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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 2171
            1.6.2  Rederive new axioms from old: ax5 , ax6 , ax9from9o , ax11 , ax12from12o   ax4 2181
            1.6.3  Legacy theorems using obsolete axioms   ax17o 2193
      1.7  Existential uniqueness
      1.8  Other axiomatizations related to classical predicate calculus
            1.8.1  Predicate calculus with all distinct variables   ax-7d 2331
            1.8.2  Aristotelian logic: Assertic syllogisms   barbara 2337
            1.8.3  Intuitionistic logic   axi4 2361
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 2370
            2.1.2  Class abstractions (a.k.a. class builders)   cab 2375
            2.1.3  Class form not-free predicate   wnfc 2512
            2.1.4  Negated equality and membership   wne 2552
                  2.1.4.1  Negated equality   nne 2556
                  2.1.4.2  Negated membership   neleq1 2645
            2.1.5  Restricted quantification   wral 2651
            2.1.6  The universal class   cvv 2901
            2.1.7  Conditional equality (experimental)   wcdeq 3089
            2.1.8  Russell's Paradox   ru 3105
            2.1.9  Proper substitution of classes for sets   wsbc 3106
            2.1.10  Proper substitution of classes for sets into classes   csb 3196
            2.1.11  Define basic set operations and relations   cdif 3262
            2.1.12  Subclasses and subsets   df-ss 3279
            2.1.13  The difference, union, and intersection of two classes   difeq1 3403
                  2.1.13.1  The difference of two classes   difeq1 3403
                  2.1.13.2  The union of two classes   elun 3433
                  2.1.13.3  The intersection of two classes   elin 3475
                  2.1.13.4  Combinations of difference, union, and intersection of two classes   unabs 3516
                  2.1.13.5  Class abstractions with difference, union, and intersection of two classes   symdif2 3552
                  2.1.13.6  Restricted uniqueness with difference, union, and intersection   reuss2 3566
            2.1.14  The empty set   c0 3573
            2.1.15  "Weak deduction theorem" for set theory   cif 3684
            2.1.16  Power classes   cpw 3744
            2.1.17  Unordered and ordered pairs   csn 3759
            2.1.18  The union of a class   cuni 3959
            2.1.19  The intersection of a class   cint 3994
            2.1.20  Indexed union and intersection   ciun 4037
            2.1.21  Disjointness   wdisj 4125
            2.1.22  Binary relations   wbr 4155
            2.1.23  Ordered-pair class abstractions (class builders)   copab 4208
            2.1.24  Transitive classes   wtr 4245
      2.2  ZF Set Theory - add the Axiom of Replacement
            2.2.1  Introduce the Axiom of Replacement   ax-rep 4263
            2.2.2  Derive the Axiom of Separation   axsep 4272
            2.2.3  Derive the Null Set Axiom   zfnuleu 4278
            2.2.4  Theorems requiring subset and intersection existence   nalset 4283
            2.2.5  Theorems requiring empty set existence   class2set 4310
      2.3  ZF Set Theory - add the Axiom of Power Sets
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 4320
            2.3.2  Derive the Axiom of Pairing   zfpair 4344
            2.3.3  Ordered pair theorem   opnz 4375
            2.3.4  Ordered-pair class abstractions (cont.)   opabid 4404
            2.3.5  Power class of union and intersection   pwin 4430
            2.3.6  Epsilon and identity relations   cep 4435
            2.3.7  Partial and complete ordering   wpo 4444
            2.3.8  Founded and well-ordering relations   wfr 4481
            2.3.9  Ordinals   word 4523
      2.4  ZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 4643
            2.4.2  Ordinals (continued)   ordon 4705
            2.4.3  Transfinite induction   tfi 4775
            2.4.4  The natural numbers (i.e. finite ordinals)   com 4787
            2.4.5  Peano's postulates   peano1 4806
            2.4.6  Finite induction (for finite ordinals)   find 4812
            2.4.7  Relations   cxp 4818
            2.4.8  Definite description binder (inverted iota)   cio 5358
            2.4.9  Functions   wfun 5390
            2.4.10  Operations   co 6022
            2.4.11  "Maps to" notation   elmpt2cl 6229
            2.4.12  Function operation   cof 6244
            2.4.13  First and second members of an ordered pair   c1st 6288
            2.4.14  Special "Maps to" operations   mpt2xopn0yelv 6402
            2.4.15  Function transposition   ctpos 6416
            2.4.16  Curry and uncurry   ccur 6455
            2.4.17  Proper subset relation   crpss 6459
            2.4.18  Iota properties   fvopab5 6472
            2.4.19  Cantor's Theorem   canth 6477
            2.4.20  Undefined values and restricted iota (description binder)   cund 6479
            2.4.21  Functions on ordinals; strictly monotone ordinal functions   iunon 6538
            2.4.22  "Strong" transfinite recursion   crecs 6570
            2.4.23  Recursive definition generator   crdg 6605
            2.4.24  Finite recursion   frfnom 6630
            2.4.25  Abian's "most fundamental" fixed point theorem   abianfplem 6653
            2.4.26  Ordinal arithmetic   c1o 6655
            2.4.27  Natural number arithmetic   nna0 6785
            2.4.28  Equivalence relations and classes   wer 6840
            2.4.29  The mapping operation   cmap 6956
            2.4.30  Infinite Cartesian products   cixp 7001
            2.4.31  Equinumerosity   cen 7044
            2.4.32  Schroeder-Bernstein Theorem   sbthlem1 7155
            2.4.33  Equinumerosity (cont.)   xpf1o 7207
            2.4.34  Pigeonhole Principle   phplem1 7224
            2.4.35  Finite sets   onomeneq 7234
            2.4.36  Finite intersections   cfi 7352
            2.4.37  Hall's marriage theorem   marypha1lem 7375
            2.4.38  Supremum   csup 7382
            2.4.39  Ordinal isomorphism, Hartog's theorem   coi 7413
            2.4.40  Hartogs function, order types, weak dominance   char 7459
      2.5  ZF Set Theory - add the Axiom of Regularity
            2.5.1  Introduce the Axiom of Regularity   ax-reg 7495
            2.5.2  Axiom of Infinity equivalents   inf0 7511
      2.6  ZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-inf 7528
            2.6.2  Existence of omega (the set of natural numbers)   omex 7533
            2.6.3  Cantor normal form   ccnf 7551
            2.6.4  Transitive closure   trcl 7599
            2.6.5  Rank   cr1 7623
            2.6.6  Scott's trick; collection principle; Hilbert's epsilon   scottex 7744
            2.6.7  Cardinal numbers   ccrd 7757
            2.6.8  Axiom of Choice equivalents   wac 7931
            2.6.9  Cardinal number arithmetic   ccda 7982
            2.6.10  The Ackermann bijection   ackbij2lem1 8034
            2.6.11  Cofinality (without Axiom of Choice)   cflem 8061
            2.6.12  Eight inequivalent definitions of finite set   sornom 8092
            2.6.13  Hereditarily size-limited sets without Choice   itunifval 8231
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 8274
            3.2.2  AC equivalents: well-ordering, Zorn's lemma   numthcor 8309
            3.2.3  Cardinal number theorems using Axiom of Choice   cardval 8356
            3.2.4  Cardinal number arithmetic using Axiom of Choice   iunctb 8384
            3.2.5  Cofinality using Axiom of Choice   alephreg 8392
      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 8492
            4.1.2  Weak universes   cwun 8510
            4.1.3  Tarski's classes   ctsk 8558
            4.1.4  Grothendieck's universes   cgru 8600
      4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
            4.2.1  Introduce the Tarski-Grothendieck Axiom   ax-groth 8633
            4.2.2  Derive the Power Set, Infinity and Choice Axioms   grothpw 8636
            4.2.3  Tarski map function   ctskm 8647
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 8654
            5.1.2  Final derivation of real and complex number postulates   axaddf 8955
            5.1.3  Real and complex number postulates restated as axioms   ax-cnex 8981
      5.2  Derive the basic properties from the field axioms
            5.2.1  Some deductions from the field axioms for complex numbers   cnex 9006
            5.2.2  Infinity and the extended real number system   cpnf 9052
            5.2.3  Restate the ordering postulates with extended real "less than"   axlttri 9082
            5.2.4  Ordering on reals   lttr 9087
            5.2.5  Initial properties of the complex numbers   mul12 9166
      5.3  Real and complex numbers - basic operations
            5.3.1  Addition   add12 9213
            5.3.2  Subtraction   cmin 9225
            5.3.3  Multiplication   muladd 9400
            5.3.4  Ordering on reals (cont.)   gt0ne0 9427
            5.3.5  Reciprocals   ixi 9585
            5.3.6  Division   cdiv 9611
            5.3.7  Ordering on reals (cont.)   elimgt0 9780
            5.3.8  Completeness Axiom and Suprema   fimaxre 9889
            5.3.9  Imaginary and complex number properties   inelr 9924
            5.3.10  Function operation analogue theorems   ofsubeq0 9931
      5.4  Integer sets
            5.4.1  Natural numbers (as a subset of complex numbers)   cn 9934
            5.4.2  Principle of mathematical induction   nnind 9952
            5.4.3  Decimal representation of numbers   c2 9983
            5.4.4  Some properties of specific numbers   0p1e1 10027
            5.4.5  The Archimedean property   nnunb 10151
            5.4.6  Nonnegative integers (as a subset of complex numbers)   cn0 10155
            5.4.7  Integers (as a subset of complex numbers)   cz 10216
            5.4.8  Decimal arithmetic   cdc 10316
            5.4.9  Upper partititions of integers   cuz 10422
            5.4.10  Well-ordering principle for bounded-below sets of integers   uzwo3 10503
            5.4.11  Rational numbers (as a subset of complex numbers)   cq 10508
            5.4.12  Existence of the set of complex numbers   rpnnen1lem1 10534
      5.5  Order sets
            5.5.1  Positive reals (as a subset of complex numbers)   crp 10546
            5.5.2  Infinity and the extended real number system (cont.)   cxne 10641
            5.5.3  Supremum on the extended reals   xrsupexmnf 10817
            5.5.4  Real number intervals   cioo 10850
            5.5.5  Finite intervals of integers   cfz 10977
            5.5.6  Half-open integer ranges   cfzo 11067
      5.6  Elementary integer functions
            5.6.1  The floor (greatest integer) function   cfl 11130
            5.6.2  The modulo (remainder) operation   cmo 11179
            5.6.3  The infinite sequence builder "seq"   om2uz0i 11216
            5.6.4  Integer powers   cexp 11311
            5.6.5  Ordered pair theorem for nonnegative integers   nn0le2msqi 11489
            5.6.6  Factorial function   cfa 11495
            5.6.7  The binomial coefficient operation   cbc 11522
            5.6.8  The ` # ` (finite set size) function   chash 11547
                  5.6.8.1  Finite induction on the size of the first component of a binary relation   brfi1indlem 11643
            5.6.9  Words over a set   cword 11646
            5.6.10  Longer string literals   cs2 11734
      5.7  Elementary real and complex functions
            5.7.1  The "shift" operation   cshi 11810
            5.7.2  Real and imaginary parts; conjugate   ccj 11830
            5.7.3  Square root; absolute value   csqr 11967
      5.8  Elementary limits and convergence
            5.8.1  Superior limit (lim sup)   clsp 12193
            5.8.2  Limits   cli 12207
            5.8.3  Finite and infinite sums   csu 12408
            5.8.4  The binomial theorem   binomlem 12537
            5.8.5  The inclusion/exclusion principle   incexclem 12545
            5.8.6  Infinite sums (cont.)   isumshft 12548
            5.8.7  Miscellaneous converging and diverging sequences   divrcnv 12561
            5.8.8  Arithmetic series   arisum 12568
            5.8.9  Geometric series   expcnv 12572
            5.8.10  Ratio test for infinite series convergence   cvgrat 12589
            5.8.11  Mertens' theorem   mertenslem1 12590
      5.9  Elementary trigonometry
            5.9.1  The exponential, sine, and cosine functions   ce 12593
            5.9.2  _e is irrational   eirrlem 12732
      5.10  Cardinality of real and complex number subsets
            5.10.1  Countability of integers and rationals   xpnnen 12737
            5.10.2  The reals are uncountable   rpnnen2lem1 12743
PART 6  ELEMENTARY NUMBER THEORY
      6.1  Elementary properties of divisibility
            6.1.1  Irrationality of square root of 2   sqr2irrlem 12776
            6.1.2  Some Number sets are chains of proper subsets   nthruc 12779
            6.1.3  The divides relation   cdivides 12781
            6.1.4  The division algorithm   divalglem0 12842
            6.1.5  Bit sequences   cbits 12860
            6.1.6  The greatest common divisor operator   cgcd 12935
            6.1.7  Bézout's identity   bezoutlem1 12967
            6.1.8  Algorithms   nn0seqcvgd 12990
            6.1.9  Euclid's Algorithm   eucalgval2 13001
      6.2  Elementary prime number theory
            6.2.1  Elementary properties   cprime 13008
            6.2.2  Properties of the canonical representation of a rational   cnumer 13054
            6.2.3  Euler's theorem   codz 13081
            6.2.4  Pythagorean Triples   coprimeprodsq 13112
            6.2.5  The prime count function   cpc 13139
            6.2.6  Pocklington's theorem   prmpwdvds 13201
            6.2.7  Infinite primes theorem   unbenlem 13205
            6.2.8  Sum of prime reciprocals   prmreclem1 13213
            6.2.9  Fundamental theorem of arithmetic   1arithlem1 13220
            6.2.10  Lagrange's four-square theorem   cgz 13226
            6.2.11  Van der Waerden's theorem   cvdwa 13262
            6.2.12  Ramsey's theorem   cram 13296
            6.2.13  Decimal arithmetic (cont.)   dec2dvds 13328
            6.2.14  Specific prime numbers   4nprm 13356
            6.2.15  Very large primes   1259lem1 13379
PART 7  BASIC STRUCTURES
      7.1  Extensible structures
            7.1.1  Basic definitions   cstr 13394
            7.1.2  Slot definitions   cplusg 13458
            7.1.3  Definition of the structure product   crest 13577
            7.1.4  Definition of the structure quotient   cordt 13650
      7.2  Moore spaces
            7.2.1  Moore closures   mrcflem 13760
            7.2.2  Independent sets in a Moore system   mrisval 13784
            7.2.3  Algebraic closure systems   isacs 13805
PART 8  BASIC CATEGORY THEORY
      8.1  Categories
            8.1.1  Categories   ccat 13818
            8.1.2  Opposite category   coppc 13866
            8.1.3  Monomorphisms and epimorphisms   cmon 13883
            8.1.4  Sections, inverses, isomorphisms   csect 13899
            8.1.5  Subcategories   cssc 13936
            8.1.6  Functors   cfunc 13980
            8.1.7  Full & faithful functors   cful 14028
            8.1.8  Natural transformations and the functor category   cnat 14067
      8.2  Arrows (disjointified hom-sets)
            8.2.1  Identity and composition for arrows   cida 14137
      8.3  Examples of categories
            8.3.1  The category of sets   csetc 14159
            8.3.2  The category of categories   ccatc 14178
      8.4  Categorical constructions
            8.4.1  Product of categories   cxpc 14194
            8.4.2  Functor evaluation   cevlf 14235
            8.4.3  Hom functor   chof 14274
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 14326
            9.2.2  Lattices   clat 14403
            9.2.3  The dual of an ordered set   codu 14484
            9.2.4  Subset order structures   cipo 14506
            9.2.5  Distributive lattices   latmass 14543
            9.2.6  Posets and lattices as relations   cps 14553
            9.2.7  Directed sets, nets   cdir 14602
PART 10  BASIC ALGEBRAIC STRUCTURES
      10.1  Monoids
            10.1.1  Definition and basic properties   cmnd 14613
            10.1.2  Monoid homomorphisms and submonoids   cmhm 14665
            10.1.3  Ordered group sum operation   gsumvallem1 14700
            10.1.4  Free monoids   cfrmd 14721
      10.2  Groups
            10.2.1  Definition and basic properties   df-grp 14741
            10.2.2  Subgroups and Quotient groups   csubg 14867
            10.2.3  Elementary theory of group homomorphisms   cghm 14932
            10.2.4  Isomorphisms of groups   cgim 14973
            10.2.5  Group actions   cga 14995
            10.2.6  Symmetry groups and Cayley's Theorem   csymg 15021
            10.2.7  Centralizers and centers   ccntz 15043
            10.2.8  The opposite group   coppg 15070
            10.2.9  p-Groups and Sylow groups; Sylow's theorems   cod 15092
            10.2.10  Direct products   clsm 15197
            10.2.11  Free groups   cefg 15267
      10.3  Abelian groups
            10.3.1  Definition and basic properties   ccmn 15341
            10.3.2  Cyclic groups   ccyg 15416
            10.3.3  Group sum operation   gsumval3a 15441
            10.3.4  Internal direct products   cdprd 15483
            10.3.5  The Fundamental Theorem of Abelian Groups   ablfacrplem 15552
      10.4  Rings
            10.4.1  Multiplicative Group   cmgp 15577
            10.4.2  Definition and basic properties   crg 15589
            10.4.3  Opposite ring   coppr 15656
            10.4.4  Divisibility   cdsr 15672
            10.4.5  Ring homomorphisms   crh 15746
      10.5  Division rings and fields
            10.5.1  Definition and basic properties   cdr 15764
            10.5.2  Subrings of a ring   csubrg 15793
            10.5.3  Absolute value (abstract algebra)   cabv 15833
            10.5.4  Star rings   cstf 15860
      10.6  Left modules
            10.6.1  Definition and basic properties   clmod 15879
            10.6.2  Subspaces and spans in a left module   clss 15937
            10.6.3  Homomorphisms and isomorphisms of left modules   clmhm 16024
            10.6.4  Subspace sum; bases for a left module   clbs 16075
      10.7  Vector spaces
            10.7.1  Definition and basic properties   clvec 16103
      10.8  Ideals
            10.8.1  The subring algebra; ideals   csra 16169
            10.8.2  Two-sided ideals and quotient rings   c2idl 16231
            10.8.3  Principal ideal rings. Divisibility in the integers   clpidl 16241
            10.8.4  Nonzero rings   cnzr 16257
            10.8.5  Left regular elements. More kinds of rings   crlreg 16268
      10.9  Associative algebras
            10.9.1  Definition and basic properties   casa 16298
      10.10  Abstract multivariate polynomials
            10.10.1  Definition and basic properties   cmps 16335
            10.10.2  Polynomial evaluation   evlslem4 16493
            10.10.3  Univariate polynomials   cps1 16498
      10.11  The complex numbers as an extensible structure
            10.11.1  Definition and basic properties   cxmt 16614
            10.11.2  Algebraic constructions based on the complexes   czrh 16703
      10.12  Hilbert spaces
            10.12.1  Definition and basic properties   cphl 16780
            10.12.2  Orthocomplements and closed subspaces   cocv 16812
            10.12.3  Orthogonal projection and orthonormal bases   cpj 16852
PART 11  BASIC TOPOLOGY
      11.1  Topology
            11.1.1  Topological spaces   ctop 16883
            11.1.2  TopBases for topologies   isbasisg 16937
            11.1.3  Examples of topologies   distop 16985
            11.1.4  Closure and interior   ccld 17005
            11.1.5  Neighborhoods   cnei 17086
            11.1.6  Limit points and perfect sets   clp 17123
            11.1.7  Subspace topologies   restrcl 17145
            11.1.8  Order topology   ordtbaslem 17176
            11.1.9  Limits and continuity in topological spaces   ccn 17212
            11.1.10  Separated spaces: T0, T1, T2 (Hausdorff) ...   ct0 17294
            11.1.11  Compactness   ccmp 17373
            11.1.12  Connectedness   ccon 17397
            11.1.13  First- and second-countability   c1stc 17423
            11.1.14  Local topological properties   clly 17450
            11.1.15  Compactly generated spaces   ckgen 17488
            11.1.16  Product topologies   ctx 17515
            11.1.17  Continuous function-builders   cnmptid 17616
            11.1.18  Quotient maps and quotient topology   ckq 17648
            11.1.19  Homeomorphisms   chmeo 17708
      11.2  Filters and filter bases
            11.2.1  Filter bases   elmptrab 17782
            11.2.2  Filters   cfil 17800
            11.2.3  Ultrafilters   cufil 17854
            11.2.4  Filter limits   cfm 17888
            11.2.5  Extension by continuity   ccnext 18013
            11.2.6  Topological groups   ctmd 18023
            11.2.7  Infinite group sum on topological groups   ctsu 18078
            11.2.8  Topological rings, fields, vector spaces   ctrg 18108
      11.3  Uniform Stuctures and Spaces
            11.3.1  Uniform structures   cust 18152
            11.3.2  The topology induced by an uniform structure   cutop 18183
            11.3.3  Uniform Spaces   cuss 18206
            11.3.4  Uniform continuity   cucn 18228
            11.3.5  Cauchy filters in uniform spaces   ccfilu 18239
            11.3.6  Complete uniform spaces   ccusp 18250
      11.4  Metric spaces
            11.4.1  Basic metric space properties   cxme 18258
            11.4.2  Metric space balls   blfval 18323
            11.4.3  Open sets of a metric space   mopnval 18360
            11.4.4  Continuity in metric spaces   metcnp3 18462
            11.4.5  The uniform structure generated by a metric   metuval 18471
            11.4.6  Examples of metric spaces   dscmet 18493
            11.4.7  Normed algebraic structures   cnm 18497
            11.4.8  Normed space homomorphisms (bounded linear operators)   cnmo 18612
            11.4.9  Topology on the reals   qtopbaslem 18665
            11.4.10  Topological definitions using the reals   cii 18778
            11.4.11  Path homotopy   chtpy 18865
            11.4.12  The fundamental group   cpco 18898
      11.5  Complex metric vector spaces
            11.5.1  Complex left modules   cclm 18960
            11.5.2  Complex pre-Hilbert space   ccph 19002
            11.5.3  Convergence and completeness   ccfil 19078
            11.5.4  Baire's Category Theorem   bcthlem1 19148
            11.5.5  Banach spaces and complex Hilbert spaces   ccms 19156
            11.5.6  Minimizing Vector Theorem   minveclem1 19194
            11.5.7  Projection Theorem   pjthlem1 19207
PART 12  BASIC REAL AND COMPLEX ANALYSIS
      12.1  Continuity
            12.1.1  Intermediate value theorem   pmltpclem1 19214
      12.2  Integrals
            12.2.1  Lebesgue measure   covol 19228
            12.2.2  Lebesgue integration   cmbf 19375
      12.3  Derivatives
            12.3.1  Real and complex differentiation   climc 19618
PART 13  BASIC REAL AND COMPLEX FUNCTIONS
      13.1  Polynomials
            13.1.1  Abstract polynomials, continued   evlslem6 19803
            13.1.2  Polynomial degrees   cmdg 19845
            13.1.3  The division algorithm for univariate polynomials   cmn1 19917
            13.1.4  Elementary properties of complex polynomials   cply 19972
            13.1.5  The division algorithm for polynomials   cquot 20076
            13.1.6  Algebraic numbers   caa 20100
            13.1.7  Liouville's approximation theorem   aalioulem1 20118
      13.2  Sequences and series
            13.2.1  Taylor polynomials and Taylor's theorem   ctayl 20138
            13.2.2  Uniform convergence   culm 20161
            13.2.3  Power series   pserval 20195
      13.3  Basic trigonometry
            13.3.1  The exponential, sine, and cosine functions (cont.)   efcn 20228
            13.3.2  Properties of pi = 3.14159...   pilem1 20236
            13.3.3  Mapping of the exponential function   efgh 20312
            13.3.4  The natural logarithm on complex numbers   clog 20321
            13.3.5  Theorems of Pythagoras, isosceles triangles, and intersecting chords   angval 20512
            13.3.6  Solutions of quadratic, cubic, and quartic equations   quad2 20548
            13.3.7  Inverse trigonometric functions   casin 20571
            13.3.8  The Birthday Problem   log2ublem1 20655
            13.3.9  Areas in R^2   carea 20663
            13.3.10  More miscellaneous converging sequences   rlimcnp 20673
            13.3.11  Inequality of arithmetic and geometric means   cvxcl 20692
            13.3.12  Euler-Mascheroni constant   cem 20699
      13.4  Basic number theory
            13.4.1  Wilson's theorem   wilthlem1 20720
            13.4.2  The Fundamental Theorem of Algebra   ftalem1 20724
            13.4.3  The Basel problem (ζ(2) = π2/6)   basellem1 20732
            13.4.4  Number-theoretical functions   ccht 20742
            13.4.5  Perfect Number Theorem   mersenne 20880
            13.4.6  Characters of Z/nZ   cdchr 20885
            13.4.7  Bertrand's postulate   bcctr 20928
            13.4.8  Legendre symbol   clgs 20947
            13.4.9  Quadratic reciprocity   lgseisenlem1 21002
            13.4.10  All primes 4n+1 are the sum of two squares   2sqlem1 21016
            13.4.11  Chebyshev's Weak Prime Number Theorem, Dirichlet's Theorem   chebbnd1lem1 21032
            13.4.12  The Prime Number Theorem   mudivsum 21093
            13.4.13  Ostrowski's theorem   abvcxp 21178
PART 14  GRAPH THEORY
      14.1  Undirected graphs - basics
            14.1.1  Undirected hypergraphs   cuhg 21203
            14.1.2  Undirected multigraphs   cumg 21216
            14.1.3  Undirected simple graphs   cuslg 21233
                  14.1.3.1  Undirected simple graphs - basics   cuslg 21233
                  14.1.3.2  Undirected simple graphs - examples   usgraexvlem 21282
                  14.1.3.3  Finite undirected simple graphs   fiusgraedgfi 21289
            14.1.4  Neighbors, complete graphs and universal vertices   cnbgra 21298
                  14.1.4.1  Neighbors   nbgraop 21304
                  14.1.4.2  Complete graphs   iscusgra 21333
                  14.1.4.3  Universal vertices   isuvtx 21365
            14.1.5  Walks, paths and cycles   cwalk 21374
                  14.1.5.1  Walks and trails   wlks 21392
                  14.1.5.2  Paths and simple paths   pths 21422
                  14.1.5.3  Circuits and cycles   crcts 21459
                  14.1.5.4  Connected graphs   cconngra 21506
            14.1.6  Vertex Degree   cvdg 21514
      14.2  Eulerian paths and the Konigsberg Bridge problem
            14.2.1  Eulerian paths   ceup 21534
            14.2.2  The Konigsberg Bridge problem   vdeg0i 21554
PART 15  GUIDES AND MISCELLANEA
      15.1  Guides (conventions, explanations, and examples)
            15.1.1  Conventions   conventions 21560
            15.1.2  Natural deduction   natded 21561
            15.1.3  Natural deduction examples   ex-natded5.2 21562
            15.1.4  Definitional examples   ex-or 21579
      15.2  Humor
            15.2.1  April Fool's theorem   avril1 21607
      15.3  (Future - to be reviewed and classified)
            15.3.1  Planar incidence geometry   cplig 21613
            15.3.2  Algebra preliminaries   crpm 21618
            15.3.3  Transitive closure   ctcl 21620
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 21624
            16.1.2  Definition and basic properties of Abelian groups   cablo 21719
            16.1.3  Subgroups   csubgo 21739
            16.1.4  Operation properties   cass 21750
            16.1.5  Group-like structures   cmagm 21756
            16.1.6  Examples of Abelian groups   ablosn 21785
            16.1.7  Group homomorphism and isomorphism   cghom 21795
      16.2  Additional material on rings and fields
            16.2.1  Definition and basic properties   crngo 21813
            16.2.2  Examples of rings   cnrngo 21841
            16.2.3  Division Rings   cdrng 21843
            16.2.4  Star Fields   csfld 21846
            16.2.5  Fields and Rings   ccm2 21848
PART 17  COMPLEX TOPOLOGICAL VECTOR SPACES (DEPRECATED)
      17.1  Complex vector spaces
            17.1.1  Definition and basic properties   cvc 21874
            17.1.2  Examples of complex vector spaces   cncvc 21912
      17.2  Normed complex vector spaces
            17.2.1  Definition and basic properties   cnv 21913
            17.2.2  Examples of normed complex vector spaces   cnnv 22018
            17.2.3  Induced metric of a normed complex vector space   imsval 22027
            17.2.4  Inner product   cdip 22046
            17.2.5  Subspaces   css 22070
      17.3  Operators on complex vector spaces
            17.3.1  Definitions and basic properties   clno 22091
      17.4  Inner product (pre-Hilbert) spaces
            17.4.1  Definition and basic properties   ccphlo 22163
            17.4.2  Examples of pre-Hilbert spaces   cncph 22170
            17.4.3  Properties of pre-Hilbert spaces   isph 22173
      17.5  Complex Banach spaces
            17.5.1  Definition and basic properties   ccbn 22214
            17.5.2  Examples of complex Banach spaces   cnbn 22221
            17.5.3  Uniform Boundedness Theorem   ubthlem1 22222
            17.5.4  Minimizing Vector Theorem   minvecolem1 22226
      17.6  Complex Hilbert spaces
            17.6.1  Definition and basic properties   chlo 22237
            17.6.2  Standard axioms for a complex Hilbert space   hlex 22250
            17.6.3  Examples of complex Hilbert spaces   cnchl 22268
            17.6.4  Subspaces   ssphl 22269
            17.6.5  Hellinger-Toeplitz Theorem   htthlem 22270
PART 18  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      18.1  Axiomatization of complex pre-Hilbert spaces
            18.1.1  Basic Hilbert space definitions   chil 22272
            18.1.2  Preliminary ZFC lemmas   df-hnorm 22321
            18.1.3  Derive the Hilbert space axioms from ZFC set theory   axhilex-zf 22334
            18.1.4  Introduce the vector space axioms for a Hilbert space   ax-hilex 22352
            18.1.5  Vector operations   hvmulex 22364
            18.1.6  Inner product postulates for a Hilbert space   ax-hfi 22431
      18.2  Inner product and norms
            18.2.1  Inner product   his5 22438
            18.2.2  Norms   dfhnorm2 22474
            18.2.3  Relate Hilbert space to normed complex vector spaces   hilablo 22512
            18.2.4  Bunjakovaskij-Cauchy-Schwarz inequality   bcsiALT 22531
      18.3  Cauchy sequences and completeness axiom
            18.3.1  Cauchy sequences and limits   hcau 22536
            18.3.2  Derivation of the completeness axiom from ZF set theory   hilmet 22546
            18.3.3  Completeness postulate for a Hilbert space   ax-hcompl 22554
            18.3.4  Relate Hilbert space to ZFC pre-Hilbert and Hilbert spaces   hhcms 22555
      18.4  Subspaces and projections
            18.4.1  Subspaces   df-sh 22559
            18.4.2  Closed subspaces   df-ch 22574
            18.4.3  Orthocomplements   df-oc 22604
            18.4.4  Subspace sum, span, lattice join, lattice supremum   df-shs 22660
            18.4.5  Projection theorem   pjhthlem1 22743
            18.4.6  Projectors   df-pjh 22747
      18.5  Properties of Hilbert subspaces
            18.5.1  Orthomodular law   omlsilem 22754
            18.5.2  Projectors (cont.)   pjhtheu2 22768
            18.5.3  Hilbert lattice operations   sh0le 22792
            18.5.4  Span (cont.) and one-dimensional subspaces   spansn0 22893
            18.5.5  Commutes relation for Hilbert lattice elements   df-cm 22935
            18.5.6  Foulis-Holland theorem   fh1 22970
            18.5.7  Quantum Logic Explorer axioms   qlax1i 22979
            18.5.8  Orthogonal subspaces   chscllem1 22989
            18.5.9  Orthoarguesian laws 5OA and 3OA   5oalem1 23006
            18.5.10  Projectors (cont.)   pjorthi 23021
            18.5.11  Mayet's equation E_3   mayete3i 23080
      18.6  Operators on Hilbert spaces
            18.6.1  Operator sum, difference, and scalar multiplication   df-hosum 23083
            18.6.2  Zero and identity operators   df-h0op 23101
            18.6.3  Operations on Hilbert space operators   hoaddcl 23111
            18.6.4  Linear, continuous, bounded, Hermitian, unitary operators and norms   df-nmop 23192
            18.6.5  Linear and continuous functionals and norms   df-nmfn 23198
            18.6.6  Adjoint   df-adjh 23202
            18.6.7  Dirac bra-ket notation   df-bra 23203
            18.6.8  Positive operators   df-leop 23205
            18.6.9  Eigenvectors, eigenvalues, spectrum   df-eigvec 23206
            18.6.10  Theorems about operators and functionals   nmopval 23209
            18.6.11  Riesz lemma   riesz3i 23415
            18.6.12  Adjoints (cont.)   cnlnadjlem1 23420
            18.6.13  Quantum computation error bound theorem   unierri 23457
            18.6.14  Dirac bra-ket notation (cont.)   branmfn 23458
            18.6.15  Positive operators (cont.)   leopg 23475
            18.6.16  Projectors as operators   pjhmopi 23499
      18.7  States on a Hilbert lattice and Godowski's equation
            18.7.1  States on a Hilbert lattice   df-st 23564
            18.7.2  Godowski's equation   golem1 23624
      18.8  Cover relation, atoms, exchange axiom, and modular symmetry
            18.8.1  Covers relation; modular pairs   df-cv 23632
            18.8.2  Atoms   df-at 23691
            18.8.3  Superposition principle   superpos 23707
            18.8.4  Atoms, exchange and covering properties, atomicity   chcv1 23708
            18.8.5  Irreducibility   chirredlem1 23743
            18.8.6  Atoms (cont.)   atcvat3i 23749
            18.8.7  Modular symmetry   mdsymlem1 23756
PART 19  SUPPLEMENTARY MATERIAL (USER'S MATHBOXES)
      19.1  Mathboxes for user contributions
            19.1.1  Mathbox guidelines   mathbox 23795
      19.2  Mathbox for Stefan Allan
      19.3  Mathbox for Thierry Arnoux
            19.3.1  Propositional Calculus - misc additions   bian1d 23800
            19.3.2  Predicate Calculus   abeq2f 23806
                  19.3.2.1  Predicate Calculus - misc additions   abeq2f 23806
                  19.3.2.2  Restricted quantification - misc additions   reximddv 23808
                  19.3.2.3  Substitution (without distinct variables) - misc additions   clelsb3f 23817
                  19.3.2.4  Existential "at most one" - misc additions   mo5f 23818
                  19.3.2.5  Existential uniqueness - misc additions   2reuswap2 23821
                  19.3.2.6  Restricted "at most one" - misc additions   rmoxfrdOLD 23825
            19.3.3  General Set Theory   ceqsexv2d 23831
                  19.3.3.1  Class abstractions (a.k.a. class builders)   ceqsexv2d 23831
                  19.3.3.2  Image Sets   abrexdomjm 23834
                  19.3.3.3  Set relations and operations - misc additions   eqri 23840
                  19.3.3.4  Unordered pairs   elpreq 23845
                  19.3.3.5  Conditional operator - misc additions   ifeqeqx 23847
                  19.3.3.6  Indexed union - misc additions   iuneq12daf 23853
                  19.3.3.7  Disjointness - misc additions   cbvdisjf 23861
            19.3.4  Relations and Functions   dfrel4 23879
                  19.3.4.1  Relations - misc additions   dfrel4 23879
                  19.3.4.2  Functions - misc additions   fdmrn 23884
                  19.3.4.3  Isomorphisms - misc. add.   gtiso 23931
                  19.3.4.4  Disjointness (additional proof requiring functions)   disjdsct 23933
                  19.3.4.5  First and second members of an ordered pair - misc additions   df1stres 23934
                  19.3.4.6  Supremum - misc additions   supssd 23941
                  19.3.4.7  Countable Sets   nnct 23942
            19.3.5  Real and Complex Numbers   addeq0 23957
                  19.3.5.1  Complex addition - misc. additions   addeq0 23957
                  19.3.5.2  Ordering on reals - misc additions   lt2addrd 23958
                  19.3.5.3  Extended reals - misc additions   xrlelttric 23959
                  19.3.5.4  Real number intervals - misc additions   icossicc 23967
                  19.3.5.5  Finite intervals of integers - misc additions   fzssnn 23985
                  19.3.5.6  Half-open integer ranges - misc additions   fzossnn 23990
                  19.3.5.7  The ` # ` (finite set size) function - misc additions   hashresfn 23996
                  19.3.5.8  The greatest common divisor operator - misc. add   numdenneg 24000
                  19.3.5.9  Integers   ltesubnnd 24002
                  19.3.5.10  Division in the extended real number system   cxdiv 24003
            19.3.6  Structure builders   ress0g 24023
                  19.3.6.1  Structure builder restriction operator   ress0g 24023
                  19.3.6.2  Posets   tospos 24027
                  19.3.6.3  Extended reals Structure - misc additions   ax-xrssca 24030
                  19.3.6.4  The extended non-negative real numbers monoid   xrge0base 24038
            19.3.7  Algebra   sumpr 24049
                  19.3.7.1  Finitely supported group sums - misc additions   sumpr 24049
                  19.3.7.2  Rings - misc additions   dvrdir 24057
                  19.3.7.3  Ordered fields   cofld 24061
                  19.3.7.4  Ring homomorphisms - misc additions   rhmdvdsr 24074
                  19.3.7.5  The ring of integers   zzsbase 24081
                  19.3.7.6  The ordered field of reals   rebase 24087
            19.3.8  Topology   hauseqcn 24099
                  19.3.8.1  Continuity - misc additions   hauseqcn 24099
                  19.3.8.2  Topology of the closed unit   unitsscn 24100
                  19.3.8.3  Topology of ` ( RR X. RR ) `   unicls 24107
                  19.3.8.4  Order topology - misc. additions   cnvordtrestixx 24117
                  19.3.8.5  Continuity in topological spaces - misc. additions   mndpluscn 24118
                  19.3.8.6  Topology of the extended non-negative real numbers monoid   xrge0hmph 24124
                  19.3.8.7  Limits - misc additions   lmlim 24139
            19.3.9  Topology and algebraic structures   zzsnm 24146
                  19.3.9.1  The norm on the ring of the integer numbers   zzsnm 24146
                  19.3.9.2  The complete ordered field of the real numbers   recms 24147
                  19.3.9.3  Topological ` ZZ ` -modules   zlm0 24149
                  19.3.9.4  The canonical embedding of the rational numbers into a division ring   cqqh 24157
                  19.3.9.5  The canonical embedding of ` RR ` into a complete ordered field   crrh 24178
                  19.3.9.6  Canonical embeddings into ` RR `   zrhre 24183
            19.3.10  Real and complex functions   clogb 24186
                  19.3.10.1  Logarithm laws generalized to an arbitrary base - logb   clogb 24186
                  19.3.10.2  Indicator Functions   cind 24206
                  19.3.10.3  Extended sum   cesum 24222
            19.3.11  Mixed Function/Constant operation   cofc 24276
            19.3.12  Abstract measure   csiga 24288
                  19.3.12.1  Sigma-Algebra   csiga 24288
                  19.3.12.2  Generated Sigma-Algebra   csigagen 24319
                  19.3.12.3  The Borel algebra on the real numbers   cbrsiga 24333
                  19.3.12.4  Product Sigma-Algebra   csx 24340
                  19.3.12.5  Measures   cmeas 24347
                  19.3.12.6  The counting measure   cntmeas 24376
                  19.3.12.7  The Lebesgue measure - misc additions   volss 24379
                  19.3.12.8  The 'almost everywhere' relation   cae 24384
                  19.3.12.9  Measurable functions   cmbfm 24396
                  19.3.12.10  Borel Algebra on ` ( RR X. RR ) `   br2base 24415
            19.3.13  Integration   itgeq12dv 24437
                  19.3.13.1  Lebesgue integral - misc additions   itgeq12dv 24437
                  19.3.13.2  Bochner integral   citgm 24438
            19.3.14  Probability   cprb 24446
                  19.3.14.1  Probability Theory   cprb 24446
                  19.3.14.2  Conditional Probabilities   ccprob 24470
                  19.3.14.3  Real Valued Random Variables   crrv 24479
                  19.3.14.4  Preimage set mapping operator   corvc 24494
                  19.3.14.5  Distribution Functions   orvcelval 24507
                  19.3.14.6  Cumulative Distribution Functions   orvclteel 24511
                  19.3.14.7  Probabilities - example   coinfliplem 24517
                  19.3.14.8  Bertrand's Ballot Problem   ballotlemoex 24524
      19.4  Mathbox for Mario Carneiro
            19.4.1  Miscellaneous stuff   quartfull 24577
            19.4.2  Zeta function   czeta 24578
            19.4.3  Gamma function   clgam 24581
            19.4.4  Derangements and the Subfactorial   deranglem 24633
            19.4.5  The Erdős-Szekeres theorem   erdszelem1 24658
            19.4.6  The Kuratowski closure-complement theorem   kur14lem1 24673
            19.4.7  Retracts and sections   cretr 24684
            19.4.8  Path-connected and simply connected spaces   cpcon 24687
            19.4.9  Covering maps   ccvm 24723
            19.4.10  Normal numbers   snmlff 24797
            19.4.11  Godel-sets of formulas   cgoe 24801
            19.4.12  Models of ZF   cgze 24829
            19.4.13  Splitting fields   citr 24843
            19.4.14  p-adic number fields   czr 24859
      19.5  Mathbox for Paul Chapman
            19.5.1  Group homomorphism and isomorphism   ghomgrpilem1 24877
            19.5.2  Real and complex numbers (cont.)   climuzcnv 24889
            19.5.3  Miscellaneous theorems   elfzm12 24893
      19.6  Mathbox for Drahflow
      19.7  Mathbox for Scott Fenton
            19.7.1  ZFC Axioms in primitive form   axextprim 24931
            19.7.2  Untangled classes   untelirr 24938
            19.7.3  Extra propositional calculus theorems   3orel1 24945
            19.7.4  Misc. Useful Theorems   nepss 24956
            19.7.5  Properties of reals and complexes   sqdivzi 24965
            19.7.6  Product sequences   prodf 24996
            19.7.7  Non-trivial convergence   ntrivcvg 25006
            19.7.8  Complex products   cprod 25012
            19.7.9  Finite products   fprod 25048
            19.7.10  Infinite products   iprodclim 25085
            19.7.11  Falling and Rising Factorial   cfallfac 25091
            19.7.12  Factorial limits   faclimlem1 25122
            19.7.13  Greatest common divisor and divisibility   pdivsq 25128
            19.7.14  Properties of relationships   brtp 25132
            19.7.15  Properties of functions and mappings   funpsstri 25147
            19.7.16  Epsilon induction   setinds 25160
            19.7.17  Ordinal numbers   elpotr 25163
            19.7.18  Defined equality axioms   axextdfeq 25180
            19.7.19  Hypothesis builders   hbntg 25188
            19.7.20  The Predecessor Class   cpred 25193
            19.7.21  (Trans)finite Recursion Theorems   tfisg 25230
            19.7.22  Well-founded induction   tz6.26 25231
            19.7.23  Transitive closure under a relationship   ctrpred 25246
            19.7.24  Founded Induction   frmin 25268
            19.7.25  Ordering Ordinal Sequences   orderseqlem 25278
            19.7.26  Well-founded recursion   wfr3g 25281
            19.7.27  Transfinite recursion via Well-founded recursion   tfrALTlem 25302
            19.7.28  Founded Recursion   frr3g 25306
            19.7.29  Surreal Numbers   csur 25320
            19.7.30  Surreal Numbers: Ordering   sltsolem1 25348
            19.7.31  Surreal Numbers: Birthday Function   bdayfo 25355
            19.7.32  Surreal Numbers: Density   fvnobday 25362
            19.7.33  Surreal Numbers: Density   nodenselem3 25363
            19.7.34  Surreal Numbers: Upper and Lower Bounds   nobndlem1 25372
            19.7.35  Surreal Numbers: Full-Eta Property   nofulllem1 25382
            19.7.36  Symmetric difference   csymdif 25387
            19.7.37  Quantifier-free definitions   ctxp 25399
            19.7.38  Alternate ordered pairs   caltop 25517
            19.7.39  Tarskian geometry   cee 25543
            19.7.40  Tarski's axioms for geometry   axdimuniq 25568
            19.7.41  Congruence properties   cofs 25632
            19.7.42  Betweenness properties   btwntriv2 25662
            19.7.43  Segment Transportation   ctransport 25679
            19.7.44  Properties relating betweenness and congruence   cifs 25685
            19.7.45  Connectivity of betweenness   btwnconn1lem1 25737
            19.7.46  Segment less than or equal to   csegle 25756
            19.7.47  Outside of relationship   coutsideof 25769
            19.7.48  Lines and Rays   cline2 25784
            19.7.49  Bernoulli polynomials and sums of k-th powers   cbp 25808
            19.7.50  Rank theorems   rankung 25823
            19.7.51  Hereditarily Finite Sets   chf 25829
      19.8  Mathbox for Anthony Hart
            19.8.1  Propositional Calculus   tb-ax1 25844
            19.8.2  Predicate Calculus   quantriv 25866
            19.8.3  Misc. Single Axiom Systems   meran1 25877
            19.8.4  Connective Symmetry   negsym1 25883
      19.9  Mathbox for Chen-Pang He
            19.9.1  Ordinal topology   ontopbas 25894
      19.10  Mathbox for Jeff Hoffman
            19.10.1  Inferences for finite induction on generic function values   fveleq 25917
            19.10.2  gdc.mm   nnssi2 25921
      19.11  Mathbox for Wolf Lammen
      19.12  Mathbox for Brendan Leahy
      19.13  Mathbox for Jeff Hankins
            19.13.1  Miscellany   a1i13 25991
            19.13.2  Basic topological facts   topbnd 26020
            19.13.3  Topology of the real numbers   ivthALT 26031
            19.13.4  Refinements   cfne 26032
            19.13.5  Neighborhood bases determine topologies   neibastop1 26081
            19.13.6  Lattice structure of topologies   topmtcl 26085
            19.13.7  Filter bases   fgmin 26092
            19.13.8  Directed sets, nets   tailfval 26094
      19.14  Mathbox for Jeff Madsen
            19.14.1  Logic and set theory   anim12da 26105
            19.14.2  Real and complex numbers; integers   filbcmb 26135
            19.14.3  Sequences and sums   sdclem2 26139
            19.14.4  Topology   subspopn 26151
            19.14.5  Metric spaces   metf1o 26154
            19.14.6  Continuous maps and homeomorphisms   constcncf 26161
            19.14.7  Boundedness   ctotbnd 26168
            19.14.8  Isometries   cismty 26200
            19.14.9  Heine-Borel Theorem   heibor1lem 26211
            19.14.10  Banach Fixed Point Theorem   bfplem1 26224
            19.14.11  Euclidean space   crrn 26227
            19.14.12  Intervals (continued)   ismrer1 26240
            19.14.13  Groups and related structures   exidcl 26244
            19.14.14  Rings   rngonegcl 26254
            19.14.15  Ring homomorphisms   crnghom 26269
            19.14.16  Commutative rings   ccring 26298
            19.14.17  Ideals   cidl 26310
            19.14.18  Prime rings and integral domains   cprrng 26349
            19.14.19  Ideal generators   cigen 26362
      19.15  Mathbox for Rodolfo Medina
            19.15.1  Partitions   prtlem60 26381
      19.16  Mathbox for Stefan O'Rear
            19.16.1  Additional elementary logic and set theory   nelss 26425
            19.16.2  Additional theory of functions   fninfp 26428
            19.16.3  Extensions beyond function theory   gsumvsmul 26438
            19.16.4  Additional topology   elrfi 26441
            19.16.5  Characterization of closure operators. Kuratowski closure axioms   ismrcd1 26445
            19.16.6  Algebraic closure systems   cnacs 26449
            19.16.7  Miscellanea 1. Map utilities   constmap 26460
            19.16.8  Miscellanea for polynomials   ofmpteq 26469
            19.16.9  Multivariate polynomials over the integers   cmzpcl 26471
            19.16.10  Miscellanea for Diophantine sets 1   coeq0 26503
            19.16.11  Diophantine sets 1: definitions   cdioph 26506
            19.16.12  Diophantine sets 2 miscellanea   ellz1 26518
            19.16.13  Diophantine sets 2: union and intersection. Monotone Boolean algebra   diophin 26524
            19.16.14  Diophantine sets 3: construction   diophrex 26527
            19.16.15  Diophantine sets 4 miscellanea   2sbcrex 26536
            19.16.16  Diophantine sets 4: Quantification   rexrabdioph 26547
            19.16.17  Diophantine sets 5: Arithmetic sets   rabdiophlem1 26554
            19.16.18  Diophantine sets 6 miscellanea   fz1ssnn 26564
            19.16.19  Diophantine sets 6: reusability. renumbering of variables   eldioph4b 26565
            19.16.20  Pigeonhole Principle and cardinality helpers   fphpd 26570
            19.16.21  A non-closed set of reals is infinite   rencldnfilem 26574
            19.16.22  Miscellanea for Lagrange's theorem   icodiamlt 26576
            19.16.23  Lagrange's rational approximation theorem   irrapxlem1 26578
            19.16.24  Pell equations 1: A nontrivial solution always exists   pellexlem1 26585
            19.16.25  Pell equations 2: Algebraic number theory of the solution set   csquarenn 26592
            19.16.26  Pell equations 3: characterizing fundamental solution   infmrgelbi 26634
            19.16.27  Logarithm laws generalized to an arbitrary base   reglogcl 26646
            19.16.28  Pell equations 4: the positive solution group is infinite cyclic   pellfund14 26654
            19.16.29  X and Y sequences 1: Definition and recurrence laws   crmx 26656
            19.16.30  Ordering and induction lemmas for the integers   monotuz 26697
            19.16.31  X and Y sequences 2: Order properties   rmxypos 26705
            19.16.32  Congruential equations   congtr 26723
            19.16.33  Alternating congruential equations   acongid 26733
            19.16.34  Additional theorems on integer divisibility   bezoutr 26743
            19.16.35  X and Y sequences 3: Divisibility properties   jm2.18 26752
            19.16.36  X and Y sequences 4: Diophantine representability of Y   jm2.27a 26769
            19.16.37  X and Y sequences 5: Diophantine representability of X, ^, _C   rmxdiophlem 26779
            19.16.38  Uncategorized stuff not associated with a major project   setindtr 26788
            19.16.39  More equivalents of the Axiom of Choice   axac10 26797
            19.16.40  Finitely generated left modules   clfig 26836
            19.16.41  Noetherian left modules I   clnm 26844
            19.16.42  Addenda for structure powers   pwssplit0 26858
            19.16.43  Direct sum of left modules   cdsmm 26868
            19.16.44  Free modules   cfrlm 26883
            19.16.45  Every set admits a group structure iff choice   unxpwdom3 26927
            19.16.46  Independent sets and families   clindf 26945
            19.16.47  Characterization of free modules   lmimlbs 26977
            19.16.48  Noetherian rings and left modules II   clnr 26984
            19.16.49  Hilbert's Basis Theorem   cldgis 26996
            19.16.50  Additional material on polynomials [DEPRECATED]   cmnc 27006
            19.16.51  Degree and minimal polynomial of algebraic numbers   cdgraa 27016
            19.16.52  Algebraic integers I   citgo 27033
            19.16.53  Finite cardinality [SO]   en1uniel 27051
            19.16.54  Words in monoids and ordered group sum   issubmd 27054
            19.16.55  Transpositions in the symmetric group   cpmtr 27055
            19.16.56  The sign of a permutation   cpsgn 27085
            19.16.57  The matrix algebra   cmmul 27110
            19.16.58  The determinant   cmdat 27154
            19.16.59  Endomorphism algebra   cmend 27160
            19.16.60  Subfields   csdrg 27174
            19.16.61  Cyclic groups and order   idomrootle 27182
            19.16.62  Cyclotomic polynomials   ccytp 27192
            19.16.63  Miscellaneous topology   fgraphopab 27200
      19.17  Mathbox for Steve Rodriguez
            19.17.1  Miscellanea   iso0 27207
            19.17.2  Function operations   caofcan 27211
            19.17.3  Calculus   lhe4.4ex1a 27217
      19.18  Mathbox for Andrew Salmon
            19.18.1  Principia Mathematica * 10   pm10.12 27224
            19.18.2  Principia Mathematica * 11   2alanimi 27238
            19.18.3  Predicate Calculus   sbeqal1 27268
            19.18.4  Principia Mathematica * 13 and * 14   pm13.13a 27278
            19.18.5  Set Theory   elnev 27309
            19.18.6  Arithmetic   addcomgi 27331
            19.18.7  Geometry   cplusr 27332
      19.19  Mathbox for Glauco Siliprandi
            19.19.1  Miscellanea   ssrexf 27354
            19.19.2  Finite multiplication of numbers and finite multiplication of functions   fmul01 27380
            19.19.3  Limits   clim1fr1 27397
            19.19.4  Derivatives   dvsinexp 27410
            19.19.5  Integrals   ioovolcl 27412
            19.19.6  Stone Weierstrass theorem - real version   stoweidlem1 27420
            19.19.7  Wallis' product for π   wallispilem1 27484
            19.19.8  Stirling's approximation formula for ` n ` factorial   stirlinglem1 27493
      19.20  Mathbox for Saveliy Skresanov
            19.20.1  Ceva's theorem   sigarval 27510
      19.21  Mathbox for Jarvin Udandy
      19.22  Mathbox for Alexander van der Vekens
            19.22.1  Double restricted existential uniqueness   r19.32 27615
                  19.22.1.1  Restricted quantification (extension)   r19.32 27615
                  19.22.1.2  The empty set (extension)   raaan2 27623
                  19.22.1.3  Restricted uniqueness and "at most one" quantification   rmoimi 27624
                  19.22.1.4  Analogs to Existential uniqueness (double quantification)   2reurex 27629
            19.22.2  Alternative definitions of function's and operation's values   wdfat 27641
                  19.22.2.1  Restricted quantification (extension)   ralbinrald 27647
                  19.22.2.2  The universal class (extension)   nvelim 27648
                  19.22.2.3  Introduce the Axiom of Power Sets (extension)   alneu 27649
                  19.22.2.4  Relations (extension)   sbcrel 27651
                  19.22.2.5  Functions (extension)   sbcfun 27657
                  19.22.2.6  Predicate "defined at"   dfateq12d 27664
                  19.22.2.7  Alternative definition of the value of a function   dfafv2 27667
                  19.22.2.8  Alternative definition of the value of an operation   aoveq123d 27713
            19.22.3  Graph theory   cfrgra 27743
                  19.22.3.1  Friendship graphs   cfrgra 27743
      19.23  Mathbox for David A. Wheeler
            19.23.1  Natural deduction   19.8ad 27808
            19.23.2  Greater than, greater than or equal to.   cge-real 27811
            19.23.3  Hyperbolic trig functions   csinh 27821
            19.23.4  Reciprocal trig functions (sec, csc, cot)   csec 27832
            19.23.5  Identities for "if"   ifnmfalse 27854
            19.23.6  Not-member-of   AnelBC 27855
            19.23.7  Decimal point   cdp2 27856
            19.23.8  Signum (sgn or sign) function   csgn 27864
            19.23.9  Ceiling function   ccei 27874
            19.23.10  Logarithms generalized to arbitrary base using ` logb `   ene0 27878
            19.23.11  Logarithm laws generalized to an arbitrary base - log_   clog_ 27881
            19.23.12  Miscellaneous   5m4e1 27883
      19.24  Mathbox for Alan Sare
            19.24.1  Supplementary "adant" deductions   ad4ant13 27886
            19.24.2  Supplementary unification deductions   biimp 27912
            19.24.3  Conventional Metamath proofs, some derived from VD proofs   iidn3 27928
            19.24.4  What is Virtual Deduction?   wvd1 28003
            19.24.5  Virtual Deduction Theorems   df-vd1 28004
            19.24.6  Theorems proved using virtual deduction   trsspwALT 28274
            19.24.7  Theorems proved using virtual deduction with mmj2 assistance   simplbi2VD 28301
            19.24.8  Virtual Deduction transcriptions of textbook proofs   sb5ALTVD 28368
            19.24.9  Theorems proved using conjunction-form virtual deduction   elpwgdedVD 28372
            19.24.10  Theorems with VD proofs in conventional notation derived from VD proofs   suctrALT3 28379
            19.24.11  Theorems with a proof in conventional notation automatically derived   notnot2ALT2 28382
      19.25  Mathbox for Jonathan Ben-Naim
            19.25.1  First order logic and set theory   bnj170 28402
            19.25.2  Well founded induction and recursion   bnj110 28569
            19.25.3  The existence of a minimal element in certain classes   bnj69 28719
            19.25.4  Well-founded induction   bnj1204 28721
            19.25.5  Well-founded recursion, part 1 of 3   bnj60 28771
            19.25.6  Well-founded recursion, part 2 of 3   bnj1500 28777
            19.25.7  Well-founded recursion, part 3 of 3   bnj1522 28781
      19.26  Mathbox for Norm Megill
            19.26.1  Experiments to study ax-7 unbundling   ax-7v 28782
                  19.26.1.1  Theorems derived from ax-7v (suffixes NEW7 and AUX7)   ax-7v 28782
                  19.26.1.2  Theorems derived from ax-7 (suffix OLD7)   ax-7OLD7 28996
            19.26.2  Miscellanea   cnaddcom 29088
            19.26.3  Atoms, hyperplanes, and covering in a left vector space (or module)   clsa 29091
            19.26.4  Functionals and kernels of a left vector space (or module)   clfn 29174
            19.26.5  Opposite rings and dual vector spaces   cld 29240
            19.26.6  Ortholattices and orthomodular lattices   cops 29289
            19.26.7  Atomic lattices with covering property   ccvr 29379
            19.26.8  Hilbert lattices   chlt 29467
            19.26.9  Projective geometries based on Hilbert lattices   clln 29607
            19.26.10  Construction of a vector space from a Hilbert lattice   cdlema1N 29907
            19.26.11  Construction of involution and inner product from a Hilbert lattice   clpoN 31597

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