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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 1296
            1.2.10  Logical 'xor'   wxo 1313
            1.2.11  True and false constants   wtru 1325
            1.2.12  Truth tables   truantru 1345
            1.2.13  Auxiliary theorems for Alan Sare's virtual deduction tool, part 1   ee22 1371
            1.2.14  Half-adders and full adders in propositional calculus   whad 1387
      1.3  Other axiomatizations of classical propositional calculus
            1.3.1  Derive the Lukasiewicz axioms from Meredith's sole axiom   meredith 1413
            1.3.2  Derive the standard axioms from the Lukasiewicz axioms   luklem1 1432
            1.3.3  Derive Nicod's axiom from the standard axioms   nic-dfim 1443
            1.3.4  Derive the Lukasiewicz axioms from Nicod's axiom   nic-imp 1449
            1.3.5  Derive Nicod's Axiom from Lukasiewicz's First Sheffer Stroke Axiom   lukshef-ax1 1468
            1.3.6  Derive the Lukasiewicz Axioms from the Tarski-Bernays-Wajsberg Axioms   tbw-bijust 1472
            1.3.7  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's First CO Axiom   merco1 1487
            1.3.8  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's Second CO Axiom   merco2 1510
            1.3.9  Derive the Lukasiewicz axioms from the The Russell-Bernays Axioms   rb-bijust 1523
            1.3.10  Stoic logic indemonstrables (Chrysippus of Soli)   mpto1 1542
      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 1549
            1.4.2  Rule scheme ax-gen (Generalization)   ax-gen 1555
            1.4.3  Axiom scheme ax-5 (Quantified Implication)   ax-5 1566
            1.4.4  Axiom scheme ax-17 (Distinctness) - first use of $d   ax-17 1626
            1.4.5  Equality predicate; define substitution   cv 1651
            1.4.6  Axiom scheme ax-9 (Existence)   ax-9 1666
            1.4.7  Axiom scheme ax-8 (Equality)   ax-8 1687
            1.4.8  Membership predicate   wcel 1725
            1.4.9  Axiom scheme ax-13 (Left Equality for Binary Predicate)   ax-13 1727
            1.4.10  Axiom scheme ax-14 (Right Equality for Binary Predicate)   ax-14 1729
            1.4.11  Logical redundancy of ax-6 , ax-7 , ax-11 , ax-12   ax9dgen 1731
      1.5  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes)
            1.5.1  Axiom scheme ax-6 (Quantified Negation)   ax-6 1744
            1.5.2  Axiom scheme ax-7 (Quantifier Commutation)   ax-7 1749
            1.5.3  Axiom scheme ax-11 (Substitution)   ax-11 1761
            1.5.4  Axiom scheme ax-12 (Quantified Equality)   ax-12 1950
      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 2212
            1.6.2  Rederive new axioms from old: ax5 , ax6 , ax9from9o , ax11 , ax12from12o   ax4 2222
            1.6.3  Legacy theorems using obsolete axioms   ax17o 2234
      1.7  Existential uniqueness
      1.8  Other axiomatizations related to classical predicate calculus
            1.8.1  Predicate calculus with all distinct variables   ax-7d 2372
            1.8.2  Aristotelian logic: Assertic syllogisms   barbara 2378
            1.8.3  Intuitionistic logic   axia1 2402
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 2417
            2.1.2  Class abstractions (a.k.a. class builders)   cab 2422
            2.1.3  Class form not-free predicate   wnfc 2559
            2.1.4  Negated equality and membership   wne 2599
                  2.1.4.1  Negated equality   neii 2603
                  2.1.4.2  Negated membership   neli 2697
            2.1.5  Restricted quantification   wral 2705
            2.1.6  The universal class   cvv 2956
            2.1.7  Conditional equality (experimental)   wcdeq 3144
            2.1.8  Russell's Paradox   ru 3160
            2.1.9  Proper substitution of classes for sets   wsbc 3161
            2.1.10  Proper substitution of classes for sets into classes   csb 3251
            2.1.11  Define basic set operations and relations   cdif 3317
            2.1.12  Subclasses and subsets   df-ss 3334
            2.1.13  The difference, union, and intersection of two classes   difeq1 3458
                  2.1.13.1  The difference of two classes   difeq1 3458
                  2.1.13.2  The union of two classes   elun 3488
                  2.1.13.3  The intersection of two classes   elin 3530
                  2.1.13.4  Combinations of difference, union, and intersection of two classes   unabs 3571
                  2.1.13.5  Class abstractions with difference, union, and intersection of two classes   symdif2 3607
                  2.1.13.6  Restricted uniqueness with difference, union, and intersection   reuss2 3621
            2.1.14  The empty set   c0 3628
            2.1.15  "Weak deduction theorem" for set theory   cif 3739
            2.1.16  Power classes   cpw 3799
            2.1.17  Unordered and ordered pairs   csn 3814
            2.1.18  The union of a class   cuni 4015
            2.1.19  The intersection of a class   cint 4050
            2.1.20  Indexed union and intersection   ciun 4093
            2.1.21  Disjointness   wdisj 4182
            2.1.22  Binary relations   wbr 4212
            2.1.23  Ordered-pair class abstractions (class builders)   copab 4265
            2.1.24  Transitive classes   wtr 4302
      2.2  ZF Set Theory - add the Axiom of Replacement
            2.2.1  Introduce the Axiom of Replacement   ax-rep 4320
            2.2.2  Derive the Axiom of Separation   axsep 4329
            2.2.3  Derive the Null Set Axiom   zfnuleu 4335
            2.2.4  Theorems requiring subset and intersection existence   nalset 4340
            2.2.5  Theorems requiring empty set existence   class2set 4367
      2.3  ZF Set Theory - add the Axiom of Power Sets
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 4377
            2.3.2  Derive the Axiom of Pairing   zfpair 4401
            2.3.3  Ordered pair theorem   opnz 4432
            2.3.4  Ordered-pair class abstractions (cont.)   opabid 4461
            2.3.5  Power class of union and intersection   pwin 4487
            2.3.6  Epsilon and identity relations   cep 4492
            2.3.7  Partial and complete ordering   wpo 4501
            2.3.8  Founded and well-ordering relations   wfr 4538
            2.3.9  Ordinals   word 4580
      2.4  ZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 4701
            2.4.2  Ordinals (continued)   ordon 4763
            2.4.3  Transfinite induction   tfi 4833
            2.4.4  The natural numbers (i.e. finite ordinals)   com 4845
            2.4.5  Peano's postulates   peano1 4864
            2.4.6  Finite induction (for finite ordinals)   find 4870
            2.4.7  Relations   cxp 4876
            2.4.8  Definite description binder (inverted iota)   cio 5416
            2.4.9  Functions   wfun 5448
            2.4.10  Operations   co 6081
            2.4.11  "Maps to" notation   elmpt2cl 6288
            2.4.12  Function operation   cof 6303
            2.4.13  First and second members of an ordered pair   c1st 6347
            2.4.14  Special "Maps to" operations   mpt2xopn0yelv 6464
            2.4.15  Function transposition   ctpos 6478
            2.4.16  Curry and uncurry   ccur 6517
            2.4.17  Proper subset relation   crpss 6521
            2.4.18  Iota properties   fvopab5 6534
            2.4.19  Cantor's Theorem   canth 6539
            2.4.20  Undefined values and restricted iota (description binder)   cund 6541
            2.4.21  Functions on ordinals; strictly monotone ordinal functions   iunon 6600
            2.4.22  "Strong" transfinite recursion   crecs 6632
            2.4.23  Recursive definition generator   crdg 6667
            2.4.24  Finite recursion   frfnom 6692
            2.4.25  Abian's "most fundamental" fixed point theorem   abianfplem 6715
            2.4.26  Ordinal arithmetic   c1o 6717
            2.4.27  Natural number arithmetic   nna0 6847
            2.4.28  Equivalence relations and classes   wer 6902
            2.4.29  The mapping operation   cmap 7018
            2.4.30  Infinite Cartesian products   cixp 7063
            2.4.31  Equinumerosity   cen 7106
            2.4.32  Schroeder-Bernstein Theorem   sbthlem1 7217
            2.4.33  Equinumerosity (cont.)   xpf1o 7269
            2.4.34  Pigeonhole Principle   phplem1 7286
            2.4.35  Finite sets   onomeneq 7296
            2.4.36  Finite intersections   cfi 7415
            2.4.37  Hall's marriage theorem   marypha1lem 7438
            2.4.38  Supremum   csup 7445
            2.4.39  Ordinal isomorphism, Hartog's theorem   coi 7478
            2.4.40  Hartogs function, order types, weak dominance   char 7524
      2.5  ZF Set Theory - add the Axiom of Regularity
            2.5.1  Introduce the Axiom of Regularity   ax-reg 7560
            2.5.2  Axiom of Infinity equivalents   inf0 7576
      2.6  ZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-inf 7593
            2.6.2  Existence of omega (the set of natural numbers)   omex 7598
            2.6.3  Cantor normal form   ccnf 7616
            2.6.4  Transitive closure   trcl 7664
            2.6.5  Rank   cr1 7688
            2.6.6  Scott's trick; collection principle; Hilbert's epsilon   scottex 7809
            2.6.7  Cardinal numbers   ccrd 7822
            2.6.8  Axiom of Choice equivalents   wac 7996
            2.6.9  Cardinal number arithmetic   ccda 8047
            2.6.10  The Ackermann bijection   ackbij2lem1 8099
            2.6.11  Cofinality (without Axiom of Choice)   cflem 8126
            2.6.12  Eight inequivalent definitions of finite set   sornom 8157
            2.6.13  Hereditarily size-limited sets without Choice   itunifval 8296
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 8339
            3.2.2  AC equivalents: well-ordering, Zorn's lemma   numthcor 8374
            3.2.3  Cardinal number theorems using Axiom of Choice   cardval 8421
            3.2.4  Cardinal number arithmetic using Axiom of Choice   iunctb 8449
            3.2.5  Cofinality using Axiom of Choice   alephreg 8457
      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 8557
            4.1.2  Weak universes   cwun 8575
            4.1.3  Tarski's classes   ctsk 8623
            4.1.4  Grothendieck's universes   cgru 8665
      4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
            4.2.1  Introduce the Tarski-Grothendieck Axiom   ax-groth 8698
            4.2.2  Derive the Power Set, Infinity and Choice Axioms   grothpw 8701
            4.2.3  Tarski map function   ctskm 8712
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 8719
            5.1.2  Final derivation of real and complex number postulates   axaddf 9020
            5.1.3  Real and complex number postulates restated as axioms   ax-cnex 9046
      5.2  Derive the basic properties from the field axioms
            5.2.1  Some deductions from the field axioms for complex numbers   cnex 9071
            5.2.2  Infinity and the extended real number system   cpnf 9117
            5.2.3  Restate the ordering postulates with extended real "less than"   axlttri 9147
            5.2.4  Ordering on reals   lttr 9152
            5.2.5  Initial properties of the complex numbers   mul12 9232
      5.3  Real and complex numbers - basic operations
            5.3.1  Addition   add12 9279
            5.3.2  Subtraction   cmin 9291
            5.3.3  Multiplication   muladd 9466
            5.3.4  Ordering on reals (cont.)   gt0ne0 9493
            5.3.5  Reciprocals   ixi 9651
            5.3.6  Division   cdiv 9677
            5.3.7  Ordering on reals (cont.)   elimgt0 9846
            5.3.8  Completeness Axiom and Suprema   fimaxre 9955
            5.3.9  Imaginary and complex number properties   inelr 9990
            5.3.10  Function operation analogue theorems   ofsubeq0 9997
      5.4  Integer sets
            5.4.1  Natural numbers (as a subset of complex numbers)   cn 10000
            5.4.2  Principle of mathematical induction   nnind 10018
            5.4.3  Decimal representation of numbers   c2 10049
            5.4.4  Some properties of specific numbers   0p1e1 10093
            5.4.5  The Archimedean property   nnunb 10217
            5.4.6  Nonnegative integers (as a subset of complex numbers)   cn0 10221
            5.4.7  Integers (as a subset of complex numbers)   cz 10282
            5.4.8  Decimal arithmetic   cdc 10382
            5.4.9  Upper partititions of integers   cuz 10488
            5.4.10  Well-ordering principle for bounded-below sets of integers   uzwo3 10569
            5.4.11  Rational numbers (as a subset of complex numbers)   cq 10574
            5.4.12  Existence of the set of complex numbers   rpnnen1lem1 10600
      5.5  Order sets
            5.5.1  Positive reals (as a subset of complex numbers)   crp 10612
            5.5.2  Infinity and the extended real number system (cont.)   cxne 10707
            5.5.3  Supremum on the extended reals   xrsupexmnf 10883
            5.5.4  Real number intervals   cioo 10916
            5.5.5  Finite intervals of integers   cfz 11043
            5.5.6  Half-open integer ranges   cfzo 11135
      5.6  Elementary integer functions
            5.6.1  The floor (greatest integer) function   cfl 11201
            5.6.2  The modulo (remainder) operation   cmo 11250
            5.6.3  The infinite sequence builder "seq"   om2uz0i 11287
            5.6.4  Integer powers   cexp 11382
            5.6.5  Ordered pair theorem for nonnegative integers   nn0le2msqi 11560
            5.6.6  Factorial function   cfa 11566
            5.6.7  The binomial coefficient operation   cbc 11593
            5.6.8  The ` # ` (finite set size) function   chash 11618
                  5.6.8.1  Finite induction on the size of the first component of a binary relation   brfi1indlem 11714
            5.6.9  Words over a set   cword 11717
            5.6.10  Longer string literals   cs2 11805
      5.7  Elementary real and complex functions
            5.7.1  The "shift" operation   cshi 11881
            5.7.2  Real and imaginary parts; conjugate   ccj 11901
            5.7.3  Square root; absolute value   csqr 12038
      5.8  Elementary limits and convergence
            5.8.1  Superior limit (lim sup)   clsp 12264
            5.8.2  Limits   cli 12278
            5.8.3  Finite and infinite sums   csu 12479
            5.8.4  The binomial theorem   binomlem 12608
            5.8.5  The inclusion/exclusion principle   incexclem 12616
            5.8.6  Infinite sums (cont.)   isumshft 12619
            5.8.7  Miscellaneous converging and diverging sequences   divrcnv 12632
            5.8.8  Arithmetic series   arisum 12639
            5.8.9  Geometric series   expcnv 12643
            5.8.10  Ratio test for infinite series convergence   cvgrat 12660
            5.8.11  Mertens' theorem   mertenslem1 12661
      5.9  Elementary trigonometry
            5.9.1  The exponential, sine, and cosine functions   ce 12664
            5.9.2  _e is irrational   eirrlem 12803
      5.10  Cardinality of real and complex number subsets
            5.10.1  Countability of integers and rationals   xpnnen 12808
            5.10.2  The reals are uncountable   rpnnen2lem1 12814
PART 6  ELEMENTARY NUMBER THEORY
      6.1  Elementary properties of divisibility
            6.1.1  Irrationality of square root of 2   sqr2irrlem 12847
            6.1.2  Some Number sets are chains of proper subsets   nthruc 12850
            6.1.3  The divides relation   cdivides 12852
            6.1.4  The division algorithm   divalglem0 12913
            6.1.5  Bit sequences   cbits 12931
            6.1.6  The greatest common divisor operator   cgcd 13006
            6.1.7  Bézout's identity   bezoutlem1 13038
            6.1.8  Algorithms   nn0seqcvgd 13061
            6.1.9  Euclid's Algorithm   eucalgval2 13072
      6.2  Elementary prime number theory
            6.2.1  Elementary properties   cprime 13079
            6.2.2  Properties of the canonical representation of a rational   cnumer 13125
            6.2.3  Euler's theorem   codz 13152
            6.2.4  Pythagorean Triples   coprimeprodsq 13183
            6.2.5  The prime count function   cpc 13210
            6.2.6  Pocklington's theorem   prmpwdvds 13272
            6.2.7  Infinite primes theorem   unbenlem 13276
            6.2.8  Sum of prime reciprocals   prmreclem1 13284
            6.2.9  Fundamental theorem of arithmetic   1arithlem1 13291
            6.2.10  Lagrange's four-square theorem   cgz 13297
            6.2.11  Van der Waerden's theorem   cvdwa 13333
            6.2.12  Ramsey's theorem   cram 13367
            6.2.13  Decimal arithmetic (cont.)   dec2dvds 13399
            6.2.14  Specific prime numbers   4nprm 13427
            6.2.15  Very large primes   1259lem1 13450
PART 7  BASIC STRUCTURES
      7.1  Extensible structures
            7.1.1  Basic definitions   cstr 13465
            7.1.2  Slot definitions   cplusg 13529
            7.1.3  Definition of the structure product   crest 13648
            7.1.4  Definition of the structure quotient   cordt 13721
      7.2  Moore spaces
            7.2.1  Moore closures   mrcflem 13831
            7.2.2  Independent sets in a Moore system   mrisval 13855
            7.2.3  Algebraic closure systems   isacs 13876
PART 8  BASIC CATEGORY THEORY
      8.1  Categories
            8.1.1  Categories   ccat 13889
            8.1.2  Opposite category   coppc 13937
            8.1.3  Monomorphisms and epimorphisms   cmon 13954
            8.1.4  Sections, inverses, isomorphisms   csect 13970
            8.1.5  Subcategories   cssc 14007
            8.1.6  Functors   cfunc 14051
            8.1.7  Full & faithful functors   cful 14099
            8.1.8  Natural transformations and the functor category   cnat 14138
      8.2  Arrows (disjointified hom-sets)
            8.2.1  Identity and composition for arrows   cida 14208
      8.3  Examples of categories
            8.3.1  The category of sets   csetc 14230
            8.3.2  The category of categories   ccatc 14249
      8.4  Categorical constructions
            8.4.1  Product of categories   cxpc 14265
            8.4.2  Functor evaluation   cevlf 14306
            8.4.3  Hom functor   chof 14345
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 14397
            9.2.2  Lattices   clat 14474
            9.2.3  The dual of an ordered set   codu 14555
            9.2.4  Subset order structures   cipo 14577
            9.2.5  Distributive lattices   latmass 14614
            9.2.6  Posets and lattices as relations   cps 14624
            9.2.7  Directed sets, nets   cdir 14673
PART 10  BASIC ALGEBRAIC STRUCTURES
      10.1  Monoids
            10.1.1  Definition and basic properties   cmnd 14684
            10.1.2  Monoid homomorphisms and submonoids   cmhm 14736
            10.1.3  Ordered group sum operation   gsumvallem1 14771
            10.1.4  Free monoids   cfrmd 14792
      10.2  Groups
            10.2.1  Definition and basic properties   df-grp 14812
            10.2.2  Subgroups and Quotient groups   csubg 14938
            10.2.3  Elementary theory of group homomorphisms   cghm 15003
            10.2.4  Isomorphisms of groups   cgim 15044
            10.2.5  Group actions   cga 15066
            10.2.6  Symmetry groups and Cayley's Theorem   csymg 15092
            10.2.7  Centralizers and centers   ccntz 15114
            10.2.8  The opposite group   coppg 15141
            10.2.9  p-Groups and Sylow groups; Sylow's theorems   cod 15163
            10.2.10  Direct products   clsm 15268
            10.2.11  Free groups   cefg 15338
      10.3  Abelian groups
            10.3.1  Definition and basic properties   ccmn 15412
            10.3.2  Cyclic groups   ccyg 15487
            10.3.3  Group sum operation   gsumval3a 15512
            10.3.4  Internal direct products   cdprd 15554
            10.3.5  The Fundamental Theorem of Abelian Groups   ablfacrplem 15623
      10.4  Rings
            10.4.1  Multiplicative Group   cmgp 15648
            10.4.2  Definition and basic properties   crg 15660
            10.4.3  Opposite ring   coppr 15727
            10.4.4  Divisibility   cdsr 15743
            10.4.5  Ring homomorphisms   crh 15817
      10.5  Division rings and fields
            10.5.1  Definition and basic properties   cdr 15835
            10.5.2  Subrings of a ring   csubrg 15864
            10.5.3  Absolute value (abstract algebra)   cabv 15904
            10.5.4  Star rings   cstf 15931
      10.6  Left modules
            10.6.1  Definition and basic properties   clmod 15950
            10.6.2  Subspaces and spans in a left module   clss 16008
            10.6.3  Homomorphisms and isomorphisms of left modules   clmhm 16095
            10.6.4  Subspace sum; bases for a left module   clbs 16146
      10.7  Vector spaces
            10.7.1  Definition and basic properties   clvec 16174
      10.8  Ideals
            10.8.1  The subring algebra; ideals   csra 16240
            10.8.2  Two-sided ideals and quotient rings   c2idl 16302
            10.8.3  Principal ideal rings. Divisibility in the integers   clpidl 16312
            10.8.4  Nonzero rings   cnzr 16328
            10.8.5  Left regular elements. More kinds of rings   crlreg 16339
      10.9  Associative algebras
            10.9.1  Definition and basic properties   casa 16369
      10.10  Abstract multivariate polynomials
            10.10.1  Definition and basic properties   cmps 16406
            10.10.2  Polynomial evaluation   evlslem4 16564
            10.10.3  Univariate polynomials   cps1 16569
      10.11  The complex numbers as an extensible structure
            10.11.1  Definition and basic properties   cpsmet 16685
            10.11.2  Algebraic constructions based on the complexes   czrh 16778
      10.12  Hilbert spaces
            10.12.1  Definition and basic properties   cphl 16855
            10.12.2  Orthocomplements and closed subspaces   cocv 16887
            10.12.3  Orthogonal projection and orthonormal bases   cpj 16927
PART 11  BASIC TOPOLOGY
      11.1  Topology
            11.1.1  Topological spaces   ctop 16958
            11.1.2  TopBases for topologies   isbasisg 17012
            11.1.3  Examples of topologies   distop 17060
            11.1.4  Closure and interior   ccld 17080
            11.1.5  Neighborhoods   cnei 17161
            11.1.6  Limit points and perfect sets   clp 17198
            11.1.7  Subspace topologies   restrcl 17221
            11.1.8  Order topology   ordtbaslem 17252
            11.1.9  Limits and continuity in topological spaces   ccn 17288
            11.1.10  Separated spaces: T0, T1, T2 (Hausdorff) ...   ct0 17370
            11.1.11  Compactness   ccmp 17449
            11.1.12  Bolzano-Weierstrass theorem   bwth 17473
            11.1.13  Connectedness   ccon 17474
            11.1.14  First- and second-countability   c1stc 17500
            11.1.15  Local topological properties   clly 17527
            11.1.16  Compactly generated spaces   ckgen 17565
            11.1.17  Product topologies   ctx 17592
            11.1.18  Continuous function-builders   cnmptid 17693
            11.1.19  Quotient maps and quotient topology   ckq 17725
            11.1.20  Homeomorphisms   chmeo 17785
      11.2  Filters and filter bases
            11.2.1  Filter bases   elmptrab 17859
            11.2.2  Filters   cfil 17877
            11.2.3  Ultrafilters   cufil 17931
            11.2.4  Filter limits   cfm 17965
            11.2.5  Extension by continuity   ccnext 18090
            11.2.6  Topological groups   ctmd 18100
            11.2.7  Infinite group sum on topological groups   ctsu 18155
            11.2.8  Topological rings, fields, vector spaces   ctrg 18185
      11.3  Uniform Stuctures and Spaces
            11.3.1  Uniform structures   cust 18229
            11.3.2  The topology induced by an uniform structure   cutop 18260
            11.3.3  Uniform Spaces   cuss 18283
            11.3.4  Uniform continuity   cucn 18305
            11.3.5  Cauchy filters in uniform spaces   ccfilu 18316
            11.3.6  Complete uniform spaces   ccusp 18327
      11.4  Metric spaces
            11.4.1  Pseudometric spaces   ispsmet 18335
            11.4.2  Basic metric space properties   cxme 18347
            11.4.3  Metric space balls   blfvalps 18413
            11.4.4  Open sets of a metric space   mopnval 18468
            11.4.5  Continuity in metric spaces   metcnp3 18570
            11.4.6  The uniform structure generated by a metric   metuvalOLD 18579
            11.4.7  Examples of metric spaces   dscmet 18620
            11.4.8  Normed algebraic structures   cnm 18624
            11.4.9  Normed space homomorphisms (bounded linear operators)   cnmo 18739
            11.4.10  Topology on the reals   qtopbaslem 18792
            11.4.11  Topological definitions using the reals   cii 18905
            11.4.12  Path homotopy   chtpy 18992
            11.4.13  The fundamental group   cpco 19025
      11.5  Complex metric vector spaces
            11.5.1  Complex left modules   cclm 19087
            11.5.2  Complex pre-Hilbert space   ccph 19129
            11.5.3  Convergence and completeness   ccfil 19205
            11.5.4  Baire's Category Theorem   bcthlem1 19277
            11.5.5  Banach spaces and complex Hilbert spaces   ccms 19285
            11.5.6  Minimizing Vector Theorem   minveclem1 19325
            11.5.7  Projection Theorem   pjthlem1 19338
PART 12  BASIC REAL AND COMPLEX ANALYSIS
      12.1  Continuity
            12.1.1  Intermediate value theorem   pmltpclem1 19345
      12.2  Integrals
            12.2.1  Lebesgue measure   covol 19359
            12.2.2  Lebesgue integration   cmbf 19506
      12.3  Derivatives
            12.3.1  Real and complex differentiation   climc 19749
PART 13  BASIC REAL AND COMPLEX FUNCTIONS
      13.1  Polynomials
            13.1.1  Abstract polynomials, continued   evlslem6 19934
            13.1.2  Polynomial degrees   cmdg 19976
            13.1.3  The division algorithm for univariate polynomials   cmn1 20048
            13.1.4  Elementary properties of complex polynomials   cply 20103
            13.1.5  The division algorithm for polynomials   cquot 20207
            13.1.6  Algebraic numbers   caa 20231
            13.1.7  Liouville's approximation theorem   aalioulem1 20249
      13.2  Sequences and series
            13.2.1  Taylor polynomials and Taylor's theorem   ctayl 20269
            13.2.2  Uniform convergence   culm 20292
            13.2.3  Power series   pserval 20326
      13.3  Basic trigonometry
            13.3.1  The exponential, sine, and cosine functions (cont.)   efcn 20359
            13.3.2  Properties of pi = 3.14159...   pilem1 20367
            13.3.3  Mapping of the exponential function   efgh 20443
            13.3.4  The natural logarithm on complex numbers   clog 20452
            13.3.5  Theorems of Pythagoras, isosceles triangles, and intersecting chords   angval 20643
            13.3.6  Solutions of quadratic, cubic, and quartic equations   quad2 20679
            13.3.7  Inverse trigonometric functions   casin 20702
            13.3.8  The Birthday Problem   log2ublem1 20786
            13.3.9  Areas in R^2   carea 20794
            13.3.10  More miscellaneous converging sequences   rlimcnp 20804
            13.3.11  Inequality of arithmetic and geometric means   cvxcl 20823
            13.3.12  Euler-Mascheroni constant   cem 20830
      13.4  Basic number theory
            13.4.1  Wilson's theorem   wilthlem1 20851
            13.4.2  The Fundamental Theorem of Algebra   ftalem1 20855
            13.4.3  The Basel problem (ζ(2) = π2/6)   basellem1 20863
            13.4.4  Number-theoretical functions   ccht 20873
            13.4.5  Perfect Number Theorem   mersenne 21011
            13.4.6  Characters of Z/nZ   cdchr 21016
            13.4.7  Bertrand's postulate   bcctr 21059
            13.4.8  Legendre symbol   clgs 21078
            13.4.9  Quadratic reciprocity   lgseisenlem1 21133
            13.4.10  All primes 4n+1 are the sum of two squares   2sqlem1 21147
            13.4.11  Chebyshev's Weak Prime Number Theorem, Dirichlet's Theorem   chebbnd1lem1 21163
            13.4.12  The Prime Number Theorem   mudivsum 21224
            13.4.13  Ostrowski's theorem   abvcxp 21309
PART 14  GRAPH THEORY
      14.1  Undirected graphs - basics
            14.1.1  Undirected hypergraphs   cuhg 21334
            14.1.2  Undirected multigraphs   cumg 21347
            14.1.3  Undirected simple graphs   cuslg 21364
                  14.1.3.1  Undirected simple graphs - basics   cuslg 21364
                  14.1.3.2  Undirected simple graphs - examples   usgraexvlem 21414
                  14.1.3.3  Finite undirected simple graphs   fiusgraedgfi 21421
            14.1.4  Neighbors, complete graphs and universal vertices   cnbgra 21430
                  14.1.4.1  Neighbors   nbgraop 21436
                  14.1.4.2  Complete graphs   iscusgra 21465
                  14.1.4.3  Universal vertices   isuvtx 21497
            14.1.5  Walks, paths and cycles   cwalk 21506
                  14.1.5.1  Walks and trails   wlks 21526
                  14.1.5.2  Paths and simple paths   pths 21566
                  14.1.5.3  Circuits and cycles   crcts 21609
                  14.1.5.4  Connected graphs   cconngra 21656
            14.1.6  Vertex Degree   cvdg 21664
      14.2  Eulerian paths and the Konigsberg Bridge problem
            14.2.1  Eulerian paths   ceup 21684
            14.2.2  The Konigsberg Bridge problem   vdeg0i 21704
PART 15  GUIDES AND MISCELLANEA
      15.1  Guides (conventions, explanations, and examples)
            15.1.1  Conventions   conventions 21710
            15.1.2  Natural deduction   natded 21711
            15.1.3  Natural deduction examples   ex-natded5.2 21712
            15.1.4  Definitional examples   ex-or 21729
      15.2  Humor
            15.2.1  April Fool's theorem   avril1 21757
      15.3  (Future - to be reviewed and classified)
            15.3.1  Planar incidence geometry   cplig 21763
            15.3.2  Algebra preliminaries   crpm 21768
            15.3.3  Transitive closure   ctcl 21770
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 21774
            16.1.2  Definition and basic properties of Abelian groups   cablo 21869
            16.1.3  Subgroups   csubgo 21889
            16.1.4  Operation properties   cass 21900
            16.1.5  Group-like structures   cmagm 21906
            16.1.6  Examples of Abelian groups   ablosn 21935
            16.1.7  Group homomorphism and isomorphism   cghom 21945
      16.2  Additional material on rings and fields
            16.2.1  Definition and basic properties   crngo 21963
            16.2.2  Examples of rings   cnrngo 21991
            16.2.3  Division Rings   cdrng 21993
            16.2.4  Star Fields   csfld 21996
            16.2.5  Fields and Rings   ccm2 21998
PART 17  COMPLEX TOPOLOGICAL VECTOR SPACES (DEPRECATED)
      17.1  Complex vector spaces
            17.1.1  Definition and basic properties   cvc 22024
            17.1.2  Examples of complex vector spaces   cncvc 22062
      17.2  Normed complex vector spaces
            17.2.1  Definition and basic properties   cnv 22063
            17.2.2  Examples of normed complex vector spaces   cnnv 22168
            17.2.3  Induced metric of a normed complex vector space   imsval 22177
            17.2.4  Inner product   cdip 22196
            17.2.5  Subspaces   css 22220
      17.3  Operators on complex vector spaces
            17.3.1  Definitions and basic properties   clno 22241
      17.4  Inner product (pre-Hilbert) spaces
            17.4.1  Definition and basic properties   ccphlo 22313
            17.4.2  Examples of pre-Hilbert spaces   cncph 22320
            17.4.3  Properties of pre-Hilbert spaces   isph 22323
      17.5  Complex Banach spaces
            17.5.1  Definition and basic properties   ccbn 22364
            17.5.2  Examples of complex Banach spaces   cnbn 22371
            17.5.3  Uniform Boundedness Theorem   ubthlem1 22372
            17.5.4  Minimizing Vector Theorem   minvecolem1 22376
      17.6  Complex Hilbert spaces
            17.6.1  Definition and basic properties   chlo 22387
            17.6.2  Standard axioms for a complex Hilbert space   hlex 22400
            17.6.3  Examples of complex Hilbert spaces   cnchl 22418
            17.6.4  Subspaces   ssphl 22419
            17.6.5  Hellinger-Toeplitz Theorem   htthlem 22420
PART 18  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      18.1  Axiomatization of complex pre-Hilbert spaces
            18.1.1  Basic Hilbert space definitions   chil 22422
            18.1.2  Preliminary ZFC lemmas   df-hnorm 22471
            18.1.3  Derive the Hilbert space axioms from ZFC set theory   axhilex-zf 22484
            18.1.4  Introduce the vector space axioms for a Hilbert space   ax-hilex 22502
            18.1.5  Vector operations   hvmulex 22514
            18.1.6  Inner product postulates for a Hilbert space   ax-hfi 22581
      18.2  Inner product and norms
            18.2.1  Inner product   his5 22588
            18.2.2  Norms   dfhnorm2 22624
            18.2.3  Relate Hilbert space to normed complex vector spaces   hilablo 22662
            18.2.4  Bunjakovaskij-Cauchy-Schwarz inequality   bcsiALT 22681
      18.3  Cauchy sequences and completeness axiom
            18.3.1  Cauchy sequences and limits   hcau 22686
            18.3.2  Derivation of the completeness axiom from ZF set theory   hilmet 22696
            18.3.3  Completeness postulate for a Hilbert space   ax-hcompl 22704
            18.3.4  Relate Hilbert space to ZFC pre-Hilbert and Hilbert spaces   hhcms 22705
      18.4  Subspaces and projections
            18.4.1  Subspaces   df-sh 22709
            18.4.2  Closed subspaces   df-ch 22724
            18.4.3  Orthocomplements   df-oc 22754
            18.4.4  Subspace sum, span, lattice join, lattice supremum   df-shs 22810
            18.4.5  Projection theorem   pjhthlem1 22893
            18.4.6  Projectors   df-pjh 22897
      18.5  Properties of Hilbert subspaces
            18.5.1  Orthomodular law   omlsilem 22904
            18.5.2  Projectors (cont.)   pjhtheu2 22918
            18.5.3  Hilbert lattice operations   sh0le 22942
            18.5.4  Span (cont.) and one-dimensional subspaces   spansn0 23043
            18.5.5  Commutes relation for Hilbert lattice elements   df-cm 23085
            18.5.6  Foulis-Holland theorem   fh1 23120
            18.5.7  Quantum Logic Explorer axioms   qlax1i 23129
            18.5.8  Orthogonal subspaces   chscllem1 23139
            18.5.9  Orthoarguesian laws 5OA and 3OA   5oalem1 23156
            18.5.10  Projectors (cont.)   pjorthi 23171
            18.5.11  Mayet's equation E_3   mayete3i 23230
      18.6  Operators on Hilbert spaces
            18.6.1  Operator sum, difference, and scalar multiplication   df-hosum 23233
            18.6.2  Zero and identity operators   df-h0op 23251
            18.6.3  Operations on Hilbert space operators   hoaddcl 23261
            18.6.4  Linear, continuous, bounded, Hermitian, unitary operators and norms   df-nmop 23342
            18.6.5  Linear and continuous functionals and norms   df-nmfn 23348
            18.6.6  Adjoint   df-adjh 23352
            18.6.7  Dirac bra-ket notation   df-bra 23353
            18.6.8  Positive operators   df-leop 23355
            18.6.9  Eigenvectors, eigenvalues, spectrum   df-eigvec 23356
            18.6.10  Theorems about operators and functionals   nmopval 23359
            18.6.11  Riesz lemma   riesz3i 23565
            18.6.12  Adjoints (cont.)   cnlnadjlem1 23570
            18.6.13  Quantum computation error bound theorem   unierri 23607
            18.6.14  Dirac bra-ket notation (cont.)   branmfn 23608
            18.6.15  Positive operators (cont.)   leopg 23625
            18.6.16  Projectors as operators   pjhmopi 23649
      18.7  States on a Hilbert lattice and Godowski's equation
            18.7.1  States on a Hilbert lattice   df-st 23714
            18.7.2  Godowski's equation   golem1 23774
      18.8  Cover relation, atoms, exchange axiom, and modular symmetry
            18.8.1  Covers relation; modular pairs   df-cv 23782
            18.8.2  Atoms   df-at 23841
            18.8.3  Superposition principle   superpos 23857
            18.8.4  Atoms, exchange and covering properties, atomicity   chcv1 23858
            18.8.5  Irreducibility   chirredlem1 23893
            18.8.6  Atoms (cont.)   atcvat3i 23899
            18.8.7  Modular symmetry   mdsymlem1 23906
PART 19  SUPPLEMENTARY MATERIAL (USER'S MATHBOXES)
      19.1  Mathboxes for user contributions
            19.1.1  Mathbox guidelines   mathbox 23945
      19.2  Mathbox for Stefan Allan
      19.3  Mathbox for Thierry Arnoux
            19.3.1  Propositional Calculus - misc additions   bian1d 23950
            19.3.2  Predicate Calculus   abeq2f 23960
                  19.3.2.1  Predicate Calculus - misc additions   abeq2f 23960
                  19.3.2.2  Restricted quantification - misc additions   reximddv 23962
                  19.3.2.3  Substitution (without distinct variables) - misc additions   clelsb3f 23971
                  19.3.2.4  Existential "at most one" - misc additions   mo5f 23972
                  19.3.2.5  Existential uniqueness - misc additions   2reuswap2 23975
                  19.3.2.6  Restricted "at most one" - misc additions   rmoxfrdOLD 23979
            19.3.3  General Set Theory   ceqsexv2d 23985
                  19.3.3.1  Class abstractions (a.k.a. class builders)   ceqsexv2d 23985
                  19.3.3.2  Image Sets   abrexdomjm 23988
                  19.3.3.3  Set relations and operations - misc additions   eqri 23994
                  19.3.3.4  Unordered pairs   elpreq 23999
                  19.3.3.5  Conditional operator - misc additions   ifeqeqx 24001
                  19.3.3.6  Indexed union - misc additions   iuneq12daf 24007
                  19.3.3.7  Disjointness - misc additions   cbvdisjf 24015
            19.3.4  Relations and Functions   dfrel4 24034
                  19.3.4.1  Relations - misc additions   dfrel4 24034
                  19.3.4.2  Functions - misc additions   fdmrn 24039
                  19.3.4.3  Isomorphisms - misc. add.   gtiso 24088
                  19.3.4.4  Disjointness (additional proof requiring functions)   disjdsct 24090
                  19.3.4.5  First and second members of an ordered pair - misc additions   df1stres 24091
                  19.3.4.6  Supremum - misc additions   supssd 24098
                  19.3.4.7  Countable Sets   nnct 24099
            19.3.5  Real and Complex Numbers   addeq0 24114
                  19.3.5.1  Complex addition - misc. additions   addeq0 24114
                  19.3.5.2  Ordering on reals - misc additions   lt2addrd 24115
                  19.3.5.3  Extended reals - misc additions   xgepnf 24116
                  19.3.5.4  Real number intervals - misc additions   icossicc 24129
                  19.3.5.5  Finite intervals of integers - misc additions   fzssnn 24147
                  19.3.5.6  Half-open integer ranges - misc additions   iundisjfi 24152
                  19.3.5.7  The ` # ` (finite set size) function - misc additions   hashresfn 24156
                  19.3.5.8  The greatest common divisor operator - misc. add   numdenneg 24160
                  19.3.5.9  Integers   ltesubnnd 24162
                  19.3.5.10  Division in the extended real number system   cxdiv 24163
            19.3.6  Structure builders   ress0g 24182
                  19.3.6.1  Structure builder restriction operator   ress0g 24182
                  19.3.6.2  Posets   tospos 24186
                  19.3.6.3  Complete lattices   clatp0ex 24193
                  19.3.6.4  Extended reals Structure - misc additions   ax-xrssca 24195
                  19.3.6.5  The extended non-negative real numbers monoid   xrge0base 24207
            19.3.7  Algebra   sumpr 24218
                  19.3.7.1  Finitely supported group sums - misc additions   sumpr 24218
                  19.3.7.2  Rings - misc additions   dvrdir 24226
                  19.3.7.3  Ordered groups   cogrp 24231
                  19.3.7.4  Ordered fields   cofld 24233
                  19.3.7.5  The Archimedean property for generic algebraic structures   cinftm 24246
                  19.3.7.6  Ring homomorphisms - misc additions   rhmdvdsr 24256
                  19.3.7.7  The ring of integers   zzsbase 24263
                  19.3.7.8  The ordered field of reals   rebase 24269
            19.3.8  Topology   cmetid 24281
                  19.3.8.1  Pseudometrics   cmetid 24281
                  19.3.8.2  Continuity - misc additions   hauseqcn 24293
                  19.3.8.3  Topology of the closed unit   unitsscn 24294
                  19.3.8.4  Topology of ` ( RR X. RR ) `   unicls 24301
                  19.3.8.5  Order topology - misc. additions   cnvordtrestixx 24311
                  19.3.8.6  Continuity in topological spaces - misc. additions   mndpluscn 24312
                  19.3.8.7  Topology of the extended non-negative real numbers monoid   xrge0hmph 24318
                  19.3.8.8  Limits - misc additions   lmlim 24333
            19.3.9  Uniform Stuctures and Spaces   chcmp 24340
                  19.3.9.1  Hausdorff Completion   chcmp 24340
            19.3.10  Topology and algebraic structures   zzsnm 24342
                  19.3.10.1  The norm on the ring of the integer numbers   zzsnm 24342
                  19.3.10.2  The complete ordered field of the real numbers   recms 24343
                  19.3.10.3  Topological ` ZZ ` -modules   zlm0 24346
                  19.3.10.4  The canonical embedding of the rational numbers into a division ring   cqqh 24356
                  19.3.10.5  The canonical embedding of ` RR ` into a complete ordered field   crrh 24377
                  19.3.10.6  Embedding into ` RR* `   cxrh 24382
                  19.3.10.7  Canonical embeddings into ` RR `   zrhre 24385
            19.3.11  Real and complex functions   clogb 24388
                  19.3.11.1  Logarithm laws generalized to an arbitrary base - logb   clogb 24388
                  19.3.11.2  Indicator Functions   cind 24408
                  19.3.11.3  Extended sum   cesum 24424
            19.3.12  Mixed Function/Constant operation   cofc 24478
            19.3.13  Abstract measure   csiga 24490
                  19.3.13.1  Sigma-Algebra   csiga 24490
                  19.3.13.2  Generated Sigma-Algebra   csigagen 24521
                  19.3.13.3  The Borel algebra on the real numbers   cbrsiga 24535
                  19.3.13.4  Product Sigma-Algebra   csx 24542
                  19.3.13.5  Measures   cmeas 24549
                  19.3.13.6  The counting measure   cntmeas 24580
                  19.3.13.7  The Lebesgue measure - misc additions   volss 24583
                  19.3.13.8  The 'almost everywhere' relation   cae 24588
                  19.3.13.9  Measurable functions   cmbfm 24600
                  19.3.13.10  Borel Algebra on ` ( RR X. RR ) `   br2base 24619
            19.3.14  Integration   itgeq12dv 24641
                  19.3.14.1  Lebesgue integral - misc additions   itgeq12dv 24641
                  19.3.14.2  Bochner integral   citgm 24642
            19.3.15  Probability   cprb 24665
                  19.3.15.1  Probability Theory   cprb 24665
                  19.3.15.2  Conditional Probabilities   ccprob 24689
                  19.3.15.3  Real Valued Random Variables   crrv 24698
                  19.3.15.4  Preimage set mapping operator   corvc 24713
                  19.3.15.5  Distribution Functions   orvcelval 24726
                  19.3.15.6  Cumulative Distribution Functions   orvclteel 24730
                  19.3.15.7  Probabilities - example   coinfliplem 24736
                  19.3.15.8  Bertrand's Ballot Problem   ballotlemoex 24743
      19.4  Mathbox for Mario Carneiro
            19.4.1  Miscellaneous stuff   quartfull 24796
            19.4.2  Zeta function   czeta 24797
            19.4.3  Gamma function   clgam 24800
            19.4.4  Derangements and the Subfactorial   deranglem 24852
            19.4.5  The Erdős-Szekeres theorem   erdszelem1 24877
            19.4.6  The Kuratowski closure-complement theorem   kur14lem1 24892
            19.4.7  Retracts and sections   cretr 24903
            19.4.8  Path-connected and simply connected spaces   cpcon 24906
            19.4.9  Covering maps   ccvm 24942
            19.4.10  Normal numbers   snmlff 25016
            19.4.11  Godel-sets of formulas   cgoe 25020
            19.4.12  Models of ZF   cgze 25048
            19.4.13  Splitting fields   citr 25062
            19.4.14  p-adic number fields   czr 25078
      19.5  Mathbox for Paul Chapman
            19.5.1  Group homomorphism and isomorphism   ghomgrpilem1 25096
            19.5.2  Real and complex numbers (cont.)   climuzcnv 25108
            19.5.3  Miscellaneous theorems   elfzm12 25112
      19.6  Mathbox for Drahflow
      19.7  Mathbox for Scott Fenton
            19.7.1  ZFC Axioms in primitive form   axextprim 25150
            19.7.2  Untangled classes   untelirr 25157
            19.7.3  Extra propositional calculus theorems   3orel1 25164
            19.7.4  Misc. Useful Theorems   nepss 25175
            19.7.5  Properties of reals and complexes   sqdivzi 25184
            19.7.6  Product sequences   prodf 25215
            19.7.7  Non-trivial convergence   ntrivcvg 25225
            19.7.8  Complex products   cprod 25231
            19.7.9  Finite products   fprod 25267
            19.7.10  Infinite products   iprodclim 25311
            19.7.11  Falling and Rising Factorial   cfallfac 25320
            19.7.12  Factorial limits   faclimlem1 25362
            19.7.13  Greatest common divisor and divisibility   pdivsq 25368
            19.7.14  Properties of relationships   brtp 25372
            19.7.15  Properties of functions and mappings   funpsstri 25389
            19.7.16  Epsilon induction   setinds 25405
            19.7.17  Ordinal numbers   elpotr 25408
            19.7.18  Defined equality axioms   axextdfeq 25425
            19.7.19  Hypothesis builders   hbntg 25433
            19.7.20  The Predecessor Class   cpred 25438
            19.7.21  (Trans)finite Recursion Theorems   tfisg 25479
            19.7.22  Well-founded induction   tz6.26 25480
            19.7.23  Transitive closure under a relationship   ctrpred 25495
            19.7.24  Founded Induction   frmin 25517
            19.7.25  Ordering Ordinal Sequences   orderseqlem 25527
            19.7.26  Well-founded recursion   cwrecs 25530
            19.7.27  Transfinite recursion via Well-founded recursion   tfrALTlem 25557
            19.7.28  Well-founded zero, successor, and limits   cwsuc 25561
            19.7.29  Founded Recursion   frr3g 25581
            19.7.30  Surreal Numbers   csur 25595
            19.7.31  Surreal Numbers: Ordering   sltsolem1 25623
            19.7.32  Surreal Numbers: Birthday Function   bdayfo 25630
            19.7.33  Surreal Numbers: Density   fvnobday 25637
            19.7.34  Surreal Numbers: Density   nodenselem3 25638
            19.7.35  Surreal Numbers: Upper and Lower Bounds   nobndlem1 25647
            19.7.36  Surreal Numbers: Full-Eta Property   nofulllem1 25657
            19.7.37  Symmetric difference   csymdif 25662
            19.7.38  Quantifier-free definitions   ctxp 25674
            19.7.39  Alternate ordered pairs   caltop 25801
            19.7.40  Tarskian geometry   cee 25827
            19.7.41  Tarski's axioms for geometry   axdimuniq 25852
            19.7.42  Congruence properties   cofs 25916
            19.7.43  Betweenness properties   btwntriv2 25946
            19.7.44  Segment Transportation   ctransport 25963
            19.7.45  Properties relating betweenness and congruence   cifs 25969
            19.7.46  Connectivity of betweenness   btwnconn1lem1 26021
            19.7.47  Segment less than or equal to   csegle 26040
            19.7.48  Outside of relationship   coutsideof 26053
            19.7.49  Lines and Rays   cline2 26068
            19.7.50  Bernoulli polynomials and sums of k-th powers   cbp 26092
            19.7.51  Rank theorems   rankung 26107
            19.7.52  Hereditarily Finite Sets   chf 26113
      19.8  Mathbox for Anthony Hart
            19.8.1  Propositional Calculus   tb-ax1 26128
            19.8.2  Predicate Calculus   quantriv 26150
            19.8.3  Misc. Single Axiom Systems   meran1 26161
            19.8.4  Connective Symmetry   negsym1 26167
      19.9  Mathbox for Chen-Pang He
            19.9.1  Ordinal topology   ontopbas 26178
      19.10  Mathbox for Jeff Hoffman
            19.10.1  Inferences for finite induction on generic function values   fveleq 26201
            19.10.2  gdc.mm   nnssi2 26205
      19.11  Mathbox for Wolf Lammen
      19.12  Mathbox for Brendan Leahy
      19.13  Mathbox for Jeff Hankins
            19.13.1  Miscellany   a1i13 26298
            19.13.2  Basic topological facts   topbnd 26327
            19.13.3  Topology of the real numbers   ivthALT 26338
            19.13.4  Refinements   cfne 26339
            19.13.5  Neighborhood bases determine topologies   neibastop1 26388
            19.13.6  Lattice structure of topologies   topmtcl 26392
            19.13.7  Filter bases   fgmin 26399
            19.13.8  Directed sets, nets   tailfval 26401
      19.14  Mathbox for Jeff Madsen
            19.14.1  Logic and set theory   anim12da 26412
            19.14.2  Real and complex numbers; integers   filbcmb 26442
            19.14.3  Sequences and sums   sdclem2 26446
            19.14.4  Topology   subspopn 26458
            19.14.5  Metric spaces   metf1o 26461
            19.14.6  Continuous maps and homeomorphisms   constcncf 26468
            19.14.7  Boundedness   ctotbnd 26475
            19.14.8  Isometries   cismty 26507
            19.14.9  Heine-Borel Theorem   heibor1lem 26518
            19.14.10  Banach Fixed Point Theorem   bfplem1 26531
            19.14.11  Euclidean space   crrn 26534
            19.14.12  Intervals (continued)   ismrer1 26547
            19.14.13  Groups and related structures   exidcl 26551
            19.14.14  Rings   rngonegcl 26561
            19.14.15  Ring homomorphisms   crnghom 26576
            19.14.16  Commutative rings   ccring 26605
            19.14.17  Ideals   cidl 26617
            19.14.18  Prime rings and integral domains   cprrng 26656
            19.14.19  Ideal generators   cigen 26669
      19.15  Mathbox for Rodolfo Medina
            19.15.1  Partitions   prtlem60 26688
      19.16  Mathbox for Stefan O'Rear
            19.16.1  Additional elementary logic and set theory   nelss 26732
            19.16.2  Additional theory of functions   fninfp 26735
            19.16.3  Extensions beyond function theory   gsumvsmul 26745
            19.16.4  Additional topology   elrfi 26748
            19.16.5  Characterization of closure operators. Kuratowski closure axioms   ismrcd1 26752
            19.16.6  Algebraic closure systems   cnacs 26756
            19.16.7  Miscellanea 1. Map utilities   constmap 26767
            19.16.8  Miscellanea for polynomials   ofmpteq 26776
            19.16.9  Multivariate polynomials over the integers   cmzpcl 26778
            19.16.10  Miscellanea for Diophantine sets 1   coeq0 26810
            19.16.11  Diophantine sets 1: definitions   cdioph 26813
            19.16.12  Diophantine sets 2 miscellanea   ellz1 26825
            19.16.13  Diophantine sets 2: union and intersection. Monotone Boolean algebra   diophin 26831
            19.16.14  Diophantine sets 3: construction   diophrex 26834
            19.16.15  Diophantine sets 4 miscellanea   2sbcrex 26843
            19.16.16  Diophantine sets 4: Quantification   rexrabdioph 26854
            19.16.17  Diophantine sets 5: Arithmetic sets   rabdiophlem1 26861
            19.16.18  Diophantine sets 6 miscellanea   fz1ssnn 26871
            19.16.19  Diophantine sets 6: reusability. renumbering of variables   eldioph4b 26872
            19.16.20  Pigeonhole Principle and cardinality helpers   fphpd 26877
            19.16.21  A non-closed set of reals is infinite   rencldnfilem 26881
            19.16.22  Miscellanea for Lagrange's theorem   icodiamlt 26883
            19.16.23  Lagrange's rational approximation theorem   irrapxlem1 26885
            19.16.24  Pell equations 1: A nontrivial solution always exists   pellexlem1 26892
            19.16.25  Pell equations 2: Algebraic number theory of the solution set   csquarenn 26899
            19.16.26  Pell equations 3: characterizing fundamental solution   infmrgelbi 26941
            19.16.27  Logarithm laws generalized to an arbitrary base   reglogcl 26953
            19.16.28  Pell equations 4: the positive solution group is infinite cyclic   pellfund14 26961
            19.16.29  X and Y sequences 1: Definition and recurrence laws   crmx 26963
            19.16.30  Ordering and induction lemmas for the integers   monotuz 27004
            19.16.31  X and Y sequences 2: Order properties   rmxypos 27012
            19.16.32  Congruential equations   congtr 27030
            19.16.33  Alternating congruential equations   acongid 27040
            19.16.34  Additional theorems on integer divisibility   bezoutr 27050
            19.16.35  X and Y sequences 3: Divisibility properties   jm2.18 27059
            19.16.36  X and Y sequences 4: Diophantine representability of Y   jm2.27a 27076
            19.16.37  X and Y sequences 5: Diophantine representability of X, ^, _C   rmxdiophlem 27086
            19.16.38  Uncategorized stuff not associated with a major project   setindtr 27095
            19.16.39  More equivalents of the Axiom of Choice   axac10 27104
            19.16.40  Finitely generated left modules   clfig 27142
            19.16.41  Noetherian left modules I   clnm 27150
            19.16.42  Addenda for structure powers   pwssplit0 27164
            19.16.43  Direct sum of left modules   cdsmm 27174
            19.16.44  Free modules   cfrlm 27189
            19.16.45  Every set admits a group structure iff choice   unxpwdom3 27233
            19.16.46  Independent sets and families   clindf 27251
            19.16.47  Characterization of free modules   lmimlbs 27283
            19.16.48  Noetherian rings and left modules II   clnr 27290
            19.16.49  Hilbert's Basis Theorem   cldgis 27302
            19.16.50  Additional material on polynomials [DEPRECATED]   cmnc 27312
            19.16.51  Degree and minimal polynomial of algebraic numbers   cdgraa 27322
            19.16.52  Algebraic integers I   citgo 27339
            19.16.53  Finite cardinality [SO]   en1uniel 27357
            19.16.54  Words in monoids and ordered group sum   issubmd 27360
            19.16.55  Transpositions in the symmetric group   cpmtr 27361
            19.16.56  The sign of a permutation   cpsgn 27391
            19.16.57  The matrix algebra   cmmul 27416
            19.16.58  The determinant   cmdat 27460
            19.16.59  Endomorphism algebra   cmend 27466
            19.16.60  Subfields   csdrg 27480
            19.16.61  Cyclic groups and order   idomrootle 27488
            19.16.62  Cyclotomic polynomials   ccytp 27498
            19.16.63  Miscellaneous topology   fgraphopab 27506
      19.17  Mathbox for Steve Rodriguez
            19.17.1  Miscellanea   iso0 27513
            19.17.2  Function operations   caofcan 27517
            19.17.3  Calculus   lhe4.4ex1a 27523
      19.18  Mathbox for Andrew Salmon
            19.18.1  Principia Mathematica * 10   pm10.12 27530
            19.18.2  Principia Mathematica * 11   2alanimi 27544
            19.18.3  Predicate Calculus   sbeqal1 27574
            19.18.4  Principia Mathematica * 13 and * 14   pm13.13a 27584
            19.18.5  Set Theory   elnev 27615
            19.18.6  Arithmetic   addcomgi 27637
            19.18.7  Geometry   cplusr 27638
      19.19  Mathbox for Glauco Siliprandi
            19.19.1  Miscellanea   ssrexf 27660
            19.19.2  Finite multiplication of numbers and finite multiplication of functions   fmul01 27686
            19.19.3  Limits   clim1fr1 27703
            19.19.4  Derivatives   dvsinexp 27716
            19.19.5  Integrals   ioovolcl 27718
            19.19.6  Stone Weierstrass theorem - real version   stoweidlem1 27726
            19.19.7  Wallis' product for π   wallispilem1 27790
            19.19.8  Stirling's approximation formula for ` n ` factorial   stirlinglem1 27799
      19.20  Mathbox for Saveliy Skresanov
            19.20.1  Ceva's theorem   sigarval 27816
      19.21  Mathbox for Jarvin Udandy
      19.22  Mathbox for Alexander van der Vekens
            19.22.1  Double restricted existential uniqueness   r19.32 27921
                  19.22.1.1  Restricted quantification (extension)   r19.32 27921
                  19.22.1.2  The empty set (extension)   raaan2 27929
                  19.22.1.3  Restricted uniqueness and "at most one" quantification   rmoimi 27930
                  19.22.1.4  Analogs to Existential uniqueness (double quantification)   2reurex 27935
            19.22.2  Alternative definitions of function's and operation's values   wdfat 27947
                  19.22.2.1  Restricted quantification (extension)   ralbinrald 27953
                  19.22.2.2  The universal class (extension)   nvelim 27954
                  19.22.2.3  Introduce the Axiom of Power Sets (extension)   alneu 27955
                  19.22.2.4  Relations (extension)   sbcrel 27957
                  19.22.2.5  Functions (extension)   sbcfun 27963
                  19.22.2.6  Predicate "defined at"   dfateq12d 27969
                  19.22.2.7  Alternative definition of the value of a function   dfafv2 27972
                  19.22.2.8  Alternative definition of the value of an operation   aoveq123d 28018
            19.22.3  Auxiliary theorems for graph theory   jaoi3 28048
                  19.22.3.1  Logical disjunction and conjunction   jaoi3 28048
                  19.22.3.2  Abbreviated conjunction and disjunction of three wff's   3an4anass 28049
                  19.22.3.3  Negated equality and membership - extension   eqneqall 28050
                  19.22.3.4  "Weak deduction theorem" for set theory - extension   ifeqda 28052
                  19.22.3.5  Power classes - extension   3xpexg 28054
                  19.22.3.6  Unordered and ordered pairs - extension   nelprd 28055
                  19.22.3.7  Indexed union and intersection - extension   iunxprg 28068
                  19.22.3.8  Binary relations - extension   breqn0 28069
                  19.22.3.9  Ordered-pair class abstractions - extension   elopaelxp 28070
                  19.22.3.10  Introduce the Axiom of Union - extension   ralxfrd2 28072
                  19.22.3.11  Relations - extension   resisresindm 28074
                  19.22.3.12  Functions - extension   f0bi 28075
                  19.22.3.13  Operations - extension   oprabv 28085
                  19.22.3.14  Equinumerosity - extension   resfnfinfin 28086
                  19.22.3.15  Subtraction - extension   cnm1cn 28088
                  19.22.3.16  Multiplication - extension   kcnktkm1cn 28089
                  19.22.3.17  Ordering on reals (cont.) - extension   leaddsuble 28091
                  19.22.3.18  Nonnegative integers (as a subset of complex numbers) - extension   0mnnnnn0 28095
                  19.22.3.19  Upper partititions of integers - extension   1eluzge0 28100
                  19.22.3.20  Finite intervals of integers - extension   ssfz12 28104
                  19.22.3.21  Half-open integer ranges - extension   elfzonn0 28122
                  19.22.3.22  The floor (greatest integer) function - extension   nn0nndivcl 28141
                  19.22.3.23  The modulo (remainder) operation - extension   modvalr 28149
                  19.22.3.24  The ` # ` (finite set size) function - extension   hashimarn 28163
                  19.22.3.25  Words over a set - extension   iswrd0i 28169
                  19.22.3.26  Words over a set - extension (concatenations)   elfzelfzccat 28177
                  19.22.3.27  Words over a set - extension (subwords)   swrdltnd 28181
                  19.22.3.28  Words over a set - extension (subwords of subwords)   swrd0swrd 28197
                  19.22.3.29  Words over a set - extension (subwords of concatenations)   swrdccat3a0 28203
                  19.22.3.30  Prime numbers: elementary properties - extension   prmgt1 28223
                  19.22.3.31  Words over a set - extension (cyclic shift)   ccsh 28230
            19.22.4  Graph theory   uhgraedgrnv 28292
                  19.22.4.1  Undirected hypergraphs   uhgraedgrnv 28292
                  19.22.4.2  Undirected simple graphs   usisuhgra 28293
                  19.22.4.3  Neighbors, complete graphs and universal vertices   nbfiusgrafi 28294
                  19.22.4.4  Walks, Paths and Cycles   wlkelwrd 28295
                  19.22.4.5  Walks/paths of length 2 as ordered triples   c2wlkot 28321
                  19.22.4.6  Vertex Degree   usgfidegfi 28360
                  19.22.4.7  Regular graphs   crgra 28365
                  19.22.4.8  Friendship graphs   cfrgra 28378
      19.23  Mathbox for David A. Wheeler
            19.23.1  Natural deduction   19.8ad 28460
            19.23.2  Greater than, greater than or equal to.   cge-real 28463
            19.23.3  Hyperbolic trig functions   csinh 28473
            19.23.4  Reciprocal trig functions (sec, csc, cot)   csec 28484
            19.23.5  Identities for "if"   ifnmfalse 28506
            19.23.6  Not-member-of   AnelBC 28507
            19.23.7  Decimal point   cdp2 28508
            19.23.8  Signum (sgn or sign) function   csgn 28516
            19.23.9  Ceiling function   ccei 28526
            19.23.10  Logarithms generalized to arbitrary base using ` logb `   ene0 28530
            19.23.11  Logarithm laws generalized to an arbitrary base - log_   clog_ 28533
            19.23.12  Miscellaneous   5m4e1 28535
      19.24  Mathbox for Alan Sare
            19.24.1  Supplementary "adant" deductions   ad4ant13 28538
            19.24.2  Supplementary unification deductions   biimp 28564
            19.24.3  Conventional Metamath proofs, some derived from VD proofs   iidn3 28583
            19.24.4  What is Virtual Deduction?   wvd1 28660
            19.24.5  Virtual Deduction Theorems   df-vd1 28661
            19.24.6  Theorems proved using virtual deduction   trsspwALT 28931
            19.24.7  Theorems proved using virtual deduction with mmj2 assistance   simplbi2VD 28958
            19.24.8  Virtual Deduction transcriptions of textbook proofs   sb5ALTVD 29025
            19.24.9  Theorems proved using conjunction-form virtual deduction   elpwgdedVD 29029
            19.24.10  Theorems with VD proofs in conventional notation derived from VD proofs   suctrALT3 29036
            19.24.11  Theorems with a proof in conventional notation automatically derived   notnot2ALT2 29039
      19.25  Mathbox for Jonathan Ben-Naim
            19.25.1  First order logic and set theory   bnj170 29062
            19.25.2  Well founded induction and recursion   bnj110 29229
            19.25.3  The existence of a minimal element in certain classes   bnj69 29379
            19.25.4  Well-founded induction   bnj1204 29381
            19.25.5  Well-founded recursion, part 1 of 3   bnj60 29431
            19.25.6  Well-founded recursion, part 2 of 3   bnj1500 29437
            19.25.7  Well-founded recursion, part 3 of 3   bnj1522 29441
      19.26  Mathbox for Norm Megill
            19.26.1  Experiments to study ax-7 unbundling   ax-7v 29442
                  19.26.1.1  Theorems derived from ax-7v (suffixes NEW7 and AUX7)   ax-7v 29442
                  19.26.1.2  Theorems derived from ax-7 (suffix OLD7)   ax-7OLD7 29678
            19.26.2  Miscellanea   cnaddcom 29769
            19.26.3  Atoms, hyperplanes, and covering in a left vector space (or module)   clsa 29772
            19.26.4  Functionals and kernels of a left vector space (or module)   clfn 29855
            19.26.5  Opposite rings and dual vector spaces   cld 29921
            19.26.6  Ortholattices and orthomodular lattices   cops 29970
            19.26.7  Atomic lattices with covering property   ccvr 30060
            19.26.8  Hilbert lattices   chlt 30148
            19.26.9  Projective geometries based on Hilbert lattices   clln 30288
            19.26.10  Construction of a vector space from a Hilbert lattice   cdlema1N 30588
            19.26.11  Construction of involution and inner product from a Hilbert lattice   clpoN 32278

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