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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 2189
            1.6.2  Rederive new axioms from old: ax5 , ax6 , ax9from9o , ax11 , ax12from12o   ax4 2199
            1.6.3  Legacy theorems using obsolete axioms   ax17o 2211
      1.7  Existential uniqueness
      1.8  Other axiomatizations related to classical predicate calculus
            1.8.1  Predicate calculus with all distinct variables   ax-7d 2349
            1.8.2  Aristotelian logic: Assertic syllogisms   barbara 2355
            1.8.3  Intuitionistic logic   axi4 2379
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 2389
            2.1.2  Class abstractions (a.k.a. class builders)   cab 2394
            2.1.3  Class form not-free predicate   wnfc 2531
            2.1.4  Negated equality and membership   wne 2571
                  2.1.4.1  Negated equality   nne 2575
                  2.1.4.2  Negated membership   neleq1 2664
            2.1.5  Restricted quantification   wral 2670
            2.1.6  The universal class   cvv 2920
            2.1.7  Conditional equality (experimental)   wcdeq 3108
            2.1.8  Russell's Paradox   ru 3124
            2.1.9  Proper substitution of classes for sets   wsbc 3125
            2.1.10  Proper substitution of classes for sets into classes   csb 3215
            2.1.11  Define basic set operations and relations   cdif 3281
            2.1.12  Subclasses and subsets   df-ss 3298
            2.1.13  The difference, union, and intersection of two classes   difeq1 3422
                  2.1.13.1  The difference of two classes   difeq1 3422
                  2.1.13.2  The union of two classes   elun 3452
                  2.1.13.3  The intersection of two classes   elin 3494
                  2.1.13.4  Combinations of difference, union, and intersection of two classes   unabs 3535
                  2.1.13.5  Class abstractions with difference, union, and intersection of two classes   symdif2 3571
                  2.1.13.6  Restricted uniqueness with difference, union, and intersection   reuss2 3585
            2.1.14  The empty set   c0 3592
            2.1.15  "Weak deduction theorem" for set theory   cif 3703
            2.1.16  Power classes   cpw 3763
            2.1.17  Unordered and ordered pairs   csn 3778
            2.1.18  The union of a class   cuni 3979
            2.1.19  The intersection of a class   cint 4014
            2.1.20  Indexed union and intersection   ciun 4057
            2.1.21  Disjointness   wdisj 4146
            2.1.22  Binary relations   wbr 4176
            2.1.23  Ordered-pair class abstractions (class builders)   copab 4229
            2.1.24  Transitive classes   wtr 4266
      2.2  ZF Set Theory - add the Axiom of Replacement
            2.2.1  Introduce the Axiom of Replacement   ax-rep 4284
            2.2.2  Derive the Axiom of Separation   axsep 4293
            2.2.3  Derive the Null Set Axiom   zfnuleu 4299
            2.2.4  Theorems requiring subset and intersection existence   nalset 4304
            2.2.5  Theorems requiring empty set existence   class2set 4331
      2.3  ZF Set Theory - add the Axiom of Power Sets
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 4341
            2.3.2  Derive the Axiom of Pairing   zfpair 4365
            2.3.3  Ordered pair theorem   opnz 4396
            2.3.4  Ordered-pair class abstractions (cont.)   opabid 4425
            2.3.5  Power class of union and intersection   pwin 4451
            2.3.6  Epsilon and identity relations   cep 4456
            2.3.7  Partial and complete ordering   wpo 4465
            2.3.8  Founded and well-ordering relations   wfr 4502
            2.3.9  Ordinals   word 4544
      2.4  ZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 4664
            2.4.2  Ordinals (continued)   ordon 4726
            2.4.3  Transfinite induction   tfi 4796
            2.4.4  The natural numbers (i.e. finite ordinals)   com 4808
            2.4.5  Peano's postulates   peano1 4827
            2.4.6  Finite induction (for finite ordinals)   find 4833
            2.4.7  Relations   cxp 4839
            2.4.8  Definite description binder (inverted iota)   cio 5379
            2.4.9  Functions   wfun 5411
            2.4.10  Operations   co 6044
            2.4.11  "Maps to" notation   elmpt2cl 6251
            2.4.12  Function operation   cof 6266
            2.4.13  First and second members of an ordered pair   c1st 6310
            2.4.14  Special "Maps to" operations   mpt2xopn0yelv 6427
            2.4.15  Function transposition   ctpos 6441
            2.4.16  Curry and uncurry   ccur 6480
            2.4.17  Proper subset relation   crpss 6484
            2.4.18  Iota properties   fvopab5 6497
            2.4.19  Cantor's Theorem   canth 6502
            2.4.20  Undefined values and restricted iota (description binder)   cund 6504
            2.4.21  Functions on ordinals; strictly monotone ordinal functions   iunon 6563
            2.4.22  "Strong" transfinite recursion   crecs 6595
            2.4.23  Recursive definition generator   crdg 6630
            2.4.24  Finite recursion   frfnom 6655
            2.4.25  Abian's "most fundamental" fixed point theorem   abianfplem 6678
            2.4.26  Ordinal arithmetic   c1o 6680
            2.4.27  Natural number arithmetic   nna0 6810
            2.4.28  Equivalence relations and classes   wer 6865
            2.4.29  The mapping operation   cmap 6981
            2.4.30  Infinite Cartesian products   cixp 7026
            2.4.31  Equinumerosity   cen 7069
            2.4.32  Schroeder-Bernstein Theorem   sbthlem1 7180
            2.4.33  Equinumerosity (cont.)   xpf1o 7232
            2.4.34  Pigeonhole Principle   phplem1 7249
            2.4.35  Finite sets   onomeneq 7259
            2.4.36  Finite intersections   cfi 7377
            2.4.37  Hall's marriage theorem   marypha1lem 7400
            2.4.38  Supremum   csup 7407
            2.4.39  Ordinal isomorphism, Hartog's theorem   coi 7438
            2.4.40  Hartogs function, order types, weak dominance   char 7484
      2.5  ZF Set Theory - add the Axiom of Regularity
            2.5.1  Introduce the Axiom of Regularity   ax-reg 7520
            2.5.2  Axiom of Infinity equivalents   inf0 7536
      2.6  ZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-inf 7553
            2.6.2  Existence of omega (the set of natural numbers)   omex 7558
            2.6.3  Cantor normal form   ccnf 7576
            2.6.4  Transitive closure   trcl 7624
            2.6.5  Rank   cr1 7648
            2.6.6  Scott's trick; collection principle; Hilbert's epsilon   scottex 7769
            2.6.7  Cardinal numbers   ccrd 7782
            2.6.8  Axiom of Choice equivalents   wac 7956
            2.6.9  Cardinal number arithmetic   ccda 8007
            2.6.10  The Ackermann bijection   ackbij2lem1 8059
            2.6.11  Cofinality (without Axiom of Choice)   cflem 8086
            2.6.12  Eight inequivalent definitions of finite set   sornom 8117
            2.6.13  Hereditarily size-limited sets without Choice   itunifval 8256
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 8299
            3.2.2  AC equivalents: well-ordering, Zorn's lemma   numthcor 8334
            3.2.3  Cardinal number theorems using Axiom of Choice   cardval 8381
            3.2.4  Cardinal number arithmetic using Axiom of Choice   iunctb 8409
            3.2.5  Cofinality using Axiom of Choice   alephreg 8417
      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 8517
            4.1.2  Weak universes   cwun 8535
            4.1.3  Tarski's classes   ctsk 8583
            4.1.4  Grothendieck's universes   cgru 8625
      4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
            4.2.1  Introduce the Tarski-Grothendieck Axiom   ax-groth 8658
            4.2.2  Derive the Power Set, Infinity and Choice Axioms   grothpw 8661
            4.2.3  Tarski map function   ctskm 8672
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 8679
            5.1.2  Final derivation of real and complex number postulates   axaddf 8980
            5.1.3  Real and complex number postulates restated as axioms   ax-cnex 9006
      5.2  Derive the basic properties from the field axioms
            5.2.1  Some deductions from the field axioms for complex numbers   cnex 9031
            5.2.2  Infinity and the extended real number system   cpnf 9077
            5.2.3  Restate the ordering postulates with extended real "less than"   axlttri 9107
            5.2.4  Ordering on reals   lttr 9112
            5.2.5  Initial properties of the complex numbers   mul12 9192
      5.3  Real and complex numbers - basic operations
            5.3.1  Addition   add12 9239
            5.3.2  Subtraction   cmin 9251
            5.3.3  Multiplication   muladd 9426
            5.3.4  Ordering on reals (cont.)   gt0ne0 9453
            5.3.5  Reciprocals   ixi 9611
            5.3.6  Division   cdiv 9637
            5.3.7  Ordering on reals (cont.)   elimgt0 9806
            5.3.8  Completeness Axiom and Suprema   fimaxre 9915
            5.3.9  Imaginary and complex number properties   inelr 9950
            5.3.10  Function operation analogue theorems   ofsubeq0 9957
      5.4  Integer sets
            5.4.1  Natural numbers (as a subset of complex numbers)   cn 9960
            5.4.2  Principle of mathematical induction   nnind 9978
            5.4.3  Decimal representation of numbers   c2 10009
            5.4.4  Some properties of specific numbers   0p1e1 10053
            5.4.5  The Archimedean property   nnunb 10177
            5.4.6  Nonnegative integers (as a subset of complex numbers)   cn0 10181
            5.4.7  Integers (as a subset of complex numbers)   cz 10242
            5.4.8  Decimal arithmetic   cdc 10342
            5.4.9  Upper partititions of integers   cuz 10448
            5.4.10  Well-ordering principle for bounded-below sets of integers   uzwo3 10529
            5.4.11  Rational numbers (as a subset of complex numbers)   cq 10534
            5.4.12  Existence of the set of complex numbers   rpnnen1lem1 10560
      5.5  Order sets
            5.5.1  Positive reals (as a subset of complex numbers)   crp 10572
            5.5.2  Infinity and the extended real number system (cont.)   cxne 10667
            5.5.3  Supremum on the extended reals   xrsupexmnf 10843
            5.5.4  Real number intervals   cioo 10876
            5.5.5  Finite intervals of integers   cfz 11003
            5.5.6  Half-open integer ranges   cfzo 11094
      5.6  Elementary integer functions
            5.6.1  The floor (greatest integer) function   cfl 11160
            5.6.2  The modulo (remainder) operation   cmo 11209
            5.6.3  The infinite sequence builder "seq"   om2uz0i 11246
            5.6.4  Integer powers   cexp 11341
            5.6.5  Ordered pair theorem for nonnegative integers   nn0le2msqi 11519
            5.6.6  Factorial function   cfa 11525
            5.6.7  The binomial coefficient operation   cbc 11552
            5.6.8  The ` # ` (finite set size) function   chash 11577
                  5.6.8.1  Finite induction on the size of the first component of a binary relation   brfi1indlem 11673
            5.6.9  Words over a set   cword 11676
            5.6.10  Longer string literals   cs2 11764
      5.7  Elementary real and complex functions
            5.7.1  The "shift" operation   cshi 11840
            5.7.2  Real and imaginary parts; conjugate   ccj 11860
            5.7.3  Square root; absolute value   csqr 11997
      5.8  Elementary limits and convergence
            5.8.1  Superior limit (lim sup)   clsp 12223
            5.8.2  Limits   cli 12237
            5.8.3  Finite and infinite sums   csu 12438
            5.8.4  The binomial theorem   binomlem 12567
            5.8.5  The inclusion/exclusion principle   incexclem 12575
            5.8.6  Infinite sums (cont.)   isumshft 12578
            5.8.7  Miscellaneous converging and diverging sequences   divrcnv 12591
            5.8.8  Arithmetic series   arisum 12598
            5.8.9  Geometric series   expcnv 12602
            5.8.10  Ratio test for infinite series convergence   cvgrat 12619
            5.8.11  Mertens' theorem   mertenslem1 12620
      5.9  Elementary trigonometry
            5.9.1  The exponential, sine, and cosine functions   ce 12623
            5.9.2  _e is irrational   eirrlem 12762
      5.10  Cardinality of real and complex number subsets
            5.10.1  Countability of integers and rationals   xpnnen 12767
            5.10.2  The reals are uncountable   rpnnen2lem1 12773
PART 6  ELEMENTARY NUMBER THEORY
      6.1  Elementary properties of divisibility
            6.1.1  Irrationality of square root of 2   sqr2irrlem 12806
            6.1.2  Some Number sets are chains of proper subsets   nthruc 12809
            6.1.3  The divides relation   cdivides 12811
            6.1.4  The division algorithm   divalglem0 12872
            6.1.5  Bit sequences   cbits 12890
            6.1.6  The greatest common divisor operator   cgcd 12965
            6.1.7  Bézout's identity   bezoutlem1 12997
            6.1.8  Algorithms   nn0seqcvgd 13020
            6.1.9  Euclid's Algorithm   eucalgval2 13031
      6.2  Elementary prime number theory
            6.2.1  Elementary properties   cprime 13038
            6.2.2  Properties of the canonical representation of a rational   cnumer 13084
            6.2.3  Euler's theorem   codz 13111
            6.2.4  Pythagorean Triples   coprimeprodsq 13142
            6.2.5  The prime count function   cpc 13169
            6.2.6  Pocklington's theorem   prmpwdvds 13231
            6.2.7  Infinite primes theorem   unbenlem 13235
            6.2.8  Sum of prime reciprocals   prmreclem1 13243
            6.2.9  Fundamental theorem of arithmetic   1arithlem1 13250
            6.2.10  Lagrange's four-square theorem   cgz 13256
            6.2.11  Van der Waerden's theorem   cvdwa 13292
            6.2.12  Ramsey's theorem   cram 13326
            6.2.13  Decimal arithmetic (cont.)   dec2dvds 13358
            6.2.14  Specific prime numbers   4nprm 13386
            6.2.15  Very large primes   1259lem1 13409
PART 7  BASIC STRUCTURES
      7.1  Extensible structures
            7.1.1  Basic definitions   cstr 13424
            7.1.2  Slot definitions   cplusg 13488
            7.1.3  Definition of the structure product   crest 13607
            7.1.4  Definition of the structure quotient   cordt 13680
      7.2  Moore spaces
            7.2.1  Moore closures   mrcflem 13790
            7.2.2  Independent sets in a Moore system   mrisval 13814
            7.2.3  Algebraic closure systems   isacs 13835
PART 8  BASIC CATEGORY THEORY
      8.1  Categories
            8.1.1  Categories   ccat 13848
            8.1.2  Opposite category   coppc 13896
            8.1.3  Monomorphisms and epimorphisms   cmon 13913
            8.1.4  Sections, inverses, isomorphisms   csect 13929
            8.1.5  Subcategories   cssc 13966
            8.1.6  Functors   cfunc 14010
            8.1.7  Full & faithful functors   cful 14058
            8.1.8  Natural transformations and the functor category   cnat 14097
      8.2  Arrows (disjointified hom-sets)
            8.2.1  Identity and composition for arrows   cida 14167
      8.3  Examples of categories
            8.3.1  The category of sets   csetc 14189
            8.3.2  The category of categories   ccatc 14208
      8.4  Categorical constructions
            8.4.1  Product of categories   cxpc 14224
            8.4.2  Functor evaluation   cevlf 14265
            8.4.3  Hom functor   chof 14304
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 14356
            9.2.2  Lattices   clat 14433
            9.2.3  The dual of an ordered set   codu 14514
            9.2.4  Subset order structures   cipo 14536
            9.2.5  Distributive lattices   latmass 14573
            9.2.6  Posets and lattices as relations   cps 14583
            9.2.7  Directed sets, nets   cdir 14632
PART 10  BASIC ALGEBRAIC STRUCTURES
      10.1  Monoids
            10.1.1  Definition and basic properties   cmnd 14643
            10.1.2  Monoid homomorphisms and submonoids   cmhm 14695
            10.1.3  Ordered group sum operation   gsumvallem1 14730
            10.1.4  Free monoids   cfrmd 14751
      10.2  Groups
            10.2.1  Definition and basic properties   df-grp 14771
            10.2.2  Subgroups and Quotient groups   csubg 14897
            10.2.3  Elementary theory of group homomorphisms   cghm 14962
            10.2.4  Isomorphisms of groups   cgim 15003
            10.2.5  Group actions   cga 15025
            10.2.6  Symmetry groups and Cayley's Theorem   csymg 15051
            10.2.7  Centralizers and centers   ccntz 15073
            10.2.8  The opposite group   coppg 15100
            10.2.9  p-Groups and Sylow groups; Sylow's theorems   cod 15122
            10.2.10  Direct products   clsm 15227
            10.2.11  Free groups   cefg 15297
      10.3  Abelian groups
            10.3.1  Definition and basic properties   ccmn 15371
            10.3.2  Cyclic groups   ccyg 15446
            10.3.3  Group sum operation   gsumval3a 15471
            10.3.4  Internal direct products   cdprd 15513
            10.3.5  The Fundamental Theorem of Abelian Groups   ablfacrplem 15582
      10.4  Rings
            10.4.1  Multiplicative Group   cmgp 15607
            10.4.2  Definition and basic properties   crg 15619
            10.4.3  Opposite ring   coppr 15686
            10.4.4  Divisibility   cdsr 15702
            10.4.5  Ring homomorphisms   crh 15776
      10.5  Division rings and fields
            10.5.1  Definition and basic properties   cdr 15794
            10.5.2  Subrings of a ring   csubrg 15823
            10.5.3  Absolute value (abstract algebra)   cabv 15863
            10.5.4  Star rings   cstf 15890
      10.6  Left modules
            10.6.1  Definition and basic properties   clmod 15909
            10.6.2  Subspaces and spans in a left module   clss 15967
            10.6.3  Homomorphisms and isomorphisms of left modules   clmhm 16054
            10.6.4  Subspace sum; bases for a left module   clbs 16105
      10.7  Vector spaces
            10.7.1  Definition and basic properties   clvec 16133
      10.8  Ideals
            10.8.1  The subring algebra; ideals   csra 16199
            10.8.2  Two-sided ideals and quotient rings   c2idl 16261
            10.8.3  Principal ideal rings. Divisibility in the integers   clpidl 16271
            10.8.4  Nonzero rings   cnzr 16287
            10.8.5  Left regular elements. More kinds of rings   crlreg 16298
      10.9  Associative algebras
            10.9.1  Definition and basic properties   casa 16328
      10.10  Abstract multivariate polynomials
            10.10.1  Definition and basic properties   cmps 16365
            10.10.2  Polynomial evaluation   evlslem4 16523
            10.10.3  Univariate polynomials   cps1 16528
      10.11  The complex numbers as an extensible structure
            10.11.1  Definition and basic properties   cpsmet 16644
            10.11.2  Algebraic constructions based on the complexes   czrh 16737
      10.12  Hilbert spaces
            10.12.1  Definition and basic properties   cphl 16814
            10.12.2  Orthocomplements and closed subspaces   cocv 16846
            10.12.3  Orthogonal projection and orthonormal bases   cpj 16886
PART 11  BASIC TOPOLOGY
      11.1  Topology
            11.1.1  Topological spaces   ctop 16917
            11.1.2  TopBases for topologies   isbasisg 16971
            11.1.3  Examples of topologies   distop 17019
            11.1.4  Closure and interior   ccld 17039
            11.1.5  Neighborhoods   cnei 17120
            11.1.6  Limit points and perfect sets   clp 17157
            11.1.7  Subspace topologies   restrcl 17179
            11.1.8  Order topology   ordtbaslem 17210
            11.1.9  Limits and continuity in topological spaces   ccn 17246
            11.1.10  Separated spaces: T0, T1, T2 (Hausdorff) ...   ct0 17328
            11.1.11  Compactness   ccmp 17407
            11.1.12  Connectedness   ccon 17431
            11.1.13  First- and second-countability   c1stc 17457
            11.1.14  Local topological properties   clly 17484
            11.1.15  Compactly generated spaces   ckgen 17522
            11.1.16  Product topologies   ctx 17549
            11.1.17  Continuous function-builders   cnmptid 17650
            11.1.18  Quotient maps and quotient topology   ckq 17682
            11.1.19  Homeomorphisms   chmeo 17742
      11.2  Filters and filter bases
            11.2.1  Filter bases   elmptrab 17816
            11.2.2  Filters   cfil 17834
            11.2.3  Ultrafilters   cufil 17888
            11.2.4  Filter limits   cfm 17922
            11.2.5  Extension by continuity   ccnext 18047
            11.2.6  Topological groups   ctmd 18057
            11.2.7  Infinite group sum on topological groups   ctsu 18112
            11.2.8  Topological rings, fields, vector spaces   ctrg 18142
      11.3  Uniform Stuctures and Spaces
            11.3.1  Uniform structures   cust 18186
            11.3.2  The topology induced by an uniform structure   cutop 18217
            11.3.3  Uniform Spaces   cuss 18240
            11.3.4  Uniform continuity   cucn 18262
            11.3.5  Cauchy filters in uniform spaces   ccfilu 18273
            11.3.6  Complete uniform spaces   ccusp 18284
      11.4  Metric spaces
            11.4.1  Pseudometric spaces   ispsmet 18292
            11.4.2  Basic metric space properties   cxme 18304
            11.4.3  Metric space balls   blfvalps 18370
            11.4.4  Open sets of a metric space   mopnval 18425
            11.4.5  Continuity in metric spaces   metcnp3 18527
            11.4.6  The uniform structure generated by a metric   metuvalOLD 18536
            11.4.7  Examples of metric spaces   dscmet 18577
            11.4.8  Normed algebraic structures   cnm 18581
            11.4.9  Normed space homomorphisms (bounded linear operators)   cnmo 18696
            11.4.10  Topology on the reals   qtopbaslem 18749
            11.4.11  Topological definitions using the reals   cii 18862
            11.4.12  Path homotopy   chtpy 18949
            11.4.13  The fundamental group   cpco 18982
      11.5  Complex metric vector spaces
            11.5.1  Complex left modules   cclm 19044
            11.5.2  Complex pre-Hilbert space   ccph 19086
            11.5.3  Convergence and completeness   ccfil 19162
            11.5.4  Baire's Category Theorem   bcthlem1 19234
            11.5.5  Banach spaces and complex Hilbert spaces   ccms 19242
            11.5.6  Minimizing Vector Theorem   minveclem1 19282
            11.5.7  Projection Theorem   pjthlem1 19295
PART 12  BASIC REAL AND COMPLEX ANALYSIS
      12.1  Continuity
            12.1.1  Intermediate value theorem   pmltpclem1 19302
      12.2  Integrals
            12.2.1  Lebesgue measure   covol 19316
            12.2.2  Lebesgue integration   cmbf 19463
      12.3  Derivatives
            12.3.1  Real and complex differentiation   climc 19706
PART 13  BASIC REAL AND COMPLEX FUNCTIONS
      13.1  Polynomials
            13.1.1  Abstract polynomials, continued   evlslem6 19891
            13.1.2  Polynomial degrees   cmdg 19933
            13.1.3  The division algorithm for univariate polynomials   cmn1 20005
            13.1.4  Elementary properties of complex polynomials   cply 20060
            13.1.5  The division algorithm for polynomials   cquot 20164
            13.1.6  Algebraic numbers   caa 20188
            13.1.7  Liouville's approximation theorem   aalioulem1 20206
      13.2  Sequences and series
            13.2.1  Taylor polynomials and Taylor's theorem   ctayl 20226
            13.2.2  Uniform convergence   culm 20249
            13.2.3  Power series   pserval 20283
      13.3  Basic trigonometry
            13.3.1  The exponential, sine, and cosine functions (cont.)   efcn 20316
            13.3.2  Properties of pi = 3.14159...   pilem1 20324
            13.3.3  Mapping of the exponential function   efgh 20400
            13.3.4  The natural logarithm on complex numbers   clog 20409
            13.3.5  Theorems of Pythagoras, isosceles triangles, and intersecting chords   angval 20600
            13.3.6  Solutions of quadratic, cubic, and quartic equations   quad2 20636
            13.3.7  Inverse trigonometric functions   casin 20659
            13.3.8  The Birthday Problem   log2ublem1 20743
            13.3.9  Areas in R^2   carea 20751
            13.3.10  More miscellaneous converging sequences   rlimcnp 20761
            13.3.11  Inequality of arithmetic and geometric means   cvxcl 20780
            13.3.12  Euler-Mascheroni constant   cem 20787
      13.4  Basic number theory
            13.4.1  Wilson's theorem   wilthlem1 20808
            13.4.2  The Fundamental Theorem of Algebra   ftalem1 20812
            13.4.3  The Basel problem (ζ(2) = π2/6)   basellem1 20820
            13.4.4  Number-theoretical functions   ccht 20830
            13.4.5  Perfect Number Theorem   mersenne 20968
            13.4.6  Characters of Z/nZ   cdchr 20973
            13.4.7  Bertrand's postulate   bcctr 21016
            13.4.8  Legendre symbol   clgs 21035
            13.4.9  Quadratic reciprocity   lgseisenlem1 21090
            13.4.10  All primes 4n+1 are the sum of two squares   2sqlem1 21104
            13.4.11  Chebyshev's Weak Prime Number Theorem, Dirichlet's Theorem   chebbnd1lem1 21120
            13.4.12  The Prime Number Theorem   mudivsum 21181
            13.4.13  Ostrowski's theorem   abvcxp 21266
PART 14  GRAPH THEORY
      14.1  Undirected graphs - basics
            14.1.1  Undirected hypergraphs   cuhg 21291
            14.1.2  Undirected multigraphs   cumg 21304
            14.1.3  Undirected simple graphs   cuslg 21321
                  14.1.3.1  Undirected simple graphs - basics   cuslg 21321
                  14.1.3.2  Undirected simple graphs - examples   usgraexvlem 21371
                  14.1.3.3  Finite undirected simple graphs   fiusgraedgfi 21378
            14.1.4  Neighbors, complete graphs and universal vertices   cnbgra 21387
                  14.1.4.1  Neighbors   nbgraop 21393
                  14.1.4.2  Complete graphs   iscusgra 21422
                  14.1.4.3  Universal vertices   isuvtx 21454
            14.1.5  Walks, paths and cycles   cwalk 21463
                  14.1.5.1  Walks and trails   wlks 21483
                  14.1.5.2  Paths and simple paths   pths 21523
                  14.1.5.3  Circuits and cycles   crcts 21566
                  14.1.5.4  Connected graphs   cconngra 21613
            14.1.6  Vertex Degree   cvdg 21621
      14.2  Eulerian paths and the Konigsberg Bridge problem
            14.2.1  Eulerian paths   ceup 21641
            14.2.2  The Konigsberg Bridge problem   vdeg0i 21661
PART 15  GUIDES AND MISCELLANEA
      15.1  Guides (conventions, explanations, and examples)
            15.1.1  Conventions   conventions 21667
            15.1.2  Natural deduction   natded 21668
            15.1.3  Natural deduction examples   ex-natded5.2 21669
            15.1.4  Definitional examples   ex-or 21686
      15.2  Humor
            15.2.1  April Fool's theorem   avril1 21714
      15.3  (Future - to be reviewed and classified)
            15.3.1  Planar incidence geometry   cplig 21720
            15.3.2  Algebra preliminaries   crpm 21725
            15.3.3  Transitive closure   ctcl 21727
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 21731
            16.1.2  Definition and basic properties of Abelian groups   cablo 21826
            16.1.3  Subgroups   csubgo 21846
            16.1.4  Operation properties   cass 21857
            16.1.5  Group-like structures   cmagm 21863
            16.1.6  Examples of Abelian groups   ablosn 21892
            16.1.7  Group homomorphism and isomorphism   cghom 21902
      16.2  Additional material on rings and fields
            16.2.1  Definition and basic properties   crngo 21920
            16.2.2  Examples of rings   cnrngo 21948
            16.2.3  Division Rings   cdrng 21950
            16.2.4  Star Fields   csfld 21953
            16.2.5  Fields and Rings   ccm2 21955
PART 17  COMPLEX TOPOLOGICAL VECTOR SPACES (DEPRECATED)
      17.1  Complex vector spaces
            17.1.1  Definition and basic properties   cvc 21981
            17.1.2  Examples of complex vector spaces   cncvc 22019
      17.2  Normed complex vector spaces
            17.2.1  Definition and basic properties   cnv 22020
            17.2.2  Examples of normed complex vector spaces   cnnv 22125
            17.2.3  Induced metric of a normed complex vector space   imsval 22134
            17.2.4  Inner product   cdip 22153
            17.2.5  Subspaces   css 22177
      17.3  Operators on complex vector spaces
            17.3.1  Definitions and basic properties   clno 22198
      17.4  Inner product (pre-Hilbert) spaces
            17.4.1  Definition and basic properties   ccphlo 22270
            17.4.2  Examples of pre-Hilbert spaces   cncph 22277
            17.4.3  Properties of pre-Hilbert spaces   isph 22280
      17.5  Complex Banach spaces
            17.5.1  Definition and basic properties   ccbn 22321
            17.5.2  Examples of complex Banach spaces   cnbn 22328
            17.5.3  Uniform Boundedness Theorem   ubthlem1 22329
            17.5.4  Minimizing Vector Theorem   minvecolem1 22333
      17.6  Complex Hilbert spaces
            17.6.1  Definition and basic properties   chlo 22344
            17.6.2  Standard axioms for a complex Hilbert space   hlex 22357
            17.6.3  Examples of complex Hilbert spaces   cnchl 22375
            17.6.4  Subspaces   ssphl 22376
            17.6.5  Hellinger-Toeplitz Theorem   htthlem 22377
PART 18  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      18.1  Axiomatization of complex pre-Hilbert spaces
            18.1.1  Basic Hilbert space definitions   chil 22379
            18.1.2  Preliminary ZFC lemmas   df-hnorm 22428
            18.1.3  Derive the Hilbert space axioms from ZFC set theory   axhilex-zf 22441
            18.1.4  Introduce the vector space axioms for a Hilbert space   ax-hilex 22459
            18.1.5  Vector operations   hvmulex 22471
            18.1.6  Inner product postulates for a Hilbert space   ax-hfi 22538
      18.2  Inner product and norms
            18.2.1  Inner product   his5 22545
            18.2.2  Norms   dfhnorm2 22581
            18.2.3  Relate Hilbert space to normed complex vector spaces   hilablo 22619
            18.2.4  Bunjakovaskij-Cauchy-Schwarz inequality   bcsiALT 22638
      18.3  Cauchy sequences and completeness axiom
            18.3.1  Cauchy sequences and limits   hcau 22643
            18.3.2  Derivation of the completeness axiom from ZF set theory   hilmet 22653
            18.3.3  Completeness postulate for a Hilbert space   ax-hcompl 22661
            18.3.4  Relate Hilbert space to ZFC pre-Hilbert and Hilbert spaces   hhcms 22662
      18.4  Subspaces and projections
            18.4.1  Subspaces   df-sh 22666
            18.4.2  Closed subspaces   df-ch 22681
            18.4.3  Orthocomplements   df-oc 22711
            18.4.4  Subspace sum, span, lattice join, lattice supremum   df-shs 22767
            18.4.5  Projection theorem   pjhthlem1 22850
            18.4.6  Projectors   df-pjh 22854
      18.5  Properties of Hilbert subspaces
            18.5.1  Orthomodular law   omlsilem 22861
            18.5.2  Projectors (cont.)   pjhtheu2 22875
            18.5.3  Hilbert lattice operations   sh0le 22899
            18.5.4  Span (cont.) and one-dimensional subspaces   spansn0 23000
            18.5.5  Commutes relation for Hilbert lattice elements   df-cm 23042
            18.5.6  Foulis-Holland theorem   fh1 23077
            18.5.7  Quantum Logic Explorer axioms   qlax1i 23086
            18.5.8  Orthogonal subspaces   chscllem1 23096
            18.5.9  Orthoarguesian laws 5OA and 3OA   5oalem1 23113
            18.5.10  Projectors (cont.)   pjorthi 23128
            18.5.11  Mayet's equation E_3   mayete3i 23187
      18.6  Operators on Hilbert spaces
            18.6.1  Operator sum, difference, and scalar multiplication   df-hosum 23190
            18.6.2  Zero and identity operators   df-h0op 23208
            18.6.3  Operations on Hilbert space operators   hoaddcl 23218
            18.6.4  Linear, continuous, bounded, Hermitian, unitary operators and norms   df-nmop 23299
            18.6.5  Linear and continuous functionals and norms   df-nmfn 23305
            18.6.6  Adjoint   df-adjh 23309
            18.6.7  Dirac bra-ket notation   df-bra 23310
            18.6.8  Positive operators   df-leop 23312
            18.6.9  Eigenvectors, eigenvalues, spectrum   df-eigvec 23313
            18.6.10  Theorems about operators and functionals   nmopval 23316
            18.6.11  Riesz lemma   riesz3i 23522
            18.6.12  Adjoints (cont.)   cnlnadjlem1 23527
            18.6.13  Quantum computation error bound theorem   unierri 23564
            18.6.14  Dirac bra-ket notation (cont.)   branmfn 23565
            18.6.15  Positive operators (cont.)   leopg 23582
            18.6.16  Projectors as operators   pjhmopi 23606
      18.7  States on a Hilbert lattice and Godowski's equation
            18.7.1  States on a Hilbert lattice   df-st 23671
            18.7.2  Godowski's equation   golem1 23731
      18.8  Cover relation, atoms, exchange axiom, and modular symmetry
            18.8.1  Covers relation; modular pairs   df-cv 23739
            18.8.2  Atoms   df-at 23798
            18.8.3  Superposition principle   superpos 23814
            18.8.4  Atoms, exchange and covering properties, atomicity   chcv1 23815
            18.8.5  Irreducibility   chirredlem1 23850
            18.8.6  Atoms (cont.)   atcvat3i 23856
            18.8.7  Modular symmetry   mdsymlem1 23863
PART 19  SUPPLEMENTARY MATERIAL (USER'S MATHBOXES)
      19.1  Mathboxes for user contributions
            19.1.1  Mathbox guidelines   mathbox 23902
      19.2  Mathbox for Stefan Allan
      19.3  Mathbox for Thierry Arnoux
            19.3.1  Propositional Calculus - misc additions   bian1d 23907
            19.3.2  Predicate Calculus   abeq2f 23917
                  19.3.2.1  Predicate Calculus - misc additions   abeq2f 23917
                  19.3.2.2  Restricted quantification - misc additions   reximddv 23919
                  19.3.2.3  Substitution (without distinct variables) - misc additions   clelsb3f 23928
                  19.3.2.4  Existential "at most one" - misc additions   mo5f 23929
                  19.3.2.5  Existential uniqueness - misc additions   2reuswap2 23932
                  19.3.2.6  Restricted "at most one" - misc additions   rmoxfrdOLD 23936
            19.3.3  General Set Theory   ceqsexv2d 23942
                  19.3.3.1  Class abstractions (a.k.a. class builders)   ceqsexv2d 23942
                  19.3.3.2  Image Sets   abrexdomjm 23945
                  19.3.3.3  Set relations and operations - misc additions   eqri 23951
                  19.3.3.4  Unordered pairs   elpreq 23956
                  19.3.3.5  Conditional operator - misc additions   ifeqeqx 23958
                  19.3.3.6  Indexed union - misc additions   iuneq12daf 23964
                  19.3.3.7  Disjointness - misc additions   cbvdisjf 23972
            19.3.4  Relations and Functions   dfrel4 23991
                  19.3.4.1  Relations - misc additions   dfrel4 23991
                  19.3.4.2  Functions - misc additions   fdmrn 23996
                  19.3.4.3  Isomorphisms - misc. add.   gtiso 24045
                  19.3.4.4  Disjointness (additional proof requiring functions)   disjdsct 24047
                  19.3.4.5  First and second members of an ordered pair - misc additions   df1stres 24048
                  19.3.4.6  Supremum - misc additions   supssd 24055
                  19.3.4.7  Countable Sets   nnct 24056
            19.3.5  Real and Complex Numbers   addeq0 24071
                  19.3.5.1  Complex addition - misc. additions   addeq0 24071
                  19.3.5.2  Ordering on reals - misc additions   lt2addrd 24072
                  19.3.5.3  Extended reals - misc additions   xgepnf 24073
                  19.3.5.4  Real number intervals - misc additions   icossicc 24086
                  19.3.5.5  Finite intervals of integers - misc additions   fzssnn 24104
                  19.3.5.6  Half-open integer ranges - misc additions   iundisjfi 24109
                  19.3.5.7  The ` # ` (finite set size) function - misc additions   hashresfn 24113
                  19.3.5.8  The greatest common divisor operator - misc. add   numdenneg 24117
                  19.3.5.9  Integers   ltesubnnd 24119
                  19.3.5.10  Division in the extended real number system   cxdiv 24120
            19.3.6  Structure builders   ress0g 24139
                  19.3.6.1  Structure builder restriction operator   ress0g 24139
                  19.3.6.2  Posets   tospos 24143
                  19.3.6.3  Complete lattices   clatp0ex 24150
                  19.3.6.4  Extended reals Structure - misc additions   ax-xrssca 24152
                  19.3.6.5  The extended non-negative real numbers monoid   xrge0base 24164
            19.3.7  Algebra   sumpr 24175
                  19.3.7.1  Finitely supported group sums - misc additions   sumpr 24175
                  19.3.7.2  Rings - misc additions   dvrdir 24183
                  19.3.7.3  Ordered groups   cogrp 24188
                  19.3.7.4  Ordered fields   cofld 24190
                  19.3.7.5  The Archimedean property for generic algebraic structures   cinftm 24203
                  19.3.7.6  Ring homomorphisms - misc additions   rhmdvdsr 24213
                  19.3.7.7  The ring of integers   zzsbase 24220
                  19.3.7.8  The ordered field of reals   rebase 24226
            19.3.8  Topology   cmetid 24238
                  19.3.8.1  Pseudometrics   cmetid 24238
                  19.3.8.2  Continuity - misc additions   hauseqcn 24250
                  19.3.8.3  Topology of the closed unit   unitsscn 24251
                  19.3.8.4  Topology of ` ( RR X. RR ) `   unicls 24258
                  19.3.8.5  Order topology - misc. additions   cnvordtrestixx 24268
                  19.3.8.6  Continuity in topological spaces - misc. additions   mndpluscn 24269
                  19.3.8.7  Topology of the extended non-negative real numbers monoid   xrge0hmph 24275
                  19.3.8.8  Limits - misc additions   lmlim 24290
            19.3.9  Uniform Stuctures and Spaces   chcmp 24297
                  19.3.9.1  Hausdorff Completion   chcmp 24297
            19.3.10  Topology and algebraic structures   zzsnm 24299
                  19.3.10.1  The norm on the ring of the integer numbers   zzsnm 24299
                  19.3.10.2  The complete ordered field of the real numbers   recms 24300
                  19.3.10.3  Topological ` ZZ ` -modules   zlm0 24303
                  19.3.10.4  The canonical embedding of the rational numbers into a division ring   cqqh 24313
                  19.3.10.5  The canonical embedding of ` RR ` into a complete ordered field   crrh 24334
                  19.3.10.6  Embedding into ` RR* `   cxrh 24339
                  19.3.10.7  Canonical embeddings into ` RR `   zrhre 24342
            19.3.11  Real and complex functions   clogb 24345
                  19.3.11.1  Logarithm laws generalized to an arbitrary base - logb   clogb 24345
                  19.3.11.2  Indicator Functions   cind 24365
                  19.3.11.3  Extended sum   cesum 24381
            19.3.12  Mixed Function/Constant operation   cofc 24435
            19.3.13  Abstract measure   csiga 24447
                  19.3.13.1  Sigma-Algebra   csiga 24447
                  19.3.13.2  Generated Sigma-Algebra   csigagen 24478
                  19.3.13.3  The Borel algebra on the real numbers   cbrsiga 24492
                  19.3.13.4  Product Sigma-Algebra   csx 24499
                  19.3.13.5  Measures   cmeas 24506
                  19.3.13.6  The counting measure   cntmeas 24537
                  19.3.13.7  The Lebesgue measure - misc additions   volss 24540
                  19.3.13.8  The 'almost everywhere' relation   cae 24545
                  19.3.13.9  Measurable functions   cmbfm 24557
                  19.3.13.10  Borel Algebra on ` ( RR X. RR ) `   br2base 24576
            19.3.14  Integration   itgeq12dv 24598
                  19.3.14.1  Lebesgue integral - misc additions   itgeq12dv 24598
                  19.3.14.2  Bochner integral   citgm 24599
            19.3.15  Probability   cprb 24622
                  19.3.15.1  Probability Theory   cprb 24622
                  19.3.15.2  Conditional Probabilities   ccprob 24646
                  19.3.15.3  Real Valued Random Variables   crrv 24655
                  19.3.15.4  Preimage set mapping operator   corvc 24670
                  19.3.15.5  Distribution Functions   orvcelval 24683
                  19.3.15.6  Cumulative Distribution Functions   orvclteel 24687
                  19.3.15.7  Probabilities - example   coinfliplem 24693
                  19.3.15.8  Bertrand's Ballot Problem   ballotlemoex 24700
      19.4  Mathbox for Mario Carneiro
            19.4.1  Miscellaneous stuff   quartfull 24753
            19.4.2  Zeta function   czeta 24754
            19.4.3  Gamma function   clgam 24757
            19.4.4  Derangements and the Subfactorial   deranglem 24809
            19.4.5  The Erdős-Szekeres theorem   erdszelem1 24834
            19.4.6  The Kuratowski closure-complement theorem   kur14lem1 24849
            19.4.7  Retracts and sections   cretr 24860
            19.4.8  Path-connected and simply connected spaces   cpcon 24863
            19.4.9  Covering maps   ccvm 24899
            19.4.10  Normal numbers   snmlff 24973
            19.4.11  Godel-sets of formulas   cgoe 24977
            19.4.12  Models of ZF   cgze 25005
            19.4.13  Splitting fields   citr 25019
            19.4.14  p-adic number fields   czr 25035
      19.5  Mathbox for Paul Chapman
            19.5.1  Group homomorphism and isomorphism   ghomgrpilem1 25053
            19.5.2  Real and complex numbers (cont.)   climuzcnv 25065
            19.5.3  Miscellaneous theorems   elfzm12 25069
      19.6  Mathbox for Drahflow
      19.7  Mathbox for Scott Fenton
            19.7.1  ZFC Axioms in primitive form   axextprim 25107
            19.7.2  Untangled classes   untelirr 25114
            19.7.3  Extra propositional calculus theorems   3orel1 25121
            19.7.4  Misc. Useful Theorems   nepss 25132
            19.7.5  Properties of reals and complexes   sqdivzi 25141
            19.7.6  Product sequences   prodf 25172
            19.7.7  Non-trivial convergence   ntrivcvg 25182
            19.7.8  Complex products   cprod 25188
            19.7.9  Finite products   fprod 25224
            19.7.10  Infinite products   iprodclim 25268
            19.7.11  Falling and Rising Factorial   cfallfac 25277
            19.7.12  Factorial limits   faclimlem1 25314
            19.7.13  Greatest common divisor and divisibility   pdivsq 25320
            19.7.14  Properties of relationships   brtp 25324
            19.7.15  Properties of functions and mappings   funpsstri 25339
            19.7.16  Epsilon induction   setinds 25352
            19.7.17  Ordinal numbers   elpotr 25355
            19.7.18  Defined equality axioms   axextdfeq 25372
            19.7.19  Hypothesis builders   hbntg 25380
            19.7.20  The Predecessor Class   cpred 25385
            19.7.21  (Trans)finite Recursion Theorems   tfisg 25422
            19.7.22  Well-founded induction   tz6.26 25423
            19.7.23  Transitive closure under a relationship   ctrpred 25438
            19.7.24  Founded Induction   frmin 25460
            19.7.25  Ordering Ordinal Sequences   orderseqlem 25470
            19.7.26  Well-founded recursion   wfr3g 25473
            19.7.27  Transfinite recursion via Well-founded recursion   tfrALTlem 25494
            19.7.28  Founded Recursion   frr3g 25498
            19.7.29  Surreal Numbers   csur 25512
            19.7.30  Surreal Numbers: Ordering   sltsolem1 25540
            19.7.31  Surreal Numbers: Birthday Function   bdayfo 25547
            19.7.32  Surreal Numbers: Density   fvnobday 25554
            19.7.33  Surreal Numbers: Density   nodenselem3 25555
            19.7.34  Surreal Numbers: Upper and Lower Bounds   nobndlem1 25564
            19.7.35  Surreal Numbers: Full-Eta Property   nofulllem1 25574
            19.7.36  Symmetric difference   csymdif 25579
            19.7.37  Quantifier-free definitions   ctxp 25591
            19.7.38  Alternate ordered pairs   caltop 25709
            19.7.39  Tarskian geometry   cee 25735
            19.7.40  Tarski's axioms for geometry   axdimuniq 25760
            19.7.41  Congruence properties   cofs 25824
            19.7.42  Betweenness properties   btwntriv2 25854
            19.7.43  Segment Transportation   ctransport 25871
            19.7.44  Properties relating betweenness and congruence   cifs 25877
            19.7.45  Connectivity of betweenness   btwnconn1lem1 25929
            19.7.46  Segment less than or equal to   csegle 25948
            19.7.47  Outside of relationship   coutsideof 25961
            19.7.48  Lines and Rays   cline2 25976
            19.7.49  Bernoulli polynomials and sums of k-th powers   cbp 26000
            19.7.50  Rank theorems   rankung 26015
            19.7.51  Hereditarily Finite Sets   chf 26021
      19.8  Mathbox for Anthony Hart
            19.8.1  Propositional Calculus   tb-ax1 26036
            19.8.2  Predicate Calculus   quantriv 26058
            19.8.3  Misc. Single Axiom Systems   meran1 26069
            19.8.4  Connective Symmetry   negsym1 26075
      19.9  Mathbox for Chen-Pang He
            19.9.1  Ordinal topology   ontopbas 26086
      19.10  Mathbox for Jeff Hoffman
            19.10.1  Inferences for finite induction on generic function values   fveleq 26109
            19.10.2  gdc.mm   nnssi2 26113
      19.11  Mathbox for Wolf Lammen
      19.12  Mathbox for Brendan Leahy
      19.13  Mathbox for Jeff Hankins
            19.13.1  Miscellany   a1i13 26192
            19.13.2  Basic topological facts   topbnd 26221
            19.13.3  Topology of the real numbers   ivthALT 26232
            19.13.4  Refinements   cfne 26233
            19.13.5  Neighborhood bases determine topologies   neibastop1 26282
            19.13.6  Lattice structure of topologies   topmtcl 26286
            19.13.7  Filter bases   fgmin 26293
            19.13.8  Directed sets, nets   tailfval 26295
      19.14  Mathbox for Jeff Madsen
            19.14.1  Logic and set theory   anim12da 26306
            19.14.2  Real and complex numbers; integers   filbcmb 26336
            19.14.3  Sequences and sums   sdclem2 26340
            19.14.4  Topology   subspopn 26352
            19.14.5  Metric spaces   metf1o 26355
            19.14.6  Continuous maps and homeomorphisms   constcncf 26362
            19.14.7  Boundedness   ctotbnd 26369
            19.14.8  Isometries   cismty 26401
            19.14.9  Heine-Borel Theorem   heibor1lem 26412
            19.14.10  Banach Fixed Point Theorem   bfplem1 26425
            19.14.11  Euclidean space   crrn 26428
            19.14.12  Intervals (continued)   ismrer1 26441
            19.14.13  Groups and related structures   exidcl 26445
            19.14.14  Rings   rngonegcl 26455
            19.14.15  Ring homomorphisms   crnghom 26470
            19.14.16  Commutative rings   ccring 26499
            19.14.17  Ideals   cidl 26511
            19.14.18  Prime rings and integral domains   cprrng 26550
            19.14.19  Ideal generators   cigen 26563
      19.15  Mathbox for Rodolfo Medina
            19.15.1  Partitions   prtlem60 26582
      19.16  Mathbox for Stefan O'Rear
            19.16.1  Additional elementary logic and set theory   nelss 26626
            19.16.2  Additional theory of functions   fninfp 26629
            19.16.3  Extensions beyond function theory   gsumvsmul 26639
            19.16.4  Additional topology   elrfi 26642
            19.16.5  Characterization of closure operators. Kuratowski closure axioms   ismrcd1 26646
            19.16.6  Algebraic closure systems   cnacs 26650
            19.16.7  Miscellanea 1. Map utilities   constmap 26661
            19.16.8  Miscellanea for polynomials   ofmpteq 26670
            19.16.9  Multivariate polynomials over the integers   cmzpcl 26672
            19.16.10  Miscellanea for Diophantine sets 1   coeq0 26704
            19.16.11  Diophantine sets 1: definitions   cdioph 26707
            19.16.12  Diophantine sets 2 miscellanea   ellz1 26719
            19.16.13  Diophantine sets 2: union and intersection. Monotone Boolean algebra   diophin 26725
            19.16.14  Diophantine sets 3: construction   diophrex 26728
            19.16.15  Diophantine sets 4 miscellanea   2sbcrex 26737
            19.16.16  Diophantine sets 4: Quantification   rexrabdioph 26748
            19.16.17  Diophantine sets 5: Arithmetic sets   rabdiophlem1 26755
            19.16.18  Diophantine sets 6 miscellanea   fz1ssnn 26765
            19.16.19  Diophantine sets 6: reusability. renumbering of variables   eldioph4b 26766
            19.16.20  Pigeonhole Principle and cardinality helpers   fphpd 26771
            19.16.21  A non-closed set of reals is infinite   rencldnfilem 26775
            19.16.22  Miscellanea for Lagrange's theorem   icodiamlt 26777
            19.16.23  Lagrange's rational approximation theorem   irrapxlem1 26779
            19.16.24  Pell equations 1: A nontrivial solution always exists   pellexlem1 26786
            19.16.25  Pell equations 2: Algebraic number theory of the solution set   csquarenn 26793
            19.16.26  Pell equations 3: characterizing fundamental solution   infmrgelbi 26835
            19.16.27  Logarithm laws generalized to an arbitrary base   reglogcl 26847
            19.16.28  Pell equations 4: the positive solution group is infinite cyclic   pellfund14 26855
            19.16.29  X and Y sequences 1: Definition and recurrence laws   crmx 26857
            19.16.30  Ordering and induction lemmas for the integers   monotuz 26898
            19.16.31  X and Y sequences 2: Order properties   rmxypos 26906
            19.16.32  Congruential equations   congtr 26924
            19.16.33  Alternating congruential equations   acongid 26934
            19.16.34  Additional theorems on integer divisibility   bezoutr 26944
            19.16.35  X and Y sequences 3: Divisibility properties   jm2.18 26953
            19.16.36  X and Y sequences 4: Diophantine representability of Y   jm2.27a 26970
            19.16.37  X and Y sequences 5: Diophantine representability of X, ^, _C   rmxdiophlem 26980
            19.16.38  Uncategorized stuff not associated with a major project   setindtr 26989
            19.16.39  More equivalents of the Axiom of Choice   axac10 26998
            19.16.40  Finitely generated left modules   clfig 27037
            19.16.41  Noetherian left modules I   clnm 27045
            19.16.42  Addenda for structure powers   pwssplit0 27059
            19.16.43  Direct sum of left modules   cdsmm 27069
            19.16.44  Free modules   cfrlm 27084
            19.16.45  Every set admits a group structure iff choice   unxpwdom3 27128
            19.16.46  Independent sets and families   clindf 27146
            19.16.47  Characterization of free modules   lmimlbs 27178
            19.16.48  Noetherian rings and left modules II   clnr 27185
            19.16.49  Hilbert's Basis Theorem   cldgis 27197
            19.16.50  Additional material on polynomials [DEPRECATED]   cmnc 27207
            19.16.51  Degree and minimal polynomial of algebraic numbers   cdgraa 27217
            19.16.52  Algebraic integers I   citgo 27234
            19.16.53  Finite cardinality [SO]   en1uniel 27252
            19.16.54  Words in monoids and ordered group sum   issubmd 27255
            19.16.55  Transpositions in the symmetric group   cpmtr 27256
            19.16.56  The sign of a permutation   cpsgn 27286
            19.16.57  The matrix algebra   cmmul 27311
            19.16.58  The determinant   cmdat 27355
            19.16.59  Endomorphism algebra   cmend 27361
            19.16.60  Subfields   csdrg 27375
            19.16.61  Cyclic groups and order   idomrootle 27383
            19.16.62  Cyclotomic polynomials   ccytp 27393
            19.16.63  Miscellaneous topology   fgraphopab 27401
      19.17  Mathbox for Steve Rodriguez
            19.17.1  Miscellanea   iso0 27408
            19.17.2  Function operations   caofcan 27412
            19.17.3  Calculus   lhe4.4ex1a 27418
      19.18  Mathbox for Andrew Salmon
            19.18.1  Principia Mathematica * 10   pm10.12 27425
            19.18.2  Principia Mathematica * 11   2alanimi 27439
            19.18.3  Predicate Calculus   sbeqal1 27469
            19.18.4  Principia Mathematica * 13 and * 14   pm13.13a 27479
            19.18.5  Set Theory   elnev 27510
            19.18.6  Arithmetic   addcomgi 27532
            19.18.7  Geometry   cplusr 27533
      19.19  Mathbox for Glauco Siliprandi
            19.19.1  Miscellanea   ssrexf 27555
            19.19.2  Finite multiplication of numbers and finite multiplication of functions   fmul01 27581
            19.19.3  Limits   clim1fr1 27598
            19.19.4  Derivatives   dvsinexp 27611
            19.19.5  Integrals   ioovolcl 27613
            19.19.6  Stone Weierstrass theorem - real version   stoweidlem1 27621
            19.19.7  Wallis' product for π   wallispilem1 27685
            19.19.8  Stirling's approximation formula for ` n ` factorial   stirlinglem1 27694
      19.20  Mathbox for Saveliy Skresanov
            19.20.1  Ceva's theorem   sigarval 27711
      19.21  Mathbox for Jarvin Udandy
      19.22  Mathbox for Alexander van der Vekens
            19.22.1  Double restricted existential uniqueness   r19.32 27816
                  19.22.1.1  Restricted quantification (extension)   r19.32 27816
                  19.22.1.2  The empty set (extension)   raaan2 27824
                  19.22.1.3  Restricted uniqueness and "at most one" quantification   rmoimi 27825
                  19.22.1.4  Analogs to Existential uniqueness (double quantification)   2reurex 27830
            19.22.2  Alternative definitions of function's and operation's values   wdfat 27842
                  19.22.2.1  Restricted quantification (extension)   ralbinrald 27848
                  19.22.2.2  The universal class (extension)   nvelim 27849
                  19.22.2.3  Introduce the Axiom of Power Sets (extension)   alneu 27850
                  19.22.2.4  Relations (extension)   sbcrel 27852
                  19.22.2.5  Functions (extension)   sbcfun 27858
                  19.22.2.6  Predicate "defined at"   dfateq12d 27864
                  19.22.2.7  Alternative definition of the value of a function   dfafv2 27867
                  19.22.2.8  Alternative definition of the value of an operation   aoveq123d 27913
            19.22.3  Auxiliary theorems for graph theory   eqneqall 27943
                  19.22.3.1  Negated equality and membership - extension   eqneqall 27943
                  19.22.3.2  "Weak deduction theorem" for set theory - extension   2if2 27945
                  19.22.3.3  Power classes - extension   3xpexg 27946
                  19.22.3.4  Unordered and ordered pairs - extension   nelprd 27947
                  19.22.3.5  Indexed union and intersection - extension   iunxprg 27960
                  19.22.3.6  Relations - extension   resisresindm 27961
                  19.22.3.7  Functions - extension   2f1fvneq 27962
                  19.22.3.8  Equinumerosity - extension   resfnfinfin 27970
                  19.22.3.9  Subtraction - extension   cnm1cn 27972
                  19.22.3.10  Multiplication - extension   kcnktkm1cn 27973
                  19.22.3.11  Ordering on reals (cont.) - extension   leaddsuble 27974
                  19.22.3.12  Nonnegative integers (as a subset of complex numbers) - extension   0mnnnnn0 27975
                  19.22.3.13  Finite intervals of integers - extension   ssfz12 27980
                  19.22.3.14  Half-open integer ranges (extension)   fzo0ss1 27989
                  19.22.3.15  The ` # ` (finite set size) function - extension   hashimarn 27998
                  19.22.3.16  Words over a set - extension   iswrd0i 28003
                  19.22.3.17  Words over a set - extension (subwords of subwords)   swrd0swrd 28013
                  19.22.3.18  Words over a set - extension (subwords of concatenations)   swrdccat3a0 28019
            19.22.4  Graph theory   uhgraedgrnv 28036
                  19.22.4.1  Undirected hypergraphs   uhgraedgrnv 28036
                  19.22.4.2  Undirected simple graphs   usisuhgra 28037
                  19.22.4.3  Neighbors, complete graphs and universal vertices   nbfiusgrafi 28038
                  19.22.4.4  Walks, Paths and Cycles   usgra2pthspth 28039
                  19.22.4.5  Walks/paths of length 2 as ordered triples   c2wlkot 28055
                  19.22.4.6  Vertex Degree   usgfidegfi 28094
                  19.22.4.7  Friendship graphs   cfrgra 28096
      19.23  Mathbox for David A. Wheeler
            19.23.1  Natural deduction   19.8ad 28178
            19.23.2  Greater than, greater than or equal to.   cge-real 28181
            19.23.3  Hyperbolic trig functions   csinh 28191
            19.23.4  Reciprocal trig functions (sec, csc, cot)   csec 28202
            19.23.5  Identities for "if"   ifnmfalse 28224
            19.23.6  Not-member-of   AnelBC 28225
            19.23.7  Decimal point   cdp2 28226
            19.23.8  Signum (sgn or sign) function   csgn 28234
            19.23.9  Ceiling function   ccei 28244
            19.23.10  Logarithms generalized to arbitrary base using ` logb `   ene0 28248
            19.23.11  Logarithm laws generalized to an arbitrary base - log_   clog_ 28251
            19.23.12  Miscellaneous   5m4e1 28253
      19.24  Mathbox for Alan Sare
            19.24.1  Supplementary "adant" deductions   ad4ant13 28256
            19.24.2  Supplementary unification deductions   biimp 28282
            19.24.3  Conventional Metamath proofs, some derived from VD proofs   iidn3 28298
            19.24.4  What is Virtual Deduction?   wvd1 28373
            19.24.5  Virtual Deduction Theorems   df-vd1 28374
            19.24.6  Theorems proved using virtual deduction   trsspwALT 28644
            19.24.7  Theorems proved using virtual deduction with mmj2 assistance   simplbi2VD 28671
            19.24.8  Virtual Deduction transcriptions of textbook proofs   sb5ALTVD 28738
            19.24.9  Theorems proved using conjunction-form virtual deduction   elpwgdedVD 28742
            19.24.10  Theorems with VD proofs in conventional notation derived from VD proofs   suctrALT3 28749
            19.24.11  Theorems with a proof in conventional notation automatically derived   notnot2ALT2 28752
      19.25  Mathbox for Jonathan Ben-Naim
            19.25.1  First order logic and set theory   bnj170 28772
            19.25.2  Well founded induction and recursion   bnj110 28939
            19.25.3  The existence of a minimal element in certain classes   bnj69 29089
            19.25.4  Well-founded induction   bnj1204 29091
            19.25.5  Well-founded recursion, part 1 of 3   bnj60 29141
            19.25.6  Well-founded recursion, part 2 of 3   bnj1500 29147
            19.25.7  Well-founded recursion, part 3 of 3   bnj1522 29151
      19.26  Mathbox for Norm Megill
            19.26.1  Experiments to study ax-7 unbundling   ax-7v 29152
                  19.26.1.1  Theorems derived from ax-7v (suffixes NEW7 and AUX7)   ax-7v 29152
                  19.26.1.2  Theorems derived from ax-7 (suffix OLD7)   ax-7OLD7 29366
            19.26.2  Miscellanea   cnaddcom 29458
            19.26.3  Atoms, hyperplanes, and covering in a left vector space (or module)   clsa 29461
            19.26.4  Functionals and kernels of a left vector space (or module)   clfn 29544
            19.26.5  Opposite rings and dual vector spaces   cld 29610
            19.26.6  Ortholattices and orthomodular lattices   cops 29659
            19.26.7  Atomic lattices with covering property   ccvr 29749
            19.26.8  Hilbert lattices   chlt 29837
            19.26.9  Projective geometries based on Hilbert lattices   clln 29977
            19.26.10  Construction of a vector space from a Hilbert lattice   cdlema1N 30277
            19.26.11  Construction of involution and inner product from a Hilbert lattice   clpoN 31967

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