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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 2211
            1.6.2  Rederive new axioms from old: ax5 , ax6 , ax9from9o , ax11 , ax12from12o   ax4 2221
            1.6.3  Legacy theorems using obsolete axioms   ax17o 2233
      1.7  Existential uniqueness
      1.8  Other axiomatizations related to classical predicate calculus
            1.8.1  Predicate calculus with all distinct variables   ax-7d 2371
            1.8.2  Aristotelian logic: Assertic syllogisms   barbara 2377
            1.8.3  Intuitionistic logic   axia1 2401
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 2416
            2.1.2  Class abstractions (a.k.a. class builders)   cab 2421
            2.1.3  Class form not-free predicate   wnfc 2558
            2.1.4  Negated equality and membership   wne 2598
                  2.1.4.1  Negated equality   nne 2602
                  2.1.4.2  Negated membership   neleq1 2691
            2.1.5  Restricted quantification   wral 2697
            2.1.6  The universal class   cvv 2948
            2.1.7  Conditional equality (experimental)   wcdeq 3136
            2.1.8  Russell's Paradox   ru 3152
            2.1.9  Proper substitution of classes for sets   wsbc 3153
            2.1.10  Proper substitution of classes for sets into classes   csb 3243
            2.1.11  Define basic set operations and relations   cdif 3309
            2.1.12  Subclasses and subsets   df-ss 3326
            2.1.13  The difference, union, and intersection of two classes   difeq1 3450
                  2.1.13.1  The difference of two classes   difeq1 3450
                  2.1.13.2  The union of two classes   elun 3480
                  2.1.13.3  The intersection of two classes   elin 3522
                  2.1.13.4  Combinations of difference, union, and intersection of two classes   unabs 3563
                  2.1.13.5  Class abstractions with difference, union, and intersection of two classes   symdif2 3599
                  2.1.13.6  Restricted uniqueness with difference, union, and intersection   reuss2 3613
            2.1.14  The empty set   c0 3620
            2.1.15  "Weak deduction theorem" for set theory   cif 3731
            2.1.16  Power classes   cpw 3791
            2.1.17  Unordered and ordered pairs   csn 3806
            2.1.18  The union of a class   cuni 4007
            2.1.19  The intersection of a class   cint 4042
            2.1.20  Indexed union and intersection   ciun 4085
            2.1.21  Disjointness   wdisj 4174
            2.1.22  Binary relations   wbr 4204
            2.1.23  Ordered-pair class abstractions (class builders)   copab 4257
            2.1.24  Transitive classes   wtr 4294
      2.2  ZF Set Theory - add the Axiom of Replacement
            2.2.1  Introduce the Axiom of Replacement   ax-rep 4312
            2.2.2  Derive the Axiom of Separation   axsep 4321
            2.2.3  Derive the Null Set Axiom   zfnuleu 4327
            2.2.4  Theorems requiring subset and intersection existence   nalset 4332
            2.2.5  Theorems requiring empty set existence   class2set 4359
      2.3  ZF Set Theory - add the Axiom of Power Sets
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 4369
            2.3.2  Derive the Axiom of Pairing   zfpair 4393
            2.3.3  Ordered pair theorem   opnz 4424
            2.3.4  Ordered-pair class abstractions (cont.)   opabid 4453
            2.3.5  Power class of union and intersection   pwin 4479
            2.3.6  Epsilon and identity relations   cep 4484
            2.3.7  Partial and complete ordering   wpo 4493
            2.3.8  Founded and well-ordering relations   wfr 4530
            2.3.9  Ordinals   word 4572
      2.4  ZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 4693
            2.4.2  Ordinals (continued)   ordon 4755
            2.4.3  Transfinite induction   tfi 4825
            2.4.4  The natural numbers (i.e. finite ordinals)   com 4837
            2.4.5  Peano's postulates   peano1 4856
            2.4.6  Finite induction (for finite ordinals)   find 4862
            2.4.7  Relations   cxp 4868
            2.4.8  Definite description binder (inverted iota)   cio 5408
            2.4.9  Functions   wfun 5440
            2.4.10  Operations   co 6073
            2.4.11  "Maps to" notation   elmpt2cl 6280
            2.4.12  Function operation   cof 6295
            2.4.13  First and second members of an ordered pair   c1st 6339
            2.4.14  Special "Maps to" operations   mpt2xopn0yelv 6456
            2.4.15  Function transposition   ctpos 6470
            2.4.16  Curry and uncurry   ccur 6509
            2.4.17  Proper subset relation   crpss 6513
            2.4.18  Iota properties   fvopab5 6526
            2.4.19  Cantor's Theorem   canth 6531
            2.4.20  Undefined values and restricted iota (description binder)   cund 6533
            2.4.21  Functions on ordinals; strictly monotone ordinal functions   iunon 6592
            2.4.22  "Strong" transfinite recursion   crecs 6624
            2.4.23  Recursive definition generator   crdg 6659
            2.4.24  Finite recursion   frfnom 6684
            2.4.25  Abian's "most fundamental" fixed point theorem   abianfplem 6707
            2.4.26  Ordinal arithmetic   c1o 6709
            2.4.27  Natural number arithmetic   nna0 6839
            2.4.28  Equivalence relations and classes   wer 6894
            2.4.29  The mapping operation   cmap 7010
            2.4.30  Infinite Cartesian products   cixp 7055
            2.4.31  Equinumerosity   cen 7098
            2.4.32  Schroeder-Bernstein Theorem   sbthlem1 7209
            2.4.33  Equinumerosity (cont.)   xpf1o 7261
            2.4.34  Pigeonhole Principle   phplem1 7278
            2.4.35  Finite sets   onomeneq 7288
            2.4.36  Finite intersections   cfi 7407
            2.4.37  Hall's marriage theorem   marypha1lem 7430
            2.4.38  Supremum   csup 7437
            2.4.39  Ordinal isomorphism, Hartog's theorem   coi 7470
            2.4.40  Hartogs function, order types, weak dominance   char 7516
      2.5  ZF Set Theory - add the Axiom of Regularity
            2.5.1  Introduce the Axiom of Regularity   ax-reg 7552
            2.5.2  Axiom of Infinity equivalents   inf0 7568
      2.6  ZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-inf 7585
            2.6.2  Existence of omega (the set of natural numbers)   omex 7590
            2.6.3  Cantor normal form   ccnf 7608
            2.6.4  Transitive closure   trcl 7656
            2.6.5  Rank   cr1 7680
            2.6.6  Scott's trick; collection principle; Hilbert's epsilon   scottex 7801
            2.6.7  Cardinal numbers   ccrd 7814
            2.6.8  Axiom of Choice equivalents   wac 7988
            2.6.9  Cardinal number arithmetic   ccda 8039
            2.6.10  The Ackermann bijection   ackbij2lem1 8091
            2.6.11  Cofinality (without Axiom of Choice)   cflem 8118
            2.6.12  Eight inequivalent definitions of finite set   sornom 8149
            2.6.13  Hereditarily size-limited sets without Choice   itunifval 8288
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 8331
            3.2.2  AC equivalents: well-ordering, Zorn's lemma   numthcor 8366
            3.2.3  Cardinal number theorems using Axiom of Choice   cardval 8413
            3.2.4  Cardinal number arithmetic using Axiom of Choice   iunctb 8441
            3.2.5  Cofinality using Axiom of Choice   alephreg 8449
      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 8549
            4.1.2  Weak universes   cwun 8567
            4.1.3  Tarski's classes   ctsk 8615
            4.1.4  Grothendieck's universes   cgru 8657
      4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
            4.2.1  Introduce the Tarski-Grothendieck Axiom   ax-groth 8690
            4.2.2  Derive the Power Set, Infinity and Choice Axioms   grothpw 8693
            4.2.3  Tarski map function   ctskm 8704
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 8711
            5.1.2  Final derivation of real and complex number postulates   axaddf 9012
            5.1.3  Real and complex number postulates restated as axioms   ax-cnex 9038
      5.2  Derive the basic properties from the field axioms
            5.2.1  Some deductions from the field axioms for complex numbers   cnex 9063
            5.2.2  Infinity and the extended real number system   cpnf 9109
            5.2.3  Restate the ordering postulates with extended real "less than"   axlttri 9139
            5.2.4  Ordering on reals   lttr 9144
            5.2.5  Initial properties of the complex numbers   mul12 9224
      5.3  Real and complex numbers - basic operations
            5.3.1  Addition   add12 9271
            5.3.2  Subtraction   cmin 9283
            5.3.3  Multiplication   muladd 9458
            5.3.4  Ordering on reals (cont.)   gt0ne0 9485
            5.3.5  Reciprocals   ixi 9643
            5.3.6  Division   cdiv 9669
            5.3.7  Ordering on reals (cont.)   elimgt0 9838
            5.3.8  Completeness Axiom and Suprema   fimaxre 9947
            5.3.9  Imaginary and complex number properties   inelr 9982
            5.3.10  Function operation analogue theorems   ofsubeq0 9989
      5.4  Integer sets
            5.4.1  Natural numbers (as a subset of complex numbers)   cn 9992
            5.4.2  Principle of mathematical induction   nnind 10010
            5.4.3  Decimal representation of numbers   c2 10041
            5.4.4  Some properties of specific numbers   0p1e1 10085
            5.4.5  The Archimedean property   nnunb 10209
            5.4.6  Nonnegative integers (as a subset of complex numbers)   cn0 10213
            5.4.7  Integers (as a subset of complex numbers)   cz 10274
            5.4.8  Decimal arithmetic   cdc 10374
            5.4.9  Upper partititions of integers   cuz 10480
            5.4.10  Well-ordering principle for bounded-below sets of integers   uzwo3 10561
            5.4.11  Rational numbers (as a subset of complex numbers)   cq 10566
            5.4.12  Existence of the set of complex numbers   rpnnen1lem1 10592
      5.5  Order sets
            5.5.1  Positive reals (as a subset of complex numbers)   crp 10604
            5.5.2  Infinity and the extended real number system (cont.)   cxne 10699
            5.5.3  Supremum on the extended reals   xrsupexmnf 10875
            5.5.4  Real number intervals   cioo 10908
            5.5.5  Finite intervals of integers   cfz 11035
            5.5.6  Half-open integer ranges   cfzo 11127
      5.6  Elementary integer functions
            5.6.1  The floor (greatest integer) function   cfl 11193
            5.6.2  The modulo (remainder) operation   cmo 11242
            5.6.3  The infinite sequence builder "seq"   om2uz0i 11279
            5.6.4  Integer powers   cexp 11374
            5.6.5  Ordered pair theorem for nonnegative integers   nn0le2msqi 11552
            5.6.6  Factorial function   cfa 11558
            5.6.7  The binomial coefficient operation   cbc 11585
            5.6.8  The ` # ` (finite set size) function   chash 11610
                  5.6.8.1  Finite induction on the size of the first component of a binary relation   brfi1indlem 11706
            5.6.9  Words over a set   cword 11709
            5.6.10  Longer string literals   cs2 11797
      5.7  Elementary real and complex functions
            5.7.1  The "shift" operation   cshi 11873
            5.7.2  Real and imaginary parts; conjugate   ccj 11893
            5.7.3  Square root; absolute value   csqr 12030
      5.8  Elementary limits and convergence
            5.8.1  Superior limit (lim sup)   clsp 12256
            5.8.2  Limits   cli 12270
            5.8.3  Finite and infinite sums   csu 12471
            5.8.4  The binomial theorem   binomlem 12600
            5.8.5  The inclusion/exclusion principle   incexclem 12608
            5.8.6  Infinite sums (cont.)   isumshft 12611
            5.8.7  Miscellaneous converging and diverging sequences   divrcnv 12624
            5.8.8  Arithmetic series   arisum 12631
            5.8.9  Geometric series   expcnv 12635
            5.8.10  Ratio test for infinite series convergence   cvgrat 12652
            5.8.11  Mertens' theorem   mertenslem1 12653
      5.9  Elementary trigonometry
            5.9.1  The exponential, sine, and cosine functions   ce 12656
            5.9.2  _e is irrational   eirrlem 12795
      5.10  Cardinality of real and complex number subsets
            5.10.1  Countability of integers and rationals   xpnnen 12800
            5.10.2  The reals are uncountable   rpnnen2lem1 12806
PART 6  ELEMENTARY NUMBER THEORY
      6.1  Elementary properties of divisibility
            6.1.1  Irrationality of square root of 2   sqr2irrlem 12839
            6.1.2  Some Number sets are chains of proper subsets   nthruc 12842
            6.1.3  The divides relation   cdivides 12844
            6.1.4  The division algorithm   divalglem0 12905
            6.1.5  Bit sequences   cbits 12923
            6.1.6  The greatest common divisor operator   cgcd 12998
            6.1.7  Bézout's identity   bezoutlem1 13030
            6.1.8  Algorithms   nn0seqcvgd 13053
            6.1.9  Euclid's Algorithm   eucalgval2 13064
      6.2  Elementary prime number theory
            6.2.1  Elementary properties   cprime 13071
            6.2.2  Properties of the canonical representation of a rational   cnumer 13117
            6.2.3  Euler's theorem   codz 13144
            6.2.4  Pythagorean Triples   coprimeprodsq 13175
            6.2.5  The prime count function   cpc 13202
            6.2.6  Pocklington's theorem   prmpwdvds 13264
            6.2.7  Infinite primes theorem   unbenlem 13268
            6.2.8  Sum of prime reciprocals   prmreclem1 13276
            6.2.9  Fundamental theorem of arithmetic   1arithlem1 13283
            6.2.10  Lagrange's four-square theorem   cgz 13289
            6.2.11  Van der Waerden's theorem   cvdwa 13325
            6.2.12  Ramsey's theorem   cram 13359
            6.2.13  Decimal arithmetic (cont.)   dec2dvds 13391
            6.2.14  Specific prime numbers   4nprm 13419
            6.2.15  Very large primes   1259lem1 13442
PART 7  BASIC STRUCTURES
      7.1  Extensible structures
            7.1.1  Basic definitions   cstr 13457
            7.1.2  Slot definitions   cplusg 13521
            7.1.3  Definition of the structure product   crest 13640
            7.1.4  Definition of the structure quotient   cordt 13713
      7.2  Moore spaces
            7.2.1  Moore closures   mrcflem 13823
            7.2.2  Independent sets in a Moore system   mrisval 13847
            7.2.3  Algebraic closure systems   isacs 13868
PART 8  BASIC CATEGORY THEORY
      8.1  Categories
            8.1.1  Categories   ccat 13881
            8.1.2  Opposite category   coppc 13929
            8.1.3  Monomorphisms and epimorphisms   cmon 13946
            8.1.4  Sections, inverses, isomorphisms   csect 13962
            8.1.5  Subcategories   cssc 13999
            8.1.6  Functors   cfunc 14043
            8.1.7  Full & faithful functors   cful 14091
            8.1.8  Natural transformations and the functor category   cnat 14130
      8.2  Arrows (disjointified hom-sets)
            8.2.1  Identity and composition for arrows   cida 14200
      8.3  Examples of categories
            8.3.1  The category of sets   csetc 14222
            8.3.2  The category of categories   ccatc 14241
      8.4  Categorical constructions
            8.4.1  Product of categories   cxpc 14257
            8.4.2  Functor evaluation   cevlf 14298
            8.4.3  Hom functor   chof 14337
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 14389
            9.2.2  Lattices   clat 14466
            9.2.3  The dual of an ordered set   codu 14547
            9.2.4  Subset order structures   cipo 14569
            9.2.5  Distributive lattices   latmass 14606
            9.2.6  Posets and lattices as relations   cps 14616
            9.2.7  Directed sets, nets   cdir 14665
PART 10  BASIC ALGEBRAIC STRUCTURES
      10.1  Monoids
            10.1.1  Definition and basic properties   cmnd 14676
            10.1.2  Monoid homomorphisms and submonoids   cmhm 14728
            10.1.3  Ordered group sum operation   gsumvallem1 14763
            10.1.4  Free monoids   cfrmd 14784
      10.2  Groups
            10.2.1  Definition and basic properties   df-grp 14804
            10.2.2  Subgroups and Quotient groups   csubg 14930
            10.2.3  Elementary theory of group homomorphisms   cghm 14995
            10.2.4  Isomorphisms of groups   cgim 15036
            10.2.5  Group actions   cga 15058
            10.2.6  Symmetry groups and Cayley's Theorem   csymg 15084
            10.2.7  Centralizers and centers   ccntz 15106
            10.2.8  The opposite group   coppg 15133
            10.2.9  p-Groups and Sylow groups; Sylow's theorems   cod 15155
            10.2.10  Direct products   clsm 15260
            10.2.11  Free groups   cefg 15330
      10.3  Abelian groups
            10.3.1  Definition and basic properties   ccmn 15404
            10.3.2  Cyclic groups   ccyg 15479
            10.3.3  Group sum operation   gsumval3a 15504
            10.3.4  Internal direct products   cdprd 15546
            10.3.5  The Fundamental Theorem of Abelian Groups   ablfacrplem 15615
      10.4  Rings
            10.4.1  Multiplicative Group   cmgp 15640
            10.4.2  Definition and basic properties   crg 15652
            10.4.3  Opposite ring   coppr 15719
            10.4.4  Divisibility   cdsr 15735
            10.4.5  Ring homomorphisms   crh 15809
      10.5  Division rings and fields
            10.5.1  Definition and basic properties   cdr 15827
            10.5.2  Subrings of a ring   csubrg 15856
            10.5.3  Absolute value (abstract algebra)   cabv 15896
            10.5.4  Star rings   cstf 15923
      10.6  Left modules
            10.6.1  Definition and basic properties   clmod 15942
            10.6.2  Subspaces and spans in a left module   clss 16000
            10.6.3  Homomorphisms and isomorphisms of left modules   clmhm 16087
            10.6.4  Subspace sum; bases for a left module   clbs 16138
      10.7  Vector spaces
            10.7.1  Definition and basic properties   clvec 16166
      10.8  Ideals
            10.8.1  The subring algebra; ideals   csra 16232
            10.8.2  Two-sided ideals and quotient rings   c2idl 16294
            10.8.3  Principal ideal rings. Divisibility in the integers   clpidl 16304
            10.8.4  Nonzero rings   cnzr 16320
            10.8.5  Left regular elements. More kinds of rings   crlreg 16331
      10.9  Associative algebras
            10.9.1  Definition and basic properties   casa 16361
      10.10  Abstract multivariate polynomials
            10.10.1  Definition and basic properties   cmps 16398
            10.10.2  Polynomial evaluation   evlslem4 16556
            10.10.3  Univariate polynomials   cps1 16561
      10.11  The complex numbers as an extensible structure
            10.11.1  Definition and basic properties   cpsmet 16677
            10.11.2  Algebraic constructions based on the complexes   czrh 16770
      10.12  Hilbert spaces
            10.12.1  Definition and basic properties   cphl 16847
            10.12.2  Orthocomplements and closed subspaces   cocv 16879
            10.12.3  Orthogonal projection and orthonormal bases   cpj 16919
PART 11  BASIC TOPOLOGY
      11.1  Topology
            11.1.1  Topological spaces   ctop 16950
            11.1.2  TopBases for topologies   isbasisg 17004
            11.1.3  Examples of topologies   distop 17052
            11.1.4  Closure and interior   ccld 17072
            11.1.5  Neighborhoods   cnei 17153
            11.1.6  Limit points and perfect sets   clp 17190
            11.1.7  Subspace topologies   restrcl 17213
            11.1.8  Order topology   ordtbaslem 17244
            11.1.9  Limits and continuity in topological spaces   ccn 17280
            11.1.10  Separated spaces: T0, T1, T2 (Hausdorff) ...   ct0 17362
            11.1.11  Compactness   ccmp 17441
            11.1.12  Bolzano-Weierstrass theorem   bwth 17465
            11.1.13  Connectedness   ccon 17466
            11.1.14  First- and second-countability   c1stc 17492
            11.1.15  Local topological properties   clly 17519
            11.1.16  Compactly generated spaces   ckgen 17557
            11.1.17  Product topologies   ctx 17584
            11.1.18  Continuous function-builders   cnmptid 17685
            11.1.19  Quotient maps and quotient topology   ckq 17717
            11.1.20  Homeomorphisms   chmeo 17777
      11.2  Filters and filter bases
            11.2.1  Filter bases   elmptrab 17851
            11.2.2  Filters   cfil 17869
            11.2.3  Ultrafilters   cufil 17923
            11.2.4  Filter limits   cfm 17957
            11.2.5  Extension by continuity   ccnext 18082
            11.2.6  Topological groups   ctmd 18092
            11.2.7  Infinite group sum on topological groups   ctsu 18147
            11.2.8  Topological rings, fields, vector spaces   ctrg 18177
      11.3  Uniform Stuctures and Spaces
            11.3.1  Uniform structures   cust 18221
            11.3.2  The topology induced by an uniform structure   cutop 18252
            11.3.3  Uniform Spaces   cuss 18275
            11.3.4  Uniform continuity   cucn 18297
            11.3.5  Cauchy filters in uniform spaces   ccfilu 18308
            11.3.6  Complete uniform spaces   ccusp 18319
      11.4  Metric spaces
            11.4.1  Pseudometric spaces   ispsmet 18327
            11.4.2  Basic metric space properties   cxme 18339
            11.4.3  Metric space balls   blfvalps 18405
            11.4.4  Open sets of a metric space   mopnval 18460
            11.4.5  Continuity in metric spaces   metcnp3 18562
            11.4.6  The uniform structure generated by a metric   metuvalOLD 18571
            11.4.7  Examples of metric spaces   dscmet 18612
            11.4.8  Normed algebraic structures   cnm 18616
            11.4.9  Normed space homomorphisms (bounded linear operators)   cnmo 18731
            11.4.10  Topology on the reals   qtopbaslem 18784
            11.4.11  Topological definitions using the reals   cii 18897
            11.4.12  Path homotopy   chtpy 18984
            11.4.13  The fundamental group   cpco 19017
      11.5  Complex metric vector spaces
            11.5.1  Complex left modules   cclm 19079
            11.5.2  Complex pre-Hilbert space   ccph 19121
            11.5.3  Convergence and completeness   ccfil 19197
            11.5.4  Baire's Category Theorem   bcthlem1 19269
            11.5.5  Banach spaces and complex Hilbert spaces   ccms 19277
            11.5.6  Minimizing Vector Theorem   minveclem1 19317
            11.5.7  Projection Theorem   pjthlem1 19330
PART 12  BASIC REAL AND COMPLEX ANALYSIS
      12.1  Continuity
            12.1.1  Intermediate value theorem   pmltpclem1 19337
      12.2  Integrals
            12.2.1  Lebesgue measure   covol 19351
            12.2.2  Lebesgue integration   cmbf 19498
      12.3  Derivatives
            12.3.1  Real and complex differentiation   climc 19741
PART 13  BASIC REAL AND COMPLEX FUNCTIONS
      13.1  Polynomials
            13.1.1  Abstract polynomials, continued   evlslem6 19926
            13.1.2  Polynomial degrees   cmdg 19968
            13.1.3  The division algorithm for univariate polynomials   cmn1 20040
            13.1.4  Elementary properties of complex polynomials   cply 20095
            13.1.5  The division algorithm for polynomials   cquot 20199
            13.1.6  Algebraic numbers   caa 20223
            13.1.7  Liouville's approximation theorem   aalioulem1 20241
      13.2  Sequences and series
            13.2.1  Taylor polynomials and Taylor's theorem   ctayl 20261
            13.2.2  Uniform convergence   culm 20284
            13.2.3  Power series   pserval 20318
      13.3  Basic trigonometry
            13.3.1  The exponential, sine, and cosine functions (cont.)   efcn 20351
            13.3.2  Properties of pi = 3.14159...   pilem1 20359
            13.3.3  Mapping of the exponential function   efgh 20435
            13.3.4  The natural logarithm on complex numbers   clog 20444
            13.3.5  Theorems of Pythagoras, isosceles triangles, and intersecting chords   angval 20635
            13.3.6  Solutions of quadratic, cubic, and quartic equations   quad2 20671
            13.3.7  Inverse trigonometric functions   casin 20694
            13.3.8  The Birthday Problem   log2ublem1 20778
            13.3.9  Areas in R^2   carea 20786
            13.3.10  More miscellaneous converging sequences   rlimcnp 20796
            13.3.11  Inequality of arithmetic and geometric means   cvxcl 20815
            13.3.12  Euler-Mascheroni constant   cem 20822
      13.4  Basic number theory
            13.4.1  Wilson's theorem   wilthlem1 20843
            13.4.2  The Fundamental Theorem of Algebra   ftalem1 20847
            13.4.3  The Basel problem (ζ(2) = π2/6)   basellem1 20855
            13.4.4  Number-theoretical functions   ccht 20865
            13.4.5  Perfect Number Theorem   mersenne 21003
            13.4.6  Characters of Z/nZ   cdchr 21008
            13.4.7  Bertrand's postulate   bcctr 21051
            13.4.8  Legendre symbol   clgs 21070
            13.4.9  Quadratic reciprocity   lgseisenlem1 21125
            13.4.10  All primes 4n+1 are the sum of two squares   2sqlem1 21139
            13.4.11  Chebyshev's Weak Prime Number Theorem, Dirichlet's Theorem   chebbnd1lem1 21155
            13.4.12  The Prime Number Theorem   mudivsum 21216
            13.4.13  Ostrowski's theorem   abvcxp 21301
PART 14  GRAPH THEORY
      14.1  Undirected graphs - basics
            14.1.1  Undirected hypergraphs   cuhg 21326
            14.1.2  Undirected multigraphs   cumg 21339
            14.1.3  Undirected simple graphs   cuslg 21356
                  14.1.3.1  Undirected simple graphs - basics   cuslg 21356
                  14.1.3.2  Undirected simple graphs - examples   usgraexvlem 21406
                  14.1.3.3  Finite undirected simple graphs   fiusgraedgfi 21413
            14.1.4  Neighbors, complete graphs and universal vertices   cnbgra 21422
                  14.1.4.1  Neighbors   nbgraop 21428
                  14.1.4.2  Complete graphs   iscusgra 21457
                  14.1.4.3  Universal vertices   isuvtx 21489
            14.1.5  Walks, paths and cycles   cwalk 21498
                  14.1.5.1  Walks and trails   wlks 21518
                  14.1.5.2  Paths and simple paths   pths 21558
                  14.1.5.3  Circuits and cycles   crcts 21601
                  14.1.5.4  Connected graphs   cconngra 21648
            14.1.6  Vertex Degree   cvdg 21656
      14.2  Eulerian paths and the Konigsberg Bridge problem
            14.2.1  Eulerian paths   ceup 21676
            14.2.2  The Konigsberg Bridge problem   vdeg0i 21696
PART 15  GUIDES AND MISCELLANEA
      15.1  Guides (conventions, explanations, and examples)
            15.1.1  Conventions   conventions 21702
            15.1.2  Natural deduction   natded 21703
            15.1.3  Natural deduction examples   ex-natded5.2 21704
            15.1.4  Definitional examples   ex-or 21721
      15.2  Humor
            15.2.1  April Fool's theorem   avril1 21749
      15.3  (Future - to be reviewed and classified)
            15.3.1  Planar incidence geometry   cplig 21755
            15.3.2  Algebra preliminaries   crpm 21760
            15.3.3  Transitive closure   ctcl 21762
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 21766
            16.1.2  Definition and basic properties of Abelian groups   cablo 21861
            16.1.3  Subgroups   csubgo 21881
            16.1.4  Operation properties   cass 21892
            16.1.5  Group-like structures   cmagm 21898
            16.1.6  Examples of Abelian groups   ablosn 21927
            16.1.7  Group homomorphism and isomorphism   cghom 21937
      16.2  Additional material on rings and fields
            16.2.1  Definition and basic properties   crngo 21955
            16.2.2  Examples of rings   cnrngo 21983
            16.2.3  Division Rings   cdrng 21985
            16.2.4  Star Fields   csfld 21988
            16.2.5  Fields and Rings   ccm2 21990
PART 17  COMPLEX TOPOLOGICAL VECTOR SPACES (DEPRECATED)
      17.1  Complex vector spaces
            17.1.1  Definition and basic properties   cvc 22016
            17.1.2  Examples of complex vector spaces   cncvc 22054
      17.2  Normed complex vector spaces
            17.2.1  Definition and basic properties   cnv 22055
            17.2.2  Examples of normed complex vector spaces   cnnv 22160
            17.2.3  Induced metric of a normed complex vector space   imsval 22169
            17.2.4  Inner product   cdip 22188
            17.2.5  Subspaces   css 22212
      17.3  Operators on complex vector spaces
            17.3.1  Definitions and basic properties   clno 22233
      17.4  Inner product (pre-Hilbert) spaces
            17.4.1  Definition and basic properties   ccphlo 22305
            17.4.2  Examples of pre-Hilbert spaces   cncph 22312
            17.4.3  Properties of pre-Hilbert spaces   isph 22315
      17.5  Complex Banach spaces
            17.5.1  Definition and basic properties   ccbn 22356
            17.5.2  Examples of complex Banach spaces   cnbn 22363
            17.5.3  Uniform Boundedness Theorem   ubthlem1 22364
            17.5.4  Minimizing Vector Theorem   minvecolem1 22368
      17.6  Complex Hilbert spaces
            17.6.1  Definition and basic properties   chlo 22379
            17.6.2  Standard axioms for a complex Hilbert space   hlex 22392
            17.6.3  Examples of complex Hilbert spaces   cnchl 22410
            17.6.4  Subspaces   ssphl 22411
            17.6.5  Hellinger-Toeplitz Theorem   htthlem 22412
PART 18  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      18.1  Axiomatization of complex pre-Hilbert spaces
            18.1.1  Basic Hilbert space definitions   chil 22414
            18.1.2  Preliminary ZFC lemmas   df-hnorm 22463
            18.1.3  Derive the Hilbert space axioms from ZFC set theory   axhilex-zf 22476
            18.1.4  Introduce the vector space axioms for a Hilbert space   ax-hilex 22494
            18.1.5  Vector operations   hvmulex 22506
            18.1.6  Inner product postulates for a Hilbert space   ax-hfi 22573
      18.2  Inner product and norms
            18.2.1  Inner product   his5 22580
            18.2.2  Norms   dfhnorm2 22616
            18.2.3  Relate Hilbert space to normed complex vector spaces   hilablo 22654
            18.2.4  Bunjakovaskij-Cauchy-Schwarz inequality   bcsiALT 22673
      18.3  Cauchy sequences and completeness axiom
            18.3.1  Cauchy sequences and limits   hcau 22678
            18.3.2  Derivation of the completeness axiom from ZF set theory   hilmet 22688
            18.3.3  Completeness postulate for a Hilbert space   ax-hcompl 22696
            18.3.4  Relate Hilbert space to ZFC pre-Hilbert and Hilbert spaces   hhcms 22697
      18.4  Subspaces and projections
            18.4.1  Subspaces   df-sh 22701
            18.4.2  Closed subspaces   df-ch 22716
            18.4.3  Orthocomplements   df-oc 22746
            18.4.4  Subspace sum, span, lattice join, lattice supremum   df-shs 22802
            18.4.5  Projection theorem   pjhthlem1 22885
            18.4.6  Projectors   df-pjh 22889
      18.5  Properties of Hilbert subspaces
            18.5.1  Orthomodular law   omlsilem 22896
            18.5.2  Projectors (cont.)   pjhtheu2 22910
            18.5.3  Hilbert lattice operations   sh0le 22934
            18.5.4  Span (cont.) and one-dimensional subspaces   spansn0 23035
            18.5.5  Commutes relation for Hilbert lattice elements   df-cm 23077
            18.5.6  Foulis-Holland theorem   fh1 23112
            18.5.7  Quantum Logic Explorer axioms   qlax1i 23121
            18.5.8  Orthogonal subspaces   chscllem1 23131
            18.5.9  Orthoarguesian laws 5OA and 3OA   5oalem1 23148
            18.5.10  Projectors (cont.)   pjorthi 23163
            18.5.11  Mayet's equation E_3   mayete3i 23222
      18.6  Operators on Hilbert spaces
            18.6.1  Operator sum, difference, and scalar multiplication   df-hosum 23225
            18.6.2  Zero and identity operators   df-h0op 23243
            18.6.3  Operations on Hilbert space operators   hoaddcl 23253
            18.6.4  Linear, continuous, bounded, Hermitian, unitary operators and norms   df-nmop 23334
            18.6.5  Linear and continuous functionals and norms   df-nmfn 23340
            18.6.6  Adjoint   df-adjh 23344
            18.6.7  Dirac bra-ket notation   df-bra 23345
            18.6.8  Positive operators   df-leop 23347
            18.6.9  Eigenvectors, eigenvalues, spectrum   df-eigvec 23348
            18.6.10  Theorems about operators and functionals   nmopval 23351
            18.6.11  Riesz lemma   riesz3i 23557
            18.6.12  Adjoints (cont.)   cnlnadjlem1 23562
            18.6.13  Quantum computation error bound theorem   unierri 23599
            18.6.14  Dirac bra-ket notation (cont.)   branmfn 23600
            18.6.15  Positive operators (cont.)   leopg 23617
            18.6.16  Projectors as operators   pjhmopi 23641
      18.7  States on a Hilbert lattice and Godowski's equation
            18.7.1  States on a Hilbert lattice   df-st 23706
            18.7.2  Godowski's equation   golem1 23766
      18.8  Cover relation, atoms, exchange axiom, and modular symmetry
            18.8.1  Covers relation; modular pairs   df-cv 23774
            18.8.2  Atoms   df-at 23833
            18.8.3  Superposition principle   superpos 23849
            18.8.4  Atoms, exchange and covering properties, atomicity   chcv1 23850
            18.8.5  Irreducibility   chirredlem1 23885
            18.8.6  Atoms (cont.)   atcvat3i 23891
            18.8.7  Modular symmetry   mdsymlem1 23898
PART 19  SUPPLEMENTARY MATERIAL (USER'S MATHBOXES)
      19.1  Mathboxes for user contributions
            19.1.1  Mathbox guidelines   mathbox 23937
      19.2  Mathbox for Stefan Allan
      19.3  Mathbox for Thierry Arnoux
            19.3.1  Propositional Calculus - misc additions   bian1d 23942
            19.3.2  Predicate Calculus   abeq2f 23952
                  19.3.2.1  Predicate Calculus - misc additions   abeq2f 23952
                  19.3.2.2  Restricted quantification - misc additions   reximddv 23954
                  19.3.2.3  Substitution (without distinct variables) - misc additions   clelsb3f 23963
                  19.3.2.4  Existential "at most one" - misc additions   mo5f 23964
                  19.3.2.5  Existential uniqueness - misc additions   2reuswap2 23967
                  19.3.2.6  Restricted "at most one" - misc additions   rmoxfrdOLD 23971
            19.3.3  General Set Theory   ceqsexv2d 23977
                  19.3.3.1  Class abstractions (a.k.a. class builders)   ceqsexv2d 23977
                  19.3.3.2  Image Sets   abrexdomjm 23980
                  19.3.3.3  Set relations and operations - misc additions   eqri 23986
                  19.3.3.4  Unordered pairs   elpreq 23991
                  19.3.3.5  Conditional operator - misc additions   ifeqeqx 23993
                  19.3.3.6  Indexed union - misc additions   iuneq12daf 23999
                  19.3.3.7  Disjointness - misc additions   cbvdisjf 24007
            19.3.4  Relations and Functions   dfrel4 24026
                  19.3.4.1  Relations - misc additions   dfrel4 24026
                  19.3.4.2  Functions - misc additions   fdmrn 24031
                  19.3.4.3  Isomorphisms - misc. add.   gtiso 24080
                  19.3.4.4  Disjointness (additional proof requiring functions)   disjdsct 24082
                  19.3.4.5  First and second members of an ordered pair - misc additions   df1stres 24083
                  19.3.4.6  Supremum - misc additions   supssd 24090
                  19.3.4.7  Countable Sets   nnct 24091
            19.3.5  Real and Complex Numbers   addeq0 24106
                  19.3.5.1  Complex addition - misc. additions   addeq0 24106
                  19.3.5.2  Ordering on reals - misc additions   lt2addrd 24107
                  19.3.5.3  Extended reals - misc additions   xgepnf 24108
                  19.3.5.4  Real number intervals - misc additions   icossicc 24121
                  19.3.5.5  Finite intervals of integers - misc additions   fzssnn 24139
                  19.3.5.6  Half-open integer ranges - misc additions   iundisjfi 24144
                  19.3.5.7  The ` # ` (finite set size) function - misc additions   hashresfn 24148
                  19.3.5.8  The greatest common divisor operator - misc. add   numdenneg 24152
                  19.3.5.9  Integers   ltesubnnd 24154
                  19.3.5.10  Division in the extended real number system   cxdiv 24155
            19.3.6  Structure builders   ress0g 24174
                  19.3.6.1  Structure builder restriction operator   ress0g 24174
                  19.3.6.2  Posets   tospos 24178
                  19.3.6.3  Complete lattices   clatp0ex 24185
                  19.3.6.4  Extended reals Structure - misc additions   ax-xrssca 24187
                  19.3.6.5  The extended non-negative real numbers monoid   xrge0base 24199
            19.3.7  Algebra   sumpr 24210
                  19.3.7.1  Finitely supported group sums - misc additions   sumpr 24210
                  19.3.7.2  Rings - misc additions   dvrdir 24218
                  19.3.7.3  Ordered groups   cogrp 24223
                  19.3.7.4  Ordered fields   cofld 24225
                  19.3.7.5  The Archimedean property for generic algebraic structures   cinftm 24238
                  19.3.7.6  Ring homomorphisms - misc additions   rhmdvdsr 24248
                  19.3.7.7  The ring of integers   zzsbase 24255
                  19.3.7.8  The ordered field of reals   rebase 24261
            19.3.8  Topology   cmetid 24273
                  19.3.8.1  Pseudometrics   cmetid 24273
                  19.3.8.2  Continuity - misc additions   hauseqcn 24285
                  19.3.8.3  Topology of the closed unit   unitsscn 24286
                  19.3.8.4  Topology of ` ( RR X. RR ) `   unicls 24293
                  19.3.8.5  Order topology - misc. additions   cnvordtrestixx 24303
                  19.3.8.6  Continuity in topological spaces - misc. additions   mndpluscn 24304
                  19.3.8.7  Topology of the extended non-negative real numbers monoid   xrge0hmph 24310
                  19.3.8.8  Limits - misc additions   lmlim 24325
            19.3.9  Uniform Stuctures and Spaces   chcmp 24332
                  19.3.9.1  Hausdorff Completion   chcmp 24332
            19.3.10  Topology and algebraic structures   zzsnm 24334
                  19.3.10.1  The norm on the ring of the integer numbers   zzsnm 24334
                  19.3.10.2  The complete ordered field of the real numbers   recms 24335
                  19.3.10.3  Topological ` ZZ ` -modules   zlm0 24338
                  19.3.10.4  The canonical embedding of the rational numbers into a division ring   cqqh 24348
                  19.3.10.5  The canonical embedding of ` RR ` into a complete ordered field   crrh 24369
                  19.3.10.6  Embedding into ` RR* `   cxrh 24374
                  19.3.10.7  Canonical embeddings into ` RR `   zrhre 24377
            19.3.11  Real and complex functions   clogb 24380
                  19.3.11.1  Logarithm laws generalized to an arbitrary base - logb   clogb 24380
                  19.3.11.2  Indicator Functions   cind 24400
                  19.3.11.3  Extended sum   cesum 24416
            19.3.12  Mixed Function/Constant operation   cofc 24470
            19.3.13  Abstract measure   csiga 24482
                  19.3.13.1  Sigma-Algebra   csiga 24482
                  19.3.13.2  Generated Sigma-Algebra   csigagen 24513
                  19.3.13.3  The Borel algebra on the real numbers   cbrsiga 24527
                  19.3.13.4  Product Sigma-Algebra   csx 24534
                  19.3.13.5  Measures   cmeas 24541
                  19.3.13.6  The counting measure   cntmeas 24572
                  19.3.13.7  The Lebesgue measure - misc additions   volss 24575
                  19.3.13.8  The 'almost everywhere' relation   cae 24580
                  19.3.13.9  Measurable functions   cmbfm 24592
                  19.3.13.10  Borel Algebra on ` ( RR X. RR ) `   br2base 24611
            19.3.14  Integration   itgeq12dv 24633
                  19.3.14.1  Lebesgue integral - misc additions   itgeq12dv 24633
                  19.3.14.2  Bochner integral   citgm 24634
            19.3.15  Probability   cprb 24657
                  19.3.15.1  Probability Theory   cprb 24657
                  19.3.15.2  Conditional Probabilities   ccprob 24681
                  19.3.15.3  Real Valued Random Variables   crrv 24690
                  19.3.15.4  Preimage set mapping operator   corvc 24705
                  19.3.15.5  Distribution Functions   orvcelval 24718
                  19.3.15.6  Cumulative Distribution Functions   orvclteel 24722
                  19.3.15.7  Probabilities - example   coinfliplem 24728
                  19.3.15.8  Bertrand's Ballot Problem   ballotlemoex 24735
      19.4  Mathbox for Mario Carneiro
            19.4.1  Miscellaneous stuff   quartfull 24788
            19.4.2  Zeta function   czeta 24789
            19.4.3  Gamma function   clgam 24792
            19.4.4  Derangements and the Subfactorial   deranglem 24844
            19.4.5  The Erdős-Szekeres theorem   erdszelem1 24869
            19.4.6  The Kuratowski closure-complement theorem   kur14lem1 24884
            19.4.7  Retracts and sections   cretr 24895
            19.4.8  Path-connected and simply connected spaces   cpcon 24898
            19.4.9  Covering maps   ccvm 24934
            19.4.10  Normal numbers   snmlff 25008
            19.4.11  Godel-sets of formulas   cgoe 25012
            19.4.12  Models of ZF   cgze 25040
            19.4.13  Splitting fields   citr 25054
            19.4.14  p-adic number fields   czr 25070
      19.5  Mathbox for Paul Chapman
            19.5.1  Group homomorphism and isomorphism   ghomgrpilem1 25088
            19.5.2  Real and complex numbers (cont.)   climuzcnv 25100
            19.5.3  Miscellaneous theorems   elfzm12 25104
      19.6  Mathbox for Drahflow
      19.7  Mathbox for Scott Fenton
            19.7.1  ZFC Axioms in primitive form   axextprim 25142
            19.7.2  Untangled classes   untelirr 25149
            19.7.3  Extra propositional calculus theorems   3orel1 25156
            19.7.4  Misc. Useful Theorems   nepss 25167
            19.7.5  Properties of reals and complexes   sqdivzi 25176
            19.7.6  Product sequences   prodf 25207
            19.7.7  Non-trivial convergence   ntrivcvg 25217
            19.7.8  Complex products   cprod 25223
            19.7.9  Finite products   fprod 25259
            19.7.10  Infinite products   iprodclim 25303
            19.7.11  Falling and Rising Factorial   cfallfac 25312
            19.7.12  Factorial limits   faclimlem1 25354
            19.7.13  Greatest common divisor and divisibility   pdivsq 25360
            19.7.14  Properties of relationships   brtp 25364
            19.7.15  Properties of functions and mappings   funpsstri 25381
            19.7.16  Epsilon induction   setinds 25397
            19.7.17  Ordinal numbers   elpotr 25400
            19.7.18  Defined equality axioms   axextdfeq 25417
            19.7.19  Hypothesis builders   hbntg 25425
            19.7.20  The Predecessor Class   cpred 25430
            19.7.21  (Trans)finite Recursion Theorems   tfisg 25471
            19.7.22  Well-founded induction   tz6.26 25472
            19.7.23  Transitive closure under a relationship   ctrpred 25487
            19.7.24  Founded Induction   frmin 25509
            19.7.25  Ordering Ordinal Sequences   orderseqlem 25519
            19.7.26  Well-founded recursion   cwrecs 25522
            19.7.27  Transfinite recursion via Well-founded recursion   tfrALTlem 25549
            19.7.28  Well-founded zero, successor, and limits   cwsuc 25553
            19.7.29  Founded Recursion   frr3g 25573
            19.7.30  Surreal Numbers   csur 25587
            19.7.31  Surreal Numbers: Ordering   sltsolem1 25615
            19.7.32  Surreal Numbers: Birthday Function   bdayfo 25622
            19.7.33  Surreal Numbers: Density   fvnobday 25629
            19.7.34  Surreal Numbers: Density   nodenselem3 25630
            19.7.35  Surreal Numbers: Upper and Lower Bounds   nobndlem1 25639
            19.7.36  Surreal Numbers: Full-Eta Property   nofulllem1 25649
            19.7.37  Symmetric difference   csymdif 25654
            19.7.38  Quantifier-free definitions   ctxp 25666
            19.7.39  Alternate ordered pairs   caltop 25793
            19.7.40  Tarskian geometry   cee 25819
            19.7.41  Tarski's axioms for geometry   axdimuniq 25844
            19.7.42  Congruence properties   cofs 25908
            19.7.43  Betweenness properties   btwntriv2 25938
            19.7.44  Segment Transportation   ctransport 25955
            19.7.45  Properties relating betweenness and congruence   cifs 25961
            19.7.46  Connectivity of betweenness   btwnconn1lem1 26013
            19.7.47  Segment less than or equal to   csegle 26032
            19.7.48  Outside of relationship   coutsideof 26045
            19.7.49  Lines and Rays   cline2 26060
            19.7.50  Bernoulli polynomials and sums of k-th powers   cbp 26084
            19.7.51  Rank theorems   rankung 26099
            19.7.52  Hereditarily Finite Sets   chf 26105
      19.8  Mathbox for Anthony Hart
            19.8.1  Propositional Calculus   tb-ax1 26120
            19.8.2  Predicate Calculus   quantriv 26142
            19.8.3  Misc. Single Axiom Systems   meran1 26153
            19.8.4  Connective Symmetry   negsym1 26159
      19.9  Mathbox for Chen-Pang He
            19.9.1  Ordinal topology   ontopbas 26170
      19.10  Mathbox for Jeff Hoffman
            19.10.1  Inferences for finite induction on generic function values   fveleq 26193
            19.10.2  gdc.mm   nnssi2 26197
      19.11  Mathbox for Wolf Lammen
      19.12  Mathbox for Brendan Leahy
      19.13  Mathbox for Jeff Hankins
            19.13.1  Miscellany   a1i13 26289
            19.13.2  Basic topological facts   topbnd 26318
            19.13.3  Topology of the real numbers   ivthALT 26329
            19.13.4  Refinements   cfne 26330
            19.13.5  Neighborhood bases determine topologies   neibastop1 26379
            19.13.6  Lattice structure of topologies   topmtcl 26383
            19.13.7  Filter bases   fgmin 26390
            19.13.8  Directed sets, nets   tailfval 26392
      19.14  Mathbox for Jeff Madsen
            19.14.1  Logic and set theory   anim12da 26403
            19.14.2  Real and complex numbers; integers   filbcmb 26433
            19.14.3  Sequences and sums   sdclem2 26437
            19.14.4  Topology   subspopn 26449
            19.14.5  Metric spaces   metf1o 26452
            19.14.6  Continuous maps and homeomorphisms   constcncf 26459
            19.14.7  Boundedness   ctotbnd 26466
            19.14.8  Isometries   cismty 26498
            19.14.9  Heine-Borel Theorem   heibor1lem 26509
            19.14.10  Banach Fixed Point Theorem   bfplem1 26522
            19.14.11  Euclidean space   crrn 26525
            19.14.12  Intervals (continued)   ismrer1 26538
            19.14.13  Groups and related structures   exidcl 26542
            19.14.14  Rings   rngonegcl 26552
            19.14.15  Ring homomorphisms   crnghom 26567
            19.14.16  Commutative rings   ccring 26596
            19.14.17  Ideals   cidl 26608
            19.14.18  Prime rings and integral domains   cprrng 26647
            19.14.19  Ideal generators   cigen 26660
      19.15  Mathbox for Rodolfo Medina
            19.15.1  Partitions   prtlem60 26679
      19.16  Mathbox for Stefan O'Rear
            19.16.1  Additional elementary logic and set theory   nelss 26723
            19.16.2  Additional theory of functions   fninfp 26726
            19.16.3  Extensions beyond function theory   gsumvsmul 26736
            19.16.4  Additional topology   elrfi 26739
            19.16.5  Characterization of closure operators. Kuratowski closure axioms   ismrcd1 26743
            19.16.6  Algebraic closure systems   cnacs 26747
            19.16.7  Miscellanea 1. Map utilities   constmap 26758
            19.16.8  Miscellanea for polynomials   ofmpteq 26767
            19.16.9  Multivariate polynomials over the integers   cmzpcl 26769
            19.16.10  Miscellanea for Diophantine sets 1   coeq0 26801
            19.16.11  Diophantine sets 1: definitions   cdioph 26804
            19.16.12  Diophantine sets 2 miscellanea   ellz1 26816
            19.16.13  Diophantine sets 2: union and intersection. Monotone Boolean algebra   diophin 26822
            19.16.14  Diophantine sets 3: construction   diophrex 26825
            19.16.15  Diophantine sets 4 miscellanea   2sbcrex 26834
            19.16.16  Diophantine sets 4: Quantification   rexrabdioph 26845
            19.16.17  Diophantine sets 5: Arithmetic sets   rabdiophlem1 26852
            19.16.18  Diophantine sets 6 miscellanea   fz1ssnn 26862
            19.16.19  Diophantine sets 6: reusability. renumbering of variables   eldioph4b 26863
            19.16.20  Pigeonhole Principle and cardinality helpers   fphpd 26868
            19.16.21  A non-closed set of reals is infinite   rencldnfilem 26872
            19.16.22  Miscellanea for Lagrange's theorem   icodiamlt 26874
            19.16.23  Lagrange's rational approximation theorem   irrapxlem1 26876
            19.16.24  Pell equations 1: A nontrivial solution always exists   pellexlem1 26883
            19.16.25  Pell equations 2: Algebraic number theory of the solution set   csquarenn 26890
            19.16.26  Pell equations 3: characterizing fundamental solution   infmrgelbi 26932
            19.16.27  Logarithm laws generalized to an arbitrary base   reglogcl 26944
            19.16.28  Pell equations 4: the positive solution group is infinite cyclic   pellfund14 26952
            19.16.29  X and Y sequences 1: Definition and recurrence laws   crmx 26954
            19.16.30  Ordering and induction lemmas for the integers   monotuz 26995
            19.16.31  X and Y sequences 2: Order properties   rmxypos 27003
            19.16.32  Congruential equations   congtr 27021
            19.16.33  Alternating congruential equations   acongid 27031
            19.16.34  Additional theorems on integer divisibility   bezoutr 27041
            19.16.35  X and Y sequences 3: Divisibility properties   jm2.18 27050
            19.16.36  X and Y sequences 4: Diophantine representability of Y   jm2.27a 27067
            19.16.37  X and Y sequences 5: Diophantine representability of X, ^, _C   rmxdiophlem 27077
            19.16.38  Uncategorized stuff not associated with a major project   setindtr 27086
            19.16.39  More equivalents of the Axiom of Choice   axac10 27095
            19.16.40  Finitely generated left modules   clfig 27133
            19.16.41  Noetherian left modules I   clnm 27141
            19.16.42  Addenda for structure powers   pwssplit0 27155
            19.16.43  Direct sum of left modules   cdsmm 27165
            19.16.44  Free modules   cfrlm 27180
            19.16.45  Every set admits a group structure iff choice   unxpwdom3 27224
            19.16.46  Independent sets and families   clindf 27242
            19.16.47  Characterization of free modules   lmimlbs 27274
            19.16.48  Noetherian rings and left modules II   clnr 27281
            19.16.49  Hilbert's Basis Theorem   cldgis 27293
            19.16.50  Additional material on polynomials [DEPRECATED]   cmnc 27303
            19.16.51  Degree and minimal polynomial of algebraic numbers   cdgraa 27313
            19.16.52  Algebraic integers I   citgo 27330
            19.16.53  Finite cardinality [SO]   en1uniel 27348
            19.16.54  Words in monoids and ordered group sum   issubmd 27351
            19.16.55  Transpositions in the symmetric group   cpmtr 27352
            19.16.56  The sign of a permutation   cpsgn 27382
            19.16.57  The matrix algebra   cmmul 27407
            19.16.58  The determinant   cmdat 27451
            19.16.59  Endomorphism algebra   cmend 27457
            19.16.60  Subfields   csdrg 27471
            19.16.61  Cyclic groups and order   idomrootle 27479
            19.16.62  Cyclotomic polynomials   ccytp 27489
            19.16.63  Miscellaneous topology   fgraphopab 27497
      19.17  Mathbox for Steve Rodriguez
            19.17.1  Miscellanea   iso0 27504
            19.17.2  Function operations   caofcan 27508
            19.17.3  Calculus   lhe4.4ex1a 27514
      19.18  Mathbox for Andrew Salmon
            19.18.1  Principia Mathematica * 10   pm10.12 27521
            19.18.2  Principia Mathematica * 11   2alanimi 27535
            19.18.3  Predicate Calculus   sbeqal1 27565
            19.18.4  Principia Mathematica * 13 and * 14   pm13.13a 27575
            19.18.5  Set Theory   elnev 27606
            19.18.6  Arithmetic   addcomgi 27628
            19.18.7  Geometry   cplusr 27629
      19.19  Mathbox for Glauco Siliprandi
            19.19.1  Miscellanea   ssrexf 27651
            19.19.2  Finite multiplication of numbers and finite multiplication of functions   fmul01 27677
            19.19.3  Limits   clim1fr1 27694
            19.19.4  Derivatives   dvsinexp 27707
            19.19.5  Integrals   ioovolcl 27709
            19.19.6  Stone Weierstrass theorem - real version   stoweidlem1 27717
            19.19.7  Wallis' product for π   wallispilem1 27781
            19.19.8  Stirling's approximation formula for ` n ` factorial   stirlinglem1 27790
      19.20  Mathbox for Saveliy Skresanov
            19.20.1  Ceva's theorem   sigarval 27807
      19.21  Mathbox for Jarvin Udandy
      19.22  Mathbox for Alexander van der Vekens
            19.22.1  Double restricted existential uniqueness   r19.32 27912
                  19.22.1.1  Restricted quantification (extension)   r19.32 27912
                  19.22.1.2  The empty set (extension)   raaan2 27920
                  19.22.1.3  Restricted uniqueness and "at most one" quantification   rmoimi 27921
                  19.22.1.4  Analogs to Existential uniqueness (double quantification)   2reurex 27926
            19.22.2  Alternative definitions of function's and operation's values   wdfat 27938
                  19.22.2.1  Restricted quantification (extension)   ralbinrald 27944
                  19.22.2.2  The universal class (extension)   nvelim 27945
                  19.22.2.3  Introduce the Axiom of Power Sets (extension)   alneu 27946
                  19.22.2.4  Relations (extension)   sbcrel 27948
                  19.22.2.5  Functions (extension)   sbcfun 27954
                  19.22.2.6  Predicate "defined at"   dfateq12d 27960
                  19.22.2.7  Alternative definition of the value of a function   dfafv2 27963
                  19.22.2.8  Alternative definition of the value of an operation   aoveq123d 28009
            19.22.3  Auxiliary theorems for graph theory   jaoi3 28039
                  19.22.3.1  Logical disjunction and conjunction   jaoi3 28039
                  19.22.3.2  Negated equality and membership - extension   eqneqall 28040
                  19.22.3.3  "Weak deduction theorem" for set theory - extension   ifeqda 28042
                  19.22.3.4  Power classes - extension   3xpexg 28044
                  19.22.3.5  Unordered and ordered pairs - extension   nelprd 28045
                  19.22.3.6  Indexed union and intersection - extension   iunxprg 28058
                  19.22.3.7  Introduce the Axiom of Union - extension   ralxfrd2 28059
                  19.22.3.8  Relations - extension   resisresindm 28061
                  19.22.3.9  Functions - extension   fvifeq 28062
                  19.22.3.10  Equinumerosity - extension   resfnfinfin 28071
                  19.22.3.11  Subtraction - extension   cnm1cn 28073
                  19.22.3.12  Multiplication - extension   kcnktkm1cn 28074
                  19.22.3.13  Ordering on reals (cont.) - extension   leaddsuble 28076
                  19.22.3.14  Nonnegative integers (as a subset of complex numbers) - extension   0mnnnnn0 28080
                  19.22.3.15  Upper partititions of integers   1eluzge0 28085
                  19.22.3.16  Finite intervals of integers - extension   ssfz12 28088
                  19.22.3.17  Half-open integer ranges - extension   elfzonn0 28105
                  19.22.3.18  The floor (greatest integer) function - extension   nn0nndivcl 28119
                  19.22.3.19  The modulo (remainder) operation - extension   modvalr 28127
                  19.22.3.20  The ` # ` (finite set size) function - extension   hashimarn 28141
                  19.22.3.21  Words over a set - extension   iswrd0i 28146
                  19.22.3.22  Words over a set - extension (concatenations)   elfzelfzccat 28150
                  19.22.3.23  Words over a set - extension (subwords)   swrdltnd 28153
                  19.22.3.24  Words over a set - extension (subwords of subwords)   swrd0swrd 28163
                  19.22.3.25  Words over a set - extension (subwords of concatenations)   swrdccat3a0 28169
                  19.22.3.26  Prime numbers: elementary properties - extension   prmgt1 28189
                  19.22.3.27  Words over a set - extension (cyclic shift)   ccsh 28196
            19.22.4  Graph theory   uhgraedgrnv 28255
                  19.22.4.1  Undirected hypergraphs   uhgraedgrnv 28255
                  19.22.4.2  Undirected simple graphs   usisuhgra 28256
                  19.22.4.3  Neighbors, complete graphs and universal vertices   nbfiusgrafi 28257
                  19.22.4.4  Walks, Paths and Cycles   usgra2pthspth 28258
                  19.22.4.5  Walks/paths of length 2 as ordered triples   c2wlkot 28274
                  19.22.4.6  Vertex Degree   usgfidegfi 28313
                  19.22.4.7  Friendship graphs   cfrgra 28315
      19.23  Mathbox for David A. Wheeler
            19.23.1  Natural deduction   19.8ad 28397
            19.23.2  Greater than, greater than or equal to.   cge-real 28400
            19.23.3  Hyperbolic trig functions   csinh 28410
            19.23.4  Reciprocal trig functions (sec, csc, cot)   csec 28421
            19.23.5  Identities for "if"   ifnmfalse 28443
            19.23.6  Not-member-of   AnelBC 28444
            19.23.7  Decimal point   cdp2 28445
            19.23.8  Signum (sgn or sign) function   csgn 28453
            19.23.9  Ceiling function   ccei 28463
            19.23.10  Logarithms generalized to arbitrary base using ` logb `   ene0 28467
            19.23.11  Logarithm laws generalized to an arbitrary base - log_   clog_ 28470
            19.23.12  Miscellaneous   5m4e1 28472
      19.24  Mathbox for Alan Sare
            19.24.1  Supplementary "adant" deductions   ad4ant13 28475
            19.24.2  Supplementary unification deductions   biimp 28501
            19.24.3  Conventional Metamath proofs, some derived from VD proofs   iidn3 28520
            19.24.4  What is Virtual Deduction?   wvd1 28597
            19.24.5  Virtual Deduction Theorems   df-vd1 28598
            19.24.6  Theorems proved using virtual deduction   trsspwALT 28868
            19.24.7  Theorems proved using virtual deduction with mmj2 assistance   simplbi2VD 28895
            19.24.8  Virtual Deduction transcriptions of textbook proofs   sb5ALTVD 28962
            19.24.9  Theorems proved using conjunction-form virtual deduction   elpwgdedVD 28966
            19.24.10  Theorems with VD proofs in conventional notation derived from VD proofs   suctrALT3 28973
            19.24.11  Theorems with a proof in conventional notation automatically derived   notnot2ALT2 28976
      19.25  Mathbox for Jonathan Ben-Naim
            19.25.1  First order logic and set theory   bnj170 28999
            19.25.2  Well founded induction and recursion   bnj110 29166
            19.25.3  The existence of a minimal element in certain classes   bnj69 29316
            19.25.4  Well-founded induction   bnj1204 29318
            19.25.5  Well-founded recursion, part 1 of 3   bnj60 29368
            19.25.6  Well-founded recursion, part 2 of 3   bnj1500 29374
            19.25.7  Well-founded recursion, part 3 of 3   bnj1522 29378
      19.26  Mathbox for Norm Megill
            19.26.1  Experiments to study ax-7 unbundling   ax-7v 29379
                  19.26.1.1  Theorems derived from ax-7v (suffixes NEW7 and AUX7)   ax-7v 29379
                  19.26.1.2  Theorems derived from ax-7 (suffix OLD7)   ax-7OLD7 29615
            19.26.2  Miscellanea   cnaddcom 29706
            19.26.3  Atoms, hyperplanes, and covering in a left vector space (or module)   clsa 29709
            19.26.4  Functionals and kernels of a left vector space (or module)   clfn 29792
            19.26.5  Opposite rings and dual vector spaces   cld 29858
            19.26.6  Ortholattices and orthomodular lattices   cops 29907
            19.26.7  Atomic lattices with covering property   ccvr 29997
            19.26.8  Hilbert lattices   chlt 30085
            19.26.9  Projective geometries based on Hilbert lattices   clln 30225
            19.26.10  Construction of a vector space from a Hilbert lattice   cdlema1N 30525
            19.26.11  Construction of involution and inner product from a Hilbert lattice   clpoN 32215

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