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Table of Contents Summary
PART 1  CLASSICAL FIRST ORDER LOGIC WITH EQUALITY
      1.1  Pre-logic
      1.2  Propositional calculus
      1.3  Other axiomatizations of 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  Predicate calculus with equality: Older axiomatization (1 rule, 14 schemes)
      1.7  Existential uniqueness
      1.8  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  Elementary real and complex functions
      5.8  Elementary limits and convergence
      5.9  Elementary trigonometry
      5.10  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  Presets and directed sets using extensible structures
      9.2  Posets and lattices using extensible structures
PART 10  BASIC ALGEBRAIC STRUCTURES
      10.1  Monoids
      10.2  Groups
      10.3  Abelian groups
      10.4  Rings
      10.5  Division rings and fields
      10.6  Left modules
      10.7  Vector spaces
      10.8  Ideals
      10.9  Associative algebras
      10.10  Abstract multivariate polynomials
      10.11  The complex numbers as an extensible structure
      10.12  Hilbert spaces
PART 11  BASIC TOPOLOGY
      11.1  Topology
      11.2  Filters and filter bases
      11.3  Uniform Stuctures and Spaces
      11.4  Metric spaces
      11.5  Complex metric vector spaces
PART 12  BASIC REAL AND COMPLEX ANALYSIS
      12.1  Continuity
      12.2  Integrals
      12.3  Derivatives
PART 13  BASIC REAL AND COMPLEX FUNCTIONS
      13.1  Polynomials
      13.2  Sequences and series
      13.3  Basic trigonometry
      13.4  Basic number theory
PART 14  GRAPH THEORY
      14.1  Undirected graphs - basics
      14.2  Eulerian paths and the Konigsberg Bridge problem
PART 15  GUIDES AND MISCELLANEA
      15.1  Guides (conventions, explanations, and examples)
      15.2  Humor
      15.3  (Future - to be reviewed and classified)
PART 16  ADDITIONAL MATERIAL ON GROUPS, RINGS, AND FIELDS (DEPRECATED)
      16.1  Additional material on group theory
      16.2  Additional material on rings and fields
PART 17  COMPLEX TOPOLOGICAL VECTOR SPACES (DEPRECATED)
      17.1  Complex vector spaces
      17.2  Normed complex vector spaces
      17.3  Operators on complex vector spaces
      17.4  Inner product (pre-Hilbert) spaces
      17.5  Complex Banach spaces
      17.6  Complex Hilbert spaces
PART 18  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      18.1  Axiomatization of complex pre-Hilbert spaces
      18.2  Inner product and norms
      18.3  Cauchy sequences and completeness axiom
      18.4  Subspaces and projections
      18.5  Properties of Hilbert subspaces
      18.6  Operators on Hilbert spaces
      18.7  States on a Hilbert lattice and Godowski's equation
      18.8  Cover relation, atoms, exchange axiom, and modular symmetry
PART 19  SUPPLEMENTARY MATERIAL (USER'S MATHBOXES)
      19.1  Mathboxes for user contributions
      19.2  Mathbox for Stefan Allan
      19.3  Mathbox for Thierry Arnoux
      19.4  Mathbox for Mario Carneiro
      19.5  Mathbox for Paul Chapman
      19.6  Mathbox for Drahflow
      19.7  Mathbox for Scott Fenton
      19.8  Mathbox for Anthony Hart
      19.9  Mathbox for Chen-Pang He
      19.10  Mathbox for Jeff Hoffman
      19.11  Mathbox for Wolf Lammen
      19.12  Mathbox for Brendan Leahy
      19.13  Mathbox for Jeff Hankins
      19.14  Mathbox for Jeff Madsen
      19.15  Mathbox for Rodolfo Medina
      19.16  Mathbox for Stefan O'Rear
      19.17  Mathbox for Steve Rodriguez
      19.18  Mathbox for Andrew Salmon
      19.19  Mathbox for Glauco Siliprandi
      19.20  Mathbox for Saveliy Skresanov
      19.21  Mathbox for Jarvin Udandy
      19.22  Mathbox for Alexander van der Vekens
      19.23  Mathbox for David A. Wheeler
      19.24  Mathbox for Alan Sare
      19.25  Mathbox for Jonathan Ben-Naim
      19.26  Mathbox for Norm Megill

Detailed Table of Contents
PART 1  CLASSICAL FIRST ORDER LOGIC WITH EQUALITY
      1.1  Pre-logic
            1.1.1  Inferences for assisting proof development   dummylink 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-1 5
            1.2.3  Logical implication   mp2 9
            1.2.4  Logical negation   con4d 99
            1.2.5  Logical equivalence   wb 177
            1.2.6  Logical disjunction and conjunction   wo 358
            1.2.7  Miscellaneous theorems of propositional calculus   pm5.21nd 869
            1.2.8  Abbreviated conjunction and disjunction of three wff's   w3o 935
            1.2.9  Logical 'nand' (Sheffer stroke)   wnan 1293
            1.2.10  Logical 'xor'   wxo 1310
            1.2.11  True and false constants   wtru 1322
            1.2.12  Truth tables   truantru 1342
            1.2.13  Auxiliary theorems for Alan Sare's virtual deduction tool, part 1   ee22 1368
            1.2.14  Half-adders and full adders in propositional calculus   whad 1384
      1.3  Other axiomatizations of classical propositional calculus
            1.3.1  Derive the Lukasiewicz axioms from Meredith's sole axiom   meredith 1410
            1.3.2  Derive the standard axioms from the Lukasiewicz axioms   luklem1 1429
            1.3.3  Derive Nicod's axiom from the standard axioms   nic-dfim 1440
            1.3.4  Derive the Lukasiewicz axioms from Nicod's axiom   nic-imp 1446
            1.3.5  Derive Nicod's Axiom from Lukasiewicz's First Sheffer Stroke Axiom   lukshef-ax1 1465
            1.3.6  Derive the Lukasiewicz Axioms from the Tarski-Bernays-Wajsberg Axioms   tbw-bijust 1469
            1.3.7  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's First CO Axiom   merco1 1484
            1.3.8  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's Second CO Axiom   merco2 1507
            1.3.9  Derive the Lukasiewicz axioms from the The Russell-Bernays Axioms   rb-bijust 1520
            1.3.10  Stoic logic indemonstrables (Chrysippus of Soli)   mpto1 1539
      1.4  Predicate calculus with equality: Tarski's system S2 (1 rule, 6 schemes)
            1.4.1  Universal quantifier; define "exists" and "not free"   wal 1546
            1.4.2  Rule scheme ax-gen (Generalization)   ax-gen 1552
            1.4.3  Axiom scheme ax-5 (Quantified Implication)   ax-5 1563
            1.4.4  Axiom scheme ax-17 (Distinctness) - first use of $d   ax-17 1623
            1.4.5  Equality predicate; define substitution   cv 1648
            1.4.6  Axiom scheme ax-9 (Existence)   ax-9 1661
            1.4.7  Axiom scheme ax-8 (Equality)   ax-8 1682
            1.4.8  Membership predicate   wcel 1717
            1.4.9  Axiom schemes ax-13 (Left Equality for Binary Predicate)   ax-13 1719
            1.4.10  Axiom schemes ax-14 (Right Equality for Binary Predicate)   ax-14 1721
            1.4.11  Logical redundancy of ax-6 , ax-7 , ax-11 , ax-12   ax9dgen 1723
      1.5  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes)
            1.5.1  Axiom scheme ax-6 (Quantified Negation)   ax-6 1736
            1.5.2  Axiom scheme ax-7 (Quantifier Commutation)   ax-7 1741
            1.5.3  Axiom scheme ax-11 (Substitution)   ax-11 1753
            1.5.4  Axiom scheme ax-12 (Quantified Equality)   ax-12 1939
      1.6  Predicate calculus with equality: Older axiomatization (1 rule, 14 schemes)
            1.6.1  Obsolete schemes ax-5o ax-4 ax-6o ax-9o ax-10o ax-10 ax-11o ax-12o ax-15 ax-16   ax-4 2169
            1.6.2  Rederive new axioms from old: ax5 , ax6 , ax9from9o , ax11 , ax12from12o   ax4 2179
            1.6.3  Legacy theorems using obsolete axioms   ax17o 2191
      1.7  Existential uniqueness
      1.8  Other axiomatizations related to classical predicate calculus
            1.8.1  Predicate calculus with all distinct variables   ax-7d 2329
            1.8.2  Aristotelian logic: Assertic syllogisms   barbara 2335
            1.8.3  Intuitionistic logic   axi4 2359
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 2368
            2.1.2  Class abstractions (a.k.a. class builders)   cab 2373
            2.1.3  Class form not-free predicate   wnfc 2510
            2.1.4  Negated equality and membership   wne 2550
                  2.1.4.1  Negated equality   nne 2554
                  2.1.4.2  Negated membership   neleq1 2643
            2.1.5  Restricted quantification   wral 2649
            2.1.6  The universal class   cvv 2899
            2.1.7  Conditional equality (experimental)   wcdeq 3087
            2.1.8  Russell's Paradox   ru 3103
            2.1.9  Proper substitution of classes for sets   wsbc 3104
            2.1.10  Proper substitution of classes for sets into classes   csb 3194
            2.1.11  Define basic set operations and relations   cdif 3260
            2.1.12  Subclasses and subsets   df-ss 3277
            2.1.13  The difference, union, and intersection of two classes   difeq1 3401
                  2.1.13.1  The difference of two classes   difeq1 3401
                  2.1.13.2  The union of two classes   elun 3431
                  2.1.13.3  The intersection of two classes   elin 3473
                  2.1.13.4  Combinations of difference, union, and intersection of two classes   unabs 3514
                  2.1.13.5  Class abstractions with difference, union, and intersection of two classes   symdif2 3550
                  2.1.13.6  Restricted uniqueness with difference, union, and intersection   reuss2 3564
            2.1.14  The empty set   c0 3571
            2.1.15  "Weak deduction theorem" for set theory   cif 3682
            2.1.16  Power classes   cpw 3742
            2.1.17  Unordered and ordered pairs   csn 3757
            2.1.18  The union of a class   cuni 3957
            2.1.19  The intersection of a class   cint 3992
            2.1.20  Indexed union and intersection   ciun 4035
            2.1.21  Disjointness   wdisj 4123
            2.1.22  Binary relations   wbr 4153
            2.1.23  Ordered-pair class abstractions (class builders)   copab 4206
            2.1.24  Transitive classes   wtr 4243
      2.2  ZF Set Theory - add the Axiom of Replacement
            2.2.1  Introduce the Axiom of Replacement   ax-rep 4261
            2.2.2  Derive the Axiom of Separation   axsep 4270
            2.2.3  Derive the Null Set Axiom   zfnuleu 4276
            2.2.4  Theorems requiring subset and intersection existence   nalset 4281
            2.2.5  Theorems requiring empty set existence   class2set 4308
      2.3  ZF Set Theory - add the Axiom of Power Sets
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 4318
            2.3.2  Derive the Axiom of Pairing   zfpair 4342
            2.3.3  Ordered pair theorem   opnz 4373
            2.3.4  Ordered-pair class abstractions (cont.)   opabid 4402
            2.3.5  Power class of union and intersection   pwin 4428
            2.3.6  Epsilon and identity relations   cep 4433
            2.3.7  Partial and complete ordering   wpo 4442
            2.3.8  Founded and well-ordering relations   wfr 4479
            2.3.9  Ordinals   word 4521
      2.4  ZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 4641
            2.4.2  Ordinals (continued)   ordon 4703
            2.4.3  Transfinite induction   tfi 4773
            2.4.4  The natural numbers (i.e. finite ordinals)   com 4785
            2.4.5  Peano's postulates   peano1 4804
            2.4.6  Finite induction (for finite ordinals)   find 4810
            2.4.7  Relations   cxp 4816
            2.4.8  Definite description binder (inverted iota)   cio 5356
            2.4.9  Functions   wfun 5388
            2.4.10  Operations   co 6020
            2.4.11  "Maps to" notation   elmpt2cl 6227
            2.4.12  Function operation   cof 6242
            2.4.13  First and second members of an ordered pair   c1st 6286
            2.4.14  Special "Maps to" operations   mpt2xopn0yelv 6400
            2.4.15  Function transposition   ctpos 6414
            2.4.16  Curry and uncurry   ccur 6453
            2.4.17  Proper subset relation   crpss 6457
            2.4.18  Iota properties   fvopab5 6470
            2.4.19  Cantor's Theorem   canth 6475
            2.4.20  Undefined values and restricted iota (description binder)   cund 6477
            2.4.21  Functions on ordinals; strictly monotone ordinal functions   iunon 6536
            2.4.22  "Strong" transfinite recursion   crecs 6568
            2.4.23  Recursive definition generator   crdg 6603
            2.4.24  Finite recursion   frfnom 6628
            2.4.25  Abian's "most fundamental" fixed point theorem   abianfplem 6651
            2.4.26  Ordinal arithmetic   c1o 6653
            2.4.27  Natural number arithmetic   nna0 6783
            2.4.28  Equivalence relations and classes   wer 6838
            2.4.29  The mapping operation   cmap 6954
            2.4.30  Infinite Cartesian products   cixp 6999
            2.4.31  Equinumerosity   cen 7042
            2.4.32  Schroeder-Bernstein Theorem   sbthlem1 7153
            2.4.33  Equinumerosity (cont.)   xpf1o 7205
            2.4.34  Pigeonhole Principle   phplem1 7222
            2.4.35  Finite sets   onomeneq 7232
            2.4.36  Finite intersections   cfi 7350
            2.4.37  Hall's marriage theorem   marypha1lem 7373
            2.4.38  Supremum   csup 7380
            2.4.39  Ordinal isomorphism, Hartog's theorem   coi 7411
            2.4.40  Hartogs function, order types, weak dominance   char 7457
      2.5  ZF Set Theory - add the Axiom of Regularity
            2.5.1  Introduce the Axiom of Regularity   ax-reg 7493
            2.5.2  Axiom of Infinity equivalents   inf0 7509
      2.6  ZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-inf 7526
            2.6.2  Existence of omega (the set of natural numbers)   omex 7531
            2.6.3  Cantor normal form   ccnf 7549
            2.6.4  Transitive closure   trcl 7597
            2.6.5  Rank   cr1 7621
            2.6.6  Scott's trick; collection principle; Hilbert's epsilon   scottex 7742
            2.6.7  Cardinal numbers   ccrd 7755
            2.6.8  Axiom of Choice equivalents   wac 7929
            2.6.9  Cardinal number arithmetic   ccda 7980
            2.6.10  The Ackermann bijection   ackbij2lem1 8032
            2.6.11  Cofinality (without Axiom of Choice)   cflem 8059
            2.6.12  Eight inequivalent definitions of finite set   sornom 8090
            2.6.13  Hereditarily size-limited sets without Choice   itunifval 8229
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
            3.2.1  Introduce the Axiom of Choice   ax-ac 8272
            3.2.2  AC equivalents: well-ordering, Zorn's lemma   numthcor 8307
            3.2.3  Cardinal number theorems using Axiom of Choice   cardval 8354
            3.2.4  Cardinal number arithmetic using Axiom of Choice   iunctb 8382
            3.2.5  Cofinality using Axiom of Choice   alephreg 8390
      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
            4.1.1  Weakly and strongly inaccessible cardinals   cwina 8490
            4.1.2  Weak universes   cwun 8508
            4.1.3  Tarski's classes   ctsk 8556
            4.1.4  Grothendieck's universes   cgru 8598
      4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
            4.2.1  Introduce the Tarski-Grothendieck Axiom   ax-groth 8631
            4.2.2  Derive the Power Set, Infinity and Choice Axioms   grothpw 8634
            4.2.3  Tarski map function   ctskm 8645
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 8652
            5.1.2  Final derivation of real and complex number postulates   axaddf 8953
            5.1.3  Real and complex number postulates restated as axioms   ax-cnex 8979
      5.2  Derive the basic properties from the field axioms
            5.2.1  Some deductions from the field axioms for complex numbers   cnex 9004
            5.2.2  Infinity and the extended real number system   cpnf 9050
            5.2.3  Restate the ordering postulates with extended real "less than"   axlttri 9080
            5.2.4  Ordering on reals   lttr 9085
            5.2.5  Initial properties of the complex numbers   mul12 9164
      5.3  Real and complex numbers - basic operations
            5.3.1  Addition   add12 9211
            5.3.2  Subtraction   cmin 9223
            5.3.3  Multiplication   muladd 9398
            5.3.4  Ordering on reals (cont.)   gt0ne0 9425
            5.3.5  Reciprocals   ixi 9583
            5.3.6  Division   cdiv 9609
            5.3.7  Ordering on reals (cont.)   elimgt0 9778
            5.3.8  Completeness Axiom and Suprema   fimaxre 9887
            5.3.9  Imaginary and complex number properties   inelr 9922
            5.3.10  Function operation analogue theorems   ofsubeq0 9929
      5.4  Integer sets
            5.4.1  Natural numbers (as a subset of complex numbers)   cn 9932
            5.4.2  Principle of mathematical induction   nnind 9950
            5.4.3  Decimal representation of numbers   c2 9981
            5.4.4  Some properties of specific numbers   0p1e1 10025
            5.4.5  The Archimedean property   nnunb 10149
            5.4.6  Nonnegative integers (as a subset of complex numbers)   cn0 10153
            5.4.7  Integers (as a subset of complex numbers)   cz 10214
            5.4.8  Decimal arithmetic   cdc 10314
            5.4.9  Upper partititions of integers   cuz 10420
            5.4.10  Well-ordering principle for bounded-below sets of integers   uzwo3 10501
            5.4.11  Rational numbers (as a subset of complex numbers)   cq 10506
            5.4.12  Existence of the set of complex numbers   rpnnen1lem1 10532
      5.5  Order sets
            5.5.1  Positive reals (as a subset of complex numbers)   crp 10544
            5.5.2  Infinity and the extended real number system (cont.)   cxne 10639
            5.5.3  Supremum on the extended reals   xrsupexmnf 10815
            5.5.4  Real number intervals   cioo 10848
            5.5.5  Finite intervals of integers   cfz 10975
            5.5.6  Half-open integer ranges   cfzo 11065
      5.6  Elementary integer functions
            5.6.1  The floor (greatest integer) function   cfl 11128
            5.6.2  The modulo (remainder) operation   cmo 11177
            5.6.3  The infinite sequence builder "seq"   om2uz0i 11214
            5.6.4  Integer powers   cexp 11309
            5.6.5  Ordered pair theorem for nonnegative integers   nn0le2msqi 11487
            5.6.6  Factorial function   cfa 11493
            5.6.7  The binomial coefficient operation   cbc 11520
            5.6.8  The ` # ` (finite set size) function   chash 11545
                  5.6.8.1  Finite induction on the size of the first component of a binary relation   brfi1indlem 11641
            5.6.9  Words over a set   cword 11644
            5.6.10  Longer string literals   cs2 11732
      5.7  Elementary real and complex functions
            5.7.1  The "shift" operation   cshi 11808
            5.7.2  Real and imaginary parts; conjugate   ccj 11828
            5.7.3  Square root; absolute value   csqr 11965
      5.8  Elementary limits and convergence
            5.8.1  Superior limit (lim sup)   clsp 12191
            5.8.2  Limits   cli 12205
            5.8.3  Finite and infinite sums   csu 12406
            5.8.4  The binomial theorem   binomlem 12535
            5.8.5  The inclusion/exclusion principle   incexclem 12543
            5.8.6  Infinite sums (cont.)   isumshft 12546
            5.8.7  Miscellaneous converging and diverging sequences   divrcnv 12559
            5.8.8  Arithmetic series   arisum 12566
            5.8.9  Geometric series   expcnv 12570
            5.8.10  Ratio test for infinite series convergence   cvgrat 12587
            5.8.11  Mertens' theorem   mertenslem1 12588
      5.9  Elementary trigonometry
            5.9.1  The exponential, sine, and cosine functions   ce 12591
            5.9.2  _e is irrational   eirrlem 12730
      5.10  Cardinality of real and complex number subsets
            5.10.1  Countability of integers and rationals   xpnnen 12735
            5.10.2  The reals are uncountable   rpnnen2lem1 12741
PART 6  ELEMENTARY NUMBER THEORY
      6.1  Elementary properties of divisibility
            6.1.1  Irrationality of square root of 2   sqr2irrlem 12774
            6.1.2  Some Number sets are chains of proper subsets   nthruc 12777
            6.1.3  The divides relation   cdivides 12779
            6.1.4  The division algorithm   divalglem0 12840
            6.1.5  Bit sequences   cbits 12858
            6.1.6  The greatest common divisor operator   cgcd 12933
            6.1.7  Bézout's identity   bezoutlem1 12965
            6.1.8  Algorithms   nn0seqcvgd 12988
            6.1.9  Euclid's Algorithm   eucalgval2 12999
      6.2  Elementary prime number theory
            6.2.1  Elementary properties   cprime 13006
            6.2.2  Properties of the canonical representation of a rational   cnumer 13052
            6.2.3  Euler's theorem   codz 13079
            6.2.4  Pythagorean Triples   coprimeprodsq 13110
            6.2.5  The prime count function   cpc 13137
            6.2.6  Pocklington's theorem   prmpwdvds 13199
            6.2.7  Infinite primes theorem   unbenlem 13203
            6.2.8  Sum of prime reciprocals   prmreclem1 13211
            6.2.9  Fundamental theorem of arithmetic   1arithlem1 13218
            6.2.10  Lagrange's four-square theorem   cgz 13224
            6.2.11  Van der Waerden's theorem   cvdwa 13260
            6.2.12  Ramsey's theorem   cram 13294
            6.2.13  Decimal arithmetic (cont.)   dec2dvds 13326
            6.2.14  Specific prime numbers   4nprm 13354
            6.2.15  Very large primes   1259lem1 13377
PART 7  BASIC STRUCTURES
      7.1  Extensible structures
            7.1.1  Basic definitions   cstr 13392
            7.1.2  Slot definitions   cplusg 13456
            7.1.3  Definition of the structure product   crest 13575
            7.1.4  Definition of the structure quotient   cordt 13648
      7.2  Moore spaces
            7.2.1  Moore closures   mrcflem 13758
            7.2.2  Independent sets in a Moore system   mrisval 13782
            7.2.3  Algebraic closure systems   isacs 13803
PART 8  BASIC CATEGORY THEORY
      8.1  Categories
            8.1.1  Categories   ccat 13816
            8.1.2  Opposite category   coppc 13864
            8.1.3  Monomorphisms and epimorphisms   cmon 13881
            8.1.4  Sections, inverses, isomorphisms   csect 13897
            8.1.5  Subcategories   cssc 13934
            8.1.6  Functors   cfunc 13978
            8.1.7  Full & faithful functors   cful 14026
            8.1.8  Natural transformations and the functor category   cnat 14065
      8.2  Arrows (disjointified hom-sets)
            8.2.1  Identity and composition for arrows   cida 14135
      8.3  Examples of categories
            8.3.1  The category of sets   csetc 14157
            8.3.2  The category of categories   ccatc 14176
      8.4  Categorical constructions
            8.4.1  Product of categories   cxpc 14192
            8.4.2  Functor evaluation   cevlf 14233
            8.4.3  Hom functor   chof 14272
PART 9  BASIC ORDER THEORY
      9.1  Presets and directed sets using extensible structures
      9.2  Posets and lattices using extensible structures
            9.2.1  Posets   cpo 14324
            9.2.2  Lattices   clat 14401
            9.2.3  The dual of an ordered set   codu 14482
            9.2.4  Subset order structures   cipo 14504
            9.2.5  Distributive lattices   latmass 14541
            9.2.6  Posets and lattices as relations   cps 14551
            9.2.7  Directed sets, nets   cdir 14600
PART 10  BASIC ALGEBRAIC STRUCTURES
      10.1  Monoids
            10.1.1  Definition and basic properties   cmnd 14611
            10.1.2  Monoid homomorphisms and submonoids   cmhm 14663
            10.1.3  Ordered group sum operation   gsumvallem1 14698
            10.1.4  Free monoids   cfrmd 14719
      10.2  Groups
            10.2.1  Definition and basic properties   df-grp 14739
            10.2.2  Subgroups and Quotient groups   csubg 14865
            10.2.3  Elementary theory of group homomorphisms   cghm 14930
            10.2.4  Isomorphisms of groups   cgim 14971
            10.2.5  Group actions   cga 14993
            10.2.6  Symmetry groups and Cayley's Theorem   csymg 15019
            10.2.7  Centralizers and centers   ccntz 15041
            10.2.8  The opposite group   coppg 15068
            10.2.9  p-Groups and Sylow groups; Sylow's theorems   cod 15090
            10.2.10  Direct products   clsm 15195
            10.2.11  Free groups   cefg 15265
      10.3  Abelian groups
            10.3.1  Definition and basic properties   ccmn 15339
            10.3.2  Cyclic groups   ccyg 15414
            10.3.3  Group sum operation   gsumval3a 15439
            10.3.4  Internal direct products   cdprd 15481
            10.3.5  The Fundamental Theorem of Abelian Groups   ablfacrplem 15550
      10.4  Rings
            10.4.1  Multiplicative Group   cmgp 15575
            10.4.2  Definition and basic properties   crg 15587
            10.4.3  Opposite ring   coppr 15654
            10.4.4  Divisibility   cdsr 15670
            10.4.5  Ring homomorphisms   crh 15744
      10.5  Division rings and fields
            10.5.1  Definition and basic properties   cdr 15762
            10.5.2  Subrings of a ring   csubrg 15791
            10.5.3  Absolute value (abstract algebra)   cabv 15831
            10.5.4  Star rings   cstf 15858
      10.6  Left modules
            10.6.1  Definition and basic properties   clmod 15877
            10.6.2  Subspaces and spans in a left module   clss 15935
            10.6.3  Homomorphisms and isomorphisms of left modules   clmhm 16022
            10.6.4  Subspace sum; bases for a left module   clbs 16073
      10.7  Vector spaces
            10.7.1  Definition and basic properties   clvec 16101
      10.8  Ideals
            10.8.1  The subring algebra; ideals   csra 16167
            10.8.2  Two-sided ideals and quotient rings   c2idl 16229
            10.8.3  Principal ideal rings. Divisibility in the integers   clpidl 16239
            10.8.4  Nonzero rings   cnzr 16255
            10.8.5  Left regular elements. More kinds of rings   crlreg 16266
      10.9  Associative algebras
            10.9.1  Definition and basic properties   casa 16296
      10.10  Abstract multivariate polynomials
            10.10.1  Definition and basic properties   cmps 16333
            10.10.2  Polynomial evaluation   evlslem4 16491
            10.10.3  Univariate polynomials   cps1 16496
      10.11  The complex numbers as an extensible structure
            10.11.1  Definition and basic properties   cxmt 16612
            10.11.2  Algebraic constructions based on the complexes   czrh 16701
      10.12  Hilbert spaces
            10.12.1  Definition and basic properties   cphl 16778
            10.12.2  Orthocomplements and closed subspaces   cocv 16810
            10.12.3  Orthogonal projection and orthonormal bases   cpj 16850
PART 11  BASIC TOPOLOGY
      11.1  Topology
            11.1.1  Topological spaces   ctop 16881
            11.1.2  TopBases for topologies   isbasisg 16935
            11.1.3  Examples of topologies   distop 16983
            11.1.4  Closure and interior   ccld 17003
            11.1.5  Neighborhoods   cnei 17084
            11.1.6  Limit points and perfect sets   clp 17121
            11.1.7  Subspace topologies   restrcl 17143
            11.1.8  Order topology   ordtbaslem 17174
            11.1.9  Limits and continuity in topological spaces   ccn 17210
            11.1.10  Separated spaces: T0, T1, T2 (Hausdorff) ...   ct0 17292
            11.1.11  Compactness   ccmp 17371
            11.1.12  Connectedness   ccon 17395
            11.1.13  First- and second-countability   c1stc 17421
            11.1.14  Local topological properties   clly 17448
            11.1.15  Compactly generated spaces   ckgen 17486
            11.1.16  Product topologies   ctx 17513
            11.1.17  Continuous function-builders   cnmptid 17614
            11.1.18  Quotient maps and quotient topology   ckq 17646
            11.1.19  Homeomorphisms   chmeo 17706
      11.2  Filters and filter bases
            11.2.1  Filter bases   elmptrab 17780
            11.2.2  Filters   cfil 17798
            11.2.3  Ultrafilters   cufil 17852
            11.2.4  Filter limits   cfm 17886
            11.2.5  Extension by continuity   ccnext 18011
            11.2.6  Topological groups   ctmd 18021
            11.2.7  Infinite group sum on topological groups   ctsu 18076
            11.2.8  Topological rings, fields, vector spaces   ctrg 18106
      11.3  Uniform Stuctures and Spaces
            11.3.1  Uniform structures   cust 18150
            11.3.2  The topology induced by an uniform structure   cutop 18181
            11.3.3  Uniform Spaces   cuss 18204
            11.3.4  Uniform continuity   cucn 18226
            11.3.5  Cauchy filters in uniform spaces   ccfilu 18237
            11.3.6  Complete uniform spaces   ccusp 18248
      11.4  Metric spaces
            11.4.1  Basic metric space properties   cxme 18256
            11.4.2  Metric space balls   blfval 18321
            11.4.3  Open sets of a metric space   mopnval 18358
            11.4.4  Continuity in metric spaces   metcnp3 18460
            11.4.5  The uniform structure generated by a metric   metuval 18469
            11.4.6  Examples of metric spaces   dscmet 18491
            11.4.7  Normed algebraic structures   cnm 18495
            11.4.8  Normed space homomorphisms (bounded linear operators)   cnmo 18610
            11.4.9  Topology on the reals   qtopbaslem 18663
            11.4.10  Topological definitions using the reals   cii 18776
            11.4.11  Path homotopy   chtpy 18863
            11.4.12  The fundamental group   cpco 18896
      11.5  Complex metric vector spaces
            11.5.1  Complex left modules   cclm 18958
            11.5.2  Complex pre-Hilbert space   ccph 19000
            11.5.3  Convergence and completeness   ccfil 19076
            11.5.4  Baire's Category Theorem   bcthlem1 19146
            11.5.5  Banach spaces and complex Hilbert spaces   ccms 19154
            11.5.6  Minimizing Vector Theorem   minveclem1 19192
            11.5.7  Projection Theorem   pjthlem1 19205
PART 12  BASIC REAL AND COMPLEX ANALYSIS
      12.1  Continuity
            12.1.1  Intermediate value theorem   pmltpclem1 19212
      12.2  Integrals
            12.2.1  Lebesgue measure   covol 19226
            12.2.2  Lebesgue integration   cmbf 19373
      12.3  Derivatives
            12.3.1  Real and complex differentiation   climc 19616
PART 13  BASIC REAL AND COMPLEX FUNCTIONS
      13.1  Polynomials
            13.1.1  Abstract polynomials, continued   evlslem6 19801
            13.1.2  Polynomial degrees   cmdg 19843
            13.1.3  The division algorithm for univariate polynomials   cmn1 19915
            13.1.4  Elementary properties of complex polynomials   cply 19970
            13.1.5  The division algorithm for polynomials   cquot 20074
            13.1.6  Algebraic numbers   caa 20098
            13.1.7  Liouville's approximation theorem   aalioulem1 20116
      13.2  Sequences and series
            13.2.1  Taylor polynomials and Taylor's theorem   ctayl 20136
            13.2.2  Uniform convergence   culm 20159
            13.2.3  Power series   pserval 20193
      13.3  Basic trigonometry
            13.3.1  The exponential, sine, and cosine functions (cont.)   efcn 20226
            13.3.2  Properties of pi = 3.14159...   pilem1 20234
            13.3.3  Mapping of the exponential function   efgh 20310
            13.3.4  The natural logarithm on complex numbers   clog 20319
            13.3.5  Theorems of Pythagoras, isosceles triangles, and intersecting chords   angval 20510
            13.3.6  Solutions of quadratic, cubic, and quartic equations   quad2 20546
            13.3.7  Inverse trigonometric functions   casin 20569
            13.3.8  The Birthday Problem   log2ublem1 20653
            13.3.9  Areas in R^2   carea 20661
            13.3.10  More miscellaneous converging sequences   rlimcnp 20671
            13.3.11  Inequality of arithmetic and geometric means   cvxcl 20690
            13.3.12  Euler-Mascheroni constant   cem 20697
      13.4  Basic number theory
            13.4.1  Wilson's theorem   wilthlem1 20718
            13.4.2  The Fundamental Theorem of Algebra   ftalem1 20722
            13.4.3  The Basel problem (ζ(2) = π2/6)   basellem1 20730
            13.4.4  Number-theoretical functions   ccht 20740
            13.4.5  Perfect Number Theorem   mersenne 20878
            13.4.6  Characters of Z/nZ   cdchr 20883
            13.4.7  Bertrand's postulate   bcctr 20926
            13.4.8  Legendre symbol   clgs 20945
            13.4.9  Quadratic reciprocity   lgseisenlem1 21000
            13.4.10  All primes 4n+1 are the sum of two squares   2sqlem1 21014
            13.4.11  Chebyshev's Weak Prime Number Theorem, Dirichlet's Theorem   chebbnd1lem1 21030
            13.4.12  The Prime Number Theorem   mudivsum 21091
            13.4.13  Ostrowski's theorem   abvcxp 21176
PART 14  GRAPH THEORY
      14.1  Undirected graphs - basics
            14.1.1  Undirected hypergraphs   cuhg 21201
            14.1.2  Undirected multigraphs   cumg 21214
            14.1.3  Undirected simple graphs   cuslg 21231
                  14.1.3.1  Undirected simple graphs - basics   cuslg 21231
                  14.1.3.2  Undirected simple graphs - examples   usgraexvlem 21280
                  14.1.3.3  Finite undirected simple graphs   fiusgraedgfi 21287
            14.1.4  Neighbors, complete graphs and universal vertices   cnbgra 21296
                  14.1.4.1  Neighbors   nbgraop 21302
                  14.1.4.2  Complete graphs   iscusgra 21331
                  14.1.4.3  Universal vertices   isuvtx 21363
            14.1.5  Walks, paths and cycles   cwalk 21372
                  14.1.5.1  Walks and trails   wlks 21390
                  14.1.5.2  Paths and simple paths   pths 21420
                  14.1.5.3  Circuits and cycles   crcts 21457
                  14.1.5.4  Connected graphs   cconngra 21504
            14.1.6  Vertex Degree   cvdg 21512
      14.2  Eulerian paths and the Konigsberg Bridge problem
            14.2.1  Eulerian paths   ceup 21532
            14.2.2  The Konigsberg Bridge problem   vdeg0i 21552
PART 15  GUIDES AND MISCELLANEA
      15.1  Guides (conventions, explanations, and examples)
            15.1.1  Conventions   conventions 21558
            15.1.2  Natural deduction   natded 21559
            15.1.3  Natural deduction examples   ex-natded5.2 21560
            15.1.4  Definitional examples   ex-or 21577
      15.2  Humor
            15.2.1  April Fool's theorem   avril1 21605
      15.3  (Future - to be reviewed and classified)
            15.3.1  Planar incidence geometry   cplig 21611
            15.3.2  Algebra preliminaries   crpm 21616
            15.3.3  Transitive closure   ctcl 21618
PART 16  ADDITIONAL MATERIAL ON GROUPS, RINGS, AND FIELDS (DEPRECATED)
      16.1  Additional material on group theory
            16.1.1  Definitions and basic properties for groups   cgr 21622
            16.1.2  Definition and basic properties of Abelian groups   cablo 21717
            16.1.3  Subgroups   csubgo 21737
            16.1.4  Operation properties   cass 21748
            16.1.5  Group-like structures   cmagm 21754
            16.1.6  Examples of Abelian groups   ablosn 21783
            16.1.7  Group homomorphism and isomorphism   cghom 21793
      16.2  Additional material on rings and fields
            16.2.1  Definition and basic properties   crngo 21811
            16.2.2  Examples of rings   cnrngo 21839
            16.2.3  Division Rings   cdrng 21841
            16.2.4  Star Fields   csfld 21844
            16.2.5  Fields and Rings   ccm2 21846
PART 17  COMPLEX TOPOLOGICAL VECTOR SPACES (DEPRECATED)
      17.1  Complex vector spaces
            17.1.1  Definition and basic properties   cvc 21872
            17.1.2  Examples of complex vector spaces   cncvc 21910
      17.2  Normed complex vector spaces
            17.2.1  Definition and basic properties   cnv 21911
            17.2.2  Examples of normed complex vector spaces   cnnv 22016
            17.2.3  Induced metric of a normed complex vector space   imsval 22025
            17.2.4  Inner product   cdip 22044
            17.2.5  Subspaces   css 22068
      17.3  Operators on complex vector spaces
            17.3.1  Definitions and basic properties   clno 22089
      17.4  Inner product (pre-Hilbert) spaces
            17.4.1  Definition and basic properties   ccphlo 22161
            17.4.2  Examples of pre-Hilbert spaces   cncph 22168
            17.4.3  Properties of pre-Hilbert spaces   isph 22171
      17.5  Complex Banach spaces
            17.5.1  Definition and basic properties   ccbn 22212
            17.5.2  Examples of complex Banach spaces   cnbn 22219
            17.5.3  Uniform Boundedness Theorem   ubthlem1 22220
            17.5.4  Minimizing Vector Theorem   minvecolem1 22224
      17.6  Complex Hilbert spaces
            17.6.1  Definition and basic properties   chlo 22235
            17.6.2  Standard axioms for a complex Hilbert space   hlex 22248
            17.6.3  Examples of complex Hilbert spaces   cnchl 22266
            17.6.4  Subspaces   ssphl 22267
            17.6.5  Hellinger-Toeplitz Theorem   htthlem 22268
PART 18  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      18.1  Axiomatization of complex pre-Hilbert spaces
            18.1.1  Basic Hilbert space definitions   chil 22270
            18.1.2  Preliminary ZFC lemmas   df-hnorm 22319
            18.1.3  Derive the Hilbert space axioms from ZFC set theory   axhilex-zf 22332
            18.1.4  Introduce the vector space axioms for a Hilbert space   ax-hilex 22350
            18.1.5  Vector operations   hvmulex 22362
            18.1.6  Inner product postulates for a Hilbert space   ax-hfi 22429
      18.2  Inner product and norms
            18.2.1  Inner product   his5 22436
            18.2.2  Norms   dfhnorm2 22472
            18.2.3  Relate Hilbert space to normed complex vector spaces   hilablo 22510
            18.2.4  Bunjakovaskij-Cauchy-Schwarz inequality   bcsiALT 22529
      18.3  Cauchy sequences and completeness axiom
            18.3.1  Cauchy sequences and limits   hcau 22534
            18.3.2  Derivation of the completeness axiom from ZF set theory   hilmet 22544
            18.3.3  Completeness postulate for a Hilbert space   ax-hcompl 22552
            18.3.4  Relate Hilbert space to ZFC pre-Hilbert and Hilbert spaces   hhcms 22553
      18.4  Subspaces and projections
            18.4.1  Subspaces   df-sh 22557
            18.4.2  Closed subspaces   df-ch 22572
            18.4.3  Orthocomplements   df-oc 22602
            18.4.4  Subspace sum, span, lattice join, lattice supremum   df-shs 22658
            18.4.5  Projection theorem   pjhthlem1 22741
            18.4.6  Projectors   df-pjh 22745
      18.5  Properties of Hilbert subspaces
            18.5.1  Orthomodular law   omlsilem 22752
            18.5.2  Projectors (cont.)   pjhtheu2 22766
            18.5.3  Hilbert lattice operations   sh0le 22790
            18.5.4  Span (cont.) and one-dimensional subspaces   spansn0 22891
            18.5.5  Commutes relation for Hilbert lattice elements   df-cm 22933
            18.5.6  Foulis-Holland theorem   fh1 22968
            18.5.7  Quantum Logic Explorer axioms   qlax1i 22977
            18.5.8  Orthogonal subspaces   chscllem1 22987
            18.5.9  Orthoarguesian laws 5OA and 3OA   5oalem1 23004
            18.5.10  Projectors (cont.)   pjorthi 23019
            18.5.11  Mayet's equation E_3   mayete3i 23078
      18.6  Operators on Hilbert spaces
            18.6.1  Operator sum, difference, and scalar multiplication   df-hosum 23081
            18.6.2  Zero and identity operators   df-h0op 23099
            18.6.3  Operations on Hilbert space operators   hoaddcl 23109
            18.6.4  Linear, continuous, bounded, Hermitian, unitary operators and norms   df-nmop 23190
            18.6.5  Linear and continuous functionals and norms   df-nmfn 23196
            18.6.6  Adjoint   df-adjh 23200
            18.6.7  Dirac bra-ket notation   df-bra 23201
            18.6.8  Positive operators   df-leop 23203
            18.6.9  Eigenvectors, eigenvalues, spectrum   df-eigvec 23204
            18.6.10  Theorems about operators and functionals   nmopval 23207
            18.6.11  Riesz lemma   riesz3i 23413
            18.6.12  Adjoints (cont.)   cnlnadjlem1 23418
            18.6.13  Quantum computation error bound theorem   unierri 23455
            18.6.14  Dirac bra-ket notation (cont.)   branmfn 23456
            18.6.15  Positive operators (cont.)   leopg 23473
            18.6.16  Projectors as operators   pjhmopi 23497
      18.7  States on a Hilbert lattice and Godowski's equation
            18.7.1  States on a Hilbert lattice   df-st 23562
            18.7.2  Godowski's equation   golem1 23622
      18.8  Cover relation, atoms, exchange axiom, and modular symmetry
            18.8.1  Covers relation; modular pairs   df-cv 23630
            18.8.2  Atoms   df-at 23689
            18.8.3  Superposition principle   superpos 23705
            18.8.4  Atoms, exchange and covering properties, atomicity   chcv1 23706
            18.8.5  Irreducibility   chirredlem1 23741
            18.8.6  Atoms (cont.)   atcvat3i 23747
            18.8.7  Modular symmetry   mdsymlem1 23754
PART 19  SUPPLEMENTARY MATERIAL (USER'S MATHBOXES)
      19.1  Mathboxes for user contributions
            19.1.1  Mathbox guidelines   mathbox 23793
      19.2  Mathbox for Stefan Allan
      19.3  Mathbox for Thierry Arnoux
            19.3.1  Propositional Calculus - misc additions   bian1d 23798
            19.3.2  Predicate Calculus   abeq2f 23804
                  19.3.2.1  Predicate Calculus - misc additions   abeq2f 23804
                  19.3.2.2  Restricted quantification - misc additions   reximddv 23806
                  19.3.2.3  Substitution (without distinct variables) - misc additions   clelsb3f 23815
                  19.3.2.4  Existential "at most one" - misc additions   mo5f 23816
                  19.3.2.5  Existential uniqueness - misc additions   2reuswap2 23819
                  19.3.2.6  Restricted "at most one" - misc additions   rmoxfrdOLD 23823
            19.3.3  General Set Theory   ceqsexv2d 23829
                  19.3.3.1  Class abstractions (a.k.a. class builders)   ceqsexv2d 23829
                  19.3.3.2  Image Sets   abrexdomjm 23832
                  19.3.3.3  Set relations and operations - misc additions   eqri 23838
                  19.3.3.4  Unordered pairs   elpreq 23843
                  19.3.3.5  Conditional operator - misc additions   ifeqeqx 23845
                  19.3.3.6  Indexed union - misc additions   iuneq12daf 23851
                  19.3.3.7  Disjointness - misc additions   cbvdisjf 23859
            19.3.4  Relations and Functions   dfrel4 23877
                  19.3.4.1  Relations - misc additions   dfrel4 23877
                  19.3.4.2  Functions - misc additions   fdmrn 23882
                  19.3.4.3  Isomorphisms - misc. add.   gtiso 23929
                  19.3.4.4  Disjointness (additional proof requiring functions)   disjdsct 23931
                  19.3.4.5  First and second members of an ordered pair - misc additions   df1stres 23932
                  19.3.4.6  Supremum - misc additions   supssd 23939
                  19.3.4.7  Countable Sets   nnct 23940
            19.3.5  Real and Complex Numbers   addeq0 23955
                  19.3.5.1  Complex addition - misc. additions   addeq0 23955
                  19.3.5.2  Ordering on reals - misc additions   lt2addrd 23956
                  19.3.5.3  Extended reals - misc additions   xrlelttric 23957
                  19.3.5.4  Real number intervals - misc additions   icossicc 23965
                  19.3.5.5  Finite intervals of integers - misc additions   fzssnn 23983
                  19.3.5.6  Half-open integer ranges - misc additions   fzossnn 23988
                  19.3.5.7  The ` # ` (finite set size) function - misc additions   hashresfn 23994
                  19.3.5.8  The greatest common divisor operator - misc. add   numdenneg 23998
                  19.3.5.9  Integers   ltesubnnd 24000
                  19.3.5.10  Division in the extended real number system   cxdiv 24001
            19.3.6  Structure builders   ress0g 24021
                  19.3.6.1  Structure builder restriction operator   ress0g 24021
                  19.3.6.2  Posets   tospos 24025
                  19.3.6.3  Extended reals Structure - misc additions   ax-xrssca 24028
                  19.3.6.4  The extended non-negative real numbers monoid   xrge0base 24036
            19.3.7  Algebra   sumpr 24047
                  19.3.7.1  Finitely supported group sums - misc additions   sumpr 24047
                  19.3.7.2  Rings - misc additions   dvrdir 24055
                  19.3.7.3  Ordered fields   cofld 24059
                  19.3.7.4  Ring homomorphisms - misc additions   rhmdvdsr 24072
                  19.3.7.5  The ring of integers   zzsbase 24079
                  19.3.7.6  The ordered field of reals   rebase 24085
            19.3.8  Topology   hauseqcn 24097
                  19.3.8.1  Continuity - misc additions   hauseqcn 24097
                  19.3.8.2  Topology of the closed unit   unitsscn 24098
                  19.3.8.3  Topology of ` ( RR X. RR ) `   unicls 24105
                  19.3.8.4  Order topology - misc. additions   cnvordtrestixx 24115
                  19.3.8.5  Continuity in topological spaces - misc. additions   mndpluscn 24116
                  19.3.8.6  Topology of the extended non-negative real numbers monoid   xrge0hmph 24122
                  19.3.8.7  Limits - misc additions   lmlim 24137
            19.3.9  Topology and algebraic structures   zzsnm 24144
                  19.3.9.1  The norm on the ring of the integer numbers   zzsnm 24144
                  19.3.9.2  The complete ordered field of the real numbers   recms 24145
                  19.3.9.3  Topological ` ZZ ` -modules   zlm0 24147
                  19.3.9.4  The canonical embedding of the rational numbers into a division ring   cqqh 24155
                  19.3.9.5  The canonical embedding of ` RR ` into a complete ordered field   crrh 24176
                  19.3.9.6  Canonical embeddings into ` RR `   zrhre 24181
            19.3.10  Real and complex functions   clogb 24184
                  19.3.10.1  Logarithm laws generalized to an arbitrary base - logb   clogb 24184
                  19.3.10.2  Indicator Functions   cind 24204
                  19.3.10.3  Extended sum   cesum 24220
            19.3.11  Mixed Function/Constant operation   cofc 24274
            19.3.12  Abstract measure   csiga 24286
                  19.3.12.1  Sigma-Algebra   csiga 24286
                  19.3.12.2  Generated Sigma-Algebra   csigagen 24317
                  19.3.12.3  The Borel algebra on the real numbers   cbrsiga 24331
                  19.3.12.4  Product Sigma-Algebra   csx 24338
                  19.3.12.5  Measures   cmeas 24345
                  19.3.12.6  The counting measure   cntmeas 24374
                  19.3.12.7  The Lebesgue measure - misc additions   volss 24377
                  19.3.12.8  The 'almost everywhere' relation   cae 24382
                  19.3.12.9  Measurable functions   cmbfm 24394
                  19.3.12.10  Borel Algebra on ` ( RR X. RR ) `   br2base 24413
            19.3.13  Integration   itgeq12dv 24435
                  19.3.13.1  Lebesgue integral - misc additions   itgeq12dv 24435
                  19.3.13.2  Bochner integral   citgm 24436
            19.3.14  Probability   cprb 24444
                  19.3.14.1  Probability Theory   cprb 24444
                  19.3.14.2  Conditional Probabilities   ccprob 24468
                  19.3.14.3  Real Valued Random Variables   crrv 24477
                  19.3.14.4  Preimage set mapping operator   corvc 24492
                  19.3.14.5  Distribution Functions   orvcelval 24505
                  19.3.14.6  Cumulative Distribution Functions   orvclteel 24509
                  19.3.14.7  Probabilities - example   coinfliplem 24515
                  19.3.14.8  Bertrand's Ballot Problem   ballotlemoex 24522
      19.4  Mathbox for Mario Carneiro
            19.4.1  Miscellaneous stuff   quartfull 24575
            19.4.2  Zeta function   czeta 24576
            19.4.3  Gamma function   clgam 24579
            19.4.4  Derangements and the Subfactorial   deranglem 24631
            19.4.5  The Erdős-Szekeres theorem   erdszelem1 24656
            19.4.6  The Kuratowski closure-complement theorem   kur14lem1 24671
            19.4.7  Retracts and sections   cretr 24682
            19.4.8  Path-connected and simply connected spaces   cpcon 24685
            19.4.9  Covering maps   ccvm 24721
            19.4.10  Normal numbers   snmlff 24795
            19.4.11  Godel-sets of formulas   cgoe 24799
            19.4.12  Models of ZF   cgze 24827
            19.4.13  Splitting fields   citr 24841
            19.4.14  p-adic number fields   czr 24857
      19.5  Mathbox for Paul Chapman
            19.5.1  Group homomorphism and isomorphism   ghomgrpilem1 24875
            19.5.2  Real and complex numbers (cont.)   climuzcnv 24887
            19.5.3  Miscellaneous theorems   elfzm12 24891
      19.6  Mathbox for Drahflow
      19.7  Mathbox for Scott Fenton
            19.7.1  ZFC Axioms in primitive form   axextprim 24929
            19.7.2  Untangled classes   untelirr 24936
            19.7.3  Extra propositional calculus theorems   3orel1 24943
            19.7.4  Misc. Useful Theorems   nepss 24954
            19.7.5  Properties of reals and complexes   sqdivzi 24963
            19.7.6  Product sequences   prodf 24994
            19.7.7  Non-trivial convergence   ntrivcvg 25004
            19.7.8  Complex products   cprod 25010
            19.7.9  Finite products   fprod 25046
            19.7.10  Infinite products   iprodclim 25083
            19.7.11  Falling and Rising Factorial   cfallfac 25089
            19.7.12  Factorial limits   faclimlem1 25120
            19.7.13  Greatest common divisor and divisibility   pdivsq 25126
            19.7.14  Properties of relationships   brtp 25130
            19.7.15  Properties of functions and mappings   funpsstri 25145
            19.7.16  Epsilon induction   setinds 25158
            19.7.17  Ordinal numbers   elpotr 25161
            19.7.18  Defined equality axioms   axextdfeq 25178
            19.7.19  Hypothesis builders   hbntg 25186
            19.7.20  The Predecessor Class   cpred 25191
            19.7.21  (Trans)finite Recursion Theorems   tfisg 25228
            19.7.22  Well-founded induction   tz6.26 25229
            19.7.23  Transitive closure under a relationship   ctrpred 25244
            19.7.24  Founded Induction   frmin 25266
            19.7.25  Ordering Ordinal Sequences   orderseqlem 25276
            19.7.26  Well-founded recursion   wfr3g 25279
            19.7.27  Transfinite recursion via Well-founded recursion   tfrALTlem 25300
            19.7.28  Founded Recursion   frr3g 25304
            19.7.29  Surreal Numbers   csur 25318
            19.7.30  Surreal Numbers: Ordering   sltsolem1 25346
            19.7.31  Surreal Numbers: Birthday Function   bdayfo 25353
            19.7.32  Surreal Numbers: Density   fvnobday 25360
            19.7.33  Surreal Numbers: Density   nodenselem3 25361
            19.7.34  Surreal Numbers: Upper and Lower Bounds   nobndlem1 25370
            19.7.35  Surreal Numbers: Full-Eta Property   nofulllem1 25380
            19.7.36  Symmetric difference   csymdif 25385
            19.7.37  Quantifier-free definitions   ctxp 25397
            19.7.38  Alternate ordered pairs   caltop 25515
            19.7.39  Tarskian geometry   cee 25541
            19.7.40  Tarski's axioms for geometry   axdimuniq 25566
            19.7.41  Congruence properties   cofs 25630
            19.7.42  Betweenness properties   btwntriv2 25660
            19.7.43  Segment Transportation   ctransport 25677
            19.7.44  Properties relating betweenness and congruence   cifs 25683
            19.7.45  Connectivity of betweenness   btwnconn1lem1 25735
            19.7.46  Segment less than or equal to   csegle 25754
            19.7.47  Outside of relationship   coutsideof 25767
            19.7.48  Lines and Rays   cline2 25782
            19.7.49  Bernoulli polynomials and sums of k-th powers   cbp 25806
            19.7.50  Rank theorems   rankung 25821
            19.7.51  Hereditarily Finite Sets   chf 25827
      19.8  Mathbox for Anthony Hart
            19.8.1  Propositional Calculus   tb-ax1 25842
            19.8.2  Predicate Calculus   quantriv 25864
            19.8.3  Misc. Single Axiom Systems   meran1 25875
            19.8.4  Connective Symmetry   negsym1 25881
      19.9  Mathbox for Chen-Pang He
            19.9.1  Ordinal topology   ontopbas 25892
      19.10  Mathbox for Jeff Hoffman
            19.10.1  Inferences for finite induction on generic function values   fveleq 25915
            19.10.2  gdc.mm   nnssi2 25919
      19.11  Mathbox for Wolf Lammen
      19.12  Mathbox for Brendan Leahy
      19.13  Mathbox for Jeff Hankins
            19.13.1  Miscellany   a1i13 25989
            19.13.2  Basic topological facts   topbnd 26018
            19.13.3  Topology of the real numbers   ivthALT 26029
            19.13.4  Refinements   cfne 26030
            19.13.5  Neighborhood bases determine topologies   neibastop1 26079
            19.13.6  Lattice structure of topologies   topmtcl 26083
            19.13.7  Filter bases   fgmin 26090
            19.13.8  Directed sets, nets   tailfval 26092
      19.14  Mathbox for Jeff Madsen
            19.14.1  Logic and set theory   anim12da 26103
            19.14.2  Real and complex numbers; integers   filbcmb 26133
            19.14.3  Sequences and sums   sdclem2 26137
            19.14.4  Topology   subspopn 26149
            19.14.5  Metric spaces   metf1o 26152
            19.14.6  Continuous maps and homeomorphisms   constcncf 26159
            19.14.7  Boundedness   ctotbnd 26166
            19.14.8  Isometries   cismty 26198
            19.14.9  Heine-Borel Theorem   heibor1lem 26209
            19.14.10  Banach Fixed Point Theorem   bfplem1 26222
            19.14.11  Euclidean space   crrn 26225
            19.14.12  Intervals (continued)   ismrer1 26238
            19.14.13  Groups and related structures   exidcl 26242
            19.14.14  Rings   rngonegcl 26252
            19.14.15  Ring homomorphisms   crnghom 26267
            19.14.16  Commutative rings   ccring 26296
            19.14.17  Ideals   cidl 26308
            19.14.18  Prime rings and integral domains   cprrng 26347
            19.14.19  Ideal generators   cigen 26360
      19.15  Mathbox for Rodolfo Medina
            19.15.1  Partitions   prtlem60 26379
      19.16  Mathbox for Stefan O'Rear
            19.16.1  Additional elementary logic and set theory   nelss 26423
            19.16.2  Additional theory of functions   fninfp 26426
            19.16.3  Extensions beyond function theory   gsumvsmul 26436
            19.16.4  Additional topology   elrfi 26439
            19.16.5  Characterization of closure operators. Kuratowski closure axioms   ismrcd1 26443
            19.16.6  Algebraic closure systems   cnacs 26447
            19.16.7  Miscellanea 1. Map utilities   constmap 26458
            19.16.8  Miscellanea for polynomials   ofmpteq 26467
            19.16.9  Multivariate polynomials over the integers   cmzpcl 26469
            19.16.10  Miscellanea for Diophantine sets 1   coeq0 26501
            19.16.11  Diophantine sets 1: definitions   cdioph 26504
            19.16.12  Diophantine sets 2 miscellanea   ellz1 26516
            19.16.13  Diophantine sets 2: union and intersection. Monotone Boolean algebra   diophin 26522
            19.16.14  Diophantine sets 3: construction   diophrex 26525
            19.16.15  Diophantine sets 4 miscellanea   2sbcrex 26534
            19.16.16  Diophantine sets 4: Quantification   rexrabdioph 26545
            19.16.17  Diophantine sets 5: Arithmetic sets   rabdiophlem1 26552
            19.16.18  Diophantine sets 6 miscellanea   fz1ssnn 26562
            19.16.19  Diophantine sets 6: reusability. renumbering of variables   eldioph4b 26563
            19.16.20  Pigeonhole Principle and cardinality helpers   fphpd 26568
            19.16.21  A non-closed set of reals is infinite   rencldnfilem 26572
            19.16.22  Miscellanea for Lagrange's theorem   icodiamlt 26574
            19.16.23  Lagrange's rational approximation theorem   irrapxlem1 26576
            19.16.24  Pell equations 1: A nontrivial solution always exists   pellexlem1 26583
            19.16.25  Pell equations 2: Algebraic number theory of the solution set   csquarenn 26590
            19.16.26  Pell equations 3: characterizing fundamental solution   infmrgelbi 26632
            19.16.27  Logarithm laws generalized to an arbitrary base   reglogcl 26644
            19.16.28  Pell equations 4: the positive solution group is infinite cyclic   pellfund14 26652
            19.16.29  X and Y sequences 1: Definition and recurrence laws   crmx 26654
            19.16.30  Ordering and induction lemmas for the integers   monotuz 26695
            19.16.31  X and Y sequences 2: Order properties   rmxypos 26703
            19.16.32  Congruential equations   congtr 26721
            19.16.33  Alternating congruential equations   acongid 26731
            19.16.34  Additional theorems on integer divisibility   bezoutr 26741
            19.16.35  X and Y sequences 3: Divisibility properties   jm2.18 26750
            19.16.36  X and Y sequences 4: Diophantine representability of Y   jm2.27a 26767
            19.16.37  X and Y sequences 5: Diophantine representability of X, ^, _C   rmxdiophlem 26777
            19.16.38  Uncategorized stuff not associated with a major project   setindtr 26786
            19.16.39  More equivalents of the Axiom of Choice   axac10 26795
            19.16.40  Finitely generated left modules   clfig 26834
            19.16.41  Noetherian left modules I   clnm 26842
            19.16.42  Addenda for structure powers   pwssplit0 26856
            19.16.43  Direct sum of left modules   cdsmm 26866
            19.16.44  Free modules   cfrlm 26881
            19.16.45  Every set admits a group structure iff choice   unxpwdom3 26925
            19.16.46  Independent sets and families   clindf 26943
            19.16.47  Characterization of free modules   lmimlbs 26975
            19.16.48  Noetherian rings and left modules II   clnr 26982
            19.16.49  Hilbert's Basis Theorem   cldgis 26994
            19.16.50  Additional material on polynomials [DEPRECATED]   cmnc 27004
            19.16.51  Degree and minimal polynomial of algebraic numbers   cdgraa 27014
            19.16.52  Algebraic integers I   citgo 27031
            19.16.53  Finite cardinality [SO]   en1uniel 27049
            19.16.54  Words in monoids and ordered group sum   issubmd 27052
            19.16.55  Transpositions in the symmetric group   cpmtr 27053
            19.16.56  The sign of a permutation   cpsgn 27083
            19.16.57  The matrix algebra   cmmul 27108
            19.16.58  The determinant   cmdat 27152
            19.16.59  Endomorphism algebra   cmend 27158
            19.16.60  Subfields   csdrg 27172
            19.16.61  Cyclic groups and order   idomrootle 27180
            19.16.62  Cyclotomic polynomials   ccytp 27190
            19.16.63  Miscellaneous topology   fgraphopab 27198
      19.17  Mathbox for Steve Rodriguez
            19.17.1  Miscellanea   iso0 27205
            19.17.2  Function operations   caofcan 27209
            19.17.3  Calculus   lhe4.4ex1a 27215
      19.18  Mathbox for Andrew Salmon
            19.18.1  Principia Mathematica * 10   pm10.12 27222
            19.18.2  Principia Mathematica * 11   2alanimi 27236
            19.18.3  Predicate Calculus   sbeqal1 27266
            19.18.4  Principia Mathematica * 13 and * 14   pm13.13a 27276
            19.18.5  Set Theory   elnev 27307
            19.18.6  Arithmetic   addcomgi 27329
            19.18.7  Geometry   cplusr 27330
      19.19  Mathbox for Glauco Siliprandi
            19.19.1  Miscellanea   ssrexf 27352
            19.19.2  Finite multiplication of numbers and finite multiplication of functions   fmul01 27378
            19.19.3  Limits   clim1fr1 27395
            19.19.4  Derivatives   dvsinexp 27408
            19.19.5  Integrals   ioovolcl 27410
            19.19.6  Stone Weierstrass theorem - real version   stoweidlem1 27418
            19.19.7  Wallis' product for π   wallispilem1 27482
            19.19.8  Stirling's approximation formula for ` n ` factorial   stirlinglem1 27491
      19.20  Mathbox for Saveliy Skresanov
            19.20.1  Ceva's theorem   sigarval 27508
      19.21  Mathbox for Jarvin Udandy
      19.22  Mathbox for Alexander van der Vekens
            19.22.1  Double restricted existential uniqueness   r19.32 27613
                  19.22.1.1  Restricted quantification (extension)   r19.32 27613
                  19.22.1.2  The empty set (extension)   raaan2 27621
                  19.22.1.3  Restricted uniqueness and "at most one" quantification   rmoimi 27622
                  19.22.1.4  Analogs to Existential uniqueness (double quantification)   2reurex 27627
            19.22.2  Alternative definitions of function's and operation's values   wdfat 27639
                  19.22.2.1  Restricted quantification (extension)   ralbinrald 27645
                  19.22.2.2  The universal class (extension)   nvelim 27646
                  19.22.2.3  Introduce the Axiom of Power Sets (extension)   alneu 27647
                  19.22.2.4  Relations (extension)   sbcrel 27649
                  19.22.2.5  Functions (extension)   sbcfun 27655
                  19.22.2.6  Predicate "defined at"   dfateq12d 27662
                  19.22.2.7  Alternative definition of the value of a function   dfafv2 27665
                  19.22.2.8  Alternative definition of the value of an operation   aoveq123d 27711
            19.22.3  Graph theory   cfrgra 27741
                  19.22.3.1  Friendship graphs   cfrgra 27741
      19.23  Mathbox for David A. Wheeler
            19.23.1  Natural deduction   19.8ad 27806
            19.23.2  Greater than, greater than or equal to.   cge-real 27809
            19.23.3  Hyperbolic trig functions   csinh 27819
            19.23.4  Reciprocal trig functions (sec, csc, cot)   csec 27830
            19.23.5  Identities for "if"   ifnmfalse 27852
            19.23.6  Not-member-of   AnelBC 27853
            19.23.7  Decimal point   cdp2 27854
            19.23.8  Signum (sgn or sign) function   csgn 27862
            19.23.9  Ceiling function   ccei 27872
            19.23.10  Logarithms generalized to arbitrary base using ` logb `   ene0 27876
            19.23.11  Logarithm laws generalized to an arbitrary base - log_   clog_ 27879
            19.23.12  Miscellaneous   5m4e1 27881
      19.24  Mathbox for Alan Sare
            19.24.1  Supplementary "adant" deductions   ad4ant13 27884
            19.24.2  Supplementary unification deductions   biimp 27910
            19.24.3  Conventional Metamath proofs, some derived from VD proofs   iidn3 27926
            19.24.4  What is Virtual Deduction?   wvd1 28001
            19.24.5  Virtual Deduction Theorems   df-vd1 28002
            19.24.6  Theorems proved using virtual deduction   trsspwALT 28272
            19.24.7  Theorems proved using virtual deduction with mmj2 assistance   simplbi2VD 28299
            19.24.8  Virtual Deduction transcriptions of textbook proofs   sb5ALTVD 28366
            19.24.9  Theorems proved using conjunction-form virtual deduction   elpwgdedVD 28370
            19.24.10  Theorems with VD proofs in conventional notation derived from VD proofs   suctrALT3 28377
            19.24.11  Theorems with a proof in conventional notation automatically derived   notnot2ALT2 28380
      19.25  Mathbox for Jonathan Ben-Naim
            19.25.1  First order logic and set theory   bnj170 28400
            19.25.2  Well founded induction and recursion   bnj110 28567
            19.25.3  The existence of a minimal element in certain classes   bnj69 28717
            19.25.4  Well-founded induction   bnj1204 28719
            19.25.5  Well-founded recursion, part 1 of 3   bnj60 28769
            19.25.6  Well-founded recursion, part 2 of 3   bnj1500 28775
            19.25.7  Well-founded recursion, part 3 of 3   bnj1522 28779
      19.26  Mathbox for Norm Megill
            19.26.1  Experiments to study ax-7 unbundling   ax-7v 28780
                  19.26.1.1  Theorems derived from ax-7v (suffixes NEW7 and AUX7)   ax-7v 28780
                  19.26.1.2  Theorems derived from ax-7 (suffix OLD7)   ax-7OLD7 28994
            19.26.2  Miscellanea   cnaddcom 29086
            19.26.3  Atoms, hyperplanes, and covering in a left vector space (or module)   clsa 29089
            19.26.4  Functionals and kernels of a left vector space (or module)   clfn 29172
            19.26.5  Opposite rings and dual vector spaces   cld 29238
            19.26.6  Ortholattices and orthomodular lattices   cops 29287
            19.26.7  Atomic lattices with covering property   ccvr 29377
            19.26.8  Hilbert lattices   chlt 29465
            19.26.9  Projective geometries based on Hilbert lattices   clln 29605
            19.26.10  Construction of a vector space from a Hilbert lattice   cdlema1N 29905
            19.26.11  Construction of involution and inner product from a Hilbert lattice   clpoN 31595

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