705
            21.49.5  ZF Set Theory - add the Axiom of Power Sets   opth1neg 48706
                  21.49.5.1  Ordered pair theorem   opth1neg 48706
                  21.49.5.2  Ordered-pair class abstractions (cont.)   brab2dd 48708
                  21.49.5.3  Functions   mof0 48711
                  21.49.5.4  Operations   fvconstr 48729
            21.49.6  ZF Set Theory - add the Axiom of Union   fmpodg 48732
                  21.49.6.1  Operations in maps-to notation (continued)   fmpodg 48732
                  21.49.6.2  Equinumerosity   fvconst0ci 48734
            21.49.7  Order sets   iccin 48738
                  21.49.7.1  Real number intervals   iccin 48738
            21.49.8  Moore spaces   mreuniss 48741
            ​*21.49.9  Topology   clduni 48742
                  21.49.9.1  Closure and interior   clduni 48742
                  21.49.9.2  Neighborhoods   neircl 48746
                  21.49.9.3  Subspace topologies   restcls2lem 48754
                  21.49.9.4  Limits and continuity in topological spaces   cnneiima 48758
                  21.49.9.5  Topological definitions using the reals   iooii 48759
                  21.49.9.6  Separated sets   sepnsepolem1 48763
                  21.49.9.7  Separated spaces: T0, T1, T2 (Hausdorff) ...   isnrm4 48772
            21.49.10  Preordered sets and directed sets using extensible structures   isprsd 48797
            21.49.11  Posets and lattices using extensible structures   lubeldm2 48798
                  21.49.11.1  Posets   lubeldm2 48798
                  21.49.11.2  Lattices   toslat 48816
                  21.49.11.3  Subset order structures   intubeu 48818
            21.49.12  Rings   elmgpcntrd 48839
                  21.49.12.1  Multiplicative Group   elmgpcntrd 48839
            21.49.13  Associative algebras   asclelbas 48840
                  21.49.13.1  Definition and basic properties   asclelbas 48840
            21.49.14  Categories   catprslem 48844
                  21.49.14.1  Categories   catprslem 48844
                  21.49.14.2  Monomorphisms and epimorphisms   idmon 48850
                  21.49.14.3  Sections, inverses, isomorphisms   isisod 48852
                  21.49.14.4  Functors   funcrcl2 48854
                  21.49.14.5  Universal property   upciclem1 48861
                  21.49.14.6  Natural transformations and the functor category   isnatd 48886
                  21.49.14.7  Product of categories   xpcfucbas 48889
                  21.49.14.8  Functor composition bifunctors   fucofulem1 48896
            21.49.15  Examples of categories   cthinc 48944
                  21.49.15.1  Thin categories   cthinc 48944
                  21.49.15.2  Preordered sets as thin categories   cprstc 48988
                  21.49.15.3  Monoids as categories   cmndtc 49011
      ​21.50  Mathbox for Emmett Weisz
            ​*21.50.1  Miscellaneous Theorems   nfintd 49029
            21.50.2  Set Recursion   csetrecs 49039
                  ​*21.50.2.1  Basic Properties of Set Recursion   csetrecs 49039
                  21.50.2.2  Examples and properties of set recursion   elsetrecslem 49055
            ​*21.50.3  Construction of Games and Surreal Numbers   cpg 49065
      ​​*21.51  Mathbox for David A. Wheeler
            21.51.1  Natural deduction   sbidd 49074
            ​*21.51.2  Greater than, greater than or equal to.   cge-real 49076
            ​*21.51.3  Hyperbolic trigonometric functions   csinh 49086
            ​*21.51.4  Reciprocal trigonometric functions (sec, csc, cot)   csec 49097
            ​*21.51.5  Identities for "if"   ifnmfalse 49119
            ​*21.51.6  Logarithms generalized to arbitrary base using ` logb `   logb2aval 49120
            ​*21.51.7  Logarithm laws generalized to an arbitrary base - log_   clog- 49121
            ​*21.51.8  Formally define notions such as reflexivity   wreflexive 49123
            ​*21.51.9  Algebra helpers   mvlraddi 49127
            ​*21.51.10  Algebra helper examples   i2linesi 49134
            ​*21.51.11  Formal methods "surprises"   alimp-surprise 49136
            ​*21.51.12  Allsome quantifier   walsi 49142
            ​*21.51.13  Miscellaneous   5m4e1 49153
            21.51.14  Theorems about algebraic numbers   aacllem 49157
      ​21.52  Mathbox for Kunhao Zheng
            21.52.1  Weighted AM-GM inequality   amgmwlem 49158