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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  Metric spaces
      11.4  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  GUIDES AND MISCELLANEA
      14.1  Guides (conventions, explanations, and examples)
      14.2  Humor
      14.3  (Future - to be reviewed and classified)
PART 15  ADDITIONAL MATERIAL ON GROUPS, RINGS, AND FIELDS (DEPRECATED)
      15.1  Additional material on group theory
      15.2  Additional material on rings and fields
PART 16  COMPLEX TOPOLOGICAL VECTOR SPACES (DEPRECATED)
      16.1  Complex vector spaces
      16.2  Normed complex vector spaces
      16.3  Operators on complex vector spaces
      16.4  Inner product (pre-Hilbert) spaces
      16.5  Complex Banach spaces
      16.6  Complex Hilbert spaces
PART 17  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      17.1  Axiomatization of complex pre-Hilbert spaces
      17.2  Inner product and norms
      17.3  Cauchy sequences and completeness axiom
      17.4  Subspaces and projections
      17.5  Properties of Hilbert subspaces
      17.6  Operators on Hilbert spaces
      17.7  States on a Hilbert lattice and Godowski's equation
      17.8  Cover relation, atoms, exchange axiom, and modular symmetry
PART 18  SUPPLEMENTARY MATERIAL (USER'S MATHBOXES)
      18.1  Mathboxes for user contributions
      18.2  Mathbox for Stefan Allan
      18.3  Mathbox for Thierry Arnoux
      18.4  Mathbox for Mario Carneiro
      18.5  Mathbox for Paul Chapman
      18.6  Mathbox for Drahflow
      18.7  Mathbox for Scott Fenton
      18.8  Mathbox for Anthony Hart
      18.9  Mathbox for Chen-Pang He
      18.10  Mathbox for Jeff Hoffman
      18.11  Mathbox for Wolf Lammen
      18.12  Mathbox for Brendan Leahy
      18.13  Mathbox for Frédéric Liné
      18.14  Mathbox for Jeff Hankins
      18.15  Mathbox for Jeff Madsen
      18.16  Mathbox for Rodolfo Medina
      18.17  Mathbox for Stefan O'Rear
      18.18  Mathbox for Steve Rodriguez
      18.19  Mathbox for Andrew Salmon
      18.20  Mathbox for Glauco Siliprandi
      18.21  Mathbox for Saveliy Skresanov
      18.22  Mathbox for Jarvin Udandy
      18.23  Mathbox for Alexander van der Vekens
      18.24  Mathbox for David A. Wheeler
      18.25  Mathbox for Alan Sare
      18.26  Mathbox for Jonathan Ben-Naim
      18.27  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   mp2b 9
            1.2.4  Logical negation   con4d 97
            1.2.5  Logical equivalence   wb 176
            1.2.6  Logical disjunction and conjunction   wo 357
            1.2.7  Miscellaneous theorems of propositional calculus   pm5.21nd 868
            1.2.8  Abbreviated conjunction and disjunction of three wff's   w3o 933
            1.2.9  Logical 'nand' (Sheffer stroke)   wnan 1287
            1.2.10  Logical 'xor'   wxo 1295
            1.2.11  True and false constants   wtru 1307
            1.2.12  Truth tables   truantru 1326
            1.2.13  Auxiliary theorems for Alan Sare's virtual deduction tool, part 1   ee22 1352
            1.2.14  Half-adders and full adders in propositional calculus   whad 1368
      1.3  Other axiomatizations of classical propositional calculus
            1.3.1  Derive the Lukasiewicz axioms from Meredith's sole axiom   meredith 1394
            1.3.2  Derive the standard axioms from the Lukasiewicz axioms   luklem1 1413
            1.3.3  Derive Nicod's axiom from the standard axioms   nic-dfim 1424
            1.3.4  Derive the Lukasiewicz axioms from Nicod's axiom   nic-imp 1430
            1.3.5  Derive Nicod's Axiom from Lukasiewicz's First Sheffer Stroke Axiom   lukshef-ax1 1449
            1.3.6  Derive the Lukasiewicz Axioms from the Tarski-Bernays-Wajsberg Axioms   tbw-bijust 1453
            1.3.7  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's First CO Axiom   merco1 1468
            1.3.8  Derive the Tarski-Bernays-Wajsberg axioms from Meredith's Second CO Axiom   merco2 1491
            1.3.9  Derive the Lukasiewicz axioms from the The Russell-Bernays Axioms   rb-bijust 1504
            1.3.10  Stoic logic indemonstrables (Chrysippus of Soli)   mpto1 1523
      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 1529
            1.4.2  Rule scheme ax-gen (Generalization)   ax-gen 1535
            1.4.3  Axiom scheme ax-5 (Quantified Implication)   ax-5 1546
            1.4.4  Axiom scheme ax-17 (Distinctness) - first use of $d   ax-17 1605
            1.4.5  Equality predicate; define substitution   cv 1624
            1.4.6  Axiom scheme ax-9 (Existence)   ax-9 1637
            1.4.7  Axiom scheme ax-8 (Equality)   ax-8 1645
            1.4.8  Membership predicate   wcel 1686
            1.4.9  Axiom schemes ax-13 (Left Membership Equality)   ax-13 1688
            1.4.10  Axiom schemes ax-14 (Right Membership Equality)   ax-14 1690
            1.4.11  Logical redundancy of ax-6 , ax-7 , ax-11 , ax-12   ax9dgen 1692
      1.5  Predicate calculus with equality: Auxiliary axiom schemes (4 schemes)
            1.5.1  Axiom scheme ax-6 (Quantified Negation)   ax-6 1705
            1.5.2  Axiom scheme ax-7 (Quantifier Commutation)   ax-7 1710
            1.5.3  Axiom scheme ax-11 (Substitution)   ax-11 1717
            1.5.4  Axiom scheme ax-12 (Quantified Equality)   ax-12 1868
      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 2076
            1.6.2  Rederive new axioms from old: theorems ax5 , ax6 , ax9from9o , ax11 , ax12   ax4 2086
            1.6.3  Legacy theorems using obsolete axioms   ax17o 2098
      1.7  Existential uniqueness
      1.8  Other axiomatizations related to classical predicate calculus
            1.8.1  Predicate calculus with all distinct variables   ax-7d 2236
            1.8.2  Aristotelian logic: Assertic syllogisms   barbara 2242
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 2266
            2.1.2  Class abstractions (a.k.a. class builders)   cab 2271
            2.1.3  Class form not-free predicate   wnfc 2408
            2.1.4  Negated equality and membership   wne 2448
            2.1.5  Restricted quantification   wral 2545
            2.1.6  The universal class   cvv 2790
            2.1.7  Conditional equality (experimental)   wcdeq 2976
            2.1.8  Russell's Paradox   ru 2992
            2.1.9  Proper substitution of classes for sets   wsbc 2993
            2.1.10  Proper substitution of classes for sets into classes   csb 3083
            2.1.11  Define basic set operations and relations   cdif 3151
            2.1.12  Subclasses and subsets   df-ss 3168
            2.1.13  The difference, union, and intersection of two classes   difeq1 3289
            2.1.14  The empty set   c0 3457
            2.1.15  "Weak deduction theorem" for set theory   cif 3567
            2.1.16  Power classes   cpw 3627
            2.1.17  Unordered and ordered pairs   csn 3642
            2.1.18  The union of a class   cuni 3829
            2.1.19  The intersection of a class   cint 3864
            2.1.20  Indexed union and intersection   ciun 3907
            2.1.21  Disjointness   wdisj 3995
            2.1.22  Binary relations   wbr 4025
            2.1.23  Ordered-pair class abstractions (class builders)   copab 4078
            2.1.24  Transitive classes   wtr 4115
      2.2  ZF Set Theory - add the Axiom of Replacement
            2.2.1  Introduce the Axiom of Replacement   ax-rep 4133
            2.2.2  Derive the Axiom of Separation   axsep 4142
            2.2.3  Derive the Null Set Axiom   zfnuleu 4148
            2.2.4  Theorems requiring subset and intersection existence   nalset 4153
            2.2.5  Theorems requiring empty set existence   class2set 4180
      2.3  ZF Set Theory - add the Axiom of Power Sets
            2.3.1  Introduce the Axiom of Power Sets   ax-pow 4190
            2.3.2  Derive the Axiom of Pairing   zfpair 4214
            2.3.3  Ordered pair theorem   opnz 4244
            2.3.4  Ordered-pair class abstractions (cont.)   opabid 4273
            2.3.5  Power class of union and intersection   pwin 4299
            2.3.6  Epsilon and identity relations   cep 4305
            2.3.7  Partial and complete ordering   wpo 4314
            2.3.8  Founded and well-ordering relations   wfr 4351
            2.3.9  Ordinals   word 4393
      2.4  ZF Set Theory - add the Axiom of Union
            2.4.1  Introduce the Axiom of Union   ax-un 4514
            2.4.2  Ordinals (continued)   ordon 4576
            2.4.3  Transfinite induction   tfi 4646
            2.4.4  The natural numbers (i.e. finite ordinals)   com 4658
            2.4.5  Peano's postulates   peano1 4677
            2.4.6  Finite induction (for finite ordinals)   find 4683
            2.4.7  Relations   cxp 4689
            2.4.8  Definite description binder (inverted iota)   cio 5219
            2.4.9  Functions   wfun 5251
            2.4.10  Operations   co 5860
            2.4.11  "Maps to" notation   elmpt2cl 6063
            2.4.12  Function operation   cof 6078
            2.4.13  First and second members of an ordered pair   c1st 6122
            2.4.14  Function transposition   ctpos 6235
            2.4.15  Curry and uncurry   ccur 6274
            2.4.16  Proper subset relation   crpss 6278
            2.4.17  Iota properties   fvopab5 6291
            2.4.18  Cantor's Theorem   canth 6296
            2.4.19  Undefined values and restricted iota (description binder)   cund 6298
            2.4.20  Functions on ordinals; strictly monotone ordinal functions   iunon 6357
            2.4.21  "Strong" transfinite recursion   crecs 6389
            2.4.22  Recursive definition generator   crdg 6424
            2.4.23  Finite recursion   frfnom 6449
            2.4.24  Abian's "most fundamental" fixed point theorem   abianfplem 6472
            2.4.25  Ordinal arithmetic   c1o 6474
            2.4.26  Natural number arithmetic   nna0 6604
            2.4.27  Equivalence relations and classes   wer 6659
            2.4.28  The mapping operation   cmap 6774
            2.4.29  Infinite Cartesian products   cixp 6819
            2.4.30  Equinumerosity   cen 6862
            2.4.31  Schroeder-Bernstein Theorem   sbthlem1 6973
            2.4.32  Equinumerosity (cont.)   xpf1o 7025
            2.4.33  Pigeonhole Principle   phplem1 7042
            2.4.34  Finite sets   onomeneq 7052
            2.4.35  Finite intersections   cfi 7166
            2.4.36  Hall's marriage theorem   marypha1lem 7188
            2.4.37  Supremum   csup 7195
            2.4.38  Ordinal isomorphism, Hartog's theorem   coi 7226
            2.4.39  Hartogs function, order types, weak dominance   char 7272
      2.5  ZF Set Theory - add the Axiom of Regularity
            2.5.1  Introduce the Axiom of Regularity   ax-reg 7308
            2.5.2  Axiom of Infinity equivalents   inf0 7324
      2.6  ZF Set Theory - add the Axiom of Infinity
            2.6.1  Introduce the Axiom of Infinity   ax-inf 7341
            2.6.2  Existence of omega (the set of natural numbers)   omex 7346
            2.6.3  Cantor normal form   ccnf 7364
            2.6.4  Transitive closure   trcl 7412
            2.6.5  Rank   cr1 7436
            2.6.6  Scott's trick; collection principle; Hilbert's epsilon   scottex 7557
            2.6.7  Cardinal numbers   ccrd 7570
            2.6.8  Axiom of Choice equivalents   wac 7744
            2.6.9  Cardinal number arithmetic   ccda 7795
            2.6.10  The Ackermann bijection   ackbij2lem1 7847
            2.6.11  Cofinality (without Axiom of Choice)   cflem 7874
            2.6.12  Eight inequivalent definitions of finite set   sornom 7905
            2.6.13  Hereditarily size-limited sets without Choice   itunifval 8044
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 8087
            3.2.2  AC equivalents: well-ordering, Zorn's lemma   numthcor 8123
            3.2.3  Cardinal number theorems using Axiom of Choice   cardval 8170
            3.2.4  Cardinal number arithmetic using Axiom of Choice   iunctb 8198
            3.2.5  Cofinality using Axiom of Choice   alephreg 8206
      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 8306
            4.1.2  Weak universes   cwun 8324
            4.1.3  Tarski's classes   ctsk 8372
            4.1.4  Grothendieck's universes   cgru 8414
      4.2  ZFC Set Theory plus the Tarski-Grothendieck Axiom
            4.2.1  Introduce the Tarski-Grothendieck Axiom   ax-groth 8447
            4.2.2  Derive the Power Set, Infinity and Choice Axioms   grothpw 8450
            4.2.3  Tarski map function   ctskm 8461
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 8468
            5.1.2  Final derivation of real and complex number postulates   axaddf 8769
            5.1.3  Real and complex number postulates restated as axioms   ax-cnex 8795
      5.2  Derive the basic properties from the field axioms
            5.2.1  Some deductions from the field axioms for complex numbers   cnex 8820
            5.2.2  Infinity and the extended real number system   cpnf 8866
            5.2.3  Restate the ordering postulates with extended real "less than"   axlttri 8896
            5.2.4  Ordering on reals   lttr 8901
            5.2.5  Initial properties of the complex numbers   mul12 8980
      5.3  Real and complex numbers - basic operations
            5.3.1  Addition   add12 9027
            5.3.2  Subtraction   cmin 9039
            5.3.3  Multiplication   muladd 9214
            5.3.4  Ordering on reals (cont.)   gt0ne0 9241
            5.3.5  Reciprocals   ixi 9399
            5.3.6  Division   cdiv 9425
            5.3.7  Ordering on reals (cont.)   elimgt0 9594
            5.3.8  Completeness Axiom and Suprema   fimaxre 9703
            5.3.9  Imaginary and complex number properties   inelr 9738
            5.3.10  Function operation analogue theorems   ofsubeq0 9745
      5.4  Integer sets
            5.4.1  Natural numbers (as a subset of complex numbers)   cn 9748
            5.4.2  Principle of mathematical induction   nnind 9766
            5.4.3  Decimal representation of numbers   c2 9797
            5.4.4  Some properties of specific numbers   0p1e1 9841
            5.4.5  The Archimedean property   nnunb 9963
            5.4.6  Nonnegative integers (as a subset of complex numbers)   cn0 9967
            5.4.7  Integers (as a subset of complex numbers)   cz 10026
            5.4.8  Decimal arithmetic   cdc 10126
            5.4.9  Upper partititions of integers   cuz 10232
            5.4.10  Well-ordering principle for bounded-below sets of integers   uzwo3 10313
            5.4.11  Rational numbers (as a subset of complex numbers)   cq 10318
            5.4.12  Existence of the set of complex numbers   rpnnen1lem1 10344
      5.5  Order sets
            5.5.1  Positive reals (as a subset of complex numbers)   crp 10356
            5.5.2  Infinity and the extended real number system (cont.)   cxne 10451
            5.5.3  Supremum on the extended reals   xrsupexmnf 10625
            5.5.4  Real number intervals   cioo 10658
            5.5.5  Finite intervals of integers   cfz 10784
            5.5.6  Half-open integer ranges   cfzo 10872
      5.6  Elementary integer functions
            5.6.1  The floor (greatest integer) function   cfl 10926
            5.6.2  The modulo (remainder) operation   cmo 10975
            5.6.3  The infinite sequence builder "seq"   om2uz0i 11012
            5.6.4  Integer powers   cexp 11106
            5.6.5  Ordered pair theorem for nonnegative integers   nn0le2msqi 11284
            5.6.6  Factorial function   cfa 11290
            5.6.7  The binomial coefficient operation   cbc 11317
            5.6.8  The ` # ` (finite set size) function   chash 11339
            5.6.9  Words over a set   cword 11405
            5.6.10  Longer string literals   cs2 11493
      5.7  Elementary real and complex functions
            5.7.1  The "shift" operation   cshi 11563
            5.7.2  Real and imaginary parts; conjugate   ccj 11583
            5.7.3  Square root; absolute value   csqr 11720
      5.8  Elementary limits and convergence
            5.8.1  Superior limit (lim sup)   clsp 11946
            5.8.2  Limits   cli 11960
            5.8.3  Finite and infinite sums   csu 12160
            5.8.4  The binomial theorem   binomlem 12289
            5.8.5  The inclusion/exclusion principle   incexclem 12297
            5.8.6  Infinite sums (cont.)   isumshft 12300
            5.8.7  Miscellaneous converging and diverging sequences   divrcnv 12313
            5.8.8  Arithmetic series   arisum 12320
            5.8.9  Geometric series   expcnv 12324
            5.8.10  Ratio test for infinite series convergence   cvgrat 12341
            5.8.11  Mertens' theorem   mertenslem1 12342
      5.9  Elementary trigonometry
            5.9.1  The exponential, sine, and cosine functions   ce 12345
            5.9.2  _e is irrational   eirrlem 12484
      5.10  Cardinality of real and complex number subsets
            5.10.1  Countability of integers and rationals   xpnnen 12489
            5.10.2  The reals are uncountable   rpnnen2lem1 12495
PART 6  ELEMENTARY NUMBER THEORY
      6.1  Elementary properties of divisibility
            6.1.1  Irrationality of square root of 2   sqr2irrlem 12528
            6.1.2  Some Number sets are chains of proper subsets   nthruc 12531
            6.1.3  The divides relation   cdivides 12533
            6.1.4  The division algorithm   divalglem0 12594
            6.1.5  Bit sequences   cbits 12612
            6.1.6  The greatest common divisor operator   cgcd 12687
            6.1.7  Bézout's identity   bezoutlem1 12719
            6.1.8  Algorithms   nn0seqcvgd 12742
            6.1.9  Euclid's Algorithm   eucalgval2 12753
      6.2  Elementary prime number theory
            6.2.1  Elementary properties   cprime 12760
            6.2.2  Properties of the canonical representation of a rational   cnumer 12806
            6.2.3  Euler's theorem   codz 12833
            6.2.4  Pythagorean Triples   coprimeprodsq 12864
            6.2.5  The prime count function   cpc 12891
            6.2.6  Pocklington's theorem   prmpwdvds 12953
            6.2.7  Infinite primes theorem   unbenlem 12957
            6.2.8  Sum of prime reciprocals   prmreclem1 12965
            6.2.9  Fundamental theorem of arithmetic   1arithlem1 12972
            6.2.10  Lagrange's four-square theorem   cgz 12978
            6.2.11  Van der Waerden's theorem   cvdwa 13014
            6.2.12  Ramsey's theorem   cram 13048
            6.2.13  Decimal arithmetic (cont.)   dec2dvds 13080
            6.2.14  Specific prime numbers   4nprm 13108
            6.2.15  Very large primes   1259lem1 13131
PART 7  BASIC STRUCTURES
      7.1  Extensible structures
            7.1.1  Basic definitions   cstr 13146
            7.1.2  Slot definitions   cplusg 13210
            7.1.3  Definition of the structure product   crest 13327
            7.1.4  Definition of the structure quotient   cordt 13400
      7.2  Moore spaces
            7.2.1  Moore closures   mrcflem 13510
            7.2.2  Independent sets in a Moore system   mrisval 13534
            7.2.3  Algebraic closure systems   isacs 13555
PART 8  BASIC CATEGORY THEORY
      8.1  Categories
            8.1.1  Categories   ccat 13568
            8.1.2  Opposite category   coppc 13616
            8.1.3  Monomorphisms and epimorphisms   cmon 13633
            8.1.4  Sections, inverses, isomorphisms   csect 13649
            8.1.5  Subcategories   cssc 13686
            8.1.6  Functors   cfunc 13730
            8.1.7  Full & faithful functors   cful 13778
            8.1.8  Natural transformations and the functor category   cnat 13817
      8.2  Arrows (disjointified hom-sets)
            8.2.1  Identity and composition for arrows   cida 13887
      8.3  Examples of categories
            8.3.1  The category of sets   csetc 13909
            8.3.2  The category of categories   ccatc 13928
      8.4  Categorical constructions
            8.4.1  Product of categories   cxpc 13944
            8.4.2  Functor evaluation   cevlf 13985
            8.4.3  Hom functor   chof 14024
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 14076
            9.2.2  Lattices   clat 14153
            9.2.3  The dual of an ordered set   codu 14234
            9.2.4  Subset order structures   cipo 14256
            9.2.5  Distributive lattices   latmass 14293
            9.2.6  Posets and lattices as relations   cps 14303
            9.2.7  Directed sets, nets   cdir 14352
PART 10  BASIC ALGEBRAIC STRUCTURES
      10.1  Monoids
            10.1.1  Definition and basic properties   cmnd 14363
            10.1.2  Monoid homomorphisms and submonoids   cmhm 14415
            10.1.3  Ordered group sum operation   gsumvallem1 14450
            10.1.4  Free monoids   cfrmd 14471
      10.2  Groups
            10.2.1  Definition and basic properties   df-grp 14491
            10.2.2  Subgroups and Quotient groups   csubg 14617
            10.2.3  Elementary theory of group homomorphisms   cghm 14682
            10.2.4  Isomorphisms of groups   cgim 14723
            10.2.5  Group actions   cga 14745
            10.2.6  Symmetry groups and Cayley's Theorem   csymg 14771
            10.2.7  Centralizers and centers   ccntz 14793
            10.2.8  The opposite group   coppg 14820
            10.2.9  p-Groups and Sylow groups; Sylow's theorems   cod 14842
            10.2.10  Direct products   clsm 14947
            10.2.11  Free groups   cefg 15017
      10.3  Abelian groups
            10.3.1  Definition and basic properties   ccmn 15091
            10.3.2  Cyclic groups   ccyg 15166
            10.3.3  Group sum operation   gsumval3a 15191
            10.3.4  Internal direct products   cdprd 15233
            10.3.5  The Fundamental Theorem of Abelian Groups   ablfacrplem 15302
      10.4  Rings
            10.4.1  Multiplicative Group   cmgp 15327
            10.4.2  Definition and basic properties   crg 15339
            10.4.3  Opposite ring   coppr 15406
            10.4.4  Divisibility   cdsr 15422
            10.4.5  Ring homomorphisms   crh 15496
      10.5  Division rings and fields
            10.5.1  Definition and basic properties   cdr 15514
            10.5.2  Subrings of a ring   csubrg 15543
            10.5.3  Absolute value (abstract algebra)   cabv 15583
            10.5.4  Star rings   cstf 15610
      10.6  Left modules
            10.6.1  Definition and basic properties   clmod 15629
            10.6.2  Subspaces and spans in a left module   clss 15691
            10.6.3  Homomorphisms and isomorphisms of left modules   clmhm 15778
            10.6.4  Subspace sum; bases for a left module   clbs 15829
      10.7  Vector spaces
            10.7.1  Definition and basic properties   clvec 15857
      10.8  Ideals
            10.8.1  The subring algebra; ideals   csra 15923
            10.8.2  Two-sided ideals and quotient rings   c2idl 15985
            10.8.3  Principal ideal rings. Divisibility in the integers   clpidl 15995
            10.8.4  Nonzero rings   cnzr 16011
            10.8.5  Left regular elements. More kinds of rings   crlreg 16022
      10.9  Associative algebras
            10.9.1  Definition and basic properties   casa 16052
      10.10  Abstract multivariate polynomials
            10.10.1  Definition and basic properties   cmps 16089
            10.10.2  Polynomial evaluation   evlslem4 16247
            10.10.3  Univariate polynomials   cps1 16252
      10.11  The complex numbers as an extensible structure
            10.11.1  Definition and basic properties   cxmt 16371
            10.11.2  Algebraic constructions based on the complexes   czrh 16453
      10.12  Hilbert spaces
            10.12.1  Definition and basic properties   cphl 16530
            10.12.2  Orthocomplements and closed subspaces   cocv 16562
            10.12.3  Orthogonal projection and orthonormal bases   cpj 16602
PART 11  BASIC TOPOLOGY
      11.1  Topology
            11.1.1  Topological spaces   ctop 16633
            11.1.2  TopBases for topologies   isbasisg 16687
            11.1.3  Examples of topologies   distop 16735
            11.1.4  Closure and interior   ccld 16755
            11.1.5  Neighborhoods   cnei 16836
            11.1.6  Limit points and perfect sets   clp 16868
            11.1.7  Subspace topologies   restrcl 16890
            11.1.8  Order topology   ordtbaslem 16920
            11.1.9  Limits and continuity in topological spaces   ccn 16956
            11.1.10  Separated spaces: T0, T1, T2 (Hausdorff) ...   ct0 17036
            11.1.11  Compactness   ccmp 17115
            11.1.12  Connectedness   ccon 17139
            11.1.13  First- and second-countability   c1stc 17165
            11.1.14  Local topological properties   clly 17192
            11.1.15  Compactly generated spaces   ckgen 17230
            11.1.16  Product topologies   ctx 17257
            11.1.17  Continuous function-builders   cnmptid 17357
            11.1.18  Quotient maps and quotient topology   ckq 17386
            11.1.19  Homeomorphisms   chmeo 17446
      11.2  Filters and filter bases
            11.2.1  Filter bases   cfbas 17520
            11.2.2  Filters   cfil 17542
            11.2.3  Ultrafilters   cufil 17596
            11.2.4  Filter limits   cfm 17630
            11.2.5  Topological groups   ctmd 17755
            11.2.6  Infinite group sum on topological groups   ctsu 17810
            11.2.7  Topological rings, fields, vector spaces   ctrg 17840
      11.3  Metric spaces
            11.3.1  Basic metric space properties   cxme 17884
            11.3.2  Metric space balls   blfval 17949
            11.3.3  Open sets of a metric space   mopnval 17986
            11.3.4  Continuity in metric spaces   metcnp3 18088
            11.3.5  Examples of metric spaces   dscmet 18097
            11.3.6  Normed algebraic structures   cnm 18101
            11.3.7  Normed space homomorphisms (bounded linear operators)   cnmo 18216
            11.3.8  Topology on the reals   qtopbaslem 18269
            11.3.9  Topological definitions using the reals   cii 18381
            11.3.10  Path homotopy   chtpy 18467
            11.3.11  The fundamental group   cpco 18500
      11.4  Complex metric vector spaces
            11.4.1  Complex left modules   cclm 18562
            11.4.2  Complex pre-Hilbert space   ccph 18604
            11.4.3  Convergence and completeness   ccfil 18680
            11.4.4  Baire's Category Theorem   bcthlem1 18748
            11.4.5  Banach spaces and complex Hilbert spaces   ccms 18756
            11.4.6  Minimizing Vector Theorem   minveclem1 18790
            11.4.7  Projection Theorem   pjthlem1 18803
PART 12  BASIC REAL AND COMPLEX ANALYSIS
      12.1  Continuity
            12.1.1  Intermediate value theorem   pmltpclem1 18810
      12.2  Integrals
            12.2.1  Lebesgue measure   covol 18824
            12.2.2  Lebesgue integration   cmbf 18971
      12.3  Derivatives
            12.3.1  Real and complex differentiation   climc 19214
PART 13  BASIC REAL AND COMPLEX FUNCTIONS
      13.1  Polynomials
            13.1.1  Abstract polynomials, continued   evlslem6 19399
            13.1.2  Polynomial degrees   cmdg 19441
            13.1.3  The division algorithm for univariate polynomials   cmn1 19513
            13.1.4  Elementary properties of complex polynomials   cply 19568
            13.1.5  The division algorithm for polynomials   cquot 19672
            13.1.6  Algebraic numbers   caa 19696
            13.1.7  Liouville's approximation theorem   aalioulem1 19714
      13.2  Sequences and series
            13.2.1  Taylor polynomials and Taylor's theorem   ctayl 19734
            13.2.2  Uniform convergence   culm 19757
            13.2.3  Power series   pserval 19788
      13.3  Basic trigonometry
            13.3.1  The exponential, sine, and cosine functions (cont.)   efcn 19821
            13.3.2  Properties of pi = 3.14159...   pilem1 19829
            13.3.3  Mapping of the exponential function   efgh 19905
            13.3.4  The natural logarithm on complex numbers   clog 19914
            13.3.5  Theorems of Pythagoras, isosceles triangles, and intersecting chords   angval 20101
            13.3.6  Solutions of quadratic, cubic, and quartic equations   quad2 20137
            13.3.7  Inverse trigonometric functions   casin 20160
            13.3.8  The Birthday Problem   log2ublem1 20244
            13.3.9  Areas in R^2   carea 20252
            13.3.10  More miscellaneous converging sequences   rlimcnp 20262
            13.3.11  Inequality of arithmetic and geometric means   cvxcl 20281
            13.3.12  Euler-Mascheroni constant   cem 20288
      13.4  Basic number theory
            13.4.1  Wilson's theorem   wilthlem1 20308
            13.4.2  The Fundamental Theorem of Algebra   ftalem1 20312
            13.4.3  The Basel problem (ζ(2) = π2/6)   basellem1 20320
            13.4.4  Number-theoretical functions   ccht 20330
            13.4.5  Perfect Number Theorem   mersenne 20468
            13.4.6  Characters of Z/nZ   cdchr 20473
            13.4.7  Bertrand's postulate   bcctr 20516
            13.4.8  Legendre symbol   clgs 20535
            13.4.9  Quadratic reciprocity   lgseisenlem1 20590
            13.4.10  All primes 4n+1 are the sum of two squares   2sqlem1 20604
            13.4.11  Chebyshev's Weak Prime Number Theorem, Dirichlet's Theorem   chebbnd1lem1 20620
            13.4.12  The Prime Number Theorem   mudivsum 20681
            13.4.13  Ostrowski's theorem   abvcxp 20766
PART 14  GUIDES AND MISCELLANEA
      14.1  Guides (conventions, explanations, and examples)
            14.1.1  Conventions   conventions 20791
            14.1.2  Natural deduction   natded 20792
            14.1.3  Natural deduction examples   ex-natded5.2 20793
            14.1.4  Definitional examples   ex-or 20810
      14.2  Humor
            14.2.1  April Fool's theorem   avril1 20838
      14.3  (Future - to be reviewed and classified)
            14.3.1  Planar incidence geometry   cplig 20844
            14.3.2  Algebra preliminaries   crpm 20849
            14.3.3  Transitive closure   ctcl 20851
PART 15  ADDITIONAL MATERIAL ON GROUPS, RINGS, AND FIELDS (DEPRECATED)
      15.1  Additional material on group theory
            15.1.1  Definitions and basic properties for groups   cgr 20855
            15.1.2  Definition and basic properties of Abelian groups   cablo 20950
            15.1.3  Subgroups   csubgo 20970
            15.1.4  Operation properties   cass 20981
            15.1.5  Group-like structures   cmagm 20987
            15.1.6  Examples of Abelian groups   ablosn 21016
            15.1.7  Group homomorphism and isomorphism   cghom 21026
      15.2  Additional material on rings and fields
            15.2.1  Definition and basic properties   crngo 21044
            15.2.2  Examples of rings   cnrngo 21072
            15.2.3  Division Rings   cdrng 21074
            15.2.4  Star Fields   csfld 21077
            15.2.5  Fields and Rings   ccm2 21079
PART 16  COMPLEX TOPOLOGICAL VECTOR SPACES (DEPRECATED)
      16.1  Complex vector spaces
            16.1.1  Definition and basic properties   cvc 21103
            16.1.2  Examples of complex vector spaces   cncvc 21141
      16.2  Normed complex vector spaces
            16.2.1  Definition and basic properties   cnv 21142
            16.2.2  Examples of normed complex vector spaces   cnnv 21247
            16.2.3  Induced metric of a normed complex vector space   imsval 21256
            16.2.4  Inner product   cdip 21275
            16.2.5  Subspaces   css 21299
      16.3  Operators on complex vector spaces
            16.3.1  Definitions and basic properties   clno 21320
      16.4  Inner product (pre-Hilbert) spaces
            16.4.1  Definition and basic properties   ccphlo 21392
            16.4.2  Examples of pre-Hilbert spaces   cncph 21399
            16.4.3  Properties of pre-Hilbert spaces   isph 21402
      16.5  Complex Banach spaces
            16.5.1  Definition and basic properties   ccbn 21443
            16.5.2  Examples of complex Banach spaces   cnbn 21450
            16.5.3  Uniform Boundedness Theorem   ubthlem1 21451
            16.5.4  Minimizing Vector Theorem   minvecolem1 21455
      16.6  Complex Hilbert spaces
            16.6.1  Definition and basic properties   chlo 21466
            16.6.2  Standard axioms for a complex Hilbert space   hlex 21479
            16.6.3  Examples of complex Hilbert spaces   cnchl 21497
            16.6.4  Subspaces   ssphl 21498
            16.6.5  Hellinger-Toeplitz Theorem   htthlem 21499
PART 17  COMPLEX HILBERT SPACE EXPLORER (DEPRECATED)
      17.1  Axiomatization of complex pre-Hilbert spaces
            17.1.1  Basic Hilbert space definitions   chil 21501
            17.1.2  Preliminary ZFC lemmas   df-hnorm 21550
            17.1.3  Derive the Hilbert space axioms from ZFC set theory   axhilex-zf 21563
            17.1.4  Introduce the vector space axioms for a Hilbert space   ax-hilex 21581
            17.1.5  Vector operations   hvmulex 21593
            17.1.6  Inner product postulates for a Hilbert space   ax-hfi 21660
      17.2  Inner product and norms
            17.2.1  Inner product   his5 21667
            17.2.2  Norms   dfhnorm2 21703
            17.2.3  Relate Hilbert space to normed complex vector spaces   hilablo 21741
            17.2.4  Bunjakovaskij-Cauchy-Schwarz inequality   bcsiALT 21760
      17.3  Cauchy sequences and completeness axiom
            17.3.1  Cauchy sequences and limits   hcau 21765
            17.3.2  Derivation of the completeness axiom from ZF set theory   hilmet 21775
            17.3.3  Completeness postulate for a Hilbert space   ax-hcompl 21783
            17.3.4  Relate Hilbert space to ZFC pre-Hilbert and Hilbert spaces   hhcms 21784
      17.4  Subspaces and projections
            17.4.1  Subspaces   df-sh 21788
            17.4.2  Closed subspaces   df-ch 21803
            17.4.3  Orthocomplements   df-oc 21833
            17.4.4  Subspace sum, span, lattice join, lattice supremum   df-shs 21889
            17.4.5  Projection theorem   pjhthlem1 21972
            17.4.6  Projectors   df-pjh 21976
      17.5  Properties of Hilbert subspaces
            17.5.1  Orthomodular law   omlsilem 21983
            17.5.2  Projectors (cont.)   pjhtheu2 21997
            17.5.3  Hilbert lattice operations   sh0le 22021
            17.5.4  Span (cont.) and one-dimensional subspaces   spansn0 22122
            17.5.5  Commutes relation for Hilbert lattice elements   df-cm 22164
            17.5.6  Foulis-Holland theorem   fh1 22199
            17.5.7  Quantum Logic Explorer axioms   qlax1i 22208
            17.5.8  Orthogonal subspaces   chscllem1 22218
            17.5.9  Orthoarguesian laws 5OA and 3OA   5oalem1 22235
            17.5.10  Projectors (cont.)   pjorthi 22250
            17.5.11  Mayet's equation E_3   mayete3i 22309
      17.6  Operators on Hilbert spaces
            17.6.1  Operator sum, difference, and scalar multiplication   df-hosum 22312
            17.6.2  Zero and identity operators   df-h0op 22330
            17.6.3  Operations on Hilbert space operators   hoaddcl 22340
            17.6.4  Linear, continuous, bounded, Hermitian, unitary operators and norms   df-nmop 22421
            17.6.5  Linear and continuous functionals and norms   df-nmfn 22427
            17.6.6  Adjoint   df-adjh 22431
            17.6.7  Dirac bra-ket notation   df-bra 22432
            17.6.8  Positive operators   df-leop 22434
            17.6.9  Eigenvectors, eigenvalues, spectrum   df-eigvec 22435
            17.6.10  Theorems about operators and functionals   nmopval 22438
            17.6.11  Riesz lemma   riesz3i 22644
            17.6.12  Adjoints (cont.)   cnlnadjlem1 22649
            17.6.13  Quantum computation error bound theorem   unierri 22686
            17.6.14  Dirac bra-ket notation (cont.)   branmfn 22687
            17.6.15  Positive operators (cont.)   leopg 22704
            17.6.16  Projectors as operators   pjhmopi 22728
      17.7  States on a Hilbert lattice and Godowski's equation
            17.7.1  States on a Hilbert lattice   df-st 22793
            17.7.2  Godowski's equation   golem1 22853
      17.8  Cover relation, atoms, exchange axiom, and modular symmetry
            17.8.1  Covers relation; modular pairs   df-cv 22861
            17.8.2  Atoms   df-at 22920
            17.8.3  Superposition principle   superpos 22936
            17.8.4  Atoms, exchange and covering properties, atomicity   chcv1 22937
            17.8.5  Irreducibility   chirredlem1 22972
            17.8.6  Atoms (cont.)   atcvat3i 22978
            17.8.7  Modular symmetry   mdsymlem1 22985
PART 18  SUPPLEMENTARY MATERIAL (USER'S MATHBOXES)
      18.1  Mathboxes for user contributions
            18.1.1  Mathbox guidelines   mathbox 23024
      18.2  Mathbox for Stefan Allan
      18.3  Mathbox for Thierry Arnoux
            18.3.1  Bertrand's Ballot Problem   ballotlemoex 23046
            18.3.2  Division in the extended real number system   cxdiv 23102
            18.3.3  Propositional Calculus - misc additions   bisimpd 23122
            18.3.4  Subclass relations - misc additions   ssrd 23127
            18.3.5  Restricted Quantification - misc additions   abeq2f 23131
            18.3.6  Substitution (without distinct variables) - misc additions   sbcss12g 23143
            18.3.7  Existential Uniqueness - misc additions   mo5f 23145
            18.3.8  Conditional operator - misc additions   ifbieq12d2 23151
            18.3.9  Indexed union - misc additions   iuneq12daf 23156
            18.3.10  Miscellaneous   ceqsexv2d 23164
            18.3.11  Functions and relations - misc additions   xpdisjres 23199
            18.3.12  First and second members of an ordered pair - misc additions   df1stres 23245
            18.3.13  Supremum - misc additions   supssd 23250
            18.3.14  Ordering on reals - misc additions   lt2addrd 23251
            18.3.15  Extended reals - misc additions   xrlelttric 23252
            18.3.16  Real number intervals - misc additions   icossicc 23260
            18.3.17  Finite intervals of integers - misc additions   fzssnn 23278
            18.3.18  Half-open integer ranges - misc additions   fzossnn 23280
            18.3.19  Closed unit   unitsscn 23282
            18.3.20  Topology of ` ( RR X. RR ) `   tpr2tp 23289
            18.3.21  Order topology - misc. additions   cnvordtrestixx 23299
            18.3.22  Continuity in topological spaces - misc. additions   ressplusf 23300
            18.3.23  Extended reals Structure - misc additions   xaddeq0 23306
            18.3.24  The extended non-negative real numbers monoid   xrge0base 23312
            18.3.25  Countable Sets   nnct 23337
            18.3.26  Disjointness - misc additions   cbvdisjf 23352
            18.3.27  Limits - misc additions   lmlim 23373
            18.3.28  Finitely supported group sums - misc additions   gsumsn2 23380
            18.3.29  Logarithm laws generalized to an arbitrary base - logb   clogb 23392
            18.3.30  Extended sum   cesum 23412
            18.3.31  Mixed Function/Constant operation   cofc 23458
            18.3.32  Sigma-Algebra   csiga 23470
            18.3.33  Generated Sigma-Algebra   csigagen 23501
            18.3.34  The Borel Algebra on real numbers   cbrsiga 23514
            18.3.35  Product Sigma-Algebra   csx 23521
            18.3.36  Measures   cmeas 23528
            18.3.37  Measurable functions   cmbfm 23557
            18.3.38  Borel Algebra on ` ( RR X. RR ) `   br2base 23576
            18.3.39  Integration with respect to a Measure   cibfm 23585
            18.3.40  Indicator Functions   cind 23596
            18.3.41  Probability Theory   cprb 23612
            18.3.42  Conditional Probabilities   ccprob 23636
            18.3.43  Real Valued Random Variables   crrv 23645
            18.3.44  Preimage set mapping operator   corvc 23658
            18.3.45  Distribution Functions   orvcelval 23671
            18.3.46  Cumulative Distribution Functions   orvclteel 23675
            18.3.47  Probabilities - example   coinfliplem 23681
      18.4  Mathbox for Mario Carneiro
            18.4.1  Miscellaneous stuff   quartfull 23688
            18.4.2  Zeta function   czeta 23689
            18.4.3  Gamma function   clgam 23692
            18.4.4  Derangements and the Subfactorial   deranglem 23699
            18.4.5  The Erdős-Szekeres theorem   erdszelem1 23724
            18.4.6  The Kuratowski closure-complement theorem   kur14lem1 23739
            18.4.7  Retracts and sections   cretr 23750
            18.4.8  Path-connected and simply connected spaces   cpcon 23752
            18.4.9  Covering maps   ccvm 23788
            18.4.10  Undirected multigraphs   cumg 23862
            18.4.11  Normal numbers   snmlff 23914
            18.4.12  Godel-sets of formulas   cgoe 23918
            18.4.13  Models of ZF   cgze 23946
            18.4.14  Splitting fields   citr 23960
            18.4.15  p-adic number fields   czr 23976
      18.5  Mathbox for Paul Chapman
            18.5.1  Group homomorphism and isomorphism   ghomgrpilem1 23994
            18.5.2  Real and complex numbers (cont.)   climuzcnv 24006
            18.5.3  Miscellaneous theorems   elfzm12 24010
      18.6  Mathbox for Drahflow
      18.7  Mathbox for Scott Fenton
            18.7.1  ZFC Axioms in primitive form   axextprim 24049
            18.7.2  Untangled classes   untelirr 24056
            18.7.3  Extra propositional calculus theorems   3orel1 24063
            18.7.4  Misc. Useful Theorems   nepss 24074
            18.7.5  Properties of reals and complexes   sqdivzi 24081
            18.7.6  Greatest common divisor and divisibility   pdivsq 24104
            18.7.7  Properties of relationships   brtp 24108
            18.7.8  Properties of functions and mappings   funpsstri 24123
            18.7.9  Epsilon induction   setinds 24136
            18.7.10  Ordinal numbers   elpotr 24139
            18.7.11  Defined equality axioms   axextdfeq 24156
            18.7.12  Hypothesis builders   hbntg 24164
            18.7.13  The Predecessor Class   cpred 24169
            18.7.14  (Trans)finite Recursion Theorems   tfisg 24206
            18.7.15  Well-founded induction   tz6.26 24207
            18.7.16  Transitive closure under a relationship   ctrpred 24222
            18.7.17  Founded Induction   frmin 24244
            18.7.18  Ordering Ordinal Sequences   orderseqlem 24254
            18.7.19  Well-founded recursion   wfr3g 24257
            18.7.20  Transfinite recursion via Well-founded recursion   tfrALTlem 24278
            18.7.21  Founded Recursion   frr3g 24282
            18.7.22  Surreal Numbers   csur 24296
            18.7.23  Surreal Numbers: Ordering   sltsolem1 24324
            18.7.24  Surreal Numbers: Birthday Function   bdayfo 24331
            18.7.25  Surreal Numbers: Density   fvnobday 24338
            18.7.26  Surreal Numbers: Density   nodenselem3 24339
            18.7.27  Surreal Numbers: Upper and Lower Bounds   nobndlem1 24348
            18.7.28  Surreal Numbers: Full-Eta Property   nofulllem1 24358
            18.7.29  Symmetric difference   csymdif 24363
            18.7.30  Quantifier-free definitions   ctxp 24375
            18.7.31  Alternate ordered pairs   caltop 24492
            18.7.32  Tarskian geometry   cee 24518
            18.7.33  Tarski's axioms for geometry   axdimuniq 24543
            18.7.34  Congruence properties   cofs 24607
            18.7.35  Betweenness properties   btwntriv2 24637
            18.7.36  Segment Transportation   ctransport 24654
            18.7.37  Properties relating betweenness and congruence   cifs 24660
            18.7.38  Connectivity of betweenness   btwnconn1lem1 24712
            18.7.39  Segment less than or equal to   csegle 24731
            18.7.40  Outside of relationship   coutsideof 24744
            18.7.41  Lines and Rays   cline2 24759
            18.7.42  Bernoulli polynomials and sums of k-th powers   cbp 24783
            18.7.43  Rank theorems   rankung 24798
            18.7.44  Hereditarily Finite Sets   chf 24804
      18.8  Mathbox for Anthony Hart
            18.8.1  Propositional Calculus   tb-ax1 24819
            18.8.2  Predicate Calculus   quantriv 24841
            18.8.3  Misc. Single Axiom Systems   meran1 24852
            18.8.4  Connective Symmetry   negsym1 24858
      18.9  Mathbox for Chen-Pang He
            18.9.1  Ordinal topology   ontopbas 24869
      18.10  Mathbox for Jeff Hoffman
            18.10.1  Inferences for finite induction on generic function values   fveleq 24892
            18.10.2  gdc.mm   nnssi2 24896
      18.11  Mathbox for Wolf Lammen
      18.12  Mathbox for Brendan Leahy
      18.13  Mathbox for Frédéric Liné
            18.13.1  Theorems from other workspaces   tpssg 24943
            18.13.2  Propositional and predicate calculus   neleq12d 24944
            18.13.3  Linear temporal logic   wbox 24981
            18.13.4  Operations   ssoprab2g 25043
            18.13.5  General Set Theory   uninqs 25050
            18.13.6  The "maps to" notation   cmpfunOLD 25153
            18.13.7  Cartesian Products   cpro 25161
            18.13.8  Operations on subsets and functions   ccst 25183
            18.13.9  Arithmetic   3timesi 25189
            18.13.10  Lattice (algebraic definition)   clatalg 25192
            18.13.11  Currying and Partial Mappings   ccur1 25205
            18.13.12  Order theory (Extensible Structure Builder)   corhom 25218
            18.13.13  Order theory   cpresetrel 25226
            18.13.14  Finite composites ( i.e. finite sums, products ... )   cprd 25309
            18.13.15  Operation properties   ccm1 25342
            18.13.16  Groups and related structures   ridlideq 25346
            18.13.17  Free structures   csubsmg 25394
            18.13.18  Translations   trdom2 25402
            18.13.19  Fields and Rings   com2i 25427
            18.13.20  Ideals   cidln 25454
            18.13.21  Generic modules and vector spaces (New Structure builder)   cact 25458
            18.13.22  Generic modules and vector spaces   cvec 25460
            18.13.23  Real vector spaces   cvr 25500
            18.13.24  Matrices   cmmat 25504
            18.13.25  Affine spaces   craffsp 25510
            18.13.26  Intervals of reals and extended reals   bsi 25512
            18.13.27  Topology   topnem 25523
            18.13.28  Continuous functions   cnrsfin 25536
            18.13.29  Homeomorphisms   dmhmph 25544
            18.13.30  Initial and final topologies   intopcoaconlem3b 25549
            18.13.31  Filters   efilcp 25563
            18.13.32  Limits   plimfil 25569
            18.13.33  Uniform spaces   cunifsp 25596
            18.13.34  Separated spaces: T0, T1, T2 (Hausdorff) ...   hst1 25598
            18.13.35  Compactness   indcomp 25600
            18.13.36  Connectedness   singempcon 25604
            18.13.37  Topological fields   ctopfld 25608
            18.13.38  Standard topology on RR   intrn 25610
            18.13.39  Standard topology of intervals of RR   stoi 25612
            18.13.40  Cantor's set   cntrset 25613
            18.13.41  Pre-calculus and Cartesian geometry   dmse1 25614
            18.13.42  Extended Real numbers   nolimf 25630
            18.13.43  ( RR ^ N ) and ( CC ^ N )   cplcv 25655
            18.13.44  Calculus   cintvl 25707
            18.13.45  Directed multi graphs   cmgra 25719
            18.13.46  Category and deductive system underlying "structure"   calg 25722
            18.13.47  Deductive systems   cded 25745
            18.13.48  Categories   ccatOLD 25763
            18.13.49  Homsets   chomOLD 25796
            18.13.50  Monomorphisms, Epimorphisms, Isomorphisms   cepiOLD 25814
            18.13.51  Functors   cfuncOLD 25842
            18.13.52  Subcategories   csubcat 25854
            18.13.53  Terminal and initial objects   ciobj 25871
            18.13.54  Sources and sinks   csrce 25876
            18.13.55  Limits and co-limits   clmct 25885
            18.13.56  Product and sum of two objects   cprodo 25888
            18.13.57  Tarski's classes   ctar 25892
            18.13.58  Category Set   ccmrcase 25921
            18.13.59  Grammars, Logics, Machines and Automata   ckln 25991
            18.13.60  Words   cwrd 25992
            18.13.61  Planar geometry   cpoints 26067
      18.14  Mathbox for Jeff Hankins
            18.14.1  Miscellany   a1i13 26211
            18.14.2  Basic topological facts   topbnd 26253
            18.14.3  Topology of the real numbers   reconnOLD 26266
            18.14.4  Refinements   cfne 26270
            18.14.5  Neighborhood bases determine topologies   neibastop1 26319
            18.14.6  Lattice structure of topologies   topmtcl 26323
            18.14.7  Filter bases   fgmin 26330
            18.14.8  Directed sets, nets   tailfval 26332
      18.15  Mathbox for Jeff Madsen
            18.15.1  Logic and set theory   anim12da 26343
            18.15.2  Real and complex numbers; integers   fimaxreOLD 26441
            18.15.3  Sequences and sums   sdclem2 26463
            18.15.4  Topology   unopnOLD 26475
            18.15.5  Metric spaces   metf1o 26480
            18.15.6  Continuous maps and homeomorphisms   constcncf 26489
            18.15.7  Product topologies   txtopiOLD 26497
            18.15.8  Boundedness   ctotbnd 26501
            18.15.9  Isometries   cismty 26533
            18.15.10  Heine-Borel Theorem   heibor1lem 26544
            18.15.11  Banach Fixed Point Theorem   bfplem1 26557
            18.15.12  Euclidean space   crrn 26560
            18.15.13  Intervals (continued)   ismrer1 26573
            18.15.14  Groups and related structures   exidcl 26577
            18.15.15  Rings   rngonegcl 26587
            18.15.16  Ring homomorphisms   crnghom 26602
            18.15.17  Commutative rings   ccring 26631
            18.15.18  Ideals   cidl 26643
            18.15.19  Prime rings and integral domains   cprrng 26682
            18.15.20  Ideal generators   cigen 26695
      18.16  Mathbox for Rodolfo Medina
            18.16.1  Partitions   prtlem60 26714
      18.17  Mathbox for Stefan O'Rear
            18.17.1  Additional elementary logic and set theory   nelss 26762
            18.17.2  Additional theory of functions   fninfp 26765
            18.17.3  Extensions beyond function theory   gsumvsmul 26775
            18.17.4  Additional topology   elrfi 26780
            18.17.5  Characterization of closure operators. Kuratowski closure axioms   ismrcd1 26784
            18.17.6  Algebraic closure systems   cnacs 26788
            18.17.7  Miscellanea 1. Map utilities   constmap 26799
            18.17.8  Miscellanea for polynomials   ofmpteq 26808
            18.17.9  Multivariate polynomials over the integers   cmzpcl 26810
            18.17.10  Miscellanea for Diophantine sets 1   coeq0 26842
            18.17.11  Diophantine sets 1: definitions   cdioph 26845
            18.17.12  Diophantine sets 2 miscellanea   ellz1 26857
            18.17.13  Diophantine sets 2: union and intersection. Monotone Boolean algebra   diophin 26863
            18.17.14  Diophantine sets 3: construction   diophrex 26866
            18.17.15  Diophantine sets 4 miscellanea   2sbcrex 26875
            18.17.16  Diophantine sets 4: Quantification   rexrabdioph 26886
            18.17.17  Diophantine sets 5: Arithmetic sets   rabdiophlem1 26893
            18.17.18  Diophantine sets 6 miscellanea   fz1ssnn 26903
            18.17.19  Diophantine sets 6: reusability. renumbering of variables   eldioph4b 26905
            18.17.20  Pigeonhole Principle and cardinality helpers   fphpd 26910
            18.17.21  A non-closed set of reals is infinite   rencldnfilem 26914
            18.17.22  Miscellanea for Lagrange's theorem   icodiamlt 26916
            18.17.23  Lagrange's rational approximation theorem   irrapxlem1 26918
            18.17.24  Pell equations 1: A nontrivial solution always exists   pellexlem1 26925
            18.17.25  Pell equations 2: Algebraic number theory of the solution set   csquarenn 26932
            18.17.26  Pell equations 3: characterizing fundamental solution   infmrgelbi 26974
            18.17.27  Logarithm laws generalized to an arbitrary base   reglogcl 26986
            18.17.28  Pell equations 4: the positive solution group is infinite cyclic   pellfund14 26994
            18.17.29  X and Y sequences 1: Definition and recurrence laws   crmx 26996
            18.17.30  Ordering and induction lemmas for the integers   monotuz 27037
            18.17.31  X and Y sequences 2: Order properties   rmxypos 27045
            18.17.32  Congruential equations   congtr 27063
            18.17.33  Alternating congruential equations   acongid 27073
            18.17.34  Additional theorems on integer divisibility   bezoutr 27083
            18.17.35  X and Y sequences 3: Divisibility properties   jm2.18 27092
            18.17.36  X and Y sequences 4: Diophantine representability of Y   jm2.27a 27109
            18.17.37  X and Y sequences 5: Diophantine representability of X, ^, _C   rmxdiophlem 27119
            18.17.38  Uncategorized stuff not associated with a major project   setindtr 27128
            18.17.39  More equivalents of the Axiom of Choice   axac10 27137
            18.17.40  Finitely generated left modules   clfig 27176
            18.17.41  Noetherian left modules I   clnm 27184
            18.17.42  Addenda for structure powers   pwssplit0 27198
            18.17.43  Direct sum of left modules   cdsmm 27208
            18.17.44  Free modules   cfrlm 27223
            18.17.45  Every set admits a group structure iff choice   unxpwdom3 27267
            18.17.46  Independent sets and families   clindf 27285
            18.17.47  Characterization of free modules   lmimlbs 27317
            18.17.48  Noetherian rings and left modules II   clnr 27324
            18.17.49  Hilbert's Basis Theorem   cldgis 27336
            18.17.50  Additional material on polynomials [DEPRECATED]   cmnc 27346
            18.17.51  Degree and minimal polynomial of algebraic numbers   cdgraa 27356
            18.17.52  Algebraic integers I   citgo 27373
            18.17.53  Finite cardinality [SO]   en1uniel 27391
            18.17.54  Words in monoids and ordered group sum   issubmd 27394
            18.17.55  Transpositions in the symmetric group   cpmtr 27395
            18.17.56  The sign of a permutation   cpsgn 27425
            18.17.57  The matrix algebra   cmmul 27450
            18.17.58  The determinant   cmdat 27494
            18.17.59  Endomorphism algebra   cmend 27500
            18.17.60  Subfields   csdrg 27514
            18.17.61  Cyclic groups and order   idomrootle 27522
            18.17.62  Cyclotomic polynomials   ccytp 27532
            18.17.63  Miscellaneous topology   fgraphopab 27540
      18.18  Mathbox for Steve Rodriguez
            18.18.1  Miscellanea   iso0 27547
            18.18.2  Function operations   caofcan 27551
            18.18.3  Calculus   lhe4.4ex1a 27557
      18.19  Mathbox for Andrew Salmon
            18.19.1  Principia Mathematica * 10   pm10.12 27564
            18.19.2  Principia Mathematica * 11   2alanimi 27578
            18.19.3  Predicate Calculus   sbeqal1 27608
            18.19.4  Principia Mathematica * 13 and * 14   pm13.13a 27618
            18.19.5  Set Theory   elnev 27649
            18.19.6  Arithmetic   addcomgi 27672
            18.19.7  Geometry   cplusr 27673
      18.20  Mathbox for Glauco Siliprandi
            18.20.1  Miscellanea   ssrexf 27695
            18.20.2  Finite multiplication of numbers and finite multiplication of functions   fmul01 27721
            18.20.3  Limits   clim1fr1 27738
            18.20.4  Derivatives   dvsinexp 27751
            18.20.5  Integrals   ioovolcl 27753
            18.20.6  Stone Weierstrass theorem - real version   stoweidlem1 27761
            18.20.7  Wallis' product for π   wallispilem1 27825
            18.20.8  Stirling's approximation formula for ` n ` factorial   stirlinglem1 27834
      18.21  Mathbox for Saveliy Skresanov
            18.21.1  Ceva's theorem   sigarval 27851
      18.22  Mathbox for Jarvin Udandy
      18.23  Mathbox for Alexander van der Vekens
            18.23.1  Double restricted existential uniqueness   r19.32 27956
                  18.23.1.1  Restricted quantification (extension)   r19.32 27956
                  18.23.1.2  The empty set (extension)   raaan2 27964
                  18.23.1.3  Restricted uniqueness and "at most one" quantification   rmoimi 27965
                  18.23.1.4  Analogs to Existential uniqueness (double quantification)   2reurex 27970
            18.23.2  Alternative definitions of function's and operation's values   wdfat 27982
                  18.23.2.1  Restricted quantification (extension)   ralbinrald 27988
                  18.23.2.2  The universal class (extension)   nvelim 27989
                  18.23.2.3  Relations (extension)   sbcrel 27990
                  18.23.2.4  Functions (extension)   sbcfun 27996
                  18.23.2.5  Predicate "defined at"   dfateq12d 28003
                  18.23.2.6  Alternative definition of the value of a function   dfafv2 28006
                  18.23.2.7  Alternative definition of the value of an operation   aoveq123d 28049
            18.23.3  Graph theory   difprsneq 28079
                  18.23.3.1  Unordered and ordered pairs (extension)   difprsneq 28079
                  18.23.3.2  Functions (extension)   f1oprg 28086
                  18.23.3.3  Operations (Extension)   nssdmovg 28088
                  18.23.3.4  "Maps to" notation (Extension)   mpt2xopn0yelv 28090
                  18.23.3.5  The ` # ` (finite set size) function (extension)   elprchashprn2 28099
                  18.23.3.6  Longer string literals (extension)   s2prop 28100
                  18.23.3.7  Undirected simple graphs   cuslg 28105
                  18.23.3.8  Undirected simple graphs (examples)   usgra1v 28137
                  18.23.3.9  Neighbors, complete graphs and universal vertices   cnbgra 28145
                  18.23.3.10  Friendship graphs   cfrgra 28180
      18.24  Mathbox for David A. Wheeler
            18.24.1  Natural deduction   19.8ad 28198
            18.24.2  Greater than, greater than or equal to.   cge-real 28201
            18.24.3  Hyperbolic trig functions   csinh 28211
            18.24.4  Reciprocal trig functions (sec, csc, cot)   csec 28222
            18.24.5  Identities for "if"   ifnmfalse 28244
            18.24.6  Not-member-of   AnelBC 28245
            18.24.7  Decimal point   cdp2 28246
            18.24.8  Signum (sgn or sign) function   csgn 28254
            18.24.9  Ceiling function   ccei 28264
            18.24.10  Logarithms generalized to arbitrary base using ` logb `   ene0 28268
            18.24.11  Logarithm laws generalized to an arbitrary base - log_   clog_ 28271
            18.24.12  Miscellaneous   5m4e1 28273
      18.25  Mathbox for Alan Sare
            18.25.1  Supplementary "adant" deductions   ad4ant13 28276
            18.25.2  Supplementary unification deductions   biimp 28302
            18.25.3  Conventional Metamath proofs, some derived from VD proofs   iidn3 28318
            18.25.4  What is Virtual Deduction?   wvd1 28393
            18.25.5  Virtual Deduction Theorems   df-vd1 28394
            18.25.6  Theorems proved using virtual deduction   trsspwALT 28665
            18.25.7  Theorems proved using virtual deduction with mmj2 assistance   simplbi2VD 28695
            18.25.8  Virtual Deduction transcriptions of textbook proofs   sb5ALTVD 28762
            18.25.9  Theorems proved using conjunction-form virtual deduction   elpwgdedVD 28766
            18.25.10  Theorems with VD proofs in conventional notation derived from VD proofs   suctrALT3 28773
            18.25.11  Theorems with a proof in conventional notation automatically derived   notnot2ALT2 28776
      18.26  Mathbox for Jonathan Ben-Naim
            18.26.1  First order logic and set theory   bnj170 28796
            18.26.2  Well founded induction and recursion   bnj110 28963
            18.26.3  The existence of a minimal element in certain classes   bnj69 29113
            18.26.4  Well-founded induction   bnj1204 29115
            18.26.5  Well-founded recursion, part 1 of 3   bnj60 29165
            18.26.6  Well-founded recursion, part 2 of 3   bnj1500 29171
            18.26.7  Well-founded recursion, part 3 of 3   bnj1522 29175
      18.27  Mathbox for Norm Megill
            18.27.1  Obsolete experiments to study ax-12o   ax12-2 29176
            18.27.2  Miscellanea   cnaddcom 29234
            18.27.3  Atoms, hyperplanes, and covering in a left vector space (or module)   clsa 29237
            18.27.4  Functionals and kernels of a left vector space (or module)   clfn 29320
            18.27.5  Opposite rings and dual vector spaces   cld 29386
            18.27.6  Ortholattices and orthomodular lattices   cops 29435
            18.27.7  Atomic lattices with covering property   ccvr 29525
            18.27.8  Hilbert lattices   chlt 29613
            18.27.9  Projective geometries based on Hilbert lattices   clln 29753
            18.27.10  Construction of a vector space from a Hilbert lattice   cdlema1N 30053
            18.27.11  Construction of involution and inner product from a Hilbert lattice   clpoN 31743

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