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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 1662
            1.4.7  Axiom scheme ax-8 (Equality)   ax-8 1683
            1.4.8  Membership predicate   wcel 1721
            1.4.9  Axiom scheme ax-13 (Left Equality for Binary Predicate)   ax-13 1723
            1.4.10  Axiom scheme ax-14 (Right Equality for Binary Predicate)   ax-14 1725
            1.4.11  Logical redundancy of ax-6 , ax-7 , ax-11 , ax-12   ax9dgen 1727
      1.5  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes)
            1.5.1  Axiom scheme ax-6 (Quantified Negation)   ax-6 1740
            1.5.2  Axiom scheme ax-7 (Quantifier Commutation)   ax-7 1745
            1.5.3  Axiom scheme ax-11 (Substitution)   ax-11 1757
            1.5.4  Axiom scheme ax-12 (Quantified Equality)   ax-12 1946
      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 2185
            1.6.2  Rederive new axioms from old: ax5 , ax6 , ax9from9o , ax11 , ax12from12o   ax4 2195
            1.6.3  Legacy theorems using obsolete axioms   ax17o 2207
      1.7  Existential uniqueness
      1.8  Other axiomatizations related to classical predicate calculus
            1.8.1  Predicate calculus with all distinct variables   ax-7d 2345
            1.8.2  Aristotelian logic: Assertic syllogisms   barbara 2351
            1.8.3  Intuitionistic logic   axi4 2375
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 2385
            2.1.2  Class abstractions (a.k.a. class builders)   cab 2390
            2.1.3  Class form not-free predicate   wnfc 2527
            2.1.4  Negated equality and membership   wne 2567
                  2.1.4.1  Negated equality   nne 2571
                  2.1.4.2  Negated membership   neleq1 2660
            2.1.5  Restricted quantification   wral 2666
            2.1.6  The universal class   cvv 2916
            2.1.7  Conditional equality (experimental)   wcdeq 3104
            2.1.8  Russell's Paradox   ru 3120
            2.1.9  Proper substitution of classes for sets   wsbc 3121
            2.1.10  Proper substitution of classes for sets into classes   csb 3211
            2.1.11  Define basic set operations and relations   cdif 3277
            2.1.12  Subclasses and subsets   df-ss 3294
            2.1.13  The difference, union, and intersection of two classes   difeq1 3418
                  2.1.13.1  The difference of two classes   difeq1 3418
                  2.1.13.2  The union of two classes   elun 3448
                  2.1.13.3  The intersection of two classes   elin 3490
                  2.1.13.4  Combinations of difference, union, and intersection of two classes   unabs 3531
                  2.1.13.5  Class abstractions with difference, union, and intersection of two classes   symdif2 3567
                  2.1.13.6  Restricted uniqueness with difference, union, and intersection   reuss2 3581
            2.1.14  The empty set   c0 3588
            2.1.15  "Weak deduction theorem" for set theory   cif 3699
            2.1.16  Power classes   cpw 3759
            2.1.17  Unordered and ordered pairs   csn 3774
            2.1.18  The union of a class   cuni 3975
            2.1.19  The intersection of a class   cint 4010
            2.1.20  Indexed union and intersection   ciun 4053
            2.1.21  Disjointness   wdisj 4142
            2.1.22  Binary relations   wbr 4172
            2.1.23  Ordered-pair class abstractions (class builders)   copab 4225
            2.1.24  Transitive classes   wtr 4262
      2.2  ZF Set Theory - add the Axiom of Replacement
            2.2.1  Introduce the Axiom of Replacement   ax-rep 4280
            2.2.2  Derive the Axiom of Separation   axsep 4289
            2.2.3  Derive the Null Set Axiom   zfnuleu 4295
            2.2.4  Theorems requiring subset and intersection existence   nalset 4300
            2.2.5  Theorems requiring empty set existence   class2set 4327
      2.3  ZF Set Theory - add the Axiom of Power Sets
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 4337
            2.3.2  Derive the Axiom of Pairing   zfpair 4361
            2.3.3  Ordered pair theorem   opnz 4392
            2.3.4  Ordered-pair class abstractions (cont.)   opabid 4421
            2.3.5  Power class of union and intersection   pwin 4447
            2.3.6  Epsilon and identity relations   cep 4452
            2.3.7  Partial and complete ordering   wpo 4461
            2.3.8  Founded and well-ordering relations   wfr 4498
            2.3.9  Ordinals   word 4540
      2.4  ZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 4660
            2.4.2  Ordinals (continued)   ordon 4722
            2.4.3  Transfinite induction   tfi 4792
            2.4.4  The natural numbers (i.e. finite ordinals)   com 4804
            2.4.5  Peano's postulates   peano1 4823
            2.4.6  Finite induction (for finite ordinals)   find 4829
            2.4.7  Relations   cxp 4835
            2.4.8  Definite description binder (inverted iota)   cio 5375
            2.4.9  Functions   wfun 5407
            2.4.10  Operations   co 6040
            2.4.11  "Maps to" notation   elmpt2cl 6247
            2.4.12  Function operation   cof 6262
            2.4.13  First and second members of an ordered pair   c1st 6306
            2.4.14  Special "Maps to" operations   mpt2xopn0yelv 6423
            2.4.15  Function transposition   ctpos 6437
            2.4.16  Curry and uncurry   ccur 6476
            2.4.17  Proper subset relation   crpss 6480
            2.4.18  Iota properties   fvopab5 6493
            2.4.19  Cantor's Theorem   canth 6498
            2.4.20  Undefined values and restricted iota (description binder)   cund 6500
            2.4.21  Functions on ordinals; strictly monotone ordinal functions   iunon 6559
            2.4.22  "Strong" transfinite recursion   crecs 6591
            2.4.23  Recursive definition generator   crdg 6626
            2.4.24  Finite recursion   frfnom 6651
            2.4.25  Abian's "most fundamental" fixed point theorem   abianfplem 6674
            2.4.26  Ordinal arithmetic   c1o 6676
            2.4.27  Natural number arithmetic   nna0 6806
            2.4.28  Equivalence relations and classes   wer 6861
            2.4.29  The mapping operation   cmap 6977
            2.4.30  Infinite Cartesian products   cixp 7022
            2.4.31  Equinumerosity   cen 7065
            2.4.32  Schroeder-Bernstein Theorem   sbthlem1 7176
            2.4.33  Equinumerosity (cont.)   xpf1o 7228
            2.4.34  Pigeonhole Principle   phplem1 7245
            2.4.35  Finite sets   onomeneq 7255
            2.4.36  Finite intersections   cfi 7373
            2.4.37  Hall's marriage theorem   marypha1lem 7396
            2.4.38  Supremum   csup 7403
            2.4.39  Ordinal isomorphism, Hartog's theorem   coi 7434
            2.4.40  Hartogs function, order types, weak dominance   char 7480
      2.5  ZF Set Theory - add the Axiom of Regularity
            2.5.1  Introduce the Axiom of Regularity   ax-reg 7516
            2.5.2  Axiom of Infinity equivalents   inf0 7532
      2.6  ZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-inf 7549
            2.6.2  Existence of omega (the set of natural numbers)   omex 7554
            2.6.3  Cantor normal form   ccnf 7572
            2.6.4  Transitive closure   trcl 7620
            2.6.5  Rank   cr1 7644
            2.6.6  Scott's trick; collection principle; Hilbert's epsilon   scottex 7765
            2.6.7  Cardinal numbers   ccrd 7778
            2.6.8  Axiom of Choice equivalents   wac 7952
            2.6.9  Cardinal number arithmetic   ccda 8003
            2.6.10  The Ackermann bijection   ackbij2lem1 8055
            2.6.11  Cofinality (without Axiom of Choice)   cflem 8082
            2.6.12  Eight inequivalent definitions of finite set   sornom 8113
            2.6.13  Hereditarily size-limited sets without Choice   itunifval 8252
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 8295
            3.2.2  AC equivalents: well-ordering, Zorn's lemma   numthcor 8330
            3.2.3  Cardinal number theorems using Axiom of Choice   cardval 8377
            3.2.4  Cardinal number arithmetic using Axiom of Choice   iunctb 8405
            3.2.5  Cofinality using Axiom of Choice   alephreg 8413
      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 8513
            4.1.2  Weak universes   cwun 8531
            4.1.3  Tarski's classes   ctsk 8579
            4.1.4  Grothendieck's universes   cgru 8621
      4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
            4.2.1  Introduce the Tarski-Grothendieck Axiom   ax-groth 8654
            4.2.2  Derive the Power Set, Infinity and Choice Axioms   grothpw 8657
            4.2.3  Tarski map function   ctskm 8668
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 8675
            5.1.2  Final derivation of real and complex number postulates   axaddf 8976
            5.1.3  Real and complex number postulates restated as axioms   ax-cnex 9002
      5.2  Derive the basic properties from the field axioms
            5.2.1  Some deductions from the field axioms for complex numbers   cnex 9027
            5.2.2  Infinity and the extended real number system   cpnf 9073
            5.2.3  Restate the ordering postulates with extended real "less than"   axlttri 9103
            5.2.4  Ordering on reals   lttr 9108
            5.2.5  Initial properties of the complex numbers   mul12 9188
      5.3  Real and complex numbers - basic operations
            5.3.1  Addition   add12 9235
            5.3.2  Subtraction   cmin 9247
            5.3.3  Multiplication   muladd 9422
            5.3.4  Ordering on reals (cont.)   gt0ne0 9449
            5.3.5  Reciprocals   ixi 9607
            5.3.6  Division   cdiv 9633
            5.3.7  Ordering on reals (cont.)   elimgt0 9802
            5.3.8  Completeness Axiom and Suprema   fimaxre 9911
            5.3.9  Imaginary and complex number properties   inelr 9946
            5.3.10  Function operation analogue theorems   ofsubeq0 9953
      5.4  Integer sets
            5.4.1  Natural numbers (as a subset of complex numbers)   cn 9956
            5.4.2  Principle of mathematical induction   nnind 9974
            5.4.3  Decimal representation of numbers   c2 10005
            5.4.4  Some properties of specific numbers   0p1e1 10049
            5.4.5  The Archimedean property   nnunb 10173
            5.4.6  Nonnegative integers (as a subset of complex numbers)   cn0 10177
            5.4.7  Integers (as a subset of complex numbers)   cz 10238
            5.4.8  Decimal arithmetic   cdc 10338
            5.4.9  Upper partititions of integers   cuz 10444
            5.4.10  Well-ordering principle for bounded-below sets of integers   uzwo3 10525
            5.4.11  Rational numbers (as a subset of complex numbers)   cq 10530
            5.4.12  Existence of the set of complex numbers   rpnnen1lem1 10556
      5.5  Order sets
            5.5.1  Positive reals (as a subset of complex numbers)   crp 10568
            5.5.2  Infinity and the extended real number system (cont.)   cxne 10663
            5.5.3  Supremum on the extended reals   xrsupexmnf 10839
            5.5.4  Real number intervals   cioo 10872
            5.5.5  Finite intervals of integers   cfz 10999
            5.5.6  Half-open integer ranges   cfzo 11090
      5.6  Elementary integer functions
            5.6.1  The floor (greatest integer) function   cfl 11156
            5.6.2  The modulo (remainder) operation   cmo 11205
            5.6.3  The infinite sequence builder "seq"   om2uz0i 11242
            5.6.4  Integer powers   cexp 11337
            5.6.5  Ordered pair theorem for nonnegative integers   nn0le2msqi 11515
            5.6.6  Factorial function   cfa 11521
            5.6.7  The binomial coefficient operation   cbc 11548
            5.6.8  The ` # ` (finite set size) function   chash 11573
                  5.6.8.1  Finite induction on the size of the first component of a binary relation   brfi1indlem 11669
            5.6.9  Words over a set   cword 11672
            5.6.10  Longer string literals   cs2 11760
      5.7  Elementary real and complex functions
            5.7.1  The "shift" operation   cshi 11836
            5.7.2  Real and imaginary parts; conjugate   ccj 11856
            5.7.3  Square root; absolute value   csqr 11993
      5.8  Elementary limits and convergence
            5.8.1  Superior limit (lim sup)   clsp 12219
            5.8.2  Limits   cli 12233
            5.8.3  Finite and infinite sums   csu 12434
            5.8.4  The binomial theorem   binomlem 12563
            5.8.5  The inclusion/exclusion principle   incexclem 12571
            5.8.6  Infinite sums (cont.)   isumshft 12574
            5.8.7  Miscellaneous converging and diverging sequences   divrcnv 12587
            5.8.8  Arithmetic series   arisum 12594
            5.8.9  Geometric series   expcnv 12598
            5.8.10  Ratio test for infinite series convergence   cvgrat 12615
            5.8.11  Mertens' theorem   mertenslem1 12616
      5.9  Elementary trigonometry
            5.9.1  The exponential, sine, and cosine functions   ce 12619
            5.9.2  _e is irrational   eirrlem 12758
      5.10  Cardinality of real and complex number subsets
            5.10.1  Countability of integers and rationals   xpnnen 12763
            5.10.2  The reals are uncountable   rpnnen2lem1 12769
PART 6  ELEMENTARY NUMBER THEORY
      6.1  Elementary properties of divisibility
            6.1.1  Irrationality of square root of 2   sqr2irrlem 12802
            6.1.2  Some Number sets are chains of proper subsets   nthruc 12805
            6.1.3  The divides relation   cdivides 12807
            6.1.4  The division algorithm   divalglem0 12868
            6.1.5  Bit sequences   cbits 12886
            6.1.6  The greatest common divisor operator   cgcd 12961
            6.1.7  Bézout's identity   bezoutlem1 12993
            6.1.8  Algorithms   nn0seqcvgd 13016
            6.1.9  Euclid's Algorithm   eucalgval2 13027
      6.2  Elementary prime number theory
            6.2.1  Elementary properties   cprime 13034
            6.2.2  Properties of the canonical representation of a rational   cnumer 13080
            6.2.3  Euler's theorem   codz 13107
            6.2.4  Pythagorean Triples   coprimeprodsq 13138
            6.2.5  The prime count function   cpc 13165
            6.2.6  Pocklington's theorem   prmpwdvds 13227
            6.2.7  Infinite primes theorem   unbenlem 13231
            6.2.8  Sum of prime reciprocals   prmreclem1 13239
            6.2.9  Fundamental theorem of arithmetic   1arithlem1 13246
            6.2.10  Lagrange's four-square theorem   cgz 13252
            6.2.11  Van der Waerden's theorem   cvdwa 13288
            6.2.12  Ramsey's theorem   cram 13322
            6.2.13  Decimal arithmetic (cont.)   dec2dvds 13354
            6.2.14  Specific prime numbers   4nprm 13382
            6.2.15  Very large primes   1259lem1 13405
PART 7  BASIC STRUCTURES
      7.1  Extensible structures
            7.1.1  Basic definitions   cstr 13420
            7.1.2  Slot definitions   cplusg 13484
            7.1.3  Definition of the structure product   crest 13603
            7.1.4  Definition of the structure quotient   cordt 13676
      7.2  Moore spaces
            7.2.1  Moore closures   mrcflem 13786
            7.2.2  Independent sets in a Moore system   mrisval 13810
            7.2.3  Algebraic closure systems   isacs 13831
PART 8  BASIC CATEGORY THEORY
      8.1  Categories
            8.1.1  Categories   ccat 13844
            8.1.2  Opposite category   coppc 13892
            8.1.3  Monomorphisms and epimorphisms   cmon 13909
            8.1.4  Sections, inverses, isomorphisms   csect 13925
            8.1.5  Subcategories   cssc 13962
            8.1.6  Functors   cfunc 14006
            8.1.7  Full & faithful functors   cful 14054
            8.1.8  Natural transformations and the functor category   cnat 14093
      8.2  Arrows (disjointified hom-sets)
            8.2.1  Identity and composition for arrows   cida 14163
      8.3  Examples of categories
            8.3.1  The category of sets   csetc 14185
            8.3.2  The category of categories   ccatc 14204
      8.4  Categorical constructions
            8.4.1  Product of categories   cxpc 14220
            8.4.2  Functor evaluation   cevlf 14261
            8.4.3  Hom functor   chof 14300
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 14352
            9.2.2  Lattices   clat 14429
            9.2.3  The dual of an ordered set   codu 14510
            9.2.4  Subset order structures   cipo 14532
            9.2.5  Distributive lattices   latmass 14569
            9.2.6  Posets and lattices as relations   cps 14579
            9.2.7  Directed sets, nets   cdir 14628
PART 10  BASIC ALGEBRAIC STRUCTURES
      10.1  Monoids
            10.1.1  Definition and basic properties   cmnd 14639
            10.1.2  Monoid homomorphisms and submonoids   cmhm 14691
            10.1.3  Ordered group sum operation   gsumvallem1 14726
            10.1.4  Free monoids   cfrmd 14747
      10.2  Groups
            10.2.1  Definition and basic properties   df-grp 14767
            10.2.2  Subgroups and Quotient groups   csubg 14893
            10.2.3  Elementary theory of group homomorphisms   cghm 14958
            10.2.4  Isomorphisms of groups   cgim 14999
            10.2.5  Group actions   cga 15021
            10.2.6  Symmetry groups and Cayley's Theorem   csymg 15047
            10.2.7  Centralizers and centers   ccntz 15069
            10.2.8  The opposite group   coppg 15096
            10.2.9  p-Groups and Sylow groups; Sylow's theorems   cod 15118
            10.2.10  Direct products   clsm 15223
            10.2.11  Free groups   cefg 15293
      10.3  Abelian groups
            10.3.1  Definition and basic properties   ccmn 15367
            10.3.2  Cyclic groups   ccyg 15442
            10.3.3  Group sum operation   gsumval3a 15467
            10.3.4  Internal direct products   cdprd 15509
            10.3.5  The Fundamental Theorem of Abelian Groups   ablfacrplem 15578
      10.4  Rings
            10.4.1  Multiplicative Group   cmgp 15603
            10.4.2  Definition and basic properties   crg 15615
            10.4.3  Opposite ring   coppr 15682
            10.4.4  Divisibility   cdsr 15698
            10.4.5  Ring homomorphisms   crh 15772
      10.5  Division rings and fields
            10.5.1  Definition and basic properties   cdr 15790
            10.5.2  Subrings of a ring   csubrg 15819
            10.5.3  Absolute value (abstract algebra)   cabv 15859
            10.5.4  Star rings   cstf 15886
      10.6  Left modules
            10.6.1  Definition and basic properties   clmod 15905
            10.6.2  Subspaces and spans in a left module   clss 15963
            10.6.3  Homomorphisms and isomorphisms of left modules   clmhm 16050
            10.6.4  Subspace sum; bases for a left module   clbs 16101
      10.7  Vector spaces
            10.7.1  Definition and basic properties   clvec 16129
      10.8  Ideals
            10.8.1  The subring algebra; ideals   csra 16195
            10.8.2  Two-sided ideals and quotient rings   c2idl 16257
            10.8.3  Principal ideal rings. Divisibility in the integers   clpidl 16267
            10.8.4  Nonzero rings   cnzr 16283
            10.8.5  Left regular elements. More kinds of rings   crlreg 16294
      10.9  Associative algebras
            10.9.1  Definition and basic properties   casa 16324
      10.10  Abstract multivariate polynomials
            10.10.1  Definition and basic properties   cmps 16361
            10.10.2  Polynomial evaluation   evlslem4 16519
            10.10.3  Univariate polynomials   cps1 16524
      10.11  The complex numbers as an extensible structure
            10.11.1  Definition and basic properties   cpsmet 16640
            10.11.2  Algebraic constructions based on the complexes   czrh 16733
      10.12  Hilbert spaces
            10.12.1  Definition and basic properties   cphl 16810
            10.12.2  Orthocomplements and closed subspaces   cocv 16842
            10.12.3  Orthogonal projection and orthonormal bases   cpj 16882
PART 11  BASIC TOPOLOGY
      11.1  Topology
            11.1.1  Topological spaces   ctop 16913
            11.1.2  TopBases for topologies   isbasisg 16967
            11.1.3  Examples of topologies   distop 17015
            11.1.4  Closure and interior   ccld 17035
            11.1.5  Neighborhoods   cnei 17116
            11.1.6  Limit points and perfect sets   clp 17153
            11.1.7  Subspace topologies   restrcl 17175
            11.1.8  Order topology   ordtbaslem 17206
            11.1.9  Limits and continuity in topological spaces   ccn 17242
            11.1.10  Separated spaces: T0, T1, T2 (Hausdorff) ...   ct0 17324
            11.1.11  Compactness   ccmp 17403
            11.1.12  Connectedness   ccon 17427
            11.1.13  First- and second-countability   c1stc 17453
            11.1.14  Local topological properties   clly 17480
            11.1.15  Compactly generated spaces   ckgen 17518
            11.1.16  Product topologies   ctx 17545
            11.1.17  Continuous function-builders   cnmptid 17646
            11.1.18  Quotient maps and quotient topology   ckq 17678
            11.1.19  Homeomorphisms   chmeo 17738
      11.2  Filters and filter bases
            11.2.1  Filter bases   elmptrab 17812
            11.2.2  Filters   cfil 17830
            11.2.3  Ultrafilters   cufil 17884
            11.2.4  Filter limits   cfm 17918
            11.2.5  Extension by continuity   ccnext 18043
            11.2.6  Topological groups   ctmd 18053
            11.2.7  Infinite group sum on topological groups   ctsu 18108
            11.2.8  Topological rings, fields, vector spaces   ctrg 18138
      11.3  Uniform Stuctures and Spaces
            11.3.1  Uniform structures   cust 18182
            11.3.2  The topology induced by an uniform structure   cutop 18213
            11.3.3  Uniform Spaces   cuss 18236
            11.3.4  Uniform continuity   cucn 18258
            11.3.5  Cauchy filters in uniform spaces   ccfilu 18269
            11.3.6  Complete uniform spaces   ccusp 18280
      11.4  Metric spaces
            11.4.1  Pseudometric spaces   ispsmet 18288
            11.4.2  Basic metric space properties   cxme 18300
            11.4.3  Metric space balls   blfvalps 18366
            11.4.4  Open sets of a metric space   mopnval 18421
            11.4.5  Continuity in metric spaces   metcnp3 18523
            11.4.6  The uniform structure generated by a metric   metuvalOLD 18532
            11.4.7  Examples of metric spaces   dscmet 18573
            11.4.8  Normed algebraic structures   cnm 18577
            11.4.9  Normed space homomorphisms (bounded linear operators)   cnmo 18692
            11.4.10  Topology on the reals   qtopbaslem 18745
            11.4.11  Topological definitions using the reals   cii 18858
            11.4.12  Path homotopy   chtpy 18945
            11.4.13  The fundamental group   cpco 18978
      11.5  Complex metric vector spaces
            11.5.1  Complex left modules   cclm 19040
            11.5.2  Complex pre-Hilbert space   ccph 19082
            11.5.3  Convergence and completeness   ccfil 19158
            11.5.4  Baire's Category Theorem   bcthlem1 19230
            11.5.5  Banach spaces and complex Hilbert spaces   ccms 19238
            11.5.6  Minimizing Vector Theorem   minveclem1 19278
            11.5.7  Projection Theorem   pjthlem1 19291
PART 12  BASIC REAL AND COMPLEX ANALYSIS
      12.1  Continuity
            12.1.1  Intermediate value theorem   pmltpclem1 19298
      12.2  Integrals
            12.2.1  Lebesgue measure   covol 19312
            12.2.2  Lebesgue integration   cmbf 19459
      12.3  Derivatives
            12.3.1  Real and complex differentiation   climc 19702
PART 13  BASIC REAL AND COMPLEX FUNCTIONS
      13.1  Polynomials
            13.1.1  Abstract polynomials, continued   evlslem6 19887
            13.1.2  Polynomial degrees   cmdg 19929
            13.1.3  The division algorithm for univariate polynomials   cmn1 20001
            13.1.4  Elementary properties of complex polynomials   cply 20056
            13.1.5  The division algorithm for polynomials   cquot 20160
            13.1.6  Algebraic numbers   caa 20184
            13.1.7  Liouville's approximation theorem   aalioulem1 20202
      13.2  Sequences and series
            13.2.1  Taylor polynomials and Taylor's theorem   ctayl 20222
            13.2.2  Uniform convergence   culm 20245
            13.2.3  Power series   pserval 20279
      13.3  Basic trigonometry
            13.3.1  The exponential, sine, and cosine functions (cont.)   efcn 20312
            13.3.2  Properties of pi = 3.14159...   pilem1 20320
            13.3.3  Mapping of the exponential function   efgh 20396
            13.3.4  The natural logarithm on complex numbers   clog 20405
            13.3.5  Theorems of Pythagoras, isosceles triangles, and intersecting chords   angval 20596
            13.3.6  Solutions of quadratic, cubic, and quartic equations   quad2 20632
            13.3.7  Inverse trigonometric functions   casin 20655
            13.3.8  The Birthday Problem   log2ublem1 20739
            13.3.9  Areas in R^2   carea 20747
            13.3.10  More miscellaneous converging sequences   rlimcnp 20757
            13.3.11  Inequality of arithmetic and geometric means   cvxcl 20776
            13.3.12  Euler-Mascheroni constant   cem 20783
      13.4  Basic number theory
            13.4.1  Wilson's theorem   wilthlem1 20804
            13.4.2  The Fundamental Theorem of Algebra   ftalem1 20808
            13.4.3  The Basel problem (ζ(2) = π2/6)   basellem1 20816
            13.4.4  Number-theoretical functions   ccht 20826
            13.4.5  Perfect Number Theorem   mersenne 20964
            13.4.6  Characters of Z/nZ   cdchr 20969
            13.4.7  Bertrand's postulate   bcctr 21012
            13.4.8  Legendre symbol   clgs 21031
            13.4.9  Quadratic reciprocity   lgseisenlem1 21086
            13.4.10  All primes 4n+1 are the sum of two squares   2sqlem1 21100
            13.4.11  Chebyshev's Weak Prime Number Theorem, Dirichlet's Theorem   chebbnd1lem1 21116
            13.4.12  The Prime Number Theorem   mudivsum 21177
            13.4.13  Ostrowski's theorem   abvcxp 21262
PART 14  GRAPH THEORY
      14.1  Undirected graphs - basics
            14.1.1  Undirected hypergraphs   cuhg 21287
            14.1.2  Undirected multigraphs   cumg 21300
            14.1.3  Undirected simple graphs   cuslg 21317
                  14.1.3.1  Undirected simple graphs - basics   cuslg 21317
                  14.1.3.2  Undirected simple graphs - examples   usgraexvlem 21367
                  14.1.3.3  Finite undirected simple graphs   fiusgraedgfi 21374
            14.1.4  Neighbors, complete graphs and universal vertices   cnbgra 21383
                  14.1.4.1  Neighbors   nbgraop 21389
                  14.1.4.2  Complete graphs   iscusgra 21418
                  14.1.4.3  Universal vertices   isuvtx 21450
            14.1.5  Walks, paths and cycles   cwalk 21459
                  14.1.5.1  Walks and trails   wlks 21479
                  14.1.5.2  Paths and simple paths   pths 21519
                  14.1.5.3  Circuits and cycles   crcts 21562
                  14.1.5.4  Connected graphs   cconngra 21609
            14.1.6  Vertex Degree   cvdg 21617
      14.2  Eulerian paths and the Konigsberg Bridge problem
            14.2.1  Eulerian paths   ceup 21637
            14.2.2  The Konigsberg Bridge problem   vdeg0i 21657
PART 15  GUIDES AND MISCELLANEA
      15.1  Guides (conventions, explanations, and examples)
            15.1.1  Conventions   conventions 21663
            15.1.2  Natural deduction   natded 21664
            15.1.3  Natural deduction examples   ex-natded5.2 21665
            15.1.4  Definitional examples   ex-or 21682
      15.2  Humor
            15.2.1  April Fool's theorem   avril1 21710
      15.3  (Future - to be reviewed and classified)
            15.3.1  Planar incidence geometry   cplig 21716
            15.3.2  Algebra preliminaries   crpm 21721
            15.3.3  Transitive closure   ctcl 21723
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 21727
            16.1.2  Definition and basic properties of Abelian groups   cablo 21822
            16.1.3  Subgroups   csubgo 21842
            16.1.4  Operation properties   cass 21853
            16.1.5  Group-like structures   cmagm 21859
            16.1.6  Examples of Abelian groups   ablosn 21888
            16.1.7  Group homomorphism and isomorphism   cghom 21898
      16.2  Additional material on rings and fields
            16.2.1  Definition and basic properties   crngo 21916
            16.2.2  Examples of rings   cnrngo 21944
            16.2.3  Division Rings   cdrng 21946
            16.2.4  Star Fields   csfld 21949
            16.2.5  Fields and Rings   ccm2 21951
PART 17  COMPLEX TOPOLOGICAL VECTOR SPACES (DEPRECATED)
      17.1  Complex vector spaces
            17.1.1  Definition and basic properties   cvc 21977
            17.1.2  Examples of complex vector spaces   cncvc 22015
      17.2  Normed complex vector spaces
            17.2.1  Definition and basic properties   cnv 22016
            17.2.2  Examples of normed complex vector spaces   cnnv 22121
            17.2.3  Induced metric of a normed complex vector space   imsval 22130
            17.2.4  Inner product   cdip 22149
            17.2.5  Subspaces   css 22173
      17.3  Operators on complex vector spaces
            17.3.1  Definitions and basic properties   clno 22194
      17.4  Inner product (pre-Hilbert) spaces
            17.4.1  Definition and basic properties   ccphlo 22266
            17.4.2  Examples of pre-Hilbert spaces   cncph 22273
            17.4.3  Properties of pre-Hilbert spaces   isph 22276
      17.5  Complex Banach spaces
            17.5.1  Definition and basic properties   ccbn 22317
            17.5.2  Examples of complex Banach spaces   cnbn 22324
            17.5.3  Uniform Boundedness Theorem   ubthlem1 22325
            17.5.4  Minimizing Vector Theorem   minvecolem1 22329
      17.6  Complex Hilbert spaces
            17.6.1  Definition and basic properties   chlo 22340
            17.6.2  Standard axioms for a complex Hilbert space   hlex 22353
            17.6.3  Examples of complex Hilbert spaces   cnchl 22371
            17.6.4  Subspaces   ssphl 22372
            17.6.5  Hellinger-Toeplitz Theorem   htthlem 22373
PART 18  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      18.1  Axiomatization of complex pre-Hilbert spaces
            18.1.1  Basic Hilbert space definitions   chil 22375
            18.1.2  Preliminary ZFC lemmas   df-hnorm 22424
            18.1.3  Derive the Hilbert space axioms from ZFC set theory   axhilex-zf 22437
            18.1.4  Introduce the vector space axioms for a Hilbert space   ax-hilex 22455
            18.1.5  Vector operations   hvmulex 22467
            18.1.6  Inner product postulates for a Hilbert space   ax-hfi 22534
      18.2  Inner product and norms
            18.2.1  Inner product   his5 22541
            18.2.2  Norms   dfhnorm2 22577
            18.2.3  Relate Hilbert space to normed complex vector spaces   hilablo 22615
            18.2.4  Bunjakovaskij-Cauchy-Schwarz inequality   bcsiALT 22634
      18.3  Cauchy sequences and completeness axiom
            18.3.1  Cauchy sequences and limits   hcau 22639
            18.3.2  Derivation of the completeness axiom from ZF set theory   hilmet 22649
            18.3.3  Completeness postulate for a Hilbert space   ax-hcompl 22657
            18.3.4  Relate Hilbert space to ZFC pre-Hilbert and Hilbert spaces   hhcms 22658
      18.4  Subspaces and projections
            18.4.1  Subspaces   df-sh 22662
            18.4.2  Closed subspaces   df-ch 22677
            18.4.3  Orthocomplements   df-oc 22707
            18.4.4  Subspace sum, span, lattice join, lattice supremum   df-shs 22763
            18.4.5  Projection theorem   pjhthlem1 22846
            18.4.6  Projectors   df-pjh 22850
      18.5  Properties of Hilbert subspaces
            18.5.1  Orthomodular law   omlsilem 22857
            18.5.2  Projectors (cont.)   pjhtheu2 22871
            18.5.3  Hilbert lattice operations   sh0le 22895
            18.5.4  Span (cont.) and one-dimensional subspaces   spansn0 22996
            18.5.5  Commutes relation for Hilbert lattice elements   df-cm 23038
            18.5.6  Foulis-Holland theorem   fh1 23073
            18.5.7  Quantum Logic Explorer axioms   qlax1i 23082
            18.5.8  Orthogonal subspaces   chscllem1 23092
            18.5.9  Orthoarguesian laws 5OA and 3OA   5oalem1 23109
            18.5.10  Projectors (cont.)   pjorthi 23124
            18.5.11  Mayet's equation E_3   mayete3i 23183
      18.6  Operators on Hilbert spaces
            18.6.1  Operator sum, difference, and scalar multiplication   df-hosum 23186
            18.6.2  Zero and identity operators   df-h0op 23204
            18.6.3  Operations on Hilbert space operators   hoaddcl 23214
            18.6.4  Linear, continuous, bounded, Hermitian, unitary operators and norms   df-nmop 23295
            18.6.5  Linear and continuous functionals and norms   df-nmfn 23301
            18.6.6  Adjoint   df-adjh 23305
            18.6.7  Dirac bra-ket notation   df-bra 23306
            18.6.8  Positive operators   df-leop 23308
            18.6.9  Eigenvectors, eigenvalues, spectrum   df-eigvec 23309
            18.6.10  Theorems about operators and functionals   nmopval 23312
            18.6.11  Riesz lemma   riesz3i 23518
            18.6.12  Adjoints (cont.)   cnlnadjlem1 23523
            18.6.13  Quantum computation error bound theorem   unierri 23560
            18.6.14  Dirac bra-ket notation (cont.)   branmfn 23561
            18.6.15  Positive operators (cont.)   leopg 23578
            18.6.16  Projectors as operators   pjhmopi 23602
      18.7  States on a Hilbert lattice and Godowski's equation
            18.7.1  States on a Hilbert lattice   df-st 23667
            18.7.2  Godowski's equation   golem1 23727
      18.8  Cover relation, atoms, exchange axiom, and modular symmetry
            18.8.1  Covers relation; modular pairs   df-cv 23735
            18.8.2  Atoms   df-at 23794
            18.8.3  Superposition principle   superpos 23810
            18.8.4  Atoms, exchange and covering properties, atomicity   chcv1 23811
            18.8.5  Irreducibility   chirredlem1 23846
            18.8.6  Atoms (cont.)   atcvat3i 23852
            18.8.7  Modular symmetry   mdsymlem1 23859
PART 19  SUPPLEMENTARY MATERIAL (USER'S MATHBOXES)
      19.1  Mathboxes for user contributions
            19.1.1  Mathbox guidelines   mathbox 23898
      19.2  Mathbox for Stefan Allan
      19.3  Mathbox for Thierry Arnoux
            19.3.1  Propositional Calculus - misc additions   bian1d 23903
            19.3.2  Predicate Calculus   abeq2f 23913
                  19.3.2.1  Predicate Calculus - misc additions   abeq2f 23913
                  19.3.2.2  Restricted quantification - misc additions   reximddv 23915
                  19.3.2.3  Substitution (without distinct variables) - misc additions   clelsb3f 23924
                  19.3.2.4  Existential "at most one" - misc additions   mo5f 23925
                  19.3.2.5  Existential uniqueness - misc additions   2reuswap2 23928
                  19.3.2.6  Restricted "at most one" - misc additions   rmoxfrdOLD 23932
            19.3.3  General Set Theory   ceqsexv2d 23938
                  19.3.3.1  Class abstractions (a.k.a. class builders)   ceqsexv2d 23938
                  19.3.3.2  Image Sets   abrexdomjm 23941
                  19.3.3.3  Set relations and operations - misc additions   eqri 23947
                  19.3.3.4  Unordered pairs   elpreq 23952
                  19.3.3.5  Conditional operator - misc additions   ifeqeqx 23954
                  19.3.3.6  Indexed union - misc additions   iuneq12daf 23960
                  19.3.3.7  Disjointness - misc additions   cbvdisjf 23968
            19.3.4  Relations and Functions   dfrel4 23987
                  19.3.4.1  Relations - misc additions   dfrel4 23987
                  19.3.4.2  Functions - misc additions   fdmrn 23992
                  19.3.4.3  Isomorphisms - misc. add.   gtiso 24041
                  19.3.4.4  Disjointness (additional proof requiring functions)   disjdsct 24043
                  19.3.4.5  First and second members of an ordered pair - misc additions   df1stres 24044
                  19.3.4.6  Supremum - misc additions   supssd 24051
                  19.3.4.7  Countable Sets   nnct 24052
            19.3.5  Real and Complex Numbers   addeq0 24067
                  19.3.5.1  Complex addition - misc. additions   addeq0 24067
                  19.3.5.2  Ordering on reals - misc additions   lt2addrd 24068
                  19.3.5.3  Extended reals - misc additions   xgepnf 24069
                  19.3.5.4  Real number intervals - misc additions   icossicc 24082
                  19.3.5.5  Finite intervals of integers - misc additions   fzssnn 24100
                  19.3.5.6  Half-open integer ranges - misc additions   iundisjfi 24105
                  19.3.5.7  The ` # ` (finite set size) function - misc additions   hashresfn 24109
                  19.3.5.8  The greatest common divisor operator - misc. add   numdenneg 24113
                  19.3.5.9  Integers   ltesubnnd 24115
                  19.3.5.10  Division in the extended real number system   cxdiv 24116
            19.3.6  Structure builders   ress0g 24135
                  19.3.6.1  Structure builder restriction operator   ress0g 24135
                  19.3.6.2  Posets   tospos 24139
                  19.3.6.3  Complete lattices   clatp0ex 24146
                  19.3.6.4  Extended reals Structure - misc additions   ax-xrssca 24148
                  19.3.6.5  The extended non-negative real numbers monoid   xrge0base 24160
            19.3.7  Algebra   sumpr 24171
                  19.3.7.1  Finitely supported group sums - misc additions   sumpr 24171
                  19.3.7.2  Rings - misc additions   dvrdir 24179
                  19.3.7.3  Ordered groups   cogrp 24184
                  19.3.7.4  Ordered fields   cofld 24186
                  19.3.7.5  The Archimedean property for generic algebraic structures   cinftm 24199
                  19.3.7.6  Ring homomorphisms - misc additions   rhmdvdsr 24209
                  19.3.7.7  The ring of integers   zzsbase 24216
                  19.3.7.8  The ordered field of reals   rebase 24222
            19.3.8  Topology   cmetid 24234
                  19.3.8.1  Pseudometrics   cmetid 24234
                  19.3.8.2  Continuity - misc additions   hauseqcn 24246
                  19.3.8.3  Topology of the closed unit   unitsscn 24247
                  19.3.8.4  Topology of ` ( RR X. RR ) `   unicls 24254
                  19.3.8.5  Order topology - misc. additions   cnvordtrestixx 24264
                  19.3.8.6  Continuity in topological spaces - misc. additions   mndpluscn 24265
                  19.3.8.7  Topology of the extended non-negative real numbers monoid   xrge0hmph 24271
                  19.3.8.8  Limits - misc additions   lmlim 24286
            19.3.9  Uniform Stuctures and Spaces   chcmp 24293
                  19.3.9.1  Hausdorff Completion   chcmp 24293
            19.3.10  Topology and algebraic structures   zzsnm 24295
                  19.3.10.1  The norm on the ring of the integer numbers   zzsnm 24295
                  19.3.10.2  The complete ordered field of the real numbers   recms 24296
                  19.3.10.3  Topological ` ZZ ` -modules   zlm0 24299
                  19.3.10.4  The canonical embedding of the rational numbers into a division ring   cqqh 24309
                  19.3.10.5  The canonical embedding of ` RR ` into a complete ordered field   crrh 24330
                  19.3.10.6  Embedding into ` RR* `   cxrh 24335
                  19.3.10.7  Canonical embeddings into ` RR `   zrhre 24338
            19.3.11  Real and complex functions   clogb 24341
                  19.3.11.1  Logarithm laws generalized to an arbitrary base - logb   clogb 24341
                  19.3.11.2  Indicator Functions   cind 24361
                  19.3.11.3  Extended sum   cesum 24377
            19.3.12  Mixed Function/Constant operation   cofc 24431
            19.3.13  Abstract measure   csiga 24443
                  19.3.13.1  Sigma-Algebra   csiga 24443
                  19.3.13.2  Generated Sigma-Algebra   csigagen 24474
                  19.3.13.3  The Borel algebra on the real numbers   cbrsiga 24488
                  19.3.13.4  Product Sigma-Algebra   csx 24495
                  19.3.13.5  Measures   cmeas 24502
                  19.3.13.6  The counting measure   cntmeas 24533
                  19.3.13.7  The Lebesgue measure - misc additions   volss 24536
                  19.3.13.8  The 'almost everywhere' relation   cae 24541
                  19.3.13.9  Measurable functions   cmbfm 24553
                  19.3.13.10  Borel Algebra on ` ( RR X. RR ) `   br2base 24572
            19.3.14  Integration   itgeq12dv 24594
                  19.3.14.1  Lebesgue integral - misc additions   itgeq12dv 24594
                  19.3.14.2  Bochner integral   citgm 24595
            19.3.15  Probability   cprb 24618
                  19.3.15.1  Probability Theory   cprb 24618
                  19.3.15.2  Conditional Probabilities   ccprob 24642
                  19.3.15.3  Real Valued Random Variables   crrv 24651
                  19.3.15.4  Preimage set mapping operator   corvc 24666
                  19.3.15.5  Distribution Functions   orvcelval 24679
                  19.3.15.6  Cumulative Distribution Functions   orvclteel 24683
                  19.3.15.7  Probabilities - example   coinfliplem 24689
                  19.3.15.8  Bertrand's Ballot Problem   ballotlemoex 24696
      19.4  Mathbox for Mario Carneiro
            19.4.1  Miscellaneous stuff   quartfull 24749
            19.4.2  Zeta function   czeta 24750
            19.4.3  Gamma function   clgam 24753
            19.4.4  Derangements and the Subfactorial   deranglem 24805
            19.4.5  The Erdős-Szekeres theorem   erdszelem1 24830
            19.4.6  The Kuratowski closure-complement theorem   kur14lem1 24845
            19.4.7  Retracts and sections   cretr 24856
            19.4.8  Path-connected and simply connected spaces   cpcon 24859
            19.4.9  Covering maps   ccvm 24895
            19.4.10  Normal numbers   snmlff 24969
            19.4.11  Godel-sets of formulas   cgoe 24973
            19.4.12  Models of ZF   cgze 25001
            19.4.13  Splitting fields   citr 25015
            19.4.14  p-adic number fields   czr 25031
      19.5  Mathbox for Paul Chapman
            19.5.1  Group homomorphism and isomorphism   ghomgrpilem1 25049
            19.5.2  Real and complex numbers (cont.)   climuzcnv 25061
            19.5.3  Miscellaneous theorems   elfzm12 25065
      19.6  Mathbox for Drahflow
      19.7  Mathbox for Scott Fenton
            19.7.1  ZFC Axioms in primitive form   axextprim 25103
            19.7.2  Untangled classes   untelirr 25110
            19.7.3  Extra propositional calculus theorems   3orel1 25117
            19.7.4  Misc. Useful Theorems   nepss 25128
            19.7.5  Properties of reals and complexes   sqdivzi 25137
            19.7.6  Product sequences   prodf 25168
            19.7.7  Non-trivial convergence   ntrivcvg 25178
            19.7.8  Complex products   cprod 25184
            19.7.9  Finite products   fprod 25220
            19.7.10  Infinite products   iprodclim 25264
            19.7.11  Falling and Rising Factorial   cfallfac 25273
            19.7.12  Factorial limits   faclimlem1 25310
            19.7.13  Greatest common divisor and divisibility   pdivsq 25316
            19.7.14  Properties of relationships   brtp 25320
            19.7.15  Properties of functions and mappings   funpsstri 25335
            19.7.16  Epsilon induction   setinds 25348
            19.7.17  Ordinal numbers   elpotr 25351
            19.7.18  Defined equality axioms   axextdfeq 25368
            19.7.19  Hypothesis builders   hbntg 25376
            19.7.20  The Predecessor Class   cpred 25381
            19.7.21  (Trans)finite Recursion Theorems   tfisg 25418
            19.7.22  Well-founded induction   tz6.26 25419
            19.7.23  Transitive closure under a relationship   ctrpred 25434
            19.7.24  Founded Induction   frmin 25456
            19.7.25  Ordering Ordinal Sequences   orderseqlem 25466
            19.7.26  Well-founded recursion   wfr3g 25469
            19.7.27  Transfinite recursion via Well-founded recursion   tfrALTlem 25490
            19.7.28  Founded Recursion   frr3g 25494
            19.7.29  Surreal Numbers   csur 25508
            19.7.30  Surreal Numbers: Ordering   sltsolem1 25536
            19.7.31  Surreal Numbers: Birthday Function   bdayfo 25543
            19.7.32  Surreal Numbers: Density   fvnobday 25550
            19.7.33  Surreal Numbers: Density   nodenselem3 25551
            19.7.34  Surreal Numbers: Upper and Lower Bounds   nobndlem1 25560
            19.7.35  Surreal Numbers: Full-Eta Property   nofulllem1 25570
            19.7.36  Symmetric difference   csymdif 25575
            19.7.37  Quantifier-free definitions   ctxp 25587
            19.7.38  Alternate ordered pairs   caltop 25705
            19.7.39  Tarskian geometry   cee 25731
            19.7.40  Tarski's axioms for geometry   axdimuniq 25756
            19.7.41  Congruence properties   cofs 25820
            19.7.42  Betweenness properties   btwntriv2 25850
            19.7.43  Segment Transportation   ctransport 25867
            19.7.44  Properties relating betweenness and congruence   cifs 25873
            19.7.45  Connectivity of betweenness   btwnconn1lem1 25925
            19.7.46  Segment less than or equal to   csegle 25944
            19.7.47  Outside of relationship   coutsideof 25957
            19.7.48  Lines and Rays   cline2 25972
            19.7.49  Bernoulli polynomials and sums of k-th powers   cbp 25996
            19.7.50  Rank theorems   rankung 26011
            19.7.51  Hereditarily Finite Sets   chf 26017
      19.8  Mathbox for Anthony Hart
            19.8.1  Propositional Calculus   tb-ax1 26032
            19.8.2  Predicate Calculus   quantriv 26054
            19.8.3  Misc. Single Axiom Systems   meran1 26065
            19.8.4  Connective Symmetry   negsym1 26071
      19.9  Mathbox for Chen-Pang He
            19.9.1  Ordinal topology   ontopbas 26082
      19.10  Mathbox for Jeff Hoffman
            19.10.1  Inferences for finite induction on generic function values   fveleq 26105
            19.10.2  gdc.mm   nnssi2 26109
      19.11  Mathbox for Wolf Lammen
      19.12  Mathbox for Brendan Leahy
      19.13  Mathbox for Jeff Hankins
            19.13.1  Miscellany   a1i13 26188
            19.13.2  Basic topological facts   topbnd 26217
            19.13.3  Topology of the real numbers   ivthALT 26228
            19.13.4  Refinements   cfne 26229
            19.13.5  Neighborhood bases determine topologies   neibastop1 26278
            19.13.6  Lattice structure of topologies   topmtcl 26282
            19.13.7  Filter bases   fgmin 26289
            19.13.8  Directed sets, nets   tailfval 26291
      19.14  Mathbox for Jeff Madsen
            19.14.1  Logic and set theory   anim12da 26302
            19.14.2  Real and complex numbers; integers   filbcmb 26332
            19.14.3  Sequences and sums   sdclem2 26336
            19.14.4  Topology   subspopn 26348
            19.14.5  Metric spaces   metf1o 26351
            19.14.6  Continuous maps and homeomorphisms   constcncf 26358
            19.14.7  Boundedness   ctotbnd 26365
            19.14.8  Isometries   cismty 26397
            19.14.9  Heine-Borel Theorem   heibor1lem 26408
            19.14.10  Banach Fixed Point Theorem   bfplem1 26421
            19.14.11  Euclidean space   crrn 26424
            19.14.12  Intervals (continued)   ismrer1 26437
            19.14.13  Groups and related structures   exidcl 26441
            19.14.14  Rings   rngonegcl 26451
            19.14.15  Ring homomorphisms   crnghom 26466
            19.14.16  Commutative rings   ccring 26495
            19.14.17  Ideals   cidl 26507
            19.14.18  Prime rings and integral domains   cprrng 26546
            19.14.19  Ideal generators   cigen 26559
      19.15  Mathbox for Rodolfo Medina
            19.15.1  Partitions   prtlem60 26578
      19.16  Mathbox for Stefan O'Rear
            19.16.1  Additional elementary logic and set theory   nelss 26622
            19.16.2  Additional theory of functions   fninfp 26625
            19.16.3  Extensions beyond function theory   gsumvsmul 26635
            19.16.4  Additional topology   elrfi 26638
            19.16.5  Characterization of closure operators. Kuratowski closure axioms   ismrcd1 26642
            19.16.6  Algebraic closure systems   cnacs 26646
            19.16.7  Miscellanea 1. Map utilities   constmap 26657
            19.16.8  Miscellanea for polynomials   ofmpteq 26666
            19.16.9  Multivariate polynomials over the integers   cmzpcl 26668
            19.16.10  Miscellanea for Diophantine sets 1   coeq0 26700
            19.16.11  Diophantine sets 1: definitions   cdioph 26703
            19.16.12  Diophantine sets 2 miscellanea   ellz1 26715
            19.16.13  Diophantine sets 2: union and intersection. Monotone Boolean algebra   diophin 26721
            19.16.14  Diophantine sets 3: construction   diophrex 26724
            19.16.15  Diophantine sets 4 miscellanea   2sbcrex 26733
            19.16.16  Diophantine sets 4: Quantification   rexrabdioph 26744
            19.16.17  Diophantine sets 5: Arithmetic sets   rabdiophlem1 26751
            19.16.18  Diophantine sets 6 miscellanea   fz1ssnn 26761
            19.16.19  Diophantine sets 6: reusability. renumbering of variables   eldioph4b 26762
            19.16.20  Pigeonhole Principle and cardinality helpers   fphpd 26767
            19.16.21  A non-closed set of reals is infinite   rencldnfilem 26771
            19.16.22  Miscellanea for Lagrange's theorem   icodiamlt 26773
            19.16.23  Lagrange's rational approximation theorem   irrapxlem1 26775
            19.16.24  Pell equations 1: A nontrivial solution always exists   pellexlem1 26782
            19.16.25  Pell equations 2: Algebraic number theory of the solution set   csquarenn 26789
            19.16.26  Pell equations 3: characterizing fundamental solution   infmrgelbi 26831
            19.16.27  Logarithm laws generalized to an arbitrary base   reglogcl 26843
            19.16.28  Pell equations 4: the positive solution group is infinite cyclic   pellfund14 26851
            19.16.29  X and Y sequences 1: Definition and recurrence laws   crmx 26853
            19.16.30  Ordering and induction lemmas for the integers   monotuz 26894
            19.16.31  X and Y sequences 2: Order properties   rmxypos 26902
            19.16.32  Congruential equations   congtr 26920
            19.16.33  Alternating congruential equations   acongid 26930
            19.16.34  Additional theorems on integer divisibility   bezoutr 26940
            19.16.35  X and Y sequences 3: Divisibility properties   jm2.18 26949
            19.16.36  X and Y sequences 4: Diophantine representability of Y   jm2.27a 26966
            19.16.37  X and Y sequences 5: Diophantine representability of X, ^, _C   rmxdiophlem 26976
            19.16.38  Uncategorized stuff not associated with a major project   setindtr 26985
            19.16.39  More equivalents of the Axiom of Choice   axac10 26994
            19.16.40  Finitely generated left modules   clfig 27033
            19.16.41  Noetherian left modules I   clnm 27041
            19.16.42  Addenda for structure powers   pwssplit0 27055
            19.16.43  Direct sum of left modules   cdsmm 27065
            19.16.44  Free modules   cfrlm 27080
            19.16.45  Every set admits a group structure iff choice   unxpwdom3 27124
            19.16.46  Independent sets and families   clindf 27142
            19.16.47  Characterization of free modules   lmimlbs 27174
            19.16.48  Noetherian rings and left modules II   clnr 27181
            19.16.49  Hilbert's Basis Theorem   cldgis 27193
            19.16.50  Additional material on polynomials [DEPRECATED]   cmnc 27203
            19.16.51  Degree and minimal polynomial of algebraic numbers   cdgraa 27213
            19.16.52  Algebraic integers I   citgo 27230
            19.16.53  Finite cardinality [SO]   en1uniel 27248
            19.16.54  Words in monoids and ordered group sum   issubmd 27251
            19.16.55  Transpositions in the symmetric group   cpmtr 27252
            19.16.56  The sign of a permutation   cpsgn 27282
            19.16.57  The matrix algebra   cmmul 27307
            19.16.58  The determinant   cmdat 27351
            19.16.59  Endomorphism algebra   cmend 27357
            19.16.60  Subfields   csdrg 27371
            19.16.61  Cyclic groups and order   idomrootle 27379
            19.16.62  Cyclotomic polynomials   ccytp 27389
            19.16.63  Miscellaneous topology   fgraphopab 27397
      19.17  Mathbox for Steve Rodriguez
            19.17.1  Miscellanea   iso0 27404
            19.17.2  Function operations   caofcan 27408
            19.17.3  Calculus   lhe4.4ex1a 27414
      19.18  Mathbox for Andrew Salmon
            19.18.1  Principia Mathematica * 10   pm10.12 27421
            19.18.2  Principia Mathematica * 11   2alanimi 27435
            19.18.3  Predicate Calculus   sbeqal1 27465
            19.18.4  Principia Mathematica * 13 and * 14   pm13.13a 27475
            19.18.5  Set Theory   elnev 27506
            19.18.6  Arithmetic   addcomgi 27528
            19.18.7  Geometry   cplusr 27529
      19.19  Mathbox for Glauco Siliprandi
            19.19.1  Miscellanea   ssrexf 27551
            19.19.2  Finite multiplication of numbers and finite multiplication of functions   fmul01 27577
            19.19.3  Limits   clim1fr1 27594
            19.19.4  Derivatives   dvsinexp 27607
            19.19.5  Integrals   ioovolcl 27609
            19.19.6  Stone Weierstrass theorem - real version   stoweidlem1 27617
            19.19.7  Wallis' product for π   wallispilem1 27681
            19.19.8  Stirling's approximation formula for ` n ` factorial   stirlinglem1 27690
      19.20  Mathbox for Saveliy Skresanov
            19.20.1  Ceva's theorem   sigarval 27707
      19.21  Mathbox for Jarvin Udandy
      19.22  Mathbox for Alexander van der Vekens
            19.22.1  Double restricted existential uniqueness   r19.32 27812
                  19.22.1.1  Restricted quantification (extension)   r19.32 27812
                  19.22.1.2  The empty set (extension)   raaan2 27820
                  19.22.1.3  Restricted uniqueness and "at most one" quantification   rmoimi 27821
                  19.22.1.4  Analogs to Existential uniqueness (double quantification)   2reurex 27826
            19.22.2  Alternative definitions of function's and operation's values   wdfat 27838
                  19.22.2.1  Restricted quantification (extension)   ralbinrald 27844
                  19.22.2.2  The universal class (extension)   nvelim 27845
                  19.22.2.3  Introduce the Axiom of Power Sets (extension)   alneu 27846
                  19.22.2.4  Relations (extension)   sbcrel 27848
                  19.22.2.5  Functions (extension)   sbcfun 27854
                  19.22.2.6  Predicate "defined at"   dfateq12d 27860
                  19.22.2.7  Alternative definition of the value of a function   dfafv2 27863
                  19.22.2.8  Alternative definition of the value of an operation   aoveq123d 27909
            19.22.3  Auxiliary theorems for graph theory   eqneqall 27939
                  19.22.3.1  Negated equality and membership - extension   eqneqall 27939
                  19.22.3.2  "Weak deduction theorem" for set theory - extension   2if2 27941
                  19.22.3.3  Power classes - extension   3xpexg 27942
                  19.22.3.4  Unordered and ordered pairs - extension   nelprd 27943
                  19.22.3.5  Indexed union and intersection - extension   iunxprg 27956
                  19.22.3.6  Relations - extension   resisresindm 27957
                  19.22.3.7  Functions - extension   2f1fvneq 27958
                  19.22.3.8  Equinumerosity - extension   resfnfinfin 27966
                  19.22.3.9  Subtraction - extension   cnm1cn 27968
                  19.22.3.10  Multiplication - extension   kcnktkm1cn 27969
                  19.22.3.11  Ordering on reals (cont.) - extension   leaddsuble 27970
                  19.22.3.12  Nonnegative integers (as a subset of complex numbers) - extension   0mnnnnn0 27971
                  19.22.3.13  Finite intervals of integers - extension   ssfz12 27976
                  19.22.3.14  Half-open integer ranges (extension)   fzo0ss1 27985
                  19.22.3.15  The ` # ` (finite set size) function - extension   hashimarn 27994
                  19.22.3.16  Words over a set - extension   iswrd0i 27999
                  19.22.3.17  Words over a set - extension (subwords of subwords)   swrd0swrd 28009
                  19.22.3.18  Words over a set - extension (subwords of concatenations)   swrdccat3a0 28015
            19.22.4  Graph theory   uhgraedgrnv 28032
                  19.22.4.1  Undirected hypergraphs   uhgraedgrnv 28032
                  19.22.4.2  Undirected simple graphs   usisuhgra 28033
                  19.22.4.3  Neighbors, complete graphs and universal vertices   nbfiusgrafi 28034
                  19.22.4.4  Walks, Paths and Cycles   usgra2pthspth 28035
                  19.22.4.5  Walks/paths of length 2 as ordered triples   c2wlkot 28051
                  19.22.4.6  Vertex Degree   usgfidegfi 28090
                  19.22.4.7  Friendship graphs   cfrgra 28092
      19.23  Mathbox for David A. Wheeler
            19.23.1  Natural deduction   19.8ad 28174
            19.23.2  Greater than, greater than or equal to.   cge-real 28177
            19.23.3  Hyperbolic trig functions   csinh 28187
            19.23.4  Reciprocal trig functions (sec, csc, cot)   csec 28198
            19.23.5  Identities for "if"   ifnmfalse 28220
            19.23.6  Not-member-of   AnelBC 28221
            19.23.7  Decimal point   cdp2 28222
            19.23.8  Signum (sgn or sign) function   csgn 28230
            19.23.9  Ceiling function   ccei 28240
            19.23.10  Logarithms generalized to arbitrary base using ` logb `   ene0 28244
            19.23.11  Logarithm laws generalized to an arbitrary base - log_   clog_ 28247
            19.23.12  Miscellaneous   5m4e1 28249
      19.24  Mathbox for Alan Sare
            19.24.1  Supplementary "adant" deductions   ad4ant13 28252
            19.24.2  Supplementary unification deductions   biimp 28278
            19.24.3  Conventional Metamath proofs, some derived from VD proofs   iidn3 28294
            19.24.4  What is Virtual Deduction?   wvd1 28369
            19.24.5  Virtual Deduction Theorems   df-vd1 28370
            19.24.6  Theorems proved using virtual deduction   trsspwALT 28640
            19.24.7  Theorems proved using virtual deduction with mmj2 assistance   simplbi2VD 28667
            19.24.8  Virtual Deduction transcriptions of textbook proofs   sb5ALTVD 28734
            19.24.9  Theorems proved using conjunction-form virtual deduction   elpwgdedVD 28738
            19.24.10  Theorems with VD proofs in conventional notation derived from VD proofs   suctrALT3 28745
            19.24.11  Theorems with a proof in conventional notation automatically derived   notnot2ALT2 28748
      19.25  Mathbox for Jonathan Ben-Naim
            19.25.1  First order logic and set theory   bnj170 28768
            19.25.2  Well founded induction and recursion   bnj110 28935
            19.25.3  The existence of a minimal element in certain classes   bnj69 29085
            19.25.4  Well-founded induction   bnj1204 29087
            19.25.5  Well-founded recursion, part 1 of 3   bnj60 29137
            19.25.6  Well-founded recursion, part 2 of 3   bnj1500 29143
            19.25.7  Well-founded recursion, part 3 of 3   bnj1522 29147
      19.26  Mathbox for Norm Megill
            19.26.1  Experiments to study ax-7 unbundling   ax-7v 29148
                  19.26.1.1  Theorems derived from ax-7v (suffixes NEW7 and AUX7)   ax-7v 29148
                  19.26.1.2  Theorems derived from ax-7 (suffix OLD7)   ax-7OLD7 29362
            19.26.2  Miscellanea   cnaddcom 29454
            19.26.3  Atoms, hyperplanes, and covering in a left vector space (or module)   clsa 29457
            19.26.4  Functionals and kernels of a left vector space (or module)   clfn 29540
            19.26.5  Opposite rings and dual vector spaces   cld 29606
            19.26.6  Ortholattices and orthomodular lattices   cops 29655
            19.26.7  Atomic lattices with covering property   ccvr 29745
            19.26.8  Hilbert lattices   chlt 29833
            19.26.9  Projective geometries based on Hilbert lattices   clln 29973
            19.26.10  Construction of a vector space from a Hilbert lattice   cdlema1N 30273
            19.26.11  Construction of involution and inner product from a Hilbert lattice   clpoN 31963

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